wcdiskmargin

Worst-case disk-based stability margins of uncertain feedback loops

Syntax

[wcDM,wcu] = wcdiskmargin(L,'siso')

[wcMM,wcu] = wcdiskmargin(L,'mimo')

[wcMMIO,wcu] = wcdiskmargin(P,C)

___ = wcdiskmargin(___,sigma)

___ = wcdiskmargin(___,opts)

[___,info] = wcdiskmargin(___)

Description

The worst-case 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 worst-case disk margins and corresponding worst-case gain and phase margins for both loop-at-a-time and multiloop variations. The function also returns the worst-case perturbation, the combination of uncertain elements that yields the weakest margins.

[wcDM,wcu] = wcdiskmargin(L,'siso') estimates the worst-case loop-at-a-time disk-based stability margins for the uncertain negative feedback loop 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 worst-case (smallest) disk margin over the modeled uncertainty in L. Disk-based 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.

[wcMM,wcu] = wcdiskmargin(L,'mimo') estimates the worst-case multiloop disk margins.

[wcMMIO,wcu] = wcdiskmargin(P,C) computes the worst-case stability margins when considering independent, concurrent variations at both the plant inputs and plant outputs the negative feedback loop of the following diagram.

___ = wcdiskmargin(___,sigma) specifies an additional skew parameter that biases the modeled gain and phase variation toward gain increase (positive sigma) or gain decrease (negative sigma). You can use this argument to test the relative sensitivity of stability margins to gain increases versus decreases. You can use this argument with any of the previous syntaxes.

___ = wcdiskmargin(___,opts) specifies additional options for the computation. Use wcOptions to create opts. You can use opts with any of the previous syntaxes.

[___,info] = wcdiskmargin(___) returns a structure with additional information about the worst-case margins and the perturbations that generate them. You can use this output argument with any of the previous syntaxes.

Examples

Worst-Case Disk Margins for Uncertain MIMO Feedback Loop

Use wcdiskmargin to compute worst-case loop-at-a-time and multiloop disk margins. This example illustrates that loop-at-a-time 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 closed-loop system of the following illustration.

P is a two-input, two-output second-order plant and C is a 2x2 static gain. Construct P in state-space form, assuming that it has an uncertain parameter and some dynamic uncertainty. Compute the worst-case disk margins at the plant output (to compute the margins at the plant input, use L = C*Pu).

p = ureal('p',10,'Percentage',10);
a = [-0.2 p;-p -0.2]; 
b = eye(2); 
c = [1 p;-p 1];
d = zeros(2,2);
P = ss(a,b,c,0);
DEL = ultidyn('DEL',[2 2],'Bound',0.1);
Pu = P*(eye(2)+DEL);

C = [1 -2;0 1];
L = Pu*C;

[wcDM,wcu] = wcdiskmargin(L,'siso');

Examine the worst-case loop-at-a-time disk margins, returned in the structure array wcDM. Each entry in this structure array contains the worst-case stability margins of the corresponding channel.

wcDM(1)  

ans = struct with fields:
           GainMargin: [0.5298 1.8875]
          PhaseMargin: [-34.1696 34.1696]
           DiskMargin: 0.6147
           LowerBound: 0.6147
           UpperBound: 0.6160
    CriticalFrequency: 0
    WorstPerturbation: [2x2 ss]

The result in wcDM(1) gives guaranteed stability margins for the specified uncertainty range. As long as the open-loop gain of the first channel changes by a factor between 0.53 and 1.88, the closed loop remains stable for all (p,DEL) values within the specified range. Similarly, the closed loop remains stable as long as the phase variation does not exceed 34° in absolute value.

Similarly, wcDM(2) shows that in the second feedback channel, the gain can vary by any factor between 0.52 and 1.93 or the phase can vary by up to 35°, and the system remains stable for such variations and the (p,DEL) uncertainty.

wcDM(2)

ans = struct with fields:
           GainMargin: [0.5167 1.9352]
          PhaseMargin: [-35.3450 35.3450]
           DiskMargin: 0.6372
           LowerBound: 0.6372
           UpperBound: 0.6386
    CriticalFrequency: -2.2950e-08
    WorstPerturbation: [2x2 ss]

The lower bound returned by wcdiskmargin is a theoretical minimum guaranteed worst-case disk margin. The upper bound corresponds to an actual perturbation in the specified uncertainty range that approaches the lower-bound prediction. The output wcu contains the values of that perturbation for each feedback channel. For example, wcu(2) is the worst combination of (alpha,DEL) for the second channel, and the disk margins for this worst combination are close to wcDM(2). In particular, DM(2).UpperBound and wcDM(1).UpperBound match.

