Worstcase diskbased stability margins of uncertain feedback loops
The worstcase disk margin is the smallest disk margin that occurs within a
specified uncertainty range. It is also the minimum guaranteed margin over the uncertainty
range. wcdiskmargin
estimates the worstcase disk margins and
corresponding worstcase gain and phase margins for both loopatatime and multiloop
variations. The function also returns the worstcase perturbation, the combination of
uncertain elements that yields the weakest margins.
[wcDM,wcu] = wcdiskmargin(L,'siso')
[wcMM,wcu] = wcdiskmargin(L,'mimo')
[wcMMIO,wcu] = wcdiskmargin(P,C)
___ = wcdiskmargin(___,E)
___ = wcdiskmargin(___,opts)
[___,info] = wcdiskmargin(___)
[
estimates the worstcase loopatatime diskbased stability margins for the uncertain
negative feedback loop wcDM
,wcu
] = wcdiskmargin(L
,'siso')feedback(L,eye(N))
, where N
is
the number of inputs and outputs in L
.
While diskmargin
computes stability margins for a nominal model, wcdiskmargin
computes
the worstcase (smallest) disk margin over the modeled uncertainty in
L
. Diskbased margin analysis provides a stronger guarantee of robust
stability than the classical gain and phase margins. For general information about disk
margins, see Stability Analysis Using Disk Margins.
___ = wcdiskmargin(___,
specifies an additional eccentricity parameter that varies the shape of the disk region used
to compute the stability margins. You can use the eccentricity argument with any of the
previous syntaxes.E
)
[___,
returns a structure with additional information about the worstcase margins and the
perturbations that generate them. You can use this output argument with any of the previous
syntaxes.info
] = wcdiskmargin(___)
Use wcdiskmargin
to compute worstcase loopatatime and multiloop disk margins. This example illustrates that loopatatime margins can give an overly optimistic assessment of the true robustness of MIMO feedback loops. Margins of individual loops can be sensitive to small perturbations within other loops.
Consider the closedloop system of the following illustration.
P is a twoinput, twooutput secondorder plant and C is a 2x2 static gain. Construct P in statespace form, assuming that it has an uncertain parameter and some dynamic uncertainty. Compute the worstcase disk margins at the plant output.
alpha = ureal('alpha',10); a = [0.2 alpha;alpha 0.2]; b = eye(2); c = [1 8;10 1]; d = zeros(2,2); P = ss(a,b,c,0); DEL = ultidyn('DEL',[2 2],'Bound',1e2); Pu = P*(eye(2)+DEL); C = [1 2;0 1]; L = Pu*C; [wcDM,wcu] = wcdiskmargin(L,'siso');
(To compute the margins at the plant input, use L = C*Pu
.) Examine the worstcase loopatatime disk margins, returned in the structure array wcDM
. Each entry in this structure array contains the worstcase stability margins of the corresponding channel.
wcDM(1)
ans = struct with fields:
GainMargin: [0.2057 4.8618]
PhaseMargin: [66.7543 66.7543]
DiskMargin: 1.3176
LowerBound: 1.3176
UpperBound: 1.3205
CriticalFrequency: 0
The result in wcDM(1)
gives guaranteed stability margins for the specified uncertainty range. As long as the openloop gain of the first channel changes by a factor no less than about 0.20 and no more than about 4.86, the closed loop remains stable for all (alpha,DEL)
values within the specified range. Similarly, the closed loop remains stable as long as the phase variation does not exceed about 66.75° in absolute value.
Similarly, wcDM(2)
shows that in the second feedback channel, the gain can vary by any factor between about 0.16 and about 6.3 or the phase can vary by up to about 72°, and the system remains stable for all uncertainties.
wcDM(2)
ans = struct with fields:
GainMargin: [0.1583 6.3187]
PhaseMargin: [72.0140 72.0140]
DiskMargin: 1.4535
LowerBound: 1.4535
UpperBound: 1.4567
CriticalFrequency: 0
The lower bound returned by wcdiskmargin
is a theoretical minimum guaranteed worstcase disk margin. The upper bound corresponds to an actual perturbation in the specified uncertainty range that approaches the lowerbound prediction. The output wcu
contains the values of that perturbation for each feedback channel. wcu(1)
is the worst combination of (alpha,DEL)
for the first channel. For L
evaluated at this combination, the actual disk margins are close to the values in wcDM(1)
, and the upper bound matches wcDM(1).UpperBound
.
