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

Authors: Mehmet Emir Koksal
Journal: Journal of Mathematics
 (2017)

---

## Abstract

The necessary and sufficient conditions where a second-order linear time-varying systemA 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.


---

## Body

## 1. 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 [1–4]. 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 Figure1, when two linear time-varying systems A and B 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 A and B are connected in cascade. If the order of connection does not affect the input-output relation of the combined system AB or BA, it is said that systems A and B are commutative.Figure 1
Cascade connection of the differential systemsA and B.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 second-order 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 fifth-order systems were studied.This work is directly focused on the commutativity of second-order linear time-varying systems with zero initial conditions. Section2 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

LetA be a second-order linear time-varying system described byA:(1a)a2ty¨At+a1ty˙At+a0tyAt=xAt;∀t≥t0,yAt0=0,y˙At0=0,wherexA(·) is the input; yA(·) is the output functions of the system; ai(t)’s, i=2,1,0, are the time-varying coefficients. They are all defined for t0,∞)→R. The (double) dot on the top represents the (second) first-order derivative with respect to time t∈R, where t≥t0, t0 being the initial time. Since A is of second order, a2t≢0; further, for the unique solvability of (1a) for the output yA(·), it is sufficient that xA(·), ai·∈P[t0,∞), which is the set of piecewise continuous mappings [t0,∞)→R [2].LetB be another second-order linear time-varying system defined byB:(1b)b2ty¨Bt+b1ty˙Bt+b0tyBt=xBt;∀t≥t0,yBt0=0,y˙Bt0=0,wherexB·, yB·, bi(t)’s are defined in a similar manner as for system A and b2t≢0.For the commutativity of systemsA and B, it is necessary and sufficient that the coefficients of B must be expressible in terms of those of A by the matrix equation(2a)b2tb1tb0t=a2t00a1ta20,5t0a0ta2-0,5t2a1t-a˙2t41c2c1c0,where c2≠0,c1,c0 are arbitrary constants; further, the coefficients of A must satisfy the following equation for the general values of t≥t0:(2b)-a20.5ddta0-4a12+3a˙22-8a1a˙2+8a˙1a2-4a2a¨216a2c1≡0.Defining(3)fAt=a2-0.5t2a1t-a˙2t4,it can be shown by routine mathematical operations that the necessary and sufficient conditions in (2a) and (2b) can be rewritten as(4a)b2tb1tb0t=a2t00a1ta20.5t0a0tfAt1c2c1c0,(4b)-a20.5ddtA0tc1≡0,where(4c)A0t=a0t-fA2t-a20.5tf˙At.If c1=0, (2b) and (4b) are automatically satisfied and hence they are redundant. But if c1≠0, these equations, together with the information a2(t)≢0, are replaced by (4c) with A0t being constant, that is, time invariant.It is naturally expected that ifA and B are commutative, similar equations to (4a), (4b), and (4c) are valid when the coefficients of B are used. In fact, the first equation in (4a) is equivalent to(5a)a2t=1c2b2t.Using this equation in the second line of equation in (4a) and solving it for a1(t), we obtain(5b)a1t=1c2b1t-c1c21.5b20.5t.Using (5a) and (5b) in (3), we express fA(t) as (5c)fAt=b2c2-0,5b2/c2b1-2c1/c21.5b20.5-1/c2b˙24=b2c2-0,52b1-b˙2/c2-2c1/c21.5b20.54=1c20.5b2-0.52b1-b˙24-c12c2=1c20.5fBt-c12c2,where(6)fBt=b2-0.5t2b1t-b˙2t4.Finally, substituting (5c) in the third line of equation of (4a) and solving it for a0(t), we obtain(7)a0t=1c2b0t-c1c20.5fBt+c122c2-c0=1c2b0t+-c1c21.5fBt+c12-2c2c02c22.Hence, writing (5a), (5b), and (7) in matrix form and letting (8a)k2k1k0=1c2-c1c21.5c12-2c2c02c22or equivalently (note c2≠0 so k2≠0)(8b)c2c1c0=1k2-k1k21.5k12-2k2k02k22,we obtain the first set of the inverse equations derived as(9a)a2ta1ta0t=b2t00b1tb20.5t0b0tfBt1k2k1k0.To find the second inverse equation, we substitute a0(t) from (7), fA(t) from (5c), and a2(t) from (5a), all into (4c); and rearranging, we obtain ∗A0t=b0c2-c1c21.5fB+c12-2c2c02c22-fBc20.5-c12c22-b20.5c20.5f˙Bc20.5=1c2b0t-fB2t-b20.5tf˙Bt+c12-4c2c04c22.Using this in (4b), eliminating a2 by (5a) and c2, c1 by (8b), and finally multiplying by -1, we proceed with(9b)-a20.5ddt1c2b0-fB2-b20.5f˙B+c12-4c2c04c22c1=0,-b20.5c20.5ddt1c2b0-fB2-b20.5f˙Bc1=0,b20.5ddtb0-fB2-b20.5f˙B-c1c21.5=0,-b20.5ddtB0tk1≡0,where(9c)B0t=b0t-fB2t-b20.5tf˙Bt.Hence, (9a) and (9b) constitute the inverse necessary and sufficient conditions in terms of the coefficients of subsystem B. 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 A0t, we write (10a)A0t=1c2B0t+c12-4c2c04c22.Solving it for B0t, we obtain (10b)B0t=c2A0t+c0-c124c2.Using (8a) and (8b), we obtain the duals of (10a) and (10b) as(11a)B0t=1k2A0t+k12-4k2k04k22,(11b)A0t=k2B0t+k0-k124k2,respectively.We remark thatfA(t) in (3) and fB(t) in (6) are similarly defined so they constitute also dual equations. We express the results we obtained by a theorem.Theorem 1.
LetA and B be two second-order linear time-varying systems described by (1a) and (1b), respectively.
The necessary and sufficient conditions where subsystemB is commutative with subsystem A are as follows:(i)
The coefficients ofB are expressed in terms of the coefficients of A as in (4a) where c2≠0, c1, c0 are some constants.(ii)
Further, the coefficients ofA and c1 satisfy (4b).
Conversely, the necessary and sufficient conditions where subsystemA is commutative with subsystem B are as follows:(iii)
The coefficients ofA are expressed in terms of the coefficients of B as in (9a) where k2≠0, k1, k0 are some constants.(iv)
Further, the coefficients ofB and k1 satisfy (9b).Conditions (i) and (ii) together are equivalent to conditions (iii) and (iv) together. There exists a unique relation between the constants c2, c1, c0 and k2, k1, k0 which are expressed by the dual equations (8a) and (8b).
Ifc1≠0 which is equivalent to k1≠0 due to (8a) and (8b), the second and fourth conditions above are replaced, respectively, by the following:(v)
A0(t) defined in (4c) is independent of time and equal to a constant.(vi)
B0(t) 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).


