Robust performance margin of uncertain multivariable system
perfmarg = robustperf(usys) [perfmarg,wcu,report,info] = robustperf(usys) [perfmarg,wcu,report,info] = robustperf(usys,opt)
The performance of a nominally stable uncertain system model
will generally degrade for specific values of its uncertain elements.
largely included for historical purposes, computes the robust performance
margin, which is one measure of the level of degradation brought on
by the modeled uncertainty.
As with other uncertain-system analysis tools, only bounds on the performance margin are computed. The exact robust performance margin is guaranteed to lie between these upper and lower bounds.
The computation used in
robustperf is a frequency-domain
calculation. Coupled with stability of the nominal system, this frequency
domain calculation determines robust performance of
If the input system
usys is a
then the analysis is performed on the frequency grid within the
Note that the stability of the nominal system is not verified by the
computation. If the input system sys is a
then the stability of the nominal system is first checked, an appropriate
frequency grid is generated (automatically), and the analysis performed
on that frequency grid. In all discussion that follows, N denotes
the number of points in the frequency grid.
usys is a
uss with M uncertain
elements. The results of
[perfmarg,perfmargunc,Report] = robustperf(usys)
are such that
perfmarg is a structure with
the following fields:
Lower bound on robust performance margin, positive scalar.
Upper bound on robust performance margin, positive scalar.
The value of frequency at which the performance degradation curve crosses the y = 1/x curve. See Generalized Robustness Analysis.
perfmargunc is a
values of uncertain elements associated with the intersection of the
performance degradation curve and the y = 1/x curve.
See Generalized Robustness Analysis. There are M field
names, which are the names of uncertain elements of
Report is a text description of the robust
performance analysis results.
usys is an array of uncertain models,
the outputs are struct arrays whose entries correspond to each model
in the array.
Create a plant with a nominal model of an integrator, and include additive unmodeled dynamics uncertainty of a level of 0.4 (this corresponds to 100% model uncertainty at 2.5 rads/s).
P = tf(1,[1 0]) + ultidyn('delta',[1 1],'bound',0.4);
Design a "proportional" controller K that puts the nominal closed-loop bandwidth at 0.8 rad/s. Roll off K at a frequency 25 times the nominal closed-loop bandwidth. Form the closed-loop sensitivity function.
BW = 0.8; K = tf(BW,[1/(25*BW) 1]); S = feedback(1,P*K);
Assess the performance margin of the closed-loop sensitivity function. Because the nominal gain of the sensitivity function is 1, and the performance degradation curve is monotonically increasing (see Generalized Robustness Analysis), the performance margin should be less than 1.
[perfmargin,punc] = robustperf(S); perfmargin perfmargin = UpperBound: 7.4305e-001 LowerBound: 7.4305e-001 CriticalFrequency: 5.3096e+000
You can verify that the upper bound of the performance margin corresponds to a point on or above the y=1/x curve. First, compute the normalized size of the value of the uncertain element, and check that this agrees with the upper bound.
nsize = actual2normalized(S.Uncertainty.delta, punc.delta) nsize = perfmargin.UpperBound ans = 7.4305e-001
Compute the system gain with that value substituted, and verify that the product of the normalized size and the system gain is greater than or equal to 1.
gain = norm(usubs(S,punc),inf,.00001); nsize*gain ans = 1.0000e+000
Finally, as a sanity check, verify that the robust performance margin is less than the robust stability margin.
[stabmargin] = robuststab(S); stabmargin stabmargin = UpperBound: 3.1251e+000 LowerBound: 3.1251e+000 DestabilizingFrequency: 4.0862e+000
While the robust stability margin is easy to describe (poles
migrating from stable region into unstable region), describing the
robust performance margin is less elementary. See the diagrams and
figures in Generalized Robustness Analysis. Rather than finding
values for uncertain elements that lead to instability, the analysis
finds values of uncertain elements "corresponding to the intersection
point of the performance degradation curve with a y=1/x hyperbola."
This characterization, mentioned above in the description of
is used often in the descriptions below.
A fourth output argument yields more specialized information, including sensitivities and frequency-by-frequency information.
[perfmarg,perfmargunc,Report,Info] = robustperf(usys)
In addition to the first 3 output arguments, described previously,
a structure with the following fields:
N-by-1 frequency vector associated with analysis.
N-by-1 struct array containing the worst uncertain element values at each frequency.
Structure of compressed data from
specify additional options for the
For example, you can control what is displayed during the computation,
turn the sensitivity computation on or off, set the step size in the
sensitivity computation, or control the option argument used in the
underlying call to
mussv. For example, you can
turn the display on and turn off the sensitivity by executing
opt = robustperfOptions('Sensitivity','off','Display','on'); [PerfMarg,Destabunc,Report,Info] = robustperf(usys,opt)
page for more information about available options.
Because the calculation is carried out with a frequency gridding,
it is possible (likely) that the true critical frequency is missing
from the frequency vector used in the analysis. This is similar to
the problem in
robuststab. However, in comparing
robuststab, the problem in
less acute. The robust performance margin, considered a function of
problem data and frequency, is typically a continuous function (unlike
the robust stability margin, described in Getting Reliable Estimates of
Robustness Margins). Hence, in robust performance margin calculations,
increasing the density of the frequency grid will always increase
the accuracy of the answers, and in the limit, answers arbitrarily
close to the actual answers are obtainable with finite frequency grids.
A rigorous robust performance analysis consists of two steps:
Verify that the nominal system is stable.
Robust performance analysis on an augmented system.
The algorithm in
robustperf follows this
in spirit, with the following limitations:
usys is a
robustperf explicitly checks the stability
of the nominal value. However, if
usys is a
assumes that the nominal value is stable, and does not perform this
The exact performance margin is guaranteed to be no
UpperBound (some uncertain elements
associated with this magnitude cause instability – one instance
is returned in the structure
instability created by
perfmargunc occurs at the
frequency value in
Similarly, the exact performance margin is guaranteed
to be no smaller than