Fractional order active disturbance rejection control method of fractional order chaotic system
By constructing a fractional active disturbance rejection controller (FOADRC) and combining it with a fractional tracking differentiator, an extended state observer, and a PID controller, the control problem of fractional chaotic systems was solved, achieving improved high precision and anti-interference capability.
Patent Information
- Application Number
- CN202511211515.0
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-27
- Publication Date
- 2025-12-05
AI Technical Summary
Fractional chaotic systems face problems such as model uncertainty, parameter fluctuations, and external disturbances in practical engineering applications, making it difficult for traditional chaos control strategies to effectively control them.
A fractional-order active disturbance rejection control (FOADRC) is constructed by combining a fractional-order tracking differentiator (FOTD), a fractional-order extended state observer (FOESO), and a fractional-order PID controller (FOPID) to provide stable control for fractional-order Lorenz chaotic systems.
It significantly improves the control accuracy and anti-interference capability of fractional-order chaotic systems, effectively suppresses chaotic phenomena, and achieves stable control of fractional-order Lorenz systems.
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Figure CN121069767A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of chaotic oscillation control, and particularly relates to a fractional order active disturbance rejection control method for a fractional order chaotic system. BACKGROUND
[0002] As a kind of dynamic system with significant nonlinear characteristics and extremely sensitive to initial conditions and parameter changes, chaotic system has been widely applied in many fields such as information security transmission, digital image processing, biomedical engineering, etc. In recent years, with the continuous deepening of scientific research, the fractional order characteristics of many actual systems have attracted widespread attention from scholars, such as high molecular viscoelastic materials, biological neural networks, and complex fluid motion systems, etc. Compared with traditional integer order chaotic systems, fractional order chaotic systems can more accurately describe the complex dynamic characteristics in these actual systems by introducing fractional calculus theory.
[0003] However, due to the nonlinear characteristics, system uncertainty and high sensitivity to external disturbances of the fractional order chaotic system itself, its control problem has become the focus and difficulty of research in this field. In-depth study of fractional order chaotic systems not only helps better understand the dynamics of actual systems, but also provides new theoretical basis for accurate control of complex systems.
[0004] Therefore, in view of the model uncertainty, parameter fluctuation and external disturbance problems commonly existing in the actual engineering application of fractional order chaotic systems, and the situation that traditional chaotic control strategies often fail to achieve the expected control effect when dealing with fractional order systems, the present application proposes a fractional order active disturbance rejection control method for fractional order chaotic systems. SUMMARY
[0005] The purpose of the present application is to provide a fractional order active disturbance rejection control method for fractional order chaotic systems, taking fractional order Lorenz chaotic system as the research object, combining fractional order tracking differentiator FOTD, fractional order extended state observer FOESO and fractional order PID controller FOPID to construct fractional order active disturbance rejection controller FOADRC, effectively making up for the shortcomings of existing control technologies, and significantly improving the control accuracy, anti-interference ability and ability to suppress chaotic phenomena of fractional order chaotic systems.
[0006] To achieve the above purpose, the present application provides a fractional order active disturbance rejection control method for fractional order chaotic systems, comprising the following steps:
[0007] Step S1, according to the fractional calculus theory, a mathematical model of the fractional order Lorenz chaotic system is constructed, the value range of each parameter of the system in the chaotic oscillation state is determined, and the dynamic characteristics are analyzed;
[0008] Step S2: To achieve accurate extraction of the fractional derivative information of the system state variables and to provide a smooth tracking signal and its derivative signal for the reference signal, a fractional tracking differentiator is designed.
[0009] Step S3: Design a fractional-order extended state observer to observe the total perturbation of the fractional-order chaotic system and output the observed values in real time;
[0010] Step S4: Compare the output of the fractional-order extended state observer with the output of the fractional-order tracking differentiator to obtain the error signal, and use it as the input of the fractional-order controller to design a fractional-order PID controller.
