Pole Placement Analysis

The pole placement analysis is used to design observer-based state-feedback controllers through linearization and pole placement techniques. The analysis linearizes the plant model at a specified operating point and designs both controller and observer gains to achieve desired closed-loop pole locations with specified damping and bandwidth characteristics.

Method Overview

The pole placement analysis in DyadControlSystems follows a systematic approach to controller design:

1. Linearization: The plant is linearized at the specified operating point with optional loop openings

2. Plant Augmentation: If integral action is requested, the plant is augmented with integrating disturbance models using add_low_frequency_disturbance

3. Discretization: If discrete-time control is specified, the plant is discretized using the chosen method

4. Controller Pole Design: Poles are modified to meet minimum damping and bandwidth constraints

5. Observer Pole Design: Observer poles are placed to be faster than controller poles

6. Controller Synthesis: State feedback gain L and observer gain K are computed using pole placement

7. Final Controller: An observer-based controller is constructed combining state feedback and state estimation

Integral Action

The analysis supports integral action through plant augmentation with integrating disturbances. This is achieved by:

• Specifying integrator_indices to indicate which control inputs should have integral action

• Providing integrator_poles to set the observer poles for the integrator states

• The plant is automatically augmented using add_low_frequency_disturbance

This approach provides disturbance rejection and eliminates steady-state errors for constant references and disturbances.

Example Definition

The following example demonstrates pole placement controller design for a DC motor system with integral action:

analysis DCMotorPolePlacementAnalysis
  extends DyadControlSystems.PolePlacementAnalysis(
    measurement = ["y"],
    control_input = ["u"], 
    loop_openings = ["y", "r"],
    min_damping = 0.707,
    controller_speed_factor = 2.0,
    observer_speed_factor = 5.0,
    integrator_indices = [1],
    integrator_poles = [-10.0],
    duration = 1.0
  )

  model = DyadExampleComponents.TestDCMotorLoadControlled()

end

using DyadExampleComponents, DyadInterface
solution = DCMotorPolePlacementAnalysis()

┌ Warning: Initialization system is overdetermined. 2 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
┌ Warning: Connected signals have different names
│  ##y#1122 -> ##u#1124
└ @ RobustAndOptimalControl ~/.julia/packages/RobustAndOptimalControl/KfD3M/src/named_systems2.jl:423

using Plots
using DyadInterface: artifacts
fig_step = artifacts(solution, :StepResponse)

fig_gang = artifacts(solution, :GangOfFour)

fig_bode = artifacts(solution, :BodePlot)

fig_pz = artifacts(solution, :PZMaps)

We can examine the controller and observer gains:

L = artifacts(solution, :ControllerGain)
K = artifacts(solution, :ObserverGain)
L, K

(3×1 DataFrame
 Row │ u1      
     │ Float64 
─────┼─────────
   1 │ 1.51995
   2 │ 0.50225
   3 │ 1.0, 1×3 DataFrame
 Row │ x1       x2       x3      
     │ Float64  Float64  Float64 
─────┼───────────────────────────
   1 │  1014.5  7157.56    510.0)

Analysis Arguments

The following arguments define a pole placement analysis:

Required Arguments

• model: The system model to be controlled.

• measurement::Vector{String}: Names of the measured output signals.

• control_input::Vector{String}: Names of the control input signals.

Controller Design Parameters

• min_damping::Real = 0.707: Minimum damping ratio for controller poles. Poles with lower damping will be modified.

• controller_speed_factor::Real = 1.0: Factor by which to speed up controller poles after damping modification.

• observer_speed_factor::Real = 5.0: Factor by which observer poles are faster than controller poles.

• min_bandwidth::Real = -1.0: Minimum bandwidth below which poles are moved to this bandwidth. Set to -1 for no limit.

• max_bandwidth::Real = -1.0: Maximum bandwidth beyond which poles are not modified. Set to -1 for no limit.

• direct_controller::Bool = false: Whether or not to use direct feedthrough in the controller.

Integral Action Parameters

• integrator_indices::Vector{Int} = []: Indices of control inputs that should be augmented with integrating disturbances for integral action. This is a vector of integers corresponding to the indices of control_input.

