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Necessary and sufficient conditions for the commutativity of linear time-varying systems are derived in the case of nonzero initial conditions. It is shown that some commutative class of linear time-varying systems may not commute with arbitrary initial conditions. In this respect, commutativity of Euler differential systems is investigated. Explicit commutativity conditions for the fifth-order systems are solved. New results about the effects of commutativity on system sensitivity and disturbance properties are presented, which is very important for network design and industrial applications where many of the systems are composed of subsystems cooperating one after another in a chain. The results are supported by examples treated either analytically or numerically.

Exhaustive study on the commutativity of time-varying systems MUHAMMET K~K S A L~This paper, which is a survey and a compact reference on the commutativity of timevarying systems, gives the complete set of necessary and sufficient commutativity conditions for systems of any order. Original results are derived on Euler's systems, and explicit commutativity conditions are presented for fourth-order systems, which have not yet appeared in the literature. IntroductionWhen two time-varying systems A and B are connected in cascade, the input-output relation of the combined system depends on the parameters of both systems and on which appears first. If both of the connections A B and BA have the same input-output pairs irrespective of the applied input, then we say that these systems are commutative systems; in this case A B and B A are equivalent, i.e., AB = BA. We note that this equivalence is with respect to the input-output relation only when the natural responses generated by the initial conditions of the subsystems A and B and hence those of A B and BA are not considered; in other words, A and B are assumed to be initially relaxed. The effect of non-zero initial states or boundary values on commutativity will be considered elsewhere.It is obvious that single-input, single-output (SISO) time-invariant systems are commutative; if T,(s) and Tb(s) represent the transfer functions of the subsystems A and B, respectively, the transfer functions T,,(s) and Tb,(s) of the cascaded systems A B and BA are always equal, i.e., T,,(s) = Tb(s)T,(s) = T,(s)Tb(s) = T,,(s). If A and B are multi-input, multi-output (MIMO) time-invariant subsystems, the commutativity requires the same number of input and output variables for both A and B; then for commutativity it is necessary and sufficient that T.(s) and Tb(s) be commutative for all s.For time-varying systems, the investigation of commutativity conditions for (MIMO) systems is extremely difficult; therefore the scope of this paper will be confined to SISO linear systems only. Although it was not previously well known how restricting the commutativity conditions were (Horowitz 1975), considerable literature has now appeared on this subject. It can readily be shown that a zero-order system is commutative with another system of any order, so this case is excluded in subsequent discussions. For first-and second-order systems the commutativity conditions are now well known (Marshall 1977, Koksal 1982, 1983, Salehi 1983.The first of the general facts derived so far about the commutativity of high-order systems is that a constant-parameter system does not commute with a time-varying system (Marshall 1977). Although some necessary and sufficient conditions were

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