Lwc = usubs(L,wcu(1)); DM = diskmargin(Lwc); DM(1)
ans = struct with fields:
GainMargin: [0.2047 4.8863]
PhaseMargin: [66.8678 66.8678]
DiskMargin: 1.3205
LowerBound: 1.3205
UpperBound: 1.3205
Frequency: 0
In practice, uncertainties as well as gain and phase variations affect both channels simultaneously. To estimate the stability margins with respect to such independent and concurrent uncertainty, examine the worstcase multiloop disk margins.
[wcMM,wcu] = wcdiskmargin(L,'mimo');
wcMM
wcMM = struct with fields:
GainMargin: [0.5980 1.6722]
PhaseMargin: [28.2392 28.2392]
DiskMargin: 0.5031
LowerBound: 0.5031
UpperBound: 0.5043
CriticalFrequency: 0
That the guaranteed margins are significantly smaller than when considering one loop at a time. This result occurs because it takes a smaller amount of gain (or phase) variation to destabilize the feedback loop when both channels are subject to concurrent and independent variations. As with the loopatatime margins, the upper bound on the worstcase margin corresponds to the worstcase (alpha,DEL)
found by wcdiskmargin
and returned in wcu
. For the multiloop margin, there is only one set of worstcase uncertainty values.
Finally, compute the multiloop margin against simultaneous variations in gain (or phase) at both the plant inputs and plant outputs. When you allow the gain (or phase) to vary in more places, it becomes easier to destabilize the feedback loop, so the margins get smaller. Thus, the multiloop margin provides the most conservative guarantee of closedloop stability for any (alpha,DEL)
in the specified ranges.
[wcMMIO,wcu] = wcdiskmargin(Pu,C); wcMMIO
wcMMIO = struct with fields:
GainMargin: [0.8198 1.2199]
PhaseMargin: [11.3134 11.3134]
DiskMargin: 0.1981
LowerBound: 0.1981
UpperBound: 0.1985
CriticalFrequency: 0
L
— Uncertain openloop responseUncertain openloop response, specified as an uncertain model such as a
uss
or ufrd
model. L
can be SISO or MIMO, as long as it has the same number of inputs and outputs.
wcdiskmargin
computes the worstcase diskbased stability margins
for the negativefeedback closedloop system
feedback(L,eye(N))
.
To compute the worstcase disk margins of the positive feedback system
feedback(L,eye(N),+1)
, use
wcdiskmargin(L)
.
When you have a controller P
and a plant C
,
you can compute the worstcase disk margins for gain (or phase) variations at the plant
inputs or outputs, as in the following diagram.
To compute margins at the plant outputs, set L = P*C
.
To compute margins at the plant inputs, set L = C*P
.
To consider variations at both the plant inputs and the plant output, use the
syntax [wcMMIO,wcu] = wcdiskmargin(P,C)
instead.
L
can be continuous time or discrete time. If
L
is a generalized statespace model (genss
)
then wcdiskmargin
uses the current value of the tunable control
design blocks in L
.
If L
is a frequencyresponse data model (such as
ufrd
), then wcdiskmargin
computes the
margins at each frequency represented in the model. The function returns the worstcase
margins at the frequency with the smallest disk margin.
If L
is a model array, then wcdiskmargin
computes margins for each model in the array.
P
— PlantPlant, specified as an uncertain model such as a uss
or
ufrd
model. P
can be SISO or MIMO, as long
as P*C
has the same number of inputs and outputs.
wcdiskmargin
computes the worstcase disk margins for a
negativefeedback closedloop system. To compute the disk margins of the system with
positive feedback, use wcdiskmargin(P,C)
.
P
can be continuous time or discrete time. If
P
is a generalized statespace model (genss
)
then wcdiskmargin
uses the current value of the tunable control
design blocks in P
.
If P
is a frequencyresponse data model (such as
frd
), then wcdiskmargin
computes the margins
at each frequency represented in the model. The function returns the worstcase margins
at the frequency with the smallest disk margin.