wcL = usubs(L,wcu(2));
DM = diskmargin(wcL);
DM(2)

ans = struct with fields:
           GainMargin: [0.5159 1.9382]
          PhaseMargin: [-35.4184 35.4184]
           DiskMargin: 0.6386
           LowerBound: 0.6386
           UpperBound: 0.6386
            Frequency: 2.2950e-08
    WorstPerturbation: [2x2 ss]

In practice, gain and phase variations affect both channels simultaneously. To estimate the stability margins with respect to such independent and concurrent variations, examine the worst-case multiloop disk margins.

wcMM = wcdiskmargin(L,'mimo')

wcMM = struct with fields:
           GainMargin: [0.8836 1.1317]
          PhaseMargin: [-7.0727 7.0727]
           DiskMargin: 0.1236
           LowerBound: 0.1236
           UpperBound: 0.1239
    CriticalFrequency: 0
    WorstPerturbation: [2x2 ss]

The multiloop margins are much weaker than when considering one loop at a time. This is because it takes a smaller amount of gain (or phase) variation to destabilize the feedback loop when both channels are subject to variations.

You can visualize how uncertainty affects the margins with wcdiskmarginplot. This plots the (disk-based) gain and phase margins as a function of frequency for the nominal and worst-case values of (alpha,DEL) as well as 20 random samples of this uncertainty. The plot shows that uncertainty weakens the margins most near DC.

wcdiskmarginplot(L,{1e-1,1e1})
legend('location','NorthWest')

Finally, compute the multiloop margin for 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 even smaller. Thus, the multiloop I/O margin provides the most comprehensive and most conservative assessment of robust stability in the face of gain or phase variations and (alpha,DEL) uncertainty.

wcMMIO = wcdiskmargin(Pu,C)

wcMMIO = struct with fields:
           GainMargin: [0.9363 1.0680]
          PhaseMargin: [-3.7680 3.7680]
           DiskMargin: 0.0658
           LowerBound: 0.0658
           UpperBound: 0.0659
    CriticalFrequency: 1.0000e-04
    WorstPerturbation: [1x1 struct]

Input Arguments

L — Uncertain open-loop response

uncertain model | model array

Uncertain open-loop 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 worst-case disk-based stability margins for the negative-feedback closed-loop system feedback(L,eye(N)).

To compute the worst-case 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 worst-case 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 state-space model (genss) then wcdiskmargin uses the current value of the tunable control design blocks in L.

If L is a frequency-response data model (such as ufrd), then wcdiskmargin computes the margins at each frequency represented in the model. The function returns the worst-case 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 — Plant

uncertain model

Plant, 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 worst-case disk margins for a negative-feedback closed-loop 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 state-space model (genss) then wcdiskmargin uses the current value of the tunable control design blocks in P.

If P is a frequency-response data model (such as frd), then wcdiskmargin computes the margins at each frequency represented in the model. The function returns the worst-case margins at the frequency with the smallest disk margin.

C — Controller

dynamic system model

Controller, 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 disk-based stability margins for a negative-feedback closed-loop 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 state-space model (genss) then wcdiskmargin uses the current value of the tunable control design blocks in C.

If C is a frequency-response data model (such as frd), then wcdiskmargin computes the margins at each frequency represented in the model. The function returns the worst-case margins at the frequency with the smallest disk margin.

sigma — Skew

0 (default) | real scalar

Skew of uncertainty region used to compute the stability margins, specified as a real scalar value. This parameter biases the uncertainty used to model gain and phase variations toward gain increase or gain decrease.

• The default sigma = 0 uses a balanced model of gain variation in a range [gmin,gmax], with gmin = 1/gmax.

• Positive sigma uses a model with more gain increase than decrease (gmin > 1/gmax).

• Negative sigma uses a model with more gain decrease than increase (gmin < 1/gmax).

Use the default sigma = 0 to get unbiased estimates of gain and phase margins. You can test relative sensitivity to gain increase and decrease by comparing the margins obtained with both positive and negative sigma values. For more detailed information about how the choice of sigma affects the margin computation, see Stability Analysis Using Disk Margins.

opts — Options for margin computation

wcOptions object

Options for worst-case computation, specified as an object you create with wcOptions. The available options include settings that let you:

• Extract frequency-dependent worst-case margins.

• Examine the sensitivity of the worst-case margins to each uncertain element.

