Two real critical constraints for real parameter margin computation
Bong Wie, Jianbo Lǚ · Journal of Guidance Control and Dynamics · 1994
A new approach to computing real parameter margins of stabilized dynamical systems with multilinear!} uncertain parameters is presented. A concept of two real critical constraints is introduced to solve the problem of determining the largest stable hypercube in parameter space that touches the stability boundary on one of its corners. The proposed approach is based on sufficient conditions for checking for critical instability only in the corner directions of the parameter space hypercube. Several examples are used to illustrate the proposed concept and approach. HIS paper is concerned with the problem of computing the structured singular values /i for uncertain dynamical systems.1 In particular, the problem of computing oo-norm real parameter margins (or real /*) is investigated, which is of much current research interest.2'14 The real /z problem is essentially the same as the problem of determining the largest stable hypercube in the uncertain parameter space. We exploit the concept of separating the real and imaginary parts of a characteristic polynomial equation. The resulting two equations are referred to as the two real critical constraints, and the vertex (or corner) property of each constraint equation is utilized. The paper is organized as follows: In Sec. II, we introduce the two real critical constraints, the two-constraint real /*, and the single-constraint real /z. In Sec. Ill, we show that the single-constraint real /* always attains its value at a corner of the parameter space hypercube of multilinearly uncertain systems at a given frequency. Section III contains the main result: a sufficient condition for the critical instability to occur at a corner of the parameter space hypercube of multilinearly uncertain systems. In Sec. IV, we apply the results of Sees. II and III to the problem of computing real parameter margins of several different types of characteristic polynomial. In Sec. V, several examples in the literature are used to illustrate the proposed concept and approach.