Abstract
In this paper, we study the problem of exponential stability of impulsive cascaded systems. In particular, we provide some sufficient conditions that guarantee the exponential stability of the cascaded systems, provided that the two subsystems are also exponentially stable. The proof of the stability of the cascaded systems is based on the second Lyapunov method and the existence of converse theorems for the stability of impulsive systems. Finally, the usefulness of our results is illustrated by its application to the problem of trajectory tracking for a wheeled robot.
Highlights
INTRODUCTIONThe present study of cascaded impulsive systems finds its motivation for two main reasons: the importance of impulsive systems and the importance of cascaded systems
The present study of cascaded impulsive systems finds its motivation for two main reasons: the importance of impulsive systems and the importance of cascaded systems.First, it is known that many biological systems, optimal control models in economics, theoretical physics, ecology, and industrial robotics have a sudden change in the form of disturbances in their states [1]–[4]
Contrary to what we mentioned above, in Theorem 5 the boundedness of the solutions is not a necessary condition to guarantee the convergence of the cascaded system; the condition imposed on the interconnection term g(t, x) between the two subsystems is more general than condition (20), and than the one mentioned in [37]
Summary
The present study of cascaded impulsive systems finds its motivation for two main reasons: the importance of impulsive systems and the importance of cascaded systems. For the case of a class of cascaded nonautonomous systems, it was shown in [37]; that a cascaded nonautonomous system is globally uniformly exponentially stable if and only if each isolated subsystem is globally uniformly exponentially stable This statement fails, for example, for asymptotic stability properties, an additional property on the boundedness of solutions can be added to recover the asymptotic stability for the entire system. In [41], a partial state feedback controller design scheme was considered for the study of the global stabilization problem for a class of cascaded nonlinear systems with a time-varying delay. The present work is an attempt to lay a foundation for the study of the exponential stability of cascaded impulsive systems and to explore the eventual benefits of their application in the impulsive control of continuous systems.
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