# Exported functions and types

In addition to the documentation for this package, we encourage the reader to explore the documentation for ControlSystems.jl and RobustAndOptimalControl.jl that contains functionality and types for basic control analysis and design, as well as the documentation of ModelingToolkit for modeling and simulation.

## Index

`ControlSystems.Simulator`

`ControlSystems.Simulator`

`ControlSystemsBase.BodemagWorkspace`

`ControlSystemsBase.DelayLtiSystem`

`ControlSystemsBase.DelayLtiSystem`

`ControlSystemsBase.LsimWorkspace`

`ControlSystemsBase.StateSpace`

`ControlSystemsBase.StaticStateSpace`

`ControlSystemsBase.StepInfo`

`ControlSystemsBase.TransferFunction`

`JuliaSimControl.AutoTuningProblem`

`JuliaSimControl.AutoTuningResult`

`JuliaSimControl.Eq29`

`JuliaSimControl.Eq32`

`JuliaSimControl.FixedGainObserver`

`JuliaSimControl.FunctionSystem`

`JuliaSimControl.FunctionSystem`

`JuliaSimControl.FunctionSystem`

`JuliaSimControl.GainMarginObjective`

`JuliaSimControl.LMI`

`JuliaSimControl.MPC.BoundsConstraint`

`JuliaSimControl.MPC.CollocationFinE`

`JuliaSimControl.MPC.CollocationFinE`

`JuliaSimControl.MPC.DifferenceCost`

`JuliaSimControl.MPC.GenericMPCProblem`

`JuliaSimControl.MPC.LQMPCProblem`

`JuliaSimControl.MPC.LinearMPCConstraints`

`JuliaSimControl.MPC.LinearMPCModel`

`JuliaSimControl.MPC.MPCConstraints`

`JuliaSimControl.MPC.MPCIntegrator`

`JuliaSimControl.MPC.MPCParameters`

`JuliaSimControl.MPC.MultipleShooting`

`JuliaSimControl.MPC.NonlinearMPCConstraints`

`JuliaSimControl.MPC.OSQPSolver`

`JuliaSimControl.MPC.Objective`

`JuliaSimControl.MPC.Objective`

`JuliaSimControl.MPC.ObjectiveInput`

`JuliaSimControl.MPC.QMPCProblem`

`JuliaSimControl.MPC.RobustMPCModel`

`JuliaSimControl.MPC.StageConstraint`

`JuliaSimControl.MPC.StageCost`

`JuliaSimControl.MPC.TerminalCost`

`JuliaSimControl.MPC.TerminalStateConstraint`

`JuliaSimControl.MPC.Trapezoidal`

`JuliaSimControl.MPC.Trapezoidal`

`JuliaSimControl.MaximumSensitivityObjective`

`JuliaSimControl.MaximumTransferObjective`

`JuliaSimControl.OperatingPoint`

`JuliaSimControl.OvershootObjective`

`JuliaSimControl.PhaseMarginObjective`

`JuliaSimControl.RiseTimeObjective`

`JuliaSimControl.SettlingTimeObjective`

`JuliaSimControl.StateFeedback`

`JuliaSimControl.StepTrackingObjective`

`JuliaSimControl.StructuredAutoTuningProblem`

`JuliaSimControl.StructuredAutoTuningResult`

`LowLevelParticleFilters.AdvancedParticleFilter`

`LowLevelParticleFilters.AuxiliaryParticleFilter`

`LowLevelParticleFilters.DAEUnscentedKalmanFilter`

`LowLevelParticleFilters.ExtendedKalmanFilter`

`LowLevelParticleFilters.KalmanFilter`

`LowLevelParticleFilters.ParticleFilter`

`LowLevelParticleFilters.TupleProduct`

`LowLevelParticleFilters.UnscentedKalmanFilter`

`LowLevelParticleFilters.UnscentedKalmanFilter`

`RobustAndOptimalControl.Disk`

`RobustAndOptimalControl.Diskmargin`

`RobustAndOptimalControl.ExtendedStateSpace`

`RobustAndOptimalControl.ExtendedStateSpace`

`RobustAndOptimalControl.LQGProblem`

`RobustAndOptimalControl.LQGProblem`

`RobustAndOptimalControl.NamedStateSpace`

`RobustAndOptimalControl.UncertainSS`

`RobustAndOptimalControl.nyquistcircles`

`RobustAndOptimalControl.δ`

`Base.step`

`CommonSolve.solve`

`CommonSolve.solve`

`CommonSolve.solve`

`CommonSolve.solve`

`CommonSolve.solve`

`ControlSystems.rlocus`

`ControlSystemsBase.G_CS`

`ControlSystemsBase.G_CS`

`ControlSystemsBase.G_PS`

`ControlSystemsBase.G_PS`

`ControlSystemsBase.add_input`

`ControlSystemsBase.add_output`

`ControlSystemsBase.append`

`ControlSystemsBase.are`

`ControlSystemsBase.are`

`ControlSystemsBase.array2mimo`

`ControlSystemsBase.balance`

`ControlSystemsBase.balance_statespace`

`ControlSystemsBase.balreal`

`ControlSystemsBase.baltrunc`

`ControlSystemsBase.bode`

`ControlSystemsBase.bodemag!`

`ControlSystemsBase.bodeplot`

`ControlSystemsBase.bodev`

`ControlSystemsBase.bodev`

`ControlSystemsBase.c2d`

`ControlSystemsBase.c2d`

`ControlSystemsBase.c2d_poly2poly`

`ControlSystemsBase.c2d_roots2poly`

`ControlSystemsBase.c2d_x0map`

`ControlSystemsBase.comp_sensitivity`

`ControlSystemsBase.covar`

`ControlSystemsBase.ctrb`

`ControlSystemsBase.d2c`

`ControlSystemsBase.dab`

`ControlSystemsBase.damp`

`ControlSystemsBase.dampreport`

`ControlSystemsBase.dcgain`

`ControlSystemsBase.delay`

`ControlSystemsBase.delaymargin`

`ControlSystemsBase.diagonalize`

`ControlSystemsBase.evalfr`

`ControlSystemsBase.extended_gangoffour`

`ControlSystemsBase.feedback`

`ControlSystemsBase.feedback`

`ControlSystemsBase.feedback2dof`

`ControlSystemsBase.feedback2dof`

`ControlSystemsBase.freqresp`

`ControlSystemsBase.freqresp!`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.freqrespv`

`ControlSystemsBase.gangoffour`

`ControlSystemsBase.gangoffourplot`

`ControlSystemsBase.gangofseven`

`ControlSystemsBase.gram`

`ControlSystemsBase.grampd`

`ControlSystemsBase.hinfnorm`

`ControlSystemsBase.impulse`

`ControlSystemsBase.innovation_form`

`ControlSystemsBase.innovation_form`

`ControlSystemsBase.input_comp_sensitivity`

`ControlSystemsBase.input_comp_sensitivity`

`ControlSystemsBase.input_names`

`ControlSystemsBase.input_sensitivity`

`ControlSystemsBase.input_sensitivity`

`ControlSystemsBase.iscontinuous`

`ControlSystemsBase.isdiscrete`

`ControlSystemsBase.isproper`

`ControlSystemsBase.isstable`

`ControlSystemsBase.kalman`

`ControlSystemsBase.laglink`

`ControlSystemsBase.leadlink`

`ControlSystemsBase.leadlinkat`

`ControlSystemsBase.leadlinkcurve`

`ControlSystemsBase.lft`

`ControlSystemsBase.linfnorm`

`ControlSystemsBase.loopshapingPI`

`ControlSystemsBase.loopshapingPID`

`ControlSystemsBase.lqr`

`ControlSystemsBase.lsim`

`ControlSystemsBase.lsim`

`ControlSystemsBase.lsim`

`ControlSystemsBase.lsim!`

`ControlSystemsBase.margin`

`ControlSystemsBase.markovparam`

`ControlSystemsBase.minreal`

`ControlSystemsBase.minreal`

`ControlSystemsBase.nicholsplot`

`ControlSystemsBase.nonlinearity`

`ControlSystemsBase.nyquist`

`ControlSystemsBase.nyquistplot`

`ControlSystemsBase.nyquistv`

`ControlSystemsBase.nyquistv`

`ControlSystemsBase.observer_controller`

`ControlSystemsBase.observer_predictor`

`ControlSystemsBase.obsv`

`ControlSystemsBase.output_comp_sensitivity`

`ControlSystemsBase.output_comp_sensitivity`

`ControlSystemsBase.output_names`

`ControlSystemsBase.output_sensitivity`

`ControlSystemsBase.output_sensitivity`

`ControlSystemsBase.pade`

`ControlSystemsBase.pade`

`ControlSystemsBase.parallel`

`ControlSystemsBase.pid`

`ControlSystemsBase.pidplots`

`ControlSystemsBase.place`

`ControlSystemsBase.placePI`

`ControlSystemsBase.poles`

`ControlSystemsBase.pzmap`

`ControlSystemsBase.reduce_sys`

`ControlSystemsBase.relative_gain_array`

`ControlSystemsBase.relative_gain_array`

`ControlSystemsBase.rgaplot`

`ControlSystemsBase.rstc`

`ControlSystemsBase.rstd`

`ControlSystemsBase.sensitivity`

`ControlSystemsBase.series`

`ControlSystemsBase.setPlotScale`

`ControlSystemsBase.sigma`

`ControlSystemsBase.sigmaplot`

`ControlSystemsBase.sigmav`

`ControlSystemsBase.sigmav`

`ControlSystemsBase.similarity_transform`

`ControlSystemsBase.sminreal`

`ControlSystemsBase.ss`

`ControlSystemsBase.ss`

`ControlSystemsBase.ssdata`

`ControlSystemsBase.ssrand`

`ControlSystemsBase.stabregionPID`

`ControlSystemsBase.starprod`

`ControlSystemsBase.state_names`

`ControlSystemsBase.stepinfo`

`ControlSystemsBase.tf`

`ControlSystemsBase.time_scale`

`ControlSystemsBase.to_sized`

`ControlSystemsBase.tzeros`

`ControlSystemsBase.zpconv`

`ControlSystemsBase.zpk`

`ControlSystemsBase.zpkdata`

`DescriptorSystems.dss`

`JuliaSimControl.ESC`

`JuliaSimControl.MPC.IpoptSolver`

`JuliaSimControl.MPC.get_xu`

`JuliaSimControl.MPC.modelfit`

`JuliaSimControl.MPC.optimize!`

`JuliaSimControl.MPC.optimize!`

`JuliaSimControl.MPC.rms`

`JuliaSimControl.MPC.rollout`

`JuliaSimControl.MPC.step!`

`JuliaSimControl.MPC.step!`

`JuliaSimControl.MPC.step!`

`JuliaSimControl.OptimizedPID`

`JuliaSimControl.PIESC`

`JuliaSimControl.app_autotuning`

`JuliaSimControl.app_modelreduction`

`JuliaSimControl.build_controlled_dynamics`

`JuliaSimControl.frequency_response_analysis`

`JuliaSimControl.frequency_response_analysis`

`JuliaSimControl.hinfsyn_lmi`

`JuliaSimControl.inverse_lqr`

`JuliaSimControl.inverse_lqr`

`JuliaSimControl.inverse_lqr`

`JuliaSimControl.inverse_lqr`

`JuliaSimControl.ispassive_lmi`

`JuliaSimControl.mussv`

`JuliaSimControl.mussv`

`JuliaSimControl.mussv`

`JuliaSimControl.mussv_DG`

`JuliaSimControl.mussv_tv`

`JuliaSimControl.mussv_tv`

`JuliaSimControl.rk4`

`JuliaSimControl.rk4`

`JuliaSimControl.spr_synthesize`

`JuliaSimControl.trim`

`LinearAlgebra.lyap`

`LinearAlgebra.norm`

`LowLevelParticleFilters.commandplot`

`LowLevelParticleFilters.correct!`

`LowLevelParticleFilters.correct!`

`LowLevelParticleFilters.debugplot`

`LowLevelParticleFilters.densityplot`

`LowLevelParticleFilters.forward_trajectory`

`LowLevelParticleFilters.forward_trajectory`

`LowLevelParticleFilters.log_likelihood_fun`

`LowLevelParticleFilters.loglik`

`LowLevelParticleFilters.logsumexp!`

`LowLevelParticleFilters.mean_trajectory`

`LowLevelParticleFilters.metropolis`

`LowLevelParticleFilters.reset!`

`LowLevelParticleFilters.simulate`

`LowLevelParticleFilters.smooth`

`LowLevelParticleFilters.smooth`

`LowLevelParticleFilters.smoothed_cov`

`LowLevelParticleFilters.smoothed_mean`

`LowLevelParticleFilters.smoothed_trajs`

`LowLevelParticleFilters.update!`

`LowLevelParticleFilters.weighted_cov`

`LowLevelParticleFilters.weighted_mean`

`LowLevelParticleFilters.weighted_mean`

`RobustAndOptimalControl.add_disturbance`

`RobustAndOptimalControl.add_input_differentiator`

`RobustAndOptimalControl.add_input_integrator`

`RobustAndOptimalControl.add_low_frequency_disturbance`

`RobustAndOptimalControl.add_low_frequency_disturbance`

`RobustAndOptimalControl.add_measurement_disturbance`

`RobustAndOptimalControl.add_output_differentiator`

`RobustAndOptimalControl.add_output_integrator`

`RobustAndOptimalControl.add_resonant_disturbance`

`RobustAndOptimalControl.add_resonant_disturbance`

`RobustAndOptimalControl.baltrunc2`

`RobustAndOptimalControl.baltrunc_coprime`

`RobustAndOptimalControl.baltrunc_unstab`

`RobustAndOptimalControl.bilinearc2d`

`RobustAndOptimalControl.bilinearc2d`

`RobustAndOptimalControl.bilinearc2d`

`RobustAndOptimalControl.bilineard2c`

`RobustAndOptimalControl.bilineard2c`

`RobustAndOptimalControl.bilineard2c`

`RobustAndOptimalControl.blocksort`

`RobustAndOptimalControl.broken_feedback`

`RobustAndOptimalControl.closedloop`

`RobustAndOptimalControl.connect`

`RobustAndOptimalControl.controller_reduction`

`RobustAndOptimalControl.controller_reduction_plot`

`RobustAndOptimalControl.controller_reduction_weight`

`RobustAndOptimalControl.dare3`

`RobustAndOptimalControl.diskmargin`

`RobustAndOptimalControl.diskmargin`

`RobustAndOptimalControl.diskmargin`

`RobustAndOptimalControl.expand_symbol`

`RobustAndOptimalControl.extended_controller`

`RobustAndOptimalControl.extended_controller`

`RobustAndOptimalControl.feedback_control`

`RobustAndOptimalControl.ff_controller`

`RobustAndOptimalControl.find_lft`

`RobustAndOptimalControl.fit_complex_perturbations`

`RobustAndOptimalControl.frequency_weighted_reduction`

`RobustAndOptimalControl.fudge_inv`

`RobustAndOptimalControl.gain_and_delay_uncertainty`

`RobustAndOptimalControl.glover_mcfarlane`

`RobustAndOptimalControl.glover_mcfarlane`

`RobustAndOptimalControl.glover_mcfarlane_2dof`

`RobustAndOptimalControl.h2norm`

`RobustAndOptimalControl.h2synthesize`

`RobustAndOptimalControl.hankelnorm`

`RobustAndOptimalControl.hanus`

`RobustAndOptimalControl.hess_form`

`RobustAndOptimalControl.hinfassumptions`

`RobustAndOptimalControl.hinfgrad`

`RobustAndOptimalControl.hinfnorm2`

`RobustAndOptimalControl.hinfpartition`

`RobustAndOptimalControl.hinfsignals`

`RobustAndOptimalControl.hinfsynthesize`

`RobustAndOptimalControl.hsvd`

`RobustAndOptimalControl.ispassive`

`RobustAndOptimalControl.loop_diskmargin`

`RobustAndOptimalControl.loop_diskmargin`

`RobustAndOptimalControl.loop_scale`

`RobustAndOptimalControl.loop_scaling`

`RobustAndOptimalControl.lqr3`

`RobustAndOptimalControl.makeweight`

`RobustAndOptimalControl.measure`

`RobustAndOptimalControl.modal_form`

`RobustAndOptimalControl.muplot`

`RobustAndOptimalControl.mvnyquistplot`

`RobustAndOptimalControl.named_ss`

`RobustAndOptimalControl.named_ss`

`RobustAndOptimalControl.named_ss`

`RobustAndOptimalControl.ncfmargin`

`RobustAndOptimalControl.neglected_delay`

`RobustAndOptimalControl.neglected_lag`

`RobustAndOptimalControl.noise_mapping`

`RobustAndOptimalControl.nugap`

`RobustAndOptimalControl.partition`

`RobustAndOptimalControl.partition`

`RobustAndOptimalControl.passivity_index`

`RobustAndOptimalControl.passivityplot`

`RobustAndOptimalControl.performance_mapping`

`RobustAndOptimalControl.robstab`

`RobustAndOptimalControl.schur_form`

`RobustAndOptimalControl.show_construction`

`RobustAndOptimalControl.sim_diskmargin`

`RobustAndOptimalControl.sim_diskmargin`

`RobustAndOptimalControl.sim_diskmargin`

`RobustAndOptimalControl.specificationplot`

`RobustAndOptimalControl.ssdata_e`

`RobustAndOptimalControl.stab_unstab`

`RobustAndOptimalControl.static_gain_compensation`

`RobustAndOptimalControl.structured_singular_value`

`RobustAndOptimalControl.structured_singular_value`

`RobustAndOptimalControl.sumblock`

`RobustAndOptimalControl.system_mapping`

`RobustAndOptimalControl.uss`

`RobustAndOptimalControl.uss`

`RobustAndOptimalControl.uss`

`RobustAndOptimalControl.uss`

`RobustAndOptimalControl.vec2sys`

`RobustAndOptimalControl.δc`

`RobustAndOptimalControl.δr`

`StatsAPI.predict!`

`StatsAPI.predict!`

# Docstrings

## JuliaSimControl

Docstrings of the MPC submodule are located under MPC.

`JuliaSimControl.Eq29`

— Type`Eq29 <: InverseLQRMethod`

`R`

is a weighting matrix, determining the relative importance of matching each input to that provided by the original controller.

`JuliaSimControl.Eq32`

— Type```
Eq32 <: InverseLQRMethod
Eq32(q1, r1)
```

`r1`

penalizes deviation from the original control signal while `q1`

penalizes deviation from the original effect the cotrol signal had on the state. In other words, the `r1`

term tries to use the same actuator configuration as the original controller, while the `q1`

term ensures that the effec of the controller is the same.

`JuliaSimControl.LMI`

— Type`LMI <: InverseLQRMethod`

`R`

is a weighting matrix, determining the relative importance of matching each input to that provided by the original controller.

`JuliaSimControl.inverse_lqr`

— Method`Q,R,S,P,L2 = inverse_lqr(method::Eq29, G::AbstractStateSpace, L)`

Solve the inverse optimal control problem that finds the LQR cost function that leads to a controller approximating state feedback controller with gain matrix `L`

. `P`

is the solution to the Riccati equation and `L2`

is the recreated feedback gain. Note: `S`

will in general not be zero, and including this cross term in the cost function may be important. If including `S`

is not possible, use the `LMI`

method to find a cost function with as small `S`

as possible.

Creates the stage-cost matrix

\[\begin{bmatrix} L^T R L & -L^T R\\ -R L & R \end{bmatrix} = \begin{bmatrix} Q & -S\\ -S^T & R \end{bmatrix} \]

Ref: "Designing MPC controllers by reverse-engineering existing LTI controllers", E. N. Hartley, J. M. Maciejowski

`JuliaSimControl.inverse_lqr`

— Method`Q,R,S,P,L2 = inverse_lqr(method::Eq32, G::AbstractStateSpace, L)`

Solve the inverse optimal control problem that finds the LQR cost function that leads to a controller approximating state feedback controller with gain matrix `L`

. `P`

is the solution to the Riccati equation and `L2`

is the recreated feedback gain. Note: `S`

will in general not be zero, and including this cross term in the cost function may be important. If including `S`

is not possible, use the `LMI`

method to find a cost function with as small `S`

as possible.

Creates the cost function

\[||Bu - BLu||^2_{q_1} + ||u - Lu||^2_{r_1}\]

Ref: "Designing MPC controllers by reverse-engineering existing LTI controllers", E. N. Hartley, J. M. Maciejowski

`JuliaSimControl.inverse_lqr`

— Method`Q,R,S,P,L2 = inverse_lqr(method::GMF)`

Solve the inverse optimal control problem that finds the LQR cost function on the form

\[x'Qx + u'Ru\]

that leads to an LQR controller approximating a Glover McFarlane controller. Using this method, `S = 0`

.

Ref: Rowe and Maciejowski, "Tuning MPC using H∞ Loop Shaping" .

**Example**

```
disc = (x) -> c2d(ss(x), 0.01)
G = tf([1, 5], [1, 2, 10]) |> disc # Plant model
W1 = tf(1,[1, 0]) |> disc # Loop shaping weight
gmf2 = glover_mcfarlane(G; W1) # Design Glover McFarlane controller
Q,R,S,P,L = inverse_lqr(GMF(gmf2)) # Get cost function
using Test
L3 = lqr(G*W1, Q, R) # Test equivalence to LQR
@test L3 ≈ -L2
```

`JuliaSimControl.inverse_lqr`

— Method`Q,R,S,P,L2 = inverse_lqr(method::LMI, G::AbstractStateSpace, L; optimizer = Hypatia.Optimizer)`

Solve the inverse optimal control problem that finds the LQR cost function that leads to a controller approximating state feedback controller with gain matrix `L`

. `P`

is the solution to the Riccati equation and `L2`

is the recreated feedback gain. Note, `L`

is supposed to be used with negative feedback, i.e., it's designed such that `u = -Lx`

.

Solves a convex LMI problem minimizing the cross term `S`

. Note: there is no guarantee that the corss term will be driven to 0 exactly. If `S`

remains large in relation to `Q`

and `R`

, `S`

must be included in the cost function for high fidelity reproduction of the original controller.

Ref: "Designing MPC controllers by reverse-engineering existing LTI controllers", E. N. Hartley, J. M. Maciejowski

`JuliaSimControl.FixedGainObserver`

— Type```
FixedGainObserver{F <: Function, x} <: AbstractFilter
FixedGainObserver(sys::AbstractStateSpace, x0, K)
```

A linear observer, similar to a Kalman filer, but with a fixed measurement feedback gain. The gain can be designed using, e.g., pole placement or solving a Riccati equation. For a robust observer, consider using `glover_mcfarlane`

followed by `inverse_lqr`

.

`JuliaSimControl.OperatingPoint`

— Type```
OperatingPoint(x, u, y)
OperatingPoint()
```

Structure representing an operating point around which a system is linearized. If no arguments are supplied, an empty operating point is created.

**Arguments:**

`x`

: State`u`

: Control input`y`

: Output

`JuliaSimControl.StateFeedback`

— Type```
StateFeedback{F <: Function, x} <: AbstractFilter
StateFeedback(discrete_dynamics, x0, nu, ny)
StateFeedback(sys::FunctionSystem, x0)
```

An observer that uses the dynamics model without any measurement feedback. This observer can be used as an oracle that has full knowledge of the state. Note, this is often an unrealistic assumption in real-world contexts and open-loop observers can not account for load disturbances. Use of this observer in a closed-loop context creates a false closed loop.

`LowLevelParticleFilters.UnscentedKalmanFilter`

— Method`UnscentedKalmanFilter(sys::FunctionSystem, R1, R2, d0=MvNormal(Matrix(R1)); p = SciMLBase.NullParameters())`

Convencience constructor for systems of type `FunctionSystem`

.

`JuliaSimControl.mussv`

— Method`mussv(M::AbstractMatrix; optimizer = Hypatia.Optimizer{eltype(M)}, bu = opnorm(M), tol = 0.001)`

Compute (an upper bound of) the structured singular value μ for diagonal Δ of complex perturbations (other structures of Δ are handled by supplying the block structure mussv(M, blocks)). `M`

is assumed to be an (n × n × N_freq) array or a matrix.

Solves a convex LMI.

**Arguments:**

`bu`

: Upper bound for bisection.`tol`

: tolerance.

See also `mussv`

, `mussv_tv`

, `mussv_DG`

.

**Extended help**

By default, the Hypatia solver (native Julia) is used to solve the problem. Other solvers that handle the this problem formulation is

- Mosek (requires license)
- SCS

The solver can be selected using the keyword argument `optimizer`

.

`JuliaSimControl.mussv`

— Method`mussv(M::AbstractStateSpace, blocks; optimizer = Hypatia.Optimizer, bu0 = 20, tol = 0.001)`

Compute (an upper bound of) the structured singular value μ of statespace model `M`

interconnected with uncertainty structure described by `blocks`

. Reference: MAE509 (LMIs in Control): Lecture 14, part C - LMIs for Robust Control with Structured Uncertainty

**Example:**

The following example illustrates how to use the structured singular value to determine how large diagonal complex uncertainty can be added at the input of a plant `P`

before the closed-loop system becomes unstable

```
julia> Δ = uss([δc(), δc()]); # Diagonal uncertainty element
julia> a = 1;
julia> P = ss([0 a; -a -1], I(2), [1 a; 0 1], 0) * (I(2) + Δ);
julia> K = ss(I(2));
julia> G = lft(P, -K);
julia> stabmargin = 1/mussv(G) # We can handle 134% of the modeled uncertainty
1.3429508196721311
julia> # Compare with the input diskmargin
diskmargin(K*system_mapping(P), -1)
Disk margin with:
Margin: 1.3469378397689664
Frequency: -0.40280561122244496
Gain margins: [-0.3469378397689664, 2.3469378397689664]
Phase margin: 84.67073122411068
Skew: -1
Worst-case perturbation: missing
```

**Extended help**

By default, the Hypatia solver is used to solve the problem. Other solvers that handle the this problem formulation is

- Mosek
- Hypatia.jl (native Julia)
- Clarabel.jl (native Julia)
- SCS (typically performs poorly for this problem)

The solver can be selected using the keyword argument `optimizer`

.

`JuliaSimControl.mussv`

— Method`mussv(G::UncertainSS; kwargs...)`

Compute (an upper bound of) the structured singular value μ of uncertain system model `G`

.

`JuliaSimControl.mussv_DG`

— Method`mussv_DG(M::AbstractMatrix; optimizer = Hypatia.Optimizer{eltype(M)}, bu = opnorm(M), tol = 0.001)`

Compute (an upper bound of) the structured singular value μ for diagonal Δ of real perturbations (other structures of Δ are not yet supported). `M`

is assumed to be an (n × n × N_freq) array or a matrix.

