PARAMETRIC UNCERTAINTIES

We come back to the problem of subsection 1.4, namely the stability of the interconnection structure M(s) — A when A = diag(5ilqi) only contains real parametric uncertainties 8,. The problem reduces to the computation of the s. s.v. g(M(s)) along the imaginary axis, i. e. for s = jw with u> Є [0, оо].

2.1.1 A PRELIMINARY RESULT

For the sake of simplicity, M(s) is assumed to be a strictly proper transfer matrix. Let (A, B, C, 0) a state-space model ofM(s). Noting that Д may be considered as a feedback constant matrix, the state-space matrix of the closed loop M(s) — A is A + ВАС, and its characteristic polynomial is:

where 8 is the vector of parametric uncertainties associated to A — diag(8ilqi). With the classical properties det(XY) = det(YX) and det(I – XY) = det(I – YX), it is straightforward to rewrite the above equation as:

Pcl{s, 8) = det(sl – A)det(I – C(sl – A^BA) (1.18)

Since M(s) = C(sl – A)~lB and Pol{s) = det(sl – A) represents the open loop characteristic polynomial (i. e. the one associated to A — 0 and thus to the nominal closed loop M(s)), the following result is obtained:

The above equation is the specialization to the case of the interconnection structure M( s) — Д of a well known result, which is the basis of the multivariable Nyquist theorem.