Robust Multi-Objective Feedback Design by Quantifier Elimination

In Anderson et al. (1975) the application of Tarski–Seidenberg decision theory (Tarski, 1951, Seidenberg, 1954) for the solution of the static output feedback stabilization problem for specific problems was first proposed. The general static output feedback stabilization problem is one of the most important open problems in feedback design. The problem can be stated mathematically as follows: find a matrix K such that all of the eigenvalues of the matrix A + BKC have negative real parts, given the matrices A,B and C. This problem has no general analytical solution. By use of the Liénard–Chipart criterion (Gantmacher, 1959), the problem can be reduced to a system of polynomial inequalities in the coefficients of the matrix K. The computational complexity and lack of software severely limited the interest in the results presented by Anderson et al. (1975). However since then some improved algorithms have been developed (Collins, 1975; Collins and Hong, 1991), and implemented (Quantifier Elimination by Partial Cylindrical Algebraic Decomposition (QEPCAD), Hong, 1992). In light of new developments in quantifier elimination theory, we explore here the application of the theory to a class of feedback design problems that is of great practical interest, that is robust multi-objective design. For a discussion of robust and multi-objective feedback design see (Dorato et al., 1992). In this study we focus on design objectives specified in the frequency domain. For the frequency domain multi-objective problems considered here there are no general analytical solutions.