Decentralized adaptive neural output feedback control of a class of large-scale time-delay systems with input saturation

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Abstract

A decentralized adaptive neural output feedback control scheme is presented for a class of large-scale time-delay systems with saturating inputs. The observer is constructed to estimate the immeasurable states of the system and the auxiliary system is designed to compensate the nonsmooth nonlinearities of input saturation constraints. Also, the control strategy is developed by the backstepping recursive method combining with neural networks (NNs) for the approximation of the unknown functions and dynamic surface control (DSC) technique for the well known ‘explosion of complexity’ problem. The advantage of this scheme is that it only relies on the output information of the system and there is no requirement for exact priori knowledge about the system parameters. It is proved that the control approach guarantees all signals in the closed-loop system uniformly ultimately bounded. Simulation results are provided to demonstrate the effectiveness and usefulness of the proposed strategy.

Introduction

In the past two decades, decentralized control of large-scale systems, especially time-delay systems, has attracted considerable attention in the control community and much significant progress has been made in this field [1], [2], [3], [4], [5], [6], [7]. Wen [1] introduced a backstepping method to design a decentralized adaptive controller for a class of linear single-input–single-output (SISO) systems. Jiang presented several decentralized output feedback control strategies for several classes of interconnected large-scale systems [2], [3], [4]. With regard to the approximation of unknown functions in nonlinear systems, it seems essentially same for the tools of NNs, fuzzy logics and wavelet series [5]. The backstepping recursive method [6] combines with them to compensate uncertainties in large-scale nonlinear systems without satisfying the matching condition. Time delay is the inherent property of a great number of engineering systems, such as chemical processes, rolling mill systems, biological processes and so on [8]. This common characteristic is particularly critical for industrial and military applications since it may degrade the control performance of the system, and even severely undermine the stability of the system. Nguang [9] and Fu [10] designed robust stabilization controllers for two classes of triangular time-delay nonlinear systems respectively by the Lyapunov-based recursive design method and Lyapunov–Krasovskii functionals. Ge [11] removed the assumption about nonlinear functions and designed control strategies for a class of strict-feedback time-delay nonlinear systems with unknown virtual control-gain functions but known their signs. This result was extended to multiple-input–multiple-output (MIMO) systems [12]. Furthermore, Wang relaxed the restriction about the unknown virtual control-gain functions with unknown upper bounds [13]. In [14], an adaptive neural control scheme was constructed for a class of strict-feedback nonlinear systems with unknown time delays. It was noted that virtual control-gain functions and their signs were both unknown. Zhang extended the work in [14] and designed an adaptive control method for a class of MIMO time-varying delaynonlinear systems with unknown dead-zones and gain signs [15]. By utilizing an integral Lyapunov function, Lyapunov–Krasovskii functionals, Nussbaum-type functions and multilayer NNs, the signals in the designed closed-loop system were proved to be semiglobally uniformly ultimately bounded. Wang [16], [17] designed adaptive control laws by combining Lyapunov–Krasovskii functionals with the backstepping method for a class of strict-feedback time-delay nonlinear systems and a class of pure-feedback nonlinear systems with unknown time-delay functions, respectively. For a class of strict-feedback time-delay systems, Hua [18] reconstructed system states and developed a robust output feedback controller. Further, considering a class of time-delay systems in the form of triangular structure with unmodeled dynamics, Hua [19] designed an observer-based neural network controller by the work in [18]. Using the state transformation from the original system to the system without input delays, Zhu [20] proposed a stabilization control scheme for a class of nonlinear systems with input and state delays. And then an output feedback control strategy was discussed in [21] and the result was extended to a kind of control approaches for a class of MIMO nonlinear systems in [22]. However, the obvious drawback of the backstepping method is the ‘explosion of complexity’ problem, that is, computational calculations increase drastically along with the growth of order of the system. To avoid the drawback of the backstepping method mentioned above, dynamic surface control (DSC) technology [23] was proposed by introducing a first-order low-pass filter at each step of the design procedure. This idea was combined with RBF neural networks for a class of SISO strict-feedback systems [24]. In the following half a decade, scores of scholars carried out remarkable research work, e.g., Li [25], Zhang [26], Tong [27], and Sun [28]. For time-delay systems, Wang designed a robust stabilization control law for a class of non-affine pure-feedback systems with unknown time-delay functions by DSC, RBF neural networks, and Lyapunov–Krasovskii functionals [29]. Zhu [30] developed an adaptive dynamic surface control scheme for the tracking problem of a class of nonlinear systems with uncertain input delay and disturbances. Li carried out a serious of work about the MIMO time-delay nonlinear systems [31], [32], [33]. Tong [34] proposed an observe-based adaptive NN-based decentralized output feedback control approach.

