# Parametric Controller Design of Hopf Bifurcation System

Authors: Jinbo Lu; Xiaorong Hou; Min Luo
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A general parametric controller design method is proposed for Hopf bifurcation of nonlinear dynamic system. This method does not increase the dimension of the system. Compared with the existing methods, the controller designed by this method has a lower controller order and a simpler structure, and it does not contain equilibrium points. The method keeps equilibrium of the origin system unchanged. Symbolic computation is used to deduce the constraints of controller, and cylindrical algebraic decomposition is used to find the stability parameter regions in parameter space of controller. The method is then employed for Hopf bifurcation control. Taking Lorenz system as an example, the controller design steps of the method and numerical simulations are discussed. Computer simulation results are presented to confirm the analytical predictions.


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## Body

## 1. Introduction

Bifurcation is a normal phenomenon of nonlinear system, and the research in bifurcation control has achieved considerable progress in the past 30 years [1]. Hopf bifurcation is an important type of bifurcation. Its practical significance lies in the notion that Hopf bifurcation is a critical state between stable and unstable. There are a lot of related research achievements [2–4], covering a lot of fields, including power system [5, 6], electronic technology [7], network technology [8], new energy [9], and optical technology [10].Hopf bifurcation research focused on phenomena analysis [11–13] and bifurcation control [14–19]. And the work of Hopf bifurcation control is designing control strategy for Hopf bifurcation system to modify the original system dynamic characteristics [16, 18, 20]. But the general applications of these designed controls are still limited. For enhancing flexibility and universality of Hopf bifurcation controller, some study of applying parametric control to Hopf bifurcation system controller design has emerged [17, 21].On the basis of nonlinear state feedback theory, [17] gives a relatively common form of Hopf bifurcation control, complete solution, and simulation of controller parameters by example. Its controller is(1)uqX,X1∗,X2∗,…,Xk∗,μ=∑i=1nAqi∏j=1kxi-xij∗+∑i=1n∑j=1kBqijxi-xij∗∏p=1kxi-xip∗+∑i=1n∑j=1kCqijxi-xij∗2∏p=1kxi-xip∗+∑i=1n∑j=1kDqijxi-xij∗2∏p=1kxi-xip∗2+⋯q=1,2,…,n,where X is system state vector, X1∗,X2∗,…,Xk∗ are k equilibrium points of system, Aqi,Bqij,Cqij, and Dqij are the controller parameters, xi is i state of system, and xij∗ and xip∗ are j and p equilibrium point value of system state xi, respectively.As can be seen from controller expression, the controller in [17] contains all of the system equilibrium points. The paper [21] has done a further study on the basis of [17]. The parameters of the controller have been simplified significantly, for a 3-dimensional system with two equilibrium points (the Rössler system). The controller in [21] is(2)u3=∑i=13Aixi-xi0+xi-xi0-+Bixi-xi0+2xi-xi0-+Bi+3xi-xi0+xi-xi0-2,where Ai, Bi, and Bi+3 are the system parameters and xi0+ and xi0- are the value of system state xi on the two equilibrium points. In addition to the controller design work, [21] used plane figure to display the controller constraints and analyzed the bifurcation cases of each connected region.Although the deep research of Hopf bifurcation control has been done in [17, 21], there are still some limitations. For example, the controller has a complicated form and higher order and includes system equilibrium points, and its parameters constraint solution is cumbersome. Besides, in [17], solution of controller parameters is not sufficient and intuitive. It just expresses several groups of controller parameters by inequality form. Except for the study of [17, 21], [22] did a further application in parametric method on the basis of [21].In view of the above questions, in this paper, the controller form is improved, and a simple and general parametric controller design method is presented. The designed controller has lower order, and there is no inclusion of system equilibrium points.The remainder structure of this paper is as follows. Section2 provides description of Hopf bifurcation system and the method and procedure of parametric controller design. Lorenz system is taken as an example in Section 3; this section designs the related parametric controller and deduces the constraints. In Section 4, numerical simulations are calculated. And in Section 5, the conclusion of this paper is given.


## 2. A Hopf Bifurcation System Description and Parametric Controller Design

### 2.1. Hopf Bifurcation System Description

Before discussing how to design a controller for Hopf bifurcation, a general formula of the Hopf bifurcation system is presented. To be more specific, consider the following general nonlinear system:(3)X˙=FX,μ=f1X,μ,f2X,μ,…,fnX,μT,where the dot denotes differentiation with respect to time t and X=x1,x2,…,xnT∈Rn is an n-dimensional state vector, while μ∈R is a scalar parameter, called bifurcation parameter (note: it can be assumed that μ is an m-dimensional vector for m≥1). F(X,μ) is a matrix of nonlinear polynomial system with Hopf bifurcations, and the following conditions must be satisfied: (1)
LetX∗ be an equilibrium (or fixed point) of the system; that is, F(X∗,μ)≡0, for any value of μ.(2)
Suppose that the Jacobian of the system evaluated at the equilibriumX∗ has eigenvalues, λj(μ), j=1,2,…,n, which may be real or complex. Assume there exists one pair of the complex conjugates λ1(μ), λ2(μ), and all of the others are in left half of complex plane.(3)
Suppose thatλ1,2(μ)=α(μ)±iβ(μ) cross the imaginary axis at μ=μ∗. When μ=μ∗, α(μ∗)=0, dα(μ∗)/dt≠0, and β(μ∗)≠0. According to the Hopf theory, a family of limit cycles will bifurcate from the equilibrium X∗ at the critical point μ∗.