See `mussv`

for more details. See also `mussv`

, `mussv_tv`

, `mussv_DG`

.

`JuliaSimControl.mussv_tv`

— Method`mussv_tv(G::AbstractStateSpace, blocks; optimizer = Hypatia.Optimizer, bu = 20, tol = 0.001)`

Compute (an upper bound of) the structured singular value margin when `G`

is interconnected with *time-varying* uncertainty structure described by `blocks`

. This value will in general be larger than the one returned by `mussv`

, but can be used to guarantee stability for *infinitely fast time-varying perturbations*, i.e., if the return value is < 1, the system is stable no matter how fast the dynamics of the perturbations change.

The result will in general be more accurate if `G`

is passed rather than a matrix `M`

, unless a very dense grid around the critical frequency is used for to calculate `M`

.

Solves a convex LMI.

**Arguments:**

`bu`

: Upper bound for bisection.`tol`

: tolerance.

`JuliaSimControl.mussv_tv`

— Method```
mussv_tv(M::AbstractArray{<:Any, 3}; optimizer = Hypatia.Optimizer, bu = 20, tol = 0.001)
mussv_tv(G::UncertainSS; optimizer = Hypatia.Optimizer, bu0 = 20, tol = 0.001)
```

Compute (an upper bound of) the structured singular value μ for diagonal, complex and time-varying Δ(t) using constant (over frequency) matrix scalings. This value will in general be larger than the one returned by `mussv`

, but can be used to guarantee stability for *infinitely fast time-varying perturbations*, i.e., if the return value is < 1, the system is stable no matter how fast the dynamics of the perturbations change.

`M`

is assumed to be an (n × n × N_freq) array or a matrix. `G`

is an `UncertainSS`

. The result will in general be more accurate if `G`

is passed rather than `M`

, unless a very dense grid around the critical frequency is used for to calculate `M`

.

Solves a convex LMI.

**Arguments:**

`bu`

: Upper bound for bisection.`tol`

: tolerance.

See also `mussv`

, `mussv_tv`

, `mussv_DG`

.

**Extended help**

The normal μ is calculated by minimizing

\[\operatorname{min}_D ||D(\omega) M(\omega) D(\omega)^{-1}||\]

where a unique $D(\omega)$ is allowed for each frequency. However, in this problem, the $D$ scalings are constant over frequency, yielding a more conservative answer, with the additional bonus of being applicable for time-varying perturbations.

This strong guarantee can be used to prove stability of nonlinear systems by formulating them as linear systems with time-varying, norm-bounded perturbations. For such systems, `mussv_tv < 1`

is a sufficient condition for stability. See Boyd et al., "Linear Matrix Inequalities in System and Control Theory" for more details.

`JuliaSimControl.hinfsyn_lmi`

— Method```
K, γ = hinfsyn_lmi(P::ExtendedStateSpace;
opt = Hypatia.Optimizer(), γrel = 1.01, ϵ = 1e-3, balance = true, perm = false,)
```

Computes an H-infinity optimal controller `K`

for an extended plant `P`

such that ||F_l(P, K)||∞ < γ (`lft(P, K)`

) for the smallest possible γ given P. This implementation solves a convex LMI problem.

**Arguments:**

`opt`

: A MathOptInterface solver instance.`γrel`

: If`γrel > 1`

, the optimal γ will be found by γ iteration after which a controller will be designed for`γ = γopt * γrel`

. It is often a good idea to design a slightly suboptimal controller, both for numerical reasons, but also since the optimal controller may contain very fast dynamics. If`γrel → ∞`

, the computed controller will approach the 𝑯₂ optimal controller. Getting a mix between 𝑯∞ and 𝑯₂ properties is another reason to choose`γrel > 1`

.`ϵ`

: A small offset to enforce strict LMI inequalities. This can be tuned if numerical issues arise.`balance`

: Perform a balancing transformation on`P`

using`ControlSystemsBase.balance_statespace`

.`perm`

: If`balance=true, perm=true`

, the balancing transform is allowed to permute the state vector. This is not allowed by default, but can improve the numerics slightly if allowed.

The Hypatia solver takes the following arguments https://github.com/chriscoey/Hypatia.jl/blob/42e4b10318570ea22adb39fec1c27d8684161cec/src/Solvers/Solvers.jl#L73

`JuliaSimControl.ispassive_lmi`

— Method`ispassive_lmi(P::AbstractStateSpace{ControlSystemsBase.Continuous}; ftype = Float64, opt = Hypatia.Optimizer{ftype}(), ϵ = 0.001, balance = true, verbose = true, silent_solver = true)`

Determine if square system `P`

is passive, i.e., $P(s) + Pᴴ(s) > 0$.

A passive system has a Nyquist curve that lies completely in the right half plane, and satisfies the following inequality (dissipation of energy)

\[\int_0^T y^T u dt > 0 ∀ T\]

The negative feedback-interconnection of two passive systems is stable and parallel connections of two passive systems as well as the inverse of a passive system are also passive. A passive controller will thus always yeild a stable feedback loop for a passive system. A series connection of two passive systems *is not* always passive.

This functions solves a convex LMI related to the KYP (positive real) lemma.

**Arguments:**

`balance`

: Balance the system before calculations?`verbose`

: Print status messages`silent_solver`

: Silence the LMI solver output

`JuliaSimControl.spr_synthesize`

— Method`K, Gcl = spr_synthesize(P0::ExtendedStateSpace{Continuous};, opt = Hypatia.Optimizer, balance = true, verbose = true, silent_solver = true, ϵ = 1e-6)`

Design a strictly positive real controller (passive) that optimizes the closed-loop H₂-norm subject to being passive.

For plants that are known to be passive, control using a passive controller is guaranteed to be stable.

The returned controller is supposed to be used with positive feedback, so `ispassive(-K)`

should hold. The resulting closed-loop system from disturbances to performance outputs is also returned, `Gcl = lft(P0, K)`

.

Implements the algorithm labeled as "Pseudocode 1" in "Synthesis of strictly positive real H2 controllers using dilated LMI", Forbes 2018

**Arguments:**

`P0`

: An`ExtendedStateSpace`

object. This object can be designed using H₂ or H∞ methods. See, e.g.,`hinfpartition`

.`opt`

: A JuMP compatible solver.`balance`

: Perform balancing of the statespace system before solving.`verbose`

: Print info?`silent_solver`

:`ϵ`

: A small numerical constant to enforce strict positive definiteness.

See also `h2synthesize`

, `hinfsynthesize`

.

`JuliaSimControl.FunctionSystem`

— Method`FunctionSystem(sys::ODESystem, u, y; z=nothing, w=nothing, kwargs...)`

Generate code for the dynamics and output of `sys`

by calling `build_controlled_dynamics`

. See `build_controlled_dynamics`

for more details.

`JuliaSimControl.build_controlled_dynamics`

— Function```
f, xout, pout = build_controlled_dynamics(sys, u; kwargs...)
f, obsf, xout, pout = build_controlled_dynamics(sys, u, y; z=nothing, w=nothing, kwargs...)
```

Build a function on the form (x,u,p,t) -> ẋ where

`x`

are the differential states`u`

are control inputs`p`

are parameters`t`

is time`kwargs`

are sent to`ModelingToolkit.build_function`

`f`

is a tuple of functions, one out of palce and one in place`(x,u,p,t) -> ẋ`

and`(ẋ,x,u,p,t) -> nothing`

`xout`

contains the order of the states included in the dynamics`pout`

contains the order of the parameters included in the dynamics

If in addition to `u`

, outputs `y`

are also specified, an additional observed function tuple is returned.

**Example**

This example forms a feedback system and builds a function of the dynamics from the reference `r`

to the output `y`

.

```
using ModelingToolkit, Test
@variables t x(t)=0 y(t)=0 u(t)=0 r(t)=0
@parameters kp=1
D = Differential(t)
eqs = [
u ~ kp * (r - y) # P controller
D(x) ~ -x + u # Plant dynamics
y ~ x # Output equation
]
@named sys = ODESystem(eqs, t)
funsys = JuliaSimControl.build_controlled_dynamics(sys, r, y; checkbounds=true)
x = zeros(funsys.nx) # sys.x
u = [1] # r
p = [1] # kp
xd = funsys(x,u,p,1)
varmap = Dict(
funsys.x[] => 1,
kp => 1,
)
@test xd == ModelingToolkit.varmap_to_vars(varmap, funsys.x)
```

`JuliaSimControl.FunctionSystem`

— Type`FunctionSystem{TE <: ControlSystemsBase.TimeEvolution, F, G}`

A structure representing the dynamical system

\[�egin{aligned} x′ &= f(x,u,p,t)\ y &= g(x,u,p,t) �nd{aligned}\]

To build a FunctionSystem for a DAE on mass-matrix form, use an `ODEFunction`

as `f`

`f = ODEFunction(dae_dyn, mass_matrix = M)`

**Fields:**

`dynamics::F`

`measurement::G`

`timeevol::TE`

: ControlSystemsBase.TimeEvolution`x`

: states`u`

: controlled inputs`y`

: measured outputs`w`

: disturbance inputs`z`

: performance outputs`p`

: parameters

`JuliaSimControl.FunctionSystem`

— Method```
FunctionSystem(f, g; kwargs...)
FunctionSystem(f, g, Ts::Real; kwargs...)
```

Constructor for `FunctionSystem`

.

**Arguments:**

`f`

: Discrete dynamics with signature (x,u,p,t)`g`

: Measurement function with signature (x,u,p,t)`Ts`

: If the sample time`Ts`

is provided, the system represents a discrete-time system, otherwise the dynamics is assumed to be continuous.`kwargs`

: Signal names

## ControlSystems and RobustAndOptimalControl

`ControlSystems.Simulator`

— Type`Simulator`

**Fields:**

```
P::StateSpace
f = (x,p,t) -> x
y = (x,t) -> y
```

`ControlSystems.Simulator`

— Method`Simulator(P::StateSpace, u = (x,t) -> 0)`

Used to simulate continuous-time systems. See function `?solve`

for additional info.

**Usage:**

```
using OrdinaryDiffEq, Plots
dt = 0.1
tfinal = 20
t = 0:dt:tfinal
P = ss(tf(1,[2,1])^2)
K = 5
reference(x,t) = [1.]
s = Simulator(P, reference)
x0 = [0.,0]
tspan = (0.0,tfinal)
sol = solve(s, x0, tspan, Tsit5())
plot(t, s.y(sol, t)[:], lab="Open loop step response")
```

`ControlSystems.rlocus`

— Method`roots, Z, K = rlocus(P::LTISystem, K = 500)`

Compute the root locus of the SISO LTISystem `P`

with a negative feedback loop and feedback gains between 0 and `K`

. `rlocus`

will use an adaptive step-size algorithm to determine the values of the feedback gains used to generate the plot.

`roots`

is a complex matrix containig the poles trajectories of the closed-loop `1+k⋅G(s)`

as a function of `k`

, `Z`

contains the zeros of the open-loop system `G(s)`

and `K`

the values of the feedback gain.

`ControlSystemsBase.lsim`

— Method`res = lsim(sys::DelayLtiSystem, u, t::AbstractArray{<:Real}; x0=fill(0.0, nstates(sys)), alg=MethodOfSteps(Tsit5()), abstol=1e-6, reltol=1e-6, force_dtmin=true, kwargs...)`

Simulate system `sys`

, over time `t`

, using input signal `u`

, with initial state `x0`

, using method `alg`

.

Arguments:

`t`

: Has to be an `AbstractVector`

with equidistant time samples (`t[i] - t[i-1]`

constant) `u`

: Function to determine control signal `ut`

at a time `t`

, on any of the following forms:

`u`

: Function to determine control signal`uₜ`

at a time`t`

, on any of the following forms:- A constant
`Number`

or`Vector`

, interpreted as a constant input. - Function
`u(x, t)`

that takes the internal state and time, note, the state representation for delay systems is not the same as for rational systems. - In-place function
`u(uₜ, x, t)`

. (Slightly more effienct)

- A constant

`alg, abstol, reltol`

and `kwargs...`

: are sent to `DelayDiffEq.solve`

.

This methods sets `force_dtmin=true`

by default to handle the discontinuity implied by, e.g., step inputs. This may lead to the solver taking a long time to solve ill-conditioned problems rather than exiting early with a warning.

Returns an instance of `SimResult`

which can be plotted directly or destructured into `y, t, x, u = res`

.

`ControlSystemsBase.lsim`

— Method`lsim(sys::HammersteinWienerSystem, u, t::AbstractArray{<:Real}; x0=fill(0.0, nstates(sys)), alg=Tsit5(), abstol=1e-6, reltol=1e-6, kwargs...)`

Simulate system `sys`

, over time `t`

, using input signal `u`

, with initial state `x0`

, using method `alg`

.

**Arguments:**

`t`

: Has to be an`AbstractVector`

with equidistant time samples (`t[i] - t[i-1]`

constant)`u`

: Function to determine control signal`uₜ`

at a time`t`

, on any of the following forms: Can be a constant`Number`

or`Vector`

, interpreted as`uₜ .= u`

, or Function`uₜ .= u(x, t)`

, or In-place function`u(uₜ, x, t)`

. (Slightly more effienct)`alg, abstol, reltol`

and`kwargs...`

: are sent to`OrdinaryDiffEq.solve`

.

Returns an instance of `SimResult`

.

`ControlSystemsBase.BodemagWorkspace`

— Method```
BodemagWorkspace(sys::LTISystem, N::Int)
BodemagWorkspace(sys::LTISystem, ω::AbstractVector)
BodemagWorkspace(R::Array{Complex{T}, 3}, mag::Array{T, 3})
BodemagWorkspace{T}(ny, nu, N)
```

Genereate a workspace object for use with the in-place function `bodemag!`

. `N`

is the number of frequency points, alternatively, the input `ω`

can be provided instead of `N`

. Note: for threaded applications, create one workspace object per thread.

**Arguments:**

`mag`

: The output array ∈ 𝐑(ny, nu, nω)`R`

: Frequency-response array ∈ 𝐂(ny, nu, nω)

`ControlSystemsBase.DelayLtiSystem`

— Type`struct DelayLtiSystem{T, S <: Real} <: LTISystem`

Represents an LTISystem with internal time-delay. See `?delay`

for a convenience constructor.

`ControlSystemsBase.DelayLtiSystem`

— Method`DelayLtiSystem{T, S}(sys::StateSpace, Tau::AbstractVector{S}=Float64[]) where {T <: Number, S <: Real}`

Create a delayed system by speciying both the system and time-delay vector. NOTE: if you want to create a system with simple input or output delays, use the Function `delay(τ)`

.

`ControlSystemsBase.LsimWorkspace`

— Method```
LsimWorkspace(sys::AbstractStateSpace, N::Int)
LsimWorkspace(sys::AbstractStateSpace, u::AbstractMatrix)
LsimWorkspace{T}(ny, nu, nx, N)
```

Genereate a workspace object for use with the in-place function `lsim!`

. `sys`

is the discrete-time system to be simulated and `N`

is the number of time steps, alternatively, the input `u`

can be provided instead of `N`

. Note: for threaded applications, create one workspace object per thread.

`ControlSystemsBase.StateSpace`

— Type`StateSpace{TE, T} <: AbstractStateSpace{TE}`

An object representing a standard state space system.

See the function `ss`

for a user facing constructor as well as the documentation page creating systems.

**Fields:**

`A::Matrix{T}`

`B::Matrix{T}`

`C::Matrix{T}`

`D::Matrix{T}`

`timeevol::TE`

`ControlSystemsBase.StaticStateSpace`

— Method`StaticStateSpace(sys::AbstractStateSpace)`

Return a `HeteroStateSpace`

where the system matrices are of type SMatrix.

*NOTE: This function is fundamentally type unstable.* For maximum performance, create the static system manually, or make use of the function-barrier technique.

`ControlSystemsBase.StepInfo`

— Type`StepInfo`

Computed using `stepinfo`

**Fields:**

`y0`

: The initial value of the step response.`yf`

: The final value of the step response.`stepsize`

: The size of the step.`peak`

: The peak value of the step response.`peaktime`

: The time at which the peak occurs.`overshoot`

: The overshoot of the step response.`settlingtime`

: The time at which the step response has settled to within`settling_th`

of the final value.`settlingtimeind::Int`

: The index at which the step response has settled to within`settling_th`

of the final value.`risetime`

: The time at which the response rises from`risetime_th[1]`

to`risetime_th[2]`

of the final value`i10::Int`

: The index at which the response reaches`risetime_th[1]`

`i90::Int`

: The index at which the response reaches`risetime_th[2]`

`res::SimResult{SR}`

: The simulation result used to compute the step response characteristics.`settling_th`

: The threshold used to compute`settlingtime`

and`settlingtimeind`

.`risetime_th`

: The thresholds used to compute`risetime`

,`i10`

, and`i90`

.

`ControlSystemsBase.TransferFunction`

— Method`F(s)`

, `F(omega, true)`

, `F(z, false)`

Notation for frequency response evaluation.

- F(s) evaluates the continuous-time transfer function F at s.
- F(omega,true) evaluates the discrete-time transfer function F at exp(im
*Ts*omega) - F(z,false) evaluates the discrete-time transfer function F at z

`Base.step`

— Method```
y, t, x = step(sys[, tfinal])
y, t, x = step(sys[, t])
```

Calculate the step response of system `sys`

. If the final time `tfinal`

or time vector `t`

is not provided, one is calculated based on the system pole locations. The return value is a structure of type `SimResult`

that can be plotted or destructured as `y, t, x = result`

.

`y`

has size `(ny, length(t), nu)`

, `x`

has size `(nx, length(t), nu)`

`ControlSystemsBase.G_CS`

— Method`G_CS(P, C)`

The closed-loop transfer function from (-) measurement noise or (+) reference to control signal. Technically, the transfer function is given by `(1 + CP)⁻¹C`

so `SC`

would be a better, but nonstandard name.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.G_PS`

— Method`G_PS(P, C)`

The closed-loop transfer function from load disturbance to plant output. Technically, the transfer function is given by `(1 + PC)⁻¹P`

so `SP`

would be a better, but nonstandard name.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.add_input`

— Function`add_input(sys::AbstractStateSpace, B2::AbstractArray, D2 = 0)`

Add inputs to `sys`

by forming

\[x' = Ax + [B \; B_2]u y = Cx + [D \; D_2]u\]

If `B2`

is an integer it will be interpreted as an index and an input matrix containing a single 1 at the specified index will be used.

Example: The following example forms an innovation model that takes innovations as inputs

```
G = ssrand(2,2,3, Ts=1)
K = kalman(G, I(G.nx), I(G.ny))
sys = add_input(G, K)
```

`ControlSystemsBase.add_output`

— Function`add_output(sys::AbstractStateSpace, C2::AbstractArray, D2 = 0)`

Add outputs to `sys`

by forming

\[x' = Ax + Bu y = [C; C_2]x + [D; D_2]u\]

If `C2`

is an integer it will be interpreted as an index and an output matrix containing a single 1 at the specified index will be used.

`ControlSystemsBase.append`

— Method`append(systems::StateSpace...), append(systems::TransferFunction...)`

Append systems in block diagonal form

`ControlSystemsBase.are`

— Method`are(::Continuous, A, B, Q, R)`

Compute 'X', the solution to the continuous-time algebraic Riccati equation, defined as A'X + XA - (XB)R^-1(B'X) + Q = 0, where R is non-singular.

In an LQR problem, `Q`

is associated with the state penalty $x'Qx$ while `R`

is associated with the control penalty $u'Ru$. See `lqr`

for more details.

Uses `MatrixEquations.arec`

. For keyword arguments, see the docstring of `ControlSystemsBase.MatrixEquations.arec`

, note that they define the input arguments in a different order.

`ControlSystemsBase.are`

— Method`are(::Discrete, A, B, Q, R; kwargs...)`

Compute `X`

, the solution to the discrete-time algebraic Riccati equation, defined as A'XA - X - (A'XB)(B'XB + R)^-1(B'XA) + Q = 0, where Q>=0 and R>0

In an LQR problem, `Q`

is associated with the state penalty $x'Qx$ while `R`

is associated with the control penalty $u'Ru$. See `lqr`

for more details.

Uses `MatrixEquations.ared`

. For keyword arguments, see the docstring of `ControlSystemsBase.MatrixEquations.ared`

, note that they define the input arguments in a different order.

`ControlSystemsBase.array2mimo`

— Method`array2mimo(M::AbstractArray{<:LTISystem})`

Take an array of `LTISystem`

s and create a single MIMO system.

`ControlSystemsBase.balance`

— Function`S, P, B = balance(A[, perm=true])`

Compute a similarity transform `T = S*P`

resulting in `B = T\A*T`

such that the row and column norms of `B`

are approximately equivalent. If `perm=false`

, the transformation will only scale `A`

using diagonal `S`

, and not permute `A`

(i.e., set `P=I`

).

`ControlSystemsBase.balance_statespace`

— Function```
A, B, C, T = balance_statespace{S}(A::Matrix{S}, B::Matrix{S}, C::Matrix{S}, perm::Bool=false)
sys, T = balance_statespace(sys::StateSpace, perm::Bool=false)
```

Computes a balancing transformation `T`

that attempts to scale the system so that the row and column norms of [T*A/T T*B; C/T 0] are approximately equal. If `perm=true`

, the states in `A`

are allowed to be reordered.

The inverse of `sysb, T = balance_statespace(sys)`

is given by `similarity_transform(sysb, T)`

This is not the same as finding a balanced realization with equal and diagonal observability and reachability gramians, see `balreal`

`ControlSystemsBase.balreal`

— Method`sysr, G, T = balreal(sys::StateSpace)`

Calculates a balanced realization of the system sys, such that the observability and reachability gramians of the balanced system are equal and diagonal `diagm(G)`

. `T`

is the similarity transform between the old state `x`

and the new state `z`

such that `Tz = x`

.

See also `gram`

, `baltrunc`

Reference: Varga A., Balancing-free square-root algorithm for computing singular perturbation approximations.

`ControlSystemsBase.baltrunc`

— Method`sysr, G, T = baltrunc(sys::StateSpace; atol = √ϵ, rtol=1e-3, n = nothing, residual = false)`

Reduces the state dimension by calculating a balanced realization of the system sys, such that the observability and reachability gramians of the balanced system are equal and diagonal `diagm(G)`

, and truncating it to order `n`

. If `n`

is not provided, it's chosen such that all states corresponding to singular values less than `atol`

and less that `rtol σmax`

are removed.

`T`

is the similarity transform between the old state `x`

and the newstate `z`

such that `Tz = x`

.

If `residual = true`

, matched static gain is achieved through "residualization", i.e., setting

\[0 = A_{21}x_{1} + A_{22}x_{2} + B_{2}u\]

where indices 1/2 correspond to the remaining/truncated states respectively.

See also `gram`

, `balreal`

Glad, Ljung, Reglerteori: Flervariabla och Olinjära metoder.

For more advanced model reduction, see RobustAndOptimalControl.jl - Model Reduction.

`ControlSystemsBase.bode`

— Method`mag, phase, w = bode(sys[, w]; unwrap=true)`

Compute the magnitude and phase parts of the frequency response of system `sys`

at frequencies `w`

. See also `bodeplot`

`mag`

and `phase`

has size `(ny, nu, length(w))`

. If `unwrap`

is true (default), the function `unwrap!`

will be applied to the phase angles. This procedure is costly and can be avoided if the unwrapping is not required.

For higher performance, see the function `bodemag!`

that computes the magnitude only.

`ControlSystemsBase.bodemag!`

— Method`mag = bodemag!(ws::BodemagWorkspace, sys::LTISystem, w::AbstractVector)`

Compute the Bode magnitude operating in-place on an instance of `BodemagWorkspace`

. Note that the returned magnitude array is aliased with `ws.mag`

. The output array `mag`

is ∈ 𝐑(ny, nu, nω).

`ControlSystemsBase.bodeplot`

— Function```
fig = bodeplot(sys, args...)
bodeplot(LTISystem[sys1, sys2...], args...; plotphase=true, kwargs...)
```

Create a Bode plot of the `LTISystem`

(s). A frequency vector `w`

can be optionally provided. To change the Magnitude scale see `setPlotScale(str)`

If `hz=true`

, the plot x-axis will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to RecipesBase.plot.

`ControlSystemsBase.bodev`

— Method`bodev(sys::LTISystem, w::AbstractVector; $(Expr(:kw, :unwrap, true)))`

For use with SISO systems where it acts the same as `bode`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.bodev`

— Method`bodev(sys::LTISystem; )`

For use with SISO systems where it acts the same as `bode`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.c2d`

— Function```
sysd = c2d(sys::AbstractStateSpace{<:Continuous}, Ts, method=:zoh; w_prewarp=0)
Gd = c2d(G::TransferFunction{<:Continuous}, Ts, method=:zoh)
```

Convert the continuous-time system `sys`

into a discrete-time system with sample time `Ts`

, using the specified `method`

(:`zoh`

, `:foh`

, `:fwdeuler`

or `:tustin`

). Note that the forward-Euler method generally requires the sample time to be very small relative to the time constants of the system.

`method = :tustin`

performs a bilinear transform with prewarp frequency `w_prewarp`

.

`w_prewarp`

: Frequency (rad/s) for pre-warping when usingthe Tustin method, has no effect for other methods.

See also `c2d_x0map`

`ControlSystemsBase.c2d`

— Function`c2d(G::DelayLtiSystem, Ts, method=:zoh)`

`ControlSystemsBase.c2d_poly2poly`

— Method`c2d_poly2poly(ro, Ts)`

returns the polynomial coefficients in discrete time given polynomial coefficients in continuous time

`ControlSystemsBase.c2d_roots2poly`

— Method`c2d_roots2poly(ro, Ts)`

returns the polynomial coefficients in discrete time given a vector of roots in continuous time

`ControlSystemsBase.c2d_x0map`

— Function`sysd, x0map = c2d_x0map(sys::AbstractStateSpace{<:Continuous}, Ts, method=:zoh; w_prewarp=0)`

Returns the discretization `sysd`

of the system `sys`

and a matrix `x0map`

that transforms the initial conditions to the discrete domain by `x0_discrete = x0map*[x0; u0]`

See `c2d`

for further details.

`ControlSystemsBase.comp_sensitivity`

— Method```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.covar`

— Method`P = covar(sys, W)`

Calculate the stationary covariance `P = E[y(t)y(t)']`

of the output `y`

of a `StateSpace`

model `sys`

driven by white Gaussian noise `w`

with covariance `E[w(t)w(τ)]=W*δ(t-τ)`

(δ is the Dirac delta).

