Over-shoot controllable neural network backstepping control method for direct current brushless motor

Abstract

The overshoot controllable neural network backstepping control method of the direct current brushless motor relates to the field of nonlinear system control, and aims to solve the problem that the existing backstepping method cannot accurately adjust the overshoot of the direct current brushless motor. Compared with other adaptive neural network backstepping control methods, the overshoot controllable neural network backstepping control method of the direct current brushless motor can not only stabilize the closed-loop system, but also can determine the overshoot of the system in advance according to the proposed parameter setting rule. Thus, the problem that the existing backstepping method cannot accurately adjust the overshoot of the direct current brushless motor is solved.

Description

Technical Field

[0001] The present invention relates to the field of nonlinear system control, in particular to an overshoot controllable neural network backstepping control method for a brushless DC motor. Background Art

[0002] Neural network backstepping control methods have been widely used in various practical nonlinear systems, such as motor servo control. When designing tracking control for nonlinear systems, the uncertainty of the nonlinear system must be considered. A commonly used control method is neural network control. This method designs the controller based on the backstepping control principle. By combining neural networks to approximate the derivatives of the unknown term function and the virtual control function in the system, it overcomes the computational complexity associated with the virtual control function derivatives during backstepping control design, significantly reducing the difficulty of designing controllers for uncertain nonlinear systems. For adaptive neural network tracking control technology, references can be made to Chinese invention patents CN114019804A, CN112192573A, and CN107577146B. Neural network control methods can effectively address system stability and disturbance immunity. Regarding transient performance, most research focuses on the convergence rate of the system state, such as convergence within a finite time, a fixed time, or a predefined time. However, overshoot is also a critical parameter of transient response. Excessive overshoot can have destructive effects on certain devices and components in the system, while zero overshoot can result in slow convergence and a long rise time. Therefore, the traditional backstepping method cannot accurately adjust the overshoot of the DC brushless motor. Summary of the Invention

[0003] The purpose of the present invention is to propose a controllable neural network backstepping control method for overshoot of a brushless DC motor in order to solve the problem that the existing backstepping method cannot accurately adjust the overshoot of the brushless DC motor.

[0004] The technical solution adopted by the present invention to solve the above technical problems is:

[0005] A method for overshoot controllable neural network backstepping control of a brushless DC motor, the method comprising:

[0006] The control signal u of the controller is used as the input of the brushless DC motor system to control the brushless DC motor system and obtain the actual speed output;

[0007] The design process of the controller and the control signal u is specifically as follows:

[0008] Step 1: Determine the state variables x1 and x2, and build a two-dimensional state space model containing unknown nonlinear terms of the system based on the state variables x1, x2, the control signal of the brushless DC motor system, and the output signal of the brushless DC motor system, so that the system speed output y tracks the target signal yd , where x1 represents the speed and x2 represents the current;

[0009] Step 2: Define error variables z1 and z2, which are expressed as:

[0010] z1=x1-y d

[0011] z2=x2-α1

[0012] Among them, α1 represents the virtual control function to be designed;

[0013] Step 3: Get the first derivative of z1 And use the instruction filter to approximate Reconstruct the unknown nonlinear terms of the system;

[0014] Step 4: Design the virtual control function α1 using the unknown nonlinear terms of the reconstructed system;

[0015] Step 5: According to the virtual control function α1, we can get z2, and then get the first-order derivative of z2. Afterwards based on And introduce the adaptive neural network, design the neural network weight update law and control signal u, the control signal u is expressed as:

[0016]

[0017] Among them, W T S(x) is a radial basis function neural network, p,q>0, p and q are the design parameters used to control the system overshoot, c2=1 / 2(LM), L and M are the self-inductance of each phase winding and the mutual inductance between the two phase windings, respectively;

[0018] Step 6: Design the Lyapunov stability criterion, and use the Lyapunov stability criterion to obtain the controller parameters, and then obtain the controller. The output of the controller is the control signal u.

[0019] Furthermore, the two-dimensional state space model containing the unknown nonlinear term of the system in step 1 is expressed as:

[0020]

[0021] y=x1,

[0022] in, is the first derivative of x1, is the first derivative of x2, c1=2k e n / J, c2=1 / 2(LM), J is the moment of inertia, k eis the back electromotive force coefficient, n is the number of pole pairs, L and M are the self-inductance of each phase winding and the mutual inductance between the two phase windings, respectively, f1(x1)=(-T L -k f x1) / J, f2(x1,x2)=-(k e nx1+Rx2) / (LM), T L is the load moment, k f is the damping coefficient, and R is the stator winding resistance of each phase winding.

[0023] Furthermore, the unknown nonlinear term of the reconstructed system is expressed as:

[0024]

[0025] in, for The estimated value of is the estimated value of z1, τ1 is the filter parameter, ξ1 is and The estimation error between .

[0026] Furthermore, the virtual control function α1 in step 4 is expressed as:

[0027]

[0028] Furthermore, the first-order derivative of z2 Expressed as:

[0029]

[0030] in, is the first-order derivative of α1, and f2 represents the uncertainty of the system.

[0031] Furthermore, the neural network weight update law is expressed as:

[0032]

[0033] Among them, τ2>0, τ2 is a filter parameter, γ>0, γ is the learning rate, for The estimated value of is the estimated value of z2.

