A design method and application of ADRC controller based on fractional-order ESO
The fractional-order model assists the expansion observer to observe and compensate disturbances, which solves the problems of complex ADRC design process and high-frequency noise sensitivity, achieves better immunity and robustness, and improves the control effect of the second-order system .
Patent Information
- Application Number
- CN202211053878.2
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2022-08-31
- Publication Date
- 2024-12-27
- Estimated Expiration
- Not applicable · inactive patent
AI Technical Summary
The existing ADRC design process is complex and difficult to apply in practice, especially the problem of high-frequency noise sensitivity, and the application of fractional order control in second-order systems is limited.
The fractional-order model assisted expansion observer (FOMESO) is used to observe and compensate the disturbance, convert the control object into a fractional-order integral series-based approximation model, and use a simple proportional controller as a feedback controller to improve the disturbance estimation capability and realize system control. Performance improvement.
Through FOMESO disturbance compensation, the sensitivity of the feedback controller to high-frequency noise is reduced, the immunity and robustness of the control system is improved, the disturbance estimation capability is enhanced, and the control system can meet higher cross-travel frequency indexes.
Smart Images
Figure CN115309055B_ABST
Abstract
Description
Technical Field
[0001] The present invention belongs to the technical field related to controller design, and more specifically, relates to a design method and application of an ADRC controller based on fractional-order ESO. Background Art
[0002] Disturbance and uncertainty are inevitably present in most control practices. How to deal with them in the control system design process is an eternal theme. ADRC (Auto Disturbance Rejection Control) is a control method proposed by Han Jingqing that is model-independent and considers a wide range of uncertainties and disturbances. ADRC regards system dynamics, as well as disturbances and uncertainties inside and outside the system as "total disturbances", expands them into disturbance state variables, estimates them through the extended state observer (ESO), and realizes system control by the feedback controller based on the ESO.
[0003] Since the nonlinear ADRC design process proposed by Han Jingqing is complicated, it is difficult to apply in practice. It was not until the linear ADRC and bandwidth parameterization design method proposed by Gao Zhiqiang et al. that the ADRC design process was simplified, making linear ADRC widely used. Fractional-order control is an extension of integer-order controllers to non-integer orders, which can achieve continuous adjustment of orders and make the controller design more refined. In recent years, the research on combining fractional-order control with ADRC has developed rapidly. Some excellent fractional-order ADRC research results show that fractional-order ADRC controllers can achieve better control effects than integer-order ADRC controllers.
[0004] For a second-order system, linear ADRC can transform the controlled object into an approximate model of double integration. The differential term that is sensitive to high-frequency noise inevitably needs to be introduced into the feedback controller to stabilize the control system. Summary of the invention
[0005] In view of the above defects or improvement needs of the prior art, the present invention provides a design method and application of an ADRC controller based on a fractional-order ESO, which observes and compensates for disturbances through a fractional-order model assisted extended observer (FOMESO), converts the control object into a fractional-order integral series form approximation model, and uses a simple proportional controller as a feedback controller to realize the control of the control system, so as to improve the disturbance estimation capability of the ESO and achieve improved system control performance.
[0006] To achieve the above object, according to one aspect of the present invention, a design method of an ADRC controller based on a fractional-order ESO is provided, the design method comprising the following steps:
[0007] (1) Construct a theoretical mathematical model of the controlled object and obtain the model parameters of the controlled object through model identification to determine the controlled object model;
[0008] (2) constructing a fractional-order model corresponding to the controlled object and determining the model parameters of the fractional-order model through an optimization algorithm so that the fractional-order model can approximately fit the frequency domain characteristics of the controlled object model;
[0009] (3) The fractional-order dynamics of the controlled object are introduced into the controlled object model, and the system dynamics, order difference, disturbance and uncertainty are taken as the total disturbance to establish the fractional-order extended state equation of the controlled object;
[0010] (4) adding the model parameters of the fractional-order model to the extended state observer, and then designing a fractional-order model-assisted extended observer for the control object;
[0011] (5) using the fractional order model to assist the total disturbance estimated by the extended observer to compensate the controlled object to generate the control input of the controlled object;
[0012] (6) Calculate the equivalent model after compensation, and then obtain the open-loop transfer function of the ADRC controller. Given the crossover frequency design index ω c , calculate and determine the parameter K of the feedback controller Pn .
