# ROBUSTNESS ANALYSIS IN THE PRESENCE OF TIME DELAYS

The problem of analyzing the robustness properties of a closed loop in the presence of classical model uncertainties (parametric uncertainties and neglected dynamics) and uncertain time delays presents a great practical interest. Indeed, when embedding control laws on a real-time computer, time delays are to be accounted for at the plant inputs (because of the time needed for computing the value of the plant input signal as a function of the plant output signal) and outputs (because of the sensors which measure the plant output signal).

The aim of this chapter is to provide an algorithm for computing a robust delay margin: see (Ferreres and Scorletti, 1998) for a complete theoretical justification. The method is presented in a qualitative way in the first section. The second section details the computational algorithm. An alternative small gain approach is presented in the third section. Both methods are compared on the missile problem in the fourth section: time delays are added at the input and outputs and uncertainties in the stability derivatives are considered.

1. INTRODUCTION TO THE PROBLEM

A frequency dependent test is introduced, whose resolution at each frequency essentially reduces to a ц problem with a specific structure (namely a one-sided skewed /і problem). If the exact value of p could be computed, the exact value of the robust delay margin could be obtained.

1.1 A ONE-SIDED SKEWED ц PROBLEM

As an extension of the classical s. s. v. v, the one-sided skewed s. s.v. v^(M) of a complex matrix M is defined (Ferreres et al., 1996b). In the stand-

ard interconnection structure M — Д, the uncertainty Д is split as Д = diag{Aі, Дг)- Ді is a mixed perturbation, while Д2 contains real (possibly repeated) scalars £[:

Д2 = diag(8ri Iki) (11.1)

Let the one-sided unit ball BA™:

BA°2s = {Д2 = diag{8llki) / 8 Є [0,1], і = 1,… ,mr} (11.2) DEFINITION 1..1

v™(M) = l/min(k / ЭД = diag(A, /гД2) with Д2 Є BA™,

A Є BA and det{I — МД) = 0)

= 0 if no (/с, Ді, Д2) exists

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