[0011] Step S5: Design a fractional-order active disturbance rejection controller to achieve stable control of the fractional-order Lorenz system.
[0012] Preferably, in step S1, a mathematical model of the fractional Lorenz chaotic system is constructed based on fractional calculus theory. The specific process is as follows:
[0013] Based on fractional calculus theory, a mathematical model of a fractional Lorenz chaotic system is constructed as follows:
[0014]
[0015] in, is a fractional differential operator, 0 < q ≤ 1 is the fractional derivative order; x1, x2, x3 are system state variables, and σ, ρ, γ are system parameters.
[0016] Preferably, in step S2, to extract the fractional derivative information of the system state variables and provide the tracking signal and its derivative signal for the reference signal, a fractional-order tracking differentiator (FOTD) is designed, as shown below:
[0017]
[0018] Among them, v d v1 is the desired signal output of the fractional-order tracking differentiator FOTD; v2 is the desired signal output of the system. d The tracking signal; v2 is the derivative of the desired signal output by the fractional-order tracking differentiator FOTD. The tracking signal; r is the fast factor; h is the filter factor; fhan(·) is the nonlinear function;
[0019] Based on the characteristic that the output curve of the tanh(·) function is smooth near zero, this paper addresses the problem of system chattering caused by the sign function sgn(·) in the fhan(·) function; the tanh(·) function is shown below:
[0020]
[0021] The improved fastest tracking function fhan1 is as follows:
[0022]
[0023] Where d, a0, a1, a2, a, z y z a Let y be an intermediate variable, and y be the state error.
[0024] Preferably, in step S3, a fractional-order extended state observer (FOESO) is designed to observe the total perturbation of the fractional-order chaotic system and output the observed values in real time. The specific process is as follows:
[0025] The fractional-order extended state observer FOESO is shown below:
[0026]
[0027] Where z1 is the estimated value of system state x1; z2 is the differential estimated value of system state x1; z3 is the estimated value of total system disturbance; β1, β2, and β3 are the gain parameters of the fractional-order extended state observer; b is the control gain; and u is the control quantity.
[0028] Preferably, in step S4, the output of the fractional-order extended state observer is compared with the output of the fractional-order tracking differentiator to obtain an error signal, which is used as the input of the fractional-order controller. The fractional-order PID controller FOPID is designed, and the specific process is as follows:
[0029] Based on the desired signal v output by the fractional-order tracking differentiator FOTD d The tracking signal v1 and the output z1 of the fractional extended state observer FOESO constitute the signal error e1, as shown below:
[0030] e1 = v1 - z1 (6);
[0031] Differentiation of the desired signal output of the fractional-order tracking differentiator FOTD The tracking differential signal v2 and the output z2 of the fractional extended state observer FOESO constitute the differential signal error e2, as shown below:
[0032] e2=v2-z2 (7);
[0033] The total system disturbance estimate z3 is obtained by using the fractional-order extended state observer FOOSO output. The total system disturbance is then compensated by feedforward. Combined with the fractional-order PID control method, the fractional-order PID controller FOPID is designed as follows:
[0034]
[0035] wherein, k p , k i , k d are control gain parameters of the fractional order PID controller FOPID respectively; u0 is used to compensate the extended state z3; b0 is a compensation factor; u is the control quantity.
[0036] Preferably, in step S5, the structure of the fractional order active disturbance rejection controller FOADRC is composed of a fractional order tracking differentiator FOTD, a fractional order extended state observer FOESO and a fractional order PID controller FOPID.
[0037] Preferably, the specific implementation process of the fractional order active disturbance rejection controller is as follows:
[0038] Firstly, input the system expected signal v d , and output v1 and v2 through the fractional order tracking differentiator FOTD;
[0039] Then, the three states z1, z2 and z3 output by the fractional order extended state observer FOESO are the estimated value of the system state x1, the differential estimated value of the system state x1 and the total disturbance of the system respectively;
[0040] wherein, the signal error e1 and the differential signal error e2 are sent to the fractional order PID controller FOPID, u0 is designed according to the fractional order PID control method, and then the total disturbance compensation is added to form the control quantity u;
[0041] Finally, the control quantity u is sent to the fractional order extended state observer FOESO and the fractional order Lorenz chaotic system respectively.
