# Decomposition of Fourth-Order Euler-Type Linear Time-Varying Differential System into Cascaded Two Second-Order Euler Commutative Pairs

Authors: Salisu Ibrahim; Abedallah Rababah
Journal: Complexity
 (2022)

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## Abstract

This paper presents decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. All necessary and sufficient conditions for the decomposition are deployed to investigate the commutativity, sensitivity, and the effect of disturbance on the fourth-order LTVS. Some systems are commutative, and some are not commutative, while some are commutative under certain conditions. Based on this fact, the commutativity of fourth-order Euler-type LTVS is investigated by introducing the commutative requirements, theories, and conditions. The fourth-order Euler-type LTVSs are investigated into commutative pairs of twice Euler-type second-order linear time-varying systems (LTVSs). The decomposition theories and conditions are derived, proved, and solved to simplify the use of commutativity for practical and industrial uses. Some fourth-order systems are sensitive toward change in initial conditions or parameters while others are not, and the effect due to disturbance also varies within systems. Furthermore, the stability and robustness of systems have so many issues. But we consider fourth-order Euler-type LTVS to observe, investigate, and tackle these issues. Lastly, the realization of fourth-order LTVS from cascaded two second-order systems can be laboratory experimented which is an open problem for future engineers to investigate. However, the theoretical results show a good agreement with the simulation results is considered in this work. Perhaps it might have unlimited physical applications in science and engineering as well as theoretical contribution. But beyond any reasonable doubt, the novelty is guaranteed because this study is the first of its kind that introduces the decomposition of the fourth-order Euler-type linear time-varying system (LTVS) as a commutative pair of two second-order Euler-type systems. Illustrative examples are presented to support the results.


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## Body

## 1. Introduction

The important feature of cascade-connected systems motivates scientists and engineers to use the stability analysis of the connected system for the modelling of many physical problems and designing engineering systems. The order of connection plays a vital role in realizing more stable systems in electronics and electrical engineering. Hence, the commutativity concept has a significant role in the engineering point of view. In 1977, Marshall in [1] investigated the commutativity for 1st order continuous LTVSs. Later on, Koksal investigated the 2nd order [2] and 3rd order [3] continuous LTVSs. Recently, Ibrahim and Koksal in [4] investigated the commutativity and sensitivity of Euler and Onsager 6th order LTVSs.Decomposition is an essential mechanism that is used in many differential systems for developing the stability of a system and resolving physical problems. It is the process of splitting a high-order linear system into lower-order commutative pairs.Decomposition formulas for 2nd order continuous-time LTVS were proved in [5] in 2016 by Koksal. The theoretical results and application for the realization of the 4th order LTVS were studied in [6] by Ibrahim and Koksal. Transitivity property of commutativity for second-order linear time-varying analogue systems was studied in [7].When it comes to fractional-order dynamic system and experimental aspects using analogue electronic devices that are reprogrammable, the decomposition of Laplace expressions into cascaded connections of small blocks makes possible a more reliable circuit design as shown in [8]. In this work, it can be noted that the orders of the blocks matter because the filter response may mitigate some frequencies that may be required by the next block.In this paper, the commutativity, decomposition, sensitivity, and the effect of disturbance on the fourth-order Euler-type LTVSs are considered. Furthermore, the cases of nonzero initial conditions (ICs) are considered.This paper is organized as follows. The mathematical materials and methods that introduce the commutativity conditions are considered in Section2. The commutative requirements and conditions along with their general solution are provided in Section 3. Results and discussions for 4th order Euler LTVS are observed in Section 4. Section 5 demonstrates and illustrates the effectiveness of the results by considering fourth-order Euler-type LTVS. Finally, the results are concluded in Section 6.


