PID Autotuning Analysis

The PID autotuning analysis is used to automatically tune the parameters of a PID controller for a given model. The analysis computes optimal PID gains that satisfy frequency-domain robustness constraints, such as maximum sensitivity (Ms), complementary sensitivity (Mt), and noise sensitivity (Mks).

Method Overview

PID autotuning in DyadControlSystems is based on step-response optimization, subject to robustness constraints on closed-loop sensitivity functions. The method performs joint optimization of a PID controller and a measurement filter of the form

K (s) = C (s) F (s) = (k_{p} + k_{i} / s + k_{d} s) \frac{1}{(s T_{f})^{2} / (4 ζ^{2}) + T_{f} s + 1}, ζ = 1

by solving

{minimize}_{C, F} \int m (e (t)) d t

subject to

\begin{aligned} | | S (s) | |_{\infty} & \leq M_{S} \\ | | T (s) | |_{\infty} & \leq M_{T} \\ | | K S (s) | |_{\infty} & \leq M_{K S} \end{aligned}

where $e (t) = r (t) - y (t)$ is the control error and $m$ is a user-defined metric (defaults to abs2). The user can select which signals to optimize, the reference value, and the filter order.

The transfer functions in the frequency-domain constraints are defined as follows:

Sensitivity: $S = 1 / (1 + P K)$
Complementary sensitivity: $T = P K / (1 + P K)$
Noise sensitivity: $K / (1 + P K)$

See control-system sensitivity and DyadControlSystems autotuning documentation for more information.

Example Definition

The following example will demonstrate autotuning of a PID controller for an active suspension system. The model of the system is available in DyadExampleComponents.

dyad

using DyadExampleComponents: ActiveSuspension

analysis ActiveSuspensionPIDAutotuningAnalysis
  extends DyadControlSystems.PIDAutotuningAnalysis(
    measurement   = "y",
    control_input = "u",
    step_input    = "u",
    step_output   = "y",
    duration = 10.0,
    Ts       = 0.01,
    Ms       = 1.2,
    Mt       = 1.2,
    Mks      = 3e4,
    ki_lb    = 2000,
    wl       = 1e-2,
    wu       = 1e3
  )

  model = DyadExampleComponents.ActiveSuspension()

end

julia

using DyadExampleComponents, DyadInterface
ActiveSuspensionPIDAutotuningAnalysis() # hide
solution = ActiveSuspensionPIDAutotuningAnalysis()

┌ Warning: Initialization system is overdetermined. 1 equations for 0 unknowns. Initialization will default to using least squares. `SCCNonlinearProblem` can only be used for initialization of fully determined systems and hence will not be used here. To suppress this warning pass warn_initialize_determined = false. To make this warning into an error, pass fully_determined = true
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/Z9mEq/src/systems/diffeqs/abstractodesystem.jl:1522
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:     2404
Number of nonzeros in Lagrangian Hessian.............:        0

Total number of variables............................:        4
                     variables with only lower bounds:        4
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:      601
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:      601

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.5712512e-02 2.75e+01 6.29e-01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   5  1.4948499e-02 2.59e+01 1.40e+03  -1.1 1.13e+05    -  1.66e-03 2.14e-05h  1
  10  8.0867914e-03 7.67e+00 6.70e+04   0.7 1.11e+04    -  6.06e-02 1.60e-04h  1
  15  3.1943462e-03 6.16e-01 1.16e+06   1.6 7.44e+03    -  4.40e-02 3.95e-01f  1
  20  2.8874972e-03 3.61e-01 5.82e+05  -0.1 2.31e+03    -  6.28e-02 1.51e-01f  1
  25  2.6192246e-03 2.43e-01 3.09e+05  -0.7 2.20e+03    -  1.54e-01 2.46e-01f  1
  30  1.0175217e-02 0.00e+00 3.66e+06   2.5 1.04e+04    -  1.68e-03 2.95e-01f  1
  35  8.1953888e-03 0.00e+00 4.11e-01  -2.6 5.88e+00    -  9.88e-01 1.00e+00h  1
  40  3.0075684e-03 0.00e+00 4.48e-02  -4.3 2.01e+03    -  1.00e+00 1.00e+00h  1
  45  1.1903039e-03 0.00e+00 3.08e-03  -5.4 6.53e+02    -  1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50  9.7690415e-04 0.00e+00 1.63e-05  -8.3 9.02e+00    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 51

                                   (scaled)                 (unscaled)
Objective...............:   9.7688560787864459e-04    9.7688560787864459e-04
Dual infeasibility......:   3.2453130769138745e-07    3.2453130769138745e-07
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   1.0961374754484757e-10    1.0961374754484757e-10
Overall NLP error.......:   3.2453130769138745e-07    3.2453130769138745e-07


