# 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