# Dynamics and Robust Control of a New Realizable Chaotic Nonlinear Model

Authors: M. Higazy; Emad E. Mahmoud; E. M. Khalil; S. Abdel-Khalek; S. M. Abo-Dahab; Hammad Alotaibi
Journal: Complexity
 (2021)

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## Abstract

We present a new viable nonlinear chaotic paradigm. This paradigm has four nonlinear terms. The essential features of the new paradigm have been investigated. Our new system is confirmed to have chaotic behaviors by calculating its Lyapunov exponents. The relations of the system states are displayed by a suggested new signal flow graph (SFG). The proposed SFG is discussed via some graph theory tools, and some of its hidden features are calculated. In addition, the system is realized via constructing its electronic circuit which helps in the real applications. Also, a robust controller for the system is designed with the aid of a genetic algorithm.


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## Body

## 1. Introduction

The first noteworthy autonomous chaotic paradigm was proposed by Lorenz in 1963 to model the dynamics of the atmospheric convection with three connected differential equations [1]. Subsequent, Rossler constructed 3D chaotic paradigm with single cross-product nonlinear term [2]. Another 3D chaotic paradigm called the Chen model was constructed by Chen and Ueta in 1999, and they proved that it is a dual and not equivalent to the Lorenz model [3]. After that, in 2002, Lü, in [4], constructed an important chaotic paradigm between Lorenz and Chen models, and then the generalized Lorenz model [5] was constructed as a link of Lorenz, Chen, and Lü paradigms. Few years ago, a novel 3D chaotic paradigm with complicated chaotic behavior and interesting features was constructed by Qi et al. in [6]. During the last ten years, constructing and studying new chaotic paradigms have attracted many researchers in various fields because the chaos phenomena have been found in several modern applications such as communication algorithms, signal processing, nonlinear networks, and chemical and biological structures. In addition, several real applications come up to several new and interesting research points as chaos control techniques [7, 8] and synchronization of chaos systems [9, 10].The 2D autonomous systems cannot have chaos features as demonstrated by Poincare–Bendixson theorem [11]. Also, in [12], the authors have proved that 3D dissipative quadratic models represented by ODEs, with total four terms at the right-hand side, cannot have chaos features. After that, in [13], Sprott suggested one quadratic nonlinear term chaotic model in which the right-hand side contains only five terms. From algebraic point of view, the system is considered the simplest chaotic model. The most interesting challenge is to design a 3D autonomous quadratic chaotic paradigm having complicated attractor entity. So, we hope to introduce a new chaotic nonlinear model with seven terms at the right-hand side and four of them are nonlinear.The following conditions are satisfied by the chaotic models [14, 15]: (i) very sensitive to starting conditions; (ii) have a single positive Lyapunov exponent. Due to the high capacity, high security, and high efficiency of dealing with chaotic systems, it has widely potentially utilized in nonlinear circuits, secure communications, lasers, neural networks, biological systems, and so on; then, studying chaotic nonlinear models is quite significant these days [16, 17].Representing the complicated paradigms by their signal flow graph is very helpful. It helps in understanding the structure and the complexity of the paradigm using the tools of the graph theory [18]. In directed graph theory, finding directed cycles in the graph is known as a common source of complexity [19]. Those cycles are significant in the context of engineering structures. Specially, they are significant providers of complexity. For instance, cycles can produce positive feedback loops [20], which drive the system to be unstable. Cycles in engineering paradigms also increase the complexity of design and analyse the context of simulation convergence [19]. There are several studies in the literature on graph complexity criteria [19]. Such criteria can either directly or indirectly be linked to the values of eigenvalues of the studied graph matrices.In real life and in process control, it is very important to control systems in order to work in a desired stable steady operating situation. All of the following are examples of the real processes that need to operate in a predetermined states, adjusting satellite orbits, controlling missile tracks in military applications, in space applications, and in industry also where it is important to control temperature, pressure, and other process variables at specific operating values. Consequently, track control methods of chaotic paradigms have taken a great attention. In [21], Yang, Chen, and Yau proposed a track controller for Lorenz structure. Their reference values were suggested to obey certain constraints which has restricted the applicability of their controller. In [22], Gao developed a track controller in such a way to avoid the constraints of the method in [21]. In [23], a novel strategy for complete and phase robust synchronizations of chaotic nonlinear systems based on single-state feedback track synchronization control technique and genetic algorithm was proposed. More studies on control of nonlinear systems can be found in [24–28]. Different control strategies can be found in [29–32].The contribution and addition to this paper is to present a new chaotic nonlinear mathematical model. This model has applications in engineering and communications, and we will prove this by making an electronic circuit for this model. Here, we follow the same strategy to design a new track controller in order to drive the proposed chaotic system to follow any desired reference points. This control technique is designed using one state variable as a feedback variable which reduces the needed sensors and make it easy to implement in real and also minimize the cost.The remaining parts of our work are organized as follows: the model description and its basic properties are investigated in five sections in Section2. The proposed SFG of the system and its discussion are studied in Section 3. Designing the electronic circuit that implements the system is presented in Section 4. The method of single-state feedback control for the proposed system is presented in Section 5. The robustness of the proposed controller is studied and discussed in Section 6. The conclusion of our study is put at the end of the paper before the list of cited references.


## 2. System Characterization and Its Basic Features

At first, we suggest a novel three-dimensional autonomous paradigm:(1)x˙1t=a1x2t−x1t+x2tx3t,x˙2t=a2x2t−a3x1tx3t,x˙3t=a4x12t−a5x3t,where x1t,x2t,x3t∈R3 and a1,a2,a3,a4,a5 are real parameters.System (1) has the following essential dynamical features.


