Empirical Methods for Determining the Stability of Certain Linear Delay Systems
In this chapter we present two methods for examining the stability of certain delay systems. These methoos make use of well-known computing packages such as MATLAB and are variations on both the Lyapunov and analytic function approaches. The first method is to consider systems where the delays are treated as parameters and the undelayed system is uniformly exponentially stable (u.e.s.). To find the “closest” parameters for which the parametric family loses u.c.s. we give a necessary and sufficient condition for a corresponding family of quadratic matrices to satisfy a Riccati matrix equation. This condition is relatively easy to verify and depends on the zeros of a quasi-polynomial defined on a compact Canesian product. In particular if the delays are commeasureable this procedure either determines the smallest delays for which the system is u.e.s. or shows that it is u.e.s. for all delays.
The second method is more general and is based on a variation of the Poisson Integral representation for H-infinity mappings. This method is more useful in determining instability than stability, but is easily implemented by standard numerical packages since it involves the computation of a parametric family of scalar functions over a fixed compact interval.
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