A FREQUENCY DOMAIN STABILITY CRITERION

The unit hypercube D is introduced as:

D = {J = [ft… <У I Si Є R and |<5*| < 1} (1.20)

Remember that the real model perturbation Д — — a— (—— I— ( . For the

sake of simplicity, we note with some abuse of notation Д Є D: this should be understood as 8 Є D.

Assume that the nominal closed loop is asymptotically stable, which is

equivalent to the assumption of an asymptotically stable transfer matrix M(s) (i. e. all eigenvalues of the state-matrix A are strictly inside the left half plane). The problem can then be formulated, either as a robustness test ("Is the closed loop of Figure 1.1 stable for all parametric uncer­tainties Si inside the unit hypercube D?"), or as the computation of a robustness measure ("What is the maximal value kmax of к for which the closed loop of Figure 1.1 is stable for all parametric uncertainties Si inside the hypercube kD?").

This last problem reduces to look for the smallest value of k, for which the closed loop becomes marginally stable (i. e. one or more poles on the imaginary axis and all other poles strictly inside the left half plane) for a parametric uncertainty inside kD. Assume indeed that S Є kD and let k increase from the zero value: since the nominal closed loop is asymp­totically stable, the closed loop becomes marginally stable for a value of S inside kD before being unstable (because of the continuity of the roots of the polynomial Pcl(s, S) as a function of the vector S of uncertain parameters).

On the other hand, equation (1.19) emphasizes the link between the singularity of the matrix I — M{juif)A and the presence of a closed loop pole on the imaginary axis at ±jw0.

As a consequence of the above discussion, the s. s.v. is introduced as follows. The complex matrix M in the following definition may be un­derstood as the value of the transfer matrix M(s) at s = juj.

DEFINITION 2..1 The s. s.v. n(M)associated to a complex matrix M and to a real model perturbation A, is defined as:

1

/x(M)

p(M) =0 if no A satisfies det(I – MA) = 0.

The idea is thus to find the minimal size model perturbation Д (or equi­valently S), which renders singular the matrix I — MA: the. v..v. v. p(M) is defined as the inverse of the size of this model perturbation.