### 2.2. Generic Parametric Controller Description

For system (3), we design a general formula of nonlinear state feedback control:(4)X˙=FX,μ+uX,μ,where(5)uX,μ=0,0,…,ujX,μ,0,…,0T,uj=gX,μfjX,μ+∑i=1,i≠jnc0ifiX,μ;here,(6)gX,μ=c1+∑k=1nck+1xk+cn+k+1xk2,and {c0i,c1,ck+1,cn+k+1∈R} are the parameters of controller. In the controller, the form of g(X,μ) is simple. g(X,μ) is the polynomial of system states and it is not including system equilibrium point (generally, the quadratic term is not needed). Comparing with [17], the designed controller is simpler and has lower order and is not including system equilibrium point. Under the control u, in order to keep all the original equilibria unchanged for controlled system (4), the following conditions must be satisfied:(7)uX∗,μ=0,uX,μ∗≠0.Remark 1.
The control equation given in (5) is not a unique control law. There are many other feasible controllers that may satisfy the system. In (5), the controller consists of system polynomial; c0i∈R also can change for c0i∈Rn. That is to say, c0i can be a one-dimensional parameter, which also can be a polynomial composed of system states. So on the premise of satisfying system requests, c0i has a variety of possible expressions, covered by all linear combinations composed of system states, and then it includes the controller expression proposed by [17].Remark 2.
There may not be only one component inu(X,μ), but usually one component can satisfy the control system requests, so we only design one component here. Besides, as you can see, no matter how many components it has, there will be no inclusion of system equilibrium points value in controller u(X,μ).Remark 3.
g
(
X
,
μ
) form is not fixed. Higher polynomial can be introduced in it. Here, in order to reduce the controller order, g(X,μ), the highest order is 2, and in most cases, 1 order g(X,μ) is enough.


### 2.3. Parametric Controller Design Steps

For explaining parametric controller design method and steps, a 3-dimension system controller is designed as an example in this paper. The controller design steps are as follows.Step 1.
According to the parametric controller equation, a 3-dimension system with Hopf bifurcation is given:(8)x˙1=f1X,μ,x˙2=f2X,μ,x˙3=f3X,μ.Here, X=[x1,x2,x3]T. Then, system (8) has parametric controller:(9)uX,μ=u1X,μ,u2X,μ,u3X,μT,(10)ujX,μ=gX,μfjX,μ+∑k=1nc0kfkX,μ;here, uj(X,μ)≠0,uk(X,μ)=0, k≠j,j=1,2,3, g(X)=c1+c2x1+c3x2+c4x3, {c0k,ci∈R,i=1,2,3,4}, and c0k and ci are controller parameters.In order to make concise expression, without loss of generality, controller (10) can be written as simplifying form:(11)u2X,μ=gX,μ·f2X,μ+c01f1X,μ.Step 2.
According to the closed-loop system expression after adding controller, we can get Jacobian matrix including controller parameters. Adding controller (11) on system (8), closed-loop expression can be written as(12)x˙1=f1X,μ,x˙2=f2X,μ+u2X,μ,x˙3=f3X,μ.At this point, the Jacobian matrix of closed-loop system can be described as (13)J=∂f1∂x1∂f1∂x2∂f1∂x3j21j22j23∂f3∂x1∂f3∂x2∂f3∂x3,where(14)j21=∂f2∂x1+∂f2∂x1gX+c2f2+c01∂f1∂x1,j22=∂f2∂x2+∂f2∂x2gX+c3f2+c01∂f1∂x2,j23=∂f2∂x3+∂f2∂x3gX+c4f2+c01∂f1∂x3.Step 3.
In accordance with the gotten Jacobian matrix, we can compute system characteristic equation at the system equilibrium point. Supposing one equilibrium point of system (8) is X0, then according to the Jacobian matrix, the characteristic equation at the system equilibrium point can be described as follows:(15)Dλ=det⁡λI-JX=X0=a0λ3+a1λ2+a2λ+a3;here,(16)a0=1,a1=-c01∂f1∂x2-∂f1∂x1-∂f2∂x2gX-∂f3∂x3-c3f2X=X0,a2=∂f1∂x1∂f2∂x2-∂f1∂x2∂f2∂x1-∂f2∂x3∂f3∂x2+∂f2∂x2∂f3∂x3gX-c2∂f1∂x2f2-c4∂f3∂x2f2+c3f2∂f3∂x3+∂f1∂x1+c01∂f3∂x3∂f1∂x2-∂f1∂x3∂f3∂x2+∂f1∂x1∂f2∂x2+∂f1∂x1∂f3∂x3-∂f1∂x2∂f2∂x1-∂f1∂x3∂f3∂x1-∂f2∂x3∂f3∂x2+∂f2∂x2∂f3∂x3X=X0,a3=-∂f1∂x1∂f2∂x2∂f3∂x3+∂f1∂x1∂f2∂x3∂f3∂x2+∂f1∂x2∂f2∂x1∂f3∂x3-∂f1∂x2∂f2∂x3∂f3∂x1-∂f1∂x3∂f2∂x1∂f3∂x2+∂f1∂x3∂f2∂x2∂f3∂x1gX+-c3∂f1∂x1∂f3∂x3+c4∂f1∂x1∂f3∂x2-c4∂f1∂x2∂f3∂x1+c2∂f1∂x2∂f3∂x3+c3∂f1∂x3∂f3∂x1-c2∂f1∂x3∂f3∂x2f2-∂f1∂x1∂f2∂x2∂f3∂x3+∂f1∂x1∂f2∂x3∂f3∂x2+∂f1∂x2∂f2∂x1∂f3∂x3-∂f1∂x2∂f2∂x3∂f3∂x1-∂f1∂x3∂f2∂x1∂f3∂x2+∂f1∂x3∂f2∂x2∂f3∂x1X=X0.Remark 4.
X
0 does not require the actual solution. It is only a formal symbol. You can see the specific condition in the subsequent computation example; X0 in Step 4 is the same as here.Step 4.
Using Hurwitz criterion, we can get the stable constraint of system and simplify it by polynomial division. According to Hurwitz criterion, the constraints including controller parameter are(17)a0>0,a1>0,a2>0,a3>0,g=a1a2-a0a3≥0.Considering a0=1, the system stable constraints can be written as(18)a1>0,a2>0,a3>0,g=a1a2-a3≥0.
In accordance with polynomial division, fork order real polynomial B(X) and m order real polynomial A(X), X∈Rn, k≥1, m≥1, A(X) can be written as(19)AX=A-XBX+LA/BX;here, A-(X) is the integral part of A(X) divided by B(X) and LA/B(X) is the residue of A(X) divided by B(X). If B(X0)=0 at X0, then LA/B(X0)=A(X0). Supposing G(X)=0 is system (8) equilibrium point equation and X0 is the equilibrium, then the system constraints can be simplified as(20)S:l1=La1/GX0>0l2=La2/GX0>0l3=La3/GX0>0l4=Lg/GX0>0.Step 5.
Using computer algebra tools such as symbol computation software and cylindrical algebraic decomposition algorithm to solve semialgebraic set composed of (20) and equilibrium point equation G(X)=0, we can get controller parameter space satisfying system requests.Remark 5.
In constraintsl4, when l4=0, Hopf bifurcation occurs, at which the system is critically stable, and its state characteristic is continuous oscillation. If l4>0, the system is stable and the state is convergent.


