robstab

Robust stability of uncertain system

Syntax

[stabmarg,wcu] = robstab(usys)

[stabmarg,wcu] = robstab(usys,w)

[stabmarg,wcu] = robstab(___,opts)

[stabmarg,wcu,info] = robstab(___)

Description

[stabmarg,wcu] = robstab(usys) calculates the robust stability margin for an uncertain system. This stability margin is relative to the uncertainty level specified in usys. A robust stability margin greater than 1 means that the system is stable for all values of its modeled uncertainty. A robust stability margin less than 1 means that the system becomes unstable for some values of the uncertain elements within their specified ranges. For example, a margin of 0.5 implies the following:

• usys remains stable as long as the uncertain element values stay within 0.5 normalized units of their nominal values.

• There is a destabilizing perturbation of size 0.5 normalized units.

The structure stabmarg contains upper and lower bounds on the actual stability margin and the critical frequency at which the stability margin is smallest. The structure wcu contains the destabilizing values of the uncertain elements.

[stabmarg,wcu] = robstab(usys,w) restricts the robust stability margin computation to the frequencies specified by w.

• If w is a cell array of the form {wmin,wmax}, then robstab restricts the stability margin computation to the interval between wmin and wmax.

• If w is a vector of frequencies, then robstab computes the robust stability margin at the specified frequencies only.

[stabmarg,wcu] = robstab(___,opts) specifies additional options for the computation. Use robOptions to create opts. You can use this syntax with any of the previous input-argument combinations.

[stabmarg,wcu,info] = robstab(___) returns a structure with additional information about the stability margins and destabilizing perturbations. See info for details about this structure. You can use this syntax with any of the previous input-argument combinations.

Examples

Robust Stability Margin of Closed-Loop System

Consider a control system whose plant contains both parametric uncertainty and dynamic uncertainty. Create a model of the plant using uncertain elements.

k = ureal('k',10,'Percent',40);
delta = ultidyn('delta',[1 1]); 
G = tf(18,[1 1.8 k]) * (1 + 0.5*delta);

Create a model of the controller, and build the closed-loop transfer function.

C = pid(2.3,3,0.38,0.001);
CL = feedback(G*C,1);

A step response plot shows that the closed-loop system is nominally stable.

step(CL.NominalValue)

Examine the robust stability of the closed-loop system.

[stabmarg,wcu] = robstab(CL);
stabmarg

stabmarg = struct with fields:
           LowerBound: 1.5960
           UpperBound: 1.5993
    CriticalFrequency: 4.8627

The LowerBound and UpperBound fields of stabmarg show the robust stability margin of the closed-loop system is around 1.6. This result means that the system can withstand about 60% more uncertainty than is specified in the uncertain elements without going unstable.

You can use uscale to scale system uncertainty by the stability margin, to examine the system response for the full range of safe uncertainties. Scale the uncertainties in CL by the robust stability margin to create a system with the maximum tolerable amount of uncertainty.

CLmaxunc = uscale(CL,stabmarg.UpperBound);
CLmaxunc.Uncertainty.delta

Uncertain LTI dynamics "delta" with 1 outputs, 1 inputs, and gain less than 1.6.

CLmaxunc.Uncertainty.k

Uncertain real parameter "k" with nominal value 10 and variability [-64,64]%.

The uncertain elements in CLmaxunc have ranges about 1.6 times the range of the original modeled uncertainty in CL.

The output wcu is a structure that contains the smallest perturbation to k and delta that make the system unstable. Confirm the instability by substituting these values into the closed-loop model and examining the pole locations.

CLunst = usubs(CL,wcu);
pole(CLunst)

ans = 8×1 complex
102 ×

  -9.9314 + 0.0000i
  -0.1027 + 0.1009i
  -0.1027 - 0.1009i
  -0.0000 + 0.0486i
  -0.0000 - 0.0486i
  -0.0115 + 0.0000i
  -0.0216 + 0.0000i
  -0.0403 + 0.0000i

The resulting system has an undamped pair of complex poles with natural frequency 4.89, which renders it unstable. The CriticalFrequency field of stabmarg contains the same value, which is the frequency at which the CL is closest to instability.

Sensitivity to Uncertain Elements

Examine the relative sensitivity of the robust stability margin to the uncertain elements of the system. Consider a model of a control system containing uncertain elements.

k = ureal('k',10,'Percent',40);
delta = ultidyn('delta',[1 1]); 
G = tf(18,[1 1.8 k]) * (1 + 0.25*delta);
C = pid(2.3,3,0.38,0.001);
CL = feedback(G*C,1);

Create an options set for robstab that enables the sensitivity calculation.

opts = robOptions('Sensitivity','On');

Calculate the robust stability margin, specifying the info output to access additional information about the calculation.

[stabmarg,wcu,info] = robstab(CL,opts);

Examine the Sensitivity field of info.

info.Sensitivity

ans = struct with fields:
    delta: 80
        k: 20

The values in this field indicate how much a change in the normalized perturbation on each element affects the stability margin. For example, the sensitivity for k is 21. This value means that a given change dk in the normalized uncertainty range of k causes a change of about 21% percent of that, or 0.21*dk, in the stability margin. The margin in this case is much more sensitive to delta, for which the margin changes by about 81% of the change in the normalized uncertainty range.