### 2.1. System (1) Generalized Hamiltonian

Smooth nonlinear system (1) is considered, given in the following formula:(2)x˙=αx∂H∂x+σx∂H∂x,where x=x1t,x2t,x3tT, Hx is smooth energy function and universally positive definite, and ∂H/∂x is a column gradient vector of Hx.By using energy function in quadratic formula, we have(3)Hx=12xTγx,where γ signifies a constant diagonal matrix, which is symmetric and positive definite, with respect to ∂H/∂x=γx, αx is antisymmetric matrix representing the vector field of the workless part, and σx is a symmetric matrix, a negative definite, representing the working or nonconservative part of the system:(4)αx=−αxT,σx=σxT.DefineHx of the model in (1) as(5)Hx=12x1t−a3x3tx2t2+x2t−a3x3t2+x3t2,x˙1tx˙2tx˙3t=Ax1t−a3x3tx2tx2t−a3x3x3t+Bx1t−a3x3tx2tx2t−a3x3x3t,where(6)A=012−a3x3ta1+x3t−12a4x1tx2t−a3x3t−12−a3x3ta1+x3t0−12a3x1t12a4x1tx2t−a3x3t12a3x1t0,B=a1a3x3tx2t12−a3x3ta1+x3t12a4x1tx2t−a3x3t12−a3x3ta1+x3t−a2a3x3t−12a3x1t12a4x1tx2t−a3x3t−12a3x1t−a5.The existence of the Hamilton in model (1) implies its ability to model natural phenomena.


### 2.2. Invariance and Symmetry of the Proposed Model

For system (1), the transformation x1t,x2t,x3t⟶−x1t,−x2t,x3t implies that the system is invariant.Then, ifx1t,x2t,x3t is a solution of model (1), then −x1t,−x2t,x3t is a solution of the same model too.


### 2.3. Dissipation of Proposed Model (1)

The divergence of proposed paradigm (1) is(7)∇·V=∂x˙1t∂x1t+∂x˙2t∂x2t+∂x˙3t∂x3t=−a1+a2−a5.Then, proposed chaotic paradigm (1) is dissipative such that(8)−a1+a2−a5<0.


### 2.4. Fixed Point and Its Stability of Proposed Model (1)

Solving the following system of equations leads to the equilibria of system (1):(9)0=a1x2t−x1t+x2tx3t,0=a2x2t−a3x1tx3t,0=a4x12t−a5x3t.System (1) has a trivial fixed point E0=0,0,0.In order to examine the stability ofE0, the Jacobian matrix of proposed model (1) at E0 is(10)JE0=−a1a100a2000−a5,the characteristic equation of JE0 is(11)λ+a1λ2+λa5−a2−a2a5=0,and as a result, the characteristic equation of JE0 has the following three eigenvalues:(12)λ1=−a1,λ2+λa5−a2−a2a5=0.By applying Routh–Hurwitz theorem, the trivial fixed point is stable if and only if(13)a1>0,a5−a2>0,and−a2a5>0.So, the constraints making the trivial fixed point stable are(14)a1>0,a5>a2,anda2a5<0,otherwise it is unstable.


### 2.5. Calculating Lyapunov Exponents for Proposed Model (1)

Proposed model (1) can be written in vector notation as(15)X˙t=hXt;η.Such thatXt=x1t,x2t,x3tT presents the vector of state space, h=h1,h2,h3T, η presents the parameters, and …T signifies matrix transpose operation. The deviations from the Xt trajectory are given in the following equation:(16)δX˙t=JijXt;ηδX,i,j=1,2,3,where Ji,j=∂hi/∂xj is the Jacobian matrix and takes the following form:(17)Ji,j=−a1a1+x3tx2t−a3x3ta2−a3x3t2a4x1t0−a5.The Lyapunov exponentsLi of the system are defined by(18)Li=limt⟶∞1tlogδxitδxi0.Numerically solve equations (14) and (15) simultaneously to find Li. Order 4 Runge–Kutta algorithm is utilized to estimate Li.The system parameters are selected as follows:a1=10, a2=5, a3=4, a4=2, and a5=3 where the initial values are chosen as x10=1, x20=2, and x30=3. The Lyapunov exponents are estimated as follows: L1=1.01, L2=0, and L3=−12.5.This implies that our proposed system (1) for this selection of the parameters a1, a2, a3, a4, and a5 is a chaotic paradigm because it has one positive Lyapunov exponent.Figure1 proves that our proposed model (1) is chaotic. Solving system (1) numerically, for specific selections of the parameter values, it is clear that it has chaotic behavior in specific domains where initially nearby trajectories break away exponentially with time. For example, Figure 1(a) displays two such numerically evaluated solutions of (1), where we give only the t,x1t plot. The chaotic behavior of those solutions is confirmed from the fact that the two initially nearby orbits break away from each other exponentially at about t≅15. As shown in Figure 1(b), model (1) has a chaotic attractor. In Figure 1(b), we plot the motion in the 3-dimensional space x1t,x2t,x5t.Figure 1
(a) Two numerically calculated solutions of (1) for a1=10, a2=5, a3=4, a4=2, and a5=3 with t0=0, x10=1, x20=2, and x30=3 (solid curve) and x10=1.001, x20=2, and x30=3.001 (dotted curve). Note the exponential separation that becomes clear at t≅15, clearing the chaotic manner of the system orbits. (b) 3-dimensional chaotic attractor in x1t,x2t,x3t space.
(a)(b)In Figure2, we fix a2=5, a3=4, a4=2, and a5=3 and vary a1∈ [1, 30]. From Figure 2, it is clear that system (1) has periodic solution when a1∈1,8.4. The chaotic attractors appear when a1∈8.4,19.3. If a1∈19.3,30, our system has fixed point solutions. Figure 2 proves that our new system contains chaotic solutions over a wide period a1∈8.4,19.3. As we did with the a1 parameter, we did with the rest of the parameters. And because we got almost the same results, we will not display the results for the rest of the parameters.Figure 2
L1, L2, and L3 versus a1, a1∈1,30 and a2=5, a3=4, a4=2, and a5=3 with the same initial conditions in Figure 1.In Figure2, we find chaotic attractors, periodic attractors, and fixed point solutions of system (1). To check this, we have solved numerically (16) (using, e.g., Mathematica 7 software) in several cases and excellent agreements are found with the results of Figure 2. For example, choosing a2=5, a3=4, a4=2, and a5=3, with the initial conditions t0=0, x10=1, x20=2, and x30=3, and when a1=7, the solution of system (1) has periodic solution trivial fixed point (see Figure 3(a)), chaotic attractor when a1=10 as in Figure 3(b). In Figure 3(c), a1=27, the solution is fixed point.Figure 3
With the same initial conditions in Figure1. (a) Periodic solution in x1t,x3t plan, a1 = 7. (b) Chaotic solution in x1t,x3t plan, a1 = 10. (c) Fixed point solution in x1t,x3t plan, a1 = 27.
(a)(b)(c)