Remark: If `sys`

is unstable then the resulting covariance is a matrix of `Inf`

s. Entries corresponding to direct feedthrough (D*W*D' .!= 0) will equal `Inf`

for continuous-time systems.

`ControlSystemsBase.ctrb`

— Method`ctrb(A, B)`

or `ctrb(sys)`

Compute the controllability matrix for the system described by `(A, B)`

or `sys`

.

Note that checking for controllability by computing the rank from `ctrb`

is not the most numerically accurate way, a better method is checking if `gram(sys, :c)`

is positive definite.

`ControlSystemsBase.d2c`

— Function`d2c(sys::AbstractStateSpace{<:Discrete}, method::Symbol = :zoh; w_prewarp=0)`

Convert discrete-time system to a continuous time system, assuming that the discrete-time system was discretized using `method`

. Available methods are `:zoh, :fwdeuler´.

`w_prewarp`

: Frequency for pre-warping when usingthe Tustin method, has no effect for other methods.

`ControlSystemsBase.dab`

— Method`X,Y = dab(A,B,C)`

Solves the Diophantine-Aryabhatta-Bezout identity

$AX + BY = C$, where $A, B, C, X$ and $Y$ are polynomials and $deg Y = deg A - 1$.

See Computer-Controlled Systems: Theory and Design, Third Edition Karl Johan Åström, Björn Wittenmark

`ControlSystemsBase.damp`

— Method`Wn, zeta, ps = damp(sys)`

Compute the natural frequencies, `Wn`

, and damping ratios, `zeta`

, of the poles, `ps`

, of `sys`

`ControlSystemsBase.dampreport`

— Method`dampreport(sys)`

Display a report of the poles, damping ratio, natural frequency, and time constant of the system `sys`

`ControlSystemsBase.dcgain`

— Function`dcgain(sys, ϵ=0)`

Compute the dcgain of system `sys`

.

equal to G(0) for continuous-time systems and G(1) for discrete-time systems.

`ϵ`

can be provided to evaluate the dcgain with a small perturbation into the stability region of the complex plane.

`ControlSystemsBase.delay`

— Method```
delay(tau)
delay(tau, Ts)
delay(T::Type{<:Number}, tau)
delay(T::Type{<:Number}, tau, Ts)
```

Create a pure time delay of length `τ`

of type `T`

.

The type `T`

defaults to `promote_type(Float64, typeof(tau))`

.

If `Ts`

is given, the delay is discretized with sampling time `Ts`

and a discrete-time StateSpace object is returned.

**Example:**

Create a LTI system with an input delay of `L`

```
L = 1
tf(1, [1, 1])*delay(L)
s = tf("s")
tf(1, [1, 1])*exp(-s*L) # Equivalent to the version above
```

`ControlSystemsBase.delaymargin`

— Method`dₘ = delaymargin(G::LTISystem)`

Return the delay margin, dₘ. For discrete-time systems, the delay margin is normalized by the sample time, i.e., the value represents the margin in number of sample times. Only supports SISO systems.

`ControlSystemsBase.diagonalize`

— Method`dsys = diagonalize(s::StateSpace, digits=12)`

Diagonalizes the system such that the A-matrix is diagonal.

`ControlSystemsBase.evalfr`

— Method`evalfr(sys, x)`

Evaluate the transfer function of the LTI system sys at the complex number s=x (continuous-time) or z=x (discrete-time).

For many values of `x`

, use `freqresp`

instead.

`ControlSystemsBase.extended_gangoffour`

— Function`extended_gangoffour(P, C, pos=true)`

Returns a single statespace system that maps

`w1`

reference or measurement noise`w2`

load disturbance

to

`z1`

control error`z2`

control input

```
z1 z2
▲ ┌─────┐ ▲ ┌─────┐
│ │ │ │ │ │
w1──+─┴─►│ C ├──┴───+─►│ P ├─┐
│ │ │ │ │ │ │
│ └─────┘ │ └─────┘ │
│ w2 │
└────────────────────────────┘
```

The returned system has the transfer-function matrix

\[\begin{bmatrix} I \\ C \end{bmatrix} (I + PC)^{-1} \begin{bmatrix} I & P \end{bmatrix}\]

or in code

```
# For SISO P
S = G[1, 1]
PS = G[1, 2]
CS = G[2, 1]
T = G[2, 2]
# For MIMO P
S = G[1:P.ny, 1:P.nu]
PS = G[1:P.ny, P.nu+1:end]
CS = G[P.ny+1:end, 1:P.nu]
T = G[P.ny+1:end, P.nu+1:end]
```

The gang of four can be plotted like so

```
Gcl = extended_gangoffour(G, C) # Form closed-loop system
bodeplot(Gcl, lab=["S" "CS" "PS" "T"], plotphase=false) |> display # Plot gang of four
```

Note, the last input of Gcl is the negative of the `PS`

and `T`

transfer functions from `gangoffour2`

. To get a transfer matrix with the same sign as `G_PS`

and `input_comp_sensitivity`

, call `extended_gangoffour(P, C, pos=false)`

. See `glover_mcfarlane`

from RobustAndOptimalControl.jl for an extended example. See also `ncfmargin`

and `feedback_control`

from RobustAndOptimalControl.jl.

`ControlSystemsBase.feedback`

— Method```
feedback(sys1::AbstractStateSpace, sys2::AbstractStateSpace;
U1=:, Y1=:, U2=:, Y2=:, W1=:, Z1=:, W2=Int[], Z2=Int[],
Wperm=:, Zperm=:, pos_feedback::Bool=false)
```

*Basic use*`feedback(sys1, sys2)`

forms the feedback interconnection

```
┌──────────────┐
◄──────────┤ sys1 │◄──── Σ ◄──────
│ │ │ │
│ └──────────────┘ -1
│ |
│ ┌──────────────┐ │
└─────►│ sys2 ├──────┘
│ │
└──────────────┘
```

*Advanced use*`feedback`

also supports more flexible use according to the figure below

```
┌──────────────┐
z1◄─────┤ sys1 │◄──────w1
┌─── y1◄─────┤ │◄──────u1 ◄─┐
│ └──────────────┘ │
│ α
│ ┌──────────────┐ │
└──► u2─────►│ sys2 ├───────►y2──┘
w2─────►│ ├───────►z2
└──────────────┘
```

`U1`

, `W1`

specifies the indices of the input signals of `sys1`

corresponding to `u1`

and `w1`

`Y1`

, `Z1`

specifies the indices of the output signals of `sys1`

corresponding to `y1`

and `z1`

`U2`

, `W2`

, `Y2`

, `Z2`

specifies the corresponding signals of `sys2`

Specify `Wperm`

and `Zperm`

to reorder the inputs (corresponding to [w1; w2]) and outputs (corresponding to [z1; z2]) in the resulting statespace model.

Negative feedback (α = -1) is the default. Specify `pos_feedback=true`

for positive feedback (α = 1).

See also `lft`

, `starprod`

, `sensitivity`

, `input_sensitivity`

, `output_sensitivity`

, `comp_sensitivity`

, `input_comp_sensitivity`

, `output_comp_sensitivity`

, `G_PS`

, `G_CS`

.

See Zhou, Doyle, Glover (1996) for similar (somewhat less symmetric) formulas.

`ControlSystemsBase.feedback`

— Method```
feedback(sys)
feedback(sys1, sys2)
```

For a general LTI-system, `feedback`

forms the negative feedback interconnection

```
>-+ sys1 +-->
| |
(-)sys2 +
```

If no second system is given, negative identity feedback is assumed

`ControlSystemsBase.feedback2dof`

— Method```
feedback2dof(P,R,S,T)
feedback2dof(B,A,R,S,T)
```

- Return
`BT/(AR+ST)`

where B and A are the numerator and denomenator polynomials of`P`

respectively - Return
`BT/(AR+ST)`

`ControlSystemsBase.feedback2dof`

— Method`feedback2dof(P::TransferFunction, C::TransferFunction, F::TransferFunction)`

Return the transfer function `P(F+C)/(1+PC)`

which is the closed-loop system with process `P`

, controller `C`

and feedforward filter `F`

from reference to control signal (by-passing `C`

).

```
+-------+
| |
+-----> F +----+
| | | |
| +-------+ |
| +-------+ | +-------+
r | - | | | | | y
+--+-----> C +----+----> P +---+-->
| | | | | |
| +-------+ +-------+ |
| |
+--------------------------------+
```

`ControlSystemsBase.freqresp!`

— Method`freqresp!(R::Array{T, 3}, sys::LTISystem, w_vec::AbstractVector{<:Real})`

In-place version of `freqresp`

that takes a pre-allocated array `R`

of size (ny, nu, nw)`

`ControlSystemsBase.freqresp`

— Method`sys_fr = freqresp(sys, w)`

Evaluate the frequency response of a linear system

`w -> C*((iw*im*I - A)^-1)*B + D`

of system `sys`

over the frequency vector `w`

.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(G::AbstractMatrix, w_vec::AbstractVector{<:Real}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(G::Number, w_vec::AbstractVector{<:Real}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.freqrespv`

— Method`freqrespv(sys::LTISystem, w_vec::AbstractVector{W}; )`

For use with SISO systems where it acts the same as `freqresp`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.gangoffour`

— Method```
S, PS, CS, T = gangoffour(P, C; minimal=true)
gangoffour(P::AbstractVector, C::AbstractVector; minimal=true)
```

Given a transfer function describing the plant `P`

and a transfer function describing the controller `C`

, computes the four transfer functions in the Gang-of-Four.

`S = 1/(1+PC)`

Sensitivity function`PS = (1+PC)\P`

Load disturbance to measurement signal`CS = (1+PC)\C`

Measurement noise to control signal`T = PC/(1+PC)`

Complementary sensitivity function

If `minimal=true`

, `minreal`

will be applied to all transfer functions.

`ControlSystemsBase.gangoffourplot`

— Method`ControlSystemsBase.gangofseven`

— Method`S, PS, CS, T, RY, RU, RE = gangofseven(P,C,F)`

Given transfer functions describing the Plant `P`

, the controller `C`

and a feed forward block `F`

, computes the four transfer functions in the Gang-of-Four and the transferfunctions corresponding to the feed forward.

`S = 1/(1+PC)`

Sensitivity function`PS = P/(1+PC)`

`CS = C/(1+PC)`

`T = PC/(1+PC)`

Complementary sensitivity function`RY = PCF/(1+PC)`

`RU = CF/(1+P*C)`

`RE = F/(1+P*C)`

`ControlSystemsBase.gram`

— Method`ControlSystemsBase.grampd`

— Method`U = grampd(sys, opt; kwargs...)`

Return a Cholesky factor `U`

of the grammian of system `sys`

. If `opt`

is `:c`

, computes the controllability grammian `G = U*U'`

. If `opt`

is `:o`

, computes the observability grammian `G = U'U`

.

Obtain a `Cholesky`

object by `Cholesky(U)`

for observability grammian

Uses `MatrixEquations.plyapc/plyapd`

. For keyword arguments, see the docstring of `ControlSystemsBase.MatrixEquations.plyapc/plyapd`

`ControlSystemsBase.hinfnorm`

— Method`Ninf, ω_peak = hinfnorm(sys; tol=1e-6)`

Compute the H∞ norm `Ninf`

of the LTI system `sys`

, together with a frequency `ω_peak`

at which the gain Ninf is achieved.

`Ninf := sup_ω σ_max[sys(iω)]`

if `G`

is stable (σ_max = largest singular value) := `Inf' if`

G` is unstable

`tol`

is an optional keyword argument for the desired relative accuracy for the computed H∞ norm (not an absolute certificate).

`sys`

is first converted to a state space model if needed.

The continuous-time L∞ norm computation implements the 'two-step algorithm' in:**N.A. Bruinsma and M. Steinbuch**, 'A fast algorithm to compute the H∞-norm of a transfer function matrix', Systems and Control Letters (1990), pp. 287-293.

For the discrete-time version, see:**P. Bongers, O. Bosgra, M. Steinbuch**, 'L∞-norm calculation for generalized state space systems in continuous and discrete time', American Control Conference, 1991.

See also `linfnorm`

.

`ControlSystemsBase.impulse`

— Method```
y, t, x = impulse(sys[, tfinal])
y, t, x = impulse(sys[, t])
```

Calculate the impulse response of system `sys`

. If the final time `tfinal`

or time vector `t`

is not provided, one is calculated based on the system pole locations. The return value is a structure of type `SimResult`

that can be plotted or destructured as `y, t, x = result`

.

`y`

has size `(ny, length(t), nu)`

, `x`

has size `(nx, length(t), nu)`

`ControlSystemsBase.innovation_form`

— Method```
sysi = innovation_form(sys, R1, R2[, R12])
sysi = innovation_form(sys; sysw=I, syse=I, R1=I, R2=I)
```

Takes a system

```
x' = Ax + Bu + w ~ R1
y = Cx + Du + e ~ R2
```

and returns the system

```
x' = Ax + Kv
y = Cx + v
```

where `v`

is the innovation sequence.

If `sysw`

(`syse`

) is given, the covariance resulting in filtering noise with `R1`

(`R2`

) through `sysw`

(`syse`

) is used as covariance.

See Stochastic Control, Chapter 4, Åström

`ControlSystemsBase.innovation_form`

— Method`sysi = innovation_form(sys, K)`

Takes a system

```
x' = Ax + Bu + Kv
y = Cx + Du + v
```

and returns the system

```
x' = Ax + Kv
y = Cx + v
```

where `v`

is the innovation sequence.

See Stochastic Control, Chapter 4, Åström

`ControlSystemsBase.input_comp_sensitivity`

— Method`input_comp_sensitivity(P,C)`

Transfer function from load disturbance to control signal.

- "Input" signifies that the transfer function is from the input of the plant.
- "Complimentary" signifies that the transfer function is to an output (in this case controller output)

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.input_names`

— Method```
input_names(P)
input_names(P, i)
```

Get a vector of strings with the names of the inputs of `P`

, or the `i`

:th name if and index is given.

`ControlSystemsBase.input_sensitivity`

— Method`input_sensitivity(P, C)`

Transfer function from load disturbance to total plant input.

- "Input" signifies that the transfer function is from the input of the plant.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.iscontinuous`

— Method`iscontinuous(sys)`

Returns `true`

for a continuous-time system `sys`

, else returns `false`

.

`ControlSystemsBase.isdiscrete`

— Method`isdiscrete(sys)`

Returns `true`

for a discrete-time system `sys`

, else returns `false`

.

`ControlSystemsBase.isproper`

— Method`isproper(tf)`

Returns `true`

if the `TransferFunction`

is proper. This means that order(den)

= order(num))

`ControlSystemsBase.isstable`

— Method`isstable(sys)`

Returns `true`

if `sys`

is stable, else returns `false`

.

`ControlSystemsBase.kalman`

— Method```
kalman(Continuous, A, C, R1, R2)
kalman(Discrete, A, C, R1, R2)
kalman(sys, R1, R2)
```

Calculate the optimal Kalman gain

The `args...; kwargs...`

are sent to the Riccati solver, allowing specification of cross-covariance etc. See `?MatrixEquations.arec/ared`

for more help.

`ControlSystemsBase.laglink`

— Method`laglink(a, M; [Ts])`

Returns a phase retarding link, the rule of thumb `a = 0.1ωc`

guarantees less than 6 degrees phase margin loss. The bode curve will go from `M`

, bend down at `a/M`

and level out at 1 for frequencies > `a`

\[\dfrac{s + a}{s + a/M} = M \dfrac{1 + s/a}{1 + sM/a}\]

`ControlSystemsBase.leadlink`

— Function`leadlink(b, N, K=1; [Ts])`

Returns a phase advancing link, the top of the phase curve is located at `ω = b√(N)`

where the link amplification is `K√(N)`

The bode curve will go from `K`

, bend up at `b`

and level out at `KN`

for frequencies > `bN`

The phase advance at `ω = b√(N)`

can be plotted as a function of `N`

with `leadlinkcurve()`

Values of `N < 1`

will give a phase retarding link.

\[KN \dfrac{s + b}{s + bN} = K \dfrac{1 + s/b}{1 + s/(bN)}\]

See also `leadlinkat`

`laglink`

`ControlSystemsBase.leadlinkat`

— Function`leadlinkat(ω, N, K=1; [Ts])`

Returns a phase advancing link, the top of the phase curve is located at `ω`

where the link amplification is `K√(N)`

The bode curve will go from `K`

, bend up at `ω/√(N)`

and level out at `KN`

for frequencies > `ω√(N)`

The phase advance at `ω`

can be plotted as a function of `N`

with `leadlinkcurve()`

Values of `N < 1`

will give a phase retarding link.

See also `leadlink`

`laglink`

`ControlSystemsBase.leadlinkcurve`

— Function`leadlinkcurve(start=1)`

Plot the phase advance as a function of `N`

for a lead link (phase advance link) If an input argument `start`

is given, the curve is plotted from `start`

to 10, else from 1 to 10.

See also `leadlink, leadlinkat`

`ControlSystemsBase.lft`

— Function`lft(G, Δ, type=:l)`

Lower and upper linear fractional transformation between systems `G`

and `Δ`

.

Specify `:l`

lor lower LFT, and `:u`

for upper LFT.

`G`

must have more inputs and outputs than `Δ`

has outputs and inputs.

For details, see Chapter 9.1 in **Zhou, K. and JC Doyle**. Essentials of robust control, Prentice hall (NJ), 1998

`ControlSystemsBase.linfnorm`

— Method`Ninf, ω_peak = linfnorm(sys; tol=1e-6)`

Compute the L∞ norm `Ninf`

of the LTI system `sys`

, together with a frequency `ω_peak`

at which the gain `Ninf`

is achieved.

`Ninf := sup_ω σ_max[sys(iω)]`

(σ_max denotes the largest singular value)

`tol`

is an optional keyword argument representing the desired relative accuracy for the computed L∞ norm (this is not an absolute certificate however).

`sys`

is first converted to a state space model if needed.

The continuous-time L∞ norm computation implements the 'two-step algorithm' in:**N.A. Bruinsma and M. Steinbuch**, 'A fast algorithm to compute the H∞-norm of a transfer function matrix', Systems and Control Letters (1990), pp. 287-293.

For the discrete-time version, see:**P. Bongers, O. Bosgra, M. Steinbuch**, 'L∞-norm calculation for generalized state space systems in continuous and discrete time', American Control Conference, 1991.

See also `hinfnorm`

.

`ControlSystemsBase.loopshapingPI`

— Method`C, kp, ki, fig, CF = loopshapingPI(P, ωp; ϕl, rl, phasemargin, form=:standard, doplot=false, Tf, F)`

Selects the parameters of a PI-controller (on parallel form) such that the Nyquist curve of `P`

at the frequency `ωp`

is moved to `rl exp(i ϕl)`

The parameters can be returned as one of several common representations chosen by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s) + T_ds)$`:series`

- $K_c(1 + 1/(τ_i s))(τ_d s + 1)$`:parallel`

- $K_p + K_i/s + K_d s$

If `phasemargin`

is supplied (in degrees), `ϕl`

is selected such that the curve is moved to an angle of `phasemargin - 180`

degrees

If no `rl`

is given, the magnitude of the curve at `ωp`

is kept the same and only the phase is affected, the same goes for `ϕl`

if no phasemargin is given.

`Tf`

: An optional time constant for second-order measurement noise filter on the form`tf(1, [Tf^2, 2*Tf/sqrt(2), 1])`

to make the controller strictly proper.`F`

: A pre-designed filter to use instead of the default second-order filter that is used if`Tf`

is given.`doplot`

plot the`gangoffourplot`

and`nyquistplot`

of the system.

See also `loopshapingPID`

, `pidplots`

, `stabregionPID`

and `placePI`

.

`ControlSystemsBase.loopshapingPID`

— Method`C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt = 1.3, ϕt=75, form=:standard, doplot=false, lb=-10, ub=10, Tf = 1/1000ω, F = nothing)`

Selects the parameters of a PID-controller such that the Nyquist curve of the loop-transfer function $L = PC$ at the frequency `ω`

is tangent to the circle where the magnitude of $T = PC / (1+PC)$ equals `Mt`

. `ϕt`

denotes the positive angle in degrees between the real axis and the tangent point.

The default values for `Mt`

and `ϕt`

are chosen to give a good design for processes with inertia, and may need tuning for simpler processes.

The gain of the resulting controller is generally increasing with increasing `ω`

and `Mt`

.

**Arguments:**

`P`

: A SISO plant.`ω`

: The specification frequency.`Mt`

: The magnitude of the complementary sensitivity function at the specification frequency, $|T(iω)|$.`ϕt`

: The`doplot`

: If true, gang of four and Nyquist plots will be returned in`fig`

.`lb`

: log10 of lower bound for`kd`

.`ub`

: log10 of upper bound for`kd`

.`Tf`

: Time constant for second-order measurement noise filter on the form`tf(1, [Tf^2, 2*Tf/sqrt(2), 1])`

to make the controller strictly proper. A practical controller typically sets this time constant slower than the default, e.g.,`Tf = 1/100ω`

or`Tf = 1/10ω`

`F`

: A pre-designed filter to use instead of the default second-order filter.

The parameters can be returned as one of several common representations chosen by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s) + T_ds)$`:series`

- $K_c(1 + 1/(τ_i s))(τ_d s + 1)$`:parallel`

- $K_p + K_i/s + K_d s$

See also `loopshapingPI`

, `pidplots`

, `stabregionPID`

and `placePI`

.

**Example:**

```
P = tf(1, [1,0,0]) # A double integrator
Mt = 1.3 # Maximum magnitude of complementary sensitivity
ω = 1 # Frequency at which the specification holds
C, kp, ki, kd, fig, CF = loopshapingPID(P, ω; Mt, ϕt = 75, doplot=true)
```

`ControlSystemsBase.lqr`

— Method```
lqr(sys, Q, R)
lqr(Continuous, A, B, Q, R, args...; kwargs...)
lqr(Discrete, A, B, Q, R, args...; kwargs...)
```

Calculate the optimal gain matrix `K`

for the state-feedback law `u = -K*x`

that minimizes the cost function:

J = integral(x'Qx + u'Ru, 0, inf) for the continuous-time model `dx = Ax + Bu`

. J = sum(x'Qx + u'Ru, 0, inf) for the discrete-time model `x[k+1] = Ax[k] + Bu[k]`

.

Solve the LQR problem for state-space system `sys`

. Works for both discrete and continuous time systems.

The `args...; kwargs...`

are sent to the Riccati solver, allowing specification of cross-covariance etc. See `?MatrixEquations.arec / ared`

for more help.

**Examples**

Continuous time

```
using LinearAlgebra # For identity matrix I
using Plots
A = [0 1; 0 0]
B = [0; 1]
C = [1 0]
sys = ss(A,B,C,0)
Q = I
R = I
L = lqr(sys,Q,R) # lqr(Continuous,A,B,Q,R) can also be used
u(x,t) = -L*x # Form control law,
t=0:0.1:5
x0 = [1,0]
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")
```

Discrete time

```
using LinearAlgebra # For identity matrix I
using Plots
Ts = 0.1
A = [1 Ts; 0 1]
B = [0;1]
C = [1 0]
sys = ss(A, B, C, 0, Ts)
Q = I
R = I
L = lqr(Discrete, A,B,Q,R) # lqr(sys,Q,R) can also be used
u(x,t) = -L*x # Form control law,
t=0:Ts:5
x0 = [1,0]
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")
```

`ControlSystemsBase.lsim!`

— Method`res = lsim!(ws::LsimWorkspace, sys::AbstractStateSpace{<:Discrete}, u, [t]; x0)`

In-place version of `lsim`

that takes a workspace object created by calling `LsimWorkspace`

. *Notice*, if `u`

is a function, `res.u === ws.u`

. If `u`

is an array, `res.u === u`

.

`ControlSystemsBase.lsim`

— Method```
result = lsim(sys, u[, t]; x0, method])
result = lsim(sys, u::Function, t; x0, method)
```

Calculate the time response of system `sys`

to input `u`

. If `x0`

is ommitted, a zero vector is used.

The result structure contains the fields `y, t, x, u`

and can be destructured automatically by iteration, e.g.,

`y, t, x, u = result`

`result::SimResult`

can also be plotted directly:

`plot(result, plotu=true, plotx=false)`

`y`

, `x`

, `u`

have time in the second dimension. Initial state `x0`

defaults to zero.

Continuous-time systems are simulated using an ODE solver if `u`

is a function (requires using ControlSystems). If `u`

is an array, the system is discretized (with `method=:zoh`

by default) before simulation. For a lower-level inteface, see `?Simulator`

and `?solve`

`u`

can be a function or a *matrix* of precalculated control signals and must have dimensions `(nu, length(t))`

. If `u`

is a function, then `u(x,i)`

(for discrete systems) or `u(x,t)`

(for continuos ones) is called to calculate the control signal at every iteration (time instance used by solver). This can be used to provide a control law such as state feedback `u(x,t) = -L*x`

calculated by `lqr`

. To simulate a unit step at `t=t₀`

, use `(x,t)-> t ≥ t₀`

, for a ramp, use `(x,t)-> t`

, for a step at `t=5`

, use (x,t)-> (t >= 5) etc.

*Note:* The function `u`

will be called once before simulating to verify that it returns an array of the correct dimensions. This can cause problems if `u`

is stateful. You can disable this check by passing `check_u = false`

.

For maximum performance, see function `lsim!`

, available for discrete-time systems only.

Usage example:

```
using ControlSystems
using LinearAlgebra: I
using Plots
A = [0 1; 0 0]
B = [0;1]
C = [1 0]
sys = ss(A,B,C,0)
Q = I
R = I
L = lqr(sys,Q,R)
u(x,t) = -L*x # Form control law
t = 0:0.1:5
x0 = [1,0]
y, t, x, uout = lsim(sys,u,t,x0=x0)
plot(t,x', lab=["Position" "Velocity"], xlabel="Time [s]")
# Alternative way of plotting
res = lsim(sys,u,t,x0=x0)
plot(res)
```

`ControlSystemsBase.margin`

— Method`wgm, gm, wpm, pm = margin(sys::LTISystem, w::Vector; full=false, allMargins=false)`

returns frequencies for gain margins, gain margins, frequencies for phase margins, phase margins

If `!allMargins`

, return only the smallest margin

If `full`

return also `fullPhase`

See also `delaymargin`

and `RobustAndOptimalControl.diskmargin`

`ControlSystemsBase.markovparam`

— Method`markovparam(sys, n)`

Compute the `n`

th markov parameter of discrete-time state-space system `sys`

. This is defined as the following:

`h(0) = D`

`h(n) = C*A^(n-1)*B`

`ControlSystemsBase.minreal`

— Function`minreal(tf::TransferFunction, eps=sqrt(eps()))`

Create a minimial representation of each transfer function in `tf`

by cancelling poles and zeros will promote system to an appropriate numeric type

`ControlSystemsBase.minreal`

— Method`minreal(sys::T; fast=false, kwargs...)`

Minimal realisation algorithm from P. Van Dooreen, The generalized eigenstructure problem in linear system theory, IEEE Transactions on Automatic Control

For information about the options, see `?ControlSystemsBase.MatrixPencils.lsminreal`

See also `sminreal`

, which is both numerically exact and substantially faster than `minreal`

, but with a much more limited potential in removing non-minimal dynamics.

`ControlSystemsBase.nicholsplot`

— Function`fig = nicholsplot{T<:LTISystem}(systems::Vector{T}, w::AbstractVector; kwargs...)`

Create a Nichols plot of the `LTISystem`

(s). A frequency vector `w`

can be optionally provided.

Keyword arguments:

```
text = true
Gains = [12, 6, 3, 1, 0.5, -0.5, -1, -3, -6, -10, -20, -40, -60]
pInc = 30
sat = 0.4
val = 0.85
fontsize = 10
```

`pInc`

determines the increment in degrees between phase lines.

`sat`

∈ [0,1] determines the saturation of the gain lines

`val`

∈ [0,1] determines the brightness of the gain lines

Additional keyword arguments are sent to the function plotting the systems and can be used to specify colors, line styles etc. using regular RecipesBase.jl syntax

This function is based on code subject to the two-clause BSD licence Copyright 2011 Will Robertson Copyright 2011 Philipp Allgeuer

`ControlSystemsBase.nonlinearity`

— Method```
nonlinearity(f)
nonlinearity(T, f)
```

Create a pure nonlinearity. `f`

is assumed to be a static (no memory) nonlinear function from $f : R -> R$.