In practice, the phenomenon of input saturation is common in industry applications, such as electric servomotors, marine vessels, robot manipulators, and so on. This problem is unavoidable in most physical actors, and it has attracted a great concern from control communities [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]. Hu, Lin and Chen designed a control approach for linear control systems subject to actuator saturation [35]. Then output feedback control methods were proposed for linear systems with actuator saturation by Wu, Lin, Zheng [36], [37]. Cao [38] discussed a fuzzy control law for nonlinear system subjected to input saturation by the T–S fuzzy model and auxiliary feedback matrix. The constrained adaptive backstepping control law was designed using filters which were employed to impose restrictions on the virtual control and the control input. Also, the constrained adaptive control strategy was investigated via the on-line approximation method for uncertain nonlinear systems [39], [40], [41]. Zhong [42] presented a novel method of designing model reference adaptive control (MRAC) for a class of SISO plants which is assumed to be stable and minimum phase with input saturation. Gao [43] investigated a NN-based strategy for a class of nonlinear systems with input saturation constraints in Brubovsky canonical form, and then numerous scholars made a contribution in this filed, such as Hu [44], Sonneveldt [45], and Chen [46], [47], [48].

From the aforementioned work, it can be seen that the previous research results mentioned about the controlled plants with time delays or input saturation constraints or both of them were for the strict-feedback systems or SISO systems. When both time delays and input saturation constraints simultaneously appear in a class of large-scale systems with non-affine pure-feedback form, the design of an output feedback control law seems more complex and challenging than the one addressed in [7], [34]. In this paper, we will consider unknown time-delay functions and interconnections among subsystems and will present a decentralized adaptive neural output feedback control scheme for a class of large-scale non-affine pure-feedback time-delay systems with input saturation. In detail, the non-affine pure-feedback system is transformed into the strict-feedback form with the help of the function transformation and unknown functions are approximated by radial basis function neural networks. Further, a state observer and an auxiliary system are constructed to estimate the immeasurable states and compensate the input saturation nonlinear characteristic, respectively. Also, a first-order low-pass filter is introduced to replace the differential operation of virtual control at each step of the backstepping design procedure. The main contributions of this paper are nontrivial and lie as follows: (1) First of all, a state observer is constructed and the control strategy only relies on the output information of the system. It is useful in industrial applications where inner states cannot be directly or not easily measured. (2) In addition, we simultaneously consider time delays and input saturations for MIMO systems in non-affine pure-feedback form and present an adaptive neural output feedback control scheme for this class of systems. (3) Last but not the least, there is no requirement for exact priori knowledge about system parameters since NNs are used to approximate unknown dynamics. DSC technique is employed in the paper, which leads to a much simpler and more practical controller than traditional backstepping-based methods.

For convenience of the analysis, the following notations are used throughout the whole paper. |·| means the absolute value of a scalar, and ||·|| representatives Euclidean norm of vectors or induced norm of matrices. λmin(·) and λmax(·) stand for the minimum and maximum eigenvalue of a matrix, respectively.

Section snippets

Problem formulation and preliminaries

Consider a class of large-scale nonlinear systems with time delays and input saturations:{ẋi,1(t)=fi,1(xi,1(t),xi,2(t))+hi,1(xi,1(tτi,1,1(t)),,xN,1(tτi,1,N(t)))+Δi,1(y1,,yN),ẋi,2(t)=fi,2(x̲i,2(t),xi,3(t))+hi,2(xi,1(tτi,2,1(t)),,xN,1(tτi,2,N(t)))+Δi,2(y1,,yN),ẋi,ni1(t)=fi,ni1(x̲i,ni1(t),xi,ni(t))+hi,ni1(xi,1(tτi,ni1,1(t)),,xN,1(tτi,ni1,N(t)))+Δi,ni1(y1,,yN),ẋi,ni(t)=fi,ni(x̲i,ni(t),ui(t))+hi,ni(xi,1(tτi,ni,1(t)),,xN,1(tτi,ni,N(t)))+Δi,ni(y1,,yN)yi(t)=xi,1(t)where xi,j(

Output feedback control law design and stability analysis

In general, the designed closed-loop control system mainly consists of two parts: (1) a class of large-scale time-delay nonlinear systems subjected to input saturations as described in Section 2 and (2) an adaptive output control law with a state observer. In this section, a state observer is designed to estimate the immeasurable variables in the system (1). Then, we will present a decentralized adaptive output feedback tracking control scheme for the system (1) that solves the control

Simulation results

In this section, two simulation examples are carried out to illustrate the effectiveness of the proposed control approach.

Conclusion

In this paper, we have shown how to design the decentralized adaptive output feedback controller for a class of large-scale time-delay systems with input constraints. The scheme was proposed by using the backstepping method combining with DSC technology, RBF neural networks, and the auxiliary system. The simplified control approach was proven to regulate tracking errors of the system to converge to a small neighborhood. Also, simulation results were provided to demonstrate the effectiveness of

Acknowledgments

The authors would like to thank anonymous reviewers and editors for their constructive suggestions. This work was sponsored in part by the National Natural Science Foundation of China (Grant nos. 61374055 and 61374114), in part by the Ph.D. Programs Foundation of Ministry of Education of China (Grant no. 20110142110036), in part by the Natural Science Foundation of Jiangsu Province (Grant nos. BK20131381 and BK20140877), in part by Jiangsu Planned Projects for Postdoctoral Research Funds (Grant

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