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Getting Reliable Estimates of Robustness Margins

This example shows how to modify the uncertainty description to avoid discontinuities and get reliable estimates of the margin of uncertainty.

Computing the smallest perturbation that causes instability at a given frequency is the cornerstone of most robustness analysis algorithms in Robust Control Toolbox™. To estimate, for example, the robust stability margin over the entire frequency range, the function robuststab performs this basic computation for a finite set of frequencies and looks at the worst case over this set.

Under most conditions, the robust stability margin is continuous with respect to frequency, so this approach gives good estimates provided you use a sufficiently dense frequency grid. However in some problems with only parameter uncertainty (UREAL), the migration of poles from stable to unstable can occur at isolated frequencies (generally unknown to you). As a result, any frequency grid that excludes these particular frequencies fail to detect worst-case perturbations and give over-optimistic stability margins.

How Discontinuities Can Hide Robustness Issues

In this example, we consider a spring-mass-damper system with 100% parameter uncertainty in the damping coefficient and 0% uncertainty in the spring coefficient. Note that all uncertainty here is of ureal type:

m = 1;
k = 1;
c = ureal('c',1,'plusminus',1);
sys = tf(1,[m c k]);

As the uncertain element c varies, the only place where the poles can migrate from stable to unstable is at s = j*1 (1 rad/sec). No amount of variation in c can cause them to migrate across the jw-axis at any other frequency.

When computing the robust stability margins with the robuststab function, almost any frequency grid will exclude f=1, leading to the incorrect conclusion that the margin of uncertainty on c is infinite.

omega = logspace(-1,1,40); % one possible grid
sysg = ufrd(sys,omega);
[stabmarg,du,rep,info] = robuststab(sysg);
stabmarg
stabmarg = 

                LowerBound: 5.0348e+03
                UpperBound: Inf
    DestabilizingFrequency: 0.1000

When we look at the mussv bounds computed by the robuststab function, the structured singular value mu is zero at all test frequencies in omega:

opt = bodeoptions;
opt.MagUnits = 'Abs';
opt.PhaseVisible = 'off';
opt.YLim = [0 1];
bodeplot(info.MussvBnds(:,1),'b*',info.MussvBnds(:,2),'g*',opt)
title('Original \mu bounds, 40 frequency points')
legend('Upper bound','Lower bound')

Figure 1: Original mu bounds, 40 frequency points.

Note that making the grid denser would not help. Only by adding f=1 to the grid will we find the true margin.

f = 1;
stabmarg = robuststab(ufrd(sys,f))
stabmarg = 

                LowerBound: 1.0000
                UpperBound: 1
    DestabilizingFrequency: 1

Modifying the Uncertainty Model to Eliminate Discontinuities

The example above shows that the robust stability margin can be a discontinuous function of frequency. In other words, it can have jumps. We can eliminate such jumps by adding a small amount of uncertain dynamics to every uncertain real parameter. This amounts to adding some dynamics to pure gains. Importantly, as the size of the added dynamics goes to zero, the estimated margin for the modified problem converges to the true margin for the original problem.

In the spring-mass-damper example, we model c as a ureal with the range [0.05,1.95] rather than [0,2], and add a ultidyn perturbation with gain bounded by 0.05. This combination covers the original uncertainty in c and introduces only 5% conservatism.

cc = ureal('cReal',1,'plusminus',0.95) + ultidyn('cUlti',[1 1],'Bound',0.05);
sysreg = usubs(sys,'c',cc);

Now let's recompute the robust stability margin:

[stabmarg,du,report,info] = robuststab(ufrd(sysreg,omega));
stabmarg
report
stabmarg = 

                LowerBound: 2.0016
                UpperBound: 2.3630
    DestabilizingFrequency: 1.0608


report =

Assuming nominal UFRD system is stable...
Uncertain system is robustly stable to modeled uncertainty.
 -- It can tolerate up to 200% of the modeled uncertainty.                                           
 -- A destabilizing combination of 236% of the modeled uncertainty was found.                        
 -- This combination causes an instability at 1.06 rad/seconds.                                      
 -- Sensitivity with respect to the uncertain elements are:                                          
     'cReal' is 35%.  Increasing 'cReal' by 25% leads to a 9% decrease in the margin.                
     'cUlti' is 63%.  Increasing 'cUlti' by 25% leads to a 16% decrease in the margin.               

Now the calculation determines that the margin is not infinite. The value 2.36 is still greater than 1 (the true margin) because the density of frequency points is not high enough.

