Discrete-time decentralized linear quadratic control for linear time-varying systems

This article addresses the problem of designing a decentralized control solution for a network of agents modeled by linear time-varying (LTV) dynamics, in a discrete-time framework. A general scheme is proposed, in which the problem is formulated as a classical linear quadratic regulator problem, for the global system, subject to a given sparsity constraint on the gain, which reflects the decentralized nature of the network. A method able to compute a sequence of well-performing stabilizing regulator gains is presented and validated resorting to simulations of two randomly generated LTV systems, one stable and the other unstable. Moreover, a tracking solution is developed, building on the solution to the regulator problem. Both methods rely on a closed-form solution, thus they can be computed very rapidly. Similarly to the centralized solution, both the presented methods require that a window of the future system dynamics is known. Both methods are validated resorting to simulations of: (i) a nonlinear network of four interconnected tanks; and (ii) a large-scale nonlinear network of interconnected tanks. When implemented to a nonlinear network, approximated by an LTV system, the proposed methods are able to compute well-performing gains that track the desired output. Finally, both algorithms are scalable, being adequate for implementation in large-scale networks.