A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems

Abstract

In this research, we developed a new discontinuous output feedback tracking controller for a class of uncertain, nonlinear, multi-input/multi-output, mechanical systems whose dynamics are first-order differentiable. A novel filter design and Lyapunov-type stability analysis are used to prove semi-global asymtotic tracking. As a by-product of the proposed framework, we also present the design of a new simple, discontinuous velocity observer that ensures global asymptotic velocity observation.

Introduction

Over the years, mechanical systems have served as an interesting benchmark for the design and validation of novel nonlinear control strategies. One problem that has attracted a good deall of interest is the output feedback control of mechanical systems. This problem is of practical importance since many commercially-available mechanical systems are not commonly equipped with velocity sensors (e.g., industrial robots); hence, full access to the system states is impossible. From a theorretical point-of-view, the challenge lies in the fact that, generally speaking, the separation principle does not hold for nonlinear systems. For these reasons, a considerable amount of work has been devoted to the problem of output feedback control of mechanical systems. The existing solutions can be distinguished by the mechanism used to estimate the unnmeasured velocity signal--model-based observers or filters (i.e., high-gain observers). Model-based observers estimate velocity by mimicking the system dynamics, while filters approximate the behavior of a differentiator over a range of frequencies (e.g., a lead compensator).

The choice of estimation mechanism is strongly influenced by the existence of uncertainty in the system model. Whereas model-based observers are usually restricted to cases where the model is exactly known, filters can provide a model-independent means of estimating velocity. Output feedback controllers using model-based observers were reported by Berghuis and Nijmeijer(1993), Erlic and Lu (1992), Lim, Dawson, and Anderson (1996), Nicosia and Tomei (1990). Output feedback controllers using various types of filters were proposed by Burg, Dawson, Hu, and de Queiroz (1996), Canudas de Wit and Fixot (1992), Kaneko and Horowitz (1997), Lee and Khalil (1997), Yuan and Stepanenko (1991), Zergeroglu, Dawson de Queiroz and Krstic (2000), Zhang, Dawson, de Queiroz, and Dixon (2000). The reader is referred to Zergeroglu et al. (2000) for a more detailed literature review. Since this paper is specifically concerned with the output feedback control of uncertain mechanical systems using a discontinuous control, some results that targeted this problem froma similar perspective are the following. In Hsu and Lizarralde (1993), a combined, variable structure/model reference adaptive control scheme was used to circumvent the lack of velocity measurements in robot manipulators. In Oh and Khalil (1995), a discontinuous control with high-gain observer was proposed to stabilize a class of multivariable nonlinear systems. This result was later extended in Oh and Khalil (1997) to the tracking problem for single-input/single-output, feedback-linearizable, minimum-phase nonlinear systems with unknown disturbance.

In this research, we propose a new output feedback discontinuous tracking controller for a general class of nonlinear, multi-input/multi-output (MIMO), mechanical (i.e., second-order) systems whose uncertain dynamics are first-order differentiable. The proposed control design is founded on ideas from our previous work in Burg et al. (1996) and Xian, Dawson, and de Queiroz (2003b). Specifically, we adapt the rationale behind the filter design in Burg et a. (1996) to the control design procedure presented in Xian et al. (2003b) with the purpose of designing an output feedback controller. What emerges is a novel second-order filter and a discontinuous control law, dependent only on position measurements, which ensure semi-global asymptotic tracking with very limited knowledge of the system dynamics. To the best of our knowledge, the proposed approach yields the first output feedback, asymptotic tracking controller for the class of uncertain MIMO systems under consideration. As a by-product of the proposed framework, we also present a new simple, discontinuous velocity observer under the condition that the system nonlinearities and control input are unknown, first-order differentiable, and bounded. The observer is shown to ensure global asymptotic observation.

Conclusion

In this research, we designed a discontinuous controller that requires only output feedback for a general class of uncertain nonlinear MIMO mechanical systems. The class of systems is restricted to those whose dynamics are first-order differentiable. The novelty of this work lies in the approach used to synthesize the control law and velocity filter. Specifically, the form of these components emerged from a new error system development and Lyapunov-type stability analysis. The resulting output feedback control law was proven to semi-globally asymptotically track any bounded, third-order differentiable reference trajectory. As a secondary result, we also presented the design of a new discontinuous velocity observer for systems where the nonlinearities and control input are unknown, first-order differentiable, and bounded. The observer was proven to ensure the global asymptotic convergence of the velocity observation error.

Conference Papers

For more details on this research, please refer the followingconference paper:

  • B. Xian, M.S. de Queiroz, D.M. Dawson, M.L. McIntyre "A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems". (Submitted to the 2004 Conference on Control Applications)

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