Method for controlling a direct current motor system

Abstract

The application relates to a control method of a direct current motor system and relates to the technical field of direct current motor control. The application is aimed at solving the problems that the traditional PID control algorithm ignores the uncertainty of the system when the system is controlled, thereby leading to low control precision of the direct current motor system, and that the state convergence speed of the direct current motor system is slow when the direct current motor system is controlled by using instruction filter backstepping technology combined with a neural network or fuzzy logic. The application not only enables the structure of the designed controller to be simple, but also improves the tracking precision of the direct current motor system, and meanwhile enables the tracking error of the direct current motor system to converge to the vicinity of the origin at a speed faster than exponential convergence.

Description

Technical Field

[0001] The invention belongs to the technical field of direct current motor control. Background Art

[0002] As a high-performance motor with a simple structure and excellent speed regulation, DC motors are widely used in various fields. Traditional PID control methods are also widely used in industrial production due to their simplicity and reliability. However, since actual DC motor models often have some nonlinearities, some system uncertainties are often ignored when applying traditional PID control algorithms, resulting in insufficient control accuracy for DC motor systems. Backstepping has therefore been proposed to address nonlinear systems. Command filter backstepping, as an improved backstepping control method, is widely used in various nonlinear systems because it eliminates the need to solve analytical expressions for the input signal derivative. While system uncertainties and unknown external disturbances can be estimated using techniques such as neural networks or fuzzy logic systems, controller design is relatively complex and the state convergence of the DC motor system is slow. Therefore, how to compensate for system uncertainties and external disturbances, improve the tracking accuracy and speed of the DC motor system, and maintain a simple controller structure remains a key issue. Summary of the Invention

[0003] The present invention aims to solve the problem that when a traditional PID control algorithm is used to control a DC motor system, the uncertainty of the system is ignored, resulting in low control accuracy of the DC motor system, and the problem that when a command filtering backstepping technology is used in combination with a neural network or fuzzy logic to control the DC motor system, the state convergence speed of the DC motor system is slow. A fast command filtering backstepping control method for a DC motor system is provided.

[0004] A control method for a DC motor system includes the following steps:

[0005] Step 1: Establish a mathematical model of the DC motor system with the actual value of the motor rotation angle x1 and the actual value of the rotation angular velocity x2 of the DC motor system as state variables, the pulse width modulation signal u as the control input of the DC motor system, and the motor rotation angle reference value y as the output;

[0006] Step 2: Using the pulse width modulation signal u as the extended state variable x3, the DC motor system is expanded into a three-dimensional uncertain nonlinear system, and a mathematical model of the three-dimensional uncertain nonlinear system is established;

[0007] Step 3: Establish the Lyapunov function V and take the first-order time derivative of the Lyapunov function V

[0008]

[0009] Among them, z1 is the actual value of the motor rotation angle x1 and the expected value of the motor rotation angle y d z2 is the error between the actual value of the angular velocity x2 and the first intermediate control variable α1, and z3 is the error between the expanded state variable x3 and the second intermediate control variable α2;

[0010] Step 4: Introduce the given control direction coefficient c2 into the mathematical model of the three-dimensional uncertain nonlinear system, so as to Rewritten as:

[0011]

[0012] Where f2(x1,x2) is the unknown friction force associated with x1 and x2, and u and y respectively d , α1 and α2 are first-order time derivatives, b2 is the unknown control direction coefficient of the DC motor system and is a non-zero constant, d2(t) is the unknown external interference term, and the unknown external interference term d2(t) includes the load, the resistance and tension applied by the outside world when the motor rotates,

[0013] Will As the unknown term of the fast command filter of the DC motor system, can be rewritten as:

[0014]

[0015] Step 5: Assume that the fast instruction filter expression of the DC motor system is:

[0016]

[0017] Among them, v1 and v2 are two state variables of the fast instruction filter, and are the first-order time derivatives of v1 and v2, z(t) is the input signal of the fast instruction filter, y v is the output signal of the fast instruction filter, λ1 and λ2 are two different fast parameters of the fast instruction filter, γ1 and γ2 are two different filter parameters of the fast instruction filter, S(x) = tanh(qx) and q is the positive parameter of the hyperbolic tangent function;

[0018] Step 6: Use z2 and α2 as the input signals of the fast instruction filter to obtain the corresponding output signal y v,1 and y v,2 , using y v,1 and y v,2Two estimation errors for building fast command filters and

[0019] The unknown term φ for constructing the fast instruction filter is:

[0020] φ=y v,1 +ξ1-c2x3,

[0021] but can be rewritten as:

[0022]

[0023] Step 7: Use the finite-time Lyapunov stability criterion according to the rewritten step 6 The expressions for α1, α2 and u are:

[0024]

[0025]

[0026]

[0027] Among them, k1 and l1 are positive parameters of z1 and S(z1) in α1, k2 and l2 are positive parameters of z2 and S(z2) in α2, k3 and l3 are positive parameters of z3 and S(z3) in u;

[0028] Step 8: Substitute α1, α2 and u into the mathematical model of the DC motor system established in step 1, and control the DC motor system through the mathematical model.