## 2. Mathematical Materials and Methods

LetC be the fourth-order Euler LTVS, described by(1)C:C4tt4y4t+C3tt3y3t+C2tt2y″t+C1tty′t+C0tyt=xt,where the input and output are xt and yt and Cit represent the coefficients of the time-varying system, which are piecewise continuous functions on t0,∞. Let the initial conditions be yt0, y′t0, y″t0, and y3t0 at the initial time t0∈R. Due to its order of 4, C4t≡0. The decomposition of C as the cascade connection of second-order systems A and B is given as(2)A:a2tyA″t+a1tyA′t+a0tyAt=xAt,B:b2tyB″t+b1tyB′t+b0tyBt=xBt,with ICs(3)yAt0,yA′t0.yBt0,yB′t0,where a2t≠0 and b2t≠0. Additionally, ai, bi, xA, xB∈Pt0,∞.The systemsA and B are called commutative, while A,B represents the commutative pair provided that the input-output relations of AB and BA are equivalent.For the cascade connectionAB in Figure 1(a), the authors in [6] obtained a 4th order LTVS for the connection AB as(4)a2b2y4+a2b1+a1b2+2a2b2′y3+a1b1+a0b2+a2b0+2a2b1′+a1b2′+a2b2″y″+a0b1+a1b0+2a2b0′+a1b1′+a2b1″y′+a0b0+a1b0′+a2b0″y=x,(5)yt0=yBt0,(6)y′t0=yB′t0,(7)y″t0=yB″t0=yAt0−b0t0yBt0−b1t0yB′t0b2t0,(8)y3t0=yB3t0=−b1t0b22t0−b2′t0b22t0yAt0+1b2t0yA′t0+b0t0b1t0b22t0+b0t0b2′t0b22t0−b0′t0b2t0yBt0+b12t0b22t0+b1t0b2′t0b22t0−b0t0b2t0−b1′t0b2t0yB′t0.Figure 1
Cascade connection of the LTVSs.
(a)(b)InterchangingA↔B and a↔b from Figure 1(b), the connection BA provides the following results:(9)b2a2y4+b2a1+b1a2+2b2a2′y3+b1a1+b0a2+b2a0+2b2a1′+b1a2′+b2a2″y″+b0a1+b1a0+2b2a0′+b1a1′b2a1″y′+a0b0+b1a0′+b2a0″y=x,(10)yt0=yAt0,(11)y′t0=yA′t0,(12)y″t0=yA″t0=yBt0−a0t0yAt0−a1t0yA′t0a2t0,(13)y3t0=yA3t0=+−a1t0a22t0−a2′t0a22t0yBt0+1a2t0yB′t0+a12t0a22t0+a1t0a2′t0a22t0−a0t0a2t0−a1′t0a2t0yA′t0+a0t0a1t0a22t0+a0t0a2′t0a22t0−a0′t0a2t0yAt0.


## 3. Commutativity Requirements

Two 2nd order LTVS subsystems A and B are called commutative if the connections AB and BA are identical. Regarding this case, the equivalence is realized iff differential equations in equations (4) and (9) are the same; in addition to the initial conditions in equations (5)–(8) and equations (10)–(13) must be the same. Solving equations (4) and (9) for b2,b1,b0 yields the matrix system(14)b2b1b0=a200a1a20.50a0f321k2k1k0,f32=14a2−0.52a1−a2′,where k2,k1,k0 are some constants. Furthermore, coefficients of the differential form of A satisfy(15)−a20.5ddta0−4a12+3a˙22−8a1a˙2+8a˙1a2−4a2a¨216a2k1=0,∀t≥t0.Commutativity with arbitrary and nonzero initial conditions can be assured by considering equations (16)–(22) (for more details, see [6]):(16)y=yB=yA,(17)y′=yB′=yA′,(18)y″=1b2−b0b2yA−b1yA′b2=1a2−a0a2yA−a1yA′a2,(19)y3=−b1b22−b2′b22yA+b0b1b22−b0′b2+b0b2′b22yB+1b2yA′+b12b22−b0b2−b1′b2+b1b2′b22,=a0a1a22−a0′a2+a0a2′a22yA+a12a22−a0a2−a1′a2+a1a2′a22yA′+−a1a22−a2′a22yB+1a2yB′,(20)−1a2+1a2k2−k0a2k2−a1k12a23/2k2+k1a2′4a23/2k2yA−k1yA′a2k2=0,(21)a1a22−k1a23/2k22−a1a22k2+k0k1a23/2k22+a1k122a22k22+a1k0a22k2+a12k12a25/2k2+a0k1a23/2k2−k1a1′2a23/2k2+a2′a22−a2′a22k2+a1k1a2′2a25/2k2−k12a2′4a22k22+k0a2′a22k2−3k1a2′28a25/2k2+k1a1″4a23/2k2yA+−1a2+k12a2k22+1a2k2+3a1k12a23/2k2−k0a2k2+3k1a2′4a23/2k2yA′=0,(22)k0+k2−1=0.


