Necessary and sufficient condition in Lyapunov robust control-multi-input case

The authors consider a linear time-invariant finite-dimensional system dx/dt=Ax+Bu with multi-input u in which the matrices A and B are in canonical controller form. It is assumed that the system is controllable and B has rank m. The authors 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. They show that it is necessary and sufficient that B ^TPB>0 and that the determinant of B^TPW be Hurwitz, where W is a block diagonal matrix. This result has application in designing robust controllers for linear uncertain systems