(Not recommended) Calculate robust stability margins of uncertain multivariable system
robuststab
is not recommended. Use robstab
instead.
[stabmarg,destabunc,report,info] = robuststab(sys) [stabmarg,destabunc,report,info] = robuststab(sys,opt)
A nominally stable uncertain system is generally unstable for specific values of its uncertain elements. Determining the values of the uncertain elements closest to their nominal values for which instability occurs is a robust stability calculation.
If the uncertain system is stable for all values of uncertain
elements within their allowable ranges (ranges for ureal
,
norm bound or positive-real constraint for ultidyn
,
radius for ucomplex
, weighted ball for ucomplexm
),
the uncertain system is robustly stable. Conversely,
if there is a combination of element values that cause instability,
and all lie within their allowable ranges, then the uncertain system
is not robustly stable.
robuststab
computes the margin of stability
robustness for an uncertain system. A stability robustness margin
greater than 1 means that the uncertain system is stable for all values
of its modeled uncertainty. A stability robustness margin less than
1 implies that certain allowable values of the uncertain elements,
within their specified ranges, lead to instability.
Numerically, a margin of 0.5 (for example) implies two things:
the uncertain system remains stable for all values of uncertain elements
that are less than 0.5 normalized units away from their nominal values
and, there is a collection of uncertain elements that are less than
or equal to 0.5 normalized units away from their nominal values that
results in instability. Similarly, a margin of 1.3 implies that the
uncertain system remains stable for all values of uncertain elements
up to 30% outside their modeled uncertain ranges. See actual2normalized
for
converting between actual and normalized deviations from the nominal
value of an uncertain element.
As with other uncertain-system analysis tools, only bounds on the exact stability margin are computed. The exact robust stability margin is guaranteed to lie in between these upper and lower bounds.
The computation used in robuststab
is a frequency-domain
calculation, which determines whether poles can migrate (due to variability
of the uncertain atoms) across the stability boundary (imaginary axis
for continuous-time, unit circle for discrete-time). Coupled with
stability of the nominal system, determining that no migration occurs
constitutes robust stability. If the input system sys
is
a ufrd
, then the analysis is performed on the frequency
grid within the ufrd
. Note that the stability of
the nominal system is not verified by the computation. If the input
system sys is a uss
, 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.
Suppose sys
is a ufrd
or uss
with M uncertain
elements. The results of
[stabmarg,destabunc,Report] = robuststab(sys)
are that stabmarg
is a structure with the
following fields
Field | Description |
---|---|
LowerBound | Lower bound on stability margin, positive scalar. If
greater than 1, then the uncertain system is guaranteed stable for
all values of the modeled uncertainty. If the nominal value of the
uncertain system is unstable, then |
UpperBound | Upper bound on stability margin, positive scalar. If less than 1, the uncertain system is not stable for all values of the modeled uncertainty. |
DestabilizingFrequency | The critical value of frequency at which instability
occurs, with uncertain elements closest to their nominal values. At
a particular value of uncertain elements (see |
destabunc
is a structure of values of uncertain
elements, closest to nominal, that cause instability. There are M field
names, which are the names of uncertain elements of sys
.
The value of each field is the corresponding value of the uncertain
element, such that when jointly combined, lead to instability. The
command pole(usubs(sys,destabunc))
shows the instability.
If A
is an uncertain element of sys, then
actual2normalized(destabunc.A,sys.Uncertainty.A)
will be less than or equal to UpperBound
,
and for at least one uncertain element of sys
,
this normalized distance will be equal to UpperBound
,
proving that UpperBound
is indeed an upper bound
on the robust stability margin.
Report
is a text description of the arguments
returned by robuststab
.
If sys
is an array of uncertain models, the
outputs are struct arrays whose entries correspond to each model in
the array.
Construct a feedback loop with a second-order plant and a PID
controller with approximate differentiation. The second-order plant
has frequency-dependent uncertainty, in the form of additive unmodeled
dynamics, introduced with an ultidyn
object and
a shaping filter.
robuststab
is used to compute the stability
margins of the closed-loop system with respect to the plant model
uncertainty.
P = tf(4,[1 .8 4]); delta = ultidyn('delta',[1 1],'SampleStateDimension',5); Pu = P + 0.25*tf([1],[.15 1])*delta; C = tf([1 1],[.1 1]) + tf(2,[1 0]); S = feedback(1,Pu*C); [stabmarg,destabunc,report,info] = robuststab(S);
You can view the stabmarg
variable.
stabmarg stabmarg = UpperBound: 0.8181 LowerBound: 0.8181 DestabilizingFrequency: 9.1321
As the margin is less than 1, the closed-loop system is not
stable for plant models covered by the uncertain model Pu
.
There is a specific plant within the uncertain behavior modeled by Pu
(actually
about 82% of the modeled uncertainty) that leads to closed-loop instability,
with the poles migrating across the stability boundary at 9.1 rads/s.