Robust Stability Margin as a Function of Frequency

Consider a model of a control system containing uncertain elements.

k = ureal('k',10,'Percent',40);
delta = ultidyn('delta',[1 1]); 
G = tf(18,[1 1.8 k]) * (1 + 0.5*delta);
C = pid(2.3,3,0.38,0.001);
CL = feedback(G*C,1);

By default, robstab computes only the weakest stability margin over all frequencies. To see how the stability margin varies with frequency, use the 'VaryFrequency' option of robOptions. For example, compute the stability margin of the system at frequency points between 0.1 and 10 rad/s.

opts = robOptions('VaryFrequency','on');
[stabmarg,wcu,info] = robstab(CL,{0.1,10},opts);
info

info = struct with fields:
                Model: 1
            Frequency: [19x1 double]
               Bounds: [19x2 double]
    WorstPerturbation: [19x1 struct]
          Sensitivity: [1x1 struct]

robstab returns the vector of frequencies in the info output, in the Frequencies field. info.Bounds contains the upper and lower bounds on the stability margin at each frequency. Use these values to plot the frequency dependence of the stability margin.

semilogx(info.Frequency,info.Bounds)
title('Stability Margin vs. Frequency')
ylabel('Margin')
xlabel('Frequency')
legend('Lower bound','Upper bound')

When you use the 'VaryFrequency' option, robstab chooses frequency points automatically. The frequencies it selects are guaranteed to include the frequency at which the stability margin is weakest (within the specified range). Display the returned frequency values to confirm that they include the critical frequency.

info.Frequency

ans = 19×1

    0.1000
    0.1061
    0.1425
    0.1914
    0.2572
    0.3455
    0.4642
    0.6236
    0.8377
    1.1253
      ⋮

stabmarg.CriticalFrequency

ans = 
4.8269

Alternatively, instead of using 'VaryFrequency', you can specify particular frequencies at which to compute the robust stability margins. info.Bounds contains the margins at all specified frequencies. However, these results are not guaranteed to include the weakest margin, which might fall between specified frequency points.

w = logspace(-1,1,25); 
[stabmarg,wcu,info] = robstab(CL,w);
semilogx(w,info.Bounds)
title('Stability Margin vs. Frequency')
ylabel('Margin')
xlabel('Frequency')
legend('Lower bound','Upper bound')

Input Arguments

usys — Dynamic system with uncertainty

uss | ufrd | genss | genfrd

Dynamic system with uncertainty, specified as a uss, ufrd, genss, or genfrd model that contains uncertain elements. For genss or genfrd models, robstab uses the current value of any tunable blocks and folds them into the known (not uncertain) part of the model.

For frequency-response models, ufrd or genfrd, robstab assumes that the system is nominally stable.

usys can also be an array of uncertain models. In that case, robstab returns the smallest margin across all models in the array, and the info output contains the index of the corresponding model.

w — Frequencies

{wmin,wmax} | vector

Frequencies at which to compute robust stability margins, specified as the cell array {wmin,wmax} or as a vector of frequency values.

• If w is a cell array of the form {wmin,wmax}, then the function computes the margins at frequencies ranging between wmin and wmax.

• If w is a vector of frequencies, then the function computes the margins at each specified frequency. For example, use logspace to generate a row vector with logarithmically spaced frequency values.

  For uss and genss models, when w is a vector, robstab(usys,w) is equivalent to robstab(ufrd(usys,w)). Therefore, usys must be nominally stable.

Specify frequencies in units of rad/TimeUnit, where TimeUnit is the TimeUnit property of the model.

opts — Options for margin computation

robOptions object

Options for computation of robust stability margins, specified as an object you create with robOptions. The available options include settings that let you:

• Extract frequency-dependent stability margins.

• Examine the sensitivity of the margins to each uncertain element.

• Improve the results of the stability-margin calculation by setting certain options for the underlying mussv calculation. In particular, setting the option 'MussvOptions' to 'mN' can reduce the gap between the lower bound and upper bound. N is the number of restarts.

For more information about all available options, see robOptions.

Example: robOptions('Sensitivity','on','MussvOptions','m3')

Output Arguments

stabmarg — Robust stability margin and critical frequency

structure

Robust stability margin and critical frequency, returned as a structure containing the following fields:

Field	Description
LowerBound	Lower bound on the actual robust stability margin of the model, returned as a scalar value. The exact stability margin is guaranteed to be no smaller than LowerBound. In other words, for all modeled uncertainty with normalized magnitude up to LowerBound, the system is guaranteed stable.
UpperBound	Upper bound on the actual robust stability margin of the model, returned as a scalar value. The exact stability margin is guaranteed to be no larger than UpperBound. In other words, there exist some uncertain-element values associated with this magnitude that cause instability. robstab returns one such instance in wcu.
CriticalFrequency	Frequency at which the stability margin is the smallest, in rad/TimeUnit, where TimeUnit is the TimeUnit property of usys.