C
— ControllerController, specified as a dynamic system model. C
can be SISO
or MIMO, as long as P*C
has the same number of inputs and outputs.
wcdiskmargin
computes the diskbased stability margins for a
negativefeedback closedloop system. To compute the disk margins of the system with
positive feedback, use wcdiskmargin(C,P)
.
C
can be continuous time or discrete time. If
C
is a generalized statespace model (genss
)
then wcdiskmargin
uses the current value of the tunable control
design blocks in C
.
If C
is a frequencyresponse data model (such as
frd
), then wcdiskmargin
computes the margins
at each frequency represented in the model. The function returns the worstcase margins
at the frequency with the smallest disk margin.
E
— EccentricityEccentricity of uncertainty region used to compute the stability margins, specified
as a real scalar value. Use this parameter to vary the shape of the disk region used to
model gain and phase variations. Varying the eccentricity parameter yields lower
estimates of the true stability margins, letting you infer a larger region of guaranteed
stability than that obtained using the default E
= 0. Some special
values of E
include:
0 — Margins based on balanced sensitivity function
1 — Margins based on sensitivity function
–1 — Margins based on complementary sensitivity function
For more detailed information about how the choice of E
affects
the margin computation, see Stability Analysis Using Disk Margins.
opts
— Options for margin computationwcOptions
objectOptions for worstcase computation, specified as an object you create with
wcOptions
. The available options include settings that let
you:
Extract frequencydependent worstcase margins.
Examine the sensitivity of the worstcase margins to each uncertain element.
Improve the results of the worstcase margin calculation by setting certain
options for the underlying mussv
calculation.
For more information about all available options, see wcOptions
.
Example: wcOptions('Sensitivity','on','MussvOptions','m3')
wcDM
— Worstcast disk margins for each feedback channelWorstcase disk margins for each feedback channel with all other loops closed,
returned as a structure for SISO feedback loops, or an Nby1
structure array for a MIMO loop with N feedback channels. The fields
of wcDM(i)
are:
GainMargin
— Minimum guaranteed gain margin of the
corresponding feedback channel, returned as a vector of the form
[gmin,gmax]
. These values mean that as long as the openloop
gain of the ith channel changes by a factor no less than
gmin
and no more than gmax
, the closed loop
remains stable for all uncertainty values within the ranges specified in
L
. If the closedloop system goes unstable for some
combination of uncertainty values, then DM(i).GainMargin = [1
1]
.
PhaseMargin
— Minimum guaranteed phase margin of the
corresponding feedback channel, returned as a vector of the form
[pm,pm]
in degrees. If the closedloop system goes unstable
for some combination of uncertainty values, then wcDM(i).PhaseMargin = [0
0]
.
DiskMargin
— Minimum guaranteed disk margin (see Stability Analysis Using Disk Margins for the
definition and interpretation of the disk margin). If the closedloop system is
unstable for some combination of uncertainelement values, then
wcDM(i).DiskMargin = 0
.
LowerBound
— Lower bound on worstcase disk margin. This
value is the same as DiskMargin
.
UpperBound
— Upper bound on worstcase disk margin. This
value is the disk margin obtained for the worst perturbation found by
wcdiskmargin
, returned as
wcu
(i)
. The actual worstcase disk margin
is no better than this value.
CriticalFrequency
— Frequency at which the disk margin for
the worst perturbation wcu(i)
is weakest, as a function of
frequency. This value is in rad/TimeUnit
, where
TimeUnit
is the TimeUnit
property of
L
.
When L = P*C
is the openloop response of a system comprising a
controller and plant with unit negative feedback in each channel,
wcDM
contains the stability margins for variations at the plant
outputs. To compute the stability margins for variations at the plant inputs, use
L = C*P
. To compute the stability margins for simultaneous,
independent variations at both the plant inputs and outputs, use wcMMIO =
wcdiskmargin(P,C)
.
When L
is a model array, wcDM
has
additional dimensions corresponding to the array dimensions of L
.