If we plot the mu bounds (reciprocal of robust stability margin) as a function of frequency, a peak clearly appears around 1 rad/s:

bodeplot(info.MussvBnds(:,1),info.MussvBnds(:,2),opt)
title('Regularized \mu, 40 frequency points, 5% added dynamics');
legend('Upper bound','Lower bound')

Figure 2: Regularized mu, 40 frequency points, 5% added dynamics

Increasing the Frequency Density for Sharper Estimates.

The sharpness of the peak of the mussv plot as well as its shape suggests that a denser frequency grid may be necessary. We'll increase the frequency-grid density by a factor of 5, and recompute:

OmegaDense = logspace(-1,1,200);
[stabmarg,du,report,info] = robuststab(ufrd(sysreg,OmegaDense));
stabmarg
report
stabmarg = 

                LowerBound: 1.0026
                UpperBound: 1.0056
    DestabilizingFrequency: 0.9885


report =

Assuming nominal UFRD system is stable...
Uncertain system is robustly stable to modeled uncertainty.
 -- It can tolerate up to 100% of the modeled uncertainty.                                           
 -- A destabilizing combination of 101% of the modeled uncertainty was found.                        
 -- This combination causes an instability at 0.988 rad/seconds.                                     
 -- Sensitivity with respect to the uncertain elements are:                                          
     'cReal' is 95%.  Increasing 'cReal' by 25% leads to a 24% decrease in the margin.               
     'cUlti' is 6%.  Increasing 'cUlti' by 25% leads to a 2% decrease in the margin.                 

The computed robust stability margin has decreased significantly. It helped to use a denser grid, and the margin estimate is now close to the true value 1. The mu plot shows a pronounced peak at 1 rad/sec:

bodeplot(info.MussvBnds(:,1),info.MussvBnds(:,2),opt)
title('Regularized \mu, 200 frequency points, 5% added dynamics');
legend('Upper bound','Lower bound')

Figure 3: Regularized mu, 200 frequency points, 5% added dynamics

For the modified uncertainty model, the robust stability margin is 1, which is equal to that of the original problem. In general, the stability margin of the modified problem is less than or equal to that of the original problem. If it is significantly less, then the answer to the question "What is the stability margin?" is very sensitive to the uncertainty model. In this case, we put more faith in the value that allows for a few percents of unmodeled dynamics. Either way, the stability margin for the modified problem is most trustworthy.

Automating Substitution of Uncertainty Models

The command complexify automates the procedure of replacing a ureal with the sum of a ureal and ultidyn. The analysis above can be repeated using complexify obtaining the same results.

sysreg = complexify(sys,0.05,'ultidyn');
[stabmargc,duc,reportc,infoc] = robuststab(ufrd(sysreg,OmegaDense));
stabmargc
reportc
bodeplot(info.MussvBnds(:,1),'b+',info.MussvBnds(:,2),'b+',...
   infoc.MussvBnds(:,1),'r',infoc.MussvBnds(:,2),'r',opt)
title('Regularized \mu, 200 frequency points, 5% added dynamics');
legend('Upper bound','Lower bound','(complexify) Upper bound',...
   '(complexify) Lower bound','Location','NorthEast');
stabmargc = 

                LowerBound: 1.0026
                UpperBound: 1.0056
    DestabilizingFrequency: 0.9885


reportc =

Assuming nominal UFRD system is stable...
Uncertain system is robustly stable to modeled uncertainty.
 -- It can tolerate up to 100% of the modeled uncertainty.                                           
 -- A destabilizing combination of 101% of the modeled uncertainty was found.                        
 -- This combination causes an instability at 0.988 rad/seconds.                                     
 -- Sensitivity with respect to the uncertain elements are:                                          
     'c' is 95%.  Increasing 'c' by 25% leads to a 24% decrease in the margin.                       
     'c_cmpxfy' is 6%.  Increasing 'c_cmpxfy' by 25% leads to a 2% decrease in the margin.           

Figure 4: Regularized mu, 200 frequency points, 5% added dynamics

The continuity of the robust stability margin, and the subsequent computational and interpretational difficulties raised by the presence of discontinuities are considered in [1]. The consequences and interpretations of the regularization illustrated in this small example are described in [2]. An extensive analysis of regularization for 2-parameter example is given in [2].

References

[1] Barmish, B.R., Khargonekar, P.P, Shi, Z.C., and R. Tempo, "Robustness margin need not be a continuous function of the problem data," Systems & Control Letters, Vol. 15, No. 2, 1990, pp. 91-98.

[2] Packard, A., and P. Pandey, "Continuity properties of the real/complex structured singular value," Vol. 38, No. 3, 1993, pp. 415-428.

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