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Improving sparsity in time and space via self-triggered sparse optimal controllers

We consider the optimal control of linear timeinvariant (LTI) systems via self-triggered sparse optimal control (SSOC) laws. Our control objective is to design an optimal control law which stabilizes the LTI system for all initial conditions, requires less sensing, minimizes communication requirements between the subsystems, minimizes the number of active actuators, and provides guaranteed closed-loop performance bounds. To achieve such control objectives, we formulate a sequence of ¿o-regularized linear-quadratic optimal control problems, wherein the objective is to optimize a cost function which involves three penalizing terms: one for maximizing the inter-execution time, another one for minimizing the number of nonzero elements of the state feedback gain, and the last one for minimizing the number of active actuators. Deriving a lower bound on inter-execution times, we propose a scheme to solve this problem. First, the relaxation is utilized to cast the problem as a semi-definite program (SDP) to compute a feedback gain while the inter-execution time is kept fixed. Second, a nonlinear equation is solved to determine the inter-execution time while the feedback gain is kept fixed. The proposed SSOC law is feasible and results in a stabilizing sequence of sparse optimal controllers. Additionally, the performance of the resulting closed-loop system does not exceed a prespecified performance bound. By numerical simulations, sparsity in time/space is improved compared to periodic time-triggered LQR design. Moreover, a tradeoff between prespecified performance bound and sparsity in time/space is observed. Finally, the paper is concluded by drawing some future directions.