Communication–computation tradeoff in distributed consensus optimization for MPC-based coordinated control under wireless communications

Abstract

This paper presents an analysis of the tradeoff between repeated communications and computations for a fast distributed computation of global decision variables in a model-predictive-control (MPC)-based coordinated control scheme. We consider a coordinated predictive control problem involving uncertain and constrained subsystem dynamics and employ a formulation that presents it as a distributed optimization problem with sets of local and global decision variables where the global variables are allowed to be optimized over a longer time interval. Considering a modified form of the dual-averaging-based distributed optimization scheme, we explore convergence bounds under ideal and non-ideal wireless communications and determine the optimal choice of communication cycles between computation steps in order to speed up the convergence per unit time of the algorithm. We apply the algorithm for a class of dynamic-policy based stochastic coordinated control problems and illustrate the results with a simulation example.

Introduction

Coordinated control of networked dynamical systems that form the interacting components of a large, complex system has been widely researched for the past several years [1], [2]. Growing ubiquity of wireless communication as a medium of interaction has, in the recent years, fueled interest in coordinated control for a variety of applications including those that involve dynamically decoupled autonomous mobile systems such as multi-vehicle systems where the subsystems need to agree on some common quantity of interest in order to achieve a shared goal (e.g., [3], [4], [5]). Coordinated operation of multi-agent systems, for instance, for consensus or synchronization, has received extensive attention from the research community [2], [6] and control schemes have been explored for agents with simple scalar dynamics to complicated multi-dimensional dynamics (e.g., [1], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Several works such as [16], [17], [4], [18] have analyzed the performance of the control schemes under various inter-subsystem communication conditions such as time-varying communication topology (e.g., [16], [17], [4]), noisy communications (e.g., [18]), delayed or failed communications (e.g., [19], [9], [10], [11]). This paper deals with a model predictive control (MPC)-based approach for a class of coordinated control problems where the subsystems directly compute the common quantities of interest through distributed optimization under wireless inter-subsystem communications.

MPC-based approaches have been extensively explored in the context of coordinated control. Several papers such as [20], [21], [22], [23], [24], [15], [25], [26] address the problem for dynamically decoupled subsystems and offer approaches in which, typically, local MPC problems are solved in each member either sequentially or in parallel together with some coordinating mechanism to ensure stability. Papers [23], [24], [25], [26], [15] deal with consensus- or synchronization-related control problems and achieve the objective without directly computing the consensus or synchronization trajectory. Ideal communications without delays or losses are assumed in the proposed schemes. A direct optimization of the consensus variable is considered in [25], [26] for an MPC-based consensus scheme using a subgradient-based distributed negotiation algorithm. The solutions in these papers are based on the standard finite-horizon MPC optimization for subsystems with time-invariant dynamics and without uncertainties and disturbances, and are obtained without incorporating computational delays and possible effects of non-ideal inter-subsystem communications. In [27], [28], the authors have considered a more general problem involving a time-varying consensus signal, and have incorporated uncertainties in subsystem dynamics and computational delays in distributed optimization by considering an extended dynamic-policy-based robust MPC formulation. The consensus signal is optimized using a dual-decomposition-based distributed algorithm which is required to give converged solutions within a pre-specified interval to ensure a desired performance.

In this paper, we consider the problem of analyzing and speeding up of the convergence of the global variables optimized with a distributed algorithm under wireless inter-member communications in an MPC-based coordinated control scheme. While we employ the dynamic-policy-based local MPC formulation similar to that in [28], for the optimization of the common variables, we consider the primal decomposition of the overall MPC problem, and propose a modified form of the dual-averaging-based distributed optimization algorithm [29] which allows us to analytically explore the convergence characteristics of the global variables under ideal and non-ideal inter-subsystem wireless communications. The modified algorithm essentially employs multi-step network-wide averaging of the dual variables every iteration so that local instances of the global variable converge closer even as the average solution converges to the optimal solution. This convergence is desirable in our MPC-based formulation of the coordinated control problem since a converged global variable ensures stability even if it is sub-optimal. Further, it improves the per-unit-time convergence of the optimization algorithm, particularly if the communication times are smaller. However, since the modified algorithm requires both more time and more communications power every iteration, a tradeoff exists and we explore an optimal design though the solution of an optimization problem.

We apply the distributed optimization algorithm for a stochastic-MPC-based coordinated control problem where we minimize the expected infinite horizon costs so as to ensure that the variance of the deviation of the overall state from the desired state is bounded by a chosen constant. We illustrate the results with numerical examples including one arising in formation flying of satellites on a two-dimensional space.

The rest of the paper is organized as follows. We give a description of the MPC-based coordinated control problem setup in Section 2. In Section 3, we discuss the modified dual-averaging-based distributed optimization technique and present our results on convergence bounds over iterations under ideal and non-ideal inter-subsystem communications. We also explore the design to speed up the per-unit-time convergence of the solution in this section. We consider the application of the algorithm with the dynamic-policy-based MPC implementation in Section 4 and present some simulation results in Section 5. Finally, we end the paper with some concluding remarks in Section 6.

Notations:I $(I_n)$ denotes an identity matrix (of size $n\times n$) and $\mathbf{1}$ denotes a vector of all ones. For a signal x(t), $x(t+i|t)$ represents the value of $x(t+i)$ predicted at time t. For a vector x, $x_{[i]}$ denotes its ith component, $\parallel x{\parallel }_p$ represent its p-norm and $\parallel x{\parallel }_{*}$ represents its dual norm ${\sup }_{\parallel u\parallel =1}{x}^{\mathrm{T}}u$ to a norm $\parallel .\parallel$. For vectors x and y, $(x;y)$ denotes ${[x^{\mathrm{T}}{y}^{\mathrm{T}}]}^{\mathrm{T}}$. For a matrix X, $X_{[i,:]}$ represents its ith row and $X_{[i,j]}$ denotes the element on its ith row and jth column. $\parallel X\parallel$ represents the spectral norm of X. $X\otimes Y$ represents the Kronecker product of X and Y, and diag($X,Y$) denotes a block diagonal matrix with blocks X and Y. For a n×n matrix W, ${\sigma }_1(W)\ge {\sigma }_2(W)\ge \dots \ge {\sigma }_n(W)$ denote its singular values and for a real symmetric n×n matrix W, ${\lambda }_1(W)\ge {\lambda }_2(W)\ge \dots \ge {\lambda }_n(W)$ denote its eigenvalues. ${\mathbb{Z}}_{+}$ and ${\mathbb{R}}_{+}$ represent the sets of non-negative integers and real numbers respectively. ${\mathbb{N}}_n$ represents the set $\{1,2,‥,n\}$. $\oplus$ denotes set addition and $\ominus$ denotes set difference. A set $\mathcal{S}$ is called a $\mathcal{C}$-set if it is compact, convex and contains the origin in its interior. Given a matrix A and a $\mathcal{C}$-set $\mathcal{X}$, $A\mathcal{X}=\{Ax|x\in \mathcal{X}\}$.