Cooperative optimization-based distributed model predictive control for constrained nonlinear large-scale systems with stability and feasibility guarantees

Highlights

• Cooperative structure for optimization of nonlinear large-scale systems.

• Take into account the interconnection between local controllers.

• Reduce the convergence time and cost values.

• Present a feasible approach.

Abstract

This paper proposes a cooperative distributed model predictive control (DMPC) to control the constrained interconnected nonlinear large-scale systems. The main contribution of this approach is its proposed novel cooperative optimization that improves the global cost function of any subsystem. Each subsystem calculates its optimal control by solving the corresponding global cost function. For each subsystem, the global cost function is defined based on a combination of cost functions of all subsystems. If the sampling time is selected appropriately, then the feasibility of the proposed approach will be guaranteed. Furthermore, the sufficient conditions for stability and consequently, for the convergence of the whole system states towards the neighborhood of the origin’s positive region are provided. The effectiveness and performance of the proposed approach are demonstrated via applying it to a nonlinear quadruple-tank system for both minimum-phase and nonminimum-phase models.

Introduction

Most practical systems like industrial plants, urban traffic, power systems, and supply chain are large-scale systems and have nonlinear dynamics consisting of several nonlinear interconnected subsystems. Distributed model predictive control (DMPC) is one of the practical methods for dealing with nonlinear large-scale systems. DMPC approaches are categorized into cooperative and non-cooperative [1]. In non-cooperative methods, each local controller computes its own optimal control input, whereas in cooperative methods, each local controller minimizes the centralized global cost function and exchanges its information via an interconnection protocol [2].

Although the optimization process may be more time-consuming and more complex in cooperative distributed approaches than in non-cooperative distributed ones, cooperative distributed approaches are still more efficient than non-cooperative ones in large-scale systems for three reasons. First, in cooperative distributed approaches, interconnections between subsystems are considered in the optimization layer, while in non-cooperative ones they are ignored. Second, in cooperative distributed approaches, the effect of control input of each subsystem is taken into account for the whole system or at least for all of its neighbors, while in non-cooperative ones, it is neglected. Third, unlike the non-cooperative distributed approaches, in cooperative ones all control inputs are optimized simultaneously [3], [4]. Dual-mode DMPC is known as a non-cooperative DMPC method according to the proximity of the states to the origin. When the states are in the neighborhood region of the origin, a linear non-cooperative DMPC algorithm is used. On the other hand, when the states are far from the origin, a nonlinear one is applied [5]. A combination of distributed estimation and DMPC architecture is proposed as another non-cooperative DMPC approach for controlling nonlinear processes [6]. The gradient projection optimizer is a nonlinear nonconvex algorithm that modifies the objective function in cooperative DMPC approaches without the need for a coordinator layer. However, there is no guarantee to achieve convergence in a reasonable time [7]. Nevertheless, some other DMPC methods require a coordination layer because local controllers are optimized independently and each local controller dispatches its information to its neighbors via a coordinator [8], [9]. On the contrary, in some other algorithms, the interconnection between neighboring subsystems is considered as an internal constraint instead of being applied to the coordinator layer [10]. The network-based cooperative DMPC approach is a combination method with multi-rate sampling that has been developed for a nonlinear uncertain large-scale system. In this approach, local controllers are interconnected using a coordination layer in a network-based coordination platform that applies an iterative algorithm to control the system [11]. The communication delay is the key concern in methods with a coordination layer. Most of these methods consider the communication delay as an external constraint [12]. Robust DMPC methods are useful approaches for the constrained interconnected large-scale systems. If the subsystems are coupled through their cost functions, these interconnections will be known as internal constraints. A non-cooperative DMPC approach based on a two-layer robust method is a proper algorithm for these constrained large-scale systems [13]. In the first layer, each subsystem receives control information of its neighboring subsystems and optimizes its local objective function that contains coupling terms. The convergence of the states is prepared in the first layer. A robust non-cooperative DMPC algorithm, which tolerates larger disturbances using the shorter prediction horizon, is presented in the second layer [14], [15]. Robust DMPC approaches are useful to control nonlinear large-scale systems whose subsystems are exposed to external disturbances and constraints of control inputs [16], [17]. Sequential DMPC algorithms are other approaches for large-scale systems that can be designed in both cooperative and non-cooperative DMPC architectures. In these approaches, each local controller optimizes its own objective function and dispatches its information through the communication channel to attain the global optimality at each sampling time. The proposed sequential DMPC operates similar to a centralized MPC. However, they are more useful for large-scale systems compared to centralized approaches [18]. Stronger interconnection between subsystems and larger sampling intervals can be taken into account in large-scale systems by using the contraction theory in DMPC approaches. [19]. One of the useful DMPC methods for uncertain systems is the two-level DMPC algorithm that uses the hierarchical framework. The control objective is compensating actuators’ faults, including uncertainties and time delays. At the first level, faults are retrieved to preserve the design specifications for the subsystem. The process of retrievement is applied at the second level by increasing the performance of the whole system. Retrievement design specifications are satisfied by applying the proposed method [20], [21]. In [22], a cooperative DMPC method is developed to control linear systems. The cooperative DMPC approach is taken into account where a centralized global cost function is optimized for each subsystem, which is defined based on the state and input information of all subsystems. Each control input is calculated separately by solving a set of linear matrix inequalities. In the presented cooperative DMPC algorithm only one control input is optimized at a time, which is a limitation for the presented method. This limitation is removed in the cooperative DMPC proposed in this paper.

In this paper, a novel cooperative DMPC (CDMPC) approach is proposed to control constrained interconnected nonlinear large-scale systems. In this approach, a novel cooperative optimization method that is its main contribution and improves the global cost function of each local controller, is proposed. Since in the proposed approach, the effect of each local controller on the cost function of other controllers is considered, the proposed approach is proper for interconnected large-scale systems. Moreover, feasibility, stability, and convergence of the proposed approach are guaranteed. The performance and effectiveness of the proposed approach are investigated for the nonminimum phase model of a large-scale system.

The remainder of paper is organized as follows. Section 2 discusses the formulation of a cooperative DMPC problem for nonlinear interconnected large-scale systems. The proposed cooperative optimization strategy is presented in Section 3. Section 4 deals with the feasibility of the proposed approach. The stability of the proposed approach is discussed in Section 5. Section 6 demonstrates the simulation results for a nonlinear quadruple-tank system. Finally, concluding remarks are presented in Section 7.