[0042] Therefore, the fractional order active disturbance rejection control method of the fractional order chaotic system adopts the above-mentioned fractional order active disturbance rejection control method, takes the fractional order Lorenz chaotic system as the research object, combines the fractional order tracking differentiator FOTD, the fractional order extended state observer FOESO and the fractional order PID controller FOPID, constructs the fractional order active disturbance rejection controller FOADRC, effectively makes up for the shortcomings of the existing control technology, significantly improves the control precision, the anti-interference ability and the ability to suppress the chaotic phenomenon of the fractional order chaotic system.
[0043] The technical solutions of the present application will be further described in detail below with reference to the drawings and embodiments. BRIEF DESCRIPTION OF DRAWINGS
[0044] Figure 1 It is a flow chart of the fractional order active disturbance rejection control method of the fractional order chaotic system of the present application;
[0045] Figure 2The diagram shows the state curves and chaotic phase diagrams of a chaotic system without any control action; where (a) is the state curve of the chaotic system; and (b) is the chaotic phase diagram of the chaotic system.
[0046] Figure 3 This is a block diagram of the fractional-order chaotic control system of the present invention;
[0047] Figure 4 This is a simulation curve comparing the system state output after adding fractional-order active disturbance rejection control in an embodiment of the present invention, 20 seconds later. Detailed Implementation
[0048] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.
[0049] like Figure 1 As shown, a fractional-order active disturbance rejection control method for a fractional-order chaotic system includes the following steps:
[0050] Step S1: Based on fractional calculus theory, construct a mathematical model of the fractional Lorenz chaotic system, determine the range of values for each parameter of the system under chaotic oscillation state, and analyze its dynamic characteristics.
[0051] Step S2: To achieve accurate extraction of fractional derivative information of system state variables and to provide a smooth tracking signal and its derivative signal for the reference signal, a fractional tracking differentiator is designed.
[0052] Step S3: Design a fractional-order extended state observer to observe the total perturbation of the fractional-order chaotic system and output the observed values in real time;
[0053] Step S4: Compare the output of the fractional-order extended state observer with the output of the fractional-order tracking differentiator to obtain the error signal, and use it as the input of the fractional-order controller to design a fractional-order PID controller.
[0054] Step S5: Design a fractional-order active disturbance rejection controller to achieve stable control of the fractional-order Lorenz system.
[0055] Example 1
[0056] Step S1: Based on fractional calculus theory, construct a mathematical model of the fractional Lorenz chaotic system, determine the range of values for each parameter of the system under chaotic oscillation state, and analyze its dynamic characteristics.
[0057] Based on fractional calculus theory, a mathematical model of a fractional Lorenz chaotic system is constructed as follows:
[0058]
[0059] in, For fractional differential operator, 0 < q ≤ 1 is the fractional derivative order; x1, x2, x3 are system state variables, and σ, ρ, γ are system parameters.
[0060] The chaotic dynamic characteristics of the fractional Lorenz chaotic system will change with the system parameters and the fractional order, and different motion states are presented, including periodic irregular oscillation, bifurcation phenomenon and chaotic behavior.
[0061] When the system parameters are σ = 10, ρ = 28 and γ = 8 / 3, the system will enter a chaotic state and show the characteristics of irregular oscillation. At this time, the simulation curves and phase diagrams of the system state variables are shown in FIG. 1. Figure 2 As shown in FIG. 1, the fractional derivative order selected by the present application is q = 0.9. It can be observed from FIG. 1 that the three state variables of the system all show non-periodic and irregular motion characteristics, and the motion trajectory of the system forms a typical chaotic attractor in the three-dimensional phase space. Figure 2
[0062] This simulation result directly reveals the chaotic dynamic characteristics of the fractional Lorenz chaotic system under specific parameters and fractional orders, and lays an important theoretical analysis and simulation basis for the subsequent design of the fractional active disturbance rejection control method.
