Relaxed ISS Small-Gain Theorems for Discrete-Time Systems

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.Comment: input-to-state stability, Lyapunov methods, small-gain conditions, discrete-time non-linear systems, large-scale interconnection

A Class of Continuous Predefined-Time Controllers

A class of continuous predefined-time controllers is designed in this paper. The control structure is built upon a class of comparison functions, whose features allow to analyze the predefined-time convergence property in the Lyapunov framework. Rather than providing exact predefined-time convergence to the equilibrium point, the proposed controller guarantees uniform predefined-time ultimate predefined boundedness of the solutions, i.e., the capability of setting an arbitrarily desired ultimate bound and an arbitrarily desired convergence time, through an appropriate selection of the controller parameters. Moreover, for a class of second-order systems, an additional analysis is carried out to show finite-gain input-output stability. The reliability of the proposed scheme is highlighted through numerical simulations in a representative example.ITESO, A.C

Backstepping Design for the Predefined-Time Stabilization of Second-Order Systems

The backstepping design of a controller which stabilizes a class of second-order systems in predefined-time is studied in this paper. The origin of a dynamical system is said to be predefined-time stable if it is fixed-time stable and an upper bound of the settling-time function can be arbitrarily chosen a priori through an appropriate selection of the system parameters. The proposed backstepping construction is based on recently proposed Lyapunov-like sufficient conditions for predefined-time stability. Different from other approaches, the proposed backstepping design allows the simultaneous construction of a Lyapunov function which meets the conditions for guaranteeing predefined-time stability. A simulation example is presented to show the behavior of a developed controller, and to show its advantages against similar schemes.ITESO, A.C