## 2.1. System (1) Generalized Hamiltonian

Smooth nonlinear system (1) is considered, given in the following formula:(2)x˙=αx∂H∂x+σx∂H∂x,where x=x1t,x2t,x3tT, Hx is smooth energy function and universally positive definite, and ∂H/∂x is a column gradient vector of Hx.By using energy function in quadratic formula, we have(3)Hx=12xTγx,where γ signifies a constant diagonal matrix, which is symmetric and positive definite, with respect to ∂H/∂x=γx, αx is antisymmetric matrix representing the vector field of the workless part, and σx is a symmetric matrix, a negative definite, representing the working or nonconservative part of the system:(4)αx=−αxT,σx=σxT.DefineHx of the model in (1) as(5)Hx=12x1t−a3x3tx2t2+x2t−a3x3t2+x3t2,x˙1tx˙2tx˙3t=Ax1t−a3x3tx2tx2t−a3x3x3t+Bx1t−a3x3tx2tx2t−a3x3x3t,where(6)A=012−a3x3ta1+x3t−12a4x1tx2t−a3x3t−12−a3x3ta1+x3t0−12a3x1t12a4x1tx2t−a3x3t12a3x1t0,B=a1a3x3tx2t12−a3x3ta1+x3t12a4x1tx2t−a3x3t12−a3x3ta1+x3t−a2a3x3t−12a3x1t12a4x1tx2t−a3x3t−12a3x1t−a5.The existence of the Hamilton in model (1) implies its ability to model natural phenomena.


## 2.2. Invariance and Symmetry of the Proposed Model

For system (1), the transformation x1t,x2t,x3t⟶−x1t,−x2t,x3t implies that the system is invariant.Then, ifx1t,x2t,x3t is a solution of model (1), then −x1t,−x2t,x3t is a solution of the same model too.


## 2.3. Dissipation of Proposed Model (1)

The divergence of proposed paradigm (1) is(7)∇·V=∂x˙1t∂x1t+∂x˙2t∂x2t+∂x˙3t∂x3t=−a1+a2−a5.Then, proposed chaotic paradigm (1) is dissipative such that(8)−a1+a2−a5<0.


## 2.4. Fixed Point and Its Stability of Proposed Model (1)

Solving the following system of equations leads to the equilibria of system (1):(9)0=a1x2t−x1t+x2tx3t,0=a2x2t−a3x1tx3t,0=a4x12t−a5x3t.System (1) has a trivial fixed point E0=0,0,0.In order to examine the stability ofE0, the Jacobian matrix of proposed model (1) at E0 is(10)JE0=−a1a100a2000−a5,the characteristic equation of JE0 is(11)λ+a1λ2+λa5−a2−a2a5=0,and as a result, the characteristic equation of JE0 has the following three eigenvalues:(12)λ1=−a1,λ2+λa5−a2−a2a5=0.By applying Routh–Hurwitz theorem, the trivial fixed point is stable if and only if(13)a1>0,a5−a2>0,and−a2a5>0.So, the constraints making the trivial fixed point stable are(14)a1>0,a5>a2,anda2a5<0,otherwise it is unstable.


## 2.5. Calculating Lyapunov Exponents for Proposed Model (1)

Proposed model (1) can be written in vector notation as(15)X˙t=hXt;η.Such thatXt=x1t,x2t,x3tT presents the vector of state space, h=h1,h2,h3T, η presents the parameters, and …T signifies matrix transpose operation. The deviations from the Xt trajectory are given in the following equation:(16)δX˙t=JijXt;ηδX,i,j=1,2,3,where Ji,j=∂hi/∂xj is the Jacobian matrix and takes the following form:(17)Ji,j=−a1a1+x3tx2t−a3x3ta2−a3x3t2a4x1t0−a5.The Lyapunov exponentsLi of the system are defined by(18)Li=limt⟶∞1tlogδxitδxi0.Numerically solve equations (14) and (15) simultaneously to find Li. Order 4 Runge–Kutta algorithm is utilized to estimate Li.The system parameters are selected as follows:a1=10, a2=5, a3=4, a4=2, and a5=3 where the initial values are chosen as x10=1, x20=2, and x30=3. The Lyapunov exponents are estimated as follows: L1=1.01, L2=0, and L3=−12.5.This implies that our proposed system (1) for this selection of the parameters a1, a2, a3, a4, and a5 is a chaotic paradigm because it has one positive Lyapunov exponent.Figure1 proves that our proposed model (1) is chaotic. Solving system (1) numerically, for specific selections of the parameter values, it is clear that it has chaotic behavior in specific domains where initially nearby trajectories break away exponentially with time. For example, Figure 1(a) displays two such numerically evaluated solutions of (1), where we give only the t,x1t plot. The chaotic behavior of those solutions is confirmed from the fact that the two initially nearby orbits break away from each other exponentially at about t≅15. As shown in Figure 1(b), model (1) has a chaotic attractor. In Figure 1(b), we plot the motion in the 3-dimensional space x1t,x2t,x5t.Figure 1
(a) Two numerically calculated solutions of (1) for a1=10, a2=5, a3=4, a4=2, and a5=3 with t0=0, x10=1, x20=2, and x30=3 (solid curve) and x10=1.001, x20=2, and x30=3.001 (dotted curve). Note the exponential separation that becomes clear at t≅15, clearing the chaotic manner of the system orbits. (b) 3-dimensional chaotic attractor in x1t,x2t,x3t space.
(a)(b)In Figure2, we fix a2=5, a3=4, a4=2, and a5=3 and vary a1∈ [1, 30]. From Figure 2, it is clear that system (1) has periodic solution when a1∈1,8.4. The chaotic attractors appear when a1∈8.4,19.3. If a1∈19.3,30, our system has fixed point solutions. Figure 2 proves that our new system contains chaotic solutions over a wide period a1∈8.4,19.3. As we did with the a1 parameter, we did with the rest of the parameters. And because we got almost the same results, we will not display the results for the rest of the parameters.Figure 2
L1, L2, and L3 versus a1, a1∈1,30 and a2=5, a3=4, a4=2, and a5=3 with the same initial conditions in Figure 1.In Figure2, we find chaotic attractors, periodic attractors, and fixed point solutions of system (1). To check this, we have solved numerically (16) (using, e.g., Mathematica 7 software) in several cases and excellent agreements are found with the results of Figure 2. For example, choosing a2=5, a3=4, a4=2, and a5=3, with the initial conditions t0=0, x10=1, x20=2, and x30=3, and when a1=7, the solution of system (1) has periodic solution trivial fixed point (see Figure 3(a)), chaotic attractor when a1=10 as in Figure 3(b). In Figure 3(c), a1=27, the solution is fixed point.Figure 3
With the same initial conditions in Figure1. (a) Periodic solution in x1t,x3t plan, a1 = 7. (b) Chaotic solution in x1t,x3t plan, a1 = 10. (c) Fixed point solution in x1t,x3t plan, a1 = 27.
(a)(b)(c)


