Sensitivity-based coordination in distributed model predictive control

Abstract

A new distributed model-predictive control method is introduced, which is based on a novel distributed optimization algorithm, relying on a sensitivity-based coordination mechanism. Coordination and therefore overall optimality is achieved by means of a linear approximation of the objective functions of neighboring controllers within the objective function of each local controller. As for most of the distributed optimization methods, an iterative solution of the distributed optimal control problems is required. An analysis of the method with respect to its convergence properties is provided. For illustration, the sensitivity-driven distributed model-predictive control (S-DMPC) method is applied to a simulated alkylation process. An almost optimal control sequence can be achieved after only one iteration in this case.

Introduction

Chemical processes are usually described by dynamic mathematical models featuring several input and output variables. Typically, industrial plants are operated by decentralized control technology based on single-input single-output PID loops enhanced by supervisory controllers [1]. The decentralized controllers are designed without an explicit consideration of the interactions between the different loops in the plant. Hence, optimization-based methods such as linear or nonlinear model predictive control (MPC) or dynamic real-time optimization (DRTO) have become important technologies, as they maintain operational constraints. We refer to the papers of Kano and Ogawa [1] and Qin and Badgwell [2] for a summary on the industrial state of the art on MPC-technology. Model-predictive controllers consider all interactions due to their centralized implementation. Typically, favorable constraint pushing and good performance can be achieved. However, the design and maintenance of large-scale centralized predictive controllers is involved. Decentralization of the established constraint-handling controllers with negligible performance deterioration is therefore desirable.

Today, an increasing activity in research on model-based decentralized and distributed control methods can be observed. This trend is driven by opportunities to achieve better computational performance [3] and to remove possible communication bottlenecks. Furthermore, reliability and maintainability could be improved compared to a monolithic centralized solution [4]. Last but not least, completely new applications requiring decentralized control have to be addressed. These new applications are positioned mainly in the field of autonomous vehicles [5], [6], [7], such as aircrafts or satellites. Typically these systems are modeled by some linear mechanical systems, where each subsystem is modeled by the same dynamics [8]. An important characteristic of this class of systems, commonly referred to as multi-agent systems (MAS), is the autonomy of the different subsystems (or agents). They are not coupled and hence do not interact. However, these agents share some common goals such as consensus [8], pattern formation (e.g. flocking) or the avoidance of collisions [5], [9]. In order to achieve these common goals, control methods are applied, that lead to a rational interaction between the agents in MAS. These methods use various modern control tools, including linear matrix inequalities [7] or control barrier functions [9]. While the systems considered can usually be described by low order models, one of the main challenges is the time-dependent communication topology. This time-dependence is caused by varying communication links due to changing distances, appearing obstacles or exogeneous disturbances. While in many papers a constant topology is assumed (e.g. [10], [11]), first results exist on time-dependent topologies (e.g. [12], [13]).

Process control problems usually feature several differences compared to MAS. In particular, they are characterized by high order dynamics with strong nonlinearities. In a faultless system, the (communication) topology is fixed. Typically, there is no complete interaction structure linking a subsystem with all others; rather, a subsystem interacts only with a few neighbors.

The subsystems of a process plant, i.e. the process units, exchange material, energy, or signals between each other. They are either controlled by a single centralized (multi-variable) controller or by a set of decentralized control systems. Typically, the design of the controllers of such a decentralized control structure does not account for the interactions between the subsystems explicitly. However, there is a desire to guarantee nominal stability, feasibility, optimality, reliability and maintainability for the control systems implemented in the plant. Distributed model predictive control (DMPC) methods are expected to contribute towards this aim because they can combine the optimality properties of centralized predictive controllers and the modularity and flexibility of decentralized control systems by means of communication, cooperation, or coordination [14].

The basis of distributed control, and in particular, distributed optimal control, has already been established in the early 1970s. Mesarovic et al. [15] presented a seminal monograph on distributed control systems. They formulated some very general principles on how a coordinated control structure can be implemented including the “interaction balance principle” and the “interaction prediction principle” [16]. At the same time, two main decomposition principles have been proposed as a starting point for the development of DMPC, namely primal decomposition [17], [18] and dual decomposition [19]. Primal decomposition is based on resource allocation, where shared resources, which are described by primal variables, are allocated by a coordinator. On the other hand, dual decomposition is driven by price coordination. Each decomposed subproblem decides on the use of its resources. However, the use of these resources is coordinated by the prices, which are the dual variables to be adjusted by the coordinator. If a DMPC scheme includes a coordinator, it is often referred to as a hierarchical control system [20].

