www.gusucode.com > robust_featured 案例源码程序 matlab代码 > robust_featured/caution_demo.m
%% Getting Reliable Estimates of Robustness Margins
% This example illustrates the pitfalls of using frequency gridding to
% compute robustness margins for systems with only real uncertain
% parameters. It presents a safer approach along with ways to mitigate
% discontinuities in the structured singular value $\mu$.
%% How Discontinuities Can Hide Robustness Issues
% 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 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. As a result, the robust stability margin is infinite at
% all frequencies except 1 rad/s, where the margin with respect to the
% specified uncertainty is 1. In other words, the robust stability margin and the
% underlying structured singular value $\mu$ are discontinuous as a function
% of frequency.
%
% The traditional approach to computing the robust stability margin is to
% pick a frequency grid and compute lower and upper bounds for $\mu$ at
% each frequency point. Under most conditions, the robust stability margin
% is continuous with respect to frequency and this approach gives good
% estimates provided you use a sufficiently dense frequency grid. However
% in problems with only |ureal| uncertainty, such as the example
% above, poles can migrate from stable to unstable only at specific
% frequencies (the points of discontinuity for $\mu$), so any frequency
% grid that excludes these particular frequencies will lead to over-optimistic
% stability margins.
%
% To see this effect, pick a frequency grid for the spring-mass-damper system
% above and compute the robust stability margins at these frequency points
% using |robstab|.
omega = logspace(-1,1,40); % one possible grid
[stabmarg,wcu,info] = robstab(sys,omega);
stabmarg
%%
% The field |info.Bounds| gives the margin lower and upper bounds at each
% frequency. Verify that the lower bound (the guaranteed margin) is large
% at all frequencies.
loglog(omega,info.Bounds(:,1))
title('Robust stability margin: 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 = robstab(sys,f)
%% Safe Computation of Robustness Margins
% Rather than specifying a frequency grid, apply |robstab| directly to the
% USS model |sys|. This uses a more advanced algorithm that is guaranteed
% to find the peak of $\mu$ even in the presence of a discontinuity. This
% approach is more accurate and often faster than frequency gridding.
[stabmarg,wcu] = robstab(sys)
%%
% This computes the correct robust stability margin (1), identifies the
% critical frequency (|f=1|), and finds the smallest destabilizing
% perturbation (setting |c=0|, as expected).
%% 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);
%%
% Recompute the robust stability margin over the frequency grid
% |omega|.
[stabmarg,~,info] = robstab(sysreg,omega);
stabmarg
%%
% Now the frequency-gridded calculation yields a margin of
% 2.36. This is still greater than 1
% (the true margin) because the density of frequency points is
% not high enough. Increase the number of points from 40 to 200
% and recompute the margin.
OmegaDense = logspace(-1,1,200);
[stabmarg,~,info] = robstab(sysreg,OmegaDense);
stabmarg
%%
% Plot the robustness margin as a function of frequency.
loglog(OmegaDense,info.Bounds(:,1),OmegaDense,info.Bounds(:,2))
title('Robust stability margin: 5% added dynamics, 200 frequency points')
legend('Lower bound','Upper bound')
%%
% The computed margin is now close to 1,
% the true margin for 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 more trustworthy.
%% Automated Regularization of Discontinuous Problems
% 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');
[stabmarg,~,info] = robstab(sysreg,OmegaDense);
stabmarg
%%
% Note that such regularization is only needed when using frequency
% gridding. Applying |robstab| directly to the original uncertain
% model |sys| yields the correct
% margin without frequency gridding or need for regularization.
%% References
% 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].
%
% [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.