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5 Stability Analysis of Polynomials with Polynomic Uncertainty

When dealing with systems with parameter uncertainty most attention is paid to robustness analysis of linear time-invariant systems. In literature the most often investigated topic of analysis of linear time-invariant systems with parametric uncertainty is the problem of stability analysis of polynomials whose coefficients depend on uncertain parameters. The aim is to verify that all roots of such a polynomial are located in some prescribed set in complex plane or to find a bound within that uncertain parameters can vary from nominal ones preserving stability. The former problem is studied in this contribution. The formulations of basic robustness problems and their first solutions for special cases are very old. For example, in the work (Neimark, 1949) some effective techniques for small number of parameters are presented. A powerful result concerning the stability analysis of polynomials with multilinear dependency of its coefficients is given in the book (Zadeh & Desoer, 1963). Also in Siljak’s book (Siljak, 1969) special classes of robust stability analysis problems with parametric uncertainty are studied. Nevertheless, the starting point of an intensive interest in this area was the celebrated Kharitonov theorem (Kharitonov, 1978) dealing with interval polynomials. This elegant theorem with surprisingly simple result is considered as the biggest achievement in control theory in last century. When analysing stability of a polynomial with some dependency of its coefficients on interval parameters the solution becomes more complicated. The Edge theorem (Bartlett et al., 1988) claims that for linear (affine) dependency it is sufficient to check polynomials on exposed edges, the Mapping theorem (Zadeh & Desoer, 1963) provides a simplified sufficient stability condition for systems with multilinear parameter dependency. To date there are only few results solving the problem of robust stability of polynomials with polynomic structure of coefficients (polynomic interval polynomials) that occur very often e.g. as characteristic polynomials in feedback control of uncertain plant with a fixed controller. None of the results is as elegant as those mentioned earlier. There are two basic approaches – algebraic and geometric. The first one is based on utilization of criteria commonly used for stability analysis of fixed polynomials – Hurwitz or Routh criterion – and their generalization for uncertain polynomials. The second one transforms the multidimensional problem in twodimensional test of frequency plot of the polynomial in