Worst-Case Performance

To be robust, your control system should meet your stability and performance requirements for all possible values of uncertain parameters. Monte Carlo parameter sampling via usample can be used for this purpose as shown in Monte Carlo Sampling of Uncertain System's Step Response, but Monte Carlo methods are inherently hit or miss. With Monte Carlo methods, you might need to take an impossibly large number of samples before you hit upon or near a worst-case parameter combination.

Robust Control Toolbox software gives you a powerful assortment of robustness analysis commands that let you directly calculate upper and lower bounds on worst-case performance without random sampling.

Worst-Case Robustness Analysis Commands
loopmargin	Comprehensive analysis of feedback loop
loopsens	Sensitivity functions of feedback loop
ncfmargin	Normalized coprime stability margin of feedback loop
robustperf	Robust performance of uncertain systems
robuststab	Stability margins of uncertain systems
wcgain	Worst-case gain of an uncertain system
wcmargin	Worst-case gain/phase margins for feedback loop
wcsens	Worst-case sensitivity functions of feedback loop

Example: ACC Two-Cart Benchmark Problem

Returning to the Example: ACC Benchmark Problem, the closed loop system is:

T=feedback(P*C,1); % Closed-loop uncertain system

This uncertain state-space model T has three uncertain parameters, k, m1, and m2, each equal to 1±20% uncertain variation. To analyze whether the closed-loop system T is robustly stable for all combinations of values for these three parameters, you can execute the commands:

[StabilityMargin,Udestab,REPORT] = robuststab(T);
REPORT

This displays the REPORT:

Uncertain System is robustly stable to modeled uncertainty.            
 -- It can tolerate up to 311% of modeled uncertainty.                 
 -- A destabilizing combination of 500% the modeled uncertainty exists,
    causing an instability at 44.3 rad/s.                              

The report tells you that the control system is robust for all parameter variations in the ±20% range, and that the smallest destabilizing combination of real variations in the values k, m1, and m2 has sizes somewhere between 311% and 500% greater than ±20%, i.e., between ±62.2% and ±100%. The value Udestab returns an estimate of the 500% destabilizing parameter variation combination:

Udestab = 
     k: 1.2174e-005
    m1: 1.2174e-005
    m2: 2.0000. 

Uncertain System Closed-Loop Bode Plots

You have a comfortable safety margin of between 311% to 500% larger than the anticipated ±20% parameter variations before the closed loop goes unstable. But how much can closed-loop performance deteriorate for parameter variations constrained to lie strictly within the anticipated ±20% range? The following code computes worst-case peak gain of T, and estimates the frequency and parameter values at which the peak gain occurs:

[PeakGain,Uwc] = wcgain(T);
Twc=usubs(T,Uwc); 
% Worst case closed-loop system T
Trand=usample(T,4); 
% 4 random samples of uncertain system T
bodemag(Twc,'r',Trand,'b-.',{.5,50}); % Do bode plot
legend('T_{wc}   - worst-case',...
'T_{rand} - random samples',3);

The resulting plot is shown in Uncertain System Closed-Loop Bode Plots.

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Modeling Uncertainty

Synthesis of Robust MIMO Controllers

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