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.
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 iswhere is system state vector, are equilibrium points of system, , and are the controller parameters, is state of system, and and are and equilibrium point value of system state , 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] iswhere , , and are the system parameters and and are the value of system state 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. Section 2 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:where the dot denotes differentiation with respect to time and is an -dimensional state vector, while is a scalar parameter, called bifurcation parameter (note: it can be assumed that is an -dimensional vector for ). is a matrix of nonlinear polynomial system with Hopf bifurcations, and the following conditions must be satisfied: (1)Let be an equilibrium (or fixed point) of the system; that is, , for any value of .(2)Suppose that the Jacobian of the system evaluated at the equilibrium has eigenvalues, , , which may be real or complex. Assume there exists one pair of the complex conjugates , , and all of the others are in left half of complex plane.(3)Suppose that cross the imaginary axis at . When , , , and . According to the Hopf theory, a family of limit cycles will bifurcate from the equilibrium at the critical point .
2.2. Generic Parametric Controller Description
For system (3), we design a general formula of nonlinear state feedback control:wherehere,and are the parameters of controller. In the controller, the form of is simple. 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 , in order to keep all the original equilibria unchanged for controlled system (4), the following conditions must be satisfied:
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; also can change for . That is to say, can be a one-dimensional parameter, which also can be a polynomial composed of system states. So on the premise of satisfying system requests, 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 in , 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 .
Remark 3. form is not fixed. Higher polynomial can be introduced in it. Here, in order to reduce the controller order, , the highest order is 2, and in most cases, 1 order 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:Here, . Then, system (8) has parametric controller:here, , , , , and and are controller parameters.
In order to make concise expression, without loss of generality, controller (10) can be written as simplifying form:
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 asAt this point, the Jacobian matrix of closed-loop system can be described as where
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 , then according to the Jacobian matrix, the characteristic equation at the system equilibrium point can be described as follows:here,
Remark 4. does not require the actual solution. It is only a formal symbol. You can see the specific condition in the subsequent computation example; 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 areConsidering , the system stable constraints can be written asIn accordance with polynomial division, for order real polynomial and order real polynomial , , , , can be written ashere, is the integral part of divided by and is the residue of divided by . If at , then . Supposing is system (8) equilibrium point equation and is the equilibrium, then the system constraints can be simplified as
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 , we can get controller parameter space satisfying system requests.
Remark 5. In constraints , when , Hopf bifurcation occurs, at which the system is critically stable, and its state characteristic is continuous oscillation. If , 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 asin which , and 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:
3.2. Controller Design
In accordance with (11), controller can be designed aswhere , , , , and are controller parameters. According to (21), controller can be rewritten asCompared 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 parameters , , and . When there is no controller, system state trajectory is chaotic. Taking system parameters in Lorenz system (21), we can obtain a new system:
In accordance with (25), the equation at system equilibrium point isFor simplifying computation process, let , , and the controller can be simplified asTaking in (27), after computation and arrangement, we can get system constraints: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: whereIn (32), if or , 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 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 see . If the system is required to be stable, parameter should satisfy . Because the system has a bifurcation when , we can choose two different conditions and for analysis in numerical simulation.
When , Lorenz system constraints (32) can be rewritten as
When , system constraints (32) can be rewritten as
4.2. Controller Parameter Region
As shown in Figure 1, 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 is found. In the region, and can stabilize system. The boundary curves of are and , and and are part of in (34). is line when ; is line when .
In order to provide a convenient explanation, we suppose the three equilibrium points of (26) are , , and , where and are a pair of symmetric equilibrium points. For parameter space solved by constraints (34), the system state at is convergent. Choosing different parameters can decide whether there is a bifurcation at system equilibrium points and : (i) there is no existing Hopf bifurcation at two equilibrium points, while the system states are stable, and at the moment the controller parameters and value should be in but not on the boundary; (ii) system states are stable at equilibrium point while at there is a Hopf bifurcation, and at the moment controller parameter value should on the boundary ; (iii) there is a Hopf bifurcation at equilibrium point while system is stable at , and at the moment controller parameter should be on the boundary ; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of and .
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 which satisfies the control requirements. The boundaries of region are lines and . They are part of in (35). is line when , and is line when .
Unlike constraints (34), there is a Hopf bifurcation at in (26) by this time. According to the different states at equilibrium points and , the cases are as follows: (i) both equilibria become stable without Hopf bifurcations; at the moment, the controller parameters and value should be selected in and should not be on the boundary; (ii) the system states are stable at equilibrium point , while there is a Hopf bifurcation at , and at the moment controller parameter value should on the boundary ; (iii) there is a Hopf bifurcation at equilibrium point , while at , system is stable, and at the moment controller parameter should be on the boundary ; (iv) there are Hopf bifurcations at both equilibrium points, and at the moment controller parameter should be on the cross point of and .
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 , , and are , , and .
When controller parameter , the system trajectory at is convergent, while according to the different cases of and , parameters and can be selected as follows: (i) when the system is stable at two equilibrium points, we can select , ; (ii) system is stable at equilibrium point , while at it has a Hopf bifurcation, and at the moment we can select , ; (iii) there is a Hopf bifurcation at equilibrium point , while system is stable at , and at the moment we can select , ; (iv) both equilibria and have Hopf bifurcations, and at the moment we can select , .
In Figure 3 the Lorenz system trajectory is convergent at when . Figures 4–11 are system simulation trajectories for different cases of and , in which, corresponding to case (i), Figure 4 is the stable convergent trajectory at , and Figure 5 is the stable convergent trajectory at . Corresponding to case (ii), Figure 6 is the stable convergent trajectory at , and Figure 7 is the system trajectory of Hopf bifurcation at . For case (iii), Figure 8 is the Hopf bifurcation trajectory at , and Figure 9 is the stable convergent trajectory at . Figures 10 and 11 are the system trajectories of Hopf bifurcation at and for case (iv).
When controller parameter , system has Hopf bifurcation at , and its trajectory is a limit cycle. And according to different cases at and , parameters and can be chosen as follows: (i) when the system is stable at two equilibrium points, we can select , ; (ii) system is stable at equilibrium point , while there is a Hopf bifurcation at , and at the moment we can select , ; (iii) there is a Hopf bifurcation at equilibrium point , while system is stable at , and at the moment we can select , ; (iv) there are Hopf bifurcations at both equilibria, and we can select , .
Figure 12 is the system Hopf bifurcation trajectory at when , and at this moment system converges to a stable limit cycle. Figures 13–20 are system simulation trajectories at and for different cases, in which, according to case (i), Figures 13 and 14 are the stable trajectories at and . In accordance with case (ii), Figure 15 is the stable trajectory at , and Figure 16 is the Hopf bifurcation trajectory at . For case (iii), Figure 17 is the Hopf bifurcation trajectory at , and Figure 18 is the stable trajectory at . In accordance with condition (iv), Figure 19 is the Hopf bifurcation trajectory at , and Figure 20 is the Hopf bifurcation trajectory at .
Figures 21 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.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is partially supported by the NSF of China Grants nos. 61374001 and 61074189.