Numerical study on effectiveness of surrogates for the matrix $l_0$-quasinorm applied to sparse feedback design

Optimal control problem formulations sometimes require the resulting controller to be sparse, i.e. to contain zero elements in gain matrix. On the one hand, sparse feedback leads to the performance drop if compared with 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 gain matrix of the controller to have special sparse structure, which is characterized by the occurence of zero rows in a matrix. In this paper various approximations to the number of nonzero rows of a matrix are considered to be applied to sparse feedback design in optimal control problems for linear systems. Along with a popular approach based on using the matrix $l_1$-norm, more complex nonconvex surrogates are involved, those surrogates being minimized via special numerical procedures. Effectiveness of the approximations is compared via numerical experiment.