## 2.1. Hopf Bifurcation System Description

Before discussing how to design a controller for Hopf bifurcation, a general formula of the Hopf bifurcation system is presented. To be more specific, consider the following general nonlinear system:(3)X˙=FX,μ=f1X,μ,f2X,μ,…,fnX,μT,where the dot denotes differentiation with respect to time t and X=x1,x2,…,xnT∈Rn is an n-dimensional state vector, while μ∈R is a scalar parameter, called bifurcation parameter (note: it can be assumed that μ is an m-dimensional vector for m≥1). F(X,μ) is a matrix of nonlinear polynomial system with Hopf bifurcations, and the following conditions must be satisfied: (1)
LetX∗ be an equilibrium (or fixed point) of the system; that is, F(X∗,μ)≡0, for any value of μ.(2)
Suppose that the Jacobian of the system evaluated at the equilibriumX∗ has eigenvalues, λj(μ), j=1,2,…,n, which may be real or complex. Assume there exists one pair of the complex conjugates λ1(μ), λ2(μ), and all of the others are in left half of complex plane.(3)
Suppose thatλ1,2(μ)=α(μ)±iβ(μ) cross the imaginary axis at μ=μ∗. When μ=μ∗, α(μ∗)=0, dα(μ∗)/dt≠0, and β(μ∗)≠0. According to the Hopf theory, a family of limit cycles will bifurcate from the equilibrium X∗ at the critical point μ∗.


## 2.2. Generic Parametric Controller Description

For system (3), we design a general formula of nonlinear state feedback control:(4)X˙=FX,μ+uX,μ,where(5)uX,μ=0,0,…,ujX,μ,0,…,0T,uj=gX,μfjX,μ+∑i=1,i≠jnc0ifiX,μ;here,(6)gX,μ=c1+∑k=1nck+1xk+cn+k+1xk2,and {c0i,c1,ck+1,cn+k+1∈R} are the parameters of controller. In the controller, the form of g(X,μ) is simple. g(X,μ) is the polynomial of system states and it is not including system equilibrium point (generally, the quadratic term is not needed). Comparing with [17], the designed controller is simpler and has lower order and is not including system equilibrium point. Under the control u, in order to keep all the original equilibria unchanged for controlled system (4), the following conditions must be satisfied:(7)uX∗,μ=0,uX,μ∗≠0.Remark 1.
The control equation given in (5) is not a unique control law. There are many other feasible controllers that may satisfy the system. In (5), the controller consists of system polynomial; c0i∈R also can change for c0i∈Rn. That is to say, c0i can be a one-dimensional parameter, which also can be a polynomial composed of system states. So on the premise of satisfying system requests, c0i has a variety of possible expressions, covered by all linear combinations composed of system states, and then it includes the controller expression proposed by [17].Remark 2.
There may not be only one component inu(X,μ), but usually one component can satisfy the control system requests, so we only design one component here. Besides, as you can see, no matter how many components it has, there will be no inclusion of system equilibrium points value in controller u(X,μ).Remark 3.
g
(
X
,
μ
) form is not fixed. Higher polynomial can be introduced in it. Here, in order to reduce the controller order, g(X,μ), the highest order is 2, and in most cases, 1 order g(X,μ) is enough.


