Elsevier

Automatica

Volume 37, Issue 3, March 2001, Pages 419-428
Automatica

Brief Paper
Nonlinear learning control for a class of nonlinear systems

https://doi.org/10.1016/S0005-1098(00)00165-5Get rights and content

Abstract

Based on the Lyapunov's direct method, a new learning control design is proposed. The proposed technique can be applied in two ways: it is either the standard backward recursive design or its extension. In the first case, the design yields a class of learning control with a difference learning law, under which the class of nonlinear systems is guaranteed to be asymptotically stable with respect to the number of trials in performing repeated tasks. However, implementation of the difference learning control requires derivative measurement of the state for guaranteed stability and performance, as required by most of the existing linear learning control laws. To overcome this difficulty, the proposed design extends the recursive design by employing a new state transformation and a new Lyapunov function, and it yields a class of learning control with a difference-differential learning law. Compared with the existing design methods most of which are based on linear analysis and design, the extension not only guarantees global stability and good performance but also removes such limitations as derivative measurement, Lipschitz condition, and resetting of initial conditions. In addition, the proposed design does not rely on the property of a system under consideration such as the input–output passivity.

Introduction

In many control applications such as robotics and automation, one of important issues is to design control systems that achieve trajectory tracking with acceptable accuracy. Often, tracking error systems are nonlinear and contain unknown parameters or time functions. If the desired trajectory is periodic or repetitive, iterative learning control can be used to improve system performance. The intuition behind this approach is periodicity of the repeated tasks, although other functions with a given and known characteristic may be learned. From trial to trial, all periodic time functions remain to be constant at any fixed instant of local time. So, a learning control if properly designed should be able to learn constants since constants are simplest form of unknowns. Through learning unknown parameters or time functions, learning control can compensate linear as well as nonlinear dynamics so that tracking performance can be enhanced.

There have been many results reported on learning control design. A recent discussion on history and various approaches of learning control can be found in Moore (1993). In model-based learning control, there are two major approaches. First, Arimoto and his coworkers (Arimoto, Kawamura, & Miyazaki 1984a, Arimoto, Kawamura, & Miyazaki 1984b; Arimoto, Kawamura, Miyazaki, & Tamaki, 1984; Kawamura, Miyazaki, & Arimoto, 1985) proposed a learning control design that updates its learning contribution from trial to trial. This approach achieves asymptotic zero tracking error by requiring derivative feedback of the state and Lipschitzian condition and by assuming the same initial conditions for all trials. Other schemes that are similar in essence to Arimoto's framework are: a high-gain, model reference adaptive control approach (Bondi, Casalino, & Gambardella, 1988); generalized inversion of input matrix (Hauser, 1987); linear high-gain robust control (Miller III, Glanz, & Kraft, 1987; Qu, Dorsey, Dawson, & Johnson, 1993) and robustness analysis under disturbance (Heinzinger, Fenwick, Paden, & Miyazaki, 1992); It was shown in Sugie and Ono (1991) that, if there is a direct transmission term from input to output, derivative measurement of the state is not needed. Removal of acceleration measurement were also achieved for second-order vector systems in Kuc, Lee, and Nam (1992) with Lipschitz condition and constant bound on the time derivation of the inertia matrix, and in Qu and Zhuang (1993) through non-differentiable nonlinear robust control. The second approach is the so-called adaptive learning control scheme in which an adaptation law is designed in a similar fashion as those in adaptive control. Learning controls designed using this method are updated not from trial to trial but continuously in time; for example, learning controllers proposed by Horowitz and his coworkers (Horowitz, Messner, & Moore, 1991; Messner, Horowitz, Kao, & Boals, 1991).

Despite of the progress accomplished, major limitations such as derivative feedback of the state, or resetting of initial condition, or Lipschitz condition, or their combinations remain for designing of a learning control for a class of nonlinear systems. The key to overcome these limitations and to account for nonlinear models of physical systems is to use nonlinear analysis and design tools. Recently, there was a new result that the asymptotic stability can be achieved using the input–output passivity of robotic systems without the limitations mentioned previously (Arimoto, 1996). However, it is required to generalize the design of a learning control so that it may not depend on the property of a system. For the extension, among various nonlinear methods, the Lyapunov second approach (Khalil, 1992; Rouche, Habets, & Laloy, 1977; Slotine & Li, 1991) stands out due to its universal applicability and physical implications. To successfully apply the Lyapunov method, one must find an appropriate Lyapunov function candidate using which control can be designed to ensure stability. In this paper, a new nonlinear learning design is presented, and it is an extension of the backward recursive (backstepping) design specifically improved for learning control design. This method allows us to extend the Arimoto's learning framework based on the Gronwall's inequality to the one that is based on the Lyapunov's direct method.

