Augmented Lagrangian Approach to Design of Structured Optimal State Feedback Gains

Abstract

Abstract-We consider the design of optimal state feedback gains subject to structural constraints on the distributed controllers. These constraints are in the form of sparsity requirements for the feedback matrix, implying that each controller has access to information from only a limited number of subsystems. The minimizer of this constrained optimal control problem is sought using the augmented Lagrangian method. Notably, this approach does not require a stabilizing structured gain to initialize the optimization algorithm. Motivated by the structure of the necessary conditions for optimality of the augmented Lagrangian, we develop an alternating descent method to determine the structured optimal gain. We also utilize the sensitivity interpretation of the Lagrange multiplier to identify favorable communication architectures for structured optimal design. Examples are provided to illustrate the effectiveness of the developed method.

ℋ<inf>2</inf>-based optimal sparse sliding mode control for networked control systems

“…Since the weighted 1 norm is not implementable (the required weights should be calculated based on the unknown feedback gain), a reweighted algorithm is proposed in [3], and further used by [14] to design sparse feedback gains. This algorithm solves weighted optimization problems iteratively in which the weights are updated inversely proportional to the strength of individual (block) entries of feedback gain in the previous iteration.…”

“…Recently, the issue of designing a control network with minimum communication links has been studied in the literature [21,23,24,22,14,15,16]. As an illustration, [14] proposes a non-convex condition which is solved numerically by exploiting a convex reweighted 1 norm approximation.…”

“…As an illustration, [14] proposes a non-convex condition which is solved numerically by exploiting a convex reweighted 1 norm approximation. Furthermore, [21] considers the problem of finding the sparsest control/observer network that satisfies the obtained stability condition.…”

“…The reference [14] uses the alternating direction of multiplier method (ADMM) to address the sparsity-promoting optimal control problem. Specifically, ADMM is utilized to identify the control structure which strikes a balance between H 2 performance of the closed-loop system and the sparsity of the controller.…”

“…[21,23,24,22,14,15,16]), in that we originally aimed at using SMC strategy and as a result the problem formulation is very different (please see the system matrices in (24)). Note also that the sparse SF obtained using the method in [14] may not necessarily be a feasible solution to the SMC problem [5]. Moreover, as seen the system matrices of the fictitious system in (25) include the topology matrix.…”