[0063] Step S2, in order to accurately extract the fractional differential information of the system state variables and provide a smooth tracking signal and its differential signal for the reference signal, a fractional tracking differentiator (FOTD) is designed, as shown in the following formula:
[0064]
[0065] Wherein, v d is the expected signal of the system; v1 is the tracking signal of the expected signal v d output by the fractional tracking differentiator FOTD; v2 is the tracking signal of the expected signal differential output by the fractional tracking differentiator FOTD; r is a fast factor; h is a filter factor; fhan(·) is a nonlinear function;
[0066] According to the smooth output curve characteristics of the tanh(·) function near zero point, the problem that the sign function sgn(·) in the fhan(·) function is easy to cause system chattering is effectively solved; wherein, the tanh(·) function is as shown in the following formula:
[0067]
[0068] When the independent variable m tends to infinity, the tanh(·) function and the sgn(·) function are close to each other, and then the improved fastest tracking function fhan1 is as shown in the following formula:
[0069]
[0070] where d, a0, a1, a2, a, z y , z a are intermediate variables, and y is the state error.
[0071] Step S3, a fractional order extended state observer (FOESO) is designed to observe the total disturbance (such as internal and external disturbances: some unknown models and disturbances, nonlinear coupling, etc.) of the fractional order chaotic system and output the observation value in real time. Therefore, the fractional order extended state observer (FOESO) is designed as follows:
[0072]
[0073] where z1 is the estimated value of the system state x1; z2 is the differential estimated value of the system state x1; z3 is the estimated value of the total disturbance (including external disturbance and model uncertainty) of the system; β1, β2, β3 are gain parameters of the fractional order extended state observer; b is the control gain; and u is the control amount.
[0074] Step S4, the output of the fractional order extended state observer is compared with the output of the fractional order tracking differentiator to obtain an error signal as the input of the fractional order PID controller (FOPID).
[0075] According to the tracking signal v1 of the expected signal v d output by the fractional order tracking differentiator (FOTD) and the z1 output by the fractional order extended state observer (FOESO), a signal error e1 is formed, as shown below:
[0076] e1=v1-z1 (6);
[0077] According to the tracking differential signal v2 of the differential of the expected signal v d output by the fractional order tracking differentiator (FOTD) and the z2 output by the fractional order extended state observer (FOESO), a differential signal error e2 is formed, as shown below:
[0078] e2=v2-z2 (7);
[0079] The total disturbance estimation value z3 output by the fractional order extended state observer (FOESO) is used to effectively feed forward compensate the total disturbance of the system, and in combination with the fractional order PID control method, a fractional order PID controller (FOPID) is designed, as shown below:
[0080]
[0081] where k p , k i, k d are the control gain parameters of the fractional order PID controller FOPID respectively; u0 is used to compensate the extended state z3; b0 is a compensation factor; u is the control amount.
[0082] Step S5, based on the chaotic dynamic characteristics of the fractional order Lorenz chaotic system under specific parameters and fractional order orders, and the fractional order tracking differentiator, the fractional order extended state observer and the fractional order PID controller (FOPID) described above, a fractional order active disturbance rejection controller is designed.
[0083] Based on the advanced control method of fractional calculus theory, the fractional order active disturbance rejection controller is designed, which can effectively deal with the problems of nonlinear characteristics, model uncertainty and external disturbance existing in the fractional order system.
[0084] In the present application, the core structure of the fractional order active disturbance rejection controller is composed of three key parts of fractional order tracking differentiator (FOTD), fractional order extended state observer (FOESO) and fractional order PID controller (FOPID). This design fully utilizes the advantages of fractional calculus, and provides a new solution for the accurate control of fractional order Lorenz chaotic system.