## 2.3. Parametric Controller Design Steps

For explaining parametric controller design method and steps, a 3-dimension system controller is designed as an example in this paper. The controller design steps are as follows.Step 1.
According to the parametric controller equation, a 3-dimension system with Hopf bifurcation is given:(8)x˙1=f1X,μ,x˙2=f2X,μ,x˙3=f3X,μ.Here, X=[x1,x2,x3]T. Then, system (8) has parametric controller:(9)uX,μ=u1X,μ,u2X,μ,u3X,μT,(10)ujX,μ=gX,μfjX,μ+∑k=1nc0kfkX,μ;here, uj(X,μ)≠0,uk(X,μ)=0, k≠j,j=1,2,3, g(X)=c1+c2x1+c3x2+c4x3, {c0k,ci∈R,i=1,2,3,4}, and c0k and ci are controller parameters.In order to make concise expression, without loss of generality, controller (10) can be written as simplifying form:(11)u2X,μ=gX,μ·f2X,μ+c01f1X,μ.Step 2.
According to the closed-loop system expression after adding controller, we can get Jacobian matrix including controller parameters. Adding controller (11) on system (8), closed-loop expression can be written as(12)x˙1=f1X,μ,x˙2=f2X,μ+u2X,μ,x˙3=f3X,μ.At this point, the Jacobian matrix of closed-loop system can be described as (13)J=∂f1∂x1∂f1∂x2∂f1∂x3j21j22j23∂f3∂x1∂f3∂x2∂f3∂x3,where(14)j21=∂f2∂x1+∂f2∂x1gX+c2f2+c01∂f1∂x1,j22=∂f2∂x2+∂f2∂x2gX+c3f2+c01∂f1∂x2,j23=∂f2∂x3+∂f2∂x3gX+c4f2+c01∂f1∂x3.Step 3.
In accordance with the gotten Jacobian matrix, we can compute system characteristic equation at the system equilibrium point. Supposing one equilibrium point of system (8) is X0, then according to the Jacobian matrix, the characteristic equation at the system equilibrium point can be described as follows:(15)Dλ=det⁡λI-JX=X0=a0λ3+a1λ2+a2λ+a3;here,(16)a0=1,a1=-c01∂f1∂x2-∂f1∂x1-∂f2∂x2gX-∂f3∂x3-c3f2X=X0,a2=∂f1∂x1∂f2∂x2-∂f1∂x2∂f2∂x1-∂f2∂x3∂f3∂x2+∂f2∂x2∂f3∂x3gX-c2∂f1∂x2f2-c4∂f3∂x2f2+c3f2∂f3∂x3+∂f1∂x1+c01∂f3∂x3∂f1∂x2-∂f1∂x3∂f3∂x2+∂f1∂x1∂f2∂x2+∂f1∂x1∂f3∂x3-∂f1∂x2∂f2∂x1-∂f1∂x3∂f3∂x1-∂f2∂x3∂f3∂x2+∂f2∂x2∂f3∂x3X=X0,a3=-∂f1∂x1∂f2∂x2∂f3∂x3+∂f1∂x1∂f2∂x3∂f3∂x2+∂f1∂x2∂f2∂x1∂f3∂x3-∂f1∂x2∂f2∂x3∂f3∂x1-∂f1∂x3∂f2∂x1∂f3∂x2+∂f1∂x3∂f2∂x2∂f3∂x1gX+-c3∂f1∂x1∂f3∂x3+c4∂f1∂x1∂f3∂x2-c4∂f1∂x2∂f3∂x1+c2∂f1∂x2∂f3∂x3+c3∂f1∂x3∂f3∂x1-c2∂f1∂x3∂f3∂x2f2-∂f1∂x1∂f2∂x2∂f3∂x3+∂f1∂x1∂f2∂x3∂f3∂x2+∂f1∂x2∂f2∂x1∂f3∂x3-∂f1∂x2∂f2∂x3∂f3∂x1-∂f1∂x3∂f2∂x1∂f3∂x2+∂f1∂x3∂f2∂x2∂f3∂x1X=X0.Remark 4.
X
0 does not require the actual solution. It is only a formal symbol. You can see the specific condition in the subsequent computation example; X0 in Step 4 is the same as here.Step 4.
Using Hurwitz criterion, we can get the stable constraint of system and simplify it by polynomial division. According to Hurwitz criterion, the constraints including controller parameter are(17)a0>0,a1>0,a2>0,a3>0,g=a1a2-a0a3≥0.Considering a0=1, the system stable constraints can be written as(18)a1>0,a2>0,a3>0,g=a1a2-a3≥0.
In accordance with polynomial division, fork order real polynomial B(X) and m order real polynomial A(X), X∈Rn, k≥1, m≥1, A(X) can be written as(19)AX=A-XBX+LA/BX;here, A-(X) is the integral part of A(X) divided by B(X) and LA/B(X) is the residue of A(X) divided by B(X). If B(X0)=0 at X0, then LA/B(X0)=A(X0). Supposing G(X)=0 is system (8) equilibrium point equation and X0 is the equilibrium, then the system constraints can be simplified as(20)S:l1=La1/GX0>0l2=La2/GX0>0l3=La3/GX0>0l4=Lg/GX0>0.Step 5.
Using computer algebra tools such as symbol computation software and cylindrical algebraic decomposition algorithm to solve semialgebraic set composed of (20) and equilibrium point equation G(X)=0, we can get controller parameter space satisfying system requests.Remark 5.
In constraintsl4, when l4=0, Hopf bifurcation occurs, at which the system is critically stable, and its state characteristic is continuous oscillation. If l4>0, the system is stable and the state is convergent.


## 3. Lorenz System Controller Design

It is well known that the Lorenz system widely exists in various research areas [23–26] and it can exhibit complex dynamics, including equilibria, limit cycles, and chaos. This section will explain the controller design and verification process by Lorenz system.


### 3.1. Lorenz System Description

According to [17], Lorenz system can be described as(21)x˙=-ax-y,y˙=bx-xz-y,z˙=xy-cz,in which a,b, and c are variable parameters. These parameters have direct effect on system characteristics. A little change of them may change the system state and make system trajectory turn from stable to bifurcation and even chaos [17]. According to (21), we can obtain system equilibrium point equation:(22)x03-bc-cx0=0.


