Maximum turn-off control for discrete-time linear systems

We aim to address the problem of finding a control input maximizing the time instants when all control channels take a zero-value (are turned off) while stabilizing the system to zero over a given horizon length. This problem is called the maximum turn-off control problem. To solve it, we reduce the problem into a block-sparse optimization problem with respect to the control input sequence, where the l2/l0$$ {\ell}_2/{\ell}_0 $$ norm of the control input sequence is the objective function that must be minimized. Because the problem is not convex, we introduce a relaxed problem based on the l2/l1$$ {\ell}_2/{\ell}_1 $$ norm, which is a convex function, and characterize the equivalence relation between the original and relaxed problems using the so-called block restricted isometry property (block-RIP). Based on the equivalence, the solution can be obtained by solving the convex relaxed problem. However, the block-RIP is not easy to interpret and verify. Thus, we propose the notion of sparse controllability Gramians, which is an extension of the controllability Gramians, and show that the block-RIP can be interpreted by the eigenvalues of the sparse controllability Gramian. This study presents an easy-to-check condition of the block-RIP. Moreover, the above control framework is extended to a model predictive control scheme. These results are demonstrated using numerical examples.