Necessary and sufficient condition in Lyapunov robust control: Multi- input case
We consider a linear time-invariant finite-dimensional system \(\dot x=Ax+Bu\) with multi-input u, in which the matrices A and B are in canonical controller form. We assume that the system is controllable and B has rank m. We study the Lyapunov equation \(PA+A^ TP+Q=0\), with \(Q>0\), and investigate the properties that P must satisfy in order that the canonical controller matrix A be Hurwitz. We show that, for the matrix A being Hurwitz, it is necessary and sufficient that \(B^ TPB>0\) and that the determinant of \(B^ TPW\) be Hurwitz, where \(W=block\) \(diag[w_ 1,...,w_ m]\), with element \(w_ i=[s^{k_ i-1},s^{k_ i-2},...,s,1]^ T\); here, the symbols \(k_ i\), \(i=1,2,...,m\), denote the Kronecker invariants with respect to the pair \(\{\) A,B\(\}\). This result has application in designing robust controllers for linear uncertain systems.
scientific article; zbMATH DE number 4116429
scientific article