For instance, if L
is a 1by3 array of twoinput, twooutput
models, then wcDM
is a 2by3 structure array.
wcDM(j,k)
contains the margins for the
j^{th} feedback channel of the k^{th}
model in the array.
wcu
— Worstcase perturbationWorst case perturbation of uncertain elements, returned as:
A structure array of dimensions Nby1 for loopatatime margins, where N is the number of feedback channels
A scalar structure for multiloop margins
The lower bound returned by wcdiskmargin
is a theoretical
minimum guaranteed worstcase disk margin. The upper bound corresponds to an actual
perturbation in the specified uncertainty range that approaches the lowerbound
prediction. wcu
contains the values of that perturbation. For
example, if the input system includes uncertain elements M
and
delta
, then wcu.M
and
wcu.delta
contain the worst perturbations found by
wcdiskmargin
. It is possible that a worse perturbation exists,
but no perturbation can yield a worse margin than the lower bound returned by
wcdiskmargin
.
Use usubs
to substitute these values for the uncertain elements
in the input system, to obtain the dynamic system that has the worstcase disk
margin.
wcMM
— Worstcase multiloop disk marginsWorstcase multiloop disk margins, returned as a structure. The gain (or phase)
margins quantify how much gain variation (or phase variation) the system can tolerate in
all feedback channels at once while remaining stable. Thus, MM
is a
single structure regardless of the number of feedback channels in the system. (For SISO
systems, MM
= DM
.) The fields of
MM
are:
GainMargin
— Minimum guaranteed multiloop gain margin,
returned as a vector of the form [gmin,gmax]
. These values mean
that as long as the gain in all loop channels changes by a factor no less than
gmin
and no more than gmax
, the closed loop
remains stable for all uncertainty values within the ranges specified in
L
. If the closedloop system goes unstable for some
combination of uncertainty values, then wcMM.GainMargin = [1
1]
.
PhaseMargin
— Minimum guaranteed multiloop phase margin,
returned as a vector of the form [pm,pm]
in degrees. If the
closedloop system goes unstable for some combination of uncertainty values, then
wcMM.PhaseMargin = [0 0]
.
DiskMargin
— Minimum guaranteed disk margin (see Stability Analysis Using Disk Margins for the
definition and interpretation of the disk margin). If the closedloop system is
unstable for some combination of uncertainelement values, then
wcMM.DiskMargin = 0
.
LowerBound
— Lower bound on worstcase disk margin. This
value is the same as DiskMargin
.
UpperBound
— Upper bound on worstcase disk margin. This
value is the disk margin obtained for the worst perturbation found by
wcdiskmargin
, returned as wcu
. The
actual worstcase multiloop disk margin is no better than this value.
CriticalFrequency
— Frequency at which the disk margin for
the worst perturbation wcu
is weakest, as a function of
frequency. This value is in rad/TimeUnit
, where
TimeUnit
is the TimeUnit
property of
L
.
When L = P*C
is the openloop response of a system comprising a
controller and plant with unit negative feedback in each channel,
MM
contains the stability margins for variations at the plant
outputs. To compute the stability margins for variations at the plant inputs, use
L = C*P
. To compute the stability margins for simultaneous,
independent variations at both the plant inputs and outputs, use wcMMIO =
wcdiskmargin(P,C)
.
When L
is a model array, MM
is a structure
array with one entry for each model in L
.
info
— Additional information about worstcase valuesAdditional information about the worstcase values, returned as a structure with the following fields:
Field  Description 

 Index of the model that has the smallest disk margin, when

 Frequency points at which
The 
 Lower and upper bounds on the actual worstcase disk margin of the
model, returned as an array. 
 Worst perturbations at each frequency point in

 Sensitivity of the worstcase disk margin to each uncertain element,
returned as a structure when the If the

wcdiskmargin
assumes negative feedback. To compute the worstcase
disk margins of a positive feedback system, use wcdiskmargin(L)
or
wcdiskmargin(P,C)
.
wcdiskmargin
models gain (and phase) variation as
ucomplex
uncertainty, combines it with the specified plant uncertainty,
and uses mussv
to compute the worstcase disk margins and perturbation.
This generalizes the diskmargin
algorithm to feedback loops with
uncertainty. For more information about diskmargin computation and interpretation, see Stability Analysis Using Disk Margins.
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