Computing Parametric LQRs For Polytopic Discrete-Time Systems

This paper addresses the problem of determining parametric linear quadratic regulators (LQRs) for polytopic discrete-time systems. Specifically, it is supposed that the matrices of the system are linear functions of a vector of parameters constrained over the simplex. It is shown that a candidate for the sought parametric LQR can be obtained by solving a semidefinite program (SDP) built through homogeneous polynomially-dependent quadratic Lyapunov functions (HPD-QLFs) of chosen degree. In particular, it is shown that the found candidate is guaranteed to approximate arbitrarily well the true parametric LQR by using a degree sufficiently large.