Parametric Robust Stability

Robust stability of LTI systemswith parametric uncertainty is a very interesting topic to study, industrial world is contained in parametric uncertainty. In industrial reality, there is not a particular system to analyze, there is a family of systems to be analyzed because the values of physical parameters are not known, we know only the lower and upper bounds of each parameter involved in the process, this is known as Parametric Uncertainty (Ackermann et al., 1993; Barmish, 1994; Bhattacharyya et al., 1995). The set of parameters involved in a system makes a Parametric Vector, the set of all vectors that can exists such that each parameter is kept within its lower and upper bounds is called a Parametric Uncertainty Box. The system we are studying is now composed of an infinite number of systems, each system corresponds to a parameter vector contained in the parametric uncertainty box. So as to test the stability of the LTI system with parametric uncertainty we have to prove that all the infinite number of systems are stable, this is called Parametric Robust Stability. The parametric robust stability problem is considerably more complicated than determine the stability of an LTI system with fixed parameters. The stability of a LTI system can be analyzed in different ways, this chapter will be analyzed by means of its characteristic polynomial, in the case of parametric uncertainty now exists a set with an infinite number of characteristic polynomials, this is known as a Family of Polynomials, and we have to test the stability of the whole family. The parametric robust stability problem in LTI systems with parametric uncertainty is solved in this chapter by means of two tools, the first is a recent stability criterion for LTI systems (Elizondo, 2001B) and the second is the mathematical tool “Sign Decomposition” (Elizondo, 1999). The recent stability criterion maps the prametric robust stability problem to a robust positivity problem of multivariable polynomic functions, sign decomposition solves this problem in necessary and sufficient conditions. By means of the recent stability criterion (Elizondo, 2001B) is possible to analyze the characteristic polynomial and determine the number of unstable roots on the right side in the complex plane. This criterion is similar to the Routh criterion although without using the traditional division of the Routh criterion. This small difference makes a big advantage when it is analized the robust stability in LTI systems with parametric uncertainty, the elements of the first column of the table (Elizondo, 2001B) they are multivariable polynomic functions and these must be positive for stability conditions. Robust positivity of amultivariable polynomial function ismore easier to prove that in the case of quotients of this class of functions, therefore, the recent criterion (Elizondo, 2001B) is easier to use than Routh criterion. There are other 1