SPECIFIC STRUCTURES OF MODEL PERTURBATIONS

In the general case of a mixed model perturbation Д, no analytical expression of the s. s.v. /x(M) is available. Nevertheless, in the special

case of a single full complex block (resp. a single repeated real scalar), the s. s.v. fi{M) coincides with the maximal singular value a{M) (resp. the real spectral radius pr(M)).

When Д is a full complex block, p{M) = o(M). The small gain theorem provides indeed a necessary and sufficient condition of stability in the context of an unstructured model perturbation. In an alternative way, the following result can be used: if A is a complex matrix, the size of the smallest unstructured complex matrix Д, which renders the matrix A + Д singular, is a{A). In the context of the initial problem, matrix M is assumed to be invertible for the sake of simplicity. The singularity of I — MA is then equivalent to the singularity of M~l – Д. As a consequence, the size of the smallest unstructured complex matrix Д, which renders the matrix I — MA singular, is:

-(M ^ = W{M) = /ЁМ) (L34)

Consider now the case of a single real repeated scalar A = 6Ir. Then p(M) = pr(M), where the real spectral radius pr(M) is the magnitude of the largest real eigenvalue of M:

pR{M) = max{\i{M) / А,(М) € R)

PR(M) is zero if M has no real eigenvalue.

To prove that p{M) = pr{M), it suffices to note that the singularity of the matrix 1-М A implies the existence of a non zero vector x satisfying:

{I – 5M)x = 0 (1.36)

which can be rewritten as:

1/6 and x are thus an eigenvalue and eigenvector of M. 6 is moreover a real scalar, whose size is to be minimized, so that д ( A ( is the magnitude of the largest real eigenvalue of M.

Remark: in the same way, it can be proved that the s. s.v. /ід(М) coincides with the spectral radius p(M) (defined as the magnitude of the largest eigenvalue of M) when the model perturbation Д is a repeated complex scalar.