## 4. Results and Discussion

In this section, the explicit commutative formulas obtained from the previous section are considered, and the coefficients of the decompositionsA and B are expressed in terms of the decomposed fourth-order Euler system C. Observe that (1) and (4) are both 4th order LTVSs, and the equivalent relation between them leads to(23)a2b2=C4t4=a22k2=C4t4⟶a2=t2C4k21/2.Comparing the coefficients of the 3rd derivatives and making use of (14) leads to(24)a1=12−C4′C4k2t2+−C41/4k1k25/4+C3C4k2−4C4k2t.Relating the coefficients of the 2nd derivatives and making use of (14) gives(25)a0=1162C4′2C43/2k23/4−4C4′2C43/2k2−4C4″C4k23/4+8C4″C4k2t2+−8C3′C4k2+3k1C4′C43/4k25/4−16C4′C4k23/4+6C3C4′C43/2k2+32C4′C4k2t+4k12k22−2C3k1C43/4k25/4+12C41/4k1k25/4−16C4k23/4−2C32C43/2k2+8C2C4k2+32C4k2−8k0k2.Considering the obtained resultsa2, a1, and a0 of equations (23)–(25), respectively, we figure out b2, b1, b0 using (14) and present them in the following explicit forms:(26)b2=k21/2c41/2t2,(27)b1=12−k2C4′C4t2+C41/4k1k21/4+C3k2C4−4C4k2t,(28)b0=1162k21/4C4′2C43/2−4k2C4′2C43/2−4k21/4C4″C4+8k2C4″C4t2+−8k2C3′C4−3k1C4′C43/4k21/4−16k21/4C4′C4+6C3k2C4′C43/2+32k2C4′C4t+8k0+2C3k1C43/4k21/4−12C41/4k1k21/4−16C4k21/4−2C32k2C43/2+8C2k2C4+32C4k2.From (1) and (4), one needs to observe the following equations:(29)tC1=a0b1+a1b0+2a2b0′+a1b1′+a2b1″,(30)C0=a0b0+a1b0′+a2b0″.Substituting the coefficientsa2, a1, a0, b2, b1, and b0 from (23)–(28) into (29) and (30), after numerous mathematical computations and comparative analysis, the following requirements are acquired:(31)C1=18−16C2−C33C42+4C2C3C4+6C32C4−32C4+C41/4k13k29/4−8C3k21/4+32C4k21/4−4C41/4k0k1k25/4,(32)C0=12565512C2+4C34C43−32C2C32C42+64C22C4−128C32C4+1024C4+8C3k13C43/2k29/4−48C41/4k13k29/4+32k0k12k22−64C4k12k27/4−12C32k12C43/2k23/2+32C2k12C4k23/2+48C3k12C4k23/2−16C4k12k23/2−32C3k0k1C43/4k25/4+192C41/4k0k1k25/4−64k02k2+256C4k2−256C2k21/4+64C32C4k21/4−1024C4k21/4.Theorem 1.
The necessary and sufficient conditions for Euler 4th-order LTVS of equation (1) to be decomposed into cascade-connected LTV commutative pairs of Euler 2nd-order are that(i)
There exist some constantsk2, k1, k0 such that the coefficients C1 and C0 can be formulated in connection with C4, C3, and C2 as in (31) and (32), respectively,(ii)
The reduced Euler 2nd order A and B must be formulated with regard to C4, C3, and C2 as in equations (23)–(28).Proof.