The proposed learning control design is applicable to the class of high-order nonlinear systems that are consisted of finite cascaded subsystems, and they include many of electrical–mechanical systems such as robots, electric motors and drivers, etc. Application results to robots (both simulation and experimentation results) can be found in Ham, Qu, and Park (1994), Ham, Qu, and Johnson (2000), Ham and Qu (1997) and Qu and Dawson (1996). However, it is the main purpose of this paper to present the nonlinear design framework of learning control in a mathematically general setting.

The proposed learning control scheme contains two parts: a feedforward/feedback part and a learning part. The latter is described either by a difference equation or by a differential-difference equation. The learning control is designed to be robust in the sense that it ensures global stability in the presence of unknown dynamics in the system so long as the unknown is bounded by a known nonlinear function of the state.

This paper is organized as follows. In Section 2, nonlinear learning control design method is introduced. The new approach is illustrated by an example in Section 3. Conclusions are made in Section 4.

Section snippets

Nonlinear learning control

In this section, a learning control design is introduced to achieve asymptotic stabilization for a class of cascaded nonlinear systems in the form that, for i=1,…,m−1,żi,j=Ai,j(z1,j,…,zi,j,t)+Bi,j(z1,j,…,zi−1,j,t)zi+1,jandżm,j=Am,j(z1,j,…,zm,j,t)+Bm,j(z1,j,…,zm−1,j,t)uj,where zi,jRn is the state of the ith subsystem, ujRn is the control variable, subscripts i and j are the indices of subsystems and learning trials, respectively.

Systems in the above class may have both unknown time-varying

Illustrative example

Consider a second-order systemż1,j=a1(t)z1,j2+z2,j,ż2,j=a2(t)(1+z21,jz2,j)+a3uj(t),where subscript j is the index of learning trials, z1,j and z2,j are state variables, and uj(t) is the control input, a1(t) and a2(t) are periodic time functions whose magnitudes are bounded by 1, and a3 is an unknown constant bounded as 1≤a3≤2.

Based on the formulation in the previous section, the state of the first subsystem should be defined to be the output tracking error as x1,j=z1,jzd, where zd is a given

Conclusion

Since most physical systems are nonlinear and perfect knowledge of their dynamics are usually unavailable, control design should be both nonlinearly based and robust. Most existing learning control scheme are based on Lipschitz condition and, in addition, they require derivative measurement of the state and resetting of initial conditions. The proposed nonlinear learning control design not only extends the backward recursive (backstepping) design but also overcomes the shortcomings of the

Chan Ho Ham received his Ph.D. from the Department of Electrical and Computer Engineering at the University of Central Florida in 1995. From 1996 to 1998, he worked as a lead engineer with the Satellite Business Division at the Hyundai Electronics Industries. He is currently working as an assistant professor with the Florida Space Institute and the Department of Mechanical, Materials, and Aerospace Engineering at the University of Central Florida. His main research interests include learning

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  • Cited by (0)

    Chan Ho Ham received his Ph.D. from the Department of Electrical and Computer Engineering at the University of Central Florida in 1995. From 1996 to 1998, he worked as a lead engineer with the Satellite Business Division at the Hyundai Electronics Industries. He is currently working as an assistant professor with the Florida Space Institute and the Department of Mechanical, Materials, and Aerospace Engineering at the University of Central Florida. His main research interests include learning control of nonlinear uncertain systems, fuzzy control, robust control and their applications to space systems and mechtronic systems.

    Zhihua Qu was born in Shanghai, China in 1963. He received his B.Sc. and M.Sc. degrees in electrical engineering from the Changsha Railway Institute in 1983 and 1986, respectively. From 1986 to 1988, he worked as a faculty member at the Changsha Railway Institute. He received his Ph.D. degree in electrical engineering from the Georgia Institute of Technology in 1990. Since then, he has been with the Department of Electrical and Computer Engineering at the University of Central Florida. Currently, he is the CAE/LINK Distinguished Professor in the College of Engineering and the Director of Electrical Engineering Program. His main research interests are nonlinear control techniques, robotics, and power systems. He has published 72 refereed journal papers in these areas and is the author of two books, Robust Control of Nonlinear Uncertain Systems by Wiley Interscience and Robust Tracking Control of Robotic Manipulators by IEEE Press. He is presently serving as an Associate Editor for Automatica and for International Journal of Robotics and Automation. He is a senior member of IEEE.

    Joseph Kaloust received his B.S.E.E, M.S.E.E, and Ph.D from the University of Central Florida in 1991, 1992, and 1995, respectively. After graduating from UCF, Joseph joined Lockheed Martin Missiles and Fire Control in Dallas, Texas as a flight dynamics and controls engineer. Dr. Kaloust joined Hope College the fall of 2000 where he is currently employed. Research area of interests include: Nonlinear control, adaptive control, robotics, and flight control systems.

    This paper was not presented at any IFAC meeting. This paper was recommended for publicaiton in revised form by Acssociate Editor H. K. Khalil under the direction of Editor Tamer Basar. This work is supported in part by U.S. National Science Foundation under grant MSS-9110034.

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