Backstepping Control of Systems with Backlash Nonlinearity
In this chapter, we consider uncertain dynamic systems preceded by unknown backlash nonlinearity. By using backstepping technique, new schemes for both state feedback and output feedback are proposed. Besides showing global stability of the system, the tr
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In this chapter, we consider uncertain dynamic systems preceded by unknown backlash nonlinearity. By using backstepping technique, new schemes for both state feedback and output feedback are proposed. Besides showing global stability of the system, the transient performance in terms of L2 norm of the tracking error is derived to be an explicit function of design parameters. For output feedback control we develop a new scheme for a class of uncertain linear systems preceded by unknown backlash nonlinearities. The controller designed by using backstepping technique consists of a new robust control law and a new estimator to estimate the unknown parameters. The result is also extended to nonlinear systems.
7.1 Introduction The development of control techniques to mitigate effects of unknown backlash has been studied for decades. Much of this interest is a consequence of its importance in present application. Interest in studying dynamic systems with backlash is motivated by their role as nonlinearities for which traditional control methods are insufficient and so requiring development of new approaches. Several adaptive control schemes have recently been proposed, see for examples [43, 44, 48, 55, 56, 120]. In [44], [55] and [56], an adaptive inverse cascaded with the plant was employed to cancel the effects of nonlinearity. In [43] a dynamic backlash model is defined to pattern a backlash rather than constructing an inverse model to mitigate the effects of the backlash. However in [43], the term multiplying the control and the uncertain parameters of the system must be within known intervals and the ‘disturbance-like’ term must be bounded by a known bound. Projection was used to handle the ‘disturbance-like’ term and unknown parameters. System stability was established and the tracking error was shown to converge to a residual. In [120], a state feedback backstepping design was developed to deal with the backlash nonlinearity, where the effect of backlash was treated as a bounded disturbance and an estimate was used to estimate its bound. J. Zhou & C. Wen: Adapt. Backstepping Ctrl. of Uncertain Systems, LNCIS 372, pp. 97–123, 2008. c Springer-Verlag Berlin Heidelberg 2008 springerlink.com
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Backstepping Control of Systems with Backlash Nonlinearity
7.2 State Feedback Control In this section, we develop two simple backstepping adaptive control schemes for the same class of nonlinear systems as in [43] and [120]. In the first scheme, a sign function is involved and this can ensure perfect tracking. To avoid possible chattering caused by the sign function, we propose an alternative smooth control law and the tracking error is still ensured to approach a prescribed bound in this case. In our design, the term multiplying the control and the system parameters are not assumed to be within known intervals. The bound of the ‘disturbancelike’ term is not required. To handle such a term, an estimator is used to estimate its bound. Besides showing global stability of the system, transient performance in terms of L2 norm of the t
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