The type `T`

defaults to `Float64`

.

NOTE: The nonlinear functionality in ControlSystemsBase.jl is currently experimental and subject to breaking changes not respecting semantic versioning. Use at your own risk.

**Example:**

Create a LTI system with a static input nonlinearity that saturates the input to [-1,1].

`tf(1, [1, 1])*nonlinearity(x->clamp(x, -1, 1))`

See also predefined nonlinearities `saturation`

, `offset`

.

Note: when composing linear systems with nonlinearities, it's often important to handle operating points correctly. See `ControlSystemsBase.offset`

for handling operating points.

`ControlSystemsBase.nyquist`

— Method`re, im, w = nyquist(sys[, w])`

Compute the real and imaginary parts of the frequency response of system `sys`

at frequencies `w`

. See also `nyquistplot`

`re`

and `im`

has size `(ny, nu, length(w))`

`ControlSystemsBase.nyquistplot`

— Function```
fig = nyquistplot(sys; Ms_circles=Float64[], Mt_circles=Float64[], unit_circle=false, hz=false, critical_point=-1, kwargs...)
nyquistplot(LTISystem[sys1, sys2...]; Ms_circles=Float64[], Mt_circles=Float64[], unit_circle=false, hz=false, critical_point=-1, kwargs...)
```

Create a Nyquist plot of the `LTISystem`

(s). A frequency vector `w`

can be optionally provided.

`unit_circle`

: if the unit circle should be displayed. The Nyquist curve crosses the unit circle at the gain corssover frequency.`Ms_circles`

: draw circles corresponding to given levels of sensitivity (circles around -1 with radii`1/Ms`

).`Ms_circles`

can be supplied as a number or a vector of numbers. A design staying outside such a circle has a phase margin of at least`2asin(1/(2Ms))`

rad and a gain margin of at least`Ms/(Ms-1)`

.`Mt_circles`

: draw circles corresponding to given levels of complementary sensitivity.`Mt_circles`

can be supplied as a number or a vector of numbers.`critical_point`

: point on real axis to mark as critical for encirclements

If `hz=true`

, the hover information will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to plot.

`ControlSystemsBase.nyquistv`

— Method`nyquistv(sys::LTISystem, w::AbstractVector; )`

For use with SISO systems where it acts the same as `nyquist`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.nyquistv`

— Method`nyquistv(sys::LTISystem; )`

For use with SISO systems where it acts the same as `nyquist`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.observer_controller`

— Method`cont = observer_controller(sys, L::AbstractMatrix, K::AbstractMatrix)`

Return the observer_controller `cont`

that is given by `ss(A - B*L - K*C + K*D*L, K, L, 0)`

Such that `feedback(sys, cont)`

produces a closed-loop system with eigenvalues given by `A-KC`

and `A-BL`

.

**Arguments:**

`sys`

: Model of system`L`

: State-feedback gain`u = -Lx`

`K`

: Observer gain

See also `observer_predictor`

and `innovation_form`

.

`ControlSystemsBase.observer_predictor`

— Method```
observer_predictor(sys::AbstractStateSpace, K; h::Int = 1)
observer_predictor(sys::AbstractStateSpace, R1, R2[, R12])
```

If `sys`

is continuous, return the observer predictor system

\[\begin{aligned} x̂' &= (A - KC)x̂ + (B-KD)u + Ky \\ ŷ &= Cx + Du \end{aligned}\]

with the input equation `[B-KD K] * [u; y]`

If `sys`

is discrete, the prediction horizon `h`

may be specified, in which case measurements up to and including time `t-h`

and inputs up to and including time `t`

are used to predict `y(t)`

.

If covariance matrices `R1, R2`

are given, the kalman gain `K`

is calculated using `kalman`

.

See also `innovation_form`

and `observer_controller`

.

`ControlSystemsBase.obsv`

— Function```
obsv(A, C, n=size(A,1))
obsv(sys, n=sys.nx)
```

Compute the observability matrix with `n`

rows for the system described by `(A, C)`

or `sys`

. Providing the optional `n > sys.nx`

returns an extended observability matrix.

Note that checking for observability by computing the rank from `obsv`

is not the most numerically accurate way, a better method is checking if `gram(sys, :o)`

is positive definite.

`ControlSystemsBase.output_comp_sensitivity`

— Method`output_comp_sensitivity(P,C)`

Transfer function from measurement noise / reference to plant output.

- "output" signifies that the transfer function is from the output of the plant.
- "Complimentary" signifies that the transfer function is to an output (in this case plant output)

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.output_names`

— Method```
output_names(P)
output_names(P, i)
```

Get a vector of strings with the names of the outputs of `P`

, or the `i`

:th name if and index is given.

`ControlSystemsBase.output_sensitivity`

— Method`output_sensitivity(P, C)`

Transfer function from measurement noise / reference to control error.

- "output" signifies that the transfer function is from the output of the plant.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.pade`

— Method`pade(G::DelayLtiSystem, N)`

Approximate all time-delays in `G`

by Padé approximations of degree `N`

.

`ControlSystemsBase.pade`

— Method`pade(τ::Real, N::Int)`

Compute the `N`

th order Padé approximation of a time-delay of length `τ`

.

`ControlSystemsBase.parallel`

— Method`parallel(sys1::LTISystem, sys2::LTISystem)`

Connect systems in parallel, equivalent to `sys2+sys1`

`ControlSystemsBase.pid`

— Function`C = pid(param_p, param_i, [param_d]; form=:standard, state_space=false, [Tf], [Ts])`

Calculates and returns a PID controller.

The `form`

can be chosen as one of the following

`:standard`

-`Kp*(1 + 1/(Ti*s) + Td*s)`

`:series`

-`Kc*(1 + 1/(τi*s))*(τd*s + 1)`

`:parallel`

-`Kp + Ki/s + Kd*s`

If `state_space`

is set to `true`

, either `kd`

has to be zero or a positive `Tf`

has to be provided for creating a filter on the input to allow for a state space realization. The filter used is `1 / (1 + s*Tf + (s*Tf)^2/2)`

, where `Tf`

can typically be chosen as `Ti/N`

for a PI controller and `Td/N`

for a PID controller, and `N`

is commonly in the range 2 to 20. The state space will be returned on controllable canonical form.

For a discrete controller a positive `Ts`

can be supplied. In this case, the continuous-time controller is discretized using the Tustin method.

**Examples**

```
C1 = pid(3.3, 1, 2) # Kd≠0 works without filter in tf form
C2 = pid(3.3, 1, 2; Tf=0.3, state_space=true) # In statespace a filter is needed
C3 = pid(2., 3, 0; Ts=0.4, state_space=true) # Discrete
```

The functions `pid_tf`

and `pid_ss`

are also exported. They take the same parameters and is what is actually called in `pid`

based on the `state_space`

parameter.

`ControlSystemsBase.pidplots`

— Method`pidplots(P, args...; params_p, params_i, params_d=0, form=:standard, ω=0, grid=false, kwargs...)`

Plots interesting figures related to closing the loop around process `P`

with a PID controller supplied in `params`

on one of the following forms:

`:standard`

-`Kp*(1 + 1/(Ti*s) + Td*s)`

`:series`

-`Kc*(1 + 1/(τi*s))*(τd*s + 1)`

`:parallel`

-`Kp + Ki/s + Kd*s`

The sent in values can be arrays to evaluate multiple different controllers, and if `grid=true`

it will be a grid search over all possible combinations of the values.

Available plots are `:gof`

for Gang of four, `:nyquist`

, `:controller`

for a bode plot of the controller TF and `:pz`

for pole-zero maps and should be supplied as additional arguments to the function.

One can also supply a frequency vector `ω`

to be used in Bode and Nyquist plots.

See also `loopshapingPI`

, `stabregionPID`

`ControlSystemsBase.place`

— Function```
place(A, B, p, opt=:c)
place(sys::StateSpace, p, opt=:c)
```

Calculate the gain matrix `K`

such that `A - BK`

has eigenvalues `p`

.

```
place(A, C, p, opt=:o)
place(sys::StateSpace, p, opt=:o)
```

Calculate the observer gain matrix `L`

such that `A - LC`

has eigenvalues `p`

.

Uses Ackermann's formula. Currently handles only SISO systems.

`ControlSystemsBase.placePI`

— Method`C, kp, ki = placePI(P, ω₀, ζ; form=:standard)`

Selects the parameters of a PI-controller such that the poles of closed loop between `P`

and `C`

are placed to match the poles of `s^2 + 2ζω₀s + ω₀^2`

.

The parameters can be returned as one of several common representations chose by `form`

, the options are

`:standard`

- $K_p(1 + 1/(T_i s))$`:series`

- $K_c(1 + 1/(τ_i s))$ (equivalent to above for PI controllers)`:parallel`

- $K_p + K_i/s$

`C`

is the returned transfer function of the controller and `params`

is a named tuple containing the parameters. The parameters can be accessed as `params.Kp`

or `params["Kp"]`

from the named tuple, or they can be unpacked using `Kp, Ti, Td = values(params)`

.

See also `loopshapingPI`

`ControlSystemsBase.poles`

— Method`poles(sys)`

Compute the poles of system `sys`

.

`ControlSystemsBase.pzmap`

— Function`fig = pzmap(fig, system, args...; kwargs...)`

Create a pole-zero map of the `LTISystem`

(s) in figure `fig`

, `args`

and `kwargs`

will be sent to the `scatter`

plot command.

To customize the unit-circle drawn for discrete systems, modify the line attributes, e.g., `linecolor=:red`

.

`ControlSystemsBase.reduce_sys`

— Method`reduce_sys(A::AbstractMatrix, B::AbstractMatrix, C::AbstractMatrix, D::AbstractMatrix, meps::AbstractFloat)`

Implements REDUCE in the Emami-Naeini & Van Dooren paper. Returns transformed A, B, C, D matrices. These are empty if there are no zeros.

`ControlSystemsBase.relative_gain_array`

— Method`relative_gain_array(A::AbstractMatrix; tol = 1.0e-15)`

Reference: "On the Relative Gain Array (RGA) with Singular and Rectangular Matrices" Jeffrey Uhlmann https://arxiv.org/pdf/1805.10312.pdf

`ControlSystemsBase.relative_gain_array`

— Method```
relative_gain_array(G, w::AbstractVector)
relative_gain_array(G, w::Number)
```

Calculate the relative gain array of `G`

at frequencies `w`

. G(iω) .* pinv(tranpose(G(iω)))

The RGA can be used to find input-output pairings for MIMO control using individially tuned loops. Pair the inputs and outputs such that the RGA(ωc) at the crossover frequency becomes as close to diagonal as possible. Avoid pairings such that RGA(0) contains negative diagonal elements.

- The sum of the absolute values of the entries in the RGA is a good measure of the "true condition number" of G, the best condition number that can be achieved by input/output scaling of
`G`

, -Glad, Ljung. - The RGA is invariant to input/output scaling of
`G`

. - If the RGA contains large entries, the system may be sensitive to model errors, -Skogestad, "Multivariable Feedback Control: Analysis and Design":
- Uncertainty in the input channels (diagonal input uncertainty). Plants with

- Element uncertainty. Large RGA-elements imply sensitivity to element-by-element uncertainty.

The relative gain array is computed using the The unit-consistent (UC) generalized inverse Reference: "On the Relative Gain Array (RGA) with Singular and Rectangular Matrices" Jeffrey Uhlmann https://arxiv.org/pdf/1805.10312.pdf

`ControlSystemsBase.rgaplot`

— Function```
rgaplot(sys, args...; hz=false)
rgaplot(LTISystem[sys1, sys2...], args...; hz=false)
```

Plot the relative-gain array entries of the `LTISystem`

(s). A frequency vector `w`

can be optionally provided.

If `hz=true`

, the plot x-axis will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to Plots.plot.

`ControlSystemsBase.rstc`

— MethodSee `?rstd`

for the discrete case

`ControlSystemsBase.rstd`

— Method```
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO,AR,AS)
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO,AR)
R,S,T = rstd(BPLUS,BMINUS,A,BM1,AM,AO)
```

Polynomial synthesis in discrete time.

Polynomial synthesis according to CCS ch 10 to design a controller $R(q) u(k) = T(q) r(k) - S(q) y(k)$

Inputs:

`BPLUS`

: Part of open loop numerator`BMINUS`

: Part of open loop numerator`A`

: Open loop denominator`BM1`

: Additional zeros`AM`

: Closed loop denominator`AO`

: Observer polynomial`AR`

: Pre-specified factor of R,

e.g integral part [1, -1]^k

`AS`

: Pre-specified factor of S,

e.g notch filter [1, 0, w^2]

Outputs: `R,S,T`

: Polynomials in controller

See function `dab`

how the solution to the Diophantine- Aryabhatta-Bezout identity is chosen.

See Computer-Controlled Systems: Theory and Design, Third Edition Karl Johan Åström, Björn Wittenmark

`ControlSystemsBase.sensitivity`

— MethodThe output sensitivity function$S_o = (I + PC)^{-1}$ is the transfer function from a reference input to control error, while the input sensitivity function $S_i = (I + CP)^{-1}$ is the transfer function from a disturbance at the plant input to the total plant input. For SISO systems, input and output sensitivity functions are equal. In general, we want to minimize the sensitivity function to improve robustness and performance, but pracitcal constraints always cause the sensitivity function to tend to 1 for high frequencies. A robust design minimizes the peak of the sensitivity function, $M_S$. The peak magnitude of $S$ is the inverse of the distance between the open-loop Nyquist curve and the critical point -1. Upper bounding the sensitivity peak $M_S$ gives lower-bounds on phase and gain margins according to

\[ϕ_m ≥ 2\text{sin}^{-1}(\frac{1}{2M_S}), g_m ≥ \frac{M_S}{M_S-1}\]

Generally, bounding $M_S$ is a better objective than looking at gain and phase margins due to the possibility of combined gain and pahse variations, which may lead to poor robustness despite large gain and pahse margins.

```
▲
│e₁
│ ┌─────┐
d₁────+──┴──► P ├─────┬──►e₄
│ └─────┘ │
│ │
│ ┌─────┐ -│
e₂◄──┴─────┤ C ◄──┬──+───d₂
└─────┘ │
│e₃
▼
```

`input_sensitivity`

is the transfer function from d₁ to e₁, (I + CP)⁻¹`output_sensitivity`

is the transfer function from d₂ to e₃, (I + PC)⁻¹`input_comp_sensitivity`

is the transfer function from d₁ to e₂, (I + CP)⁻¹CP`output_comp_sensitivity`

is the transfer function from d₂ to e₄, (I + PC)⁻¹PC`G_PS`

is the transfer function from d₁ to e₄, (1 + PC)⁻¹P`G_CS`

is the transfer function from d₂ to e₂, (1 + CP)⁻¹C

`ControlSystemsBase.series`

— Method`series(sys1::LTISystem, sys2::LTISystem)`

Connect systems in series, equivalent to `sys2*sys1`

`ControlSystemsBase.setPlotScale`

— Method`setPlotScale(str)`

Set the default scale of magnitude in `bodeplot`

and `sigmaplot`

. `str`

should be either `"dB"`

or `"log10"`

.

`ControlSystemsBase.sigma`

— Method`sv, w = sigma(sys[, w])`

Compute the singular values `sv`

of the frequency response of system `sys`

at frequencies `w`

. See also `sigmaplot`

`sv`

has size `(max(ny, nu), length(w))`

`ControlSystemsBase.sigmaplot`

— Function```
sigmaplot(sys, args...; hz=false)
sigmaplot(LTISystem[sys1, sys2...], args...; hz=false)
```

Plot the singular values of the frequency response of the `LTISystem`

(s). A frequency vector `w`

can be optionally provided.

If `hz=true`

, the plot x-axis will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to Plots.plot.

`ControlSystemsBase.sigmav`

— Method`sigmav(sys::LTISystem, w::AbstractVector; )`

For use with SISO systems where it acts the same as `sigma`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.sigmav`

— Method`sigmav(sys::LTISystem; )`

For use with SISO systems where it acts the same as `sigma`

but with the extra dimensions removed in the returned values.

`ControlSystemsBase.similarity_transform`

— Method`syst = similarity_transform(sys, T; unitary=false)`

Perform a similarity transform `T : Tx̃ = x`

on `sys`

such that

```
Ã = T⁻¹AT
B̃ = T⁻¹ B
C̃ = CT
D̃ = D
```

If `unitary=true`

, `T`

is assumed unitary and the matrix adjoint is used instead of the inverse. See also `balance_statespace`

.

`ControlSystemsBase.sminreal`

— Method`sminreal(sys)`

Compute the structurally minimal realization of the state-space system `sys`

. A structurally minimal realization is one where only states that can be determined to be uncontrollable and unobservable based on the location of 0s in `sys`

are removed.

Systems with numerical noise in the coefficients, e.g., noise on the order of `eps`

require truncation to zero to be affected by structural simplification, e.g.,

```
trunc_zero!(A) = A[abs.(A) .< 10eps(maximum(abs, A))] .= 0
trunc_zero!(sys.A); trunc_zero!(sys.B); trunc_zero!(sys.C)
sminreal(sys)
```

See also `minreal`

`ControlSystemsBase.ss`

— Method```
sys = ss(A, B, C, D) # Continuous
sys = ss(A, B, C, D, Ts) # Discrete
```

Create a state-space model `sys::StateSpace{TE, T}`

with matrix element type `T`

and TE is `Continuous`

or `<:Discrete`

.

This is a continuous-time model if `Ts`

is omitted. Otherwise, this is a discrete-time model with sampling period `Ts`

.

`D`

may be specified as `0`

in which case a zero matrix of appropriate size is constructed automatically. `sys = ss(D [, Ts])`

specifies a static gain matrix `D`

.

To associate names with states, inputs and outputs, see `named_ss`

.

`ControlSystemsBase.ssdata`

— Method`A, B, C, D = ssdata(sys)`

A destructor that outputs the statespace matrices.

`ControlSystemsBase.ssrand`

— Method`ssrand(T::Type, ny::Int, nu::Int, nstates::Int; proper=false, stable=true, Ts=nothing)`

Returns a random `StateSpace`

model with `ny`

outputs, `nu`

inputs, and `nstates`

states, whose matrix elements are normally distributed.

It is possible to specify if the system should be `proper`

or `stable`

.

Specify a sample time `Ts`

to obtain a discrete-time system.

`ControlSystemsBase.stabregionPID`

— Function`kp, ki, fig = stabregionPID(P, [ω]; kd=0, doplot=false, form=:standard)`

Segments of the curve generated by this program is the boundary of the stability region for a process with transfer function P(s) The provided derivative gain is expected on parallel form, i.e., the form kp + ki/s + kd s, but the result can be transformed to any form given by the `form`

keyword. The curve is found by analyzing

\[P(s)C(s) = -1 ⟹ \\ |PC| = |P| |C| = 1 \\ arg(P) + arg(C) = -π\]

If `P`

is a function (e.g. s -> exp(-sqrt(s)) ), the stability of feedback loops using PI-controllers can be analyzed for processes with models with arbitrary analytic functions See also `loopshapingPI`

, `loopshapingPID`

, `pidplots`

`ControlSystemsBase.starprod`

— Method`starprod(sys1, sys2, dimu, dimy)`

Compute the Redheffer star product.

`length(U1) = length(Y2) = dimu`

and `length(Y1) = length(U2) = dimy`

For details, see Chapter 9.3 in **Zhou, K. and JC Doyle**. Essentials of robust control, Prentice hall (NJ), 1998

`ControlSystemsBase.state_names`

— Method```
state_names(P)
state_names(P, i)
```

Get a vector of strings with the names of the states of `P`

, or the `i`

:th name if and index is given.

`ControlSystemsBase.stepinfo`

— Method`stepinfo(res::SimResult; y0 = nothing, yf = nothing, settling_th = 0.02, risetime_th = (0.1, 0.9))`

Compute the step response characteristics for a simulation result. The following information is computed and stored in a `StepInfo`

struct:

`y0`

: The initial value of the response`yf`

: The final value of the response`stepsize`

: The size of the step`peak`

: The peak value of the response`peaktime`

: The time at which the peak occurs`overshoot`

: The percentage overshoot of the response`undershoot`

: The percentage undershoot of the response. If the step response never reaches below the initial value, the undershoot is zero.`settlingtime`

: The time at which the response settles within`settling_th`

of the final value`settlingtimeind`

: The index at which the response settles within`settling_th`

of the final value`risetime`

: The time at which the response rises from`risetime_th[1]`

to`risetime_th[2]`

of the final value

**Arguments:**

`res`

: The result from a simulation using`step`

(or`lsim`

)`y0`

: The initial value, if not provided, the first value of the response is used.`yf`

: The final value, if not provided, the last value of the response is used. The simulation must have reached steady-state for an automatically computed value to make sense. If the simulation has not reached steady state, you may provide the final value manually.`settling_th`

: The threshold for computing the settling time. The settling time is the time at which the response settles within`settling_th`

of the final value.`risetime_th`

: The lower and upper threshold for computing the rise time. The rise time is the time at which the response rises from`risetime_th[1]`

to`risetime_th[2]`

of the final value.

**Example:**

```
G = tf([1], [1, 1, 1])
res = step(G, 15)
si = stepinfo(res)
plot(si)
```

`ControlSystemsBase.tf`

— Method```
sys = tf(num, den[, Ts])
sys = tf(gain[, Ts])
```

Create as a fraction of polynomials:

`sys::TransferFunction{SisoRational{T,TR}} = numerator/denominator`

where T is the type of the coefficients in the polynomial.

`num`

: the coefficients of the numerator polynomial. Either scalar or vector to create SISO systems

or an array of vectors to create MIMO system.

`den`

: the coefficients of the denominator polynomial. Either vector to create SISO systems

or an array of vectors to create MIMO system.

`Ts`

: Sample time if discrete time system.

The polynomial coefficients are ordered starting from the highest order term.

Other uses:

`tf(sys)`

: Convert`sys`

to`tf`

form.`tf("s")`

,`tf("z")`

: Create the continuous transferfunction`s`

.

`ControlSystemsBase.time_scale`

— Method```
time_scale(sys::AbstractStateSpace{Continuous}, a; balanced = false)
time_scale(G::TransferFunction{Continuous}, a; balanced = true)
```

Rescale the time axis (change time unit) of `sys`

.

For systems where the dominant time constants are very far from 1, e.g., in electronics, rescaling the time axis may be beneficial for numerical performance, in particular for continuous-time simulations.

Scaling of time for a function $f(t)$ with Laplace transform $F(s)$ can be stated as

\[f(at) \leftrightarrow \dfrac{1}{a} F\big(\dfrac{s}{a}\big)\]

The keyword argument `balanced`

indicates whether or not to apply a balanced scaling on the `B`

and `C`

matrices. For statespace systems, this defaults to false since it changes the state representation, only `B`

will be scaled. For transfer functions, it defaults to true.

**Example:**

The following example show how a system with a time constant on the order of one micro-second is rescaled such that the time constant becomes 1, i.e., the time unit is changed from seconds to micro-seconds.

```
Gs = tf(1, [1e-6, 1]) # micro-second time scale modeled in seconds
Gms = time_scale(Gs, 1e-6) # Change to micro-second time scale
Gms == tf(1, [1, 1]) # Gms now has micro-seconds as time unit
```

The next example illustrates how the time axis of a time-domain simulation changes by time scaling

```
t = 0:0.1:50 # original time axis
a = 10 # Scaling factor
sys1 = ssrand(1,1,5)
res1 = step(sys1, t) # Perform original simulation
sys2 = time_scale(sys, a) # Scale time
res2 = step(sys2, t ./ a) # Simulate on scaled time axis, note the `1/a`
isapprox(res1.y, res2.y, rtol=1e-3, atol=1e-3)
```

`ControlSystemsBase.to_sized`

— Method`to_sized(sys::AbstractStateSpace)`

Return a `HeteroStateSpace`

where the system matrices are of type SizedMatrix.

*NOTE: This function is fundamentally type unstable.* For maximum performance, create the sized system manually, or make use of the function-barrier technique.

`ControlSystemsBase.tzeros`

— Method`tzeros(sys)`

Compute the invariant zeros of the system `sys`

. If `sys`

is a minimal realization, these are also the transmission zeros.

`ControlSystemsBase.zpconv`

— Method`zpc(a,r,b,s)`

form `conv(a,r) + conv(b,s)`

where the lengths of the polynomials are equalized by zero-padding such that the addition can be carried out

`ControlSystemsBase.zpk`

— Method```
zpk(gain[, Ts])
zpk(num, den, k[, Ts])
zpk(sys)
```

Create transfer function on zero pole gain form. The numerator and denominator are represented by their poles and zeros.

