Variable fractional order active disturbance rejection control method applied to second-order controlled system

Abstract

The invention discloses a variable fractional order active-disturbance-rejection control method applied to a second-order controlled system, and relates to the field of second-order controlled system control, and the method comprises the steps: building a variable fractional order expansion state observer according to a state equation of the second-order controlled system; according to the variable fractional order expansion state observer, a fractional order proportional differential controller is established, and variable fractional order active disturbance rejection control applied to the second-order controlled system is completed. According to the invention, the problem of contradiction among the anti-interference performance, the anti-noise capability and the stability margin in the prior art is solved.

Description

Technical Field

[0001] The present invention relates to the field of second-order controlled system control, and in particular to a variational fractional-order auto-disturbance rejection control method applied to a second-order controlled system. Background Art

[0002] In actual industrial control systems, system uncertainty and external interference will deteriorate the control performance of the system. How to deal with system uncertainty and interference is a persistent challenge for controller design. Traditional ADRC is based on integer-order models. However, actual systems often have fractional-order characteristics. Using fractional-order calculus theory to establish a mathematical model can more realistically reflect system characteristics. Integer-order ADRC shows certain limitations when dealing with fractional-order objects.

[0003] Prior art 1 proposes fractional-order active disturbance rejection control (FOADRC) for fractional-order systems, which achieves better control effects than traditional active disturbance rejection control. However, it is difficult to obtain an accurate fractional-order model of the system, which poses a challenge to the application of fractional-order active disturbance rejection control. Prior art 2 introduces fractional-order states on the basis of integer-order models and establishes a new type of fractional-order active disturbance rejection control. This method requires a higher bandwidth to achieve a more ideal control quality, but too high a bandwidth will amplify the sampling noise of the system. In addition, the stability margin and disturbance rejection of the control system are related to the fractional order and the observer bandwidth. There is a contradiction between the stability margin, disturbance rejection and noise rejection capabilities, and parameter tuning is difficult, and even the ideal effect cannot be achieved. Prior art 3 proposes a fractional-order active disturbance rejection control based on known model information to address the above contradictions. The known model information is taken into account in the fractional-order extended state observer, which improves the stability margin and disturbance rejection capability of the system. However, this method still has the contradiction between stability margin, disturbance rejection and noise, and the establishment of the model needs to rely on known model information, which is limited in its further application in uncertain systems. Summary of the invention

[0004] In view of the above-mentioned deficiencies in the prior art, the present invention provides a variational fractional-order auto-disturbance rejection control method applied to a second-order controlled system, which solves the contradiction between the interference rejection, anti-noise capability and stability margin in the prior art.

[0005] In order to achieve the above-mentioned invention object, the technical solution adopted by the present invention is: a variational fractional-order active disturbance rejection control method applied to a second-order controlled system, comprising the following steps: S1: According to the state equation of the second-order controlled system, a variable fractional-order extended state observer is established; S2: Based on the variable fractional-order extended state observer, a fractional-order proportional-differential controller is established to complete the variable fractional-order active disturbance rejection control applied to the second-order controlled system.

[0006] Furthermore, the S1 includes the following sub-steps: S11: Convert the transfer function of the second-order controlled system into a differential equation and introduce fractional differential terms; S12: Convert the differential equation with fractional differential terms into a state equation; S13: Establish a variable fractional-order extended state observer based on the state equation.

[0007] Furthermore, the transfer function of the second-order controlled system in S11 is:

[0008] in, is the transfer function, Output for the system The representation in the complex frequency domain after Laplace transform is: Input for the system The representation in the complex frequency domain after Laplace transform is: and is a constant, is a complex variable; Convert to differential equation form:

[0009] Among them, the superscript represents the second-order derivative, the superscript represents the first-order derivative; The fractional order differential term is introduced as follows:

[0010] in, Output for the system The fractional derivative of and is a fractional order state, For disturbance, is the lumped disturbance, is a constant.

[0011] Furthermore, the differential equation introducing the fractional-order differential term in S12 is converted into a state equation as follows:

[0012] in, , and is the system state variable, , and is the fractional derivative value of the system state variable, is the lumped disturbance of the fractional derivative.

[0013] Furthermore, the variable fractional order extended state observer established in S13 is:

[0014] where, 、 and are the observed values of 、 and respectively, 、 and are the fractional derivative values of the observed values respectively, 、 and are the observer gains, is the observed value; Solving gives:

[0015] where, 、 and are the representations in the complex frequency domain after Laplace transform of 、 and respectively, 、 、 、 、 and are transfer functions;

[0016]

[0017] .

