Surrogates for the matrix l0-quasinorm in sparse feedback design: Numerical study of the efficiency

Abstract

Some formulations of the optimal control problem require the resulting controller to be sparse; i.e., to contain zero elements in the gain matrix. On one hand, sparse feedback leads to the drop of performance as compared to the optimal control; on the other hand, it confers useful properties to the system. For instance, sparse controllers allow to design distributed systems with decentralized feedback. Some sparse formulations require the gain matrix of the controller to have a special sparse structure which is characterized by the presence of zero rows in the matrix. In this paper, various approximations to the number of nonzero rows of a matrix are considered and applied to sparse feedback design in optimal control problems for linear systems. Along with a popular approach based on using the matrix $\ell_1$-norm, more complex nonconvex surrogates are proposed and discussed, those surrogates being minimized via special numerical procedures. The efficiency of the approximations is compared via numerical experiments.