Design of Optimal Sparse Feedback Gains via the Alternating Direction Method of Multipliers

Abstract

Abstract-We design sparse and block sparse feedback gains that minimize the variance amplification (i.e., the norm) of distributed systems. Our approach consists of two steps. First, we identify sparsity patterns of feedback gains by incorporating sparsity-promoting penalty functions into the optimal control problem, where the added terms penalize the number of communication links in the distributed controller. Second, we optimize feedback gains subject to structural constraints determined by the identified sparsity patterns. In the first step, the sparsity structure of feedback gains is identified using the alternating direction method of multipliers, which is a powerful algorithm well-suited to large optimization problems. This method alternates between promoting the sparsity of the controller and optimizing the closed-loop performance, which allows us to exploit the structure of the corresponding objective functions. In particular, we take advantage of the separability of the sparsity-promoting penalty functions to decompose the minimization problem into sub-problems that can be solved analytically. Several examples are provided to illustrate the effectiveness of the developed approach.

Fixed-order decentralized/distributed control of islanded inverter-interfaced microgrids

“…This question has been recently addressed by some researchers in [28,29,30,31,32]. They have focused on the problem of sparse static output/state feedback controller design where the gain between the subsystems' inputs and outputs/states is sparsified [28,29,30,31].…”

“…They have focused on the problem of sparse static output/state feedback controller design where the gain between the subsystems' inputs and outputs/states is sparsified [28,29,30,31]. In this way, the amount of information exchange between subsystems and the controller is reduced.…”

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

“…Firstly, the problem formulation in this manuscript is substantially different from the ones used for sparsely distributed state feedback (SF) and static output feedback in the literature (e.g. [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].…”

“…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.…”

Continuous Hands-off Control by CLOT Norm Minimization

“…Sparsity-promoting optimization has been applied to networked control in (Nagahara et al, 2014), where quantization errors and data rate can be reduced at the same time by sparse representation of control packets. Other examples of control applications include optimal controller placement by (Casas et al, 2012;Clason and Kunisch, 2012;Fardad et al, 2011), design of feedback gains by (Lin et al, 2013;Polyak et al, 2013), state estimation by (Charles et al, 2011), to name a few.…”