Some Robust Stability Tests

Abstract

We have already discussed the importance of stability and more generally D-stability to control system analysis and design. We saw that for linear time invariant systems, these concepts can be expressed in terms of polynomial root locations. In Chapter 3 we recalled some basic stability tests and introduced the Finite Nyquist Theorem. From our discussion in Chapter 4, there are ample reasons for investigating not only root locations of a single polynomial,but rat her root locations of families of polynomials with parameter uncertainty. One can immediately recognize that the Routh-Hurwitz criterion and the Nyquist Theorem can be used for this purpose. For the Routh-Hurwitz criterion one constructs the Routh table where the first column will now contain entries that in general are rational functions of the uncertain parameters. One still needs to guarantee that for all uncertainty val ues the first column contains positive entries. In the case of the Nyquist Theorem the image of the Nyquist path for each polynomial in the family must have the appropriate number of encircle ments of the origin. These are only two of many robust stability tests that can be employed. In fact, over the last decade there has been a tremendous amount of interest in developing efficient analytical results for polynomial family stabi lity and D-stability. The book by Barmish [6] provides an excellent reference for this work.