wcsens (Robust Control Toolbox™)

Robust Control Toolbox™

wcsens

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Calculate worst-case sensitivity and complementary sensitivity functions of plant-controller feedback loop

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.

Description
Equation

Input sensitivity ()
(I+CP)-1

Input complementary sensitivity ()
CP(I+CP)-1

Output sensitivity ()
(I+PC)-1

Output complementary sensitivity ()
PC(I+PC)-1

Input loop transfer function
CP

Output loop transfer function
PC

Description	Equation
Input sensitivity ()	(I+CP)-1
Input complementary sensitivity ()	CP(I+CP)-1
Output sensitivity ()	(I+PC)-1
Output complementary sensitivity ()	PC(I+PC)-1
Input loop transfer function	CP
Output loop transfer function	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:

Table 5-2: 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.

**Table 5-2: 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.

Table 5-3: 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 wcgopt object is 'off', the values are NaN.

**Table 5-3: 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 `wcgopt` object is `'off'`, the values are `NaN`.

wcst = wcsens(L,type) and wcst = wcsens(P,C,type) allows 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 containing a collection of strings, i.e. 'Si','To', as well as a comma separated list.

wcst = wcsens(L,type,scaling) and wcst = wcsens(P,C,type,scaling) adds a scaling to the worst-case sensitivity analysis. scaling is either the character strings 'Absolute' (default), 'Relative' or a ss/tf/zpk/frd object. The default scaling 'Absolute' calculates bounds on the maximum gain of the uncertain sensitivity function. The 'Relative' scaling 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, Similarly if scaling is a ss/tf/zpk/frd object, bounds on the maximum scaled gain of the uncertain sensitivity function are found. If scaling is 'Relative'or a ss/tf/zpk/frd object, the worst-case analysis peaks over frequency. If scaling is an object, its input/output dimensions should be 1-by-1 or dimensions compatible with P and C. type and scaling can also be combined 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 wcgopt 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 wcgopt options object, and set the Sensitivity property to 'on'.

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

Example

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.

Reference

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.

See Also
loopsens Calculate sensitivity functions of feedback loops

loopmargin Comprehensive analysis of feedback loops

robuststab Calculate stability margins of uncertain systems

usubs Substitutes values for uncertain atoms

wcgain Calculate worst-case gain of a system

wcgopt Create a worst-case options object

wcmargin Calculate worst-case margins for feedback loop

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wcnorm Block Reference

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