Parametric stabilization is easy
This paper deals with parameter depending linear control systems of the form \(\dot x=A(\lambda)x+B(\lambda)u\) where the entries of the matrices \(A(\lambda)\), \(B(\lambda)\) are real polynomial functions of the parameter vector \(\lambda\in R^ r\). If the system is controllable for each \(\lambda\in R^ r\) and \(\gamma\) is an arbitrary real, it is shown that there exists a state feedback matrix \(K(\lambda)\) with entries polynomial in \(\lambda\) such that the spectrum of the closed loop matrix \(A(\lambda)+B(\lambda)K(\lambda)\) lies within the half plane [Re \(s\leq\gamma]\) for all \(\lambda\). This main result is illustrated by the computation of the feedback matrix \(K(\lambda)\) for an example with two state and two input variables, using the symbolic manipulation system REDUCE. It is shown that it is impossible to place the poles of the system arbitrarily by feedback matrices \(K(\lambda)\) which are continuous in \(\lambda\).
parameter depending linear control systems
scientific article; zbMATH DE number 3875100