`sys::TransferFunction{SisoZpk{T,TR}} = k*numerator/denominator`

where `T`

is the type of `k`

and `TR`

the type of the zeros/poles, usually Float64 and Complex{Float64}.

`num`

: the roots of the numerator polynomial. Either scalar or vector to create SISO systems

or an array of vectors to create MIMO system.

`den`

: the roots of the denominator polynomial. Either vector to create SISO systems

or an array of vectors to create MIMO system.

`k`

: The gain of the system. Obs, this is not the same as`dcgain`

.`Ts`

: Sample time if discrete time system.

Other uses:

`zpk(sys)`

: Convert`sys`

to`zpk`

form.`zpk("s")`

: Create the transferfunction`s`

.

`ControlSystemsBase.zpkdata`

— Method`z, p, k = zpkdata(sys)`

Compute the zeros, poles, and gains of system `sys`

.

**Returns**

`z`

: Matrix{Vector{ComplexF64}}, (ny × nu)`p`

: Matrix{Vector{ComplexF64}}, (ny × nu)`k`

: Matrix{Float64}, (ny × nu)

`LinearAlgebra.lyap`

— Method`lyap(A, Q; kwargs...)`

Compute the solution `X`

to the discrete Lyapunov equation `AXA' - X + Q = 0`

.

Uses `MatrixEquations.lyapc / MatrixEquations.lyapd`

. For keyword arguments, see the docstring of `ControlSystemsBase.MatrixEquations.lyapc / ControlSystemsBase.MatrixEquations.lyapd`

`LinearAlgebra.norm`

— Function`norm(sys, p=2; tol=1e-6)`

`norm(sys)`

or `norm(sys,2)`

computes the H2 norm of the LTI system `sys`

.

`norm(sys, Inf)`

computes the H∞ norm of the LTI system `sys`

. The H∞ norm is the same as the H∞ for stable systems, and Inf for unstable systems. If the peak gain frequency is required as well, use the function `hinfnorm`

instead. See `hinfnorm`

for further documentation.

`tol`

is an optional keyword argument, used only for the computation of L∞ norms. It represents the desired relative accuracy for the computed L∞ norm (this is not an absolute certificate however).

`sys`

is first converted to a `StateSpace`

model if needed.

`RobustAndOptimalControl.Disk`

— Type`Disk`

Represents a perturbation disc in the complex plane. `Disk(0.5, 2)`

represents all perturbations in the circle centered at 1.25 with radius 0.75, or in other words, a gain margin of 2 and a pahse margin of 36.9 degrees.

A disk can be converted to a Nyquist exclusion disk by `nyquist(disk)`

and plotted using `plot(disk)`

.

**Arguments:**

`γmin`

: Lower intercept`γmax`

: Upper intercept`c`

: Center`r`

: Radius`ϕm`

: Angle of tangent line through origin.

If γmax < γmin the disk is inverted. See `diskmargin`

for disk margin computations.

`RobustAndOptimalControl.Diskmargin`

— Type`Diskmargin`

The notation follows "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet

**Fields:**

`α`

: The disk margin `ω0`

: The worst-case frequency `f0`

: The destabilizing perturbation `f0`

is a complex number with simultaneous gain and phase variation. This critical perturbation causes an instability with closed-loop pole on the imaginary axis at the critical frequency ω0 `δ0`

: The uncertain element generating f0. `γmin`

: The lower real-axis intercept of the disk (analogous to classical lower gain margin). `γmax`

: The upper real-axis intercept of the disk (analogous to classical upper gain margin). `ϕm`

: is analogous to the classical phase margin. `σ`

: The skew parameter that was used to calculate the margin

Note, `γmax`

and `ϕm`

are in smaller than the classical gain and phase margins sicne the classical margins do not consider simultaneous perturbations in gain and phase.

The "disk" margin becomes a half plane for `α = 2`

and an inverted circle for `α > 2`

. In this case, the upper gain margin is infinite. See the paper for more details, in particular figure 6.

`RobustAndOptimalControl.ExtendedStateSpace`

— Type`ExtendedStateSpace{TE, T} <: AbstractStateSpace{TE}`

A type that represents the two-input, two-output system

```
z ┌─────┐ w
◄──┤ │◄──
│ P │
◄──┤ │◄──
y └─────┘ u
```

where

`z`

denotes controlled outputs (sometimes called performance outputs)`y`

denotes measured outputs`w`

denotes external inputs, such as disturbances or references`u`

denotes control inputs

The call `lft(P, K)`

forms the (lower) linear fractional transform

```
z ┌─────┐ w
◄──┤ │◄──
│ P │
┌──┤ │◄─┐
│y └─────┘ u│
│ │
│ ┌─────┐ │
│ │ │ │
└─►│ K ├──┘
│ │
└─────┘
```

i.e., closing the lower loop around `K`

.

An `ExtendedStateSpace`

can be converted to a standard `StateSpace`

by `ss(P)`

, this will keep all inputs and outputs, effectively removing the partitioning only.

When `feedback`

is called on this type, defaults are automatically set for the feedback indices. Other functions defined for this type include

`system_mapping`

`performance_mapping`

`noise_mapping`

`lft`

`feedback`

has special overloads that sets defaults connections for`ExtendedStateSpace`

.

and the following design functions expect `ExtendedStateSpace`

as inputs

`hinfsynthesize`

`h2synthesize`

`LQGProblem`

(also accepts other types)

`RobustAndOptimalControl.ExtendedStateSpace`

— Method`se = ExtendedStateSpace(s::AbstractStateSpace; kwargs...)`

The conversion from a regular statespace object to an `ExtendedStateSpace`

creates the following system by default

\[\begin{bmatrix} A & B & B \\ C & D & D \\ C & D & D \end{bmatrix}\]

i.e., the system and performance mappings are identical, `system_mapping(se) == performance_mapping(se) == s`

. However, all matrices `B1, B2, C1, C2; D11, D12, D21, D22`

are overridable by a corresponding keyword argument. In this case, the controlled outputs are the same as measured outputs.

Related: `se = convert(ExtendedStateSpace, s::StateSpace)`

produces an `ExtendedStateSpace`

with empty `performance_mapping`

from w->z such that `ss(se) == s`

.

`RobustAndOptimalControl.LQGProblem`

— Type`G = LQGProblem(sys::ExtendedStateSpace, Q1, Q2, R1, R2; qQ=0, qR=0, SQ=nothing, SR=nothing)`

Return an LQG object that describes the closed control loop around the process `sys=ss(A,B,C,D)`

where the controller is of LQG-type. The controller is specified by weight matrices `Q1,Q2`

that penalizes state deviations and control signal variance respectively, and covariance matrices `R1,R2`

which specify state drift and measurement covariance respectively.

`sys`

is an extended statespace object where the upper channel corresponds to disturbances to performance variables (w→z), and the lower channel corresponds to inputs to outputs (u→y), such that `lft(sys, K)`

forms the closed-loop transfer function from external inputs/disturbances to performance variables.

`qQ`

and `qR`

can be set to incorporate loop-transfer recovery, i.e.,

```
L = lqr(A, B, Q1+qQ*C'C, Q2)
K = kalman(A, C, R1+qR*B*B', R2)
```

Increasing `qQ`

will add more cost in output direction, e.g., encouraging the use of cheap control, while increasing `qR`

adds fictious dynamics noise, makes the observer faster in the direction we control.

**Example**

In this example we will control a MIMO system with one unstable pole and one unstable zero. When the system has both unstable zeros and poles, there are fundamental limitations on performance. The unstable zero is in this case faster than the unstable pole, so the system is controllable. For good performance, we want as large separation between the unstable zero dynamics and the unstable poles as possible.

```
s = tf("s")
P = [1/(s+1) 2/(s+2); 1/(s+3) 1/(s-1)]
sys = ExtendedStateSpace(ss(P)) # Controlled outputs same as measured outputs and state noise affects at inputs only.
eye(n) = Matrix{Float64}(I,n,n) # For convinience
qQ = 0
qR = 0
Q1 = 10eye(2)
Q2 = 1eye(2)
R1 = 1eye(2)
R2 = 1eye(2)
G = LQGProblem(sys, Q1, Q2, R1, R2, qQ=qQ, qR=qR)
T = comp_sensitivity(G)
S = sensitivity(G)
Gcl = closedloop(G)*static_gain_compensation(G)
plot(
sigmaplot([S,T, Gcl],exp10.(range(-3, stop=3, length=1000)), lab=["S" "T" "Gry"]),
plot(step(Gcl, 5))
)
```

**Extended help**

Several functions are defined for instances of `LQGProblem`

`RobustAndOptimalControl.LQGProblem`

— Method`LQGProblem(P::ExtendedStateSpace)`

If only an `ExtendedStateSpace`

system is provided, e.g. from `hinfpartition`

, the system `P`

is assumed to correspond to the H₂ optimal control problem with

```
C1'C1 = Q1
D12'D12 = Q2
SQ = C1'D12 # Cross term
B1*B1' = R1
D21*D21' = R2
SR = B1*D21' # Cross term
```

and an `LQGProblem`

with the above covariance matrices is returned. The system description in the returned LQGProblem will have `B1 = C1 = I`

. See Ch. 14 in Robust and optimal control for reference.

**Example:**

All the following ways of obtaining the H2 optimal controller are (almost) equivalent

```
using Test
G = ss(tf(1, [10, 1]))
WS = tf(1, [1, 1e-6])
WU = makeweight(1e-2, 0.1, 100)
Gd = hinfpartition(G, WS, WU, [])
K, Gcl = h2synthesize(Gd) # First option, using H2 formulas
K2, Gcl2 = h2synthesize(Gd, 1000) # Second option, using H∞ formulas with large γ
lqg = LQGProblem(Gd) # Third option, convert to an LQGProblem and obtain controller
K3 = -observer_controller(lqg)
@test h2norm(lft(Gd, K )) ≈ 3.0568 atol=1e-3
@test h2norm(lft(Gd, K2)) ≈ 3.0568 atol=1e-3
@test h2norm(lft(Gd, K3)) ≈ 3.0568 atol=1e-3
```

`RobustAndOptimalControl.NamedStateSpace`

— TypeSee `named_ss`

for a convenient constructor.

`RobustAndOptimalControl.UncertainSS`

— Type`UncertainSS{TE} <: AbstractStateSpace{TE}`

Represents LFT_u(M, Diagonal(Δ))

`RobustAndOptimalControl.nyquistcircles`

— Type`nyquistcircles(w, centers, radii)`

Plot the nyquist curve with circles. It only makes sense to call this function if the circles represent additive uncertainty, i.e., were calculated with `relative=false`

.

See also `fit_complex_perturbations`

`RobustAndOptimalControl.δ`

— Type`δ(N=32)`

Create an uncertain element of `N`

uniformly distributed samples ∈ [-1, 1]

`ControlSystemsBase.G_CS`

— Method`G_CS(l::LQGProblem)`

`ControlSystemsBase.G_PS`

— Method`G_PS(l::LQGProblem)`

`ControlSystemsBase.input_comp_sensitivity`

— Method`input_comp_sensitivity(l::LQGProblem)`

`ControlSystemsBase.input_sensitivity`

— Method`input_sensitivity(l::LQGProblem)`

`ControlSystemsBase.output_comp_sensitivity`

— Method`output_comp_sensitivity(l::LQGProblem)`

`ControlSystemsBase.output_sensitivity`

— Method`output_sensitivity(l::LQGProblem)`

`ControlSystemsBase.ss`

— Function`ss(A, B1, B2, C1, C2, D11, D12, D21, D22 [, Ts])`

Create an `ExtendedStateSpace`

.

`DescriptorSystems.dss`

— Method`DescriptorSystems.dss(sys::AbstractStateSpace)`

Convert `sys`

to a descriptor statespace system from DescriptorSystems.jl

`RobustAndOptimalControl.add_disturbance`

— Method`add_disturbance(sys::StateSpace, Ad::Matrix, Cd::Matrix)`

See CCS pp. 144

**Arguments:**

`sys`

: System to augment`Ad`

: The dynamics of the disturbance`Cd`

: How the disturbance states affect the states of`sys`

. This matrix has the shape (sys.nx, size(Ad, 1))

See also `add_low_frequency_disturbance`

, `add_resonant_disturbance`

`RobustAndOptimalControl.add_input_differentiator`

— Function`add_input_differentiator(sys::StateSpace, ui = 1:sys.nu; goodwin=false)`

Augment the output of `sys`

with the difference `u(k+1)-u(k)`

**Arguments:**

`ui`

: An index or vector of indices indicating which inputs to differentiate.`goodwin`

: If true, the difference operator will use the Goodwin δ operator, i.e.,`(u(k+1)-u(k)) / sys.Ts`

.

The augmented system will have the matrices

`[A 0; 0 0] [B; I] [C 0; 0 -I] [D; I]`

with `length(ui)`

added states and outputs.

`RobustAndOptimalControl.add_input_integrator`

— Function`add_input_integrator(sys::StateSpace, ui = 1, ϵ = 0)`

Augment the output of `sys`

with the integral of input at index `ui`

, i.e., `y_aug = [y; ∫u[ui]]`

See also `add_low_frequency_disturbance`

`RobustAndOptimalControl.add_low_frequency_disturbance`

— Method`add_low_frequency_disturbance(sys::StateSpace, Ai::Integer; ϵ = 0)`

A disturbance affecting only state `Ai`

.

`RobustAndOptimalControl.add_low_frequency_disturbance`

— Method`add_low_frequency_disturbance(sys::StateSpace; ϵ = 0, measurement = false)`

Augment `sys`

with a low-frequency (integrating if `ϵ=0`

) disturbance model. If an integrating input disturbance is used together with an observer, the controller will have integral action.

**Arguments:**

`ϵ`

: Move the integrator pole`ϵ`

into the stable region.`measurement`

: If true, the disturbance is a measurement disturbance, otherwise it's an input diturbance.

`RobustAndOptimalControl.add_measurement_disturbance`

— Method`add_measurement_disturbance(sys::StateSpace{Continuous}, Ad::Matrix, Cd::Matrix)`

`RobustAndOptimalControl.add_output_differentiator`

— Function`add_differentiator(sys::StateSpace{<:Discrete})`

Augment the output of `sys`

with the numerical difference (discrete-time derivative) of output, i.e., `y_aug = [y; (y-y_prev)/sys.Ts]`

To add both an integrator and a differentiator to a SISO system, use

`Gd = add_output_integrator(add_output_differentiator(G), 1)`

`RobustAndOptimalControl.add_output_integrator`

— Function`add_output_integrator(sys::StateSpace{<:Discrete}, ind = 1; ϵ = 0)`

Augment the output of `sys`

with the integral of output at index `ind`

, i.e., `y_aug = [y; ∫y[ind]]`

To add both an integrator and a differentiator to a SISO system, use

`Gd = add_output_integrator(add_output_differentiator(G), 1)`

Note: numerical integration is subject to numerical drift. If the output of the system corresponds to, e.g., a velocity reference and the integral to position reference, consider methods for mitigating this drift.

`RobustAndOptimalControl.add_resonant_disturbance`

— Method`add_resonant_disturbance(sys::AbstractStateSpace, ω, ζ, Bd::AbstractArray)`

`Bd`

: The disturbance input matrix.

`RobustAndOptimalControl.add_resonant_disturbance`

— Method`add_resonant_disturbance(sys::StateSpace{Continuous}, ω, ζ, Ai::Int; measurement = false)`

Augment `sys`

with a resonant disturbance model.

**Arguments:**

`ω`

: Frequency`ζ`

: Relative damping.`Ai`

: The affected state`measurement`

: If true, the disturbace is acting on the output, this will cause the controller to have zeros at ω (roots of poly s² + 2ζωs + ω²). If false, the disturbance is acting on the input, this will cause the controller to have poles at ω (roots of poly s² + 2ζωs + ω²).

`RobustAndOptimalControl.baltrunc2`

— Method`sysr, hs = baltrunc2(sys::LTISystem; residual=false, n=missing, kwargs...)`

Compute the a balanced truncation of order `n`

and the hankel singular values

For keyword arguments, see the docstring of `DescriptorSystems.gbalmr`

, reproduced below

```
gbalmr(sys, balance = false, matchdc = false, ord = missing, offset = √ϵ,
atolhsv = 0, rtolhsv = nϵ, atolmin = atolhsv, rtolmin = rtolhsv,
atol = 0, atol1 = atol, atol2 = atol, rtol, fast = true) -> (sysr, hs)
```

Compute for a proper and stable descriptor system `sys = (A-λE,B,C,D)`

with the transfer function matrix `G(λ)`

, a reduced order realization `sysr = (Ar-λEr,Br,Cr,Dr)`

and the vector `hs`

of decreasingly ordered Hankel singular values of the system `sys`

. If `balance = true`

, a balancing-based approach is used to determine a reduced order minimal realization of the form `sysr = (Ar-λI,Br,Cr,Dr)`

. For a continuous-time system `sys`

, the resulting realization `sysr`

is balanced, i.e., the controllability and observability grammians are equal and diagonal. If additonally `matchdc = true`

, the resulting `sysr`

is computed using state rezidualization formulas (also known as *singular perturbation approximation*) which additionally preserves the DC-gain of `sys`

. In this case, the resulting realization `sysr`

is balanced (for both continuous- and discrete-time systems). If `balance = false`

, an enhanced accuracy balancing-free approach is used to determine the reduced order system `sysr`

.

If `ord = nr`

, the resulting order of `sysr`

is `min(nr,nrmin)`

, where `nrmin`

is the order of a minimal realization of `sys`

determined as the number of Hankel singular values exceeding `max(atolmin,rtolmin*HN)`

, with `HN`

, the Hankel norm of `G(λ)`

. If `ord = missing`

, the resulting order is chosen as the number of Hankel singular values exceeding `max(atolhsv,rtolhsv*HN)`

.

To check the stability of the eigenvalues of the pencil `A-λE`

, the real parts of eigenvalues must be less than `-β`

for a continuous-time system or the moduli of eigenvalues must be less than `1-β`

for a discrete-time system, where `β`

is the stability domain boundary offset. The offset `β`

to be used can be specified via the keyword parameter `offset = β`

. The default value used for `β`

is `sqrt(ϵ)`

, where `ϵ`

is the working machine precision.

The keyword arguments `atol1`

, `atol2`

, and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

, the absolute tolerance for the nonzero elements of `E`

, and the relative tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

and `E`

. The default relative tolerance is `nϵ`

, where `ϵ`

is the working *machine epsilon* and `n`

is the order of the system `sys`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

and `atol2 = atol`

.

If `E`

is singular, the uncontrollable infinite eigenvalues of the pair `(A,E)`

and the non-dynamic modes are elliminated using minimal realization techniques. The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

.

Method: For the order reduction of a standard system, the balancing-free method of [1] or the balancing-based method of [2] are used. For a descriptor system the balancing related order reduction methods of [3] are used. To preserve the DC-gain of the original system, the singular perturbation approximation method of [4] is used in conjunction with the balancing-based or balancing-free approach of [5].

References

[1] A. Varga. Efficient minimal realization procedure based on balancing. In A. El Moudni, P. Borne, and S.G. Tzafestas (Eds.), Prepr. of the IMACS Symp. on Modelling and Control of Technological Systems, Lille, France, vol. 2, pp.42-47, 1991.

[2] M. S. Tombs and I. Postlethwaite. Truncated balanced realization of a stable non-minimal state-space system. Int. J. Control, vol. 46, pp. 1319–1330, 1987.

[3] T. Stykel. Gramian based model reduction for descriptor systems. Mathematics of Control, Signals, and Systems, 16:297–319, 2004.

[4] Y. Liu Y. and B.D.O. Anderson Singular Perturbation Approximation of Balanced Systems, Int. J. Control, Vol. 50, pp. 1379-1405, 1989.

[5] Varga A. Balancing-free square-root algorithm for computing singular perturbation approximations. Proc. 30-th IEEE CDC, Brighton, Dec. 11-13, 1991, Vol. 2, pp. 1062-1065.

`RobustAndOptimalControl.baltrunc_coprime`

— Method`sysr, hs, info = baltrunc_coprime(sys; residual = false, n = missing, factorization::F = DescriptorSystems.gnlcf, kwargs...)`

Compute a balanced truncation of the left coprime factorization of `sys`

. See `baltrunc2`

for additional keyword-argument help.

Coprime-factor reduction performs a coprime factorization of the model into $P(s) = M(s)^{-1}N(s)$ where $M$ and $N$ are stable factors even if $P$ contains unstable modes. After this, the system $NM = \begin{bmatrix}N & M \end{bmatrix}$ is reduced using balanced truncation and the final reduced-order model is formed as $P_r(s) = M_r(s)^{-1}N_r(s)$. For this method, the Hankel signular values of $NM$ are reported and the reported errors are $||NM - N_rM_r||_\infty$. This method is of particular interest in closed-loop situations, where a model-reduction error $||NM - N_rM_r||_\infty$ no greater than the normalized-coprime margin of the plant and the controller, guaratees that the closed loop remains stable when either $P$ or $K$ are reduced. The normalized-coprime margin can be computed with `ncfmargin(P, K)`

(`ncfmargin`

).

**Arguments:**

`factorization`

: The function to perform the coprime factorization. A non-normalized factorization may be used by passing`RobustAndOptimalControl.DescriptorSystems.glcf`

.`kwargs`

: Are passed to`DescriptorSystems.gbalmr`

`RobustAndOptimalControl.baltrunc_unstab`

— Function`baltrunc_unstab(sys::LTISystem; residual = false, n = missing, kwargs...)`

Balanced truncation for unstable models. An additive decomposition of sys into `sys = sys_stable + sys_unstable`

is performed after which `sys_stable`

is reduced. The order `n`

must not be less than the number of unstable poles.

See `baltrunc2`

for other keyword arguments.

`RobustAndOptimalControl.bilinearc2d`

— Method`bilinearc2d(Ac::AbstractArray, Bc::AbstractArray, Cc::AbstractArray, Dc::AbstractArray, Ts::Number; tolerance=1e-12)`

Balanced Bilinear transformation in State-Space. This method computes a discrete time equivalent of a continuous-time system, such that

\[G_d(z) = s2z[G_c(s)]\]

in a manner which accomplishes the following (i) Preserves the infinity L-infinity norm over the transformation (ii) Finds a system which balances B and C, in the sense that $||B||_2=||C||_2$ (iii) Satisfies $G_c(s) = z2s[s2z[G_c(s)]]$ for some map z2s[]

`RobustAndOptimalControl.bilinearc2d`

— Method`bilinearc2d(sys::ExtendedStateSpace, Ts::Number)`

Applies a Balanced Bilinear transformation to a discrete-time extended statespace object

`RobustAndOptimalControl.bilinearc2d`

— Method`bilinearc2d(sys::StateSpace, Ts::Number)`

Applies a Balanced Bilinear transformation to a discrete-time statespace object

`RobustAndOptimalControl.bilineard2c`

— Method`bilineard2c(Ad::AbstractArray, Bd::AbstractArray, Cd::AbstractArray, Dd::AbstractArray, Ts::Number; tolerance=1e-12)`

Balanced Bilinear transformation in State-Space. This method computes a continuous time equivalent of a discrete time system, such that

`G_c(z) = z2s[G_d(z)]`

in a manner which accomplishes the following (i) Preserves the infinity L-infinity norm over the transformation (ii) Finds a system which balances B and C, in the sense that ||B||*2=||C||*2 (iii) Satisfies G*d(z) = s2z[z2s[G*d(z)]] for some map s2z[]

`RobustAndOptimalControl.bilineard2c`

— Method`bilineard2c(sys::ExtendedStateSpace)`

Applies a Balanced Bilinear transformation to continuous-time extended statespace object

`RobustAndOptimalControl.bilineard2c`

— Method`bilineard2c(sys::StateSpace)`

Applies a Balanced Bilinear transformation to continuous-time statespace object

`RobustAndOptimalControl.blocksort`

— Method`blocks, M = blocksort(P::UncertainSS)`

Returns the block structure of `P.Δ`

as well as `P.M`

permuted according to the sorted block structure. `blocks`

is a vector of vectors with the block structure of perturbation blocks as described by μ-tools, i.e.

`[-N, 0]`

denotes a repeated real block of size`N`

`[N, 0]`

denotes a repeated complex block of size`N`

`[ny, nu]`

denotes a full complex block of size`ny × nu`

`RobustAndOptimalControl.broken_feedback`

— Method`broken_feedback(L, i)`

Closes all loops in square MIMO system `L`

except for loops `i`

. Forms L1 in fig 14. of "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet

`RobustAndOptimalControl.closedloop`

— Function`closedloop(l::LQGProblem, L = lqr(l), K = kalman(l))`

Closed-loop system as defined in Glad and Ljung eq. 8.28. Note, this definition of closed loop is not the same as lft(P, K), which has B1 instead of B2 as input matrix. Use `lft(l)`

to get the system from disturbances to controlled variables `w -> z`

.

The return value will be the closed loop from reference only, other disturbance signals (B1) are ignored. See `feedback`

for a more advanced option.

Use `static_gain_compensation`

to adjust the gain from references acting on the input B2, `dcgain(closedloop(l))*static_gain_compensation(l) ≈ I`

`RobustAndOptimalControl.connect`

— Method`connect(systems, connections; w1, z1 = (:), verbose = true, kwargs...)`

Create complicated feedback interconnection.

Addition and subtraction nodes are achieved by creating a linear combination node, i.e., a system with a `D`

matrix only.

**Arguments:**

`systems`

: A vector of named systems to be connected`connections`

: a vector of pairs indicating output => input mappings.`u1`

: input mappings (alternative input argument)`y1`

: output mappings (alternative input argument)

`w1`

: external signals`z1`

: outputs (can overlap with`y1`

)`verbose`

: Issue warnings for signals that have no connection

Note: Positive feedback is used, controllers that are intended to be connected with negative feedback must thus be negated.

Example: The following complicated feedback interconnection

```
yF
┌────────────────────────────────┐
│ │
┌───────┐ │ ┌───────┐ yR ┌─────────┐ │ ┌───────┐
uF │ │ │ │ ├──────► │ yC │ uP│ │ yP
────► F ├─┴──► R │ │ C ├────+────► P ├────┬────►
│ │ │ │ ┌──► │ │ │ │
└───────┘ └───────┘ │ └─────────┘ └───────┘ │
│ │
└───────────────────────────────────┘
```

can be created by

```
F = named_ss(ssrand(1, 1, 2, proper=true), x=:xF, u=:uF, y=:yF)
R = named_ss(ssrand(1, 1, 2, proper=true), x=:xR, u=:uR, y=:yR)
C = named_ss(ssrand(1, 1, 2, proper=true), x=:xC, u=:uC, y=:yC)
P = named_ss(ssrand(1, 1, 3, proper=true), x=:xP, u=:uP, y=:yP)
addP = sumblock("uP = yF + yC") # Sum node before P
addC = sumblock("uC = yR - yP") # Sum node before C
connections = [
:yP => :yP # Output to input
:uP => :uP
:yC => :yC
:yF => :yF
:yF => :uR
:uC => :uC
:yR => :yR
]
w1 = [:uF] # External inputs
G = connect([F, R, C, P, addP, addC], connections; w1)
```

If an external input is to be connected to multiple points, use a `splitter`

to split up the signal into a set of unique names which are then used in the connections.

