An ISS-modular approach for adaptive neural control of pure-feedback systems

doi:10.1016/j.automatica.2006.01.004

Automatica

Volume 42, Issue 5, May 2006, Pages 723-731

https://doi.org/10.1016/j.automatica.2006.01.004 Get rights and content

Abstract

Controlling non-affine non-linear systems is a challenging problem in control theory. In this paper, we consider adaptive neural control of a completely non-affine pure-feedback system using radial basis function (RBF) neural networks (NN). An ISS-modular approach is presented by combining adaptive neural design with the backstepping method, input-to-state stability (ISS) analysis and the small-gain theorem. The difficulty in controlling the non-affine pure-feedback system is overcome by achieving the so-called “ISS-modularity” of the controller-estimator. Specifically, a neural controller is designed to achieve ISS for the state error subsystem with respect to the neural weight estimation errors, and a neural weight estimator is designed to achieve ISS for the weight estimation subsystem with respect to the system state errors. The stability of the entire closed-loop system is guaranteed by the small-gain theorem. The ISS-modular approach provides an effective way for controlling non-affine non-linear systems. Simulation studies are included to demonstrate the effectiveness of the proposed approach.

Introduction

In non-linear control design, the backstepping design method (Krstic, Kanellakopoulos, & Kokotovic, 1995) has been successful for special classes of non-linear systems with its constructive Lyapunov design procedures. A great deal of progress has been achieved for the control of strict-feedback systems with unknown parameters (Kokotovic and Arcak, 2001, Krstic et al., 1995) and with unknown non-linearities (Choi and Farrell, 2001, Ge et al., 2001; Ge & Wang, 2002a; Kwan & Lewis, 2000; Lewis et al., 1999, Polycarpou and Mears, 1998, Zhang et al., 2000). Nevertheless, it is noticed that relatively fewer results have been obtained for the class of pure-feedback systems, which is given in a general form as (Krstic et al., 1995) ${\dot{x}}_{i} = f_{i} ({\bar{x}}_{i}, x_{i + 1}), i = 1, \dots, n - 1,$ ${\dot{x}}_{n} = f_{n} ({\bar{x}}_{n}, u),$ $y = x_{1},$ where ${\bar{x}}_{i} = [x_{1}, \dots, x_{i}]^{T} \in R^{i}$ , $i = 1, \dots, n$ , $u \in R$ , $y \in R$ are state variables, system input and output, respectively; $f_{i} (\cdot)$ $(i = 1, \dots, n)$ are smooth non-linear functions. The pure-feedback system (1) represents a class of lower-triangular non-linear systems which has a more representative form than the strict-feedback systems. In practice, there are many systems falling into this category featured with a cascade and non-affine structure, such as biochemical process (Krstic et al., 1995), Duffing oscillator (Dong, Chen, & Chen, 1997), aircraft flight control system (Hunt & Meyer, 1997), mechanical systems (Ferrara & Giacomini, 2000), etc. A more recent example of practical pure-feedback systems is a simplified dynamic model for a reduced scale autonomous helicopter (Mahony & Lozano, 2000).

It can be seen that pure-feedback system (1) has no affine appearance of the variables to be used as virtual controls, and of the actual control u itself. The cascade and non-affine properties make it quite difficult to find the explicit virtual controls and the actual control to stabilize the pure-feedback systems using backstepping design (Krstic et al., 1995). In the literature of pure-feedback system control, parametric pure-feedback systems were mainly considered (Ferrara & Giacomini, 2000; Kanellakopoulos, Kokotovic, & Morse, 1991; Krstic et al., 1995; Seto, Annaswamy, & Baillieul, 1994). Recently, by combining the backstepping methodology with adaptive neural design, several special cases of pure-feedback systems, which are affine in control u, were investigated (Ge & Wang, 2002b; Wang & Huang, 2002). However, the problem of controlling the completely non-affine pure-feedback system (1) remains unsolved in the literature. The main difficulty for adaptive neural control of pure-feedback system (1) lies in that, when neural networks are used to approximate some desired virtual controls $α_{i}^{*}$ and desired practical control $u^{*}$ in the backstepping design, as done for lower-triangular systems (Ge and Wang, 2002a, Ge and Wang, 2002b; Kwan and Lewis, 2000, Wang and Huang, 2002, Zhang et al., 2000), it will generally involve the NN approximation of a function of u and $\dot{u}$ . As the NN approximation is one part of control u, this will lead to a circular construction of the practical controller. In Ge and Wang (2002b), Wang and Huang (2002), the circularity problem was avoided because much simpler pure-feedback systems were investigated.