[0018] Furthermore, the fractional order proportional derivative controller established in S2 is:

[0019]

[0020]

[0021] where, is the proportional controller, is the differential gain, is the proportional gain, is the equivalent controlled object transfer function, is an imaginary unit, is the system open-loop cut-off frequency, To input the reference value, for Representation in the complex frequency domain after Laplace transform.

[0022] The beneficial effects of the present invention are: (1) In view of the contradiction between the anti-disturbance performance, anti-noise capability and stability margin in the prior art, the present invention provides a variable fractional-order active disturbance rejection control method (VFOADRC). By establishing a variable fractional-order extended state observer (VFOESO) and adjusting the differential order of the extended state observer, the present invention combines the characteristics of the traditional fractional-order control scheme, namely, strong disturbance rejection capability at high fractional orders and high phase margin at low fractional orders, thereby solving the contradiction between the disturbance rejection capability and the phase margin. In addition, the present invention introduces a fractional-order differential control based on the variable fractional-order extended state observer on the basis of the proportional control rate. By adjusting the fractional-order differential gain, the system phase margin is improved, thereby achieving the improvement of the anti-disturbance performance of the system at low bandwidth.

[0023] (2) The present invention has the advantages of good anti-interference, strong robustness, high stability margin and strong anti-noise, and has strong robustness to parameter mismatch and high stability margin. When the parameter mismatch is large, the stability of the system can still be maintained. BRIEF DESCRIPTION OF THE DRAWINGS

[0024] Figure 1 The present invention is a flow chart of a variational fractional-order auto-disturbance rejection control method applied to a second-order controlled system.

[0025] Figure 2 The figure is an overall block diagram of the variational fractional-order active disturbance rejection control method applied to a second-order controlled system according to the present invention.

[0026] Figure 3 Bode plot of the open-loop transfer function of the second-order controlled system.

[0027] Figure 4 Bode plot of the closed-loop transfer function of the second-order controlled system.

[0028] Figure 5 Bode plot of the disturbance transfer function of the second-order controlled system.

[0029] Figure 6 Bode plot of the noise transfer function of the second-order controlled system. DETAILED DESCRIPTION

[0030] The present invention will be further described below in conjunction with the accompanying drawings and specific embodiments.

[0031] like Figure 1 As shown, a variational fractional-order active disturbance rejection control method applied to a second-order controlled system comprises the following steps: S1: According to the state equation of the second-order controlled system, a variable fractional-order extended state observer is established; S2: Based on the variable fractional-order extended state observer, a fractional-order proportional-differential controller is established to complete the variable fractional-order active disturbance rejection control applied to the second-order controlled system.

[0032] For the second-order system, the present invention introduces two variable fractional orders and establishes a variable fractional-order extended state observer and a fractional-order proportional differential control rate, thereby improving the system's anti-interference, phase margin and noise resistance at low bandwidth.

[0033] The method consists of two parts: the design of a variable fractional-order extended state observer and the design of a fractional-order proportional-derivative control rate.

[0034] The S1 includes the following sub-steps: S11: Convert the transfer function of the second-order controlled system into a differential equation and introduce fractional differential terms; S12: Convert the differential equation with fractional differential terms into a state equation; S13: Establish a variable fractional-order extended state observer based on the state equation.

[0035] The transfer function of the second-order controlled system in S11 is:

[0036] in, is the transfer function, Output for the system The representation in the complex frequency domain after Laplace transform is: Input for the system The representation in the complex frequency domain after Laplace transform is: and is a constant, is a complex variable; Convert to differential equation form:

[0037] Among them, the superscript represents the second-order derivative, the superscript represents the first-order derivative; The fractional order differential term is introduced as follows:

[0038] in, Output for the system The fractional derivative of and It is a fractional order state, and its value range is between 0 and 1. For disturbance, is the lumped disturbance, is a constant, .

[0039] The differential equation introducing the fractional-order differential term in S12 is converted into a state equation as follows: make , then:

[0040] in, , and is the system state variable, , and is the fractional differential value of the system state variable, Lumped disturbance The fractional derivative of .

[0041] In S13, a variable fractional-order extended state observer is established according to the state equation:

[0042] in, , and They are , and The observed value of , and are the fractional differential values ​​of the observed values, , and is the observer gain, for Observed value of The solution is:

[0043] in, , and They are , and The representation in the complex frequency domain after Laplace transform is: , , , , and is the transfer function;

[0044]

[0045] .

[0046] The fractional-order proportional differential controller established in S2 is:

[0047]

[0048]

[0049]

[0050] in, is a proportional controller, is the differential gain, is the proportional gain, is the equivalent controlled object transfer function, is an imaginary unit, is the system open-loop cut-off frequency, To input the reference value, for Representation in the complex frequency domain after Laplace transform.

