A fractional-order circuit control method and system based on full-drive model

Abstract

The present invention discloses a fractional-order circuit control method and system based on a full-drive model, which relates to the field of tracking control technology, constructs a fractional-order generalized model of a fractional-order circuit; performs equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and constructs the output of the fractional-order full-drive model; constructs a control model based on the output and the fractional-order full-drive model; constructs a compensation model based on the output and a set target value; constructs an augmented model based on the compensation model, the control model, the output and the fractional-order full-drive model; constructs the feedback part of the control model; substitutes the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit; solves the state matrix to obtain the control parameters of the control model. The present invention transforms the modeled fractional-order generalized time-varying system into a fractional-order full-drive model with equality constraints, designs an output tracking controller based on the model, provides a controller parameter solving method, and is suitable for fractional-order nonlinear time-varying circuits.

Description

Technical Field

[0001] The present invention relates to the field of tracking control technology, and in particular to a fractional-order circuit control method and system based on an all-wheel drive model. Background Art

[0002] In recent years, fractional-order circuit models have been widely used in many engineering fields, such as fuel cells, fractional-order circuit models of biological tissues, and the design of fractional-order filters. Fractional-order components can increase the freedom of fractional-order circuit models. Common fractional-order components include supercoils and supercapacitors. There has been some progress in the research of fractional-order circuit models with supercapacitors, but the research on fractional-order circuit models with supercoils is still in its infancy. The reason is that this type of fractional-order circuit model not only has the constraints of fractional-order dynamic differential equations, but also may have equality constraints, which is very challenging.

[0003] Most of the fractional-order circuit models studied so far are time-invariant systems, that is, the parameters of each component in the fractional-order circuit do not change with time. However, time-varying components are widely present in practical engineering, such as time-varying resistors. Due to the complexity of the system, it is difficult to calculate the dynamic response of the fractional-order circuit model with time-varying resistors even under linear conditions. These difficulties hinder our analysis of the various characteristics of the fractional-order circuit model.

[0004] Tracking control is a very valuable control method in the field of fractional-order circuits and systems. It can ensure that the state or output of the system follows the established target value. Due to the complexity of fractional-order time-varying fractional-order circuits, how to design an output tracking controller is still a difficult problem to be solved. Summary of the invention

[0005] The purpose of the present invention is to provide a fractional-order circuit control method and system based on a full-drive model, which transforms the modeled fractional-order generalized time-varying system into a fractional-order full-drive model with equality constraints, designs an output tracking controller based on the model, and transforms the fractional-order time-varying system into a linear time-invariant system under the action of the controller. By using the stability theory of linear time-invariant systems and the linear matrix inequality method, a controller parameter solution method is given, and is suitable for fractional-order nonlinear time-varying circuits.

[0006] To achieve the above object, the present invention provides the following solutions:

[0007] A fractional-order circuit control method based on an all-wheel drive model, the fractional-order circuit comprising a resistor module, a coil module and a power supply module connected to each other, the method comprising:

[0008] Constructing a fractional-order generalized model of the fractional-order circuit; performing equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and constructing an output of the fractional-order full-drive model;

[0009] Building a control model of the power module based on the output and the fractional-order full-drive model; building a compensation model based on the output and a set target value;

[0010] Constructing an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model;

[0011] Constructing a feedback part of the control model; substituting the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit;

[0012] The state matrix is ​​solved to obtain control parameters of the control model.

[0013] Preferably, the fractional-order generalized model is as follows:

[0014] ED α x(t)=Ax(t)+Bu(t);

[0015] Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistor module, A is the resistance parameter matrix of the resistor module, B is the first constant matrix, u(t) is the parameter matrix of the power module, and α is the fractional order.

[0016] Preferably, the output is as follows:

[0017] y(t) = Cx(t);

[0018] Where: y(t) is the output, C = [1 00], and x(t) is the current parameter matrix of the resistance module.

[0019] The present invention also provides a fractional-order circuit control system based on an all-wheel drive model, wherein the fractional-order circuit comprises a resistor module, a coil module and a power supply module connected to each other, and the system comprises:

[0020] A model building and decomposition module, used to build a fractional-order generalized model of the fractional-order circuit, perform equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and build an output of the fractional-order full-drive model;

[0021] A compensation module, configured to construct a control model of the power module based on the output and the fractional-order full-drive model, and to construct a compensation model based on the output and a set target value;

[0022] An augmentation module, configured to construct an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model;

[0023] A feedback module, used for constructing a feedback part of the control model, and substituting the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit;

[0024] A solution module is used to solve the state matrix to obtain the control parameters of the control model.

[0025] Preferably, the fractional-order generalized model is as follows:

[0026] ED α x(t)=Ax(t)+Bu(t);

[0027] Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistor module, A is the resistance parameter matrix of the resistor module, B is the first constant matrix, u(t) is the parameter matrix of the power module, and α is the fractional order.