In this paper, we consider adaptive neural control of the completely non-affine pure-feedback system (1). To overcome the aforementioned difficulty, we employ the input-to-state stability (ISS) analysis (Sontag, 1989, Sontag and Wang, 1996) and the small gain theorem (Jiang, Teel, & Praly, 1994) rather than constructing an overall Lyapunov function for the entire closed-loop. It is observed that in the adaptive neural control approaches (e.g., Ge and Wang, 2002a, Ge and Wang, 2002b, Kwan and Lewis, 2000, Zhang et al., 2000), the resulting closed-loop system commonly consists of two interconnected subsystems: the state error subsystem and the weight estimation subsystem. The interconnected structure motivates us to solve this problem using the celebrated small-gain theorem, especially the ISS-type small-gain theorem (Jiang et al., 1994), which will be shown useful to achieve the main results of this paper. By combining adaptive neural design with the ISS-type small-gain theorem, we present an ISS-modular approach for non-affine pure-feedback system control. The adaptive neural control approach is designed to achieve a significant level of “ISS-modularity” of the controller-estimator pair, i.e., to stabilize the interconnected state error subsystem and the weight estimation subsystem, any ISS neural controller can be combined with any ISS neural weight estimator, provided that the small-gain condition of the interconnected subsystems is satisfied. The neural controller is to achieve ISS with respect to the NN weight estimation errors. The neural weight estimator, in turn, will be designed to achieve ISS with respect to the system state errors. The stability of the entire closed-loop system will be guaranteed by using the small-gain theorem. By achieving the ISS-modularity of the interconnected control module and estimation module, the difficulty in controlling non-affine pure-feedback system (1) is separated into two relatively easier ones: the input-to-state stability analyses of the two subsystems, and the derivation of the entire closed-loop stability by using the small-gain theorem. The employment of ISS analysis and the small gain theorem avoids the construction of an overall Lyapunov function for the entire system, and subsequently overcomes the aforementioned circular controller construction problem.

The ISS-modular approach is inspired by the modular design in Krstic et al. (1995), which was developed for parametric strict-feedback systems. Compared with existing results for affine-in-control pure-feedback systems (Ge & Wang, 2002b; Wang & Huang, 2002), this paper presents yet another method in controlling non-affine pure-feedback system (1) with less restrictive assumptions. The ISS-modular approach provides a simple and effective way for adaptive neural control of uncertain non-linear systems. The proposed adaptive neural controller can also be directly applied to the uncertain strict-feedback systems. Since there are many practical systems falling into the category of non-linear strict-feedback and pure-feedback forms, the proposed scheme will find a wide variety of industrial applications.

The rest of the paper is organized as follows: the problem formulation as well as some preliminary results are presented in Section 2. Section 3 presents the ISS-modular approach for adaptive neural control of uncertain pure-feedback system (1). Simulation results performed on an illustrative example are included in Section 4 to demonstrate the effectiveness of the approach. Section 5 contains the conclusions.

Terminology: a continuous function $α : R_{+} \to R_{+}$ is said to belong to class $K$ if it is strictly increasing and $α (0) = 0$ . It is said to belong to class $K_{\infty}$ if $a = \infty$ and $α (r) \to \infty$ as $r \to \infty$ . A continuous function $β : R_{+} \times R_{+} \to R_{+}$ is said to belong to class $KL$ if, for each fixed s, the mapping $β (r, s)$ belongs to class $K$ with respect to r and, for each fixed r, the mapping $β (r, s)$ is decreasing with respect to s and $β (r, s) \to 0$ as $s \to \infty$ .

