On the stability of asynchronous multirate linear systems1 Amit Bhaya and Paulo R. Medeiros2 Abstract The stability of asynchronous multirate linear systems is studied. The Ritchey-Franklin stability criterion that was derived under the assumption that certain state transi- tion matrices are diagonalizable was preceeded by more general stability criteria for the so called desynchronized systems and difference inclusions in the Russian literature. These conditions are modified in this paper to formulate a general approach that does not require any diagonaliz- ability assumptions and applied to an illustrative example in order to compare with the Ritchey-Franklin condition. 1 Introduction Asynchronous or desynchronized linear systems are char- acterized by the fact that they contain elements that op- erate at rates that differ in phase or frequency. An analysis of such systems, obtaining sufficient stability conditions, was performed by Ritchey and Franklin [l] under certain diagonalizability assumptions. Sggfors and Toivonen [2] showed that the equidistribution of asynchronous sam- pling times conjectured by Ritchey and Franklin was a consequence of the famous Weyl equidistribution theo- rem, thus putting the results of Ritchey and Franklin on firmer theoretical ground. In a similar context, Krtolica et al. [3] obtained necessary and sufficient conditions for stability of linear feedback systems with random commu- nication delays. However, the authors cited above all seem to be unaware of the earlier literature [4] on the subject of asynchronous (or desynchronized) linear systems and dif- ference inclusions (see [5] and refs. therein). In particular, these results can be modified to give a rigorous analysis of the stability of asynchronous linear multirate systems. It is the objective of this paper to point out this fact and formulate the problem in terms of difference inclusions. 2 The Ritchey-Franklin model In this section, we give a brief description of the stability problem for asynchronous multirate linear systems pro- posed by Ritchey and Franklin [l], and refer the reader to this paper for details as well as the stability criterion. ‘This research was partially supported by grants from CNPq, the Brazilian National Council for Scientific and Technological Development and CAPES, the funding agency of the Brazilian Ministry of Education. 2Dept. of Electrical Engineering, COPPE, Federal University of Rio de Janeiro, P.O. Box 68504, 21945-970 Rio de Janeiro, RJ, BRAZIL, Email: AMIT.BHAYA@NA-NET.ORNL.GOV and MEDEIROS@IPEE.COPPE.UFRJ.BR The hybrid system studied in [l] includes both the plant (continuous-time, with state vector denoted zC) the con- troller (discrete-time, with state vector denoted 2,~) and sample and hold elements (with state vector denoted 2,). The overall state vector (X) thus has three parts: contin- uous states (zC), sample states (zcc) and discrete states (zd) and is written as: X = [z: zT CC:]‘. The system is subject to continuous-time state transitions which occur in the intervals between discrete-time state transitions. It is assumed that the latter occur in such a manner that the overall system is well-posed (i.e. there are no algebraic loops created by the discrete state transition matrices, de- noted Si and Di). Let the continuous-time state equation be written as i e = AZ,+ Bu, u(t) = x5. Then, from the standard theory of sampled-data systems, for any time in- terval At which contains no sampleor discrete transitions, the overall state equation can be written as follows: [ @(At) r(At) 0 xe (to X(t, + At) = 0 I, 0 XS@O) , 0 0 Id I[ 1 td(tO) where I, and Id are identity matrices of the appropriate size, @(At) = exp(AAt), and r(At) = JoA’ @(z)Bdz. The transition matrices Si for the sample states are easily found by inspection, and the discrete state transition ma- trices (Di) from the corresponding difference equations [l]. Given the information above and given a sequence of events (e.g. discrete event with transition matrix D1, followed by a continuous-time segment of length Atl, followed a sampling event with transition matrix S1, . . . ) one can write down the corresponding state transition matrix Ik = . . -&Q(Atl)D1. Ritchey and Franklin de- rive a stability criterion that is based on properties of this matrix. 3 Desynchronized linear systems - the 4K approach In a series of fundamental papers, Kleptsin, Kozyakin, Krasnoselskii and Kuznetsov [4] introduced and solved the problem of stability of various classes of discrete desyn- chronized systems, as they termed asynchronous systems. In this section, we give a brief overview of the general method, which we will refer to as the 4K method. The terminology needed is introduced below. Consider a sys- tem described by the equation: xi(c+‘) = 5 Aijxj(T:), i=1,2 )‘.‘) iv. (1) j=l TP-01 3:00 0-7803-4187-2 Proc. of the 36th IEEE CDC San Diego, Ca. Dec., 1997 0000 TA11-2 10:20 Proc. of the 36th IEEE CDC San Diego, Ca. Dec 199 0-7803-4187-2 2041-2042