[0029] Furthermore, the mathematical model expression of the DC motor system described in step 1 above is:

[0030]

[0031] in, and are the first time derivatives of x1 and x2 respectively.

[0032] Furthermore, the mathematical model expression of the three-dimensional uncertain nonlinear system described in step 2 above is:

[0033]

[0034] in, is the first derivative of x3.

[0035] Furthermore, the first-order time derivative of the Lyapunov function V described in step 3 above is The expression is as follows:

[0036]

[0037] Furthermore, the first-order time derivative expressions of z1 and z2 are as follows:

[0038]

[0039] Beneficial effects of the present invention:

[0040] This invention proposes a fast command filter backstepping control method for a DC motor system. This method utilizes a fast command filter and backstepping control method to design a fast command filter backstepping control method for a DC motor system. Compared to the problems of traditional PID control algorithms and command filter backstepping control methods, this invention not only simplifies the structure of the designed controller but also improves the tracking accuracy of the DC motor system. It also enables the tracking error of the DC motor system to converge to the origin at a rate faster than exponential convergence. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 is the output response curve of the DC motor system;

[0042] Figure 2 is the output tracking error curve of the DC motor system;

[0043] Figure 3 is the response curve of the state x2 of the DC motor system;

[0044] Figure 4 A graph of the unknown terms constructed for the filter;

[0045] Figure 5 This is the control input u curve of the DC motor system. DETAILED DESCRIPTION

[0046] The following will be combined with the accompanying 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 embodiments described 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 making creative work are within the scope of protection of the present invention. It should be noted that the embodiments of the present invention and the features in the embodiments can be combined with each other in the absence of conflict.

[0047] Specific implementation method 1: refer to Figures 1 to 5 Specifically describing this embodiment, the control method of the DC motor system described in this embodiment is characterized by comprising the following steps:

[0048] Step 1: Establish a mathematical model of the DC motor system with the actual value of the motor rotation angle x1 and the actual value of the rotation angular velocity x2 of the DC motor system as state variables, the pulse width modulation signal u as the control input of the DC motor system, and the motor rotation angle reference value y as the output;

[0049]

[0050] in, and are the first-order time derivatives of x1 and x2 respectively, f2(x1, x2) is the unknown friction force related to x1 and x2, b2 is the unknown control direction coefficient of the DC motor system and is a non-zero constant, d2(t) is the unknown external interference term, and the unknown external interference term d2(t) includes the load, the resistance and tension applied by the outside world when the motor rotates.

[0051] When the state variables x1 and x2 of the system are bounded, f2(x1,x2) and its first-order time derivative are continuous and bounded; d2 and its first-order derivative The purpose of this embodiment is to make the output y of the DC motor system converge to the expected value y of the motor rotation angle d , and the convergence speed is faster than exponential convergence.

[0052] Step 2: Use the pulse width modulation signal u as the extended state variable x3, expand the DC motor system into a three-dimensional uncertain nonlinear system, and establish a mathematical model of the three-dimensional uncertain nonlinear system:

[0053]

[0054] in, is the first derivative of x3.

[0055] Step 3: Define the actual value x1 of the motor rotation angle and the expected value y of the motor rotation angle d The error between z1=x1-y d , the error z2 between the actual value of the angular velocity x2 and the first intermediate control variable α1 = x2-α1, and the error z3 between the expanded state variable x3 and the second intermediate control variable α2 = x3-α2. Establish the Lyapunov function V:

[0056]

[0057] Take the first-order time derivative of the Lyapunov function V

[0058]

[0059] Step 4: Introduce the given control direction coefficient c2 into the mathematical model of the three-dimensional uncertain nonlinear system, so as to Rewrite:

[0060]

[0061] in, and u and y respectively d , the first-order time derivatives of α1 and α2.

