Abstract
This article investigates a novel sparsity-constrained controllability maximization problem for continuous-time linear systems. For controllability metrics, we employ the minimum eigenvalue and the determinant of the controllability Gramian. Unlike the previous problem setting based on the trace of the Gramian, these metrics are not the linear functions of decision variables and are difficult to deal with. To circumvent this issue, we adopt a parallelepiped approximation of the metrics based on their geometric properties. Since these modified optimization problems are highly nonconvex, we introduce a convex relaxation problem for its computational tractability. After a reformulation of the problem into an optimal control problem to which Pontryagin’s maximum principle is applicable, we give a sufficient condition under which the relaxed problem gives a solution of the main problem.
Highlights
N OWADAYS, control system designs that incorporate a notion of sparsity have attracted a lot of attention in the control community
It should be emphasized that optimization problems involving both of the l0 norm and the L0 norm have not been investigated in the area of sparse optimization, except for our recent study [6], to the best of our knowledge
This paper has analyzed two node scheduling problems that are related with the minimum eigenvalue of the controllability Gramian and the volume of the reachable set
Summary
N OWADAYS, control system designs that incorporate a notion of sparsity have attracted a lot of attention in the control community. We considered the L0 constraint on control inputs and formulated a node scheduling problem for continuous-time systems in [20], which is based on a controllability metric of the trace of the Gramian. This scheduling problem is analyzed in [21], which provides an explicit formula of the optimal solutions and shows that the solutions are obtained by a greedy algorithm.
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