### 3.2. Controller Design

In accordance with (11), controller can be designed as(23)u2=y˙c1+c2x+c3y+c4z+c0x˙,where c0∈R, c1∈R, c2∈R, c3∈R, and c4∈R are controller parameters. According to (21), controller can be rewritten as(24)u2=bx-xz-yc1+c2x+c3y+c4z-c0ax-y.Compared with the existing research work [17, 21], (24) has a more concise expression, and it does not include equilibrium point value.


### 3.3. System Stable and Hopf Bifurcation Condition Computation

Lorenz system state characteristics will be directly affected by system parameters. In order to verify controller performance, we select system parametersa=4, b=16, and c=1. When there is no controller, system state trajectory is chaotic. Taking system parameters in Lorenz system (21), we can obtain a new system: (25)x˙=-4x-y,y˙=16x-xz-y,z˙=xy-z.In accordance with (25), the equation at system equilibrium point is(26)x03-15x0=0.For simplifying computation process, let c0=0, c3=0, and the controller can be simplified as(27)u2x,y,z=y˙c1+c2x+c4z.Taking in (27), after computation and arrangement, we can get system constraints:(28)l1=c4x02+c2x0+c1+6,(29)l2=5+5c1+16c4x02+16c2x0-59c1-55,(30)l3=120c4+12c1+12x02+120c2x0-60c1-60,(31)l4=5c12+16c22+240c42+32c1c4+23c1-4c4+18x02+32c1c2+480c2c4-4c2x0-59c12-349c1-270.Combined with (26) at equilibrium point, constraints equations (28)~(31) constitute a semialgebra set. Using computer algebra tools, we can simplify the semialgebra set. The arranged system constraints are as follows: (32)-5≤c1<-1,c1+15c4+1>0,-15c22+c1+15c4+12>0,15c22+c1+15c4-182-164>0,φ≥0,where(33)φ=c12-15c22+255c42+30c1c416c12-240c22+3600c42+480c1c4-120c4-8c1+1.In (32), if c1=-5 or φ=0, the Hopf bifurcation will happen in system. The constraints listed in (32) can use cylindrical algebraic decomposition algorithm to calculate the controller parameter space. But the 3-dimension decomposition result is still not intuitive. In the following numerical simulation, we will assign a certain value to c1 and then mark out the controller parameter range in 2-dimension space.


## 3.1. Lorenz System Description

According to [17], Lorenz system can be described as(21)x˙=-ax-y,y˙=bx-xz-y,z˙=xy-cz,in which a,b, and c are variable parameters. These parameters have direct effect on system characteristics. A little change of them may change the system state and make system trajectory turn from stable to bifurcation and even chaos [17]. According to (21), we can obtain system equilibrium point equation:(22)x03-bc-cx0=0.


## 3.2. Controller Design

In accordance with (11), controller can be designed as(23)u2=y˙c1+c2x+c3y+c4z+c0x˙,where c0∈R, c1∈R, c2∈R, c3∈R, and c4∈R are controller parameters. According to (21), controller can be rewritten as(24)u2=bx-xz-yc1+c2x+c3y+c4z-c0ax-y.Compared with the existing research work [17, 21], (24) has a more concise expression, and it does not include equilibrium point value.


## 3.3. System Stable and Hopf Bifurcation Condition Computation

Lorenz system state characteristics will be directly affected by system parameters. In order to verify controller performance, we select system parametersa=4, b=16, and c=1. When there is no controller, system state trajectory is chaotic. Taking system parameters in Lorenz system (21), we can obtain a new system: (25)x˙=-4x-y,y˙=16x-xz-y,z˙=xy-z.In accordance with (25), the equation at system equilibrium point is(26)x03-15x0=0.For simplifying computation process, let c0=0, c3=0, and the controller can be simplified as(27)u2x,y,z=y˙c1+c2x+c4z.Taking in (27), after computation and arrangement, we can get system constraints:(28)l1=c4x02+c2x0+c1+6,(29)l2=5+5c1+16c4x02+16c2x0-59c1-55,(30)l3=120c4+12c1+12x02+120c2x0-60c1-60,(31)l4=5c12+16c22+240c42+32c1c4+23c1-4c4+18x02+32c1c2+480c2c4-4c2x0-59c12-349c1-270.Combined with (26) at equilibrium point, constraints equations (28)~(31) constitute a semialgebra set. Using computer algebra tools, we can simplify the semialgebra set. The arranged system constraints are as follows: (32)-5≤c1<-1,c1+15c4+1>0,-15c22+c1+15c4+12>0,15c22+c1+15c4-182-164>0,φ≥0,where(33)φ=c12-15c22+255c42+30c1c416c12-240c22+3600c42+480c1c4-120c4-8c1+1.In (32), if c1=-5 or φ=0, the Hopf bifurcation will happen in system. The constraints listed in (32) can use cylindrical algebraic decomposition algorithm to calculate the controller parameter space. But the 3-dimension decomposition result is still not intuitive. In the following numerical simulation, we will assign a certain value to c1 and then mark out the controller parameter range in 2-dimension space.


## 4. Simulations

### 4.1. Constraints Calculation

Through the above analysis results, we can seec1∈-5,-1. If the system is required to be stable, parameter c1 should satisfy c1∈(-5,-1). Because the system has a bifurcation when c1=-5, we can choose two different conditions c1=-3 and c1=-5 for analysis in numerical simulation.Whenc1=-3, Lorenz system constraints (32) can be rewritten as(34)c4>215,-15c22+225c42-60c4+4>0,15c22+225c42-3754c4+394>0,φ=5c22-75c42+30c4-3240c22-3600c42+1560c4-169≥0.Whenc1=-5, system constraints (32) can be rewritten as(35)c4>415,-15c22+225c42-120c4+16>0,15c22+225c42-6154c4+1054>0,φ=-3c22+45c42-30c4+5-80c22+1200c42-840c4+147≥0.


