Infinite-Dimensional Feedback Systems: the Circle Criterion and Input-to-State Stability

Abstract

An input-to-state stability theory, which subsumes results of circle criterion type, is developed in the context of a class of infinite-dimensional systems. The generic system is of Lur'e type: a feedback interconnection of a well-posed infinite-dimensional linear system and a nonlinearity. The class of nonlinearities is subject to a (generalized) sector condition and contains, as particular subclasses, both static nonlinearities and hysteresis operators of Preisach type.

Stability Analysis and Stabilization of Systems With Input Backlash

“…These conditions are given in terms of Linear Matrix Inequalities (LMI), which can be solved using standard numerical algorithms, within a time of computation which is polynomial with respect to the dimension of the data. Even if the spirit of the approach proposed is quite similar to the SISO statement in Corollary 16 in [12], it allows to deal with MIMO systems. Moreover, while both [12] and our paper address the characterization of the finite-time attractor, our approach, differently from [12], uses the knowledge of the Lyapunov function.…”

“…Even if the spirit of the approach proposed is quite similar to the SISO statement in Corollary 16 in [12], it allows to deal with MIMO systems. Moreover, while both [12] and our paper address the characterization of the finite-time attractor, our approach, differently from [12], uses the knowledge of the Lyapunov function. In this sense, our approach can be considered as complementary with the method proposed in [12].…”

“…Moreover, while both [12] and our paper address the characterization of the finite-time attractor, our approach, differently from [12], uses the knowledge of the Lyapunov function. In this sense, our approach can be considered as complementary with the method proposed in [12]. Furthermore, at the opposite of conditions developed in [24], we do not need to use the time-derivative version of the system.…”

“…However, although there is a large literature dealing with the stability analysis and the estimation of input/output properties (see e.g. [12], [20] for recent results), there are not many papers suggesting design methods for nonlinear systems with memory operators, except [13] where a dissipativity property is used, and [23] where the dynamics of the differential equation modeling a one-dimensional hysteresis is controlled.…”

On the new notion of input-to-state safety

“…There are many tools available for analyzing the robustness of systems' stability, including, H ∞ and L 2 -stability theories [17], [6], absolute stability theory [8], input-to-state stability (ISS) theory [19] and many others. However, analogous tools for systems' safety are still minimal in literature which makes it difficult to carry out robustness analysis to the aforementioned works that deal with the problem of stabilization with guaranteed safety.…”