The report
variable is specific, giving a
plain-language version of the conclusion.
report report = Uncertain System is NOT robustly stable to modeled uncertainty. -- It can tolerate up to 81.8% of modeled uncertainty. -- A destabilizing combination of 81.8% the modeled uncertainty exists, causing an instability at 9.13 rad/s. -- Sensitivity with respect to uncertain element ... 'delta' is 100%. Increasing 'delta' by 25% leads to a 25% decrease in the margin.
Because the problem has only one uncertain element, the stability margin is completely determined by this element, and hence the margin exhibits 100% sensitivity to this uncertain element.
You can verify that the destabilizing value of delta
is
indeed about 0.82 normalized units from its nominal value.
actual2normalized(S.Uncertainty.delta,destabunc.delta) ans = 0.8181
Use usubs
to substitute the specific value
into the closed-loop system. Verify that there is a closed-loop pole
near j9.1
, and plot the unit-step response of the
nominal closed-loop system, as well as the unstable closed-loop system.
Sbad = usubs(S,destabunc); pole(Sbad) ans = 1.0e+002 * -3.2318 -0.2539 -0.0000 + 0.0913i -0.0000 - 0.0913i -0.0203 + 0.0211i -0.0203 - 0.0211i -0.0106 + 0.0116i -0.0106 - 0.0116i step(S.NominalValue,'r--',Sbad,'g',4);
Finally, as an ad-hoc test, set the gain bound on the uncertain delta
to
0.81 (slightly less than the stability margin). Sample the closed-loop
system at 100 values, and compute the poles of all these systems.
S.Uncertainty.delta.Bound = 0.81; S100 = usample(S,100); p100 = pole(S100); max(real(p100(:))) ans = -6.4647e-007
As expected, all poles have negative real parts.
A fourth output argument yields more specialized information, including sensitivities and frequency-by-frequency information.
[StabMarg,Destabunc,Report,Info] = robuststab(sys)
In addition to the first 3 output arguments, described previously, Info
is
a structure with the following fields
Field | Description |
---|---|
Sensitivity | A |
Frequency | N-by-1 frequency vector associated with analysis. |
BadUncertainValues | N-by-1 struct array containing the destabilizing uncertain element values at each frequency. |
MussvBnds | A 1-by-2 |
MussvInfo | Structure of compressed data from |
Use robuststabOptions
to
specify additional options for the robuststab
computation.
For example, you can control what is displayed during the computation,
turning 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 instance, you
can turn the display on, and the sensitivity calculation off by executing
opt = robuststabOptions('Sensitivity','off','Display','on'); [StabMarg,Destabunc,Report,Info] = robuststab(sys,opt)
See the robuststabOptions
reference
page for more information about available options.
Under most conditions, the robust stability margin at each frequency is a continuous function of the problem data at that frequency. Because the problem data, in turn, is a continuous function of frequency, it follows that finite frequency grids are usually adequate in correctly assessing robust stability bounds, assuming the frequency grid is dense enough.
Nevertheless, there are simple examples that violate this. In some problems, the migration of poles from stable to unstable only occurs at a finite collection of specific frequencies (generally unknown to you). Any frequency grid that excludes these critical frequencies (and almost every grid will exclude them) will result in undetected migration and misleading results, namely stability margins of ∞.
See Getting Reliable Estimates of Robustness Margins for more information about circumventing the problem in an engineering-relevant fashion.
A rigorous robust stability analysis consists of two steps:
Verify that the nominal system is stable;
Verify that no poles cross the stability boundary as the uncertain elements vary within their ranges.
Because the stability boundary is also associated with the frequency response, the second step can be interpreted (and carried out) as a frequency domain calculation. This amounts to a classical µ-analysis problem.
The algorithm in robuststab
follows this
in spirit, with the following limitations.
If sys
is a uss
object,
then the first requirement of stability of nominal value is explicitly
checked within robuststab
. However, if sys
is
an ufrd
, then the verification of nominal stability
from the nominal frequency response data is not performed, and is
instead assumed.
In the second step (monitoring the stability boundary
for the migration of poles), rather than check all points on stability
boundary, the algorithm only detects migration of poles across the
stability boundary at the frequencies in info.Frequency
.
See Limitations for information about issues related to migration detection.
The exact stability margin is guaranteed to be no larger than UpperBound
(some
uncertain elements associated with this magnitude cause instability
– one instance is returned in the structure destabunc
).
The instability created by destabunc
occurs at
the frequency value in DestabilizingFrequency
.
Similarly, the exact stability margin is guaranteed to be no
smaller than LowerBound
. In other words, for all
modeled uncertainty with magnitude up to LowerBound
,
the system is guaranteed stable. These bounds are derived using the
upper bound for the structured singular value, which is essentially
optimally-scaled, small-gain theorem analysis.