Distributed controller design and performance optimization for discrete‐time linear systems

Abstract

SummaryThis article addresses the problem of distributed controller design for linear discrete‐time systems. The problem is posed using the classical framework of state feedback gain optimization over an infinite‐horizon quadratic cost, with an additional sparsity constraint on the gain matrix to model the distributed nature of the controller. An equivalent formulation is derived that consists in the optimization of the steady‐state solution of a matrix difference equation, and two algorithms for distributed gain computation are proposed based on it. The first method consists in a step‐by‐step optimization of said difference matrix equation, and allows for fast computation of stabilizing state feedback gains. The second algorithm optimizes the same matrix equation over a finite time window to approximate asymptotic behavior and thus minimize the infinite‐horizon quadratic cost. To assess the performance of the proposed solutions, simulation results are presented for the problem of distributed control of a quadruple‐tank process, as well as a version of that problem scaled up to 40 interconnected tanks.

Efficient Algorithm for the Computation of the Solution to a Sparse Matrix Equation in Distributed Control Theory

“…The computational efficiency of the synthesis with the proposed solution is compared to the state-of-the-art synthesis procedure. This network is widely studied [3,19,20] because it is analogous to a broad range of industrial processes. Consider N interconnected tanks, as shown in Figure 1, where N is an even integer.…”

“…The goal is to compare the computational efficiency of the gain synthesis using the solution to (11) proposed in this short communication and compare it to the state-of-the-art solution (3). In that sense, the gain synthesis is performed for both alternatives, for a range of N. Figure 2 depicts the elapsed wall-clock time for the distributed filter synthesis, resulting from the average of three simulations of a MATLAB implementation on a single thread of a server with 24GB RAM and 24 Intel Xeon hexa-core CPUs at 2.40GHz.…”

“…Equation ( 1) arises frequently in the field of distributed control and estimation theory. In fact, it is the backbone of the state-of-the-art methods for distributed filter design [2] and controller synthesis [3] for large-scale dynamical systems. However, the solution to (1), presented in these papers, relies on an inefficient closed-form matrix computation.…”

Optimal power‐constrained control of distributed systems with information constraints

“…The design of optimal distributed control laws is difficult in general because the "information structure" (i.e., "who knows what and when") decides on the convexity of the problem [3][4][5]. Intuitively, this difficulty arises because of additional sparsity constraints on the feedback control that model the availability of information to local decision makers [6]. In Witsenhausen, [4] this is demonstrated by providing an example of information structure which results in optimal nonlinear policy even for a linear system and quadratic cost function.…”

“…Remark 3. Note that constraints (6) are defined in expectation, that is, we require satisfaction of those constraints on average. This (later proven), together with assumptions on one-step delay between neigboring decision makers, implies the optimality of linear control policies for the information-constrained problem addressed here.…”