Quadratic Model Predictive Control Including Input Cardinality Constraints

Abstract

This note addresses the problem of feedback control with a constrained number of active inputs. This problem is known as sparse control. Specifically, we describe a novel quadratic model predictive control strategy that guarantees sparsity by bounding directly the $\ell _0$ -norm of the control input vector at each control horizon instant. Besides this sparsity constraint, bounded constraints are also imposed on both control input and system state. Under this scenario, we provide sufficient conditions for guaranteeing practical stability of the closed-loop. We transform the combinatorial optimization problem into an equivalent optimization problem that does not consider relaxation in the cardinality constraints. The equivalent optimization problem can be solved utilizing standard nonlinear programming toolboxes that provides the input control sequence corresponding to the global optimum.