The detailed proof of (i) of Theorem1 can be seen in equations (29) and (30), while in (ii) of Theorem 1, equations (23)–(28) are obtained from the coefficients of equations (1) and (4).
Note. Theorem 1 is applicable to the zero IC case. Regarding the case of nonzero ICs, it is covered in Theorem 2.Theorem 2.
The necessary and sufficient conditions for the decomposition of Euler 4th order LTVS C with nonzero ICs into its twin Euler 2nd order LTV commutative pairs A and B are that(i)
The requirements of Theorem1 are fulfilled.(ii)
The ICs ofA,B,C must obey equations (16) and (17).(iii)
Furthermore, the ICs ofA must obey equations (20) and (21).(iv)
In addition, the ICs ofC must be satisfied.(33)y″t0=−2t2+C328t2C42−C22t2C4−k124t2C4k23/2+C3k18t2C45/4k23/4−3k14t2C41/4k23/4+k02t2C4k2+1t2k21/4+k2t2C4+C3′2tC4−3C3C4′8tC42−2C4′tC4−3k1C4′16tC45/4k23/4+C4′tC4k21/4+C4′24C42−C4′28C42k21/4−C4″2C4+C4″4C4k21/4yAt0,(34)y3t0=−C3316t3C43+C2C34t3C42+C3t3C4−k138t3C43/4k29/4+3C3k1216t3C43/2k23/2−3k128t3C4k23/2+k0k14t3C43/4k25/4+k12t3C41/4k2−C2k14t3C45/4k23/4+3C3k18t3C45/4k23/4−k1t3C41/4k23/4−C3k04t3C43/2k2−C32t3C4k21/4+k12t3C43/4k21/4−C3k22t3C43/2−C2′2t2C4−C3C3′4t2C42+C3′2t2C4+3k1C3′8t2C45/4k23/4+C3k2C3′4t2C42+3C32C4′16t2C43+C2C4′4t2C42+5C3C4′8t2C42−3C4′t2C4−3k12C4′32t2C43/2k23/2+k1C4′2t2C45/4k2−3C3k1C4′16t2C49/4k23/4+3C4′2t2C4k21/4−11k1C4′8t2C45/4k23/4−C3C4′2t2C42k21/4−3C32k2C4′16t2C43−5C3′C4′8tC42+7C3C4′216tC43+3C4′22tC42−k1C4′216tC49/4k2+17k1C4′264tC49/4k23/4+C3C4′216tC43k21/4−C4′24tC42k21/4−3C4′38C43yAt0+3C4′316C43k21/4+C4″2tC4−C3C4″8tC42−3C4″tC4+k1C4″8tC45/4k2−7k1C4″16tC45/4k23/4−C3C4″8tC42k21/4+3C4″2tC4k21/4+3C4′C4″4C42−3C4′C4″8C42k21/4−C432C4+C434C4k21/4+3C328t2C42−C22t2C4−3C32t2C4−3C3k18t2C45/4k23/4+1t2k21/4+3k14t2C41/4k23/4+k02t2C4k2+k2t2C4−3C3C4′8tC42+3k1C4′16tC45/4k23/4+C4′tC4k21/4−C4′28C42k21/4+C4″4C4k21/4yA′t0.(v)
Fork1=0, the ICs must satisfy (22).Proof.
Theorem2 (i) is obvious; for Theorem 2 (ii), the ICs in equations (16) and (17) are obtained from equations (5), (10) and equation (6), (11), respectively. Patterning Theorem 2(iii), the ICs in equations (20) and (21) are obtained from equations (7), (12) and equations (8), (13), respectively.
Regarding equation (33), we consider equation (18); inserting in values of a2, a1, and a0 of equations (23)–(25) leads to equation (33). Equation (34) is obtained by using equation (19); substituting the values of a2, a1, and a0 of equations (23)–(25), respectively, in equation (19) provides equation (34). Item (v) results from equation (22) as the solution of equations (20) and (21).