• integrator_poles::Vector{Real} = []: Observer poles for the integrator states. Must have the same length as integrator_indices.

System Parameters

• loop_openings::Vector{String} = []: Names of loop openings to break feedback loops during linearization.

• t::Real = 0.0: The time at which to perform the linearization of the system.

Discretization Parameters

• disc::String = "cont": Discretization method. Use "cont" for continuous-time, or "zoh", "tustin", "foh" for discrete-time.

• Ts::Real = -1.0: Sampling time for discretization. Required when disc != "cont". Set to -1 for automatic selection.

Analysis Parameters

• wl::Real = -1: The lower frequency bound for analysis plots. Set to -1 for automatic selection.

• wu::Real = -1: The upper frequency bound for analysis plots. Set to -1 for automatic selection.

• num_frequencies::Int = 3000: The number of frequencies to be used in frequency-domain analysis.

• duration::Real = -1.0: The duration of the step-response analysis. Set to -1 for automatic selection.

Pole Placement Algorithm

The controller design follows this systematic approach:

1. Linearization: The plant is linearized at the specified operating point with loop openings

2. Plant Augmentation: If integrator_indices is specified, the plant is augmented with integrating disturbances using add_low_frequency_disturbance to provide integral action

3. Discretization: If disc != "cont", the plant is discretized using the specified method and sampling time

4. Pole Analysis: Original poles are computed and analyzed for damping characteristics

5. Controller Pole Design:

• Poles below min_bandwidth are moved to the minimum bandwidth (if specified)

• Poles with damping below min_damping are modified to have the specified minimum damping

• Well-damped poles are left unchanged (poles with damping ≥ min_damping)

• Poles above max_bandwidth are ignored (if specified)

• All poles are then scaled by controller_speed_factor

6. Observer Pole Design: Observer poles are set to be observer_speed_factor times faster than controller poles, with additional poles from integrator_poles for the augmented disturbance states

7. Controller Synthesis: State feedback gain L and observer gain K are computed using place

8. Final Controller: An observer-controller is created using observer_controller

Controller Design Considerations

Damping and Bandwidth

• Minimum Damping: Ensures adequate transient response by modifying poorly damped complex poles

• Bandwidth Constraints: min_bandwidth moves slow poles to improve response speed, while max_bandwidth preserves high-frequency poles for robustness

• Speed Factors: Allow independent tuning of controller and observer dynamics

Integral Action

• Disturbance Rejection: Integral action eliminates steady-state errors from constant disturbances

• Reference Tracking: Provides zero steady-state error for step references

• Observer Poles: Integrator states require separate observer pole specification for proper estimation

Artifacts

A PolePlacementAnalysis returns the following artifacts:

Standard Plots

• :StepResponse: Step response showing reference tracking, disturbance rejection, and control input response.

• :GangOfFour: Gang of four transfer functions (S, PS, CS, T) showing closed-loop sensitivity properties.

• :BodePlot: Bode plot of the open-loop transfer function showing gain and phase characteristics.

• :MarginPlot: Gain and phase margins indicating stability robustness.

• :PZMaps: Pole-zero maps of the open-loop and closed-loop systems.

Tables

• :ControllerGain: DataFrame of the state feedback gain matrix L.

• :ObserverGain: DataFrame of the observer gain matrix K.

Design Guidelines

• Avoid making fast process poles slower to avoid high sensitivity.

• Slow process zeros should be matched by closed-loop poles to avoid high sensitivity.

• Significant movement of high-frequency poles may lead to poor robustness due to high-frequency model uncertainty.

• Slow unstable zeros and fast unstable poles lead to fundamental limitations on the performance.

See "Pole placement" by Bo Bernharsson and Karl Johan Åström for more details.

Notes

• The system must be controllable and observable for pole placement to work

• Complex conjugate pairs are handled as single entities.

• High-frequency poles can be preserved by setting max_bandwidth for robustness against high-frequency uncertainty.

• For exact manual pole placement, use Dyad Studio and call place manually, see

Further Reading

• Pole Placement (Wikipedia)

• Observer-Controller Design (Wikipedia)

• Ackermann's Formula (Wikipedia)

• Integral Action Documentation

• ControlSystems.jl Documentation