## 3. System Signal Flow Graph

Representing the complex systems graphically is useful for understanding the relationship between its components. Our system components are its direct state variablesx1,x2, and x3 in addition to the combined three states x12,x1x3, and x2x3.Each state variable is represented by a vertex in the digraphG of system (1). The edge u,v∈G iff ∂v/∂u≠0. Then, the adjacency matrix of G is an image of the relationships between the system state variables of system (1). Figure 4 shows the proposed graph of the studied system.Figure 4
Proposed signal flow graph of the proposed chaotic system.The following is the weighted adjacency matrix,AG, of the system’s proposed graph:(19)AG=x1x2x3x12x1x3x2x3−a100110a1a2000100−a501100a40000−a30000100000.The eigenvalues of the matrix in (9) can be used to compute the energy of the digraph G (see [19]). To compute the energy of the proposed graph, we follow the method explained in [19] (page 6).Firstly, formulate the adjacency matrix of the system graphM where(20)Mij=1,for any edgeu,v,u≠vof the graph;0,otherwise.Then,(21)MG=x1x2x3x12x1x3x2x3000110100001000011001000010000100000.The energyEG of the graph G is calculated via the following formula [19]:(22)EG=1E∑k=1Ewk∑SVDMG,where E is the number of edges in G, wE is the edge weights, and SVDMG is a vector of singular values of matrix. For more details about graph energy, see [19].Letλ1,λ2,…,λn are the eigenvalues of the matrix MTM, with repetitions. Order these so that λ1≥λ2≥⋯≥λn≥0, assuming υi=λi, then υ1≥υ2≥⋯≥υn≥0, and they are the components of the SVD. Then,(23)SVDMG=3.5321,2.3473,1.0,1.0,0,12061.Then,EG=4.5, that can be used as measure of the system complexity.


## 4. System Realization Using NI Multisim

In this section, we realize the studied system using NI Multisim and its electronic circuit is designed as shown in Figure5. The circuit consists of three integrators symbolized as X1, X2, and X3 all of which are constructed using the 741 OP. AMP; three summers S1, S2, and A3 each of which has three inputs and one output; two multipliers symbolized with X1X3, X1X1, and X2X3 each of which has two inputs and one output; three 1 micro Farad capacitors C1, C2, and C3 where it is initially charged by 1, 2, and 3 volts, respectively, to realize the initial conditions of the studied system; and ten resistors R1, R2, …, R10, where R1, R3, R5, R7, R8, R9, and R10 are equal 10 kohms and R2, R4, and R6 are equal 100 kohms. The input and output gains of the summers are designed as follows (Table 1) in order to realize the studied system. Figure 6 shows the phase portrait of the state variable x2 with respect to the state variable x1. Figure 7 shows the phase portrait of the state variable x3 with respect to the state variable x1. Figure 8 shows the phase portrait of the state variable x3 with respect to the state variable x2.Figure 5
The electronic circuit of the proposed system.Table 1
The input/output gains of the summers.
Input A gainInput B gainInput C gainOutput gainS1−101101S2−41−51A32031Figure 6
Phase portrait of the state variablex2 versus the state variable x1.Figure 7
Phase portrait of the state variablex3 with respect to the state variable x1.Figure 8
Phase portrait of the state variablex3 with respect to the state variable x2.Table2 shows the designed values of input/output gains of the used multipliers.Table 2
The input/output gains of the multipliers.
Output gainX gainY gainS1111S2111A3111


