Closed-Loop Analysis

The Closed-Loop Analysis computes and visualizes the frequency- and time-domain properties of a feedback system by linearizing the model around an operating point. This analysis is useful for assessing stability, robustness, and performance of the closed-loop system.

Method Overview

The analysis considers the standard feedback interconnection:

              d             
     ┌─────┐  │  ┌─────┐    
r  e │     │u ▼  │     │ y  
──+─►│  C  ├──+─►│  P  ├─┬─►
  ▲  │     │     │     │ │  
 -│  └─────┘     └─────┘ │  
  │                      │  
  └──────────────────────┘

where C is the controller, P is the plant, r is the reference, d is an input disturbance, y is the measured output, and u is the control input.

The analysis linearizes the model and computes:

• Closed-loop transfer functions:

  • Sensitivity: $S=1/(1+PC)$. For MIMO systems, the output sensitivity is computed.

  • Complementary sensitivity: $T=PC/(1+PC)$

  • Controller sensitivity: $C/(1+PC)$

  • Plant sensitivity: $P/(1+PC)$

• Robustness margins:

  • Disk margin (combined gain and phase margin). For MIMO systems, the simultaneous margin at the output is computed.

  • Classical gain and phase margins

• Step responses to reference and disturbance inputs

Example Definition

This example is a continuation of the DC Motor Control tutorial. We will analyze the closed-loop properties of the transfer function from load disturbance (entering at the plant input u) to output y.

analysis DCMotorClosedLoopAnalysis
  extends DyadControlSystems.ClosedLoopAnalysis(
    measurement   = ["y"],
    control_input = ["u"],
    loop_openings = ["r"],
    wl = 1,
    wu = 1e4,
    num_frequencies = 300,
    pos_feedback = true,
    duration = 2.0
  )
  model = DyadExampleComponents.TestDCMotorLoadControlled()
end

using DyadInterface: artifacts
asol = DCMotorClosedLoopAnalysis()

ClosedLoopAnalysisSolution
Phase margin (disk based): 34.9°
Gain margin (disk based): ["0.5", "1.9"]

using Plots
plot!(artifacts(asol, :all); size=(1200,1200))

Analysis Arguments

The following arguments define a ClosedLoopAnalysis:

Required Arguments

• model: The model to be analyzed.

• measurement::Vector{String}: Name of the measurement signal analysis points (plant outputs).

• control_input::Vector{String}: Name of the control input signal analysis points.

Optional Arguments

• wl::Real = -1: Lower frequency bound for analysis (set to -1 for automatic selection).

• wu::Real = -1: Upper frequency bound for analysis (set to -1 for automatic selection).

• num_frequencies::Int = 300: Number of frequency points.

• pos_feedback::Bool = true: Whether the feedback is positive (default is true; negative feedback is often built into the model).

• duration::Real = -1.0: Duration of the step-response simulation (set to -1 for automatic selection).

Artifacts

A ClosedLoopAnalysis returns the following artifacts:

Standard Plots

• all: Combined plot showing:

  • Bode plots of $S$, $T$, $C/(1+PC)$, $P/(1+PC)$

  • Disk margin and classical margins

  • Step responses to reference and disturbance steps

Further Reading

• ControlSystems.jl documentation

• Disk margin (RobustAndOptimalControl.jl)

• Feedback control (Wikipedia)

Notes about MIMO Systems

When MIMO systems are analyzed, some sensitivity functions are drawn as Sigma plots rather than Bode plots. The disk margin is in this case computed for the output loop-transfer function $L_o=PC$, that is, the margin for simultaneous output-perturbations is analyzed. This is generally a conservative analysis. All computed sensitivity functions are computed at the plant output (using the loop-transfer function $L_o=PC$ rather than $L_i=CP$).