Maximum turn‐off control for discrete‐time linear systems. (18th July 2022)

• Record Type: Journal Article
• Authors: Iwata, Takumi; Azuma, Shun‐ichi; Ariizumi, Ryo; Asai, Toru
• Other Names: Bako Laurent guestEditor.; Mo Yilin guestEditor.; Nagahara Masaaki guestEditor.
• Is Part Of: International journal of robust and nonlinear control. Volume 33:Number 1(2023)
• Page Start: 23
• Page End: 34
• Publication Date: 2022-07-18
• Subjects: block‐sparse optimization -- block‐sparsity -- convex optimization -- discrete‐time control -- linear control -- model predictive control -- sparse control; Automatic control -- Periodicals; Control theory -- Periodicals; Nonlinear systems -- Periodicals; 629.836
• DOI: 10.1002/rnc.6283
• Languages: English
• ISSNs: 1049-8923

Abstract

Abstract: 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 ℓ 2 / ℓ 0 $$ {\ell}_2/{\ell}_0 $$ 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 ℓ 2 / ℓ 1 $$ {\ell}_2/{\ell}_1 $$ 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.