Elsevier

Automatica

Volume 38, Issue 4, April 2002, Pages 671-682
Automatica

Adaptive NN control of uncertain nonlinear pure-feedback systems

https://doi.org/10.1016/S0005-1098(01)00254-0Get rights and content

Abstract

This paper is concerned with the control of nonlinear pure-feedback systems with unknown nonlinear functions. This problem is considered difficult to be dealt with in the control literature, mainly because that the triangular structure of pure-feedback systems has no affine appearance of the variables to be used as virtual controls. To overcome this difficulty, implicit function theorem is firstly exploited to assert the existence of the continuous desired virtual controls. NN approximators are then used to approximate the continuous desired virtual controls and desired practical control. With mild assumptions on the partial derivatives of the unknown functions, the developed adaptive NN control schemes achieve semi-global uniform ultimate boundedness of all the signals in the closed-loop. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters.

Introduction

In the past decade, interest in adaptive control of nonlinear systems has been ever increasing, and many significant developments have been achieved. As a breakthrough in nonlinear control area, adaptive backstepping was introduced to achieve global stability and asymptotic tracking for a large class of nonlinear systems in the parametric strict-feedback form by Kanellakopoulos, Kokotovic, and Morse (1991). Later, the overparametrization problem was successfully eliminated in Krstic̀, Kanellakopoulos, and Kokotovic (1992) through the tuning function method. In an effort to extend the backstepping idea to larger classes of nonlinear systems, Kanellakopoulos et al. (1991) studied the adaptive control problem of parametric pure-feedback systems and obtained regionally stable results; Seto, Annaswamy, and Baillieul (1994) proposed several adaptive approaches for nonlinear systems with a triangular structure. To accommodate uncertainties, robust adaptive backstepping control has been studied for nonlinear strict-feedback systems with time-varying disturbances and static or dynamic uncertainties in Freeman and Kokotovic̀ (1996), Yao and Tomizuka (1997), Jiang and Praly (1998) and Pan and Basar (1998) (to name just a few).

On the other hand, adaptive neural control schemes have been found to be particularly useful for the control of highly uncertain, nonlinear and complex systems (see Lewis, Jagannathan, & Yeildirek, 1999; Ge, Hang, Lee, & Zhang, 2001 and the references therein). In the earlier NN control schemes, optimization techniques were mainly used to derive parameter adaptation laws with little analytical results for stability and performance. To overcome these problems, some elegant adaptive NN control approaches have been proposed based on Lyapunov's stability theory (Narendra & Parthasarathy, 1990; Polycarpou & Ioannou, 1992; Sanner & Slotine, 1992; Rovithakis & Christodoulou, 1994; Chen & Khalil, 1995; Yesidirek & Lewis, 1995; Spooner & Passino, 1996). However, one limitation of these schemes is that they can only be applied to nonlinear systems where certain types of matching conditions are required to be satisfied.

Using the idea of adaptive backstepping design (Krstic̀, Kanellakopoulos, & Kokotovic̀, 1995), several neural-based adaptive controllers (Polycarpou & Mears, 1998; Ge et al., 2001) have been investigated for some classes of nonlinear systems in the following strict-feedback form without the requirement of matching conditionsẋi=fi(x̄i)+gi(x̄i)xi+1,1⩽i⩽n−1,ẋn=fn(x̄n)+gn(x̄n)u,n⩾2,y=x1,where x̄i=[x1,…,xi]T∈Ri,i=1,…,n,u∈R,y∈R are state variables, system input and output, respectively; fi(·) and gi(·), i=1,…,n are unknown smooth functions. In Polycarpou and Mears (1998), an indirect adaptive NN control scheme was presented for system (1) with the affine terms gi(x̄i)=1,i=1,…,n−1, and gn(x̄n)=g being an unknown constant. The unknown functions fi(x̄i),i=1,…,n are firstly approximated on-line by neural networks, then a stabilizing controller is constructed based on the approximation. Through the introduction of novel integral Lyapunov functions, direct adaptive neural network control was proposed for system (1) (Ge et al., 2001), in which the possible controller singularity problem usually met in adaptive control is avoided without using projection.