A robust stability margin greater than 1 means that usys is stable for all values of its modeled uncertainty. A robust stability margin less than 1 implies that usys becomes unstable for some values of its uncertain elements within their specified ranges. For example, a margin of 0.5 implies the following:

• usys remains stable as long as the uncertain element values stay within 0.5 normalized units of their nominal values.

• There is a destabilizing perturbation of size 0.5 normalized units.

Use uscale to scale the amount of uncertainty in usys by the stability margin to examine the actual tolerable ranges of uncertainty.

If the nominal value of usys is unstable, the stability margin is 0. If usys is a ufrd or genfrd model, robstab assumes it is nominally stable.

If usys is an array of uncertain models, stabmarg contains the smallest margin across all models in the array. In that case, the info output contains the index of the corresponding model in its Model field.

wcu — Perturbations causing instability

structure

Smallest perturbations of uncertain elements that cause instability in usys, returned as a structure whose fields are the names of the uncertain elements of usys. Each field contains the actual destabilizing value for each uncertain element of usys. For example, if usys includes an uncertain matrix M and SISO uncertain dynamics delta, then wcu.M is a numeric matrix and wcu.delta is a SISO state-space model.

Use usubs(usys,wcu) to substitute these values for the uncertain elements in usys, to obtain the unstable dynamic system that deviates the least from the nominal system. Use actual2normalized to convert these actual uncertainty values to the normalized units in which the stability margin is expressed.

For ureal parameters in usys whose range is not centered around the nominal value, robstab makes the following adjustments for the purposes of its analysis:

• When the worst perturbation (the smallest destabilizing perturbation) lies outside the range of validity of the actual-to-normalized transformation (see getLimits), then robstab sets the corresponding entry of wcu to the nearest valid value. In other words, if actpert is the worst perturbation in actual units, robgain sets wcu to the nearest value inside the interval ActLims returned by getLimits.

• When there is no destabilizing perturbation, then robstab sets the corresponding entry of wcu to the nominal value of the ureal parameter.

info — Additional information about stability margins

structure

Additional information about the robust stability margins, returned as a structure with the following fields:

Field	Description
Model	Index of the model that has the weakest stability margin, when usys is an array of models.
Frequency	Frequency points at which robstab returns the robust stability margin, returned as a vector. If the 'VaryFrequency' option of robOptions is 'off', then info.Frequency is the critical frequency, the frequency at which the smallest margin occurs. If the smallest lower bound and the smallest upper bound on the stability margin occur at different frequencies, then info.Frequency is a vector containing these two frequencies. If the 'VaryFrequency' option of robOptions is 'on', then info.Frequency contains the frequencies selected by robstab. These frequencies are guaranteed to include the frequency at which the stability margin is smallest. If you specify a vector of frequencies w at which to compute the stability margins, then info.Frequency = w. When you specify a frequency vector, these frequencies are not guaranteed to include the frequency at which the stability margin is smallest. The 'VaryFrequency' option is meaningful only for uss and genss models. robstab ignores the option for ufrd and genfrd models.
Bounds	Lower and upper bounds on the actual robust stability 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	Smallest destabilizing perturbations at each frequency point in info.Frequency, returned as a structure array. The fields of info.WorstPerturbation are the names of the uncertain elements in usys, and each field contains the destabilizing value of the corresponding element at each frequency. For example, if usys 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.
Sensitivity	Sensitivity of the stability margin to each uncertain element, returned as a structure when the 'Sensitivity' option of robOptions is 'on'. The fields of info.Sensitivity are the names of the uncertain elements in usys. Each field contains a percentage that measures how much the uncertainty in the corresponding element affects the stability 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 stability margin. If the 'Sensitivity' option of robOptions is off (the default setting), then info.Sensitivity is NaN.

Algorithms

Computing the robustness margin at a particular frequency is equivalent to computing the structured singular value, μ, for some appropriate block structure (μ-analysis).

For uss and genss models, robstab(usys) and robstab(usys,{wmin,wmax}) use an algorithm that finds the smallest margin across frequency. This algorithm does not rely on frequency gridding and is not adversely affected by discontinuities of the μ structured singular value. See Getting Reliable Estimates of Robustness Margins for more information.

For ufrd and genfrd models, robstab computes the μ lower and upper bounds at each frequency point. This computation offers no guarantee between frequency points and can be inaccurate if there are discontinuities or sharp peaks in μ. The syntax robstab(uss,w), where w is a vector of frequency points, is the same as robstab(ufrd(uss,w)) and also relies on frequency gridding to compute the margin.

In general, the algorithm for state-space models is faster and safer than the frequency-gridding approach. In some cases, however, the state-space algorithm requires many μ calculations. In those cases, specifying a frequency grid as a vector w can be faster, provided that the robustness margin varies smoothly with frequency. Such smooth variation is typical for systems with dynamic uncertainty.

Version History

Introduced in R2016b