## 5. Applications to Fourth-Order Euler-Type LTVS

We apply the results obtained from the previous sections to investigate the decomposition of fourth-order Euler LTVS as a commutative pair of two second-order Euler-type systems.


### 5.1. Example 1

Considering the constantsC4=C3=C2=1, with ki′s in equation (36), C1=−15/8 and C0=299/192 are obtained by making use of equations (31) and (32), respectively. The general Euler-type fourth-order LTVS C of equation (1) becomes(35)t4y4t+t3y3t+t2y″t−158ty't+299192yt=xt.Note that equation (22) is satisfied with the constants(36)k2=3,k1=0,k0=−2.Furthermore, equations (16)–(19) lead to(37)yBt0=yAt0=yt0,(38)yB′t0=yA′t0=y′t0,(39)y″t0=−118+23yt0+32y′t0fort0=1,(40)y3t0=1116−13yt0+−58+23y′t0fort0=1.The Euler subsystemsA and B generate the equations(41)A:t23yA″t−32tyA′t+33+83243yAt=xAt,(42)B:3t2yB′t−332tyB′t+113−88yBt=xBt.Note that both subsystems A and B in the decomposition are of Euler type. Simulation was carried out with a sinusoid of frequency3 rad/sec, amplitude 2, and bias −21/10, considering ode (Bogacki–Shampine) as the solver, while maintaining a fixed step length of 0.01. Simulink outcomes are illustrated in Figure 2. The initial time and initial states chosen are t0=1 and y1=yA1=yB1=yB′1=yA′1=y′1=0.5. Equations (39) and (40) lead to y″1=1/16+1/3,y31=1/32+1/23. After satisfying all decomposition conditions, the systems AB,BA, and C produce a similar response (B=BA=C, see Figure 2). However, with a small modification in the decomposition requirement by changing yB1=yA1=y1=0.5 to yA1=y1=0.6, then equation (37) is not satisfied, and the decomposition is messed up, and it is not valid any more (see AB1,BA1,C1 in Figure 2). Observe that AB1 is slightly disturbed by the changes, while BA1 is so sensitive to ICs. Therefore, A and B should be connected in the cascade synthesis of C.Figure 2
Simulation results of example 1.


### 5.2. Example 2

In this example, the 4th order LTVS C is reconstructed by considering the 2nd order subsystem A. In line with this, consider the 2nd order LTVS A defined by(43)A:yA″t+sintyA′t+sin2t4+sint5+cost2+34yAt=xAt.For the constantk1:(44)1k22k2−k2k0−k22−0.5k2k1t−k1k2k0k1−k1−0.5k2k1+Vk12+k2−k2k0−k22+1.5k2k1tyAyA′=0,where V=1−k2+0.5k12+k2k0t+0.25k2k1t2.For the commutativity with nonzero ICs, the coefficient matrix in equation (44) must be singular at t0=0, that is, its determinant is zero if(45)k1=∓25k0+k2−1.Furthermore, equation (44) at t0=0 requires(46)k2−k2k0−k22yA−k1k2yA′=0.Inserting equation (45) into equation (46) guaranties commutativity of A and B:(47)yA′=±52yA,withyAarbitrary.Consider the casek1≠0 and choose(48)k2=1,k1=∓25,k0=1.By making use of equation (14), the pair B is obtained in terms of A:(49)B:yB″t+sint+25yB′t+sin2t4+sint5+cost2+34yBt=xBt.Considering the coefficientsb2, b1, b0 of equation (49) and the system A, that is, a2, a1, a0 from equation (43), we realize fourth-order LTVS C as(50)C:y4t+2sint+25y3t+3sin2t2+3sint5+3cost+12y″t+sin3t2+3sin2t25+3costsint+3cost5−3sint2−125y′t+sin4t16+sin3t45+34costsin2t+3cos2t4−7sin2t8+3costsint25−cost4−5sint4−316yt=xt.Regarding the decomposition with nonzero initial condition,yt0≠0, equations (16)–(19) yield(51)yBt0=yAt0=yt0,(52)yB′t0=yA′t0=y′t0=±52yAt0,(53)yB″t0=y″t0=34yt0fort0=0,(54)y3t0=−14y′t0fort0=0.Simulations are carried out with a sinusoid of frequency3rad/sec, amplitude 3, and bias −0.5. The rationale behind using Runge–Kutta as a solver was based on its numerical stability as well as accuracy. The Runge–Kutta schemes satisfy the possibility condition for any time-step interval “change in time” (see [9] and [10]). Simulink outcomes are illustrated in Figure 3. The initial time and initial conditions chosen are t0=0 and y0=yA0=yB0=−1, respectively. Computing equations (52)–(54) leads to yB′0=yA′0=y′0=±5/2, y″0=−3/4 and y3=−5/85. The responses of system C and its decompositions as AB and BA give the same output as shown in Figure 3 (C). A noise signal in the form of saw-tooth wave (signal generator) with amplitude 0.05 and frequency 0.01 rad/sec is injected between the subsystems A and B. The simulation results are shown in Figure 3 (AB and BA).Figure 3
Simulation results of example 2.


