wcsens

(Not recommended) Calculate worst-case sensitivity and complementary sensitivity functions of plant-controller feedback loop

wcsens is not recommended. Use wcgain instead. For more information, see Version History.

Syntax

wcst = wcsens(L)
wcst = wcsens(L,type)
wcst = wcsens(L,opt)
wcst = wcsens(L,type,scaling)
wcst = wcsens(L,type,scaling,opt)
wcst = wcsens(P,C)
wcst = wcsens(P,C,type)
wcst = wcsens(P,C,opt)
wcst = wcsens(P,C,type,scaling)
wcst = wcsens(P,C,type,scaling,opt)

Description

The sensitivity function, S = (I + L)–1, and the complementary sensitivity function, T = L(I + L)–1, where L is the loop gain matrix associated with the input, CP, or output, PC, are two transfer functions related to the robustness and performance of the closed-loop system. The multivariable closed-loop interconnection structure, shown below, defines the input/output sensitivity, complementary sensitivity and loop transfer functions.

Control structure feedback(P,C) for computing sensitivity functions. The structure includes disturbance inputs d1 (plant input) and d2 (controller input), and measurement outputs e1 (plant input), e2 (controller output), e3 (controller input), and e4 (plant output).

The following table gives the values of the input and output sensitivity functions for this control structure.

Description

Equation

Input sensitivity Si (closed-loop transfer function from d1 to e1)

Si = (I + CP)–1

Input complementary sensitivity Ti (closed-loop transfer function from d1 to e2)

Ti = CP(I + CP)–1

Output sensitivity So (closed-loop transfer function from d2 to e3)

So = (I + PC)–1

Output complementary sensitivity To (closed-loop transfer function from d2 to e4)

To = PC(I + PC)–1

Input loop transfer function Li

Li = CP

Output loop transfer function Lo

Lo = PC

wcst = wcsens(L) calculates the worst-case sensitivity and complementary sensitivity functions for the loop transfer matrix L in feedback in negative feedback with an identity matrix. If L is a uss object, the frequency range and number of points are chosen automatically.

wcst = wcsens(P,C) calculates the worst-case sensitivity and complementary sensitivity functions for the feedback loop C in negative feedback with P. C should only be the compensator in the feedback path, not any reference channels, if it is a 2-dof architecture (see loopsens). If P and C are ss/tf/zpk or uss objects, the frequency range and number of points are chosen automatically. wcst is a structure with the following substructures:

Fields of wcst

Field

Description

Si

Worst-case input-to-plant sensitivity function

Ti

Worst-case input-to-plant complementary sensitivity function

So

Worst-case output-to-plant sensitivity function

To

Worst-case output-to-plant complementary sensitivity function

PSi

Worst-case plant times input-to-plant sensitivity function

CSo

Worst-case compensator times output-to-plant sensitivity function

Stable

1 if nominal closed loop is stable, 0 otherwise. NaN for frd/ufrd objects.

Each sensitivity substructure is a structures with five fields MaximumGain, BadUncertainValues, System, BadSystem, Sensitivity derived from the outputs of wcgain.

Fields of Si, So, Ti, To, PSi, CSo

Field

Description

MaximumGain

struct with fields LowerBound, UpperBound and CriticalFrequency. LowerBound and UpperBound are bounds on the unweighted maximum gain of the uncertain sensitivity function. CriticalFrequency is the frequency at which the maximum gain occurs.

BadUncertainValues

Struct, containing values of uncertain elements which maximize the sensitivity gain. There are M fluidness, which are the names of uncertain elements of sensitivity function. The value of each field is the corresponding value of the uncertain element, such that when jointly combined, lead to the gain value in MaximumGain.LowerBound.

System

Uncertain sensitivity function (ufrd or uss).

BadSystem

Worst-case system based on the uncertain object values in BadUncertainValues. BadSystem is defined as BadSystem=usubs(System, BadUncertainValues).