[0028] Preferably, the output is as follows:

[0029] y(t) = Cx(t);

[0030] Where: y(t) is the output, C = [1 00], and x(t) is the current parameter matrix of the resistance module.

[0031] According to the specific embodiments provided by the present invention, the present invention discloses the following technical effects:

[0032] The present invention discloses a fractional-order circuit control method and system based on a full-drive model, wherein the fractional-order circuit includes a resistor module, a coil module and a power module connected to each other, and the method includes: constructing a fractional-order generalized model of the fractional-order circuit; performing equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and constructing the output of the fractional-order full-drive model; constructing a control model of the power module based on the output and the fractional-order full-drive model; constructing a compensation model based on the output and a set target value; constructing an augmented model based on the compensation model, the control model, the output and the fractional-order full-drive model; constructing a feedback part of the control model; substituting the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit; solving the state matrix to obtain control parameters of the control model. The present invention transforms the modeled fractional-order generalized time-varying system into a fractional-order full-drive model with equality constraints, designs an output tracking controller based on the model, provides a controller parameter solving method, and is applicable to fractional-order nonlinear time-varying circuits. BRIEF DESCRIPTION OF THE DRAWINGS

[0033] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0034] Figure 1 This is a flow chart of the fractional-order circuit control method based on the full-drive model of the present invention;

[0035] Figure 2 It is a structural diagram of a fractional-order circuit control system based on an all-wheel drive model of the present invention;

[0036] Figure 3 Schematic diagram of the fractional-order circuit structure;

[0037] Figure 4 It is the state response curve diagram when the target value is set to 0;

[0038] Figure 5 This is a schematic diagram of the comparison curve between the output and the set target value when the set target value is 1.

[0039] Explanation of symbols: 1. Model building and decomposition module; 2. Compensation module; 3. Augmentation module; 4. Feedback module; 5. Solution module; 61. First resistor; 62. Second resistor; 63. Third resistor; 71. First coil; 72. Second coil; 81. First power supply; 82. Second power supply. DETAILED DESCRIPTION

[0040] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0041] The purpose of the present invention is to provide a fractional-order circuit control method and system based on a full-drive model, which transforms the modeled fractional-order generalized time-varying system into a fractional-order full-drive model with equality constraints, designs an output tracking controller based on the model, and transforms the fractional-order time-varying system into a linear time-invariant system under the action of the controller. By using the stability theory of linear time-invariant systems and the linear matrix inequality method, a controller parameter solution method is given, and is suitable for fractional-order nonlinear time-varying circuits.

[0042] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments.

[0043] The fractional order circuit structure of the present invention is as follows: Figure 3 As shown, the fractional order circuit includes a resistor module, a coil module and a power module connected to each other, the resistor module includes a first resistor 61, a second resistor 62 and a third resistor 63, the coil module includes a first coil 71 and a second coil 72, the power module includes a first power supply 81 and a second power supply 82, the first coil 71 is connected to the first end of the first resistor 61 and the first end of the first power supply 81 respectively, the second end of the first resistor 61, the first end of the second resistor 62 and the first end of the third resistor 63 are connected, the second end of the third resistor 63, the second end of the first power supply 81 and the first end of the second power supply 82 are connected, and the second coil 72 is connected to the second end of the second resistor 63 and the second end of the second power supply 82 respectively. The first resistor 61, the second resistor 62 and the third resistor 63 are any one of a time-varying resistor and a time-invariant resistor, and the first coil 71 and the second coil 72 are any one of a super inductor and a conventional inductor.

[0044] Figure 1 FIG. 1 is a flow chart of the fractional-order circuit control method based on the full-drive model of the present invention. Figure 1 As shown, the present invention provides a fractional-order circuit control method based on a full-drive model, comprising:

[0045] Step S1, constructing a fractional-order generalized model of the fractional-order circuit; performing equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and constructing an output of the fractional-order full-drive model.

[0046] The fractional-order generalized model is as follows:

[0047] ED α x(t)=Ax(t)+Bu(t);

[0048] Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistance module, A is the resistance parameter matrix of the resistance module, B is the first constant matrix, u(t) is the parameter matrix of the power module, α is the fractional order, L1 is the inductance of the first coil, L2 is the inductance of the second coil, L1≠0, L2≠0, is the voltage value of the first coil, is the voltage value of the second coil, is the current value of the first coil, is the current value of the first coil, R1 is the resistance value of the first resistor, R2 is the resistance value of the second resistor, R3 is the resistance value of the third resistor, R1=R1(t)>0, R2=R2(t)>0, R3=R3(t)>0, x1(t) is the current value of the first resistor, x2(t) is the current value of the second resistor, x3(t) is the current value of the third resistor, t is time, e1 is the voltage value of the first power supply, and e2 is the voltage value of the first power supply.

