Stability of first order discrete time systems with time-variant communication delays in the feedback path

Abstract

This paper addresses the problem of stability in 1st order systems with time-variant communication delays in the feedback path. Sets of necessary and sets of SUA;-cient conditions are obtained and resulting stability regions are shown. I. IntroductionData networks play an increasingly important role in the development of technology in the 21St century.With improved access to high speed data networks, many engineering systems will be embedded in networks that can support fast feedback loops. This paper takes a close look at the simple case of a first order system with proportional feedback through a communication network. The problem of stability in discrete time systems with time-variant delay has been addressed in only a few publications [1]-[4], even though its importance is rising rapidly. The derived conditions for stability are either conservative and relatively easy to test [1]-[4], or they are necessary and sufficient conditions but NP hard to test PI 7 vi , 191.This paper attempts to close the gap between existing N P hard necessary and sufficient stability conditions and conservative sufficient stability conditions. This is achieved by deriving sets of sufficient conditions as well as sets of necessary conditions, thus tightly bounding the actual stability region. Time variant delays in discrete-time systemsA. The nature of a discrete time-variant delay One important characteristic of time-variant discrete delays is that it maps a countable sequence of input values into a countable sequence of sets of output values. Hence, the output set of a time-variant discrete delay operator can contain zero, one or several values.For this reason, it is necessary to add an interface at the delay output. We can distinguish between two components of a time-variant linear network connection: a delay introduced by the communication link, (i.e. the delay itself) and the interface that connects the network to the system at the receiving end. Figure 1 shows the two components of the time-variant linear network connection. The input and the output of the time-variant delay d(n) are the sample sets u(n) and U ( . ) . The symbol n describes discrete time with respect to an arbitrary sampling period. The sets U ( . ) and v(n) describe all input/output samples (scalar or otherwise) that are entering or leaving the block d(n) within a sampling period, i.e. between time instant n-1 and time instant n. The output signal is a sequence of sets satisfying the following relation: Figure 1: A discrete time-variant delay We will use the following notation: 0 M ( n ) : The cardinality of the input u(n) at time n; 0 N ( n ) : The cardinality of the output v(n) at time instant n; 0 d ( n ) : a function d : Z+ -+ Z+ where 2, is the set of nonnegative integers satisfying the inequality The function d ( n ) represents the delay the communication link introduces at time instant n for the input U ( . ) .

Sistemas de control basados en red. Modelado y diseño de estructuras de control.

“…Otro grupo de investigadores, liderado por el Dr. Peter H. Bauer en la University of Miami (EEUU) y el Dr. Kamal Premaratne en la University of Notre Dame (EEUU), ha realizado importantes contribuciones al problema de los SCBR ( [Bau98], [Bau99a], [Bau99b], [Lor00], [Bau01a], [Bau01b], [Bau01c], [Sic01]). Sus trabajos se centran en el estudio de los SCBR con retrasos variables pero conocidos, considerando el caso de retrasos arbitrariamente grandes, demostrando que estos no son necesariamente perjudiciales para la estabilidad del sistema.…”