Brief PaperAdaptive stabilization of uncertain nonholonomic systems by state and output feedback☆
Introduction
Due to Brockett's theorem (Brockett, 1983), it is well known that nonholonomic systems with restricted mobility cannot be stabilized to a desired configuration (or posture) via differentiable, or even continuous, pure-state feedback, although it is controllable. A number of approaches have been proposed for the problem, which can be classified as (i) discontinuous time-invariant stabilization (Astolfi, 1996), (ii) time-varying stabilization (Walsh & Bushnell, 1995; Samson, 1993) and (iii) hybrid stabilization (Sordalen & Canudas de Wit, 1995; Canudas de Wit, Berghuis, & Nijmeijer, 1994). See the survey paper Kolmanovsky and McClamroch (1995) for more details and references therein.
One commonly used approach for controller design of nonholonomic systems is to convert, with appropriate state and input transformations, the original systems into some canonical forms for which controller design can be carried out easier (Murray & Sastry, 1993; M'Closkey & Murray, 1992; Huo & Ge, 2001). Using the special algebra structures of the canonical forms, various feedback strategies have been proposed to stabilize nonholonomic systems in the literature (Astolfi, 1996; Ge, Sun, Lee, & Spong, 2001a; Murray, 1993; Sun, Ge, Huo, & Lee, 2001; Jiang & Nijmeijer, 1999). Recently, adaptive control strategies were proposed to stabilize the dynamic nonholonomic systems with modeling or parametric uncertainties (Colbaugh, Barany, & Glass, 1996; Ge, Wang, Lee, & Zhou, 2001b). Neural network control was applied to obtain practical point stabilization solution for a nonholonomic mobile robot with uncertainty (Fierro & Lewis, 1995). Hybrid control based on supervisory adaptive control was presented to globally asymptotically stabilize a wheeled mobile robot (Hespanha, Liberzon, & Morse, 1999). Adaptive state feedback control was considered in Do and Pan (2002) using input-to-state scaling. It should be noticed that all these papers are concerned with state-feedback control. Output feedback tracking and regulation were presented in Dixon, Dawson, Zergeroglu, and Behal (2001) for practical wheeled mobile robots. In Jiang (2000), robust exponential regulation for nonholonomic systems with input and state-driven disturbances was presented under the assumption that the bounds of the disturbances are known.
This paper addresses the problem of stabilization of a class of nonholonomic systems in chained form with drift nonlinearity and parameter uncertainties. The main contributions of this paper are listed as follows:
- (i)
adaptive state feedback stabilization using state scaling and backstepping is developed without imposing any restriction on the system order and the growth of the drift nonlinearities;
- (ii)
new adaptive switching is designed to handle the technical problem of uncontrollability at x0(t0)=0, which prevents the possible finite escape of system states, and at the same time guarantees the boundedness of all the signals in the system; and
- (iii)
output feedback stabilization of a class of uncertain nonholonomic systems is provided when only partial system states are measurable with an adaptive nonlinear observer to solve the problem of the regressor matrix involving unmeasured states.
Section snippets
Problem formulation
The nonholonomic systems in a chained form have been formulated as a canonical form for many nonlinear mechanical systems with nonholonomic constraints on velocities. The purpose of this paper is to deal with a class of perturbed canonical nonholonomic systems as follows:where are system states, , u0 and u1 are control inputs, φ0(x0)∈Rl and are vectors of
State feedback control
When full state (x0,x) is available, the class of systems which can be handled is much larger than that for output feedback control. A full characterization of the class of systems (1) is given by the following assumption on φ0(x0) and , 1⩽i⩽n. Assumption 1 For φ0, there is a known smooth function vector ϕ0 such thatFor 1⩽i⩽n, there are some known smooth function vectors , such that
The nonlinearities satisfy the triangularity
Output feedback control
When only the system output is measurable and the rest of the system states are not available for feedback, we need to estimate them. It is found that when the system uncertainties are in the linear-in-parameters (LIP) form, a more stringent condition has to be imposed to make the output feedback stabilization problem solvable. Assumption 2 x0-subsystem in system (1) owns a special structure, i.e. , where c0 is known. For each 1⩽i⩽n, there is a known smooth function vector such that
Simulation results
In this section, the bilinear model of a mobile robot with small angle measurement error (Morin, Pomet, & Samson, 1998) is simulated as it is a practical system, which is described byBy using the transformation x0=xl, x1=yl, x2=θl+ε, u0=v, u1=w, it becomeswhich is in the form as discussed in Remark 5.
Due to space limitation, state feedback control is omitted. Assuming that x2 is unmeasured, the controller design procedure in Section 4 is
Conclusion
In this paper, constructive adaptive state feedback control has been presented for stabilizing a class of uncertain nonholonomic chained systems without imposing any restriction on the system order and the growth of the drift nonlinearities. A new switching control strategy has been proposed. Observer-based output feedback control has been proposed when only partial system states are measurable. An adaptive observer has been constructed to handle the technical problem due to the presence of
Acknowledgements
The authors would like to thank all the constructive comments from the Editor, the Associate Editor and anonymous reviewers.
Shuzhi Sam 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 1992 to 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
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Shuzhi Sam 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 1992 to 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 as an Associate Professor. He has authored and co-authored over 100 international journal and conference papers, two monographs, and co-invented two patents. He has been serving as an Associate Editor, IEEE Transactions on Control Systems Technology since 1999, and a member of the Technical Committee on Intelligent Control of the IEEE Control System Society since 2000. He was the recipient of the 1999 National Technology Award, 2001 University Young Research Award, and 2002 Temasek Young Investigator Award, Singapore. He serves as a technical consultant local industry. His current research interests are Nonlinear Control, Neural Networks and Fuzzy Logic, Robotics and Real-Time Implementation.
Zhuping Wang received the B.Eng. and the M.Eng. in Department of Automatic Control in 1994 and 1997, respectively, both from Northwestern Polytechnic University, P.R. China. In 1999, she joined the National University of Singapore and finished her Ph.D. study in 2002. She is currently a research engineer in the Department of Electrical & Computer Engineering. Her research interests include control of flexible link robots and smart materials robot control, control of nonholonomic systems, adaptive nonlinear control, neural network control and robust control.
T. H. Lee received the B.A. degree with First Class Honours in the Engineering Tripos from Cambridge University, England, in 1980; and the Ph.D. degree from Yale University in 1987. He is a Professor in the Department of Electrical and Computer Engineering at the National University of Singapore. He is also currently Head of the Drives, Power and Control Systems Group in this Department; and Vice-President and Director of the Office of Research at the University.
Dr. Lee's research interests are in the areas of adaptive systems, knowledge-based control, intelligent mechatronics and computational intelligence. He currently holds Associate Editor appointments in Automatica; the IEEE Transactions in Systems, Man and Cybernetics; Control Engineering Practice (an IFAC journal); the International Journal of Systems Science (Taylor and Francis, London); and Mechatronics Journal (Oxford, Pergamon Press).
Dr. Lee was a recipient of the Cambridge University Charles Baker Prize in Engineering. He has also co-authored three research monographs, and holds four patents (two of which are in the technology area of adaptive systems, and the other two are in the area of intelligent mechatronics).
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This paper was not presented at any IFAC meeting. This paper was recommended for publication by Associate Editor Henk Nijmeijer under the direction of Editor Hassan Khalil.