Event-triggered distributed self-learning robust tracking control for uncertain nonlinear interconnected systems

Highlights

• The proposed method extends application scope of ADP theory to the filed of distributed tracking control of interconnected nonlinear systems with uncertain interconnection terms.

• A novel discounted cost function is defined for each auxiliary subsystem which can accommodate the control objectives of guaranteeing the tracking performance, optimizing sample intervals and compensating the unmatched interconnection terms simultaneously.

• The proposed triggering condition for each subsystem is designed by using the estimated worst-case control policy error of auxiliary subsystem, which can maximize the sample intervals to further reduce the computational burden and save the communication resources.

• The event-triggered mechanism designed in this paper operates in an asynchronous and distributed manner which only needs local state information.

Abstract

In this paper, a novel event-triggered self-learning robust tracking control scheme for a class of nonlinear interconnected systems with uncertain interconnections is developed based on adaptive dynamic programming (ADP). Initially, the robust tracking control problem of interconnected systems is transformed to the robust stabilization one of augmented interconnected systems. To address the uncertain interconnections as well as optimize the sample intervals, a group of auxiliary subsystems, each with a novel discounted cost function, are introduced, which enables the robust stabilization problem to be further converted to a group of two-player zero-sum differential games. Next, an event-triggered ADP algorithm is proposed to obtain the saddle point solutions, based on which the distributed event-triggered robust tracking control policies for the overall system are established. Specifically, the proposed event-triggered mechanism is asynchronous and distributed where the sample intervals are optimized to further reduce the computational burden and save the communication resources. Moreover, the tracking errors and the approximation errors of critic network weights are demonstrated to be uniformly ultimately bounded by using the Lyapunov approach. Finally, two simulation examples are provided to verify the effectiveness of the proposed method.

Introduction

During the few decades, the research of nonlinear interconnected systems has attracted considerable attention of many scholars due to its wide use in practical industrial applications, such as large-scale power systems, robot manipulators, and transportation systems [1], [2], [3]. In contrast to the centralized control method which generally requires global system information and sufficient processing capacity of centralized controller, the superiority of distributed control approach lies in that it only depend on the locally available information such that the controllers can be deployed independently in each subsystem. As an important brunch of distributed control, the distributed tracking control of interconnected systems has been investigated extensively, and many valuable results have been reported such as [4], [5], [6].

In the design of distributed tracking controllers, one main challenge is how to deal with interconnection terms, which may lead to control performance degradation or even system instability, particularly in the case of uncertain interconnections arising from modeling errors and exogenous disturbances. Commonly used methods for dealing with unknown interconnections include neural networks (NNs) adaptive method [5], sliding model method [7] and back-stepping method [8]. In addition to these mentioned methods, optimal control approaches can also be employed to address the uncertainty. As described in [9], the robust control problem of uncertain systems can be translated into the optimal control problem of nominal systems by introducing a modified cost function. As we known, adaptive dynamic programming (ADP) is a powerful self-learning method for solving optimization problem of nonlinear systems, which was proposed by Werbos in [10] and has been extensively investigated to design optimal controllers for various nonlinear systems [11], [12], [13], [14], [15]. Owing to its unique advantage of obtaining the approximate optimal solution, ADP method can be combined with the idea of robust-optimization transformation to facilitate the distributed control strategy design for the stabilization problem of interconnected systems [16], [17]. Recently, the ADP-based decentralized tracking controllers were synthesized in[18], [19], [20], [21]. Nevertheless, the aforementioned ADP algorithms were implemented under the traditional time-triggered framework without considering communication resources limitation, which may restrict their practical implementation to a certain degree.

To overcome the deficiency of time-triggered control schemes, many event-triggered control methods were proposed reduce the unnecessary communication waste and alleviate the computational burden[22], [23], [24], [25], [26], [27], [28], [29]. Under the event-triggered framework, the controller is updated only when the triggering condition is violated, which can decrease the frequency of communication between controlled plants and actuators. Increasing attention has been made to integrate the event-triggered control mechanism with ADP method to solve optimal stabilization problems[30], optimal tracking control problems [31], [32], [33], robust control problems [34], differential games[35], consensus problem [36], and so forth. Although great efforts have been made on the event-triggered control of various nonlinear systems, there are few works about interconnected nonlinear systems. More recently, the event-triggered distributed control schemes were developed to solve the stabilization problem of interconnected systems in [37], [38], [39]. However, these mentioned results can not be straightly extended to solve the tracking problem of uncertain interconnected systems. In addition, the triggering conditions were designed from the view of ensuring the stability of closed-loop systems without considering the sample intervals optimization. In [40], an improved triggering condition with optimized sample interval was presented in the event-triggered stabilization controller design of large-scale nonlinear systems. Nevertheless, the result obtained in [40] is inapplicable when the controlled plant contains uncertain interconnections. To the best knowledge of authors, there are still no results on event-triggered distributed robust tracking control design along with sample intervals optimization for interconnected system with uncertain interconnection terms, which motivates our research to some extend.

In this paper, a novel event-triggered distributed self-learning robust tracking control scheme is developed for interconnected nonlinear systems with uncertain interconnection terms based on ADP method. First, the robust tracking control problem of interconnected systems is transformed to the robust stabilization one of augmented interconnected system composed of tracking errors and the desired states through system augmentation technique. To deal with the uncertain interconnection terms and optimize the sample intervals simultaneously, the robust stabilization problem is further converted to a group of two-player zero-sum differential games by introducing a group of auxiliary subsystems and a novel discounted cost function. Then, an event-triggered ADP algorithm is developed to learn the solutions to the associated Hamilton-Jacobi-Issacs(HJI) equations where only one critic NN is utilized to approximate the optimal cost function for each auxiliary subsystem. Using the obtained approximate optimal cost function, the sampled state based control laws of auxiliary subsystems are computed, which are then utilized to design the distributed triggering conditions and establish the distributed event-triggered robust tracking control policies for the overall system. Furthermore, it is demonstrated that the developed event-triggered distributed robust tracking controller can guarantee all the signals of closed-loop system to be uniformly ultimately bounded by performing the Lyapunov analysis. Finally, two simulation examples are given to validate the propose control scheme. The main contributions of this paper can be highlighted as follows.

(1)The proposed method extends the application scope of ADP theory and event-triggered control to the filed of distributed tracking control of interconnected nonlinear systems with uncertain interconnection terms.

(2)Different from [18], [19], [41], a novel discounted cost function is defined for each auxiliary subsystem which can accommodate the control objectives of guaranteeing the tracking performance, optimizing sample intervals and compensating the unmatched interconnection terms simultaneously.

(3)Compared with [37], [38], [39], the proposed triggering condition for each subsystem is designed by using the estimated worst-case control policy error of auxiliary subsystem, which can maximize the sample intervals to further reduce the computational burden and save the communication resources. Moreover, the event-triggered mechanism designed in this paper operates in an asynchronous and distributed manner which only needs local state information.

Besides, the notations used in this paper are listed as follows. $R,$ $R^m$ and $R^{n\times m}$ denotes the set of all real numbers, the real $m$-vector and the real $n\times m$ matrix, respectively. $\Omega$ stands for a compact subset of $R^n$. For any vector $x\in R^n,$ $\parallel x\parallel ={\left(x^{T}x\right)}^{1/2}$ is the Euclidean norm of $x$. For $A\in R^{n\times m},$ its maximum and minimum eigenvalues are written as ${\lambda }_{\max }\left(A\right)$ and ${\lambda }_{\min }\left(A\right),$ respectively.