Uncertainty with Phase Information and the Stability of Uncertain Delay Systems

It is well-known that many robust stability and performance problems in control lead to computing the structured singular values of some closed-loop system transfer matrices. The standard definition of structured singular value assumes that the uncertainties are only norm bounded. However, in many applications, the uncertainties involved are constrained, for example, the uncertain parameters are nonnegative or have restricted phase angles. The structured singular value with phase information has been considered in a recent paper [2] where the block structured uncertainties are considered and a computable upper bound is derived. In this paper, we generalize the results in [2] to include repeated scalar blocks. We then apply these generalized results to the stability problem of uncertain delay systems. The stability of uncertain delay systems has received much attention recently and many sufficient conditions have been derived. Unfortunately, most of the existing conditions in the literature are extremely conservative and many conditions are actually much more conservative than the small gain condition used in one of the authors early work. A notable exception is the result in [1] where a complete solution is given for delay-independent stability using structured singular value characterization. Complete solution has not been reported for testing the stability of uncertain delay systems that are not stable independent of delay. In this paper, a sufficient stability condition is derived using the structured singular value with phase information which is obviously less conservative than the standard structured singular value test. Numerical examples show that this condition can be extremely effective. In the following, we shall briefly summarize our main results. Let n = k1 + k2 + · · ·+ km and 1 ≤ ` ≤ m. Define