## 3. Example

Let the coefficients ofA be a2=t4 and a1=t3. Then, by (3), fAt=-t/2; hence f˙At=-1/2. Substituting these in (4c) and choosing A0=-1, we find a0t=-1-t2/4. Hence, A is described by(12a)A:t4y¨At+t3y˙At-1+t24yAt=xAt.Since a0t is chosen so that (4c) is satisfied with a constant A0, the above system has second-order pairs computed by (4a) where c1 can be chosen different from zero due to condition (v) of Theorem 1. In fact, choosing c2=4, c1=-3, and c0=1, the following commutative pair B of A is obtained:(12b)b2b1b0=t400t3t20-1+t24-t214-31=4t44t3-3t2-t2+1.5t-3,B:4t4y¨Bt+4t3-3t2y˙Bt-t2-1.5t+3yBt=xBt.Using (6)(13)fBt=12t224t3-3t2-16t34=-t-0.75,where f˙B(t)=-1. Evaluating (9c), we find (14)B0=-t2+1,5t-3--t-0,752-2t2-1=-5716.Or directly from (10b)(15)B0=-14+1-916=-3-916=-5716,which is the same result in (14). The constants k2, k1, k0 are found from (8a) as (16)k2k1k0=14--389-24216=1438132.So that the coefficients of A can be inversely computed by using (9a): (17)a2a1a0=4t4004t3-3t22t20-t2+15t-3-t-0.7511438132=t4t3-34t2+34t2t24+1,5t4-34-3t8-932+132=t4t3-t24-1.Simulation results for AB and BA for inputs x1t=-6.5+10sin⁡(8πt) and x2t=2.5e-t are shown for 0.5≤t≤4 in Figure 2. For both inputs, AB and BA give the same response (AB1=BA1 and AB2=BA2), respectively. If the coefficient a0(t) of A is changed to -1-1.25t2, then (4c) is not constant any more since A0t=-1-t2; so, A will not have a second-order time-varying commutative pair unless c1=0. For c1=-3≠0 and (2a) is spoiled between the coefficients of A and B both, A will not commute with B any more. This is verified by Figure 3 which is obtained by input x1t=-6.5+10sin⁡(8πt); it is seen obviously that the responses of AB and BA are not coincident at all. All simulations are performed with the initial and final times t0=0.5 and t=4, respectively, by Simulink program of MATLAB 2010a.Figure 2
Responses of cascade connectionsAB and BA for sinusoidal (AB1=BA1) and exponential (AB2=BA2) inputs.Figure 3
Responses ofAB and BA for sinusoidal input when A and B do not satisfy commutativity conditions.


## 4. Conclusions

The commutativity conditions for a second-order linear time-varying systemB commutative with another second-order linear time-varying system A 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 A and B are in the same form whether they are expressed in terms of the coefficients of systems A or B. 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 [19–21].



---

*Source: 3070236-2017-08-27.xml*