## 5.1. Example 1

Considering the constantsC4=C3=C2=1, with ki′s in equation (36), C1=−15/8 and C0=299/192 are obtained by making use of equations (31) and (32), respectively. The general Euler-type fourth-order LTVS C of equation (1) becomes(35)t4y4t+t3y3t+t2y″t−158ty't+299192yt=xt.Note that equation (22) is satisfied with the constants(36)k2=3,k1=0,k0=−2.Furthermore, equations (16)–(19) lead to(37)yBt0=yAt0=yt0,(38)yB′t0=yA′t0=y′t0,(39)y″t0=−118+23yt0+32y′t0fort0=1,(40)y3t0=1116−13yt0+−58+23y′t0fort0=1.The Euler subsystemsA and B generate the equations(41)A:t23yA″t−32tyA′t+33+83243yAt=xAt,(42)B:3t2yB′t−332tyB′t+113−88yBt=xBt.Note that both subsystems A and B in the decomposition are of Euler type. Simulation was carried out with a sinusoid of frequency3 rad/sec, amplitude 2, and bias −21/10, considering ode (Bogacki–Shampine) as the solver, while maintaining a fixed step length of 0.01. Simulink outcomes are illustrated in Figure 2. The initial time and initial states chosen are t0=1 and y1=yA1=yB1=yB′1=yA′1=y′1=0.5. Equations (39) and (40) lead to y″1=1/16+1/3,y31=1/32+1/23. After satisfying all decomposition conditions, the systems AB,BA, and C produce a similar response (B=BA=C, see Figure 2). However, with a small modification in the decomposition requirement by changing yB1=yA1=y1=0.5 to yA1=y1=0.6, then equation (37) is not satisfied, and the decomposition is messed up, and it is not valid any more (see AB1,BA1,C1 in Figure 2). Observe that AB1 is slightly disturbed by the changes, while BA1 is so sensitive to ICs. Therefore, A and B should be connected in the cascade synthesis of C.Figure 2
Simulation results of example 1.