• Improve the results of the worst-case 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')

Output Arguments

wcDM — Worst-cast disk margins for each feedback channel

structure | structure array

Worst-case disk margins for each feedback channel with all other loops closed, returned as a structure for SISO feedback loops, or an N-by-1 structure array for a MIMO loop with N feedback channels. The fields of wcDM(i) are:

Field	Value
GainMargin	Minimum guaranteed disk-based gain margin of the corresponding feedback channel, returned as a vector of the form [gmin,gmax]. These values mean that as long as the open-loop 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 open-loop gain can change sign without loss of stability, gmin can be less than zero for large enough negative sigma. If the closed-loop system goes unstable for some combination of uncertainty values, then wcDM(i).GainMargin = [1 1].
PhaseMargin	Minimum guaranteed disk-based phase margin of the corresponding feedback channel, returned as a vector of the form [-pm,pm] in degrees. If the closed-loop 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 closed-loop system is unstable for some combination of uncertain-element values, then wcDM(i).DiskMargin = 0.
LowerBound	Lower bound on worst-case disk margin. This value is the same as DiskMargin.
UpperBound	Upper bound on worst-case disk margin. This value is the disk margin obtained for the worst perturbation found by wcdiskmargin, returned as wcu(i). The actual worst-case 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.
WorstPerturbation	Smallest gain and phase variation that drives the feedback loop unstable for the worst-case combination of uncertain elements. The perturbation is returned as a state-space (ss) model with N inputs and outputs, where N is the number of inputs and outputs in L. The system F(s) = WorstPerturbation is such that for the worst-case combination of uncertain elements of L (the values returned in wcu) the following feedback loop has a pole on the stability boundary at wcDM(i).CriticalFrequency. This state-space model is a diagonal perturbation of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop. For the loop-at-a-time margin of the kth feedback loop, only the kth entry fk(s) of wcDM(k).WorstPerturbation differs from unity. For more information on interpreting wcDM(K).WorstPerturbation, see Disk Margin and Smallest Destabilizing Perturbation This field is different from the WorstPerturbation field of the info output argument. That field contains the values of the uncertain elements of L that yield the smallest margins at each frequency.

When L = P*C is the open-loop 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 1-by-3 array of two-input, two-output models, then wcDM is a 2-by-3 structure array. wcDM(j,k) contains the margins for the j^th feedback channel of the k^th model in the array.

wcu — Perturbation yielding the weakest margins

structure array | structure

Perturbation of uncertain elements yielding the weakest margins, returned as:

• A structure array of dimensions N-by-1 for loop-at-a-time 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 worst-case disk margin. The upper bound corresponds to an actual perturbation in the specified uncertainty range that approaches the lower-bound 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 worst-case disk margin.

wcMM — Worst-case multiloop disk margins

structure

Worst-case 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, wcMM is a single structure regardless of the number of feedback channels in the system. (For SISO systems, wcMM = wcDM.) The fields of wcMM are:

Field	Value
GainMargin	Minimum guaranteed multiloop disk-based 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 closed-loop system goes unstable for some combination of uncertainty values, then wcMM.GainMargin = [1 1].
PhaseMargin	Minimum guaranteed multiloop disk-based phase margin, returned as a vector of the form [-pm,pm] in degrees. If the closed-loop 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 closed-loop system is unstable for some combination of uncertain-element values, then wcMM.DiskMargin = 0.
LowerBound	Lower bound on worst-case disk margin. This value is the same as DiskMargin.
UpperBound	Upper bound on worst-case disk margin. This value is the disk margin obtained for the worst perturbation found by wcdiskmargin, returned as wcu. The actual worst-case 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.
WorstPerturbation	Smallest gain and phase variation that drives the feedback loop unstable for the worst-case combination of uncertain elements. The perturbation is returned as a state-space (ss) model with N inputs and outputs, where N is the number of inputs and outputs in L. The system F(s) = WorstPerturbation is such that for the worst-case combination of uncertain elements of L (the values returned in wcu) the following feedback loop has a pole on the stability boundary at wcMM.CriticalFrequency. This state-space model is a diagonal perturbation of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop. For more information on interpreting wcDM(K).WorstPerturbation, see Disk Margin and Smallest Destabilizing Perturbation This field is different from the WorstPerturbation field of the info output argument. That field contains the values of the uncertain elements of L that yield the smallest margins at each frequency.

When L = P*C is the open-loop response of a system comprising a controller and plant with unit negative feedback in each channel, wcMM 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, wcMM is a structure array with one entry for each model in L.

wcMMIO — Worst-case disk margins for independent variations in all input and output channels

structure

Worst-case disk margins for independent variations in all input and output channels of the plant P, returned as a structure having the same fields as wcMM.