## 5. Single-State Feedback Control for the Proposed System

### 5.1. Controller Design

In this section, we designed a reference point-based single-state feedback controller. The suggested controller will drive the states of the proposed chaotic model to follow the desired predetermined reference values. The proposed controller’s input consists of the output and the states of the chaotic system. The output of the controller is used as an input of the controlled chaotic system.In [21], the proposed controller was depended on the sliding-mode technique and the Lorenz system states were derived to track the reference values but under some constraints which limit its real applications. In [22], the author proposed a track controller by which the Lorenz system states were derived to follow any of their predetermined reference values. In this work, a reference value single-state feedback controller is to be constructed in such a way that the proposed chaotic model states will be derived to track any desired reference values without any constraints.Here, it is proposed that the statex2 of the studied system is the only state that can be easily measured. For studied system (1), a novel reference point-based single-state feedback controller is designed as follows:(24)C1=−a1x2r+a1x1r−x2rx3r,C2=kx2−x2r−a2x2r+a3x1rx3r,C3=−a4x1r2+a5x3r,where xir is the desired reference value of the system state xi for all i∈1,2,3 and k is the controller gain.In (24), the closed-loop feedback is constructed only using the second state variable x2. Hence, it is called single-state feedback controller.The controlled system via the proposed controller becomes(25)dx1dt=a1x2−a1x1+x2x3+C1,dx2dt=a2x2−a3x1x3+C2,dx3dt=a4x12−a5x3+C3.Assume thatxir denotes the desired reference value of the state variable xi where i=1,2,3. And let Ei=xi−xir:i=1,2,3 is the errors set between the system state variable and their references. Then, putting x1=E1+x1r, x2=E2+x2r, and x3=E3+x3r in (25), the control error dynamical system can be derived as follows:(26)dE1dt=a1E2−a1E1+E2E3+x3rE2+x2rE3,dE2dt=a2+kE2−a3E1E3−a3x3rE1−a3x1rE3,dE3dt=a4E12+2a4x1rE1−a5E3.It is clear that theEo=0,0,0 is an equilibrium of error dynamical system (26) and its Jacobian matrix can be written as(27)JE=−a1a1+E3+x3rE2+x2r−a3E3−a3x3ra2+k−a3E1−a3x1r2a4E1+2a4x1r0−a5,JE0=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5.If the controller gaink can be calculated such that the eigenvalues of the Jacobian matrix 16 are all stable, the control errors E1, E2, and E3 will be stable asymptotically at its equilibrium E0=0,0,0. This imply that the system states x1, x2, and x3 will follow their desired reference values x1r, x2r, and x3r, respectively.Then, our task is to calculate the suitable value of the gaink. Our result is written in the following result.Theorem 1.
LetΩ=a3x3r2+a1a3x3r−a1a2+a1a5−a2a5−2a4x1rx2r, Φ=a3a5x3r2−a1a2a5+2a3a4x3rx1r2+a1a3a5x3r+2a1a3a4, a=a1+a5, b=−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r, and c=Ωa5+Ωa1−Φ−Ωa2. The states of studied system (1) can follow any desired predetermined values x1r, x2r, and x3r via the proposed controller 12, if the controller gain k is adjusted such that(28)k<mina1−a2+a5,Φa1a5−2a4x2rx1r,−b−b2−4ac2a.Proof.
Studied system (1) has the following characteristic equation at its equilibrium point:(29)Δλ=λ+a1−a1−x3r−x2ra3x3rλ−a2+ka3x1r−2a4x1r0λ+a5=0,Δλ=λ+a1−a1−x3r−x2ra3x3rλ−a2+ka3x1r−2a4x1r0λ+a5=0,
Letβ2=a1−k−a2+a5, β1=−a1+a5k+Ω, where Ω=a3x3r2+a1a3x3r−a1a2+a1a5−a2a5−2a4x1rx2r and(30)β0=a3a5x3r2−ka1a5−a1a2a5+2a3a4x3rx1r2+2ka4x2rx1r+a1a3a5x3r+2a1a3a4x1r2+2a2a4x2rx1r=k2a4x2rx1r−a1a5+Φ,where Φ=a3a5x3r2−a1a2a5+2a3a4x3rx1r2+a1a3a5x3r+2a1a3a4.
Then,Δλ can be written as(31)λ3+β2λ2+β1λ+β0=0.
Applying Routh–Hurwitz stability criterion, the eigenvalues of (31) have negative real parts if and only if β2>0, β0>0, and β2β1−β0>0.
Fromβ2>0 and β0>0, it is clear that(32)k<a1−a2+a5, andk<Φa1a5−2a4x2rx1r.
Note that(33)β2β1−β0=a1−k−a2+a5−a1+a5k+Ω−k2a4x2rx1r−a1a5+Φ,(34)=a1−k−a2+a5−a1+a5k+Ω−k2a4x2rx1r−a1a5+Φ=a1+a5k2+k−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r+Ωa5+Ωa1−Φ−Ωa2=ak2+bk+c,where a=a1+a5, b=−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r, and c=Ωa5+Ωa1−Φ−Ωa2.
From (33) and β2β1−β0>0, we can derive that(35)k−−b+b2−4ac2ak−−b−b2−4ac2a>0,which implies that(36)k−b−b2−4ac2a,ork>−b+b2−4ac2a0.
While concluding, to meetβ2>0, β0>0, and β2β1−β0>0, the control gain can be selected as(37)k<mina1−a2+a5,Φa1a5−2a4x2rx1r,−b−b2−4ac2a,which completes the proof.


### 5.2. Case Studies of Reference Value-Based Control of the Studied System

Case 1.
x1r=x2r=x3r=0
Applying Theorem1, the controller gain can be calculated and we can select it k=−8. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 9 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 9
State dynamics under the track control method with reference0,0,0 and the controller gain k=−8, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)Case 2.
x1r=10, x2r=−10, x3r=20
Applying Theorem1, the controller gain can be calculated and we can select it k=−50. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 10 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 10
State dynamics under the track control method with reference10,−10,20 and the controller gain k=−50, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)Case 3.
x1r=−6, x2r=5, x3r=−62
Applying Theorem1, the controller gain can be calculated and we can select it k=−60. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 11 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 11
State dynamics under the track control method with reference10,−10,20 and the controller gain k=−50, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)