[0085] As shown in Figure 3 , the fractional order active disturbance rejection controller, the input system expected signal v d is output by the fractional order tracking differentiator FOTD. The three states z1, z2 and z3 output by the fractional order extended state observer FOESO are respectively the estimated value of the system state x1, the differential estimated value of the system state x1 and the total disturbance of the system. Among them, the signal error e1 and the differential signal error e2 are sent to the fractional order PID controller FOPID, and the fractional order controller u0 is designed according to the fractional order PID control method, and then the total disturbance compensation is added to form the control amount u, which is sent to the fractional order extended state observer FOESO and the fractional order Lorenz chaotic system respectively, wherein d1 is the system disturbance.
[0086] Example 2
[0087] The initial states of the fractional order Lorenz chaotic system, the fractional order extended state observer and the fractional order tracking differentiator are selected as:
[0088] [x 10 ,x 20 ,x 30 ]=[1,1,1];
[0089] [z 10 ,z 20 ,z 30 ]=[0,0,0];
[0090] [v 10 ,v 20 ]=[0,0];
[0091] The fractional derivative order of the application is: q=0.9;
[0092] The fractional Lorenz chaotic system parameters are: sigma=10; rho=28; gamma=8 / 3;
[0093] The fractional tracking differentiator parameters are: r=1; h=0.01;
[0094] The gain parameters of the fractional extended state observer are: beta1=200; beta2=500; beta3=2000; the control gain is: b=1;
[0095] The fractional PID controller parameters are: k p =50; k i =1; k d =20; b0=1;
[0096] When the fractional active disturbance rejection controller is not added, the system state variable output is as shown in (a) of the figure, and presents periodic irregular oscillation, Figure 2 (b) of the figure shows that the system presents a chaotic attractor, and the system is in a chaotic operating state. Figure 2
[0097] The fractional active disturbance rejection controller designed in the application is put into operation at 20s, and the three state variable change curves of the fractional active disturbance rejection controller are given as shown in the figure. Figure 4 It can be seen from the figure that before the controller is put into operation, the fractional Lorenz chaotic system is in a chaotic operating state, and the system state of the fractional Lorenz chaotic system after the controller is applied no longer presents chaotic phenomenon. Figure 4
[0098] By using the fractional active disturbance rejection control technology and the PID control theory to design a suitable fractional active disturbance rejection controller, the chaotic phenomenon of the fractional Lorenz chaotic system can be well controlled, the controller design method is simple, the action time is short, the error converges fast, and the method has certain engineering practical significance.
[0099] Therefore, the fractional active disturbance rejection control method of the above fractional chaotic system is adopted, the fractional Lorenz chaotic system is taken as a research object, the fractional active disturbance rejection controller FOADRC is constructed by combining the fractional tracking differentiator FOTD, the fractional extended state observer FOESO and the fractional PID controller FOPID, the defects of the existing control technology are effectively made up, the control precision, the anti-interference ability and the ability of suppressing chaotic phenomenon of the fractional chaotic system are significantly improved.
[0100] It should be pointed out that the above embodiments are only used to illustrate the technical solutions of the present application but not to limit the present application. Although the present application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still make modifications or equivalent replacements to the technical solutions of the present application, and these modifications or equivalent replacements should not make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present application.
Claims
1. A fractional order active disturbance rejection control method for a fractional order chaotic system, characterized in that, The method comprises the following steps: Step S1, according to the fractional calculus theory, a mathematical model of the fractional Lorenz chaotic system is constructed, the value range of each parameter of the system in a chaotic oscillation state is determined, and the dynamic characteristics are analyzed; Step S2, a fractional tracking differentiator is designed to realize accurate extraction of fractional differential information of the state variables of the system and to provide a smooth tracking signal and a differential signal thereof for a reference signal; Step S3, a fractional extended state observer is designed to observe the total disturbance of the fractional chaotic system and to output an observation value in real time; Step S4, an error signal is obtained by comparing the output of the fractional extended state observer with the output of the fractional tracking differentiator, the error signal is taken as the input of a fractional PID controller, and the fractional PID controller is designed; Step S5, a fractional active disturbance rejection controller is designed to realize stable control of the fractional Lorenz system.
2. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 1, wherein, In step S1, according to the fractional calculus theory, a mathematical model of the fractional Lorenz chaotic system is constructed, and the specific process is as follows: According to the fractional calculus theory, a mathematical model of the fractional Lorenz chaotic system is constructed, as shown in the following formula: wherein, is a fractional derivative operator with 0 < q ≤ 1 being the order of the fractional derivative; x1, x2, x3 are system state variables, and σ, ρ, γ are system parameters.
3. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 2, wherein, In step S2, a fractional tracking differentiator FOTD is designed to realize extraction of fractional differential information of the state variables of the system and to provide a tracking signal and a differential signal thereof for a reference signal, as shown in the following formula: where v d is the system desired signal; v1is the tracking signal of the fractional order tracking differentiator (FOTD) output of the desired signal v d ; v2is the tracking signal of the fractional order tracking differentiator (FOTD) output of the desired signal derivative ; r is the fast factor; h is the filter factor; fhan(·) is a nonlinear function; According to the smooth output curve characteristic of the tanh(·) function near the zero point, the problem that the sign function sgn(·) in the fhan(·) function is easy to cause system chattering is solved; wherein, the tanh(·) function is as shown in the following formula: Then, the improved fastest tracking function fhan1 is as shown in the following formula: where d, a0, a1, a2, a, z y , z a are intermediate variables, and y is the state error.
4. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 1, wherein, In step S3, a fractional extended state observer FOESO is designed to observe the total disturbance of the fractional chaotic system and to output an observation value in real time, and the specific process is as follows: The fractional extended state observer FOESO is as shown in the following formula: Wherein, z1 is the estimated value of the system state x1; z2 is the differential estimated value of the system state x1; z3 is the estimated value of the total disturbance of the system; β1, β2, β3 are gain parameters of the fractional extended state observer; b is a control gain; and u is a control amount.
5. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 1, wherein, In step S4, an error signal is obtained by comparing the output of the fractional extended state observer with the output of the fractional tracking differentiator, the error signal is taken as the input of a fractional PID controller FOPID, and the specific process is as follows: The tracking signal v1 of the fractional order tracking differentiator FOTD output d The tracking signal v1 of the fractional order tracking differentiator FOTD output e1=v1-z1 (6); Fractional order tracking differentiator FOTD output of a desired signal derivative The tracking differentiated signal v2 and the z2 output by the fractional order extended state observer FOESO form a differentiated signal error e2 as follows: e2=v2-z2 (7); The fractional PID controller FOPID is designed by feeding forward compensating the total disturbance of the system through the total disturbance estimated value z3 of the fractional extended state observer FOESO and combining the fractional PID control method, as shown in the following formula: where k p , k i , k d are the control gain parameters of the fractional order PID controller FOPID, respectively; u0is used to compensate the extended state z3; b0is the compensation factor; u is the control variable.
6. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 1, wherein, In step S5, the structure of the fractional active disturbance rejection controller FOADRC is composed of a fractional tracking differentiator FOTD, a fractional extended state observer FOESO and a fractional PID controller FOPID.
7. The fractional order active disturbance rejection control method of a fractional order chaotic system according to claim 1, wherein, The specific implementation process of the fractional active disturbance rejection controller is as follows: First, the input system expects a signal v d , which is output by the fractional order tracking differentiator FOTD v1 and v2; Then, the three states z1, z2 and z3 output by the FOESO are the estimated value of the system state x1, the differential estimated value of the system state x1 and the total disturbance of the system respectively; Wherein, the signal error e1 and the differential signal error e2 are sent to the FOPID, and u0 is designed according to the FOPID control method, and then the total disturbance compensation is added to form the control quantity u; Finally, the control quantity u is sent to the FOESO and the fractional Lorenz chaotic system respectively.