Decentralized adaptive output-feedback control for a class of nonlinear large-scale systems with unknown time-varying delayed interactions

Abstract

In this paper, the authors investigate a decentralized adaptive output-feedback controller design for large-scale nonlinear systems with input saturations and time-delayed interconnections unmatched in control inputs. The interaction terms with unknown time-varying delays are bounded by unknown nonlinear bounding functions including all states of subsystems. This point is a main contribution of this paper compared with previous output-feedback control approaches which assume that the time-delayed bounding functions only depend on measurable output variables. The bounding functions are compensated by using appropriate Lyapunov–Krasovskii functionals and the function approximation technique based on neural networks. The observer dynamic surface design technique is employed to design the proposed memoryless local controller for each subsystem. In addition, we prove that all signals in the closed-loop system are semiglobally uniformly bounded and control errors converge to an adjustable neighborhood of the origin. Finally, an example is provided to illustrate the effectiveness of the proposed control system.

Introduction

Over the last decade, the increasing complexity and the large scale of uncertain modern engineering systems necessitate the design of decentralized adaptive controllers that would operate using only local states for feedback in each of the subsystems (see [1], [13], [14], [15], [17], [18], [19], [22], [32], [35], and the references therein). Furthermore, the study on time delay effects in interconnected systems becomes an important issue due to the information transmission among subsystems. It is widely known that time delays are a source of the instability of control systems [4], [22], [30]. Therefore, many researchers have investigated decentralized control problems for interconnected systems with time delays [2], [3], [7], [8], [25], [36], [39]. Decentralized adaptive control approaches have been also proposed to deal with uncertainties of time-delayed large-scale systems [5], [9], [11], [37], [38], [40]. In [32], a decentralized controller in the convex optimization context was designed for stabilization of interconnected systems with linear subsystems and nonlinear time-varying interconnections. In [2], a non-fragile state controller with additive perturbations and reduced design complexity was developed for a class of continuous-time nonlinear delayed symmetric composite systems. However, the aforementioned literatures for interconnected time-delay systems are based on full state-feedback control approaches, namely, all state variables are required to design decentralized adaptive controllers. In addition, they cannot be applied to nonlinear large-scale systems with time-delayed interactions unmatched in control inputs.

Despite the importance of these problems, there are few research results reported for them. Recently, Hua and Guan [12] proposed an output feedback stabilization method using the backstepping technique [21] for nonlinear interconnected time-delay systems where neural networks were used for compensating unknown bounds of interconnected nonlinearities. However, this result was limited to a class of systems, in which time-delayed bounding functions of interaction terms only depend on measurable output variables. Therefore, time-delayed bounding functions including all state variables should be considered for a decentralized output-feedback control of nonlinear time-delay interconnected systems.

On the other hand, a dynamic surface design technique was proposed to solve the problem of “explosion of complexity” caused by the repeated differentiations of virtual controllers in the backstepping design procedure [33]. Its main idea is introducing a first-order filtering of the synthesized virtual control law at each step of the backstepping design procedure. The dynamic surface design method was applied to adaptive systems for uncertain nonlinear functions [34], [41]. Recently, adaptive dynamic surface control schemes were presented for nonlinear systems with unknown time delays [42], [43]. These methods, however, require the information of all state variables to design the controller.

In this paper, we propose a decentralized adaptive output-feedback control approach for input-saturated nonlinear large-scale systems with unknown time-varying delayed interconnections. The considered interconnections do not satisfy the matching requirement on control inputs. The nonlinear time-delayed interaction terms are bounded by unknown nonlinear bounding functions including all state variables of subsystems. It is assumed that the time-varying delays and their first-derivatives are all unknown. The proposed memoryless local output-feedback controller for each subsystem is simply designed via the observer dynamic surface design technique. By applying the appropriate Lyapunov–Krasovskii functionals, the unknown time-varying delayed interconnection terms are eliminated. In addition, we employ the function approximation technique using neural networks to compensate uncertain nonlinear terms in the controller design procedure. The proposed approach is the first trial in the fields of the decentralized output-feedback control for nonlinear large-scale systems with time-delayed interconnections. It is shown that all the signals in the total closed-loop system are semiglobally uniformly bounded and the tracking errors can be made arbitrarily small by adjusting the design parameters. Finally, the theoretical results are illustrated via computer simulations.

The rest of this paper is organized as follows. In Section 2, we present the decentralized output-feedback control problem for large-scale nonlinear systems with time-varying delayed interconnections and input saturations. In Section 3, the decentralized memoryless adaptive output-feedback control system is proposed for uncertain interconnected time-delay systems. In addition, the stability of the proposed control system are analyzed. Simulation results are discussed in Section 4. Finally, Section 5 gives some conclusions.