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Inverse Commutativity Conditions for Second-Order Linear Time-Varying Systems
The necessary and sufficient conditions where a second-order linear time-varying system is commutative with another system of the same type have been given in the literature for both zero initial states and nonzero initial states. These conditions are mainly expressed in terms of the coefficients of the differential equation describing system . In this contribution, the inverse conditions expressed in terms of the coefficients of the differential equation describing system have been derived and shown to be of the same form of the original equations appearing in the literature.
As one of the main fields of applied mathematics, differential equations arise in acoustics, electromagnetics, electrodynamics, fluid dynamics, wave motion, wave distribution, and many other sciences and branches of engineering. There is a tremendous amount of work on the theory and techniques for solving differential equations and on their applications [1–4]. Particularly, they are used as a powerful tool for modelling, analyzing, and solving real engineering problems and for discussing the results turned up at the end of analyzing for resolution of naturel problems. For example, they are used in system and control theory, which is an interdisciplinary branch of electric-electronics engineering and applied mathematics that deal with the behavior of dynamical systems with inputs and how their behavior is modified by different combinations such as cascade and feedback connections [5–8]. When the cascade connection in system design is considered, the commutativity concept places an important role to improve different system performances [9–11].
As shown in Figure 1, when two linear time-varying systems and described by linear time-varying differential equations are connected in cascade so that the output of one appears as the input of the other, it is said that systems and are connected in cascade. If the order of connection does not affect the input-output relation of the combined system or , it is said that systems and are commutative.
Although the first paper about the commutativity appeared in 1977  which had introduced the commutativity concept for the first time and studied the commutativity of the first-order continuous-time linear time-varying systems, this paper is important for proving that a time-varying system can be commutative with another time-varying system only; further very few classes of systems can be commutative. In particular, commutativity conditions for relaxed second-order systems first appeared in 1982 . In 1984  and 1985 , commutativity conditions for third- and fourth-order continuous-time linear time-varying systems were studied, respectively. The content of the published but undistributed work  can be found in journal paper  which presents an exhaustive study on the commutativity of continuous-time linear time-varying systems. That paper is the first tutorial paper in the literature.
During the period from 1989 to 2011, no publication about commutativity had appeared in the literature. In 2011, the second basic journal publication  appeared. In this paper, commutativity in case of nonzero initial conditions, commutativity of Euler’s systems, new results about effects of commutativity, reduction of disturbance by change of connection order in a chain structure of subsystems, and the most important explicit commutativity conditions for fifth-order systems were studied.
This work is directly focused on the commutativity of second-order linear time-varying systems with zero initial conditions. Section 2 summarizes the necessary and sufficient conditions of commutativity of such systems as appearing in the literature and then presents the inverse conditions. Section 3 covers an example and its simulation results. Finally, Section 4 includes conclusions.
2. Inverse Conditions of Commutativity
Let be a second-order linear time-varying system described by :
where is the input; is the output functions of the system; ’s, , are the time-varying coefficients. They are all defined for . The (double) dot on the top represents the (second) first-order derivative with respect to time , where , being the initial time. Since is of second order, ; further, for the unique solvability of (1a) for the output , it is sufficient that , , which is the set of piecewise continuous mappings .
Let be another second-order linear time-varying system defined by :
where , , ’s are defined in a similar manner as for system and .
For the commutativity of systems and , it is necessary and sufficient that the coefficients of must be expressible in terms of those of by the matrix equationwhere are arbitrary constants; further, the coefficients of must satisfy the following equation for the general values of :Definingit can be shown by routine mathematical operations that the necessary and sufficient conditions in (2a) and (2b) can be rewritten aswhereIf , (2b) and (4b) are automatically satisfied and hence they are redundant. But if , these equations, together with the information , are replaced by (4c) with being constant, that is, time invariant.
It is naturally expected that if and are commutative, similar equations to (4a), (4b), and (4c) are valid when the coefficients of are used. In fact, the first equation in (4a) is equivalent toUsing this equation in the second line of equation in (4a) and solving it for , we obtainUsing (5a) and (5b) in (3), we express as whereFinally, substituting (5c) in the third line of equation of (4a) and solving it for , we obtainHence, writing (5a), (5b), and (7) in matrix form and letting or equivalently (note so )we obtain the first set of the inverse equations derived asTo find the second inverse equation, we substitute from (7), from (5c), and from (5a), all into (4c); and rearranging, we obtain Using this in (4b), eliminating by (5a) and , by (8b), and finally multiplying by , we proceed withwhereHence, (9a) and (9b) constitute the inverse necessary and sufficient conditions in terms of the coefficients of subsystem . These equations are the duals of (4a) and (4b), respectively; similarly, (9c) is the dual of (4c).
Theorem 1. Let and be two second-order linear time-varying systems described by (1a) and (1b), respectively.
The necessary and sufficient conditions where subsystem is commutative with subsystem are as follows:(i)The coefficients of are expressed in terms of the coefficients of as in (4a) where , , are some constants.(ii)Further, the coefficients of and satisfy (4b).Conversely, the necessary and sufficient conditions where subsystem is commutative with subsystem are as follows:(iii)The coefficients of are expressed in terms of the coefficients of as in (9a) where , , are some constants.(iv)Further, the coefficients of and satisfy (9b).Conditions (i) and (ii) together are equivalent to conditions (iii) and (iv) together. There exists a unique relation between the constants , , and , , which are expressed by the dual equations (8a) and (8b).
If which is equivalent to due to (8a) and (8b), the second and fourth conditions above are replaced, respectively, by the following:(v) defined in (4c) is independent of time and equal to a constant.(vi) defined in (9c) is independent of time and equal to a constant.Conditions (i) and (v) together are equivalent to conditions (iii) and (vi) together. Further (v) and (vi) are equivalent conditions due to (10a), (10b), (11a), and (11b).
Let the coefficients of be and . Then, by (3), ; hence . Substituting these in (4c) and choosing , we find . Hence, is described bySince is chosen so that (4c) is satisfied with a constant , the above system has second-order pairs computed by (4a) where can be chosen different from zero due to condition (v) of Theorem 1. In fact, choosing , , and , the following commutative pair of is obtained:Using (6)where . Evaluating (9c), we find Or directly from (10b)which is the same result in (14). The constants , , are found from (8a) as So that the coefficients of can be inversely computed by using (9a): Simulation results for and for inputs and are shown for in Figure 2. For both inputs, and give the same response ( and ), respectively. If the coefficient of is changed to , then (4c) is not constant any more since ; so, will not have a second-order time-varying commutative pair unless . For and (2a) is spoiled between the coefficients of and both, will not commute with any more. This is verified by Figure 3 which is obtained by input ; it is seen obviously that the responses of and are not coincident at all. All simulations are performed with the initial and final times and , respectively, by Simulink program of MATLAB 2010a.
The commutativity conditions for a second-order linear time-varying system commutative with another second-order linear time-varying system is well known in the literature. In this paper, the inverse commutativity conditions are obtained. It is shown that the commutativity conditions for any two second-order linear time-varying systems and are in the same form whether they are expressed in terms of the coefficients of systems or . The results are illustrated by an example. Inverse commutativity conditions obtained are used in transitivity property of commutativity for time-varying systems, which is the subject of future work. Further, the problem of commutativity of switched systems , which are also linear time-varying systems, can be an interesting research subject which has not been studied before. Additionally, commutativity conditions could be studied for fractional order linear differential systems and even linear systems of some fractional difference equations [19–21].
Conflicts of Interest
The author declares that they have no conflicts of interest.
This study is supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under the Project no. 115E952.
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Copyright © 2017 Mehmet Emir Koksal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.