[0062] In the above formula As the unknown term of the fast command filter of the DC motor system, can be rewritten as:

[0063]

[0064] Step 5: Assume that the fast instruction filter expression of the DC motor system is:

[0065]

[0066] Among them, v1 and v2 are two state variables of the fast instruction filter, and are the first-order time derivatives of v1 and v2, z(t) is the input signal of the fast instruction filter, y v is the output signal of the fast instruction filter, λ1 and λ2 are two different fast parameters of the fast instruction filter, γ1 and γ2 are two different filter parameters of the fast instruction filter, S(x) = tanh(qx) and q is the positive parameter of the hyperbolic tangent function.

[0067] Step 6: Use z2 and α2 as the input signals of the fast instruction filter to obtain the corresponding output signal y v,1 and y v,2 ,

[0068]

[0069]

[0070] Take advantage of y v,1 and y v,2 Two estimation errors for building fast command filters and

[0071] The unknown term φ for constructing the fast instruction filter is:

[0072] φ=y v,1 +ξ1-c2x3,

[0073] but can be rewritten as:

[0074]

[0075] Step 7: Use the finite-time Lyapunov stability criterion according to the rewritten step 6 The expressions for α1, α2 and u are:

[0076]

[0077]

[0078]

[0079] Among them, k1 and l1 are positive parameters of z1 and S(z1) in α1, k2 and l2 are positive parameters of z2 and S(z2) in α2, k3 and l3 are positive parameters of z3 and S(z3) in u;

[0080] Step 8: Substitute α1, α2 and u into the mathematical model of the DC motor system established in step 1, and control the DC motor system through the mathematical model.

[0081] The following proves that u in step 7 can make the DC motor system tracking error converge to the origin at a speed faster than exponential convergence. The proof process is as follows:

[0082] The first-order time derivative expressions of z1 and z2 are as follows:

[0083]

[0084] The error between the derivative of the input signal and the output signal of the DC motor system fast command filter satisfies:

[0085]

[0086] Substituting formulas 2, 3, and 4 into formula 1, we can obtain:

[0087]

[0088] Among them, l1>0, ε i |≤1,s i >0,

[0089]

[0090] Let z(t)=[z1,z2,z3] T , we can get:

[0091]

[0092] Among them, ρ∈(0,1), the convergence time of the system state satisfies:

[0093]

[0094] Among them, the initial value of the Lyapunov function satisfies V[z(0)]≤M z , at this time the tracking error of the system z1=x1-y d It can converge to the vicinity of the origin.

[0095] When the traditional control method is applied to the DC motor system, the derivative expression of its Lyapunov function is:

[0096]

[0097] At this time, the system tracking error satisfies the uniform and ultimately bounded condition, so V[z(t)] can be obtained to satisfy:

[0098]

[0099] Let c / b≤V(z(0))≤M z , μ≤c / ((1-ρ)b) and μ≤(c / ((1-ρ)a)) 2 , then the time for the system error to converge to a smaller neighborhood of the origin is:

[0100]

[0101] When appropriate parameters are selected to meet

[0102]

[0103] There is T s ≤T e This also shows that the convergence speed of the system error in the fast instruction filtering backstepping control method of the DC motor system is faster than that of the traditional control method.

[0104] Moreover, from z(t) we can get:

[0105]

[0106] Then the system state variables x1 and x2 belong to the following compact set:

[0107]

[0108] Where α0 = y d .

[0109] Then the large compact set Ω x Can be defined as That is, x(t)∈Ω x ,t≥0.

[0110] In actual application, the parameter values ​​of this embodiment are as follows:

[0111] b2=2, the initial values ​​of the state variables are x1(0)=0.5, x2(0)=0, the initial value of the control input is u=0, f2=0.15x1+0.2sin(x2), d2=0.4sin(0.75t), y d =sin(t).

[0112] For the designed DC motor system fast instruction filter, the fast instruction filter parameters are λ 1,1 =100,λ 2,1 =100,γ 1,1 =100,γ 2,1 =100,λ 1,2 =10,λ 2,2 =10,γ 1,2 =10 and γ 2,2 =10; initial value v of the state variable of the fast instruction filter 1,1 =v 2,1 =11.89, v 1,2 =v 2,2 =-127.43.

[0113] After equivalent transformation of the system, c2=1, and the controller parameters are k1=k2=k3=10, l1=l2=l3=8.

[0114] The coefficient q of the hyperbolic tangent function tanh(·) involved in the DC motor system fast command filter and controller is 1,1 =q 2,1 =q 1,2 =q 2,2 =5,q1=q2=q3=5.