[0051] The overall control process of the present invention is as follows Figure 2 As shown, by establishing a variable fractional-order extended state observer to observe the lumped disturbance, the observed lumped disturbance is compensated into the control rate, and a fractional-order proportional differential control rate is established based on the observer, which improves the disturbance suppression capability and has strong robustness.

[0052] In one embodiment of the present invention, the transfer function formula of the second-order controlled system, the variable fractional-order extended state observer formula and the fractional-order proportional differential controller formula can be obtained:

[0053] Open-loop transfer function of a second-order controlled system for:

[0054] Given the system open-loop cutoff frequency , we can calculate:

[0055]

[0056] in, is the phase margin of the open-loop system.

[0057] By drawing the Bode plot of the open-loop transfer function of the system, such as Figure 3 As shown, it can be found that the present invention can significantly improve the open-loop phase margin of the system.

[0058] The closed-loop transfer function of the system for:

[0059] Disturbance Transfer Function It can be calculated that:

[0060] in, For disturbance Representation in the complex frequency domain after Laplace transform; There is sampling noise When the noise transfer function for:

[0061] By drawing the Bode diagram of the closed-loop transfer function, disturbance transfer function and noise transfer function of the system, such as Figure 4-6 As shown, it can be seen that the present invention effectively solves the contradiction between the anti-disturbance capability and stability margin of the traditional fractional-order anti-disturbance control. Without increasing the bandwidth of the observer, the present invention significantly improves the anti-disturbance capability of the system, suppresses the resonant peak, provides more flexible phase margin adjustment, and enhances the noise suppression capability. By introducing two different fractional-order differentials, more flexible parameter adjustment is achieved, which greatly improves the robustness of the system to parameter disturbances. In addition, the present invention effectively reduces the resonant peak and enhances high-frequency noise suppression. By enhancing the low-frequency anti-disturbance capability and suppressing the resonant peak, the anti-disturbance capability of the system is greatly improved.

[0062] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific variations and combinations that do not deviate from the essence of the present invention based on the technical revelations disclosed by the present invention, and these variations and combinations are still within the protection scope of the invention.

Claims

1. A variational fractional-order active disturbance rejection control method for a second-order controlled system, characterized in that: The following steps are involved: S1: According to the state equation of the second-order controlled system, a variable fractional-order extended state observer is established; S2: Based on the variable fractional-order extended state observer, a fractional-order proportional-differential controller is established to complete the variable fractional-order active disturbance rejection control applied to the second-order controlled system.

2. The variational fractional-order active disturbance rejection control method for a second-order controlled system according to claim 1, characterized in that: The S1 includes the following sub-steps: S11: Convert the transfer function of the second-order controlled system into a differential equation and introduce fractional differential terms; S12: Convert the differential equation with fractional differential terms into a state equation; S13: Establish a variable fractional-order extended state observer based on the state equation.

3. The variational fractional-order active disturbance rejection control method for a second-order controlled system according to claim 2, characterized in that: The transfer function of the second-order controlled system in S11 is: in, is the transfer function, Output for the system The representation in the complex frequency domain after Laplace transform is: Input for the system The representation in the complex frequency domain after Laplace transform is: and is a constant, is a complex variable; Convert to differential equation form: Among them, the superscript represents the second-order derivative, the superscript represents the first-order derivative; The fractional order differential term is introduced as follows: in, Output for the system The fractional derivative of and is a fractional order state, For disturbance, is the lumped disturbance, is a constant.

4. The variational fractional-order auto-disturbance rejection control method for a second-order controlled system according to claim 3, characterized in that: The differential equation introducing the fractional-order differential term in S12 is converted into a state equation as follows: in, , and is the system state variable, , and is the fractional differential value of the system state variable, Lumped disturbance The fractional derivative of .

5. The variational fractional-order active disturbance rejection control method for a second-order controlled system according to claim 4, characterized in that: In S13, a variable fractional-order extended state observer is established according to the state equation: in, , and They are , and The observed value of , and are fractional differential values ​​of the observed values, , and is the observer gain, for Observed value of The solution is: in, , and They are , and The representation in the complex frequency domain after Laplace transform is: , , , , and is the transfer function; 。 6. The variational fractional-order active disturbance rejection control method for a second-order controlled system according to claim 5, characterized in that: The fractional-order proportional differential controller established in S2 is: in, is a proportional controller, is the differential gain, is the proportional gain, is the equivalent controlled object transfer function, is an imaginary unit, is the system open-loop cut-off frequency, To input the reference value, for Representation in the complex frequency domain after Laplace transform.

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