[0049] Substituting each parameter into the formula, we get:

[0050]

[0051] It can then be transformed into the following fractional-order decomposition system with equality constraints:

[0052]

[0053] x3(t)=x1(t)+x2(t);

[0054] Where:

[0055] It can be seen that x3(t) is determined by x1(t) and x2(t), so the fractional-order all-wheel drive model can be obtained as:

[0056]

[0057] Where:

[0058] The output is as follows:

[0059] y(t)=Px(t);

[0060] Where: y(t) is the output, P = [1 0 0].

[0061] The output can also be expressed as:

[0062]

[0063] Where:

[0064] Step S2, constructing a control model of the power module based on the output and the fractional-order all-wheel drive model; and constructing a compensation model based on the output and a set target value.

[0065] Design the corresponding control model so that the output y(t) tracks the set target value y g , the control model is as follows:

[0066]

[0067] Where: v(t) represents the feedback part to be designed. For the fractional-order generalized model, we have: 1) det(λE-A) is not always 0, where λ = s α , s∈C, C is the entire set of complex numbers, det is the determinant; 2) deg(det(λE-A))=rank(E), deg is the degree of the polynomial, rank(E) is the rank of the matrix E. Under the control model, the overall goal of the fractional-order circuit is when y g = 0, the fractional-order circuit is asymptotically stable and when y g ≠0, the tracking condition is met:

[0068] The compensation model is as follows:

[0069] D α x e (t) = e;

[0070] Where: x e (t) is the state of the compensation model, e = y(t)-y g is the tracking error.

[0071] Step S3, constructing an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model.

[0072] The augmented model is as follows:

[0073]

[0074] Where: I is the identity matrix of suitable dimensions.

[0075] Step S4, constructing the feedback part of the control model; substituting the feedback part into the augmented model to obtain the state matrix of the fractional-order circuit.

[0076] The feedback part is as follows:

[0077]

[0078] Where: A0 and A1 are the control parameters to be solved. is the set of real numbers.

[0079] The state matrix is ​​as follows:

[0080]

[0081] Step S5, solving the state matrix to obtain control parameters of the control model.

[0082] Let K = [A0 A1], then

[0083] Where:

[0084] At this time, we can use the stability criterion of fractional-order linear time-invariant system and the linear matrix inequality method to solve K, assuming that the matrix X = X * ∈C 3×3 >0 and matrix The following linear matrix inequality is satisfied:

[0085]

[0086] In the formula: r = exp(j(1-α)π / 2), j is the unit of imaginary number, exp is the exponential function with natural constant as base, is the conjugate of r, is the conjugate of the matrix X, For the matrix The transpose of . X* is the conjugate transposed matrix of matrix X, X is a complex matrix of dimension 3×3, matrix Y is a real matrix of dimension 2×3, Y T is the transposed matrix of matrix Y.

[0087] The control model corresponding to the above method is written into a computer simulation program, the control parameters of the control model are solved, and simulation verification is performed.

[0088] Simulation example: Take L1=L2=1H(henry), R1=R1(t)=(2+sin(t))Ω(ohm), R2=R2(t)=(2+cos(t))Ω(ohm), R3=R3(t)=(2+sin(t))Ω(ohm), α=0.8, set the target value y g =1. Substitute the above data into the formula And using the linear matrix inequality toolbox in MATLAB, the feasible solution can be obtained as follows:

[0089]

[0090]

[0091] Then we can find the parameters Right now

[0092] Under this controller, when y g = 0 when the state response curve of the fractional order circuit is as follows Figure 4 As shown, from Figure 4 It can be seen from the figure that the fractional-order circuit is asymptotically stable. g= 1, the output y(t) of the fractional-order circuit is consistent with the set target value y g (t) Figure 5 As shown, from Figure 5 It can be seen that the output of the fractional-order circuit achieves the desired tracking effect.

[0093] Figure 2 FIG. 1 is a structural diagram of a fractional-order circuit control system based on a full-drive model of the present invention. Figure 2 As shown, the present invention provides a fractional-order circuit control system based on a full-drive model, wherein the fractional-order circuit includes a resistance module, a coil module and a power supply module connected to each other, and the system includes: a model building and decomposition module 1, a compensation module 2, an augmentation module 3, a feedback module 4 and a solution module.

[0094] The model building and decomposition module 1 is used to build a fractional-order generalized model of the fractional-order circuit, perform equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and construct the output of the fractional-order full-drive model.

[0095] The compensation module 2 is used to construct a control model of the power module based on the output and the fractional-order all-wheel drive model, and to construct a compensation model based on the output and a set target value.