Section snippets

Problem formulation

For the control of pure-feedback system (1), define $g_{i} ({\bar{x}}_{i}, x_{i + 1}) = \frac{\partial f_{i} ({\bar{x}}_{i}, x_{i + 1})}{\partial x_{i + 1}}, i = 1, \dots, n - 1,$ $g_{n} ({\bar{x}}_{n}, u) = \frac{\partial f_{n} ({\bar{x}}_{n}, u)}{\partial u} .$ For simplicity of presentation, denote $x_{n + 1} = u$ .

Assumption 1

The signs of $g_{i} (\cdot, \cdot), i = 1, \dots, n$ are known, and there exist constants $0 < {\underset{̲}{g}}_{i} ⩽ {\bar{g}}_{i} < \infty$ such that (i) $| g_{i} ({\bar{x}}_{i}, x_{i + 1}) | > {\underset{̲}{g}}_{i}$ $(i = 1, \dots, n)$ , $\forall ({\bar{x}}_{i}, x_{i + 1}) \in R^{i} \times R$ ; and (ii) $| g_{i} ({\bar{x}}_{i}, x_{i + 1}) | ⩽ {\bar{g}}_{i}$ $(i = 1, \dots, n)$ , $\forall ({\bar{x}}_{i}, x_{i + 1}) \in Ω_{{\bar{x}}_{i + 1}}$ where $Ω_{{\bar{x}}_{i + 1}} \subset R^{i + 1}$ is a compact set.

Assumption 1 implies that partial derivatives $g_{i}$ $(i = 1, \dots, n)$ are strictly either positive or negative.

Adaptive neural control design

In this section, we develop an ISS-modular approach to overcome the circularity problem (as mentioned in the Introduction). In Section 3.1, an adaptive neural controller is designed to achieve ISS with respect to the NN weight estimation errors. In Section 3.2, a neural weight estimator is designed to achieve ISS with respect to the system state errors. The stability of the entire closed-loop system will be guaranteed by using the small-gain theorem in Section 3.3.

An example

To verify the effectiveness of the proposed approach, the developed adaptive NN controller is applied to the following non-linear system: ${\dot{x}}_{1} = x_{1} + x_{2} + \frac{x_{2}^{3}}{5}, {\dot{x}}_{2} = x_{1} x_{2} + u + \frac{u^{3}}{7}, y = x_{1},$ which is in the non-affine pure-feedback form (1).

The reference model is taken as the famous van der Pol oscillator (see, e.g., Vidyasagar, 1993) ${\dot{x}}_{d 1} = x_{d 2}, {\dot{x}}_{d 2} = - x_{d 1} + β (1 - x_{d 1}^{2}) x_{d 2}, y_{d} = x_{d 1},$ which yields a limit cycle trajectory when $β > 0$ ( $β = 0.2$ in this simulation), for initial states starting from points other than $(0, 0)$ .

The control

Conclusion

An “ISS-modular” approach for adaptive neural control of the non-affine pure-feedback system was presented. By achieving the ISS-modularity of the interconnected control module and estimation module, the difficult problem of non-affine pure-feedback system control was resolved by combining adaptive neural design with the backstepping method, ISS analysis and the small-gain theorem. The employment of ISS analysis and the small gain theorem avoids the construction of an overall Lyapunov function

Acknowledgments

This research was supported in part by the Hong Kong Research Grant Council under the CERG Grant CityU 1114/05E, and by the Natural Science Foundation of Guangdong Province under Grant no. 05006528. The authors would also thank the anonymous reviewers for the constructive comments which helps improve the quality and presentation of the paper.

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Cong Wang received B.E. and M.E. degrees from Department of Automatic Control, Beijing University of Aeronautic & Astronautics, China, in 1989 and 1997, respectively, and the Ph.D. degree from the Department of Electrical & Computer Engineering, the National University of Singapore in 2002. From 2001 to 2004, he did his postdoctoral research at the Department of Electronic Engineering, City University of Hong Kong. He has been with the College of Automation, the South China University of Technology since 2004, where he is currently a Professor. He has authored and co-authored over 30 international journal and conference papers. He is presently serving as an Associate Editor of IEEE Control Systems Society Conference Editorial Board. From May 2005, he serves as a program director at the Directorates for Information Sciences, the National Natural Science Foundation of China (NSFC). His research interest includes deterministic learning theory, intelligent and autonomous control, dynamical pattern recognition, and cognitive and brain sciences.

