Observer Design of Switched Positive Systems with Time-Varying Delays
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Observer Design of Switched Positive Systems with Time-Varying Delays Mei Xiang · Zhengrong Xiang
Received: 23 August 2012 / Revised: 23 January 2013 © Springer Science+Business Media New York 2013
Abstract This paper is concerned with the design of positive observers for switched positive linear systems with time-varying delays. Attention is focused on designing the positive observers such that the error switched systems are exponentially stable. Based on the average dwell time approach, sufficient conditions, which ensure the estimated error exponentially converges to zero, are formulated in a set of linear matrix inequalities (LMIs). Finally, an illustrative example is given to show the efficiency of the proposed method. Keywords Positive observers · Switched positive systems · Time-varying delays · Exponential stability · Linear matrix inequalities (LMIs) · Average dwell time
1 Introduction A switched system is a type of hybrid dynamical system that combines discrete states and continuous states. Typically, it consists of a number of subsystems and a switching signal, which defines a specific subsystem being activated during a certain interval. Many dynamical systems can be modeled as such switched systems [5, 20, 21]. Recently, switched positive systems, whose states and outputs are non-negative whenever the initial conditions and inputs are non-negative, have been investigated by many researchers due to their broad applications in communication networks [31], the viral mutation dynamics under drug treatment [9], formation flying [12], and systems theory [1, 13, 14, 29, 30]. Moreover, recently Kaczorek [15] has presented M. Xiang · Z. Xiang () School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China e-mail: [email protected] M. Xiang e-mail: [email protected]
Circuits Syst Signal Process
some results on 2-D positive switched systems, which has made much contribution to the system theory. A switched positive system means a switched system in which each subsystem itself is a positive system. It should be pointed out that studying the switched positive systems is more challenging than that of general switched systems and positive systems because, in order to obtain some results, one has to combine both of their features [2, 4, 6, 17, 18, 22, 27]. In practice, delays are universal in real engineering processes and have very complex impact on system dynamics. Hence, it is theoretically challenging and of fundamental importance to study time-delay systems. Although many results have been reported for these systems [7, 16, 24, 25, 34–36], until recently the switched positive liner systems with time delays have been investigated [23, 37]. On the other hand, in actual operation, the states of the systems are not all measurable, thus it is necessary to design state observers for the systems. The developed observer design techniques for non-positive dynamical systems may not be applicable when dealing with positive dynamical systems, since it is of
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