Worst-case norm of uncertain matrix
maxnorm = wcnorm(m) [maxnorm,wcu] = wcnorm(m) [maxnorm,wcu] = wcnorm(m,opts) [maxnorm,wcu,info] = wcnorm(m) [maxnorm,wcu,info] = wcnorm(m,opts)
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.
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.
Lower bound on worst-case norm, positive scalar.
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:
A third output argument provides information about sensitivities of the worst-case norm to the uncertain elements ranges.
[maxnorm,maxnormunc,info] = wcgain(mat)
The third output argument info is a structure with the following fields:
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 wcgainOptions object is 'off', the values are NaN.
1-by-1 scalar matrix with the value of 1. In more complicated situations (described later) the value of this field depends on the input data.
You can construct an uncertain matrix and compute the worst-case norm of the matrix, as well as its inverse. Your objective is to 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',2,'Range',[1 3]); 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 = LowerBound: 14.7199 UpperBound: 14.7327 [maxnormMi] = wcnorm(Mi) maxnormMi = LowerBound: 2.5963 UpperBound: 2.5979
The condition number of M must be less than the product of the two upper bounds for all values of the uncertain elements making up M. Conversely, the largest value of M condition number must be at least equal to the condition number of the nominal value of M. Compute these crude bounds on the worst-case value of the condition number.
condUpperBound = maxnormM.UpperBound*maxnormMi.UpperBound; condLowerBound = cond(M.NominalValue); [condLowerBound condUpperBound] ans = 5.0757 38.2743
How can you get a more accurate estimate? Recall that the condition number of an nxm matrix M can be expressed as an optimization, where a free norm-bounded matrix Δ tries to align the gains of M and M–1
If M is itself 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 then by using wcnorm to carry out the maximization.
Create a 2-by-2 ucomplexm object, with nominal value equal to zero.
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.
You can create the expression involving M, Delta and inv(M).
H = M*Delta*Mi;
Finally, consider the stopping criteria and call wcnorm. One stopping criteria for wcnorm(H) is based on the norm of the nominal value of H. During the computation, if wcnorm determines that the worst-case norm is at least
then the calculation is terminated. In our case, since H.NominalValue equals 0, the stopping criteria is governed by ABadThreshold. The default value of ABadThreshold is 5. To keep wcnorm from prematurely stopping, set ABadThreshold to 38 (based on our crude upper bound above).
opt = wcgopt('ABadThreshold',38); [maxKappa,wcu,info] = wcnorm(H,opt); maxKappa maxKappa = LowerBound: 26.9629 UpperBound: 27.9926
You can verify that wcu makes the condition number as large as maxKappa.LowerBound.
cond(usubs(M,wcu)) ans = 26.9629