David J. Hill received B.E. and B.Sc. degrees from the University of Queensland, Australia, in 1972 and 1974, respectively. He received Ph.D. degree in Electrical Engineering from the University of Newcastle, Australia, in 1976.

He is currently an Australian Research Council Federation Fellow in the Research School of Information Science and Engineering at The Australian National University. He has held academic and substantial visiting positions at the universities of Melbourne, California (Berkeley), Newcastle (Australia), Lund (Sweden), Sydney and Hong Kong (City). He holds honorary professorships at the University of Sydney, Huazhong University of Science and Technology, China, South China University of Technology and City University of Hong Kong. His research interests are in network systems, circuits and control with particular experience in stability analysis, non-linear control and applications mainly to energy and information systems. He is a Fellow of the IEEE and a Fellow of Institution of Engineers, Australia and a Foreign Member of the Royal Swedish Academy of Engineering Sciences.

Shuzhi Sam Ge received B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London. He has been with the Department of Electrical & Computer Engineering, the National University of Singapore since 1993, where he is currently a Professor. He is a Fellow of the IEEE and has (co)-authored three books: Adaptive Neural Network Control of Robotic Manipulators (World Scientific, 1998), Stable adaptive Neural Network Control (Kluwer, 2001) and Switched Linear Systems: Control and Design (Springer-Verlag, 2005), and over 200 international journal and conference papers. He has been serving as Editor of International Journal of Control, Automation and Systems since 2003, and Associate Editors for Automatica and a number of IEEE Transactions. His current research interests are control of nonlinear systems, hybrid systems, neural/fuzzy systems, sensor fusion, and system development.

Guanrong Chen received the M.Sc. degree in Computer Science from Zhongshan University, China, and the Ph.D. degree in Applied Mathematics from Texas A&M University, USA. Currently he is a Chair Professor and the Founding Director of the Centre for Chaos Control and Synchronization at the City University of Hong Kong. Since 1997, he has been a Fellow of the IEEE, awarded for his fundamental contributions to the theory and applications of chaos control and bifurcation analysis. He has (co)-authored 16 research monographs and advanced textbooks, more than 350 SCI journal papers, and about 200 refereed conference papers, published since 1981 in the fields of nonlinear system dynamics and controls.

Prof. Chen served and is serving as Chief Editor, Deputy Chief Editor, Advisory Editor and Associate Editors for eight international journals including the IEEE Transactions on Circuits and Systems, IEEE Transactions on Automatic Control, and the International Journal of Bifurcation and Chaos. He received the 1998 Harden–Simons Prize for the Outstanding Journal Paper Award from the American Society of Engineering Education, the 2001 M. Barry Carlton Best Annual Transactions Paper Award from the IEEE Aerospace and Electronic Systems Society, and the 2005 Guillemin–Cauer Best Transaction Annual Paper Award from the IEEE Circuits and Systems Society. He is Honorary Professor of the Central Queensland University, Australia, as well as Honorary Guest-Chair Professor of several Universities in China.

^☆: This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Thomas Parisini under the direction of Editor Robert R. Bitmead.

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Brief paperAn ISS-modular approach for adaptive neural control of pure-feedback systems☆

Abstract

Introduction

Section snippets

Problem formulation

Adaptive neural control design

An example

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

Automatica

Automatica

Mathematical analysis

Adaptive observer backstepping control using neural networks

IEEE Transactions on Neural Networks

Singular perturbations and input-to-state stability

IEEE Transactions on Automatic Control

Adaptive control of the uncertain Duffing oscillator

International Journal of Bifurcation and Chaos

Control of a class of mechanical systems with uncertainties via a constructive adaptive/second order VSC approach

Transactions of ASME, Journal of Dynamic Systems, Measurement and Control

Stable adaptive neural network control

Direct adaptive NN control of a class of nonlinear systems

IEEE Transactions on Neural Networks

Neural networks: A comprehensive foundation

Nonlinear control systems II

Brief paper
An ISS-modular approach for adaptive neural control of pure-feedback systems☆