## 5.2. Example 2

In this example, the 4th order LTVS C is reconstructed by considering the 2nd order subsystem A. In line with this, consider the 2nd order LTVS A defined by(43)A:yA″t+sintyA′t+sin2t4+sint5+cost2+34yAt=xAt.For the constantk1:(44)1k22k2−k2k0−k22−0.5k2k1t−k1k2k0k1−k1−0.5k2k1+Vk12+k2−k2k0−k22+1.5k2k1tyAyA′=0,where V=1−k2+0.5k12+k2k0t+0.25k2k1t2.For the commutativity with nonzero ICs, the coefficient matrix in equation (44) must be singular at t0=0, that is, its determinant is zero if(45)k1=∓25k0+k2−1.Furthermore, equation (44) at t0=0 requires(46)k2−k2k0−k22yA−k1k2yA′=0.Inserting equation (45) into equation (46) guaranties commutativity of A and B:(47)yA′=±52yA,withyAarbitrary.Consider the casek1≠0 and choose(48)k2=1,k1=∓25,k0=1.By making use of equation (14), the pair B is obtained in terms of A:(49)B:yB″t+sint+25yB′t+sin2t4+sint5+cost2+34yBt=xBt.Considering the coefficientsb2, b1, b0 of equation (49) and the system A, that is, a2, a1, a0 from equation (43), we realize fourth-order LTVS C as(50)C:y4t+2sint+25y3t+3sin2t2+3sint5+3cost+12y″t+sin3t2+3sin2t25+3costsint+3cost5−3sint2−125y′t+sin4t16+sin3t45+34costsin2t+3cos2t4−7sin2t8+3costsint25−cost4−5sint4−316yt=xt.Regarding the decomposition with nonzero initial condition,yt0≠0, equations (16)–(19) yield(51)yBt0=yAt0=yt0,(52)yB′t0=yA′t0=y′t0=±52yAt0,(53)yB″t0=y″t0=34yt0fort0=0,(54)y3t0=−14y′t0fort0=0.Simulations are carried out with a sinusoid of frequency3rad/sec, amplitude 3, and bias −0.5. The rationale behind using Runge–Kutta as a solver was based on its numerical stability as well as accuracy. The Runge–Kutta schemes satisfy the possibility condition for any time-step interval “change in time” (see [9] and [10]). Simulink outcomes are illustrated in Figure 3. The initial time and initial conditions chosen are t0=0 and y0=yA0=yB0=−1, respectively. Computing equations (52)–(54) leads to yB′0=yA′0=y′0=±5/2, y″0=−3/4 and y3=−5/85. The responses of system C and its decompositions as AB and BA give the same output as shown in Figure 3 (C). A noise signal in the form of saw-tooth wave (signal generator) with amplitude 0.05 and frequency 0.01 rad/sec is injected between the subsystems A and B. The simulation results are shown in Figure 3 (AB and BA).Figure 3
Simulation results of example 2.


## 6. Conclusion

The commutativity and decomposition of fourth-order Euler LTVS are considered as a commutative pair of two second-order Euler-type systems. The results are analyzed explicitly and depicted by simulations. Our results highlight the sensitivity of fourth-order Euler-type LTVSs as a result of changes in ICs and present the effect of disturbance due to external noise. Based on these findings, we investigated and discovered that the fourth-order Euler LTVSC possess its commutative pairs A and B which are obtained as a result of the decomposition process in Theorems 1 and 2 as well as in example 1. The commutative pairs A and B are commutative under certain conditions and can be used for the realization of fourth-order LTVS as seen in example 2. The fourth-order Euler LTVS C in equation (35) and the systems AB and BA of example 1 are sensible toward changes in ICs. Moreover, the system in equation (50) and the systems AB and BA of example 2 show great level of commutativity imbalance toward noise disturbance. Example 1 investigated the decomposition of fourth-order Euler-type LTVS, and our findings verify that fourth-order Euler-type LTVSs always have general decomposed commutative pairs which are also Euler-type systems. The Euler systems possess both constant forward feedback conjugates and non-constant feedback conjugates as commutative pairs. Also, they have less effect toward disturbance due to noise and are sensible toward change in ICs (see [4]). Example 2 presented second-order LTVS as a subsystem that is used for the realization of fourth-order LTVS. The system has a great effect toward disturbance due to noise. These special properties of Euler systems make them a case of interest and differentiate them from other systems.At last, one can easily observe that the theoretical results are in good agreement with the simulation results. Generalizing and applying these theorems in nonlinear systems, fractional systems, and partial differential systems is an open problem for future investigations. Hence, they remind system designers which connection order should be used in the cascade structure of system synthesis.



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