Number of objective function evaluations             = 52
Number of objective gradient evaluations             = 52
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 53
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 52
Number of Lagrangian Hessian evaluations             = 0
Total seconds in IPOPT                               = 27.021

EXIT: Optimal Solution Found.
┌ Warning: Initialization system is overdetermined. 1 equations for 0 unknowns. Initialization will default to using least squares. `SCCNonlinearProblem` can only be used for initialization of fully determined systems and hence will not be used here. To suppress this warning pass warn_initialize_determined = false. To make this warning into an error, pass fully_determined = true
└ @ ModelingToolkit ~/.julia/packages/ModelingToolkit/Z9mEq/src/systems/diffeqs/abstractodesystem.jl:1522
This is Ipopt version 3.14.17, running with linear solver MUMPS 5.8.0.

Number of nonzeros in equality constraint Jacobian...:        0
Number of nonzeros in inequality constraint Jacobian.:     2404
Number of nonzeros in Lagrangian Hessian.............:        0

Total number of variables............................:        4
                     variables with only lower bounds:        4
                variables with lower and upper bounds:        0
                     variables with only upper bounds:        0
Total number of equality constraints.................:        0
Total number of inequality constraints...............:      601
        inequality constraints with only lower bounds:        0
   inequality constraints with lower and upper bounds:        0
        inequality constraints with only upper bounds:      601

iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
   0  1.5712512e-02 2.75e+01 6.29e-01   0.0 0.00e+00    -  0.00e+00 0.00e+00   0
   5  1.4948499e-02 2.59e+01 1.40e+03  -1.1 1.13e+05    -  1.66e-03 2.14e-05h  1
  10  8.0867914e-03 7.67e+00 6.70e+04   0.7 1.11e+04    -  6.06e-02 1.60e-04h  1
  15  3.1943462e-03 6.16e-01 1.16e+06   1.6 7.44e+03    -  4.40e-02 3.95e-01f  1
  20  2.8874972e-03 3.61e-01 5.82e+05  -0.1 2.31e+03    -  6.28e-02 1.51e-01f  1
  25  2.6192246e-03 2.43e-01 3.09e+05  -0.7 2.20e+03    -  1.54e-01 2.46e-01f  1
  30  1.0175217e-02 0.00e+00 3.66e+06   2.5 1.04e+04    -  1.68e-03 2.95e-01f  1
  35  8.1953888e-03 0.00e+00 4.11e-01  -2.6 5.88e+00    -  9.88e-01 1.00e+00h  1
  40  3.0075684e-03 0.00e+00 4.48e-02  -4.3 2.01e+03    -  1.00e+00 1.00e+00h  1
  45  1.1903039e-03 0.00e+00 3.08e-03  -5.4 6.53e+02    -  1.00e+00 1.00e+00h  1
iter    objective    inf_pr   inf_du lg(mu)  ||d||  lg(rg) alpha_du alpha_pr  ls
  50  9.7690415e-04 0.00e+00 1.63e-05  -8.3 9.02e+00    -  1.00e+00 1.00e+00h  1

Number of Iterations....: 51

                                   (scaled)                 (unscaled)
Objective...............:   9.7688560787864459e-04    9.7688560787864459e-04
Dual infeasibility......:   3.2453130769138745e-07    3.2453130769138745e-07
Constraint violation....:   0.0000000000000000e+00    0.0000000000000000e+00
Variable bound violation:   0.0000000000000000e+00    0.0000000000000000e+00
Complementarity.........:   1.0961374754484757e-10    1.0961374754484757e-10
Overall NLP error.......:   3.2453130769138745e-07    3.2453130769138745e-07


Number of objective function evaluations             = 52
Number of objective gradient evaluations             = 52
Number of equality constraint evaluations            = 0
Number of inequality constraint evaluations          = 53
Number of equality constraint Jacobian evaluations   = 0
Number of inequality constraint Jacobian evaluations = 52
Number of Lagrangian Hessian evaluations             = 0
Total seconds in IPOPT                               = 0.487

EXIT: Optimal Solution Found.

julia

using Plots
figs  = artifacts(solution, :SensitivityFunctions)
fign  = artifacts(solution, :NoiseSensitivityAndController)
figr  = artifacts(solution, :OptimizedResponse)
figny = artifacts(solution, :NyquistPlot)

plot!(figs, ylims=(1e-2, Inf))
plot!(figny, legend=:bottomleft)

plot(figs, fign, figr, figny,
    link = :none,
    size = (600, 600),
)

We may also ask for the optimized controller parameters:

julia

artifacts(solution, :OptimizedParameters)

