Ben Niu, Jidong Liu, Ding Wang, Xudong Zhao, and Huanqing Wang

Abstract—An adaptive decentralized asymptotic tracking control scheme is developed in this paper for a class of large-scale nonlinear systems with unknown strong interconnections,unknown time-varying parameters, and disturbances. First, by employing the intrinsic properties of Gaussian functions for the interconnection terms for the first time, all extra signals in the framework of decentralized control are filtered out, thereby removing all additional assumptions imposed on the interconnections, such as upper bounding functions and matching conditions.Second, by introducing two integral bounded functions,asymptotic tracking control is realized. Moreover, the nonlinear filters with the compensation terms are introduced to circumvent the issue of “explosion of complexity”. It is shown that all the closed-loop signals are bounded and the tracking errors converge to zero asymptotically. In the end, a simulation example is carried out to demonstrate the effectiveness of the proposed approach.

I. INTRODUCTION

IN the last few decades, with the development of science and technology leading to increased interconnectedness and complexity of engineering applications, nonlinear interconnected systems which can be used to accurately model a wide range of fields such as networked control systems, power systems, and aerospace systems, etc., have got a lot of attention [1]–[11]. Decentralized control, which is a control strategy where a local controller demands only local signals, is widely adopted to deal with nonlinear interconnected systems.Contrasted with centralized control, decentralized control has the advantages of simple structure and less computational burden. However, in the area of decentralized control, there are two traditional assumptions: 1) The interconnection term must be bounded by known or partially known functions [2],[8], [12]–[20]; and 2) The matching condition [21]–[24]. In fact, the information of interconnections is often unpredictable which means these assumptions are too strict to be satisfied in many practical systems. As far as the authors know, only [25]handled a class of large-scale nonlinear systems with completely unknown interconnections until now. Nevertheless, the interconnections in [25] contain only the outputs of all subsystems, which are usually called weak interconnections. Hence, proposing a design technique to deal with completely unknown strong interconnections, in which the interconnections are assumed to be the functions of all states of the whole interconnected system, is a very meaningful research subject.

In recent years, adaptive decentralized control of uncertain large-scale nonlinear interconnected systems has become an emerging research hotspot. With the rapid development of backstepping theory, many influential achievements have been obtained in this research area such as the results in [18],[26]–[35]. By employing a bound estimation technique and two smooth functions with special characteristics, the adaptive decentralized tracking control problem with the prescribed performance was considered in [36] for a class of nonlinear interconnected systems with unknown time-varying parameters. In [37], an adaptive decentralized output feedback tracking control frame was proposed for a class of stochastic nonlinear interconnected systems with parametric uncertainties and partially unknown interactions. Besides, with the aid of graph theory, [38] studied the adaptive decentralized tracking control problem for a class of interconnected nonlinear systems with strong interconnections, where the bounding functions of the strong interconnections are assumed to exist. As we all know, asymptotic tracking control has giant potentiality in actual applications and is a desired design goal in engineering. However, it is noticed that none of the aforementioned efforts have achieved asymptotic tracking control. References [39] and [40] achieve asymptotic tracking for linear systems, but the methods of [39] and [40] are not suitable for nonlinear interconnected systems. Efforts towards asymptotic tracking for a class of uncertain large-scale stochastic nonlinear interconnected systems were made in[41], but the applicable scope of the results is limited to stochastic interconnected systems with weak interconnections.For the complexity caused by the interaction among the uncertain parameters, large-scale system structure, and strong interconnections in the controller design, no work has been done on adaptive decentralized asymptotic tracking control for uncertain large-scale interconnected systems with unknown strong interconnections until now, which mainly motivates our current research work.

On the other hand, although most contributions on adaptive decentralized control have been presented using backstepping method, there is a major drawback that needs to be solved,that is the issue of “explosion of complexity” is brought by differentiating the virtual controllers in the backstepping procedure. Fortunately, the dynamic surface control (DSC)approach was first proposed in [42] to avoid the trouble of“explosion of complexity”, where the differentiations of virtual controllers are avoided by introducing a low-pass filter in each step of backstepping. Subsequently, numerous improved DSC strategies have been developed to achieve control objectives for multiple types of nonlinear systems[43]–[53]. Nevertheless, the above proposed DSC strategies are an imperfect fit to adaptive decentralized asymptotic tracking control. Naturally, the questions come: When uncertain large-scale interconnected systems with unknown strong interconnections are studied, how to realize adaptive decentralized asymptotic tracking control while solving the issue of “explosion of complexity” with the help of DSC? If this is possible, then how to design the applicable low-pass filters? These design difficulties also inspire the research of this work.

Based on the above discussion, we take into account an important problem in the field of adaptive decentralized control, i.e., achieving asymptotic tracking control for uncertain large-scale nonlinear interconnected systems with unknown strong interconnections. In comparison with the literature considering adaptive decentralized control, the main contributions of this paper are listed below. i) Compared with a large number of existing works ([2], [7], [8], [12]–[24], [30],[32], [36], [37]) in which interconnected terms need either to satisfy matching conditions or to be bounded by known or partially known functions, the decentralized control scheme proposed in this paper removes all the two widely adopted traditional conditions of the interconnected terms by using the inherent properties of Gaussian function and thereby deals with completely unknown strong interconnections successfully. ii) Differently from the results in [18], [27]–[40], which only realize bounded tracking control, the asymptotic tracking control is realized in this paper even though the uncertain parameters, large-scale system structure and unknown strong interconnections are considered. iii) By applying the DSC technology, the inherent “explosion of complexity” problem in backstepping is eliminated.

