COMPUTATION OF THE ROBUSTNESS

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COMPUTATION OF THE ROBUSTNESS

In the context of a robust stability problem, l/fi(M(jw)) represents the size of the smallest parametric uncertainty S, which brings one closed loop pole on the imaginary axis at ±ju. The robust stability margin kmax is obtained by computing the s. s.v. along the imaginary axis:

The principle is thus to detect the crossing of one of the closed loop poles through the imaginary axis. ктах corresponds to the size of the smallest parametric uncertainty 8, which brings one closed loop pole on the imaginary axis.

Remark: several reasons exist for handling the, v..v. к n(M(ju)) rather than its inverse the multiloop stability margin (m. s.m.).

As a first point, the s. s.v. can not take an infinite value, since

the nominal closed loop is asymptotically stable, whereas the m. s.m. may be infinite (if no structured model perturbation exists, which destabil­izes the closed loop). On the other hand, the s. s.v. can be considered as an extension of classical algebraic notions, namely the spectral radius and the maximal singular value of a matrix (i. e. its spectral norm – see below).

2.2 THE GENERAL CASE

The problem of extending the approach of subsection 2.1 to the case of neglected dynamics seems a priori more complex, since Д is now a dynamic transfer matrix instead of a simple gain matrix. Nevertheless, assume that a complex matrix Д0 was found, which satisfies det(I – M(ju>)A°) = 0 at frequency u>. It suffices then to find a transfer matrix A(s) with A(ju}) = Д0. When applying A(s) to the interconnection structure, a closed loop pole is obtained on the imaginary axis at ±juj.

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