## 5.1. Controller Design

In this section, we designed a reference point-based single-state feedback controller. The suggested controller will drive the states of the proposed chaotic model to follow the desired predetermined reference values. The proposed controller’s input consists of the output and the states of the chaotic system. The output of the controller is used as an input of the controlled chaotic system.In [21], the proposed controller was depended on the sliding-mode technique and the Lorenz system states were derived to track the reference values but under some constraints which limit its real applications. In [22], the author proposed a track controller by which the Lorenz system states were derived to follow any of their predetermined reference values. In this work, a reference value single-state feedback controller is to be constructed in such a way that the proposed chaotic model states will be derived to track any desired reference values without any constraints.Here, it is proposed that the statex2 of the studied system is the only state that can be easily measured. For studied system (1), a novel reference point-based single-state feedback controller is designed as follows:(24)C1=−a1x2r+a1x1r−x2rx3r,C2=kx2−x2r−a2x2r+a3x1rx3r,C3=−a4x1r2+a5x3r,where xir is the desired reference value of the system state xi for all i∈1,2,3 and k is the controller gain.In (24), the closed-loop feedback is constructed only using the second state variable x2. Hence, it is called single-state feedback controller.The controlled system via the proposed controller becomes(25)dx1dt=a1x2−a1x1+x2x3+C1,dx2dt=a2x2−a3x1x3+C2,dx3dt=a4x12−a5x3+C3.Assume thatxir denotes the desired reference value of the state variable xi where i=1,2,3. And let Ei=xi−xir:i=1,2,3 is the errors set between the system state variable and their references. Then, putting x1=E1+x1r, x2=E2+x2r, and x3=E3+x3r in (25), the control error dynamical system can be derived as follows:(26)dE1dt=a1E2−a1E1+E2E3+x3rE2+x2rE3,dE2dt=a2+kE2−a3E1E3−a3x3rE1−a3x1rE3,dE3dt=a4E12+2a4x1rE1−a5E3.It is clear that theEo=0,0,0 is an equilibrium of error dynamical system (26) and its Jacobian matrix can be written as(27)JE=−a1a1+E3+x3rE2+x2r−a3E3−a3x3ra2+k−a3E1−a3x1r2a4E1+2a4x1r0−a5,JE0=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5.If the controller gaink can be calculated such that the eigenvalues of the Jacobian matrix 16 are all stable, the control errors E1, E2, and E3 will be stable asymptotically at its equilibrium E0=0,0,0. This imply that the system states x1, x2, and x3 will follow their desired reference values x1r, x2r, and x3r, respectively.Then, our task is to calculate the suitable value of the gaink. Our result is written in the following result.Theorem 1.
LetΩ=a3x3r2+a1a3x3r−a1a2+a1a5−a2a5−2a4x1rx2r, Φ=a3a5x3r2−a1a2a5+2a3a4x3rx1r2+a1a3a5x3r+2a1a3a4, a=a1+a5, b=−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r, and c=Ωa5+Ωa1−Φ−Ωa2. The states of studied system (1) can follow any desired predetermined values x1r, x2r, and x3r via the proposed controller 12, if the controller gain k is adjusted such that(28)k<mina1−a2+a5,Φa1a5−2a4x2rx1r,−b−b2−4ac2a.Proof.
Studied system (1) has the following characteristic equation at its equilibrium point:(29)Δλ=λ+a1−a1−x3r−x2ra3x3rλ−a2+ka3x1r−2a4x1r0λ+a5=0,Δλ=λ+a1−a1−x3r−x2ra3x3rλ−a2+ka3x1r−2a4x1r0λ+a5=0,
Letβ2=a1−k−a2+a5, β1=−a1+a5k+Ω, where Ω=a3x3r2+a1a3x3r−a1a2+a1a5−a2a5−2a4x1rx2r and(30)β0=a3a5x3r2−ka1a5−a1a2a5+2a3a4x3rx1r2+2ka4x2rx1r+a1a3a5x3r+2a1a3a4x1r2+2a2a4x2rx1r=k2a4x2rx1r−a1a5+Φ,where Φ=a3a5x3r2−a1a2a5+2a3a4x3rx1r2+a1a3a5x3r+2a1a3a4.
Then,Δλ can be written as(31)λ3+β2λ2+β1λ+β0=0.
Applying Routh–Hurwitz stability criterion, the eigenvalues of (31) have negative real parts if and only if β2>0, β0>0, and β2β1−β0>0.
Fromβ2>0 and β0>0, it is clear that(32)k<a1−a2+a5, andk<Φa1a5−2a4x2rx1r.
Note that(33)β2β1−β0=a1−k−a2+a5−a1+a5k+Ω−k2a4x2rx1r−a1a5+Φ,(34)=a1−k−a2+a5−a1+a5k+Ω−k2a4x2rx1r−a1a5+Φ=a1+a5k2+k−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r+Ωa5+Ωa1−Φ−Ωa2=ak2+bk+c,where a=a1+a5, b=−Ω−a12−a52+a1a2−a1a5+a2a5−2a4x1rx2r, and c=Ωa5+Ωa1−Φ−Ωa2.
From (33) and β2β1−β0>0, we can derive that(35)k−−b+b2−4ac2ak−−b−b2−4ac2a>0,which implies that(36)k−b−b2−4ac2a,ork>−b+b2−4ac2a0.
While concluding, to meetβ2>0, β0>0, and β2β1−β0>0, the control gain can be selected as(37)k<mina1−a2+a5,Φa1a5−2a4x2rx1r,−b−b2−4ac2a,which completes the proof.


## 5.2. Case Studies of Reference Value-Based Control of the Studied System

Case 1.
x1r=x2r=x3r=0
Applying Theorem1, the controller gain can be calculated and we can select it k=−8. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 9 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 9
State dynamics under the track control method with reference0,0,0 and the controller gain k=−8, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)Case 2.
x1r=10, x2r=−10, x3r=20
Applying Theorem1, the controller gain can be calculated and we can select it k=−50. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 10 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 10
State dynamics under the track control method with reference10,−10,20 and the controller gain k=−50, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)Case 3.
x1r=−6, x2r=5, x3r=−62
Applying Theorem1, the controller gain can be calculated and we can select it k=−60. Proposed controller (24) is applied to studied system (1) at t=30 secs. Figure 11 shows the system states before and after applying control. It is obvious that the proposed controller drives the state variables to follow their reference values after applying it immediately.Figure 11
State dynamics under the track control method with reference10,−10,20 and the controller gain k=−50, where the controller is applied after 30 secs. (a) The state variable x1; (b) the state variable x2; (c) the state variable x3; (d) track control errors E1, E2, and E3.
(a)(b)(c)(d)


## 6. Robust Controller

In this section, a tracking robust controller is to be designed. Since in practice, the uncertainties are unavoidable matter, then it is important to design a robust single-state feedback tracking controller for the studied system.