II. PROBLEM STATEMENT AND PRELIMINARIES

Consider a class of uncertain large-scale nonlinear systems with strong interconnections in the following form:

Remark 2:Lemma 3 shows a very useful property of the basis function vector of an RBF NN. As you will see later, this property is skillfully utilized to achieve decentralized control in our controller design procedure.

III. CONTROLLER DESIGN AND STABILITY ANALYSIS

A. Adaptive Decentralized Controller Design

In this subsection, the adaptive decentralized controller of each subsystem of the interconnected system (1) is constructed. To avoid tedium analysis and calculation, we first define the following symbols for unknown parameter vectors, and unknown parameters εi j:

B. Stability Analysis

In this subsection, the stability analysis and asymptotic tracking performance of the whole closed-loop system (1) are verified.

The differential of the boundary erroreij=sij?αijis

In addition,Bi1(·) andBij(·) are continuous functions and their expressions are as follows:

Consider the following Lyapunov function:

Based on the previous analysis, the main result is summarized as the following theorem.

Theorem 1:Under Assumption 1, consider the uncertain large-scale nonlinear interconnected systems (1) with any initial conditionV(0)≤q, whereqis a positive constant. Then the adaptive law (54), controller (50) and nonlinear filters(24), (39) can guarantee the following performances:

Remark 3:Unlike [58], which only removes the matching conditions imposed on the interconnection terms, the decentralized controller proposed in this paper not only removes the matching conditions imposed on the interconnection terms, but also removes the assumption that the interconnection terms are bounded by known or partially known functions.

IV. SIMULATION EXAMPLES

Example 1:To further clarify the effectiveness of the proposed adaptive decentralized control scheme, a numerical example is provided. Consider a nonlinear system consisting of three subsystems as follows:

Example 2:To illustrate the effectiveness of the proposed control approach, we consider a practical interconnected system shown in Fig. 12, which consists of two inverted pendulums fixed on two cars and connected by springs [36].The position δ of the spring is not fixed, it can move alongl0.The length of each pendulum isl0. The massm2of the car is fixed, and their movement makes the distancel2(t)?l1(t)bounded. The dynamics of the above inverted pendulums can be modeled as

Fig. 1. The block diagram of the proposed decentralized control scheme.

Fig. 2. The block diagram of stability analysis.

Fig. 3. Tracking performance of subsystem 1.

Fig. 4. Tracking performance of subsystem 2.

Fig. 5. Tracking performance of subsystem 3.

Fig. 6. All adaptive laws of subsystem 1.

Fig. 7. All adaptive laws of subsystem 2.

Fig. 8. All adaptive laws of subsystem 3.

Fig. 9. Controller of subsystem 1.

Fig. 10. Controller of subsystem 2.

Fig. 11. Controller of subsystem 3.

Fig. 12. Two inverted pendulums on carts.

Fig. 13. Tracking performance of subsystem 1.

Fig. 14. Tracking performance of subsystem 2.

Fig. 15. All adaptive laws of subsystem 1.

Fig. 16. All adaptive laws of subsystem 2.

Fig. 17. Controller of subsystem 1.

Fig. 18. Controller of subsystem 2.

whereu1andu2denote control torques,y1andy2denote the angles of the pendulum,d1(t) andd2(t) are external disturbances,m1,v,δ,l0,kandgare constants,m2(t)=3+0.01sin(t), δ(t)=1+0.01sin(t),v(t)=m1/m1+m2(t),d1(t)=0.01sin(3t) ,d2(t)=0.01sin(2t), andl1=100sin(12t), andl2=0.1(2+sin(15t)). The system parameters are assumed to be unknown. Moreover, the reference signals are set asyd1=sin(6t) andyd2=2cos(7t).

According to Theorem 1, the virtual control laws, the controllers and nonlinear filters of the subsystems are designed as

wherezi1=xi1?ydi,

wherezi2=xi2?si1,

whereei1=si1?αi1, and the adaptive laws of of the subsystems are designed as

In the simulation, we selectr11=r21=1000,r12=r13=10,r22=r23=10, β11=β21=5, η12=η22=1,k11=k12=k21=k22=100, τ11=τ21=100,m1=3,k=1,l0=2 andg=10. In addition, we use ω1(t)=ω2(t)=10e?2t, σ1(t)=10e?2t,σ2(t)=e?2t. The tracking performances are shown in Figs. 13 and 14, and we can see that our approach achieves asymptotic tracking. The responses of the designed adaptive laws are presented in Figs. 15 and 16, respectively. The responses of the designed controllers are presented in Figs. 17 and 18,respectively. Figs. 15–18 demonstrate that all the signals in the closed-loop system (77) are bounded.

V. CONCLUSION

In this paper, an adaptive decentralized asymptotic tracking control scheme has been presented for a class of large-scale nonlinear systems with unknown strong interconnections,unknown time-varying parameters, and disturbances. The main obstacle is how to realize the asymptotic tracking control task without any additional assumptions imposed on the interconnections. By fusing the inherent properties of the Gauss functions, backstepping procedure, and DSC approach,the desired adaptive decentralized controller of each subsystem is constructed such that both the boundedness of the whole closed-loop system and the asymptotic tracking performance are ensured. Finally, a practical example is given to verify the effectiveness of the proposed control algorithm.Future research will focus on adaptive decentralized finitetime control of uncertain large-scale nonlinear interconnected systems.