[0096] The augmentation module 3 is used to construct an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model.

[0097] The feedback module 4 is used to construct the feedback part of the control model, and substitute the feedback part into the augmented model to obtain the state matrix of the fractional-order circuit.

[0098] The solving module 5 is used to solve the state matrix to obtain the control parameters of the control model.

[0099] Optionally, the fractional-order generalized model is as follows:

[0100] ED α x(t)=Ax(t)+Bu(t);

[0101] Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistor module, A is the resistance parameter matrix of the resistor module, B is the first constant matrix, u(t) is the parameter matrix of the power module, and α is the fractional order.

[0102] Optionally, the output is as follows:

[0103] y(t) = Cx(t);

[0104] Where: y(t) is the output, C =

[100] .

[0105] The present invention models a fractional-order time-varying fractional-order circuit with a super coil and a time-varying resistor into a fractional-order generalized time-varying model, which has not only fractional-order differential equation constraints but also equality constraints. When the coil is a conventional coil, that is, the model fractional order is 1, the fractional order degenerates into an integer-order generalized fractional order; when the resistor is a conventional time-invariant resistor, the system degenerates into a fractional-order generalized system. Therefore, the model modeled by the present invention has a wider range of representativeness.

[0106] The control model design method of the present invention is simple. A fractional-order full-drive model is established, and the control model is designed using the fractional-order full-drive model, time-varying parameters are offset, and the fractional-order time-varying model is converted into a fractional-order linear time-invariant model, avoiding the need to find a complex Lyapunov function. The existing fractional-order linear time-invariant model system stability method can be directly used to analyze and solve the control model. Therefore, the control model design method of the present invention is simple.

[0107] In this specification, each embodiment is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the system disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the method part.

[0108] The principles and implementation methods of the present invention are described in this article using specific examples. The description of the above embodiments is only used to help understand the method and core idea of ​​the present invention. At the same time, for those skilled in the art, according to the idea of ​​the present invention, there will be changes in the specific implementation methods and application scope. In summary, the content of this specification should not be understood as limiting the present invention.

Claims

1. A fractional-order circuit control method based on an all-wheel drive model, wherein the fractional-order circuit comprises a resistor module, a coil module and a power module connected to each other, characterized in that: The method comprises: Constructing a fractional-order generalized model of the fractional-order circuit; performing equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and constructing an output of the fractional-order full-drive model; Building a control model of the power module based on the output and the fractional-order full-drive model; building a compensation model based on the output and a set target value; Constructing an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model; Constructing a feedback part of the control model; substituting the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit; The state matrix is ​​solved to obtain control parameters of the control model.

2. The fractional-order circuit control method based on the full-drive model according to claim 1 is characterized in that: The fractional-order generalized model is as follows: ED α x(t)=Ax(t)+Bu(t); Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistor module, A is the resistance parameter matrix of the resistor module, B is the first constant matrix, u(t) is the parameter matrix of the power module, and α is the fractional order.

3. The fractional-order circuit control method based on the full-drive model according to claim 1, characterized in that: The output is as follows: y(t) = Cx(t); Where: y(t) is the output, C = [100], and x(t) is the current parameter matrix of the resistance module.

4. A fractional-order circuit control system based on an all-wheel drive model, wherein the fractional-order circuit comprises a resistor module, a coil module and a power supply module connected to each other, characterized in that: The system comprises: A model building and decomposition module, used to build a fractional-order generalized model of the fractional-order circuit, perform equivalent decomposition on the fractional-order generalized model to obtain a fractional-order full-drive model, and build an output of the fractional-order full-drive model; A compensation module, configured to construct a control model of the power module based on the output and the fractional-order full-drive model, and to construct a compensation model based on the output and a set target value; An augmentation module, configured to construct an augmented model based on the compensation model, the control model, the output and the fractional-order all-wheel drive model; A feedback module, used for constructing a feedback part of the control model, and substituting the feedback part into the augmented model to obtain a state matrix of the fractional-order circuit; A solution module is used to solve the state matrix to obtain the control parameters of the control model.

5. The fractional-order circuit control system based on the full-drive model according to claim 4 is characterized in that: The fractional-order generalized model is as follows: ED α x(t)=Ax(t)+Bu(t); Where: E is the parameter matrix of the coil module, D is the Caputo fractional derivative, x(t) is the current parameter matrix of the resistor module, A is the resistance parameter matrix of the resistor module, B is the first constant matrix, u(t) is the parameter matrix of the power module, and α is the fractional order.

6. The fractional-order circuit control system based on the full-drive model according to claim 4 is characterized in that: The output is as follows: y(t) = Cx(t); Where: y(t) is the output, C = [100], and x(t) is the current parameter matrix of the resistance module.

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