Comments on ‘ Commutativity of second-order time-varying systems ’

Transactions of the Institute of Measurement and Control

2018

After introducing commutativity concept and summarizing the relevant literature, this work is focused on the commutativity of feedback conjugates. It is already known that a linear time-varying differential system describing a single input-single output dynamical system is always commutative with its constant gain feedback pairs. In this article, it is proven that among the time-varying feedback conjugates of a linear time-varying system, constant feedback conjugates are the only commutative feedback pairs and any of the time-varying feedback conjugates cannot constitute a commutative pair of a linear time-varying system.

Section: Introductionmentioning

confidence: 99%

Commutativity of systems with their feedback conjugates

Transactions of the Institute of Measurement and Control

2018

“…In Marshall, Koksal, and Koksal, the commutativity results for second‐order systems are presented with the contribution of S.V. Saleh . Explicit commutativity conditions for the third and fourth‐order systems are derived in Koksal .…”

Section: Introductionmentioning

confidence: 99%

“…E. Marshall has shown the necessary and sufficient conditions for the commutativity of first-order systems and proven that for the commutativity of two linear systems, it is required that either both systems are time-invariant (scalar constant gain systems are excluded) or both systems are time-varying. After Marshall's work, there had been several publications confined to special cases [32][33][34][35] until the first exhaustive study of commutativity of continuous-time linear time-varying systems appeared in 1988. 36 In Marshall, 31 Koksal, 32 and Koksal, 33 the commutativity results for second-order systems are presented with the contribution of S.V.…”

Section: Introductionmentioning

confidence: 99%

“…Saleh. 34 Explicit commutativity conditions for the third and fourth-order systems are derived in Koksal. 35 The general commutativity conditions for linear time-varying systems presented in 36 not only covered all the previous works as special cases, but it also forecasted open questions on commutativity conditions with non-zero initial values, role of the performance concerning sensitivity, disturbance, and noise, which are all important from engineering point of view.…”

Section: Introductionmentioning

confidence: 99%

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Commutativity of first‐order discrete‐time linear time‐varying systems

Math Methods in App Sciences

2018

After introducing the concept of commutativity for continuous‐time linear time‐varying systems, the related literature and the results obtained so far are presented. For a simple introduction of the commutativity of discrete‐time linear time‐varying systems, the problem is formulated for first‐order systems. Finally, explicit necessary and sufficient conditions for the commutativity of first‐order discrete‐time linear time‐varying systems are derived, and their advantageous use in digital system design is illustrated, which are the main objectives of the paper. The results are verified by examples which include an application in amplitude modulation for digital telecommunication.

“…After several publications about the commutativity of low-order systems [7][8][9][10], an exhaustive study was introduced in 1988 [11]. This paper had been the main reference for many years since it covers the most general necessary and sufficient conditions for the commutativity of systems of any order but without initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Decomposition of a second-order linear time-varying differential system as the series connection of two first order commutative pairs

2016

Open Mathematics

Necessary and sufficiently conditions are derived for the decomposition of a second order linear timevarying system into two cascade connected commutative first order linear time-varying subsystems. The explicit formulas describing these subsystems are presented. It is shown that a very small class of systems satisfies the stated conditions. The results are well verified by simulations. It is also shown that its cascade synthesis is less sensitive to numerical errors than the direct simulation of the system itself.