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.
mat is a
uss with M uncertain elements.
The results of
[maxnorm,maxnormunc] = wcnorm(mat)
maxnorm is a structure with the following
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
The value of each field is the corresponding value of the uncertain
element, such that when jointly combined, lead to the norm value in
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:
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
M condition number must be at least equal to
the condition number of the nominal value of
Compute these crude bounds on the worst-case value of the condition
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
be expressed as an optimization, where a free norm-bounded matrix
Δ tries to align the gains of
M is itself uncertain, then the worst-case
condition number involves further maximization over the possible values
M. Therefore, you can compute the worst-case
condition number of an uncertain matrix by using a
element, and then by using
wcnorm to carry out
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
2-by-2 matrices with the maximum singular value less than or equal
You can create the expression involving
M, Delta and
H = M*Delta*Mi;
Finally, consider the stopping criteria and call
One stopping criteria for
wcnorm(H) is based on
the norm of the nominal value of
H. During the
wcnorm determines that the worst-case
norm is at least
then the calculation is terminated. In our case, since
0, the stopping criteria is governed by
The default value of
ABadThreshold is 5. To keep
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
cond(usubs(M,wcu)) ans = 26.9629