[0013] Furthermore, in step (5), the feedback controller C n (s) Use a proportional controller.
[0014] Furthermore, the control object is a second-order control object, and its differential equation is:
[0015]
[0016] In the formula, a0 and a1 are the differential coefficients of the controlled object, b is the control gain, and w is the external disturbance; the conversion to the transfer function model is:
[0017]
[0018] Where: s is the Laplace operator, Y(s) and U(s) are the Laplace transforms of y and u respectively.
[0019] Furthermore, the expression of the fractional order model is:
[0020]
[0021] In the formula, α is the fractional order of the model, a f0 with a f1 is the coefficient of the model’s fractional-order dynamics, b f is the fractional-order model gain.
[0022] Furthermore, the expression of the model parameter optimization model established when determining the model parameters of the fractional-order model through the optimization algorithm is:
[0023] min x f opt (x)stx lb <x<x ub
[0024] Where x = [a 1f a 0f b f ] is the optimization variable vector, f opt (x) is the optimization objective function, x lb and x ub are the lower and upper bounds of x;
[0025] The optimization objective function is:
[0026]
[0027] Among them, |P(jω i )| and ∠P(jω i ) are the amplitude ratio and phase difference of P(s), respectively, |P f (jω i ) and ∠P f (jω i ) are P f (s) amplitude ratio and phase difference, ω i is a frequency logarithmically spaced vector.
[0028] Furthermore, introducing the fractional-order dynamics of the control object means converting the differential equation of the control object into:
[0029]
[0030] Where α is the fractional differential order, 0<α<1, and is the coefficient of the fractional-order differential dynamics of the controlled object y, b0 is the estimate of the controlled object gain, is the total disturbance, is an unknown disturbance, which has the form:
[0031]
[0032] Let x1 = y, x2 = y α , x3=f, and the fractional-order extended state equation of the control object is established as:
[0033]
[0034] Where: x = [x1 x2 x3]T ,
[0035] E=[0 0 1] T , C=[1 0 0].
[0036] Furthermore, the fractional-order model-assisted extended observer is referred to as FOMESO, and the structure of FOMESO is:
[0037]
[0038] Where z = [z1 z2 z3] T , z1, z2 and z3 are the state variables of FOMESO, which are the estimates of the control object states x1, x2, x3, respectively. L=[β1 β2 β3] T is the observer gain vector, and the parameters in matrices A and B are b0=b f .
[0039] Furthermore, the observer gain L = [β1 β2 β3] T Calculated by the following formula:
[0040]
[0041] Where: o For a given FOMESO bandwidth, ω o >0.
[0042] Furthermore, the equivalent model after compensation is:
[0043]
[0044] in:
[0045]
[0046] Given amplitude crossing frequency ω c , the parameter K of the feedback controller Pn Calculate as follows:
[0047] |C n (jω c ) e (jω c )|=1
[0048] Solving this equation gives the feedback controller parameter K Pn .
[0049] According to another aspect of the present invention, a computer-readable storage medium is provided, wherein the computer-readable storage medium includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the design method of an ADRC controller based on fractional-order ESO as described above.
[0050] In general, compared with the prior art, the design method and application of the ADRC controller based on fractional-order ESO provided by the present invention have the following beneficial effects:
[0051] 1. By adding the model information of the fractional-order approximation model to FOMESO, the disturbance estimation burden of FOMESO is reduced, the required observer bandwidth is low, and it has a stronger disturbance estimation ability, so that the control system has better anti-disturbance performance. At the same time, the disturbance estimation ability will increase with the increase of the observer bandwidth.
[0052] 2. The controlled object is approximately converted into a fractional-order integral series model through FOMESO disturbance compensation. The feedback controller does not need to introduce differential terms that are sensitive to high-frequency noise. A simple proportional controller can stabilize the system and is insensitive to input high-frequency noise.
[0053] 3. By selecting the appropriate cross-frequency index, the designed control system open-loop transfer function has the characteristic of horizontal phase, the control system step response overshoot is insensitive to the system gain, and the control system response is robust to the system gain.
