In this joint research project, we show that measure differential inclusions with certain maximal monotonicity conditions exhibit the convergence property. A system, which is excited by an input, is called convergent if it has a unique solution that is bounded on the whole time axis and this solution is globally asymptotically stable. Obviously, if such a solution does exist, then all other solutions converge to this solution, regardless of their initial conditions, and can be considered as a steady-state solution.

The property of convergence can be beneficial from several points of view. Firstly, in many control problems it is required that controllers are designed in such a way that all solutions of the corresponding closed-loop system “forget” their initial conditions. In this case, all solutions converge to some steady-state solution that is determined only by the input of the closed-loop system. This input can be, for example, a command signal or a signal generated by a feedforward part of the controller or, as in the observer design problem, it can be the measured signal from the observed system. Such a convergence property of a system plays an important role in many nonlinear control problems including tracking, synchronization, observer design, and the output regulation problem. Secondly, from a dynamics point of view, convergence is an interesting property because it excludes the possibility of different coexisting steady-state solutions: namely, a convergent system excited by a periodic input has a unique globally asymptotically stable periodic solution.

We present theorems which give sufficient conditions for the uniform convergence of measure differential inclusions with certain maximal monotonicity properties. The framework of measure differential inclusions allows us to describe systems with state discontinuities. Moreover, we illustrate how these convergence results for measure differential inclusions can be exploited to solve tracking problems for certain classes of non-smooth mechanical systems with friction and one-way clutches. Illustrative examples of convergent mechanical systems are discussed in detail.

Motor-load configuration and tracking control with impulsive inputs and one-way clutch

Publications:

Leine, R.I. & van de Wouw, N. Stability and Convergence of Mechanical Systems with Unilateral Constraints, Lecture Notes in Applied and Computational Mechanics Vol. 36, Berlin Heidelberg New-York, Springer-Verlag, 2008. ISBN: 978-3-540-76974-3

Leine. R.I. and van de Wouw, N: "Uniform convergence of monotone measure differential inclusions: with application to the control of mechanical systems with unilateral constraints" International Journal of Bifurcation and Chaos, Vol 15, No 5, pp 1435-1457, 2008 PDF (830kb)

Van de Wouw, N. and Leine R.I., "Tracking control for a class of measure differential inclusions", Proceedings of the 47th IEEE Conference on Decision and Control (CDC2008), Cancun, Mexico, 2008. PDF (466kb)

Van de Wouw, N. and Leine R.I., "Convergence properties of monotone measure differential inclusions." Proceedings of the 6th EUROMECH Nonlinear Dynamics Conference (ENOC2008), St. Petersburg, Russia, 2008. PDF (135kb)

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09/12/11 | Remco Leine | ZfM | ETH