Robust Adaptive Finite-Time Stabilization of Nonlinearly Parameterized Nonholonomic Systems
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Robust Adaptive Finite-Time Stabilization of Nonlinearly Parameterized Nonholonomic Systems Fangzheng Gao · Yanling Shang · Fushun Yuan
Received: 15 September 2011 / Accepted: 15 May 2012 / Published online: 26 May 2012 © Springer Science+Business Media B.V. 2012
Abstract This paper investigates the problem of adaptive finite-time stabilization of nonlinearly parameterized nonholonomic systems. By skilly using the parameter separation, input-state-scaling, and adding a power integrator techniques, an adaptive state feedback controller is obtained. Based on switching strategy to eliminate the phenomenon of uncontrollability, the proposed controller can guarantee that the system states globally finite-time converge to the origin, while other signals remain bounded. Simulation examples demonstrate the effectiveness and the robust features of the proposed approach. Keywords Nonholonomic systems · Nonlinear parameterization · Adding a power integrator · Finite-time stabilization Mathematics Subject Classification 93C10
1 Introduction In the past decades, the feedback stabilization of nonholonomic systems has attracted much attention in control field; see, for example, the survey papers [1] and the references cited therein for an interesting introduction to this expanding area. Due to the limitation of Brockett’s condition [2], it is well known that the nonholonomic systems cannot be stabilized by any static continuous state feedback, although it is controllable. To overcome this difficulty, a number of approaches have been proposed, which can be classified as discontinuous feedback, time-varying feedback and hybrid control laws [3–14]. It should be noted that the majority of these constructive methods have been developed around an important class of driftless nonholonomic systems in chained form, which was first introduced in [3]. As exF. Gao () · F. Yuan School of Mathematics and Statistics, Anyang Normal University, Anyang 455002, Henan Province, P.R. China e-mail: [email protected] Y. Shang Center of Computer, Anyang Normal University, Anyang 455002, Henan Province, P.R. China
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plained and illustrated in [1] and the references therein, many nonlinear mechanical systems with nonholonomic constraints on velocities can be transformed, either locally or globally, to chained form systems via coordinates and state-feedback transformation. For instance, we have seen such examples as tricycle-type mobile robots, cars towing several trailers, the knife edge, a vertical rolling wheel, and a rigid spacecraft with two torque actuators. In practical applications, what is important for a system design is robustness consideration against possible modeling errors and external disturbances. The robustness issue for the asymptotic and exponential stability properties has been extensively studied. For instance, a class of nonholonomic systems perturbed by strong nonlinear uncertainties was studied in [7]. Discontinuous state and output feedback controllers were designed and achieved globally exponential stability in
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