[0054] 4. Frequency domain analysis shows that the equivalent model of the control object converted by FOMESO disturbance compensation is closer to the ideal fractional-order integral series model, and the control system can meet higher crossover frequency indicators. In addition, the disturbance observation capability of fractional-order ESO is stronger, and the anti-disturbance performance of the control system is better. BRIEF DESCRIPTION OF THE DRAWINGS
[0055] Figure 1 It is a structural block diagram of an ADRC controller based on a fractional-order ESO constructed by a design method of an ADRC controller based on a fractional-order ESO provided by the present invention;
[0056] Figure 2 (a) and (b) are the Bode diagrams of the control object and the approximate fractional order model constructed by the present invention, respectively;
[0057] Figure 3 (a) and (b) are respectively the Bode diagrams of the equivalent model constructed by the present invention and the ideal equivalent model of fractional-order integral series type;
[0058] Figure 4It is a comparison diagram of the step response of the system under different system loop gains constructed by the present invention;
[0059] Figure 5 It is a comparison diagram of the step response of the system under different observer bandwidths constructed by the present invention. DETAILED DESCRIPTION
[0060] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0061] See also Figure 1 The present invention provides a design method of an ADRC controller based on a fractional-order ESO, and the design method mainly comprises the following steps:
[0062] Step 1: construct a theoretical mathematical model of the controlled object through theoretical analysis, and obtain the model parameters of the controlled object through model identification to determine the controlled object model.
[0063] The control object is a second-order control object, and its differential equation is:
[0064]
[0065] In the formula, a0 and a1 are the differential coefficients of the controlled object, b is the control gain, and w is the external disturbance. The conversion to the transfer function model is:
[0066]
[0067] Where: s is the Laplace operator, Y(s) and U(s) are the Laplace transforms of y and u respectively.
[0068] Step 2: construct a fractional-order model corresponding to the controlled object, and determine the model parameters of the fractional-order model through an optimization algorithm, so that the fractional-order model can approximately fit the frequency domain characteristics of the controlled object model.
[0069] The expression of the fractional order model is:
[0070]
[0071] In the formula, α is the fractional order of the model, a f0 with a f1 is the coefficient of the model’s fractional-order dynamics, b f is the fractional-order model gain.
[0072] Determining the model parameters of the fractional-order model by an optimization algorithm includes the following steps:
[0073] S21, design the optimization objective function, establish the optimization model, and determine the upper and lower bounds of the optimization variables.
[0074] S22, using an optimization algorithm to solve the optimization problem, and taking the set of parameters corresponding to the minimum value of the objective function as the optimization result.
[0075] The expression of the model parameter optimization model established when determining the model parameters of the fractional-order model through the optimization algorithm is:
[0076] min x f opt (x)stx lb <x<x ub
[0077] Where x = [a 1f a 0f b f ] is the optimization variable vector, f opt (x) is the optimization objective function, x lb and x ub are the lower and upper bounds of x.
[0078] The optimization objective function is:
[0079]
[0080] Where: |P(jω i ) and ∠P(jω i ) are the amplitude ratio and phase difference of P(s), respectively, |P f (jω i ) and ∠P f (jω i ) are P f (s) amplitude ratio and phase difference, ω i is an element of a frequency logarithmically spaced vector. Use the optimization algorithm to solve the optimization problem, and take the set of parameters corresponding to the minimum value of the objective function as the optimization result x opt .
[0081] The optimization result is x opt =[a 1f a 0f b f ].
[0082] Step three, the fractional-order dynamics of the controlled object are introduced into the controlled object model, and the system dynamics, order difference, disturbance and uncertainty are taken as the total disturbance to establish the fractional-order extended state equation of the controlled object.
[0083] Introducing the fractional-order dynamics of the control object means converting the differential equation of the control object into:
[0084]
[0085] Where α is the fractional differential order (0<α<1), and is the coefficient of the fractional-order differential dynamics of the controlled object y, b0 is the estimate of the controlled object gain, is the total disturbance, is an unknown disturbance, which has the form:
[0086]
[0087] Let x1 = y, x2 = y α , x3=f, and the fractional-order extended state equation of the control object is established as:
[0088]
[0089] Where: x = [x1 x2 x3] T ,
[0090] E=[0 0 1] T , C=[1 0 0].