### 4.2. Controller Parameter Region

As shown in Figure1, implicit function curves of constraints (34) equations are drawn in the 2-dimension space. According to cylindrical algebraic decomposition algorithm, we can search the space divided by these function curves; then region Qs is found. In the region, c2 and c4 can stabilize system. The boundary curves of Qs are L1 and L2, and L1 and L2 are part of φ=0 in (34). L1 is line 60c2+6015c4-1315=0 when c2≤0; L2 is line -60c2+6015c4-1315=0 when c2≥0.Figure 1
System constraints curves inc2-c4 plane when c0=0 and c1=-3.In order to provide a convenient explanation, we suppose the three equilibrium points of (26) are E0, E-, and E+, where E- and E+ are a pair of symmetric equilibrium points. For parameter space solved by constraints (34), the system state at E0 is convergent. Choosing different parameters can decide whether there is a bifurcation at system equilibrium points E- and E+: (i) there is no existing Hopf bifurcation at two equilibrium points, while the system states are stable, and at the moment the controller parameters c2 and c4 value should be in Qs but not on the boundary; (ii) system states are stable at equilibrium point E- while at E+ there is a Hopf bifurcation, and at the moment controller parameter value should on the boundary L1; (iii) there is a Hopf bifurcation at equilibrium point E- while system is stable at E+, and at the moment controller parameter should be on the boundary L2; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of L1 and L2.The implicit function curves of constraints (35) equations are shown in Figure 2. Similar to constraints (34), using cylindrical algebraic decomposition algorithm, we can find the parameter space Qh which satisfies the control requirements. The boundaries of region Qh are lines L3 and L4. They are part of φ=0 in (35). L3 is line 20c2+2015c4-715=0 when c2≤0, and L4 is line -20c2+2015c4-715=0 when c2≥0.Figure 2
System constraints curves inc2-c4 plane when c0=0 and c1=-5.Unlike constraints (34), there is a Hopf bifurcation at E0 in (26) by this time. According to the different states at equilibrium points E- and E+, the cases are as follows: (i) both equilibria become stable without Hopf bifurcations; at the moment, the controller parameters c2 and c4 value should be selected in Qh and should not be on the boundary; (ii) the system states are stable at equilibrium point E-, while there is a Hopf bifurcation at E+, and at the moment controller parameter value should on the boundary L3; (iii) there is a Hopf bifurcation at equilibrium point E-, while at E+, system is stable, and at the moment controller parameter should be on the boundary L4; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of L3 and L4.


### 4.3. Numeric Simulation Computation Near Equilibria

According to the different conditions at three equilibrium points, we compute the controller parameters below. In order to explain the validity of solved controller parameter range, we conduct the numeric simulation of state stability near equilibrium points for Lorenz system (25). The simulation results are shown in Figures 3–20. When simulating, the initial values of system equilibrium points E0, E-, and E+ are (0,1,-1), (-4,-4,16), and (4,4,16).Figure 3
The Lorenz system trajectory is stable atE0 when c1=-3.Figure 4
Stable trajectory atE-, when c1=-3, for case (i).Figure 5
Stable trajectory atE+, when c1=-3, for case (i).Figure 6
Stable trajectory atE-, when c1=-3, for case (ii).Figure 7
Hopf bifurcation atE+, when c1=-3, for case (ii).Figure 8
Hopf bifurcation atE-, when c1=-3, for case (iii).Figure 9
Stable trajectory atE+, when c1=-3, for case (iii).Figure 10
Hopf bifurcation atE-, when c1=-3, for case (iv).Figure 11
Hopf bifurcation atE+, when c1=-3, for case (iv).Figure 12
System Hopf bifurcation trajectory atE0 when c1=-5.Figure 13
Stable trajectory atE-, when c1=-5, for case (i).Figure 14
Stable trajectory atE+, when c1=-5, for case (i).Figure 15
Stable trajectory atE-, when c1=-5, for case (ii).Figure 16
Hopf bifurcation atE+, when c1=-5, for case (ii).Figure 17
Hopf bifurcation atE-, when c1=-5, for case (iii).Figure 18
Stable trajectory atE+, when c1=-5, for case (iii).Figure 19
Hopf bifurcation atE-, when c1=-5, for case (iv).Figure 20
Hopf bifurcation atE+, when c1=-5, for case (iv).When controller parameterc1=-3, the system trajectory at E0 is convergent, while according to the different cases of E- and E+, parameters c2 and c4 can be selected as follows: (i) when the system is stable at two equilibrium points, we can select c2=1, c4=1; (ii) system is stable at equilibrium point E-, while at E+ it has a Hopf bifurcation, and at the moment we can select c2=-1, c4=(1/60)13+415≈0.47487; (iii) there is a Hopf bifurcation at equilibrium point E-, while system is stable at E+, and at the moment we can select c2=1, c4=(1/60)13+415≈0.47487; (iv) both equilibria E- and E+ have Hopf bifurcations, and at the moment we can select c2=0, c4=13/60≈0.21667.In Figure3 the Lorenz system trajectory is convergent at E0 when c1=-3. Figures 4–11 are system simulation trajectories for different cases of E- and E+, in which, corresponding to case (i), Figure 4 is the stable convergent trajectory at E-, and Figure 5 is the stable convergent trajectory at E+. Corresponding to case (ii), Figure 6 is the stable convergent trajectory at E-, and Figure 7 is the system trajectory of Hopf bifurcation at E+. For case (iii), Figure 8 is the Hopf bifurcation trajectory at E-, and Figure 9 is the stable convergent trajectory at E+. Figures 10 and 11 are the system trajectories of Hopf bifurcation at E- and E+ for case (iv).When controller parameterc1=-5, system has Hopf bifurcation at E0, and its trajectory is a limit cycle. And according to different cases at E- and E+, parameters c2 and c4 can be chosen as follows: (i) when the system is stable at two equilibrium points, we can select c2=1, c4=1; (ii) system is stable at equilibrium point E-, while there is a Hopf bifurcation at E+, and at the moment we can select c2=-1, c4=(1/60)21+415≈0.60819; (iii) there is a Hopf bifurcation at equilibrium point E-, while system is stable at E+, and at the moment we can select c2=1, c4=(1/60)21+415≈0.60819; (iv) there are Hopf bifurcations at both equilibria, and we can select c2=0, c4=7/20=0.35.Figure12 is the system Hopf bifurcation trajectory at E0 when c1=-5, and at this moment system converges to a stable limit cycle. Figures 13–20 are system simulation trajectories at E- and E+ for different cases, in which, according to case (i), Figures 13 and 14 are the stable trajectories at E- and E+. In accordance with case (ii), Figure 15 is the stable trajectory at E-, and Figure 16 is the Hopf bifurcation trajectory at E+. For case (iii), Figure 17 is the Hopf bifurcation trajectory at E-, and Figure 18 is the stable trajectory at E+. In accordance with condition (iv), Figure 19 is the Hopf bifurcation trajectory at E-, and Figure 20 is the Hopf bifurcation trajectory at E+.Figures21 and 22 are the system stable trajectories, using the controllers of this paper and [17]. Curve 1 is the system trajectory under the control of [17] and curve 2 is under the control of this paper. From Figures 21 and 22, it is easy to see that both controllers can stabilize the Lorenz system at equilibrium, and the controller designed in this paper is more simple and practical.Figure 21
Stable atE- using controllers in [17] and this paper.Figure 22
Stable atE+ using controllers in [17] and this paper.


