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# mucomp

mu (structured singular value) calculation

### Syntax

[BOUND, D, G] = mucomp(Z, K, T)

### Arguments

Z

the complex n-by-n matrix for which the structured singular value is to be computed

K

the vector of length m containing the block dimensions of the structured uncertainty Δ. The uncertainty Δ is supposed to be a block diagonal matrix.

T

the vector of length m indicating the type of each uncertainty block. T(I) = 1 if the corresponding block is real T(I) = 2 if the corresponding block is complex.

BOUND

the upper bound on the structured singular value.

D, G

vectors of length n containing the diagonal entries of the diagonal matrices D and G, respectively, such that the matrix Z'*diag(D)^2*Z + sqrt(-1)*(diag(G)*Z-Z'*diag(G)) - bound^2*diag(D)^2 is negative semidefinite.

### Description

This function computes an upper bound on the structured singular value for a given square complex matrix and given block structure of the uncertainty.

The structured singular value μ(Z) is defined as the inverse of the norm of the smallest uncertainty Δ that makes det(I- ΔZ)=0. Here Δ is supposed to be a block diagonal matrix.

### Examples

K=[1,1,2,1,1];
T=[1,1,2,2,2];
Z=[-1+%i*6, 2-%i*3, 3+%i*8, 3+%i*8,-5-%i*9,-6+%i*2;
4+%i*2,-2+%i*5,-6-%i*7,-4+%i*11,8-%i*7, 12-%i;
5-%i*4,-4-%i*8, 1-%i*3,-6+%i*14,2-%i*5, 4+%i*16;
-1+%i*6, 2-%i*3, 3+%i*8, 3+%i*8,-5-%i*9,-6+%i*2;
4+%i*2,-2+%i*5,-6-%i*7,-4+%i*11,8-%i*7, 12-%i;
5-%i*4,-4-%i*8, 1-%i*3,-6+%i*14,2-%i*5, 4+%i*16];

[BOUND, D, G] = mucomp(Z, K, T)
spec(Z'*(diag(D)^2)*Z + %i*(diag(G)*Z-Z'*diag(G)) - BOUND^2*diag(D)^2)

### Used functions

This function is based on the Slicot routine AB13MD.

### References

Fan, M.K.H., Tits, A.L., and Doyle, J.C. Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics. IEEE Trans. Automatic Control, vol. AC-36, 1991, pp. 25-38. Slicot routine AB13MD.