Inverse Commutativity Conditions for Second-Order Linear Time-Varying Systems

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. The necessary and sufficient conditions where a second-order linear time-varying system A is commutative with another system B 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 A. In this contribution, the inverse conditions expressed in terms of the coefficients of the differential equation describing system B have been derived and shown to be of the same form of the original equations appearing in the literature.


Introduction
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][2][3][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][6][7][8].When the cascade connection in system design is considered, the commutativity concept places an important role to improve different system performances [9][10][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 [12] 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 secondorder systems first appeared in 1982 [13].In 1984 [14] and 1985 [15], commutativity conditions for third-and fourth-order continuous-time linear time-varying systems were studied, respectively.The content of the published but undistributed work [15] can be found in journal paper [16] 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 [17] 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 fifthorder 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.

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,  = 2, 1, 0, are the time-varying coefficients.They are all defined for [ 0 , ∞) → .The (double) dot on the top represents the (second) first-order derivative with respect to time  ∈ , where  ≥  0 ,  0 being the initial time.Since  is of second order,  2 () ̸ ≡ 0; further, for the unique solvability of (1a) for the output Let  be another second-order linear time-varying system defined by where   (⋅),   (⋅),   ()'s are defined in a similar manner as for system  and  2 () ̸ ≡ 0.
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 equation where  2 ̸ = 0,  1 ,  0 are arbitrary constants; further, the coefficients of  must satisfy the following equation for the general values of  ≥  0 : Defining it can be shown by routine mathematical operations that the necessary and sufficient conditions in (2a) and (2b) can be rewritten as where If  1 = 0, (2b) and (4b) are automatically satisfied and hence they are redundant.But if  1 ̸ = 0, these equations, together with the information  2 () ̸ ≡ 0, are replaced by (4c) with  0 () being constant, that is, time invariant.
Hence, (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).Finally, using (9c) in the above obtained intermediate equation for  0 (), we write Solving it for  0 (), we obtain Using (8a) and (8b), we obtain the duals of (10a) and (10b) as respectively.We remark that   () in ( 3) and   () in ( 6) are similarly defined so they constitute also dual equations.We express the results we obtained by a theorem.

Theorem 1. Let 𝐴 and 𝐵 be two second-order linear timevarying systems described by (1a) and (1b), respectively.
The necessary and sufficient conditions subsystem  is commutative with subsystem  are as follows: (i) The coefficients of  are expressed in terms of the coefficients of  as in (4a) where  2 ̸ = 0,  1 ,  0 are some constants.
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  2 ̸ = 0,  1 ,  0 are some constants.
Conditions (i) and (ii) together are equivalent to conditions (iii) and (iv) together.There exists a unique relation between the constants  2 ,  1 ,  0 and  2 ,  1 ,  0 which are expressed by the dual equations (8a) and (8b).
If  1 ̸ = 0 which is equivalent to  1 ̸ = 0 due to (8a) and (8b), the second and fourth conditions above are replaced, respectively, by the following: 4c) is independent of time and equal to a constant.(vi)  0 () defined in (9c) is independent of time and equal to a constant.

Conclusions
The commutativity conditions for a second-order linear time-varying system  commutative with another secondorder 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 [18], 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][20][21].

Figure 1 :
Figure 1: Cascade connection of the differential systems  and .