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Erfurt, Germany

Ito H.,Kyushu Institute of Technology | Dashkovskiy S.,FH Erfurt | Wirth F.,University of Wrzburg
Automatica | Year: 2012

This paper addresses the problem of verifying stability of networks whose subsystems admit dissipation inequalities of integral input-to-state stability (iISS). We focus on two ways of constructing a Lyapunov function satisfying a dissipation inequality of a given network. Their difference from one another is elucidated from the viewpoint of formulation, relation, fundamental limitation and capability. One is referred to as the max-type construction resulting in a Lipschitz continuous Lyapunov function. The other is the sum-type construction resulting in a continuously differentiable Lyapunov function. This paper presents geometrical conditions under which the Lyapunov construction is possible for a network comprising n<2 subsystems. Although the sum-type construction for general n>2 has not yet been reduced to a readily computable condition, we obtain a simple condition of iISS small gain in the case of n=2. It is demonstrated that the max-type construction fails to offer a Lyapunov function if the network contains subsystems which are not input-to-state stable (ISS). © 2012 Elsevier Ltd. All rights reserved.

Dashkovskiy S.,FH Erfurt | Ito H.,Kyushu Institute of Technology | Wirth F.,University of Wurzburg
European Journal of Control | Year: 2011

In this paper we consider the stability of networks consisting of nonlinear ISS systems supplied with ISS Lyapunov functions defined in dissipative form. The problem of constructing an ISS Lyapunov function for the network is addressed. Our aim is to provide a geometrical condition of a small gain type under which this construction is possible and to describe a method of an explicit construction of such an ISS Lyapunov function. In the dissipative form, the geometrical approach allows us to discuss both Lipschitz continuous construction and continuously differentiable construction of ISS Lyapunov functions. © 2011 EUCA.

Polushin I.G.,University of Western Ontario | Dashkovskiy S.N.,FH Erfurt | Takhmar A.,University of Western Ontario | Patel R.V.,University of Western Ontario
Automatica | Year: 2013

For cooperative force-reflecting teleoperation over networks, conventional passivity-based approaches have limited applicability due to nonpassive slave-slave interactions and irregular communication delays imposed by networks. In this paper, a small gain framework for stability analysis design of cooperative network-based force reflecting teleoperator systems is developed. The framework is based on a version of weak input-to-output practical stability (WIOPS) nonlinear small gain theorem that is applicable to stability analysis of large-scale network-based interconnections. Based on this result, we design a cooperative force-reflecting teleoperator system which is guaranteed to be stable in the presence of multiple network-induced communication constraints by appropriate adjustment of local control gains and/or force-reflection gains. Experimental results are presented that confirm the validity of the proposed approach. © 2012 Elsevier Ltd. All rights reserved.

Dashkovskiy S.,FH Erfurt | Kosmykov M.,University of Bremen
Automatica | Year: 2013

We consider the interconnections of arbitrary topology of a finite number of ISS hybrid systems and study whether the ISS property is maintained for the overall system. We show that if the small gain condition is satisfied, then the whole network is ISS and show how a non-smooth ISS-Lyapunov function can be explicitly constructed in this case. © 2013 Elsevier Ltd. All rights reserved.

Dashkovskiy S.,FH Erfurt | Mironchenko A.,University of Bremen
Mathematics of Control, Signals, and Systems | Year: 2013

We develop tools for investigation of input-to-state stability (ISS) of infinite-dimensional control systems. We show that for certain classes of admissible inputs, the existence of an ISS-Lyapunov function implies the ISS of a system. Then for the case of systems described by abstract equations in Banach spaces, we develop two methods of construction of local and global ISS-Lyapunov functions. We prove a linearization principle that allows a construction of a local ISS-Lyapunov function for a system, the linear approximation of which is ISS. In order to study the interconnections of nonlinear infinite-dimensional systems, we generalize the small-gain theorem to the case of infinite-dimensional systems and provide a way to construct an ISS-Lyapunov function for an entire interconnection, if ISS-Lyapunov functions for subsystems are known and the small-gain condition is satisfied. We illustrate the theory on examples of linear and semilinear reaction-diffusion equations. © 2012 Springer-Verlag London Limited.

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