`RobustAndOptimalControl.controller_reduction`

— Function`controller_reduction(P::ExtendedStateSpace, K, r, out=true; kwargs...)`

Minimize ||(K-Kᵣ) W||∞ if out=false ||W (K-Kᵣ)||∞ if out=true See Robust and Optimal Control Ch 19.1 out indicates if the weight will be applied as output or input weight.

This function expects a *positive feedback controller `K`

.

This method corresponds to the methods labelled SW1/SW2(SPA) in Andreas Varga, "Controller Reduction Using Accuracy-Enhancing Methods" SW1 is the default method, corresponding to `out=true`

.

This method does not support unstable controllers. See the reference above for alternatives. See also `stab_unstab`

and `baltrunc_unstab`

.

`RobustAndOptimalControl.controller_reduction_plot`

— Function`controller_reduction_plot(G, K)`

Plot the normalized-coprime margin (`ncfmargin`

) as a function of controller order when `baltrunc_coprime`

is used to reduce the controller. Red, orange and green bands correspond to rules of thumb for bad, okay and good coprime uncertainty margins. A value of 0 indicate an unstable closed loop.

If `G`

is an ExtendedStateSpace system, a second plot will be shown indicating the $H_∞$ norm between inputs and performance outputs $||T_{zw}||_\infty$ when the function `controller_reduction`

is used to reduce the controller.

The order of the controller can safely be reduced as long as the normalized coprime margin remains sufficiently large. If the controller contains integrators, it may be advicable to protect the integrators from the reduction, e.g., if the controller is obtained using `glover_mcfarlane`

, perform the reduction on `info.Gs, info.Ks`

rather than on `K`

, and form `Kr`

using the reduced `Ks`

.

See `glover_mcfarlane`

or the docs for an example.

`RobustAndOptimalControl.controller_reduction_weight`

— Method`controller_reduction_weight(P::ExtendedStateSpace, K)`

Lemma 19.1 See Robust and Optimal Control Ch 19.1

`RobustAndOptimalControl.dare3`

— Method`dare3(P::AbstractStateSpace, Q1::AbstractMatrix, Q2::AbstractMatrix, Q3::AbstractMatrix; full=false)`

Solve the discrete-time algebraic Riccati equation for a discrete LQR cost augmented with control differences

\[x^{T} Q_1 x + u^{T} Q_2 u + Δu^{T} Q_3 Δu, \quad Δu = u(k) - u(k-1)\]

If `full`

, the returned matrix will include the state `u(k-1)`

, otherwise the returned matrix will be of the same size as `Q1`

.

`RobustAndOptimalControl.diskmargin`

— Function```
diskmargin(L, σ = 0)
diskmargin(L, σ::Real, ω)
```

Calculate the disk margin of LTI system `L`

. `L`

is supposed to be a loop-transfer function, i.e., it should be square. If `L = PC`

the disk margin for output perturbations is computed, whereas if `L = CP`

, input perturbations are considered. If the method `diskmargin(P, C, args...)`

is used, both are computed. Note, if `L`

is MIMO, a simultaneous margin is computed, see `loop_diskmargin`

for single loop margins of MIMO systems.

The implementation and notation follows "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet.

The margins are aviable as fields of the returned objects, see `Diskmargin`

.

**Arguments:**

`L`

: A loop-transfer function.`σ`

: If little is known about the distribution of gain variations then σ = 0 is a reasonable choice as it allows for a gain increase or decrease by the same relative amount.*The choice σ < 0*is justified if the gain can decrease by a larger factor than it can increase. Similarly,*the choice σ > 0*is justified when the gain can increase by a larger factor than it can decrease.*If σ = −1*then the disk margin condition is αmax = inv(MT). This margin is related to the robust stability condition for models with multiplicative uncertainty of the form P (1 + δ). If σ = +1 then the disk margin condition is αmax = inv(MS)`kwargs`

: Are sent to the`hinfnorm`

calculation`ω`

: If a vector of frequencies is supplied, the frequency-dependent disk margin will be computed, see example below.

**Example:**

```
L = tf(25, [1,10,10,10])
dm = diskmargin(L, 0)
plot(dm) # Plot the disk margin to illustrate maximum allowed simultaneous gain and phase variations.
nyquistplot(L)
plot!(dm, nyquist=true) # plot a nyquist exclusion disk. The Nyquist curve will be tangent to this disk at `dm.ω0`
nyquistplot!(dm.f0*L) # If we perturb the system with the worst-case perturbation `f0`, the curve will pass through the critical point -1.
## Frequency-dependent margin
w = exp10.(LinRange(-2, 2, 500))
dms = diskmargin(L, 0, w)
plot(dms; lower=true, phase=true)
```

See also `ncfmargin`

and `loop_diskmargin`

.

`RobustAndOptimalControl.diskmargin`

— Method`diskmargin(P::LTISystem, C::LTISystem, σ, w::AbstractVector, args...; kwargs...)`

Simultaneuous diskmargin at outputs, inputs and input/output simultaneously of `P`

. Returns a named tuple with the fields `input, output, simultaneous_input, simultaneous_output, simultaneous`

where `input`

and `output`

represent loop-at-a-time margins, `simultaneous_input`

is the margin for simultaneous perturbations on all inputs and `simultaneous`

is the margin for perturbations on all inputs and outputs simultaneously.

Note: simultaneous margins are more conservative than single-loop margins and are likely to be much lower than the single-loop margins. Indeed, with several simultaneous perturbations, it's in general easier to make the system unstable. It's not uncommon for a simultaneous margin involving two signals to be on the order of half the size of the single-loop margins.

See also `ncfmargin`

and `loop_diskmargin`

.

`RobustAndOptimalControl.diskmargin`

— Method`diskmargin(L::LTISystem, σ::Real, ω)`

Calculate the diskmargin at a particular frequency or vector of frequencies. If `ω`

is a vector, you get a frequency-dependent diskmargin plot if you plot the returned value. See also `ncfmargin`

.

`RobustAndOptimalControl.expand_symbol`

— Method`expand_symbol(s::Symbol, n::Int)`

Takes a symbol and an integer and returns a vector of symbols with increasing numbers appended to the end. E.g., (:x, 3) -> [:x1, :x2, :x3]

The short-hand syntax `s^n`

is also available, e.g., `:x^3 == expand_symbol(:x, 3)`

.

Useful to create signal names for named systems.

`RobustAndOptimalControl.extended_controller`

— Function`extended_controller(l::LQGProblem, L = lqr(l), K = kalman(l))`

Returns an expression for the controller that is obtained when state-feedback `u = -L(xᵣ-x̂)`

is combined with a Kalman filter with gain `K`

that produces state estimates x̂. The controller is an instance of `ExtendedStateSpace`

where `C2 = -L, D21 = L`

and `B2 = K`

.

The returned system has *inputs*`[xᵣ; y]`

and outputs the control signal `u`

. If a reference model `R`

is used to generate state references `xᵣ`

, the controller from `e = ry - y -> u`

is given by

```
Ce = extended_controller(l)
Ce = named_ss(Ce; x = :xC, y = :u, u = [R.y; :y^l.ny]) # Name the inputs of Ce the same as the outputs of `R`.
connect([R, Ce]; u1 = R.y, y1 = R.y, w1 = [:ry^l.ny, :y^l.ny], z1=[:u])
```

Since the negative part of the feedback is built into the returned system, we have

```
C = observer_controller(l)
Ce = extended_controller(l)
system_mapping(Ce) == -C
```

`RobustAndOptimalControl.extended_controller`

— Method`extended_controller(K::AbstractStateSpace)`

Takes a controller and returns an `ExtendedStateSpace`

version which has augmented input `[r; y]`

and output `y`

(`z`

output is 0-dim).

`RobustAndOptimalControl.feedback_control`

— Method`G = feedback_control(P, K)`

Return the (negative feedback) closed-loop system from input of `K`

to output of `P`

while outputing also the control signal (output of `K`

), i.e., `G`

maps references to `[y; u]`

**Example:**

The following are two equivalent ways of achieving the same thing

```
G = ssrand(3,4,2)
K = ssrand(4,3,2)
Gcl1 = feedback_control(G, K) # First option
# Second option using named systems and connect
G = named_ss(G, :G)
K = named_ss(K, :K)
S = sumblock("Ku = r - Gy", n=3) # Create a sumblock that computes r - Gy for vectors of length 3
z1 = [G.y; K.y] # Output both plant and controller outputs
w1 = :r^3 # Extenal inputs are the three references into the sum block
connections = [K.y .=> G.u; G.y .=> G.y; K.u .=> K.u] # Since the sumblock uses the same names as the IO signals of G,K, we can reuse these names here
Gcl2 = connect([G, K, S], connections; z1, w1)
@test linfnorm(minreal(Gcl1 - Gcl2.sys))[1] < 1e-10 # They are the same
```

To include also an input disturbance, use

`Gcl = feedback(K, P, W2=:, Z2=:, Zperm=[(1:ny).+nu; 1:nu]) # y,u from r,d`

See also `extended_gangoffour`

.

`RobustAndOptimalControl.ff_controller`

— Function`ff_controller(l::LQGProblem, L = lqr(l), K = kalman(l))`

Return the feedforward controller $C_{ff}$ that maps references to plant inputs: $u = C_{fb}y + C_{ff}r$

See also `observer_controller`

.

`RobustAndOptimalControl.find_lft`

— Method`l, res = find_lft(sys::StateSpace{<:Any, <:StaticParticles{<:Any, N}}, δ) where N`

NOTE: This function is experimental.

Given an systems `sys`

with uncertain coefficients in the form of `StaticParticles`

, search for a lower linear fractional transformation `M`

such that `lft(M, δ) ≈ sys`

.

`δ`

can be either the source of uncertainty in `sys`

, i.e., a vector of the unique uncertain parameters that were used to create `sys`

. These should be constructed as uniform randomly distributed particles for most robust-control theory to be applicable. `δ`

can also be an integer, in which case a numer of `δ`

sources of uncertainty are automatically created. This could be used for order reduction if the number of uncertainty sources in `sys`

is large.

Note, uncertainty in `sys`

is only supported in `A`

and `B`

, `C`

and `D`

must be deterministic.

Returns `l::LFT`

that internaly contains all four blocks of `M`

as well as `δ`

. Call `ss(l,sys)`

do obtain `lft(M, δ) ≈ sys`

.

Call `Matrix(l)`

to obtain `M = [M11 M12; M21 M22]`

`RobustAndOptimalControl.fit_complex_perturbations`

— Method`centers, radii = fit_complex_perturbations(P, w; relative=true, nominal=:mean)`

For each frequency in `w`

, fit a circle in the complex plane that contains all models in the model set `P`

, represented as an `LTISystem`

with `Particles`

coefficients. Note, the resulting radii can be quite unstable if the number of particles is small, in particular if the particles are normally distributed instead of uniformly.

If `realtive = true`

, circles encompassing `|(P - Pn)/Pn|`

will be returned (multiplicative/relative uncertainty). If `realtive = false`

, circles encompassing `|P - Pn|`

will be returned (additive uncertainty).

If `nominal = :mean`

, the mean of `P`

will be used as nominal model. If `nominal = :first`

, the first particle will be used. See `MonteCarloMeasurements.with_nominal`

to set the nominal value in the first particle. If `nominal = :center`

, the middle point `(pmaximum(ri)+pminimum(ri))/2`

will be used. This typically gives the smallest circles.

See also `nyquistcircles`

to plot circles (only if relative=false).

`RobustAndOptimalControl.frequency_weighted_reduction`

— Function`sysr, hs = frequency_weighted_reduction(G, Wo, Wi, r=nothing; residual=true)`

Find Gr such that $||Wₒ(G-Gr)Wᵢ||∞$ is minimized. For a realtive reduction, set Wo = inv(G) and Wi = I.

If `residual = true`

, matched static gain is achieved through "residualization", i.e., setting

\[0 = A_{21}x_{1} + A_{22}x_{2} + B_{2}u\]

where indices 1/2 correspond to the remaining/truncated states respectively. This choice typically results in a better match in the low-frequency region and a smaller overall error.

Note: This function only handles exponentially stable models. To reduce unstable and marginally stable models, decompose the system into stable and unstable parts using `stab_unstab`

, reduce the stable part and then add the unstable part back.

**Example:**

The following example performs reduction with a frequency focus between frequencies `w1`

and `w2`

.

```
using DSP
w1 = 1e-4
w2 = 1e-1
wmax = 1
fc = DSP.analogfilter(DSP.Bandpass(w1, w2, fs=wmax), DSP.Butterworth(2))
tfc = DSP.PolynomialRatio(fc)
W = tf(DSP.coefb(tfc), DSP.coefa(tfc))
rsys, hs = frequency_weighted_reduction(sys, W, 1)
```

`RobustAndOptimalControl.fudge_inv`

— Function`fudge_inv(s::AbstractStateSpace, ε = 0.001)`

Allow inverting a proper statespace system by adding a tiny (ε) feedthrough term to the `D`

matrix. The system must still be square.

`RobustAndOptimalControl.gain_and_delay_uncertainty`

— Method`gain_and_delay_uncertainty(kmin, kmax, Lmax)`

Return a multiplicative weight to represent the uncertainty coming from neglecting the dynamics `k*exp(-s*L)`

where `k ∈ [kmin, kmax]`

and `L ≤ Lmax`

. This weight is slightly optimistic, an expression for a more exact weight appears in eq (7.27), "Multivariable Feedback Control: Analysis and Design"

See also `neglected_lag`

and `neglected_delay`

.

**Example:**

```
a = 10
P = ss([0 a; -a 0], I(2), [1 a; -a 1], 0) # Plant
W0 = gain_and_delay_uncertainty(0.5, 2, 0.005) |> ss # Weight
W = I(2) + W0*I(2) * uss([δc(), δc()]) # Create a diagonal real uncertainty weighted in frequency by W0
Ps = P*W # Uncertain plant
Psamples = rand(Ps, 500) # Sample the uncertain plant for plotting
w = exp10.(LinRange(-1, 3, 300)) # Frequency vector
bodeplot(Psamples, w)
```

`RobustAndOptimalControl.glover_mcfarlane`

— Function`K, γ, info = glover_mcfarlane(G::AbstractStateSpace{<:Discrete}, γ = 1.1; W1=1, W2=1, strictly_proper=false)`

For discrete systems, the `info`

tuple contains also feedback gains `F, L`

and observer gain `Hkf`

such that the controller on observer form is given by

\[x^+ = Ax + Bu + H_{kf} (Cx - y)\\ u = Fx + L (Cx - y)\]

Note, this controller is *not* strictly proper, i.e., it has a non-zero D matrix. The controller can be transformed to observer form for the scaled plant (`info.Gs`

) by `Ko = observer_controller(info)`

, in which case the following holds `G*K == info.Gs*Ko`

(realizations are different).

If `strictly_proper = true`

, the returned controller `K`

will have `D == 0`

. This can be advantageous in implementations where computational delays are present. In this case, `info.L == 0`

as well.

Ref discrete version: Iglesias, "The Strictly Proper Discrete-Time Controller for the Normalized Left-Coprime Factorization Robust Stabilization Problem"

`RobustAndOptimalControl.glover_mcfarlane`

— Function`K, γ, info = glover_mcfarlane(G::AbstractStateSpace, γ = 1.1; W1=1, W2=1)`

Design a controller for `G`

that maximizes the stability margin ϵ = 1/γ with normalized coprime factor uncertainty using the method of Glover and McFarlane

```
γ = 1/ϵ = ||[K;I] inv(I-G*K)*inv(M)||∞
G = inv(M + ΔM)*(N + ΔN)
```

γ is given as a relative factor above γmin and must be greater than 1, i.e., if γ = 1.1, the controller will be designed for γ = 1.1*γmin.

We want γmin (which is always ≥ 1) as small as possible, and we usually require that γmin is less than 4, corresponding to 25% allowed coprime uncertainty.

Performance modeling is incorporated in the design by calling `glover_mcfarlane`

on the shaped system `Gs = W2*G*W1`

and then forming the controller as `K = W1*Ks*W2`

. Using this formulation, traditional loop shaping can be done on `Gs = W2*G*W1`

. The plant shaping is handled internally if keyword arguments `W1, W2`

are used and the returned controller is already scaled. In this case, `Gs`

and `Ks`

are included in the `info`

named tuple for inspection.

See also `glover_mcfarlane_2dof`

to design a feedforward filter as well and `baltrunc_coprime`

for controller order reduction. When reducing the order of the calculated controller, reduce the order of `info.Ks`

and form `Kr=W1*Ksred*W2`

. Verify the robustness using `ncfmargin(info.Gs, Ksred)`

as well as `ncfmargin(G, Kr)`

.

**Example:**

Example 9.3 from the reference below.

```
using RobustAndOptimalControl, ControlSystemsBase, Plots, Test
G = tf(200, [10, 1])*tf(1, [0.05, 1])^2 |> ss
Gd = tf(100, [10, 1]) |> ss
W1 = tf([1, 2], [1, 1e-6]) |> ss
K, γ, info = glover_mcfarlane(G, 1.1; W1)
@test info.γmin ≈ 2.34 atol=0.005
Gcl = extended_gangoffour(G, K) # Form closed-loop system
fig1 = bodeplot([G, info.Gs, G*K], lab=["G" "" "G scaled" "" "Loop transfer"])
fig2 = bodeplot(Gcl, lab=["S" "KS" "PS" "T"], plotphase=false) # Plot gang of four
fig3 = plot(step(Gd*feedback(1, info.Gs), 3), lab="Initial controller")
plot!(step(Gd*feedback(1, G*K), 3), lab="Robustified")
fig4 = nyquistplot([info.Gs, G*K], ylims=(-2,1), xlims=(-2, 1),
Ms_circles = 1.5,
lab = ["Initial controller" "Robustified"],
title = "Loop transfers with and without robustified controller"
)
plot(fig1, fig2, fig3, fig4)
```

Example of controller reduction: The order of the controller designed above can be reduced maintaining at least 2/3 of the robustness margin like this

```
e,_ = ncfmargin(info.Gs, info.Ks)
Kr, hs, infor = baltrunc_coprime(info.Ks, n=info.Ks.nx)
n = findlast(RobustAndOptimalControl.error_bound(hs) .> 2e/3) # 2/3 e sets the robustness margin
Ksr, hs, infor = baltrunc_coprime(info.Ks; n)
@test ncfmargin(info.Gs, Ksr)[1] >= 2/3 * e
Kr = W1*Ksr
bodeplot([G*K, G*Kr], lab=["L original" "" "L Reduced" ""])
```

This gives a final controller `Kr`

of order 3 instead of order 5, but a very similar robustness margin. You may also call

`controller_reduction_plot(info.Gs, info.Ks)`

to help you select the controller order.

Ref: Sec 9.4.1 of Skogestad, "Multivariable Feedback Control: Analysis and Design"

**Extended help**

Skogestad gives the following general advice:

Scale the plant outputs and inputs. This is very important for most design procedures and is sometimes forgotten. In general, scaling improves the conditioning of the design problem, it enables meaningful analysis to be made of the robustness properties of the feedback system in the frequency domain, and for loop-shaping it can simplify the selection of weights. There are a variety of methods available including normalization with respect to the magnitude of the maximum or average value of the signal in question. If one is to go straight to a design the following variation has proved useful in practice:

- The outputs are scaled such that equal magnitudes of cross-coupling into each of the outputs is equally undesirable.
- Each input is scaled by a given percentage (say 10%) of its expected range of operation. That is, the inputs are scaled to reflect the relative actuator capabilities.

Order the inputs and outputs so that the plant is as diagonal as possible. The relative gain array

`relative_gain_array`

can be useful here. The purpose of this pseudo-diagonalization is to ease the design of the pre- and post-compensators which, for simplicity, will be chosen to be diagonal.Next, we discuss the selection of weights to obtain the shaped plant $G_s = W_2 G W_1$ where $W_1 = W_p W_a W_g$

Select the elements of diagonal pre- and post-compensators $W_p$ and $W_2$ so that the singular values of $W_2 G W_p$ are desirable. This would normally mean high gain at low frequencies, roll-off rates of approximately 20 dB/decade (a slope of about 1) at the desired bandwidth(s), with higher rates at high frequencies. Some trial and error is involved here. $W_2$ is usually chosen as a constant, reflecting the relative importance of the outputs to be controlled and the other measurements being fed back to the controller. For example, if there are feedback measurements of two outputs to be controlled and a velocity signal, then $W_2$ might be chosen to be

`diag([1, 1, 0.1])`

, where 0.1 is in the velocity signal channel. $W_p$ contains the dynamic shaping. Integral action, for low frequency performance; phase-advance for reducing the roll-off rates at crossover, and phase-lag to increase the roll-off rates at high frequencies should all be placed in $W_p$ if desired. The weights should be chosen so that no unstable hidden modes are created in $G_s$.Optional: Introduce an additional gain matrix $W_g$ cascaded with $W_a$ to provide control over actuator usage. $W_g$ is diagonal and is adjusted so that actuator rate limits are not exceeded for reference demands and typical disturbances on the scaled plant outputs. This requires some trial and error.

Robustly stabilize the shaped plant $G_s = W_2 G W_1$ , where $W_1 = W_p W_a W_g$, using

`glover_mcfarlane`

. First, the maximum stability margin $ϵ_{max} = 1/γ_{min}$ is calculated. If the margin is too small, $ϵmax < 0.25$, then go back and modify the weights. Otherwise, a γ-suboptimal controller is synthesized. There is usually no advantage to be gained by using the optimal controller. When $ϵ_{max}$ > 0.25 (respectively $γ_{min}$ < 4) the design is usually successful. In this case, at least 25% coprime factor uncertainty is allowed, and we also find that the shape of the open-loop singular values will not have changed much after robust stabilization. A small value of ϵmax indicates that the chosen singular value loop-shapes are incompatible with robust stability requirements. That the loop-shapes do not change much following robust stabilization if γ is small (ϵ large), is justified theoretically in McFarlane and Glover (1990).Analyze the design and if all the specifications are not met make further modifications to the weights.

Implement the controller. The configuration shown in below has been found useful when compared with the conventional set up. This is because the references do not directly excite the dynamics of $K$, which can result in large amounts of overshoot (classical derivative kick). The constant prefilter ensures a steady-state gain of 1 between r and y, assuming integral action in $W_1$ or $G$ (note, the K returned by this function has opposite sign compared to that of Skogestad, so we use negative feedback here).

Anti-windup can be added to $W_1$ but putting $W_1$ on Hanus form after the synthesis, see `hanus`

.

```
┌─────────┐ ┌────────┐ ┌────────┐
r │ │ us│ │ u │ │ y
───►│(K*W2)(0)├──+──►│ W1 ├─────►│ G ├────┬──►
│ │ │- │ │ │ │ │
└─────────┘ │ └────────┘ └────────┘ │
│ │
│ │
│ ┌────────┐ ┌────────┐ │
│ │ │ ys │ │ │
└───┤ K │◄─────┤ W2 │◄───┘
│ │ │ │
└────────┘ └────────┘
```

Keywords: `nfcsyn`

, `coprimeunc`

`RobustAndOptimalControl.glover_mcfarlane_2dof`

— Function```
K, γ, info = glover_mcfarlane_2dof(G::AbstractStateSpace{Continuous}, Tref::AbstractStateSpace{Continuous}, γ = 1.1, ρ = 1.1;
W1 = 1, Wo = I, match_dc = true, kwargs...)
```

Joint design of feedback and feedforward compensators

\[K = \begin{bmatrix} K_1 & K_2 \end{bmatrix}\]

```
┌──────┐ ┌──────┐ ┌──────┐ ┌─────┐
r │ │ │ │ │ │ │ │
──►│ Wi ├──►│ K1 ├───+───►│ W1 ├───►│ G ├─┐y
│ │ │ │ │ │ │ │ │ │
└──────┘ └──────┘ │ └──────┘ └─────┘ │
│ │
│ ┌──────┐ │
│ │ │ │
└────┤ K2 ◄────────────┘
│ │
└──────┘
```

Where the returned controller `K`

takes the measurement vector `[r; y]`

(positive feedback), i.e., it includes all blocks `Wi, K1, K2, W1`

. If `match_dc = true`

, `Wi`

is automatically computed to make sure the static gain matches `Tref`

exactly, otherwise `Wi`

is set to `I`

. The `info`

named tuple contains the feedforward filter for inspection (`info.K1 = K1*Wi`

).

**Arguments:**

`G`

: Plant model`Tref`

: Reference model`γ`

: Relative γ`ρ`

: Design parameter, typically 1 < ρ < 3. Increase to emphasize model matching at the expense of robustness.`W1`

: Pre-compensator for loop shaping.`Wo`

: Output selction matrix. If there are more measurements than controlled variables, this matrix let's you select which measurements are to be controlled.`kwargs`

: Are sent to`hinfsynthesize`

.

