Observer-based fuzzy adaptive optimal stabilization control for completely unknown nonlinear interconnected systems

Abstract

Fuzzy adaptive optimal bounded control problems are first investigated for a class of nonlinear continuous-time interconnected systems whose system internal dynamics and unmatched interconnections are completely unknown, when there exist unavailable states in the subsystems of the interconnected systems. The system states and the interconnection terms of the interconnected system are approximated by using a fuzzy state observer. The decentralized optimal controllers and observer-critic structure are designed according to adaptive dynamic programming and enforcement learning technology. The presented control methods can ensure that the system states and parameter estimation errors of the interconnected systems are ultimately uniformly bounded. A simulation example validates the effectiveness of the presented scheme.

Introduction

In recent two decades, fuzzy logic systems (FLSs) and neural networks (NNs) have attracted lots of attention in nonlinear control community, and some valuable results were developed, such as [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. To mention a few, Xie et.al [1] reduced significantly the conservatism of controller design for discrete-time Takagi–Sugeno fuzzy systems by an efficient augmented multi-indexed matrix approach. The NN adaptive control methods were investigated for non-lower triangular nonlinear systems with unmodeled dynamics [2] and uncertain stochastic nonlinear systems with dead-zone and output constraint in nonstrict-feedback form [3]. Although the above-mentioned control methods achieved great progress for nonlinear systems, the fuzzy or neural network control schemes had not considered the optimal control problem of nonlinear systems.

The optimal control problem is a core issue in the modern control theory, and the theory of optimal control has attracted a great deal of attention [14], [15], [16]. Many scholars combined approximate dynamic programming (ADP) and reinforcement learning (RL) to achieve the design of optimal control by approximating the solution of a Hamilton–Jacobi–Bellman (HJB) equation because it is tough to obtain the closed form solution of the HJB equations of nonlinear system (see [17], [18], [19], [20], [21], [22] and references herein). Infinite-time optimal tracking control problems were investigated for affine nonlinear discrete-time (DT) systems by using a kind of heuristic dynamic programming (HDP) method [17]. And then, the research of optimal control design was extended to non-affine nonlinear DT systems [18], [19]. Optimal control methods were developed for non-affine nonlinear DT systems by utilizing a globalized dual heuristic programming algorithm [18] and a DT local policy iteration ADP algorithm in [19]. Because of the difference between DT systems and continuous-time systems, the results of DT systems cannot be directly applied into the optimal problems of the continuous-time systems. In the continuous-time framework, an online optimal control scheme was proposed for affine nonlinear systems under the circumstances that the internal system dynamics are partially known [20]. To realize a truly model-free ADP algorithm, [21] presented an online adaptive optimal control scheme with completely unknown internal system dynamics for continuous-time systems. But reference [21] only considers linear systems. In order to extend to nonlinear unknown continuous-time systems, the authors in [22] proposed identifier-critic-based ADP algorithm by using a dual neural network structure, and achieved finite-time convergence of the NN parameters. All the above-mentioned adaptive control methods only considered multi-input multi-output systems, and these methods cannot be directly applied to nonlinear interconnected systems due to the interconnections among each subsystem.

Nonlinear interconnected systems, which consist of some lower order subsystems with interconnections, widely exist in engineering and science community, such as economic systems [23], power systems [24], etc. Since decentralized control schemes which only utilize the local states or outputs of each subsystem use less computation cost than centralized control schemes, many decentralized optimal control schemes have been proposed for nonlinear interconnected systems [25], [26], [27], [28], [29], [30]. For nonlinear continuous-time interconnected systems whose internal system dynamics are known, decentralized NN-based optimal stabilization controllers were designed with cost functions which reflect the bounds of the interconnections among subsystems [25]. Wang et al. [26] developed near-optimal stabilization control strategy by using a novel adaptive law without the requirement that the initial control is stabilized. Sun et al. [27] investigated the fuzzy adaptive state feedback decentralized optimal tracing control problem for a class of nonlinear large-scale systems by using fuzzy logic and the backstepping design method. The common feature of the mentioned decentralized control schemes is that the system states are assumed to be measurable. To relax the assumption, Zhao et al. presented two observer-based decentralized control methods to solve the tracking optimal control problems for the continuous-time nonlinear interconnected systems with unknown system dynamics [28] and for the similar systems with exactly known system dynamics [31]. The authors in [29] developed state feedback stabilization control and observer-based stabilization control strategy to realize the event-triggered optimal control for a class of nonlinear interconnected systems. Tong et al. [30] proposed an adaptive output feedback decentralized optimal tracking control scheme by using fuzzy logic and backstepping technology for nonlinear interconnected systems with unmeasured states and unknown system dynamics. By analyzing the above-mentioned literature, we observe that the interconnections in the references [25], [26] must satisfy the matched conditions. To remove the matched conditions, some scholars proposed different control schemes [27], [28], [29], [30], [31], [32], [33]. The authors in [27], [30] used fuzzy backstepping optimal control technologies to remove the conditions. The authors in [28], [32] relaxed the matched conditions by adding compensation terms. In [29], [31], the interconnects among the subsystems can be estimated by state observers such that the matched conditions are removed.

In many practical systems, the mathematic models are rarely known due to the complexity of the interconnected systems, and the output of the controllers is always bounded owing to the physical limitations of the actuator. In addition, optimal control can effectively improve the performances and economic benefits of the controlled systems. Motivated by the references [29], [31], an observer-based fuzzy adaptive optimal bounded control problem is investigated for nonlinear continuous-time interconnected systems. The model of the interconnected system is completely unknown, the interconnections of each subsystem are unmatched, and some states of the interconnected system are immeasurable. The system states and the interconnection terms of the interconnected system are approximated by using a fuzzy state observer. The decentralized optimal controllers and an observer-critic structure are designed according to adaptive dynamic programming and enforcement learning technology. The contributions of the paper with respect to the existing references are listed as follows. (i) The paper first applied the optimal fuzzy bounded control scheme to the unknown interconnected system. Currently, most references on continuous-time or discrete-time nonlinear interconnected systems, like [30], [32], did not considered the bounded control problems. (ii) A fuzzy state observer is designed to estimate the unknown mismatched interconnections and system dynamics of the whole interconnected system. The references [29], [31] about the interconnected systems require that the system dynamics are known. (iii) The unmatched conditions of the interconnected system can be relaxed. The unmatched conditions in the references [27], [28], [30], [31], [32], [33] are required that the norms of the interconnection terms satisfy some upper boundaries. Compared with these references, the boundaries of the interconnected terms are unknown in the paper.

The rest of this paper is organized as follows. Section 2 formulates the control problem of the interconnected system, and briefly introduces fuzzy logic theory. In Section 3, a fuzzy state observer is designed, and its stability are analyzed. The decentralized control scheme is proposed for an interconnected system, and stability of the closed-loop system are analyzed in Section 4. Simulation results are provided to verify the effectiveness of the proposed scheme in Section 5. Finally, conclusions are drawn.