Abstract
This article treats an optimal scheduling problem of control nodes in networked systems. We newly introduce both the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^0$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell ^0$</tex-math></inline-formula> constraints on control inputs to extract a time-varying small number of effective control nodes. As the cost function, we adopt the trace of the controllability Gramian to reduce the required control energy. Since the formulated optimization problem is combinatorial, 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. Finally, the proposed method is applied to a rebalancing problem of a mobility network.
Highlights
Nowadays, control system designs that incorporate a notion of sparsity have attracted a lot of attention in the control community
Such an approach finds essential information that gives a significant impact to the system of interest, and it plays an important role in many occasions in large-scale networked systems such as control node selection tackled in this paper
This paper has analyzed an optimal node scheduling that maximizes the trace of the controllability Gramian
Summary
Control system designs that incorporate a notion of sparsity have attracted a lot of attention in the control community. This scheduling problem is analyzed in [14], which provides an explicit formula of the optimal solutions and shows that the solutions are obtained by a greedy algorithm These two works on continuous-time systems mainly consider the L0 control cost, and the resulting number of activated control nodes at each time instance (i.e., the l0 control cost) is not taken into account. This paper newly considers an optimal node scheduling problem under the L0 and l0 constraints By introducing these two constraints, we can find a time-varying small number of control nodes while reducing the support of control inputs.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call