## 4.1. Constraints Calculation

Through the above analysis results, we can seec1∈-5,-1. If the system is required to be stable, parameter c1 should satisfy c1∈(-5,-1). Because the system has a bifurcation when c1=-5, we can choose two different conditions c1=-3 and c1=-5 for analysis in numerical simulation.Whenc1=-3, Lorenz system constraints (32) can be rewritten as(34)c4>215,-15c22+225c42-60c4+4>0,15c22+225c42-3754c4+394>0,φ=5c22-75c42+30c4-3240c22-3600c42+1560c4-169≥0.Whenc1=-5, system constraints (32) can be rewritten as(35)c4>415,-15c22+225c42-120c4+16>0,15c22+225c42-6154c4+1054>0,φ=-3c22+45c42-30c4+5-80c22+1200c42-840c4+147≥0.


## 4.2. Controller Parameter Region

As shown in Figure1, implicit function curves of constraints (34) equations are drawn in the 2-dimension space. According to cylindrical algebraic decomposition algorithm, we can search the space divided by these function curves; then region Qs is found. In the region, c2 and c4 can stabilize system. The boundary curves of Qs are L1 and L2, and L1 and L2 are part of φ=0 in (34). L1 is line 60c2+6015c4-1315=0 when c2≤0; L2 is line -60c2+6015c4-1315=0 when c2≥0.Figure 1
System constraints curves inc2-c4 plane when c0=0 and c1=-3.In order to provide a convenient explanation, we suppose the three equilibrium points of (26) are E0, E-, and E+, where E- and E+ are a pair of symmetric equilibrium points. For parameter space solved by constraints (34), the system state at E0 is convergent. Choosing different parameters can decide whether there is a bifurcation at system equilibrium points E- and E+: (i) there is no existing Hopf bifurcation at two equilibrium points, while the system states are stable, and at the moment the controller parameters c2 and c4 value should be in Qs but not on the boundary; (ii) system states are stable at equilibrium point E- while at E+ there is a Hopf bifurcation, and at the moment controller parameter value should on the boundary L1; (iii) there is a Hopf bifurcation at equilibrium point E- while system is stable at E+, and at the moment controller parameter should be on the boundary L2; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of L1 and L2.The implicit function curves of constraints (35) equations are shown in Figure 2. Similar to constraints (34), using cylindrical algebraic decomposition algorithm, we can find the parameter space Qh which satisfies the control requirements. The boundaries of region Qh are lines L3 and L4. They are part of φ=0 in (35). L3 is line 20c2+2015c4-715=0 when c2≤0, and L4 is line -20c2+2015c4-715=0 when c2≥0.Figure 2
System constraints curves inc2-c4 plane when c0=0 and c1=-5.Unlike constraints (34), there is a Hopf bifurcation at E0 in (26) by this time. According to the different states at equilibrium points E- and E+, the cases are as follows: (i) both equilibria become stable without Hopf bifurcations; at the moment, the controller parameters c2 and c4 value should be selected in Qh and should not be on the boundary; (ii) the system states are stable at equilibrium point E-, while there is a Hopf bifurcation at E+, and at the moment controller parameter value should on the boundary L3; (iii) there is a Hopf bifurcation at equilibrium point E-, while at E+, system is stable, and at the moment controller parameter should be on the boundary L4; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of L3 and L4.