While the nonlinear strict-feedback systems have been much investigated via backstepping design, only a few results are available in the literature for the control of nonlinear pure-feedback systems (Nam & Arapostations, 1988; Kanellakopoulos et al., 1991; Seto et al., 1994; Krstic̀ et al., 1995). The pure-feedback system represents a more general class of triangular systems which have no affine appearance of the variables to be used as virtual controls. In practice, there are many systems falling into this category, such as mechanical systems (Ferrara & Giacomini, 2000), aircraft flight control system (Hunt & Meyer, 1997), biochemical process (Krstic̀ et al., 1995), Duffing oscillator (Dong, Chen, & Chen, 1997), etc. As indicated in Krstic̀ et al. (1995), it was quite restrictive to find the explicit virtual controls to stabilize the pure-feedback systems by using integrator backstepping. In Kanellakopoulos et al. (1991); Krstic̀ et al. (1995), while excellent results are given for global stabilization of parametric strict-feedback systems, only local stability is achieved in a well defined region around origin for parametric pure-feedback systems. By imposing additional restrictions on the nonlinearities, global stability is obtained for a special case of the parametric pure-feedback systems in Seto et al. (1994). Note that in Kanellakopoulos et al. (1991); Seto et al. (1994); Krstic̀ et al. (1995), the nonlinearities are known smooth functions, and the unknown parameters occur linearly.

In this paper, adaptive NN control schemes are proposed for the following uncertain nonlinear pure-feedback systems:Σ1:ẋi=fi(x̄i,xi+1),1⩽i⩽n−2,ẋn−1=fn−1(x̄n−1)+gn−1(x̄n−1)xn,ẋn=fn(x̄n)+gn(x̄n−1)u,n⩾3,y=x1andΣ2:ẋi=fi(x̄i,xi+1),1⩽i⩽n−2,ẋn−1=fn−1(x̄n−1)+gn−1(x̄n−1)xn,ẋn=fn(x̄n)+gn(x̄n)u,n⩾3,y=x1,where x̄i=[x1,…,xi]T∈Ri,i=1,…,n,u∈R,y∈R are state variables, system input and output, respectively; fi(x̄i,xi+1)(i=1,…,n−2), fj(·) and gj(·) (j=n−1,n) are unknown smooth functions. Here the difference between Σ1 and Σ2 only lies in gn(·), i.e., gn(·)=gn(x̄n−1) in Σ1, while gn(·)=gn(x̄n) in Σ2.

Due to the difficulties for controlling non-affine systems, the systems considered in this paper are affine in control u. Moreover, they are also affine in xn in the ẋn−1 equations. To the authors’ knowledge, no effective method for this control problem exists in the literature at present stage. This is mainly due to the fact that it is very difficult to find virtual controls αi in terms of x̄i in backstepping design procedure. To overcome this difficulty, implicit function theorem is firstly exploited to assert the existence of the continuous desired virtual controls αi(x̄i) for i=1,…,n−2, then NN approximators are used to approximate the continuous desired virtual controls αi and desired practical control u as in Ge, Hang, and Zhang (1999). With the help of NN approximation, there is no need to solve the implicit functions for the explicit virtual controls and the practical controller to cancel the unknown functions in backstepping design. The idea of integrator backstepping is still employed, i.e., some of the state variables are considered as “virtual controls”, and intermediate control laws are designed in the constructive design procedures. With mild assumptions on the partial derivatives of the unknown functions fi(x̄i,xi+1) (i=1,…,n−2) as well as on gj(·) (j=n−1,n), the developed adaptive NN control scheme achieves semi-global uniform ultimate boundedness of all the signals in the closed-loop. Moreover, the output of the system is proven to converge to a small neighborhood of the desired trajectory. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters.

Section snippets

Problem formulation and preliminaries

The control objective is to design direct adaptive NN controllers for the systems such that (i) all the signals in the closed-loop remain semi-globally uniformly ultimately bounded, and (ii) the output y follows a desired trajectory yd generated from the following smooth, bounded reference model:ẋdi=fdi(xd),1⩽i⩽m,yd=xd1,where xd=[xd1,xd2,…,xdm]TRm are the states, ydR is the system output, fdi(·),i=1,2,…,m are known smooth nonlinear functions. Assume that xd remain bounded, i.e., xd∈Ωd,∀t⩾0.