Ref: Sec. 9.4.3 of Skogestad, "Multivariable Feedback Control: Analysis and Design". The reference contains valuable pointers regarding gain-scheduling implementation of the designed controller as an observer with feedback from estimated states. In order to get anti-windup protection when `W1`

contains an integrator, transform `W1`

to self-conditioned Hanus form (using `hanus`

) and implement the controller like this

```
W1h = hanus(W1) # Perform outside loop
# Each iteration
us = filter(Ks, [r; y]) # filter inputs through info.Ks (filter is a fictive function that applies the transfer function)
u = filter(W1h, [us; ua]) # filter us and u-actual (after input saturation) through W1h
ua = clamp(u, lower, upper) # Calculate ua for next iteration as the saturated value of u
```

**Example:**

```
using RobustAndOptimalControl, Plots
P = tf([1, 5], [1, 2, 10]) # Plant
W1 = tf(1,[1, 0]) |> ss # Loop shaping controller
Tref = tf(1, [1, 1])^2 |> ss # Reference model (preferably of same order as P)
K1dof, γ1, info1 = glover_mcfarlane(ss(P), 1.1; W1)
K2dof, γ2, info2 = glover_mcfarlane_2dof(ss(P), Tref, 1.1, 1.1; W1)
G1 = feedback(P*K1dof)
G2 = info2.Gcl
w = exp10.(LinRange(-2, 2, 200))
bodeplot(info2.K1, w, lab="Feedforward filter")
plot([step(G1, 15), step(G2, 15), step(Tref, 15)], lab=["1-DOF" "2-DOF" "Tref"])
```

`RobustAndOptimalControl.h2norm`

— Method`n = h2norm(sys::LTISystem; kwargs...)`

A numerically robust version of `norm`

using DescriptorSystems.jl

For keyword arguments, see the docstring of `DescriptorSystems.gh2norm`

, reproduced below

`gh2norm(sys, fast = true, offset = sqrt(ϵ), atol = 0, atol1 = atol, atol2 = atol, atolinf = atol, rtol = n*ϵ)`

Compute for a descriptor system `sys = (A-λE,B,C,D)`

the `H2`

norm of its transfer function matrix `G(λ)`

. The `H2`

norm is infinite, if `sys`

has unstable poles, or, for a continuous-time, the system has nonzero gain at infinity. To check the stability, the eigenvalues of the *pole pencil*`A-λE`

must have real parts less than `-β`

for a continuous-time system or have moduli less than `1-β`

for a discrete-time system, where `β`

is the stability domain boundary offset. The offset `β`

to be used can be specified via the keyword parameter `offset = β`

. The default value used for `β`

is `sqrt(ϵ)`

, where `ϵ`

is the working machine precision.

For a continuous-time system `sys`

with `E`

singular, a reduced order realization is determined first, without uncontrollable and unobservable nonzero finite and infinite eigenvalues of the corresponding pole pencil. The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

.

The keyword arguments `atol1`

, `atol2`

, and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

, the absolute tolerance for the nonzero elements of `E`

, and the relative tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

and `E`

. The keyword argument `atolinf`

is the absolute tolerance for the gain of `G(λ)`

at `λ = ∞`

. The used default value is `atolinf = 0`

. The default relative tolerance is `n*ϵ`

, where `ϵ`

is the working machine epsilon and `n`

is the order of the system `sys`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

and `atol2 = atol`

.

`RobustAndOptimalControl.h2synthesize`

— Function`K, Cl = h2synthesize(P::ExtendedStateSpace, γ = nothing)`

Synthesize H₂-optimal controller K and calculate the closed-loop transfer function from `w`

to `z`

. Ref: Cha. 14.5 in Robust and Optimal Control.

If `γ = nothing`

, use the formulas for H₂ in Ch 14.5. If γ is a large value, the H∞ formulas are used. As γ → ∞, these two are equivalent. The h∞ formulas do a coordinate transfromation that handles slightly more general systems so if you run into an error, it might be worth trying setting γ to something large, e.g., 1000.

`RobustAndOptimalControl.hankelnorm`

— Method`n, hsv = hankelnorm(sys::LTISystem; kwargs...)`

Compute the hankelnorm and the hankel singular values

For keyword arguments, see the docstring of `DescriptorSystems.ghanorm`

, reproduced below

`ghanorm(sys, fast = true, atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ) -> (hanorm, hs)`

Compute for a proper and stable descriptor system `sys = (A-λE,B,C,D)`

with the transfer function matrix `G(λ)`

, the Hankel norm `hanorm =`

$\small ||G(\lambda)||_H$ and the vector of Hankel singular values `hs`

of the system.

For a proper system with `E`

singular, the uncontrollable infinite eigenvalues of the pair `(A,E)`

and the non-dynamic modes are elliminated using minimal realization techniques. The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

.

The keyword arguments `atol1`

, `atol2`

, and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

, the absolute tolerance for the nonzero elements of `E`

, and the relative tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

and `E`

. The default relative tolerance is `n*ϵ`

, where `ϵ`

is the working machine epsilon and `n`

is the order of the system `sys`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

and `atol2 = atol`

.

`RobustAndOptimalControl.hanus`

— Method`Wh = hanus(W)`

Return `Wh`

on Hanus form. `Wh`

has twice the number of inputs, where the second half of the inputs are "actual inputs", e.g., potentially saturated. This is used to endow `W`

with anti-windup protection. `W`

must have an invertable `D`

matrix and be minimum phase.

Ref: Sec 9.4.5 of Skogestad, "Multivariable Feedback Control: Analysis and Design"

`RobustAndOptimalControl.hess_form`

— Method`sysm, T, HF = hess_form(sys)`

Bring `sys`

to Hessenberg form form.

The Hessenberg form is characterized by `A`

having upper Hessenberg structure. `T`

is the similarity transform applied to the system such that

`sysm ≈ similarity_transform(sys, T)`

`HF`

is the Hessenberg-factorization of `A`

.

See also `modal_form`

and `schur_form`

`RobustAndOptimalControl.hinfassumptions`

— Method`flag = hinfassumptions(P::ExtendedStateSpace; verbose=true)`

Check the assumptions for using the γ-iteration synthesis in Theorem 1.

`RobustAndOptimalControl.hinfgrad`

— Method```
∇A, ∇B, ∇C, ∇D, hn, ω = hinfgrad(sys; rtolinf=1e-8, kwargs...)
∇A, ∇B, ∇C, ∇D = hinfgrad(sys, hn, ω)
```

Compute the gradient of the H∞ norm w.r.t. the statespace matrices `A,B,C,D`

. If only a system is provided, the norm `hn`

and the peak frequency `ω`

are automatically calculated. `kwargs`

are sent to `hinfnorm2`

. Note, the default tolerance to which the norm is calculated is set smaller than default for `hinfnorm2`

, gradients will be discontinuous with any non-finite tolerance, and sensitive optimization algorithms may require even tighter tolerance.

In cases where the maximum singular value is reached at more than one frequency, a random frequency is used.

If the system is unstable, the gradients are `NaN`

. Strategies to find an initial stabilizing controllers are outlined in Apkarian and D. Noll, "Nonsmooth H∞ Synthesis" in IEEE Transactions on Automatic Control.

An `rrule`

for ChainRules is defined using this function, so `hn`

is differentiable with any AD package that derives its rules from ChainRules (only applies to the `hn`

return value, not `ω`

).

`RobustAndOptimalControl.hinfnorm2`

— Method`n, ω = hinfnorm2(sys::LTISystem; kwargs...)`

A numerically robust version of `hinfnorm`

using DescriptorSystems.jl

For keyword arguments, see the docstring of `DescriptorSystems.ghinfnorm`

, reproduced below

`ghinfnorm(sys, rtolinf = 0.001, fast = true, offset = sqrt(ϵ), atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ) -> (hinfnorm, fpeak)`

Compute for a descriptor system `sys = (A-λE,B,C,D)`

with the transfer function matrix `G(λ)`

the `H∞`

norm `hinfnorm`

(i.e., the peak gain of `G(λ)`

) and the corresponding peak frequency `fpeak`

, where the peak gain is achieved. The `H∞`

norm is infinite if the *pole pencil*`A-λE`

has unstable zeros (i.e., `sys`

has unstable poles). To check the stability, the eigenvalues of the pencil `A-λE`

must have real parts less than `-β`

for a continuous-time system or have moduli less than `1-β`

for a discrete-time system, where `β`

is the stability domain boundary offset. The offset `β`

to be used can be specified via the keyword parameter `offset = β`

. The default value used for `β`

is `sqrt(ϵ)`

, where `ϵ`

is the working machine precision.

The keyword argument `rtolinf`

specifies the relative accuracy for the computed infinity norm. The default value used for `rtolinf`

is `0.001`

.

For a continuous-time system `sys`

with `E`

singular, a reduced order realization is determined first, without uncontrollable and unobservable nonzero finite and infinite eigenvalues of the corresponding pole pencil. The rank determinations in the performed reductions are based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

.

The keyword arguments `atol1`

, `atol2`

, and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of matrices `A`

, `B`

, `C`

, `D`

, the absolute tolerance for the nonzero elements of `E`

, and the relative tolerance for the nonzero elements of `A`

, `B`

, `C`

, `D`

and `E`

. The default relative tolerance is `n*ϵ`

, where `ϵ`

is the working machine epsilon and `n`

is the order of the system `sys`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

and `atol2 = atol`

.

`RobustAndOptimalControl.hinfpartition`

— Method`P = hinfpartition(G, WS, WU, WT)`

Transform a SISO or MIMO system $G$, with weighting functions $W_S, W_U, W_T$ into an LFT with an isolated controller, and write the resulting system, $P(s)$, on a state-space form. Valid inputs for $G$ are transfer functions (with dynamics, can be both MIMO and SISO, both in tf and ss forms). Valid inputs for the weighting functions are empty arrays, numbers (static gains), and `LTISystem`

s.

Note, `system_mapping(P)`

is equal to `-G`

.

`RobustAndOptimalControl.hinfsignals`

— Method`hinfsignals(P::ExtendedStateSpace, G::LTISystem, C::LTISystem)`

Use the extended state-space model, a plant and the found controller to extract the closed loop transfer functions.

`Pcl : w → z`

: From input to the weighted functions`S : w → e`

: From input to error`CS : w → u`

: From input to control`T : w → y`

: From input to output

`RobustAndOptimalControl.hinfsynthesize`

— Method`K, γ, mats = hinfsynthesize(P::ExtendedStateSpace; gtol = 1e-4, interval = (0, 20), verbose = false, tolerance = 1.0e-10, γrel = 1.01, transform = true, ftype = Float64, check = true)`

Computes an H-infinity optimal controller `K`

for an extended plant `P`

such that $||F_l(P, K)||∞ < γ$(`lft(P, K)`

) for the smallest possible γ given `P`

. The routine is known as the γ-iteration, and is based on the paper "State-space formulae for all stabilizing controllers that satisfy an H∞-norm bound and relations to risk sensitivity" by Glover and Doyle.

**Arguments:**

`gtol`

: Tolerance for γ.`interval`

: The starting interval for the bisection.`verbose`

: Print progress?`tolerance`

: For detecting eigenvalues on the imaginary axis.`γrel`

: If`γrel > 1`

, the optimal γ will be found by γ iteration after which a controller will be designed for`γ = γopt * γrel`

. It is often a good idea to design a slightly suboptimal controller, both for numerical reasons, but also since the optimal controller may contain very fast dynamics. If`γrel → ∞`

, the computed controller will approach the 𝑯₂ optimal controller. Getting a mix between 𝑯∞ and 𝑯₂ properties is another reason to choose`γrel > 1`

.`transform`

: Apply coordiante transform in order to tolerate a wider range or problem specifications.`ftype`

: construct problem matrices in higher precision for increased numerical robustness. If the calculated controller achieves`check`

: Perform a post-design check of the γ value achieved by the calculated controller. A warning is issued if the achieved γ differs from the γ calculated during design. If this warning is issued, consider using a higher-precision number type like`ftype = BigFloat`

.

See the example folder for example usage.

`RobustAndOptimalControl.hsvd`

— Method`hsvd(sys::AbstractStateSpace)`

Return the Hankel singular values of `sys`

, computed as the eigenvalues of `QP`

Where `Q`

and `P`

are the Gramians of `sys`

.

`RobustAndOptimalControl.ispassive`

— Method`ispassive(P; kwargs...)`

Determine if square system `P`

is passive, i.e., $P(s) + Pᴴ(s) > 0$.

A passive system has a Nyquist curve that lies completely in the right half plane, and satisfies the following inequality (dissipation of energy)

\[\int_0^T y^T u dt > 0 ∀ T\]

The negative feedback-interconnection of two passive systems is stable and parallel connections of two passive systems as well as the inverse of a passive system are also passive. A passive controller will thus always yeild a stable feedback loop for a passive system. A series connection of two passive systems *is not* always passive.

See also `passivityplot`

, `passivity_index`

.

`RobustAndOptimalControl.loop_diskmargin`

— Method`loop_diskmargin(P, C, args...; kwargs...)`

Calculate the loop-at-a-time diskmargin for each output and input of `P`

. See also `diskmargin`

, `sim_diskmargin`

. Ref: "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet

`RobustAndOptimalControl.loop_diskmargin`

— Method`loop_diskmargin(L, args...; kwargs...)`

Calculate the loop-at-a-time diskmargin for each output of `L`

.

See also `diskmargin`

, `sim_diskmargin`

. Ref: "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet

`RobustAndOptimalControl.loop_scale`

— Function`loop_scale(L::LTISystem, w = 0)`

Find the optimal diagonal scaling matrix `D`

such that `D\L(iw)*D`

has a minimized condition number at frequency `w`

. Applicable to square `L`

only. Use `loop_scaling`

to obtain `D`

.

`RobustAndOptimalControl.loop_scaling`

— Function`loop_scaling(M0::Matrix, tol = 0.0001)`

Find the optimal diagonal scaling matrix `D`

such that `D\M0*D`

has a minimized condition number. Applicable to square `M0`

only. See also `structured_singular_value`

with option `dynamic=true`

. Use `loop_scale`

to find and apply the scaling to a loop-transfer function.

`RobustAndOptimalControl.lqr3`

— Method`lqr3(P::AbstractStateSpace, Q1::AbstractMatrix, Q2::AbstractMatrix, Q3::AbstractMatrix)`

Calculate the feedback gain of the discrete LQR cost function augmented with control differences

\[x^{T} Q_1 x + u^{T} Q_2 u + Δu^{T} Q_3 Δu, \quad Δu = u(k) - u(k-1)\]

`RobustAndOptimalControl.makeweight`

— Method```
makeweight(low, f_mid, high)
makeweight(low, (f_mid, gain_mid), high)
```

Create a weighting function that goes from gain `low`

at zero frequency, through gain `gain_mid`

to gain `high`

at ∞

**Arguments:**

`low`

: A number specifying the DC gain`mid`

: A number specifying the frequency at which the gain is 1, or a tuple`(freq, gain)`

. If`gain_mid`

is not specified, the geometric mean of`high`

and`low`

is used.`high`

: A number specifying the gain at ∞

```
using ControlSystemsBase, Plots
W = makeweight(10, (5,2), 1/10)
bodeplot(W)
hline!([10, 2, 1/10], l=(:black, :dash), primary=false)
vline!([5], l=(:black, :dash), primary=false)
```

`RobustAndOptimalControl.measure`

— Method`measure(s::NamedStateSpace, names)`

Return a system with specified states as measurement outputs.

`RobustAndOptimalControl.modal_form`

— Method`sysm, T, E = modal_form(sys; C1 = false)`

Bring `sys`

to modal form.

The modal form is characterized by being tridiagonal with the real values of eigenvalues of `A`

on the main diagonal and the complex parts on the first sub and super diagonals. `T`

is the similarity transform applied to the system such that

`sysm ≈ similarity_transform(sys, T)`

If `C1`

, then an additional convention for SISO systems is used, that the `C`

-matrix coefficient of real eigenvalues is 1. If `C1 = false`

, the `B`

and `C`

coefficients are chosen in a balanced fashion.

`E`

is an eigen factorization of `A`

.

See also `hess_form`

and `schur_form`

`RobustAndOptimalControl.muplot`

— Function```
muplot(sys, args...; hz=false)
muplot(LTISystem[sys1, sys2...], args...; hz=false)
```

Plot the structured singular values (assuming time-varying diagonal complex uncertainty) of the frequency response of the `LTISystem`

(s). This plot is similar to `sigmaplot`

, but scales the loop-transfer function to minimize the maximum singular value. Only applicable to square systems. A frequency vector `w`

can be optionally provided.

`hz=true`

, the plot x-axis will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to Plots.plot.

`RobustAndOptimalControl.mvnyquistplot`

— Function`fig = mvnyquistplot(sys, w; unit_circle=true, hz = false, kwargs...)`

Create a Nyquist plot of the `LTISystem`

. A frequency vector `w`

must be provided.

`unit_circle`

: if the unit circle should be displayed

If `hz=true`

, the hover information will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to plot.

**Example**

```
w = 2π .* exp10.(LinRange(-2, 2, 500))
W = makeweight(0.40, 15, 3) # frequency weight for uncertain dynamics
Pn = tf(1, [1/60, 1]) |> ss # nominal plant
d = δss(1,1) # Uncertain dynamics
Pd = Pn*(I(1) + W*d) # weighted dynamic uncertainty on the input of Pn
Pp = rand(Pd, 200) # sample the uncertain plant
Gcl = lft(Pd, ss(-1)) # closed loop system
structured_singular_value(Gcl) # larger than 1 => not robustly stable
unsafe_comparisons(true)
mvnyquistplot(Pp, w, points=true) # MV Nyquist plot encircles origin for some samples => not robustly stable
```

`RobustAndOptimalControl.named_ss`

— Method`named_ss(sys::AbstractStateSpace, name; x, y, u)`

If a single name of the system is provided, the outputs, inputs and states will be automatically named `y,u,x`

with `name`

as prefix.

`RobustAndOptimalControl.named_ss`

— Method`named_ss(sys::AbstractStateSpace{T}; x, u, y)`

Create a `NamedStateSpace`

system. This kind of system uses names rather than integer indices to refer to states, inputs and outputs.

- If a single name is provided but a vector of names is expected, this name will be used as prefix followed by a numerical index.
- If no name is provided, default names (
`x,y,u`

) will be used.

**Arguments:**

`sys`

: A system to add names to.`x`

: A list of symbols with names of the states.`u`

: A list of symbols with names of the inputs.`y`

: A list of symbols with names of the outputs.

**Example**

```
G1 = ss(1,1,1,0)
G2 = ss(1,1,1,0)
s1 = named_ss(G1, x = :x, u = :u1, y=:y1)
s2 = named_ss(G2, x = :z, u = :u2, y=:y2)
s1[:y1, :u1] # Index using symbols. Uses prefix matching if no exact match is found.
fb = feedback(s1, s2, r = :r) #
```

`RobustAndOptimalControl.named_ss`

— Method```
named_ss(sys::ExtendedStateSpace; kwargs...)
named_ss(sys::ExtendedStateSpace, name; kwargs...)
```

Assign names to an ExtendedStateSpace. If no specific names are provided for signals `z,y,w,u`

and states`x`

, names will be generated automatically.

**Arguments:**

`name`

: Prefix to add to all automatically generated names.`x`

`u`

`y`

`w`

`z`

`RobustAndOptimalControl.ncfmargin`

— Method`m, ω = ncfmargin(P, K)`

Normalized coprime factor margin, defined has the *inverse* of

\[\begin{Vmatrix} \begin{bmatrix} I \\ K \end{bmatrix} (I + PK)^{-1} \begin{bmatrix} I & P \end{bmatrix} \end{Vmatrix}_\infty\]

A margin ≥ 0.25-0.3 is a reasonable for robustness.

If controller `K`

stabilizes `P`

with margin `m`

, then `K`

will also stabilize `P̃`

if `nugap(P, P̃) < m`

.

See also `extended_gangoffour`

, `diskmargin`

, `controller_reduction_plot`

.

**Extended help**

- Robustness with respect to coprime factor uncertainty does not necessarily imply robustness with respect to input uncertainty. Skogestad p. 96 remark 4

`RobustAndOptimalControl.neglected_delay`

— Method`neglected_delay(Lmax)`

Return a multiplicative weight to represent the uncertainty coming from neglecting the dynamics `exp(-s*L)`

where `L ≤ Lmax`

. "Multivariable Feedback Control: Analysis and Design" Ch 7.4.5

See also `gain_and_delay_uncertainty`

and `neglected_lag`

.

**Example:**

```
a = 10
P = ss([0 a; -a 0], I(2), [1 a; -a 1], 0) # Plant
W0 = neglected_delay(0.005) |> ss # Weight
W = I(2) + W0*I(2) * uss([δc(), δc()]) # Create a diagonal real uncertainty weighted in frequency by W0
Ps = P*W # Uncertain plant
Psamples = rand(Ps, 500) # Sample the uncertain plant for plotting
w = exp10.(LinRange(-1, 3, 300)) # Frequency vector
bodeplot(Psamples, w)
```

`RobustAndOptimalControl.neglected_lag`

— Method`neglected_lag(τmax)`

Return a multiplicative weight to represent the uncertainty coming from neglecting the dynamics `1/(s*τ + 1)`

where `τ ≤ τmax`

. "Multivariable Feedback Control: Analysis and Design" Ch 7.4.5

See also `gain_and_delay_uncertainty`

and `neglected_delay`

.

**Example:**

```
a = 10
P = ss([0 a; -a 0], I(2), [1 a; -a 1], 0) # Plant
W0 = neglected_lag(0.05) |> ss # Weight
W = I(2) + W0*I(2) * uss([δc(), δc()]) # Create a diagonal real uncertainty weighted in frequency by W0
Ps = P*W # Uncertain plant
Psamples = rand(Ps, 100) # Sample the uncertain plant for plotting
w = exp10.(LinRange(-1, 3, 300)) # Frequency vector
sigmaplot(Psamples, w)
```

`RobustAndOptimalControl.noise_mapping`

— Function`noise_mapping(P::ExtendedStateSpace)`

Return the system from w -> y See also `performance_mapping`

, `system_mapping`

, `noise_mapping`

`RobustAndOptimalControl.nugap`

— Method`nugap(sys0::LTISystem, sys1::LTISystem; kwargs...)`

Compute the ν-gap metric between two systems. See also `ncfmargin`

.

For keyword arguments, see the docstring of `DescriptorSystems.gnugap`

, reproduced below

```
gnugap(sys1, sys2; freq = ω, rtolinf = 0.00001, fast = true, offset = sqrt(ϵ),
atol = 0, atol1 = atol, atol2 = atol, rtol = n*ϵ) -> (nugapdist, fpeak)
```

Compute the ν-gap distance `nugapdist`

between two descriptor systems `sys1 = (A1-λE1,B1,C1,D1)`

and `sys2 = (A2-λE2,B2,C2,D2)`

and the corresponding frequency `fpeak`

(in rad/TimeUnit), where the ν-gap distance achieves its peak value.

If `freq = missing`

, the resulting `nugapdist`

satisfies `0 <= nugapdist <= 1`

. The value `nugapdist = 1`

results, if the winding number is different of zero in which case `fpeak = []`

.

If `freq = ω`

, where `ω`

is a given vector of real frequency values, the resulting `nugapdist`

is a vector of pointwise ν-gap distances of the dimension of `ω`

, whose components satisfies `0 <= maximum(nugapdist) <= 1`

. In this case, `fpeak`

is the frequency for which the pointwise distance achieves its peak value. All components of `nugapdist`

are set to 1 if the winding number is different of zero in which case `fpeak = []`

.

The stability boundary offset, `β`

, to be used to assess the finite zeros which belong to the boundary of the stability domain can be specified via the keyword parameter `offset = β`

. Accordingly, for a continuous-time system, these are the finite zeros having real parts within the interval `[-β,β]`

, while for a discrete-time system, these are the finite zeros having moduli within the interval `[1-β,1+β]`

. The default value used for `β`

is `sqrt(ϵ)`

, where `ϵ`

is the working machine precision.

Pencil reduction algorithms are employed to compute range and coimage spaces which perform rank decisions based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

.

The keyword arguments `atol1`

, `atol2`

and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of `A1`

, `A2`

, `B1`

, `B2`

, `C1`

, `C2`

, `D1`

and `D2`

, the absolute tolerance for the nonzero elements of `E1`

and `E2`

, and the relative tolerance for the nonzero elements of all above matrices. The default relative tolerance is `n*ϵ`

, where `ϵ`

is the working machine epsilon and `n`

is the maximum of the orders of the systems `sys1`

and `sys2`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

, `atol2 = atol`

.

The keyword argument `rtolinf`

specifies the relative accuracy to be used to compute the ν-gap as the infinity norm of the relevant system according to [1]. The default value used for `rtolinf`

is `0.00001`

.

*Method:* The evaluation of ν-gap uses the definition proposed in [1], extended to generalized LTI (descriptor) systems. The computation of winding number is based on enhancements covering zeros on the boundary of the stability domain and infinite zeros.

*References:*

[1] G. Vinnicombe. Uncertainty and feedback: H∞ loop-shaping and the ν-gap metric. Imperial College Press, London, 2001.

`RobustAndOptimalControl.partition`

— Method`partition(P::AbstractStateSpace, nw::Int, nz::Int)`

`RobustAndOptimalControl.partition`

— Method`partition(P::AbstractStateSpace; u, y, w=!u, z=!y)`

Partition `P`

into an `ExtendedStateSpace`

.

`u`

indicates the indices of the controllable inputs.`y`

indicates the indices of the measurable outputs.`w`

is the complement of`u`

.`z`

is the complement of`y`

.

`RobustAndOptimalControl.passivity_index`

— Method`passivity_index(P; kwargs...)`

Return

\[γ = \begin{Vmatrix} (I-P)(I+P)^{-1} \end{Vmatrix}_∞\]

If $γ ≤ 1$, the system is passive. If the system has unstable zeros, $γ = ∞$

The negative feedback interconnection of two systems with passivity indices γ₁ and γ₂ is stable if $γ₁γ₂ < 1$.

A passive system has a Nyquist curve that lies completely in the right half plane, and satisfies the following inequality (dissipation of energy)

\[\int_0^T y^T u dt > 0 ∀ T\]

The negative feedback-interconnection of two passive systems is stable and parallel connections of two passive systems as well as the inverse of a passive system are also passive. A passive controller will thus always yeild a stable feedback loop for a passive system. A series connection of two passive systems *is not* always passive.

See also `ispassive`

, `passivityplot`

.

`RobustAndOptimalControl.passivityplot`

— Function```
passivityplot(sys, args...; hz=false)
passivityplot(LTISystem[sys1, sys2...], args...; hz=false)
```

Plot the passivity index of a `LTISystem`

(s). The system is passive for frequencies where the index is < 0.

A frequency vector `w`

can be optionally provided.

`hz=true`

, the plot x-axis will be displayed in Hertz, the input frequency vector is still treated as rad/s.

`kwargs`

is sent as argument to Plots.plot.

See `passivity_index`

for additional details. See also `ispassive`

, `passivity_index`

.

`RobustAndOptimalControl.performance_mapping`

— Function`performance_mapping(P::ExtendedStateSpace)`

Return the system from w -> z See also `performance_mapping`

, `system_mapping`

, `noise_mapping`

`RobustAndOptimalControl.robstab`

— Method`robstab(M0::UncertainSS, w=exp10.(LinRange(-3, 3, 1500)); kwargs...)`

Return the robust stability margin of an uncertain model, defined as the inverse of the structured singular value. Currently, only diagonal complex perturbations supported.

`RobustAndOptimalControl.schur_form`

— Method`sysm, T, SF = schur_form(sys)`

Bring `sys`

to Schur form.

The Schur form is characterized by `A`

being Schur with the real values of eigenvalues of `A`

on the main diagonal. `T`

is the similarity transform applied to the system such that

`sysm ≈ similarity_transform(sys, T)`

`SF`

is the Schur-factorization of `A`

.

See also `modal_form`

and `hess_form`

`RobustAndOptimalControl.show_construction`

— Method`show_construction([io::IO,] sys::LTISystem; name = "temp", letb = true)`

Print code to `io`

that reconstructs `sys`

.

`letb`

: If true, the code is surrounded by a let block.