[0091] Step 4: Add the model parameters of the fractional-order model to the extended state observer, and then design a fractional-order model-assisted extended observer for the control object.
[0092] Among them, given the order α (0<α<1) and bandwidth index ω of ESO o (ω o >0), design the extended state observer gain L = [β1β2β3] T , and then obtain FOMESO designed for the control object.
[0093] The FOMESO structure is:
[0094]
[0095] Where z = [z1 z2 z3] T , z1, z2 and z3 are the state variables of FOMESO, which are the estimates of the control object states x1, x2, x3, respectively. L=[β1 β2 β3] T is the observer gain vector, and the parameters in matrices A and B are b0=b f .
[0096] Observer gain L = [β1 β2 β3] T Calculated by the following formula:
[0097]
[0098] Where: o (ω o >0) is the given FOMESO bandwidth.
[0099] Step 5: Feedback controller C n (s) A proportional controller is used to compensate the controlled object by using the total disturbance estimated by the fractional order model assisted extended observer to jointly generate the control input of the controlled object.
[0100] Feedback Controller C n (s) Use a proportional (P) controller, that is:
[0101] C n (s) = K Pn
[0102] Where: K Pn is the proportional gain.
[0103] The total disturbance estimated by FOMESO is used to compensate the control object, and the control input of the control object is generated as follows:
[0104]
[0105] Where: u0 is the feedback controller C n (s)Output.
[0106] Step 6: Calculate the equivalent model after compensation, and then obtain the open-loop transfer function of the ADRC controller. Given the crossover frequency design index ω c , calculate and determine the parameter K of the feedback controller Pn .
[0107] The equivalent model after compensation is:
[0108]
[0109] in:
[0110]
[0111] Given amplitude crossing frequency ω c , the parameter K of the feedback controller Pn Calculate as follows:
[0112] |C n (jω c )e (jω c )|=1
[0113] Solving this equation gives the feedback controller parameter K Pn .
[0114] See also Figure 2 ,This implementation shows through frequency domain analysis that the equivalent model of the ,control object converted by FOMESO disturbance compensation is closer to the ,ideal fractional-order integral series model, and the control system can meet ,higher cross-over frequency index.
[0115] See also Figure 3 ,By selecting the appropriate cross-over frequency index, the designed control system open-loop transfer function has the ,characteristic of horizontal phase, the control system step response overshoot is insensitive to the ,system gain, and the response of the control system is robust to the ,system gain.
[0116] See also Figure 4 By adding the model information of the fractional-order approximation model into FOMESO, the disturbance estimation burden of FOMESO is reduced, the required observer bandwidth is low, and it has stronger disturbance estimation ability, so that the control system has better anti-disturbance performance. At the same time, the disturbance estimation ability will improve with the increase of the observer bandwidth.
[0117] In addition, the step response comparison diagram of the system under different observer bandwidths constructed in this embodiment is shown in FIG. Figure 5 shown.
[0118] The present invention will be further described below in conjunction with specific embodiments:
[0119] 1) Given the control object model parameters, a0=31.303, a1=38.917, b=171.886;
[0120] 2) Construct the corresponding fractional-order model. Given the fractional-order order α = 0.75, the fractional-order model parameter obtained by optimization is a 0f =2.690, a 1f =2.836, b f =16.047, Bode diagram of the control object model and the approximate fractional-order model.
[0121] 3) Introducing the fractional-order dynamics of the control object, the coefficient matrices of the fractional-order expanded state equation of the control object are established as follows:
[0122] E=[0 0 1] T , C = [1 0 0]
[0123] 4) Given FOMESO bandwidth ω o=100rad / s, the extended state observer gain L is calculated to be [β1 β2 β3] T , get the designed FOMESO;
[0124] 5) Given the amplitude crossover frequency design index ω c , calculate the feedback controller parameter K Pn =152.184.
[0125] The present invention also provides a computer-readable storage medium, which includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the design method of the ADRC controller based on fractional-order ESO as described above.
[0126] It will be easily understood by those skilled in the art that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.