1×7 DataFrame

Row	kp_parallel	ki_parallel	kd_parallel	Kp_standard	Ti_standard	Td_standard	Tf
	Float64	Float64	Float64	Float64	Float64	Float64	Float64
1	1239.1	5461.29	702.605	1239.1	0.226888	0.567029	0.0259857

Analysis Arguments

The following arguments define a PID autotuning analysis:

Required Arguments

model: The system model to be controlled.
measurement::String: Name of the measured output signal.
control_input::String: Name of the control input signal.
step_input::String: Name of the step input signal.
step_output::String: Name of the step output signal.
Ts::Real: Sampling time to use in the discretization.
duration::Real: Simulation duration.
Ms::Real: Maximum sensitivity constraint. Values between 1 and 2 are typical. A lower value increases robustness but may reduce performance.
Mt::Real: Maximum complementary sensitivity constraint. Values between 1 and 2 are typical. A lower value increases robustness but may reduce performance.
Mks::Real: Maximum noise sensitivity constraint. This controls the maximum amplification of measurement noise in the control signal.

Optional Arguments

ref::Real = 0.0: Reference value for the step input.
disc::String = "tustin": Discretization method.
kp_lb, ki_lb, kd_lb, Tf_lb: Lower bounds for PID parameters.
kp_ub, ki_ub, kd_ub, Tf_ub: Upper bounds for PID parameters.
kp_guess, ki_guess, kd_guess, Tf_guess: Initial guesses for PID parameters.
timeweight::Bool = false: Whether or not to use time-weighted optimization.
filter_order::Int = 2: Order of the derivative filter (1 or 2).
optimize_d::Bool = false: Whether or not to optimize the damping parameter of the second-order filter.
wl::Real: Lower frequency bound for design.
wu::Real: Upper frequency bound for design.
num_frequencies::Int: Number of frequency points.

Controller Structure

The optimized controller has the form:

K (s) = C (s) F (s) = (k_{p} + k_{i} / s + k_{d} s) F (s)

where $F (s)$ is a low-pass filter, either first- or second-order depending on filter_order.

Constraints

Sensitivity (Ms): Limits the peak of the sensitivity function $S (s)$ . Typical values are between 1 and 2.
Complementary Sensitivity (Mt): Limits the peak of $T (s)$ . Typical values are between 1 and 2.
Noise Sensitivity (Mks): Limits the peak of $K (s) S (s)$ , which determines the amplification of measurement noise in the control signal.

Lower values increase robustness but may reduce performance.

The peak $M_{S}$ in the sensitivity function $S (s)$ is associated with lower bounds on the classical gain and phase margins, $g_{m}, ϕ_{m}$ :

\begin{aligned} g_{m} & \geq \frac{M_{S}}{M_{S} - 1} \\ ϕ_{m} & \geq 2 {sin}^{- 1} (\frac{1}{2 M_{S}}) \end{aligned}

For example, a maximum sensitivity of $M_{S} = 1.3$ corresponds to a gain margin of at least 4.3 and a phase margin of at least 45 degrees. The peak constraints $M_{S}$ and $M_{T}$ corresponds to circles which the Nyquist curve must fall outside, these circles are drawn with dashed lines in the Nyquist plot artifact.

Advanced Options

Reference vs. Disturbance Step: By default, the disturbance step response is optimized. To optimize reference tracking, set step_input = :reference_input.
Controller Type: By setting parameter bounds, you can restrict the controller to P, PI, PD, or PID.
Filter Damping: With filter_order = 2 and optimize_d = true, the damping ratio of the filter is also optimized.
Penalizing Control Action: You can augment the model to penalize control effort or noise amplification.

Artifacts

A PIDAutotuningAnalysis returns the following artifacts:

Standard Plots

:SensitivityFunctions: Bode plot of sensitivity (S) and complementary sensitivity (T) functions.
:NoiseSensitivityAndController: Bode plot of noise sensitivity (KS) and controller (K).
:OptimizedResponse: Step response of the optimized closed-loop system.
:NyquistPlot: Nyquist plot of the loop transfer function.

Tables

OptimizedParameters: DataFrame of the optimized PID parameters. The parameters are available for the parallel controller structure as well as converted to the standard form $K_{p} (1 + 1 / (T_{s} s) + T_{d} s)$

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PID Autotuning Analysis ​

Method Overview ​

Example Definition ​

Analysis Arguments ​

Required Arguments ​

Optional Arguments ​

Controller Structure ​

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Advanced Options ​

Artifacts ​

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PID Autotuning Analysis

Method Overview

Example Definition

Analysis Arguments

Required Arguments

Optional Arguments

Controller Structure

Constraints

Advanced Options

Artifacts

Standard Plots

Tables

Further Reading