```
julia> sys = ss(tf(1, [1, 1]))
StateSpace{Continuous, Float64}
A =
-1.0
B =
1.0
C =
1.0
D =
0.0
Continuous-time state-space model
julia> show_construction(sys, name="Jörgen")
Jörgen = let
JörgenA = [-1.0;;]
JörgenB = [1.0;;]
JörgenC = [1.0;;]
JörgenD = [0.0;;]
ss(JörgenA, JörgenB, JörgenC, JörgenD)
end
```

`RobustAndOptimalControl.sim_diskmargin`

— Function`sim_diskmargin(L, σ::Real = 0, l=1e-3, u=1e3)`

Return the smallest simultaneous diskmargin over the grid l:u See also `ncfmargin`

.

`RobustAndOptimalControl.sim_diskmargin`

— Function`sim_diskmargin(P::LTISystem, C::LTISystem, σ::Real = 0)`

Simultaneuous diskmargin at both outputs and inputs of `P`

. Ref: "An Introduction to Disk Margins", Peter Seiler, Andrew Packard, and Pascal Gahinet https://arxiv.org/abs/2003.04771 See also `ncfmargin`

.

`RobustAndOptimalControl.sim_diskmargin`

— Method```
sim_diskmargin(L, σ::Real, w::AbstractVector)
sim_diskmargin(L, σ::Real = 0)
```

Simultaneuous diskmargin at the outputs of `L`

. Users should consider using `diskmargin`

.

`RobustAndOptimalControl.specificationplot`

— Function`specificationplot([S,CS,T], [WS,WU,WT])`

This function visualizes the control synthesis using the `hinfsynthesize`

with the three weighting functions $W_S(s), W_U(s), W_T(s)$ inverted and scaled by γ, against the corresponding transfer functions $S(s), C(s)S(s), T(s)$, to verify visually that the specifications are met. This may be run using both MIMO and SISO systems.

**Keyword args**

`wint`

:`(-3, 5)`

frequency range (log10)`wnum`

: 201 number of frequency points`hz`

: true`nsigma`

: typemax(Int) number of singular values to show`s_labels`

: `[ "σ(S)", "σ(CS)", "σ(T)",

]`

`w_labels`

: `[ "γ σ(Wₛ⁻¹)", "γ σ(Wᵤ⁻¹)", "γ σ(Wₜ⁻¹)",

]`

`colors`

:`[:blue, :red, :green]`

colors for $S$, $CS$ and $T$

`RobustAndOptimalControl.ssdata_e`

— Method`A, B1, B2, C1, C2, D11, D12, D21, D22 = ssdata_e(sys)`

`RobustAndOptimalControl.stab_unstab`

— Method`stab, unstab = stab_unstab(sys; kwargs...)`

Decompose `sys`

into `sys = stab + unstab`

where `stab`

contains all stable poles and `unstab`

contains unstable poles. See `gsdec(sys; job = "finite", smarg, fast = true, atol = 0, atol1 = atol, atol2 = atol, rtol = nϵ) -> (sys1, sys2)`

Compute for the descriptor system `sys = (A-λE,B,C,D)`

with the transfer function matrix `G(λ)`

, the additive spectral decomposition `G(λ) = G1(λ) + G2(λ)`

such that `G1(λ)`

, the transfer function matrix of the descriptor system `sys1 = (A1-λE1,B1,C1,D1)`

, has only poles in a certain domain of interest `Cg`

of the complex plane and `G2(λ)`

, the transfer function matrix of the descriptor system `sys2 = (A2-λE2,B2,C2,0)`

, has only poles outside of `Cg`

.

The keyword argument `smarg`

, if provided, specifies the stability margin for the stable eigenvalues of `A-λE`

, such that, in the continuous-time case, the stable eigenvalues have real parts less than or equal to `smarg`

, and in the discrete-time case, the stable eigenvalues have moduli less than or equal to `smarg`

. If `smarg = missing`

, the used default values are: `smarg = -sqrt(ϵ)`

, for a continuous-time system, and `smarg = 1-sqrt(ϵ)`

, for a discrete-time system), where `ϵ`

is the machine precision of the working accuracy.

The keyword argument `job`

, in conjunction with `smarg`

, defines the domain of interest `Cg`

, as follows:

for `job = "finite"`

, `Cg`

is the whole complex plane without the point at infinity, and `sys1`

has only finite poles and `sys2`

has only infinite poles (default); the resulting `A2`

is nonsingular and upper triangular, while the resulting `E2`

is nilpotent and upper triangular;

for `job = "infinite"`

, `Cg`

is the point at infinity, and `sys1`

has only infinite poles and `sys2`

has only finite poles and is the strictly proper part of `sys`

; the resulting `A1`

is nonsingular and upper triangular, while the resulting `E1`

is nilpotent and upper triangular;

for `job = "stable"`

, `Cg`

is the stability domain of eigenvalues defined by `smarg`

, and `sys1`

has only stable poles and `sys2`

has only unstable and infinite poles; the resulting pairs `(A1,E1)`

and `(A2,E2)`

are in generalized Schur form with `E1`

upper triangular and nonsingular and `E2`

upper triangular;

for `job = "unstable"`

, `Cg`

is the complement of the stability domain of the eigenvalues defined by `smarg`

, and `sys1`

has only unstable and infinite poles and `sys2`

has only stable poles; the resulting pairs `(A1,E1)`

and `(A2,E2)`

are in generalized Schur form with `E1`

upper triangular and `E2`

upper triangular and nonsingular.

The keyword arguments `atol1`

, `atol2`

, and `rtol`

, specify, respectively, the absolute tolerance for the nonzero elements of `A`

, the absolute tolerance for the nonzero elements of `E`

, and the relative tolerance for the nonzero elements of `A`

and `E`

. The default relative tolerance is `n*ϵ`

, where `ϵ`

is the working machine epsilon and `n`

is the order of the system `sys`

. The keyword argument `atol`

can be used to simultaneously set `atol1 = atol`

, `atol2 = atol`

.

The separation of the finite and infinite eigenvalues is performed using rank decisions based on rank revealing QR-decompositions with column pivoting if `fast = true`

or the more reliable SVD-decompositions if `fast = false`

. for keyword arguments (argument `job`

is set to `"stable"`

in this function).

`RobustAndOptimalControl.static_gain_compensation`

— Function```
static_gain_compensation(l::LQGProblem, L = lqr(l))
static_gain_compensation(A, B, C, D, L)
```

Find $L_r$ such that

`dcgain(closedloop(G)*Lr) ≈ I`

`RobustAndOptimalControl.structured_singular_value`

— Method`structured_singular_value(M0::UncertainSS, [w::AbstractVector]; kwargs...)`

`w`

: Frequency vector, if none is provided, the maximum μ over a grid 1e-3 : 1e3 will be returned.

`RobustAndOptimalControl.structured_singular_value`

— Method`μ = structured_singular_value(M; tol=1e-4, scalings=false, dynamic=false)`

Compute (an upper bound of) the structured singular value μ for diagonal Δ of complex perturbations (other structures of Δ are not yet supported). `M`

is assumed to be an (n × n × N_freq) array or a matrix.

We currently don't have any methods to compute a lower bound, but if all perturbations are complex the spectral radius `ρ(M)`

is always a lower bound (usually not a good one).

If `scalings = true`

, return also a `n × nf`

matrix `Dm`

with the diagonal scalings `D`

such that

```
D = Diagonal(Dm[:, i])
σ̄(D\M[:,:,i]*D)
```

is minimized.

If `dynamic = true`

, the perturbations are assumed to be time-varying `Δ(t)`

. In this case, the same scaling is used for all frequencies and the returned `D`

if `scalings=true`

is a vector `d`

such that `D = Diagonal(d)`

.

`RobustAndOptimalControl.sumblock`

— Method`sumblock(ex::String; Ts = 0, n = 1)`

Create a summation node that sums (or subtracts) vectors of length `n`

.

**Arguments:**

`Ts`

: Sample time`n`

: The length of the input and output vectors. Set`n=1`

for scalars.

**Examples:**

```
julia> sumblock("uP = vf + yL")
NamedStateSpace{Continuous, Int64}
D =
1 1
With state names:
input names: vf yL
output names: uP
julia> sumblock("x_diff = xr - xh"; n=3)
NamedStateSpace{Continuous, Int64}
D =
1 0 0 -1 0 0
0 1 0 0 -1 0
0 0 1 0 0 -1
With state names:
input names: xr1 xr2 xr3 xh1 xh2 xh3
output names: x_diff1 x_diff2 x_diff3
julia> sumblock("a = b + c - d")
NamedStateSpace{Continuous, Int64}
D =
1 1 -1
With state names:
input names: b c d
output names: a
```

`RobustAndOptimalControl.system_mapping`

— Function`system_mapping(P::ExtendedStateSpace)`

Return the system from u -> y See also `performance_mapping`

, `system_mapping`

, `noise_mapping`

`RobustAndOptimalControl.uss`

— Function`uss(d::AbstractVector{<:δ}, Ts = nothing)`

Create a diagonal uncertain statespace object with the uncertain elements `d`

on the diagonal.

`RobustAndOptimalControl.uss`

— Function`uss(D::AbstractArray, Δ, Ts = nothing)`

If only a single `D`

matrix is provided, it's treated as `D11`

if Δ is given, and as `D22`

if no Δ is provided.

`RobustAndOptimalControl.uss`

— Function`uss(D11, D12, D21, D22, Δ, Ts = nothing)`

Create an uncertain statespace object with only gin matrices.

`RobustAndOptimalControl.uss`

— Method`uss(d::δ{C, F}, Ts = nothing)`

Convert a δ object to an UncertainSS

`RobustAndOptimalControl.vec2sys`

— Function`vec2sys(v::AbstractArray, ny::Int, nu::Int, ts = nothing)`

Create a statespace system from the parameters

`v = vec(sys) = [vec(sys.A); vec(sys.B); vec(sys.C); vec(sys.D)]`

Use `vec(sys)`

to create `v`

.

This can be useful in order to convert to and from vectors for, e.g., optimization.

```
julia> sys = ss(tf(1, [1, 1]))
StateSpace{Continuous, Float64}
A =
-1.0
B =
1.0
C =
1.0
D =
0.0
Continuous-time state-space model
julia> v = vec(sys)
4-element Vector{Float64}:
-1.0
1.0
1.0
0.0
julia> sys2 = vec2sys(v, sys.ny, sys.nu)
StateSpace{Continuous, Float64}
A =
-1.0
B =
1.0
C =
1.0
D =
0.0
Continuous-time state-space model
```

`RobustAndOptimalControl.δc`

— Function`δc(val::Complex = complex(0.0), radius::Real = 1.0, name)`

Create a complex, uncertain parameter. If no name is given, a boring name will be generated automatically.

`RobustAndOptimalControl.δr`

— Function`δr(val::Real = 0.0, radius::Real = 1.0, name)`

Create a real, uncertain parameter. If no name is given, a boring name will be generated automatically.

`LowLevelParticleFilters.AdvancedParticleFilter`

— Method`AdvancedParticleFilter(Nparticles, dynamics, measurement, measurement_likelihood, dynamics_density, initial_density; p = SciMLBase.NullParameters(), kwargs...)`

See the docs for more information: https://baggepinnen.github.io/LowLevelParticleFilters.jl/stable/#AdvancedParticleFilter-1

`LowLevelParticleFilters.AuxiliaryParticleFilter`

— Method`AuxiliaryParticleFilter(args...; kwargs...)`

Takes exactly the same arguments as `ParticleFilter`

, or an instance of `ParticleFilter`

.

`LowLevelParticleFilters.DAEUnscentedKalmanFilter`

— Method`DAEUnscentedKalmanFilter(ukf; g, get_x_z, build_xz, xz0, threads=false)`

An Unscented Kalman filter for differential-algebraic systems (DAE).

Ref: "Nonlinear State Estimation of Differential Algebraic Systems", Mandela, Rengaswamy, Narasimhan

This filter is still considered experimental and subject to change without respecting semantic versioning. Use at your own risk.

**Arguments**

`ukf`

is a regular`UnscentedKalmanFilter`

that contains`dynamics(xz, u, p, t)`

that propagates the combined state`xz(k)`

to`xz(k+1)`

and a measurement function with signature`(xz, u, p, t)`

`g(x, z, u, p, t)`

is a function that should fulfill`g(x, z, u, p, t) = 0`

`get_x_z(xz) -> x, z`

is a function that decomposes`xz`

into`x`

and`z`

`build_xz(x, z)`

is the inverse of`get_x_z`

`xz0`

the initial full state.`threads`

: If true, evaluates dynamics on sigma points in parallel. This typically requires the dynamics to be non-allocating (use StaticArrays) to improve performance.

**Assumptions**

- The DAE dynamics is index 1 and can be written on the form

```
ẋ = f(x, z, u, p, t) # Differential equations
0 = g(x, z, u, p, t) # Algebraic equations
y = h(x, z, u, p, t) # Measurements
```

the measurements may be functions of both differential states `x`

and algebraic variables `z`

. Please note, the actual dynamics and measurement functions stored in the internal `ukf`

should have signatures `(xz, u, p, t)`

, i.e., they take the combined state containing both `x`

and `z`

in a single vector as dictated by the function `build_xz`

. It is only the function `g`

that is assumed to actually have the signature `g(x,z,u,p,t)`

.

`LowLevelParticleFilters.ExtendedKalmanFilter`

— Type```
ExtendedKalmanFilter(kf, dynamics, measurement)
ExtendedKalmanFilter(dynamics, measurement, R1,R2,d0=MvNormal(Matrix(R1)); nu::Int, p = SciMLBase.NullParameters(), α = 1.0, check = true)
```

A nonlinear state estimator propagating uncertainty using linearization.

The constructor to the extended Kalman filter takes dynamics and measurement functions, and either covariance matrices, or a `KalmanFilter`

. If the former constructor is used, the number of inputs to the system dynamics, `nu`

, must be explicitly provided with a keyword argument.

The filter will internally linearize the dynamics using ForwardDiff.

The dynamics and measurement function are on the following form

```
x(t+1) = dynamics(x, u, p, t) + w
y = measurement(x, u, p, t) + e
```

where `w ~ N(0, R1)`

, `e ~ N(0, R2)`

and `x(0) ~ d0`

See also `UnscentedKalmanFilter`

which is typically more accurate than `ExtendedKalmanFilter`

. See `KalmanFilter`

for detailed instructions on how to set up a Kalman filter `kf`

.

`LowLevelParticleFilters.KalmanFilter`

— Type`KalmanFilter(A,B,C,D,R1,R2,d0=MvNormal(R1); p = SciMLBase.NullParameters(), α=1)`

The matrices `A,B,C,D`

define the dynamics

```
x' = Ax + Bu + w
y = Cx + Du + e
```

where `w ~ N(0, R1)`

, `e ~ N(0, R2)`

and `x(0) ~ d0`

The matrices can be time varying such that, e.g., `A[:, :, t]`

contains the $A$ matrix at time index `t`

. They can also be given as functions on the form

`Afun(x, u, p, t) -> A`

For maximum performance, provide statically sized matrices from StaticArrays.jl

α is an optional "forgetting factor", if this is set to a value > 1, such as 1.01-1.2, the filter will, in addition to the covariance inflation due to $R_1$, exhibit "exponential forgetting" similar to a Recursive Least-Squares (RLS) estimator. It is thus possible to get a RLS-like algorithm by setting $R_1=0, R_2 = 1/α$ and $α > 1$ ($α$ is the inverse of the traditional RLS parameter $α = 1/λ$). The exact form of the covariance update is

\[R(t+1|t) = α AR(t)A^T + R_1\]

`LowLevelParticleFilters.ParticleFilter`

— Method`ParticleFilter(num_particles, dynamics, measurement, dynamics_density, measurement_density, initial_density; p = SciMLBase.NullParameters())`

See the docs for more information: https://baggepinnen.github.io/LowLevelParticleFilters.jl/stable/#Particle-filter-1

`LowLevelParticleFilters.TupleProduct`

— Type`TupleProduct(v::NTuple{N,UnivariateDistribution})`

Create a product distribution where the individual distributions are stored in a tuple. Supports mixed/hybrid Continuous and Discrete distributions

`LowLevelParticleFilters.UnscentedKalmanFilter`

— Type`UnscentedKalmanFilter(dynamics, measurement, R1, R2, d0=MvNormal(Matrix(R1)); p = SciMLBase.NullParameters(), ny, nu)`

A nonlinear state estimator propagating uncertainty using the unscented transform.

The dynamics and measurement function are on the following form

```
x' = dynamics(x, u, p, t) + w
y = measurement(x, u, p, t) + e
```

where `w ~ N(0, R1)`

, `e ~ N(0, R2)`

and `x(0) ~ d0`

The matrices `R1, R2`

can be time varying such that, e.g., `R1[:, :, t]`

contains the $R_1$ matrix at time index `t`

. They can also be given as functions on the form

`Rfun(x, u, p, t) -> R`

For maximum performance, provide statically sized matrices from StaticArrays.jl

`ny, nu`

indicate the number of outputs and inputs.

`LowLevelParticleFilters.commandplot`

— Function`commandplot(pf, u, y, p=parameters(pf); kwargs...)`

Produce a helpful plot. For customization options (`kwargs...`

), see `?pplot`

. After each time step, a command from the user is requested.

- q: quit
- s n: step
`n`

steps

`LowLevelParticleFilters.correct!`

— Function`(; ll, e, S, Sᵪ, K) = correct!(kf::AbstractKalmanFilter, u, y, p = parameters(kf), t::Integer = index(kf), R2)`

The correct step for a Kalman filter returns not only the log likelihood `ll`

and the prediction error `e`

, but also the covariance of the output `S`

, its Cholesky factor `Sᵪ`

and the Kalman gain `K`

.

If `R2`

stored in `kf`

is a function `R2(x, u, p, t)`

, this function is evaluated at the state *before* the correction is performed. The measurement noise covariance matrix `R2`

stored in the filter object can optionally be overridden by passing the argument `R2`

, in this case `R2`

must be a matrix.

`LowLevelParticleFilters.correct!`

— Function`ll, e = correct!(f, u, y, p = parameters(f), t = index(f))`

Update state/covariance/weights based on measurement `y`

, returns loglikelihood and prediction error (the error is always 0 for particle filters).

`LowLevelParticleFilters.debugplot`

— Function`debugplot(pf, u, y, p=parameters(pf); runall=false, kwargs...)`

Produce a helpful plot. For customization options (`kwargs...`

), see `?pplot`

.

`runall=false:`

if true, runs all time steps befor displaying (faster), if false, displays the plot after each time step.

The generated plot becomes quite heavy. Initially, try limiting your input to 100 time steps to verify that it doesn't crash.

`LowLevelParticleFilters.densityplot`

— Function`densityplot(x,[w])`

Plot (weighted) particles densities

`LowLevelParticleFilters.forward_trajectory`

— Function`sol = forward_trajectory(pf, u::AbstractVector, y::AbstractVector, p=parameters(pf))`

Run the particle filter for a sequence of inputs and measurements. Return a solution with `x,w,we,ll = particles, weights, expweights and loglikelihood`

If MonteCarloMeasurements.jl is loaded, you may transform the output particles to `Matrix{MonteCarloMeasurements.Particles}`

using `Particles(x,we)`

. Internally, the particles are then resampled such that they all have unit weight. This is conventient for making use of the plotting facilities of MonteCarloMeasurements.jl.

`sol`

can be plotted

`plot(sol::ParticleFilteringSolution; nbinsy=30, xreal=nothing, dim=nothing)`

`LowLevelParticleFilters.forward_trajectory`

— Function`sol = forward_trajectory(kf::AbstractKalmanFilter, u::Vector, y::Vector, p=parameters(kf))`

Run a Kalman filter forward

Returns a KalmanFilteringSolution: with the following

`x`

: predictions`xt`

: filtered estimates`R`

: predicted covariance matrices`Rt`

: filter covariances`ll`

: loglik

`sol`

can be plotted

`plot(sol::KalmanFilteringSolution; plotx = true, plotxt=true, plotu=true, ploty=true)`

`LowLevelParticleFilters.log_likelihood_fun`

— Method`ll(θ) = log_likelihood_fun(filter_from_parameters(θ::Vector)::Function, priors::Vector{Distribution}, u, y, p)`

returns function θ -> p(y|θ)p(θ)

`LowLevelParticleFilters.loglik`

— Function`ll = loglik(filter, u, y, p=parameters(filter))`

Calculate loglikelihood for entire sequences `u,y`

`LowLevelParticleFilters.logsumexp!`

— Function`ll = logsumexp!(w, we [, maxw])`

Normalizes the weight vector `w`

and returns the weighted log-likelihood

https://arxiv.org/pdf/1412.8695.pdf eq 3.8 for p(y) https://discourse.julialang.org/t/fast-logsumexp/22827/7?u=baggepinnen for stable logsumexp

`LowLevelParticleFilters.mean_trajectory`

— Method`x,ll = mean_trajectory(pf, u::Vector{Vector}, y::Vector{Vector}, p=parameters(pf))`

This Function resets the particle filter to the initial state distribution upon start

`LowLevelParticleFilters.metropolis`

— Function`metropolis(ll::Function(θ), R::Int, θ₀::Vector, draw::Function(θ) = naive_sampler(θ₀))`

Performs MCMC sampling using the marginal Metropolis (-Hastings) algorithm `draw = θ -> θ'`

samples a new parameter vector given an old parameter vector. The distribution must be symmetric, e.g., a Gaussian. `R`

is the number of iterations. See `log_likelihood_fun`

**Example:**

```
filter_from_parameters(θ) = ParticleFilter(N, dynamics, measurement, MvNormal(n,exp(θ[1])), MvNormal(p,exp(θ[2])), d0)
priors = [Normal(0,0.1),Normal(0,0.1)]
ll = log_likelihood_fun(filter_from_parameters,priors,u,y,1)
θ₀ = log.([1.,1.]) # Initial point
draw = θ -> θ .+ rand(MvNormal(0.1ones(2))) # Function that proposes new parameters (has to be symmetric)
burnin = 200 # If using threaded call, provide number of burnin iterations
# @time theta, lls = metropolis(ll, 2000, θ₀, draw) # Run single threaded
# thetam = reduce(hcat, theta)'
@time thetalls = LowLevelParticleFilters.metropolis_threaded(burnin, ll, 5000, θ₀, draw) # run on all threads, will provide (2000-burnin)*nthreads() samples
histogram(exp.(thetalls[:,1:2]), layout=3)
plot!(thetalls[:,3], subplot=3) # if threaded call, log likelihoods are in the last column
```

`LowLevelParticleFilters.reset!`

— MethodReset the filter to initial state and covariance/distribution

`LowLevelParticleFilters.simulate`

— Function```
x,u,y = simulate(f::AbstractFilter, T::Int, du::Distribution, p=parameters(f), [N]; dynamics_noise=true)
x,u,y = simulate(f::AbstractFilter, u, p=parameters(f); dynamics_noise=true)
```

Simulate dynamical system forward in time `T`

steps, or for the duration of `u`

, returns state sequence, inputs and measurements `du`

is a distribution of random inputs.

If MonteCarloMeasurements.jl is loaded, the argument `N::Int`

can be supplied, in which case `N`

simulations are done and the result is returned in the form of `Vector{MonteCarloMeasurements.Particles}`

.

`LowLevelParticleFilters.smooth`

— Function```
xb,ll = smooth(pf, M, u, y, p=parameters(pf))
xb,ll = smooth(pf, xf, wf, wef, ll, M, u, y, p=parameters(pf))
```

Perform particle smoothing using forward-filtering, backward simulation. Return smoothed particles and loglikelihood. See also `smoothed_trajs`

, `smoothed_mean`

, `smoothed_cov`

`LowLevelParticleFilters.smooth`

— Function```
xT,RT,ll = smooth(kf::KalmanFilter, u::Vector, y::Vector, p=parameters(kf))
xT,RT,ll = smooth(kf::ExtendedKalmanFilter, u::Vector, y::Vector, p=parameters(kf))
```

Returns smoothed estimates of state `x`

and covariance `R`

given all input output data `u,y`

`LowLevelParticleFilters.smoothed_cov`

— Method`smoothed_cov(xb)`

Helper function to calculate the covariance of smoothed particle trajectories

`LowLevelParticleFilters.smoothed_mean`

— Method`smoothed_mean(xb)`

Helper function to calculate the mean of smoothed particle trajectories

`LowLevelParticleFilters.smoothed_trajs`

— Method`smoothed_trajs(xb)`

Helper function to get particle trajectories as a 3-dimensions array (N,M,T) instead of matrix of vectors.

`LowLevelParticleFilters.update!`

— Function`ll, e = update!(f::AbstractFilter, u, y, p = parameters(f), t = index(f))`

Perform one step of `predict!`

and `correct!`

, returns loglikelihood and prediction error

`LowLevelParticleFilters.weighted_cov`

— Method`weighted_cov(x,we)`

Similar to `weighted_mean`

, but returns covariances

`LowLevelParticleFilters.weighted_mean`

— Method`x̂ = weighted_mean(x,we)`

Calculated weighted mean of particle trajectories. `we`

are expweights.

`LowLevelParticleFilters.weighted_mean`

— Method```
x̂ = weighted_mean(pf)
x̂ = weighted_mean(s::PFstate)
```

`StatsAPI.predict!`

— Function`predict!(f, u, p = parameters(f), t = index(f))`

Move filter state forward in time using dynamics equation and input vector `u`

.

`StatsAPI.predict!`

— Function`predict!(kf::AbstractKalmanFilter, u, p = parameters(kf), t::Integer = index(kf); R1)`

Perform the prediction step (updating the state estimate to $x(t+1|t)$). If `R1`

stored in `kf`

is a function `R1(x, u, p, t)`

, this function is evaluated at the state *before* the prediciton is performed. The dynamics noise covariance matrix `R1`

stored in `kf`

can optionally be overridden by passing the argument `R1`

, in this case `R1`

must be a matrix.