Linear positive control systems on time scales; controllability
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Linear positive control systems on time scales; controllability Zbigniew Bartosiewicz
Received: 2 May 2012 / Accepted: 21 January 2013 / Published online: 8 February 2013 © The Author(s) 2013. This article is published with open access at Springerlink.com
Abstract Positive linear systems on arbitrary time scales are studied. The theory developed in the paper unifies and extends concepts and results known for continuoustime and discrete-time systems. A necessary and sufficient condition for a linear system on a time scale to be positive is presented. Properties of positive reachable sets are investigated and characterizations of various controllability properties are presented. A modified Gram matrix of the system is used to state necessary and sufficient condition of positive reachability of a positive system on an arbitrary time scale. Keywords Positive linear control system · System on time scale · Positive accessibility · Positive reachability · Gram matrix 1 Introduction In many applications the variables that appear in a mathematical description take only positive or nonnegative values. Examples of such systems can be found in [4,12,15,17], where also a theory of linear positive systems was developed. Usually the systems that are studied fall into two separate classes: continuous-time systems and discrete-time systems. In [17], all the problems are studied twice in these two separate settings. The characterizations of properties of positive systems for these two classes are sometimes similar, or even identical, and sometimes essentially distinct. Stefan Hilger in his Ph.D. thesis [16] started the most successful attempt to unify the theories of continuous-time systems and discrete-time systems into one theory. It
This study was supported by the Bialystok University of Technology under grant No. S/WI/2/2011. Z. Bartosiewicz (B) Bialystok University of Technology, Wiejska 45A, 15-351 Bialystok, Poland e-mail: [email protected]
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is based on the concept of time scale and the calculus on time scales. A time scale is a model of time. Time may be discrete or continuous, or partly continuous and partly discrete. The concepts of standard derivative used in the case of continuous time and forward difference used in the discrete time are unified into one concept of delta derivative. This allows to consider delta differential equations on arbitrary time scales. They generalize standard differential equations and difference equations. Theory of dynamical systems on time scales was developed in [5]. Special attention was paid to linear delta differential equations. Another theory unifying discrete and continuous dynamical systems was developed in [19] The interest in control systems on time scales dates back to 2004. The first results have concerned controllability, observability and realizations of linear constantcoefficient and varying-coefficient control systems with outputs [2,3,13]. Since then the literature on control systems on time scales has been rapidly growing, including also n
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