### 6.1. Robust Track Controller Algorithm Design

Suppose the system parameters have some perturbations. The fuzzy studied system can be written in the following format:(38)dx1dt=a1+Δa1x2−a1+Δa1x1+x2x3+C1,dx2dt=a2+Δa2x2−a3+Δa3x1x3+C2,dx3dt=a4+Δa4x12−a5+Δa5x3+C3.Simplifying (38), we get(39)dx1dt=a1x2−a1x1+x2x3+Δa1x2−x1+C1,dx2dt=a2x2−a3x1x3+Δa2x2−Δa3x1x3+C2,dx3dt=a4x12−a5x3+Δa4x12−Δa5x3+C3.The uncertainties are collected in a column vectoru=u1,u2,u3T where u1=Δa1x2−x1, u2=Δa2x2−Δa3x1x3, and u3=Δa4x12−Δa5x3. Applying designed controller (24) upon system (39), the new dynamics of the tracking error will be as(40)dE1dt=a1E2−a1E1+E2E3+x3rE2+x2rE3+u1,dE2dt=a2+kE2−a3E1E3−a3x3rE1−a3x1rE3+u2,dE3dt=a4E12+2a4x1rE1−a5E3+u3.That can be written as(41)dE1dtdE2dtdE3dt=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5E1E2E3+E2E3+u1−a3E1E3+u2a4E12+u3.Let(42)E=E1E2E3,A=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5,U=U1U2U3=E2E3+u1−a3E1E3+u2a4E12+u3.Then, the Laplace transformation of the track control error systemE˙=AE+U is given as follows: Es=sI−A−1Us.Since the controller gain has no effect upon the uncertain termsU1 and U3, an improvement to the single-state feedback controller is designed as follows:(43)C1=k1x2−x2r−a1x2r+a1x1r−x2rx3r,C2=k2x2−x2r−a2x2r+a3x1rx3r,C3=k3x2−x2r−a4x1r2+a5x3r.Utilizing proposed robust controller (43) with disturbed system (39), the closed-loop track control error dynamics are given as(44)dE1dt=k1E2+a1E2−a1E1+E2E3+x3rE2+x2rE3+u1,dE2dt=k2+a2E2−a3E1E3−a3x3rE1−a3x1rE3+u2,dE3dt=k3E2+a4E12+2a4x1rE1−a5E3+u3.That can be rewritten in a compact format asE˙=BE+U which equal the system(45)dE1dtdE2dtdE3dt=−a1k1+a1+x3rx2r−a3x3ra2+k2−a3x1r2a4x1rk3−a5E1E2E3+E2E3+u1−a3E1E3+u2a4E12+u3.Then, the Laplace transform of (45) is got by Es=sI−B−1Us.Consequently, our aim now is to find the values of the gainsk1,k2,k3 in order to reduce the disturbances effects. To accomplish this task, the following fitness function is to be minimized:(46)f=∑isiI−B−1si=jωdi,where ωdi=2πfii=1,2,…, are the dominant frequencies related to the disturbances. We can calculate the dominant frequencies using the known fast Fourier transform (FFT) method [33].Genetic algorithm [34] will be used here to minimize the fitness function (46).The following is the process of finding the robust controller gainsk1, k2, and k3.Algorithm of seeking the robust controller gainsk1, k2, and k3 is as follows:(1)
Determine the dominant frequencies for the feedback state variablex2(2)
Construct fitness function (46)(3)
The nonlinear constraint is set asReλr+θ≤0, where for r=1,2,3,λr are the eigenvalues of the matrix and θ is to be set as a positive real value(4)
Utilize the GA toolbox in Matlab to choose the optimal robust controller gainsk1, k2, and k3


### 6.2. Numerical Simulation for the Robust Track Controller

In the following simulations, the disturbances are taken as(47)u1=sin2πtx2−x1+noise;u2=sin4πtx1+noise;u3=sin6πtx3;where noise=n∗random_number, n=s/10snr/20, snr is the signal to noise ratio that set as snr=20logs/n=100, and s is the standard deviation of x2.For illustration, the references are selected to bex1r=−20,x2r=40, and x3r=−60. Using FFT, the dominant frequencies of x2 are selected as f1=0.206653HZ; f2=0.319979HZ; f3=0.333311HZ; f4=0.786614HZ; f5=0.293313779HZ; f6=0.213319112HZ; f7=0.166655556; f8=0.193320445; and f9=0.219985334. Figure 12 displays the spectrum of x2 using the FFT method. The selected dominant frequencies are used to construct fitness function (46). θ is set to equal 3. Figure 13 shows the optimization process via GA. The selected gains are k1=−116.67;k2=−332;k3=−68. Figure 14 shows the system response before and after control with the corresponding track errors. The controller drives the states to follow the reference values: x1r=x2r=x3r=0. Figure 15 shows the system response before and after control with the corresponding track errors. The controller drives the states to follow the reference values: x1r=−5;x2r=5;x3r=−6.Figure 12
Amplitude versus frequencies ofx2 spectrum via FFT.Figure 13
Fitness value versus generations in GA optimization process.Figure 14
The system response before and after control withx1r=x2r=x3r=0: (a) the state variable x1 before and after control; (b) state variable x2 before and after control; (c) the state variable x3 before and after control; (d) the track errors before and after control.
(a)(b)(c)(d)Figure 15
The system response before and after control withx1r=−5;x2r=5;x3r=−6: (a) the state variable x1 before and after control; (b) the state variable x2 before and after control; (c) the state variable x3 before and after control; (d) track errors before and after control.
(a)(b)(c)(d)