Lemma 1

Direct adaptive NN control for Σ1

For the control of pure-feedback systems Σ1 and Σ2, define gi(x̄i,xi+1)≔∂fi(x̄i,xi+1)/∂xi+1,i=1,…,n−2, which are also unknown nonlinear functions.

Assumption 2

The signs of gi(x̄i,xi+1),i=1,…,n−2, gn−1(x̄n−1) and gn(x̄n−1) are known, and there exist constants gi1>gi0>0 such that (i) |gi(·)|>gi0>0,x̄n∈Rn, and (ii) |gi(·)|⩽gi1<∞,x̄n∈Ωx̄n⊂Rn where Ωx̄n is a compact region, i=1,…,n.

The above assumption implies that partial derivatives gi(·), i=1,…,n are strictly either positive or negative. Without losing

Direct adaptive NN control for Σ2

In this section, the design procedure is very similar to that of Section 3, except that integral Lyapunov function is employed in controller design to avoid the possible singularity problem caused by gn(x̄n) in the last equation of Σ2.

For system Σ2, all the assumptions on gi(·),i=1,·,n−1 are the same. The following assumption is made for gn(x̄n).

Assumption 4

The sign of gn(x̄n) is known, and there exist a constant gn0>0 and a known smooth function ḡn(x̄n) such that ḡn(x̄n)⩾|gn(x̄n)|⩾gn0,∀x̄n∈Rn. Without

Conclusion

In this paper, direct adaptive NN control schemes are presented for nonlinear pure-feedback systems with unknown nonlinear functions. Implicit function theorem is firstly exploited to assert the existence of the continuous desired virtual controls. NN approximators are then used to approximate the continuous desired virtual controls and desired practical control. With mild assumptions on the partial derivatives of the unknown functions, the developed adaptive NN control scheme achieves

Acknowledgements

The authors wish to thank Baozhong Yang for fruitful discussions on implicit function theorem.

S. S. Ge received the B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China, in 1986, and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London, in 1993. From May 1992 to June 1993, he did his postdoctoral research at Leicester University, England. He has been with the Department of Electrical & Computer Engineering, the National University of Singapore since 1993, and is

References (25)

  • Ge, S. S., Hang, C. C., Lee, T. H., & Zhang, T. (2001). Stable adaptive neural network control. Kluwer Academic (in...
  • S. Haykin

    Neural networks: A comprehensive foundation

    (1999)
  • Cited by (499)

    View all citing articles on Scopus

    S. S. Ge received the B.Sc. degree from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China, in 1986, and the Ph.D. degree and the Diploma of Imperial College (DIC) from Imperial College of Science, Technology and Medicine, University of London, in 1993. From May 1992 to June 1993, he did his postdoctoral research at Leicester University, England. He has been with the Department of Electrical & Computer Engineering, the National University of Singapore since 1993, and is currently an Associate Professor. He was a visiting staff in Laboratoire de'Automatique de Grenoble, France in 1996, the University of Melbourne, Australia in 1998–99, and University of Petroleum, China in 2001. He has authored and co-authored over 100 international journal and conference papers, one monograph and co-invented one patent. He served as an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society in 1998 and 1999, has been serving as an Associate Editor, IEEE Transactions on Control Systems Technology since June 1999, and a Member of the Technical Committee on Intelligent Control of the IEEE Control System Society since 2000. He was the winner of the 1999 National Technology Award, Singapore. He serves as a technical consultant local industry. He is a Senior Member of IEEE. His current research interests are Control of nonlinear systems, Neural Networks and Fuzzy Logic, Real-Time Implementation, Robotics and Artificial Intelligence.

    Cong Wang received the B.E. and M.E. degrees from Department of Automatic Control, Beijing University of Aeronautic & Astronautics, China, in 1989 and 1997, respectively. He has recently finished his Ph.D. studies from the Department of Electrical & Computer Engineering, National University of Singapore. He is currently a postdoctoral fellow at the Centre for Chaos Control and Synchronization, City University of Hong Kong. His research interest includes adaptive neural control, neural networks, chaos control and synchronization, and control applications.

    This paper was presented at the 5th IFAC Symposium on Nonlinear Control Systems, St. Petersburg, Russia, in July 2001. This paper was recommended for publication in revised form by Associate Editor Antonio E. de Barros Ruano under the direction of Editor Frank L. Lewis.

    1

    Present address: Department of Electronic Engineering, City University of Hong Kong.

    View full text