Claims
1. A design method for an ADRC controller based on a fractional-order ESO, characterized in that: The design method includes the following steps: (1) Construct a theoretical mathematical model of the controlled object and obtain the model parameters of the controlled object through model identification to determine the controlled object model; (2) constructing a fractional-order model corresponding to the controlled object and determining the model parameters of the fractional-order model through an optimization algorithm so that the fractional-order model can approximately fit the frequency domain characteristics of the controlled object model; (3) The fractional-order dynamics of the controlled object are introduced into the controlled object model, and the system dynamics, order difference, disturbance and uncertainty are taken as the total disturbance to establish the fractional-order extended state equation of the controlled object; (4) adding the model parameters of the fractional-order model to the extended state observer, and then designing a fractional-order model-assisted extended observer for the control object; (5) using the fractional order model to assist the total disturbance estimated by the extended observer to compensate the controlled object to generate the control input of the controlled object; (6) Calculate the equivalent model after compensation, and then obtain the open-loop transfer function of the ADRC controller. Given the crossover frequency design index ω c , calculate and determine the parameter K of the feedback controller Pn ; The expression of the fractional order model is: In the formula, α is the fractional order of the model, a f0 with a f1 is the coefficient of the model’s fractional-order dynamics, b f is the fractional order model gain; The expression of the model parameter optimization model established when determining the model parameters of the fractional-order model through the optimization algorithm is: min x f opt (x)s.t.x lb <x<x ub Where x = [a 1f a 0f b f ] is the optimization variable vector, f opt (x) is the optimization objective function, x lb and x ub are the lower and upper bounds of x; The optimization objective function is: Among them, |P(jω i )| and ∠P(jω i ) are the amplitude ratio and phase difference of P(s), respectively, |P f (jω i )| and ∠P f (jω i ) are P f (s) amplitude ratio and phase difference, ω i is a frequency logarithmically spaced vector.
2. The design method of the ADRC controller based on fractional-order ESO according to claim 1, characterized in that: In step (5), the feedback controller C n (s) Use a proportional controller.
3. The design method of the ADRC controller based on fractional-order ESO according to claim 1, characterized in that: The control object is a second-order control object, and its differential equation is: In the formula, a0 and a1 are the differential coefficients of the controlled object, b is the control gain, and w is the external disturbance; the conversion to the transfer function model is: Where: s is the Laplace operator, Y(s) and U(s) are the Laplace transforms of y and u respectively.
4. The design method of the ADRC controller based on fractional-order ESO as claimed in claim 3, characterized in that: Introducing the fractional-order dynamics of the control object means converting the differential equation of the control object into: Where α is the fractional differential order, 0<α<1, and is the coefficient of the fractional-order differential dynamics of the controlled object y, b0 is the estimate of the controlled object gain, is the total disturbance, is an unknown disturbance, which has the form: Let x1 = y, x2 = y α , x3=f, and the fractional-order extended state equation of the control object is established as: Where: x = [x1 x2 x3] T , E=[001] T ,C=[100]。 5. The design method of the ADRC controller based on fractional-order ESO according to claim 1, characterized in that: The fractional-order model-assisted extended observer is referred to as FOMESO. The structure of FOMESO is: Where z = [z1 z2 z3] T , z1, z2 and z3 are the state variables of FOMESO, which are the estimates of the control object states x1, x2, x3, respectively. L=[β1β2β3] T is the observer gain vector, and the parameters in matrices A and B are b0=b f .
6. The design method of the ADRC controller based on fractional-order ESO as claimed in claim 5, characterized in that: Observer gain L = [β1β2β3] T Calculated by the following formula: Where: o For a given FOMESO bandwidth, ω o >0.
7. The design method of the ADRC controller based on fractional-order ESO as claimed in claim 5, characterized in that: The equivalent model after compensation is: in: Given amplitude crossing frequency ω c , the parameter K of the feedback controller Pn Calculate as follows: |C n (jω c )P e (jω c )|=1 Solving this equation gives the feedback controller parameter K Pn .
8. A computer-readable storage medium, characterized in that: The computer-readable storage medium includes a stored computer program, wherein when the computer program is executed by a processor, the device where the storage medium is located is controlled to execute the design method of an ADRC controller based on a fractional-order ESO according to any one of claims 1 to 7.