On stability and robustness of nonlinear cascaded systems - Application to mechanical systems

Abstract : We present new tools for stability and robustness analysis of nonlinear dynamical systems. We provide a precise Lyapunov framework for uniform semiglobal and practical asymptotic stability. ``Semiglobal'' refers to the situation when the domain of attraction is not the whole state-space but, instead, a compact set that may be arbitrarily enlarged by a convenient tuning of parameters. ``Practical'' concerns the case when an arbitrarily small compact neighborhood of the origin (instead of the origin itself) is asymptotically stable. As opposed to many related concepts, they allow the estimate of solutions to depend on the tuning parameter and so, potentially, on the radius of the desired domain of attraction and the amplitude of the tolerated steady-state error. Compared to classical results for global asymptotic stability, this feature requires to impose an additional requirement on the bounds on the Lyapunov function. We illustrate the importance of this condition by showing that, when the latter is violated, no stability property is ensured. We also derive a converse Lyapunov result for the class of USPAS systems whose solutions' estimate is independent of the radius of the attractive ball. The generated Lyapunov function is especially designed to fit the context of cascaded systems as its gradient is bounded by a time-invariant function. With the proposed Lyapunov framework for semiglobal and practical asymptotic stability, some tools are presented that ensure the preservation of these stability properties by cascade interconnection. In general terms, similarly to existing results for global asymptotic stability, it is required that the solutions of the overall cascade be bounded and that a convenient Lyapunov function be explicitly known for the driven subsystem. In view of the converse result we establish, we relax this latter requirement in the semiglobal case for a wide class of systems. This is particularly useful when invoking averaging techniques, as illustrated by the output feedback control of the double integrator affected by a persistently exciting signal. Furthermore, in the case of uniform global practical asymptotic stability, the boundedness assumption on the solutions of the cascade is replaced by growth restriction on the interconnection term. This makes it easy to apply in specific problems. We illustrate its use by quantifying the effect of smoothing $\sign$ functions in disturbance rejection. We show that, if some (non necessarily compact) sets are globally asymptotically stable (GAS) for two subsystems taken separately, then their cross product is GAS for the corresponding cascade provided that its solutions are globally bounded. On some occasions, this requirement can be replaced by a simple growth order condition on the interconnection term (plus forward completeness). This work includes, as a special case, partial stability for cascades. As an illustration, we provide a concise proof for a recently established result of formation control of surface vessels along a straight path and with a prescribed velocity. We analyze the stability of cascaded systems with inputs by providing sufficient conditions under which integral input to state stability is preserved by cascade interconnection. These conditions are first expressed in Lyapunov terms and then in terms of estimates of the solutions of each subsystem taken separately. We illustrate the significance of our theoretical findings by solving specific open problems in the field of mechanical systems. We proceed to the robustness analysis of PID-controlled manipulators to friction, model uncertainty, actuators' dynamics, etc. Another application concerns the formation control of spacecrafts. We establish global practical asymptotic stability of the corresponding system when only bounds on the leader's anomaly are available. Finally, we show that a similar stability property can be obtained for the synchronization of two surface vessels with little information on the leader vehicle.
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Automatic. Université Paris Sud - Paris XI, 2006. English


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Submitted on : Wednesday, May 5, 2010 - 2:10:59 PM
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Antoine Chaillet. On stability and robustness of nonlinear cascaded systems - Application to mechanical systems. Automatic. Université Paris Sud - Paris XI, 2006. English. <tel-00480931>

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