wcnorm

Worst-case norm of uncertain matrix

Syntax

maxnorm = wcnorm(m)
[maxnorm,wcu] = wcnorm(m)
[maxnorm,wcu] = wcnorm(m,opts)
[maxnorm,wcu,info] = wcnorm(m)
[maxnorm,wcu,info] = wcnorm(m,opts)

Description

The norm of an uncertain matrix generally depends on the values of its uncertain elements. Determining the maximum norm over all allowable values of the uncertain elements is referred to as a worst-case norm analysis. The maximum norm is called the worst-case norm.

As with other uncertain-system analysis tools, only bounds on the worst-case norm are computed. The exact value of the worst-case norm is guaranteed to lie between these upper and lower bounds.

Basic syntax

Suppose mat is a umat or a uss with M uncertain elements. The results of

[maxnorm,maxnormunc] = wcnorm(mat) 

maxnorm is a structure with the following fields.

Field	Description
LowerBound	Lower bound on worst-case norm, positive scalar.
UpperBound	Upper bound on worst-case norm, positive scalar.

maxnormunc is a structure that includes values of uncertain elements and maximizes the matrix norm. There are M field names, which are the names of uncertain elements of mat. The value of each field is the corresponding value of the uncertain element, such that when jointly combined, lead to the norm value in maxnorm.LowerBound. The following command shows the norm:

 norm(usubs(mat,maxnormunc)) 

Basic syntax with third output argument

A third output argument provides information about sensitivities of the worst-case norm to the uncertain elements ranges.

[maxnorm,maxnormunc,info] = wcnorm(mat) 

The third output argument info is a structure with the following fields:

Field	Description
Model	Index of model with largest gain (when mat is an array of uncertain matrices)
WorstPerturbation	Structure of worst-case uncertainty values. The fields of info.WorstPerturbation are the names of the uncertain elements in mat, and each field contains the worst-case value of the corresponding element.
Sensitivity	A struct with M fields. Fieldnames are names of uncertain elements of sys. Field values are positive numbers, each entry indicating the local sensitivity of the worst-case norm in maxnorm.LowerBound to all of the individual uncertain elements’ uncertainty ranges. For instance, a value of 25 indicates that if the uncertainty range is increased by 8%, then the worst-case norm should increase by about 2%. If the Sensitivity property of the wcOptions object is 'off', the values are NaN.
BadUncertainValues	Same as WorstPerturbation. Included for compatibility with R2016a and earlier.
ArrayIndex	Same as Model. Included for compatibility with R2016a and earlier.

Examples

Worst-Case Norm and Condition Number of an Uncertain Matrix

Construct an uncertain matrix and compute the worst-case norm of the matrix and of its inverse. These computations let you accurately estimate the worst-case, or the largest, value of the condition number of the matrix.

a = ureal('a',5,'Range',[4 6]); 
b = ureal('b',3,'Range',[2 10]); 
c = ureal('c',9,'Range',[8 11]); 
d = ureal('d',1,'Range',[0 2]); 

M = [a b;c d];
Mi = inv(M);

maxnormM = wcnorm(M)

maxnormM = struct with fields:
    LowerBound: 14.7199
    UpperBound: 14.7227

maxnormMi = wcnorm(Mi)

maxnormMi = struct with fields:
    LowerBound: 2.5963
    UpperBound: 2.5968

The condition number of M must be less than the product of the two upper bounds for all values of the uncertain elements of M. Conversely, the condition number of the largest value of M must be at least equal to the condition number of the nominal value of M. Compute these bounds on the worst-case value of the condition number.

condUpperBound = maxnormM.UpperBound*maxnormMi.UpperBound; 
condLowerBound = cond(M.NominalValue); 
[condLowerBound condUpperBound]

ans = 1×2

    5.0757   38.2312

The range between these lower and upper bounds is fairly large. You can get a more accurate estimate. Recall that the condition number of an n-by-m matrix M can be expressed as an optimization, where a free norm-bounded matrix $$\Delta$$ tries to align the gains of M and inv(M):

$$\begin{array}{c}\kappa \left(M\right)=\underset{\Delta \in C^{m\times m}}{\max }\left({\sigma }_{\max }\left(M\Delta M^{-1}\right)\right)\\ {\sigma }_{\max }\left(\Delta \right)\le 1\end{array}$$

If M is uncertain, then the worst-case condition number involves further maximization over the possible values of M. Therefore, you can compute the worst-case condition number of an uncertain matrix by using a ucomplexm uncertain element and using wcnorm to carry out the maximization.

Create a 2-by-2 ucomplexm element with nominal value 0.

Delta = ucomplexm('Delta',zeros(2,2));

The range of values represented by Delta includes 2-by-2 matrices with the maximum singular value less than or equal to 1.

Create the expression involving M, Delta, and inv(M).

H = M*Delta*Mi;

opt = wcOptions('MussvOptions','m5');
[maxKappa,wcu,info] = wcnorm(H,opt);
maxKappa

maxKappa = struct with fields:
    LowerBound: 26.8406
    UpperBound: 38.2349

Verify that the values in wcu make the condition number as large as maxKappa.LowerBound.

cond(usubs(M,wcu))

ans = 26.9629

Algorithms

See wcgain.