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ANALYSIS

Closed-Loop Sensitivity Analysis ​

Download as a Dyad projectClosedLoopSensitivityExample.zipOpen in Dyad Studio

The Closed-Loop Sensitivity Analysis computes and visualizes the sensitivity function    at one or several specified analysis points in a feedback system. This is useful for assessing the robustness of the closed-loop system to disturbances and model uncertainties.

Method Overview ​

The sensitivity function is the fundamental closed-loop transfer function. In the block diagram

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

performing a sensitivity analysis at the analysis point y computes the output sensitivity function    which is the transfer function between d₂ and e₃, while an analysis in the analysis point u computes the input sensitivity function   , which is the transfer function between d₁ and e₁. For SISO systems,  . The relationship between the sensitivity function and the complementary sensitivity function is given by   .

This analysis linearizes the model and computes the sensitivity function at the specified analysis point or points, optionally breaking feedback loops as needed.

The peak value of (the norm) is related to classical robustness margins such as gain and phase margin. The margin bounds derived from this peak are shown in the Bode plot legend.

Example Definition ​

This example is a continuation of the DC Motor Control tutorial. We will analyze the sensitivity function at the plant output y.

We set w_motor to zero so the controller does not saturate at the linearization point. Since the analysis opens the loop at r, the connection to controller.u_s is broken during linearization. We provide an initial condition so the operating point is well-defined:

julia
using DyadInterface: artifacts
using Plots
asol = DCMotorClosedLoopSensitivityAnalysis()
ClosedLoopSensitivityAnalysisSolution
Analysis points: ["y"]
Loop openings: ["r"]
Frequency bounds: [1.0, 10000.0]
H∞ norm ||S(s)||: 1.9
Phase margin lower bound: 31.0°
Gain margin lower bound: 2.1
julia
fig = artifacts(asol, :BodePlot)
plot!(legend = :bottomright)

Analysis Arguments ​

The following arguments define a ClosedLoopSensitivityAnalysis:

Required Arguments ​

  • model: The model to be analyzed.

  • analysis_points::Vector{String}: Name of the analysis points where sensitivity is computed.

Optional Arguments ​

  • loop_openings::Vector{String} = []: Names of loop openings to break feedback if present.

  • wl::Real = -1.0: Lower frequency bound for the analysis (set to -1 for automatic selection).

  • wu::Real = -1.0: Upper frequency bound for the analysis (set to -1 for automatic selection).

Artifacts ​

A ClosedLoopSensitivityAnalysis returns the following artifact:

Standard Plot ​

  • BodePlot: Bode plot of the closed-loop sensitivity function . The plot includes the norm (maximum value), and the corresponding minimum gain and phase margins. For MIMO analysis, the plot shows the singular values of the sensitivity function as a function of frequency.

Further Reading ​