Sampling method for semidefinite programs with nonnegative Popov function constraints∗

An important class of optimization problems in control and signal processing involves the constraint that a Popov function is nonnegative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finitedimensional linear matrix inequality via the Kalman-Yakubovich-Popov lemma. However, the linear matrix inequality reformulation requires an auxiliary matrix variable and often results in a very large semidefinite programming problem. Several recently published methods exploit problem structure in these semidefinite programs to alleviate the computational cost associated with the large matrix variable. These algorithms are capable of solving much larger problems than general-purpose semidefinite programming packages. In this paper we address the same problem by presenting an alternative to the linear matrix inequality formulation of the nonnegative Popov function constraint. We sample the constraint to obtain an equivalent set of inequalities of low dimension, thus avoiding the large matrix variable in the linear matrix inequality formulation. Moreover, the resulting semidefinite program has constraints with low-rank structure, which allows the problems to be solved efficiently by existing semidefinite programming packages. The sampling formulation is obtained by first expressing the Popov function inequality as a sum-of-squares condition imposed on a polynomial matrix and then converting the constraint into an equivalent finite set of interpolation constraints. A complexity analysis and numerical examples are provided to demonstrate the performance improvement over existing techniques.