For variations applied simultaneously at inputs and outputs, the WorstPerturbation field is itself a structure with fields Input and Output. Each of these fields contains a state-space model such that for Fi(s) = wcMMIO.WorstPerturbation.Input and Fo(s) = wcMMIO.WorstPerturbation.Output, the system of the following diagram is marginally unstable, with a pole on the stability boundary at the frequency wcMMIO.CriticalFrequency, when P is evaluated with the worst-case uncertainty values wcu.

These state-space models Input and Output are diagonal perturbations of the form F(s) = diag(f1(s),...,fN(s)). Each fj(s) is a real-parameter dynamic system that realizes the worst-case complex gain and phase variation applied to each channel of the feedback loop.

info — Additional information about worst-case values

structure

Additional information about the worst-case values, returned as a structure with the following fields:

Field	Description
Model	Index of the model that has the smallest disk margin, when L is an array of models.
Frequency	Frequency points at which wcdiskmargin returns the minimum guaranteed margins, returned as a vector. If the 'VaryFrequency' option of wcOptions is 'off', then info.Frequency is the critical frequency, the frequency at which the worst-case disk margin occurs. If the largest lower bound and the smallest upper bound on the worst-case disk margin occur at different frequencies, then info.Frequency is a vector containing these two frequencies. If the 'VaryFrequency' option of wcOptions is 'on', then info.Frequency contains the frequencies selected by wcdiskmargin. These frequencies are guaranteed to include the frequency at which the worst-case disk margin occurs. The 'VaryFrequency' option is meaningful only for uss and genss models. wcdiskmargin ignores the option for ufrd and genfrd models.
Bounds	Lower and upper bounds on the actual worst-case disk margin of the model, returned as an array. info.Bounds(:,1) contains the lower bound at each corresponding frequency in info.Frequency, and info.Bounds(:,2) contains the corresponding upper bounds.
WorstPerturbation	Worst perturbations at each frequency point in info.Frequency, returned as a structure array. Here, worst refers to the perturbations that cause the smallest disk margin at a particular frequency. The fields of info.WorstPerturbation are the names of the uncertain elements in the input model. Each field contains the worst value of the corresponding element at each frequency. For example, if L includes an uncertain parameter p and SISO uncertain dynamics delta, then info.WorstPerturbation.p is a collection of numeric values and info.WorstPerturbation.delta is a collection of SISO state-space models. info.WorstPerturbation contains the minimum-margin values of the uncertain elements in the input system L or P. It is distinct from the WorstPerturbation field of the output structures wcDM, wcMM, and wcMMIO. Those field contain state-space models representing the smallest gain and phase variations that drive the feedback loop unstable.
Sensitivity	Sensitivity of the worst-case disk margin to each uncertain element, returned as a structure when the 'Sensitivity' option of wcOptions is 'on'. The fields of info.Sensitivity are the names of the uncertain elements in the input model. Each field contains a percentage that measures how much the uncertainty in the corresponding element affects the worst disk margin. For example, if info.Sensitivity.p is 50, then a given fractional change in the uncertainty range of p causes half as much fractional change in the worst disk margin. If the 'Sensitivity' option of wcOptions is off (the default setting), then info.Sensitivity is NaN.

Tips

• wcdiskmargin assumes negative feedback. To compute the worst-case disk margins of a positive feedback system, use wcdiskmargin(-L) or wcdiskmargin(P,-C).

• You can visualize worst-case disk margins with wcdiskmarginplot.

Algorithms

wcdiskmargin models gain (and phase) variation as umargin uncertainty, combines it with the specified plant uncertainty, and uses mussv to compute the worst-case disk margins and perturbation. This generalizes the diskmargin algorithm to feedback loops with uncertainty. For more information about disk-margin computation and interpretation, see Stability Analysis Using Disk Margins.

Version History

Introduced in R2018b

R2020a: Worst-case disk-based gain margin range can include negative gains

The wcdiskmargin command returns disk-based gain margins in the GainMargin field of its output structures wcDM, wcMM, and wcMMIO. These margins take the form [gmin,gmax], meaning that the open-loop gain can be multiplied by any factor in that range without loss of closed-loop stability. Beginning in R2020a, the lower end of the range gmin can be negative for some negative values of the skew sigma, if the closed-loop system remains stable even if the sign of the open-loop gain changes. The skew controls the bias in the disk-based gain margin toward gain decrease or increase (see Stability Analysis Using Disk Margins). Previously, the gain-margin range was always positive.