## 6.1. Robust Track Controller Algorithm Design

Suppose the system parameters have some perturbations. The fuzzy studied system can be written in the following format:(38)dx1dt=a1+Δa1x2−a1+Δa1x1+x2x3+C1,dx2dt=a2+Δa2x2−a3+Δa3x1x3+C2,dx3dt=a4+Δa4x12−a5+Δa5x3+C3.Simplifying (38), we get(39)dx1dt=a1x2−a1x1+x2x3+Δa1x2−x1+C1,dx2dt=a2x2−a3x1x3+Δa2x2−Δa3x1x3+C2,dx3dt=a4x12−a5x3+Δa4x12−Δa5x3+C3.The uncertainties are collected in a column vectoru=u1,u2,u3T where u1=Δa1x2−x1, u2=Δa2x2−Δa3x1x3, and u3=Δa4x12−Δa5x3. Applying designed controller (24) upon system (39), the new dynamics of the tracking error will be as(40)dE1dt=a1E2−a1E1+E2E3+x3rE2+x2rE3+u1,dE2dt=a2+kE2−a3E1E3−a3x3rE1−a3x1rE3+u2,dE3dt=a4E12+2a4x1rE1−a5E3+u3.That can be written as(41)dE1dtdE2dtdE3dt=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5E1E2E3+E2E3+u1−a3E1E3+u2a4E12+u3.Let(42)E=E1E2E3,A=−a1a1+x3rx2r−a3x3ra2+k−a3x1r2a4x1r0−a5,U=U1U2U3=E2E3+u1−a3E1E3+u2a4E12+u3.Then, the Laplace transformation of the track control error systemE˙=AE+U is given as follows: Es=sI−A−1Us.Since the controller gain has no effect upon the uncertain termsU1 and U3, an improvement to the single-state feedback controller is designed as follows:(43)C1=k1x2−x2r−a1x2r+a1x1r−x2rx3r,C2=k2x2−x2r−a2x2r+a3x1rx3r,C3=k3x2−x2r−a4x1r2+a5x3r.Utilizing proposed robust controller (43) with disturbed system (39), the closed-loop track control error dynamics are given as(44)dE1dt=k1E2+a1E2−a1E1+E2E3+x3rE2+x2rE3+u1,dE2dt=k2+a2E2−a3E1E3−a3x3rE1−a3x1rE3+u2,dE3dt=k3E2+a4E12+2a4x1rE1−a5E3+u3.That can be rewritten in a compact format asE˙=BE+U which equal the system(45)dE1dtdE2dtdE3dt=−a1k1+a1+x3rx2r−a3x3ra2+k2−a3x1r2a4x1rk3−a5E1E2E3+E2E3+u1−a3E1E3+u2a4E12+u3.Then, the Laplace transform of (45) is got by Es=sI−B−1Us.Consequently, our aim now is to find the values of the gainsk1,k2,k3 in order to reduce the disturbances effects. To accomplish this task, the following fitness function is to be minimized:(46)f=∑isiI−B−1si=jωdi,where ωdi=2πfii=1,2,…, are the dominant frequencies related to the disturbances. We can calculate the dominant frequencies using the known fast Fourier transform (FFT) method [33].Genetic algorithm [34] will be used here to minimize the fitness function (46).The following is the process of finding the robust controller gainsk1, k2, and k3.Algorithm of seeking the robust controller gainsk1, k2, and k3 is as follows:(1)
Determine the dominant frequencies for the feedback state variablex2(2)
Construct fitness function (46)(3)
The nonlinear constraint is set asReλr+θ≤0, where for r=1,2,3,λr are the eigenvalues of the matrix and θ is to be set as a positive real value(4)
Utilize the GA toolbox in Matlab to choose the optimal robust controller gainsk1, k2, and k3


## 6.2. Numerical Simulation for the Robust Track Controller

In the following simulations, the disturbances are taken as(47)u1=sin2πtx2−x1+noise;u2=sin4πtx1+noise;u3=sin6πtx3;where noise=n∗random_number, n=s/10snr/20, snr is the signal to noise ratio that set as snr=20logs/n=100, and s is the standard deviation of x2.For illustration, the references are selected to bex1r=−20,x2r=40, and x3r=−60. Using FFT, the dominant frequencies of x2 are selected as f1=0.206653HZ; f2=0.319979HZ; f3=0.333311HZ; f4=0.786614HZ; f5=0.293313779HZ; f6=0.213319112HZ; f7=0.166655556; f8=0.193320445; and f9=0.219985334. Figure 12 displays the spectrum of x2 using the FFT method. The selected dominant frequencies are used to construct fitness function (46). θ is set to equal 3. Figure 13 shows the optimization process via GA. The selected gains are k1=−116.67;k2=−332;k3=−68. Figure 14 shows the system response before and after control with the corresponding track errors. The controller drives the states to follow the reference values: x1r=x2r=x3r=0. Figure 15 shows the system response before and after control with the corresponding track errors. The controller drives the states to follow the reference values: x1r=−5;x2r=5;x3r=−6.Figure 12
Amplitude versus frequencies ofx2 spectrum via FFT.Figure 13
Fitness value versus generations in GA optimization process.Figure 14
The system response before and after control withx1r=x2r=x3r=0: (a) the state variable x1 before and after control; (b) state variable x2 before and after control; (c) the state variable x3 before and after control; (d) the track errors before and after control.
(a)(b)(c)(d)Figure 15
The system response before and after control withx1r=−5;x2r=5;x3r=−6: (a) the state variable x1 before and after control; (b) the state variable x2 before and after control; (c) the state variable x3 before and after control; (d) track errors before and after control.
(a)(b)(c)(d)


## 7. Conclusion

In this work, the dynamics of a new proposed chaotic model have been studied. The new proposed model chaotic behaviors have been proved by calculating its Lyapunov exponents. The proposed model has been studied via discussing its symmetry and invariance, dissipativity, and the stability of its fixed point plus its Lyapunov exponents. The proposed model has been confirmed to be chaotic. The proposed chaotic model complexity has been measured using its SFG. The proposed chaotic model has been confirmed to be realizable via constructing areal electronic circuit that simulate its behavior. In addition, a new single-state feedback controller has been designed in order to operate the system such that its states can follow a predetermined reference value. The robustness of the proposed controller has been discussed. The proposed robust controller has been constructed with the aid of the genetic algorithm. For future work, we suggest utilizing the proposed model in real applications such as secure communication applications.



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*Source: 6692369-2021-01-16.xml*