Sensitivity

Struct with M fields, fieldnames are names of uncertain elements of system. Values of fields are positive numbers, each entry indicating the local sensitivity of the maximum gain to all of the individual uncertain elements uncertainty ranges. For instance, a value of 50 indicates that if the uncertainty range is enlarged by 8%, then the maximum gain should increase by about 4%. If the 'Sensitivity' property of the wcgainOptions object is 'off', the values are NaN.

wcst = wcsens(L,type) and wcst = wcsens(P,C,type) allow selection of individual Sensitivity and Complementary Sensitivity functions, type, as 'Si','Ti','So','To','PSi','CSo' corresponding to the sensitivity and complementary sensitivity functions. Setting type to 'S' or 'T' selects all sensitivity functions ('Si','So','PSi','CSo') or all complementary sensitivity functions ('Ti','To'). Similarly, setting type to 'Input' or 'Output' selects all input Sensitivity functions ('Si','Ti','PSi') or all output sensitivity functions ('So,'To','CSo'). 'All' selects all six Sensitivity functions for analysis (default). type may also be a cell array containing multiple function types, such as {'Si','To'}.

wcst = wcsens(L,type,scaling) and wcst = wcsens(P,C,type,scaling) add a scaling to the worst-case sensitivity analysis. scaling is one of the following:

  • 'Absolute' (default) — Calculates bounds on the maximum gain of the uncertain sensitivity function.

  • 'Relative' — Finds bounds on the maximum relative gain of the uncertain sensitivity function. That is, the maximum relative gain is the largest ratio of the worst-case gain and the nominal gain evaluated at each frequency point in the analysis.

  • LTI model (ss, tf, zpk, or frd) — Calculates bounds on the maximum scaled gain of the uncertain sensitivity function. The input/output dimensions of the LTI model must be either 1-by-1, or compatible with P and C.

You can also combine type and scaling in a cell array, e.g.

wcst = wcsens(P,C,{'Ti','So'},'Abs','Si','Rel','PSi',wt)

wcst = wcsens(P,C,opt) or wcst = wcsens(P,C,type,scaling,opt) specifies options for the worst-case gain calculation as defined by opt. (See wcgainOptions for more details on the options for wcsens.)

The sensitivity of the worst-case sensitivity calculations to the individual uncertain components can be determined using the options object opt. To compute the sensitivities to the individual uncertain components, create a wcgainOptions options object, and set the Sensitivity property to 'on'.

opt = wcgainOptions('Sensitivity','on'); 
wcst = wcsens(P,C,opt)

Examples

The following constructs a feedback loop with a first order plant and a proportional-integral controller. The time constant is uncertain and the model also includes an multiplicative uncertainty. The nominal (input) sensitivity function has a peak of 1.09 at omega = 1.55 rad/sec. Since the plant and controller are single-input / single-output, the input/output sensitivity functions are the same.

  delta = ultidyn('delta',[1 1]); 
  tau = ureal('tau',5,'range',[4 6]); 
  P = tf(1,[tau 1])*(1+0.25*delta); 
  C=tf([4 4],[1 0]); 
  looptransfer = loopsens(P,C); 
  Snom = looptransfer.Si.NominalValue; 
  norm(Snom,inf) 
  ans = 
    1.0864

wcsens is then used to compute the worst-case sensitivity function as the uncertainty ranges over its possible values. More information about the fields in wcst.Si can be found in the wcgain help. The badsystem field of wcst.Si contains the worst case sensitivity function. This worst case sensitivity has a peak of 1.52 at omega = 1.02 rad/sec. The maxgainunc field of wcst.Si contains the perturbation that corresponds to this worst case sensitivity function.

wcst = wcsens(P,C)      
wcst = 
        Si: [1x1 struct] 
        Ti: [1x1 struct] 
        So: [1x1 struct] 
        To: [1x1 struct] 
       PSi: [1x1 struct] 
       CSo: [1x1 struct] 
    Stable: 1 
Swc = wcst.Si.BadSystem; 
omega = logspace(-1,1,50); 
bodemag(Snom,'-',Swc,'-.',omega); 
legend('Nominal Sensitivity','Worst-Case Sensitivity',... 
  'Location','SouthEast') 
norm(Swc,inf) 
ans = 
    1.5075

For multi-input/multi-output systems the various input/output sensitivity functions will, in general, be different.

References

J. Shin, G.J. Balas, and A.K. Packard, “Worst case analysis of the X-38 crew return vehicle flight control system,” AIAA Journal of Guidance, Dynamics and Control, vol. 24, no. 2, March-April 2001, pp. 261-269.

Version History

Introduced before R2006a

collapse all

See Also

| | | |