Necessary and sufficient condition in Lyapunov robust control: Multi-input case

We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A ^T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB ^T PB>0 and that the determinant ofB ^T PW be Hurwitz, whereW=block diagw ₁,...,w _m ], with elementw _i =s ^k _i –1,s ^k _i –2,...,s, 1] ^T ; here, the symbolsk _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.