Various survey papers exist on distributed control. In 1975, Singh et al. [21] presented an early review on practical distributed control methods for interconnected systems, where all architectures presented feature both, hierarchical and distributed elements. A number of variations of dual decomposition [19] including the three-level method of Tamura [22] are reviewed by these authors. They also present indirect methods suggested by Smith and Sage [23] and by Takahara [24], the “pseudo-model coordination method” [25], as well as some suboptimal methods. In 1977, Mahmoud [26] presented a very comprehensive overview. While the main subject of that article covers multilevel optimization techniques, it also deals with early progress in multilevel systems identification as well as the application to water resource systems. In 1978, Sandell et al. [27] presented another survey on the topics of model simplifications, stability analysis of interconnected systems and decentralized control methods. In a recent survey, Rawlings and Stewart [14] summarize the present status of research in the field of coordinated optimization-based control, as well as opportunities and challenges for future research. The different control architectures to be considered in hierarchical and distributed control have been reviewed recently by Scattolini [20].

The main research focus in hierarchical and distributed model predictive control is geared towards linear systems. Wakasa et al. [28], for example, apply dual decomposition to SISO linear time-invariant systems, the subsystems of which are only coupled by their outputs. The output constraints correspond to the formation of a flock of multiple vehicles. The dual problem is solved using a subgradient optimization algorithm. Necoara and Suykens [29] propose a proximal center-based dual decomposition method, which is based on a smoothing of the Lagrange function to result in a continuous convex objective function. The method is successfully applied to a system of coupled linear oscillators [30]. Venkat et al. [31] propose a DMPC method for linear discrete-time systems based on primal decomposition, which they call “feasible cooperation-based control”. The objective function of the complete system is included in all subsystem controllers to ensure coordination. In each iteration, the cooperative method generates a feasible solution of the optimal control problem. Based on this idea, Stewart et al. [32] have developed a distributed controller, which guarantees stability even in the suboptimal case. Not only the full objective function but also the full system dynamics have to be considered by each of the distributed controllers in this scheme, however. Another DMPC formulation based on resource allocation, i.e. a primal decomposition approach, is proposed by Marcos et al. [33]. Their method can be applied to a system described by linear step-response models. The method is successfully applied to a fluid catalytic cracking process, where fast convergence can be observed.

So far, there exist only few results related to nonlinear DMPC. Talukdar et al. [34] propose a cooperative distributed model predictive control method for nonlinear systems. They assume that the subsystems are only able to communicate and cooperate with neighboring subsystems. The method is applied to the IEEE 118 bus test case in order to study how to prevent cascading failures of that network. Liu et al. [35] propose a Lyapunov-based DMPC method. An additional equation ensuring a decrease of the Lyapunov function is added to the model to ensure closed-loop stability. In this method, only the decision variables are distributed, while the controllers of all the subsystems consider the full state vector and the objective function of the complete system. While in [35] the coordination of two Lyapunov-based model predictive controllers is described, the method is generalized in [36] for an arbitrary number of controllers in a sequential or iterative controller architecture. The authors successfully apply their method to a simulated alkylation process, which will also be picked up as an illustrative example in this paper. Dunbar [37] proposes a distributed MPC method and applies the method to a system of coupled nonlinear oscillators. Magni and Scattolini [38] introduce another stabilizing decentralized MPC scheme for nonlinear discrete-time systems. Since the method does not involve any coordination or cooperation scheme, the resulting control system is suboptimal.

In this paper, we will introduce a new sensitivity-driven distributed model predictive control (S-DMPC) scheme, which is based on a novel distributed dynamic optimization algorithm [39] motivated by so-called “goal-interaction operators” [15]. We assume the objective function of the complete system to be separable. The objective functions of the subsystems are modified using information on the complete system to achieve optimality of the distributed control scheme. The modification of the objective function of a given subsystem incorporates a linearization of the objective functions of the neighboring subsystems. Hence, coordination of the subsystem controllers is based on first order sensitivities. Each of the distributed controllers considers only a part of the full objective function and a reduced set of constraints and decision variables. In this paper, we focus on linear systems, though the idea also carries over to the nonlinear case. We prove, that the distributed optimization method converges under given assumptions to the solution of the complete system. The resulting DMPC method is applied to a simulated chemical process to illustrate its capabilities. In this case study, fast convergence can be observed: Almost optimal control performance can be achieved after only one iteration.

The remainder of this paper is organized as follows: In Section 2, we state the optimal control problem to be solved. Section 3 presents the distributed optimization method and a convergence analysis. Section 4 presents the formulation of closed-loop S-DMPC using the distributed optimization method of Section 3. In Section 5, the linear S-DMPC method is applied to a simulated nonlinear alkylation process. Finally, we conclude the paper with a summary and an outlook in Section 6.