## 4.3. Numeric Simulation Computation Near Equilibria

According to the different conditions at three equilibrium points, we compute the controller parameters below. In order to explain the validity of solved controller parameter range, we conduct the numeric simulation of state stability near equilibrium points for Lorenz system (25). The simulation results are shown in Figures 3–20. When simulating, the initial values of system equilibrium points E0, E-, and E+ are (0,1,-1), (-4,-4,16), and (4,4,16).Figure 3
The Lorenz system trajectory is stable atE0 when c1=-3.Figure 4
Stable trajectory atE-, when c1=-3, for case (i).Figure 5
Stable trajectory atE+, when c1=-3, for case (i).Figure 6
Stable trajectory atE-, when c1=-3, for case (ii).Figure 7
Hopf bifurcation atE+, when c1=-3, for case (ii).Figure 8
Hopf bifurcation atE-, when c1=-3, for case (iii).Figure 9
Stable trajectory atE+, when c1=-3, for case (iii).Figure 10
Hopf bifurcation atE-, when c1=-3, for case (iv).Figure 11
Hopf bifurcation atE+, when c1=-3, for case (iv).Figure 12
System Hopf bifurcation trajectory atE0 when c1=-5.Figure 13
Stable trajectory atE-, when c1=-5, for case (i).Figure 14
Stable trajectory atE+, when c1=-5, for case (i).Figure 15
Stable trajectory atE-, when c1=-5, for case (ii).Figure 16
Hopf bifurcation atE+, when c1=-5, for case (ii).Figure 17
Hopf bifurcation atE-, when c1=-5, for case (iii).Figure 18
Stable trajectory atE+, when c1=-5, for case (iii).Figure 19
Hopf bifurcation atE-, when c1=-5, for case (iv).Figure 20
Hopf bifurcation atE+, when c1=-5, for case (iv).When controller parameterc1=-3, the system trajectory at E0 is convergent, while according to the different cases of E- and E+, parameters c2 and c4 can be selected as follows: (i) when the system is stable at two equilibrium points, we can select c2=1, c4=1; (ii) system is stable at equilibrium point E-, while at E+ it has a Hopf bifurcation, and at the moment we can select c2=-1, c4=(1/60)13+415≈0.47487; (iii) there is a Hopf bifurcation at equilibrium point E-, while system is stable at E+, and at the moment we can select c2=1, c4=(1/60)13+415≈0.47487; (iv) both equilibria E- and E+ have Hopf bifurcations, and at the moment we can select c2=0, c4=13/60≈0.21667.In Figure3 the Lorenz system trajectory is convergent at E0 when c1=-3. Figures 4–11 are system simulation trajectories for different cases of E- and E+, in which, corresponding to case (i), Figure 4 is the stable convergent trajectory at E-, and Figure 5 is the stable convergent trajectory at E+. Corresponding to case (ii), Figure 6 is the stable convergent trajectory at E-, and Figure 7 is the system trajectory of Hopf bifurcation at E+. For case (iii), Figure 8 is the Hopf bifurcation trajectory at E-, and Figure 9 is the stable convergent trajectory at E+. Figures 10 and 11 are the system trajectories of Hopf bifurcation at E- and E+ for case (iv).When controller parameterc1=-5, system has Hopf bifurcation at E0, and its trajectory is a limit cycle. And according to different cases at E- and E+, parameters c2 and c4 can be chosen as follows: (i) when the system is stable at two equilibrium points, we can select c2=1, c4=1; (ii) system is stable at equilibrium point E-, while there is a Hopf bifurcation at E+, and at the moment we can select c2=-1, c4=(1/60)21+415≈0.60819; (iii) there is a Hopf bifurcation at equilibrium point E-, while system is stable at E+, and at the moment we can select c2=1, c4=(1/60)21+415≈0.60819; (iv) there are Hopf bifurcations at both equilibria, and we can select c2=0, c4=7/20=0.35.Figure12 is the system Hopf bifurcation trajectory at E0 when c1=-5, and at this moment system converges to a stable limit cycle. Figures 13–20 are system simulation trajectories at E- and E+ for different cases, in which, according to case (i), Figures 13 and 14 are the stable trajectories at E- and E+. In accordance with case (ii), Figure 15 is the stable trajectory at E-, and Figure 16 is the Hopf bifurcation trajectory at E+. For case (iii), Figure 17 is the Hopf bifurcation trajectory at E-, and Figure 18 is the stable trajectory at E+. In accordance with condition (iv), Figure 19 is the Hopf bifurcation trajectory at E-, and Figure 20 is the Hopf bifurcation trajectory at E+.Figures21 and 22 are the system stable trajectories, using the controllers of this paper and [17]. Curve 1 is the system trajectory under the control of [17] and curve 2 is under the control of this paper. From Figures 21 and 22, it is easy to see that both controllers can stabilize the Lorenz system at equilibrium, and the controller designed in this paper is more simple and practical.Figure 21
Stable atE- using controllers in [17] and this paper.Figure 22
Stable atE+ using controllers in [17] and this paper.


## 5. Conclusions

Aiming at the limitations of existing controller such as complex controller form, high order, including system equilibrium points in controller, and complicated solving method for parameters constraints, in this paper, a new parametric controller design method is proposed by researching and analyzing present Hopf bifurcation nonlinear system parametric controller design methods. This paper presents the design principle analysis and computation steps of this new parametric controller, designs a corresponding parametric controller for a Lorenz system example, deduces the constraints, and explains the controller method validity by numeric simulation. From simulation, compared with the other methods, it is easy to see that this new parametric controller method has simple form, lower order, not including any equilibrium point value.



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*Source: 584954-2015-12-31.xml*