[0115] However, in order to compare the estimation effect of the unknown term constructed by the filter, it is also necessary to build a fast instruction filter whose filter parameter is λ 1,α1 =100,λ 2,α1 =100,γ 1,α1 =100,γ 2,α1 =100, the initial value of its state variable is v 1,α1 =v 2,α1 =-11.89, the coefficient q of the hyperbolic tangent function tanh(·) 1,α1 = 5. The system sampling interval is 0.001 seconds.

[0116] Although the present invention is described herein with reference to specific embodiments, it should be understood that these embodiments are merely illustrative of the principles and applications of the invention. It should be understood that many modifications may be made to the illustrative embodiments, and that other arrangements may be devised, without departing from the spirit and scope of the invention as defined by the appended claims. It should be understood that the various dependent claims and features described herein may be combined in ways other than those described in the original claims. It should also be understood that features described in conjunction with individual embodiments may be used in conjunction with other described embodiments.

Claims

1. A control method for a DC motor system, characterized in that: The following steps are involved: Step 1: Establish a mathematical model of the DC motor system with the actual value of the motor rotation angle x1 and the actual value of the rotation angular velocity x2 of the DC motor system as state variables, the pulse width modulation signal u as the control input of the DC motor system, and the motor rotation angle reference value y as the output; Step 2: Using the pulse width modulation signal u as the extended state variable x3, the DC motor system is expanded into a three-dimensional uncertain nonlinear system, and a mathematical model of the three-dimensional uncertain nonlinear system is established; Step 3: Establish the Lyapunov function V and take the first-order time derivative of the Lyapunov function V Among them, z1 is the actual value of the motor rotation angle x1 and the expected value of the motor rotation angle y d z2 is the error between the actual value of the angular velocity x2 and the first intermediate control variable α1, and z3 is the error between the expanded state variable x3 and the second intermediate control variable α2; Step 4: Introduce the given control direction coefficient c2 into the mathematical model of the three-dimensional uncertain nonlinear system, so as to Rewritten as: Where f2(x1,x2) is the unknown friction force associated with x1 and x2, and u and y respectively d , α1 and α2 are first-order time derivatives, b2 is the unknown control direction coefficient of the DC motor system and is a non-zero constant, d2(t) is the unknown external interference term, and the unknown external interference term d2(t) includes the load, the resistance and tension applied by the outside world when the motor rotates, Will As the unknown term of the fast command filter of the DC motor system, can be rewritten as: Step 5: Assume that the fast instruction filter expression of the DC motor system is: Among them, v1 and v2 are two state variables of the fast instruction filter, and are the first-order time derivatives of v1 and v2, z(t) is the input signal of the fast instruction filter, y v is the output signal of the fast instruction filter, λ1 and λ2 are two different fast parameters of the fast instruction filter, γ1 and γ2 are two different filter parameters of the fast instruction filter, S(x) = tanh(qx) and q is the positive parameter of the hyperbolic tangent function; Step 6: Use z2 and α2 as the input signals of the fast instruction filter to obtain the corresponding output signal y v,1 and y v,2 , using y v,1 and y v,2 Two estimation errors for building fast command filters and The unknown term φ for constructing the fast instruction filter is: φ=y v,1 +ξ1-c2x3, but can be rewritten as: Step 7: Use the finite-time Lyapunov stability criterion according to the rewritten step 6 The expressions for α1, α2 and u are: Among them, k1 and l1 are positive parameters of z1 and S(z1) in α1, k2 and l2 are positive parameters of z2 and S(z2) in α2, k3 and l3 are positive parameters of z3 and S(z3) in u; Step 8: Substitute α1, α2 and u into the mathematical model of the DC motor system established in step 1, and control the DC motor system through the mathematical model.

2. The control method of the DC motor system according to claim 1, characterized in that: The mathematical model expression of the DC motor system in step 1 is: in, and are the first time derivatives of x1 and x2 respectively.

3. The control method of the DC motor system according to claim 2, characterized in that: The mathematical model expression of the three-dimensional uncertain nonlinear system in step 2 is: in, is the first derivative of x3.

4. The control method of the DC motor system according to claim 1, characterized in that: The first-order time derivative of the Lyapunov function V described in step 3 The expression is as follows:

5. The control method of the DC motor system according to claim 1, characterized in that: The first-order time derivative expressions of z1 and z2 are as follows:

Images