[0034] Furthermore, the Lyapunov stability criterion is specifically:

[0035] If there exists a positive constant p,q,0≤δ<∞, and a Lyapunov function V(e1)=e1=x1-y d The following conditions are met:

[0036]

[0037] Then the solution of the system is actually stable under the predetermined system overshoot σ;

[0038] also,

[0039] (1) If p = 2k, q = k 2 / c1, where k>0, then σ≈0;

[0040] (2) If p = 2α, q = (α 2 +β 2 ) / c1, where α, β>0, then

[0041] Among them, ρ>0 is a bounded constant, e1 is the error, and are the first and second order derivatives of the Lyapunov function, k, α, β are design parameters, y d (0) is the tracking error at time 0.

[0042] Furthermore, the k e =6.6v / krpm, n=2.

[0043] Furthermore, the J=173 g·cm 2 , LM=0.8mH.

[0044] Furthermore, the k f =0.

[0045] The beneficial effects of the present invention are:

[0046] Compared to other adaptive neural network backstepping control methods, this application not only stabilizes the closed-loop system but also predetermines the system's overshoot based on the proposed parameter setting rules. This solves the problem that existing backstepping methods cannot accurately adjust the overshoot of brushless DC motors. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 This is the tracking response curve diagram of the system in this application when σ≈5%;

[0048] Figure 2 This is the tracking error curve when σ≈5% for the system of this application;

[0049] Figure 3 This is the control signal u curve when σ≈5% of the system in this application;

[0050] Figure 4 This is the tracking response curve diagram of the system in this application when σ≈0%;

[0051] Figure 5This is the tracking error curve when σ≈0% of the system in this application;

[0052] Figure 6 This is a curve diagram of the control signal u when σ≈0% of the system of this application. DETAILED DESCRIPTION

[0053] It should be noted that, unless there is any conflict, the various embodiments disclosed in this application can be combined with each other.

[0054] Specific implementation method 1: refer to Figure 1 Specifically describing this embodiment, the overshoot controllable neural network backstepping control method of the brushless DC motor described in this embodiment is as follows:

[0055] Design the control input u, and input the designed control input u into the brushless DC motor system to obtain the actual speed output;

[0056] The specific steps of designing the control input u are:

[0057] The control signal u of the controller is used as the input of the brushless DC motor system to control the brushless DC motor system and obtain the actual speed output;

[0058] The design process of the controller and the control signal u is specifically as follows:

[0059] Step 1: Determine the state variables x1 and x2, and build a two-dimensional state space model containing unknown nonlinear terms of the system based on the state variables x1, x2, the control signal of the brushless DC motor system, and the output signal of the brushless DC motor system, so that the system speed output y tracks the target signal y d , where x1 represents the speed and x2 represents the current;

[0060] Step 2: Define error variables z1 and z2, which are expressed as:

[0061] z1=x1-y d

[0062] z2=x2-α1

[0063] Among them, α1 represents the virtual control function to be designed;

[0064] Step 3: Get the first derivative of z1 And use the instruction filter to approximate Reconstruct the unknown nonlinear terms of the system;

[0065] Step 4: Design the virtual control function α1 using the unknown nonlinear terms of the reconstructed system;

[0066] Step 5: According to the virtual control function α1, we can get z2, and then get the first-order derivative of z2. Afterwards based on And introduce the adaptive neural network, design the neural network weight update law and control signal u, the control signal u is expressed as:

[0067]

[0068] Among them, W T S(x) is a radial basis function neural network, p,q>0, p and q are the design parameters used to control the system overshoot, c2=1 / 2(LM), L and M are the self-inductance of each phase winding and the mutual inductance between the two phase windings, respectively;

[0069] Step 6: Design the Lyapunov stability criterion, and use the Lyapunov stability criterion to obtain the controller parameters, and then obtain the controller. The output of the controller is the control signal u.

[0070] The Lyapunov stability criterion is a second-order Lyapunov stability criterion, which is expressed as:

[0071] Consider the following system:

[0072]

[0073] y=x1

[0074] in represents the system state vector, and x(0)=[0,0] T ; is a Lipschitz continuous function; Represents the system output, and the controller is designed to track the signal y d ; Target signal y d and its derivatives is bounded and continuous and

[0075] Second-order Lyapunov stability criterion: If there exists a positive constant p,q,0≤δ<∞, and a Lyapunov function V(e1)=e1=x1-y d The following conditions must be met:

[0076]

[0077] Then the solution of system (1) is actually stable under the condition that the system overshoot is predetermined to be σ.

[0078] also,

[0079] (1) If p = 2k, q = k 2 / c1, where k>0, then σ≈0;

[0080] (2) If p = 2α, q = (α 2 +β 2 ) / c1, where α, β>0, then

[0081] The following will prove that criterion 1 can ensure that the system tracking error is actually stable under the condition of predetermined overshoot. The proof process is as follows:

[0082] Proof: Let From (2), we can see that:

[0083]

[0084] Removing the absolute value sign of inequality (3) yields:

[0085]

[0086]

[0087] where pη1 and are local Lipschitz functions with respect to η1 because of their differentiability, and qe1(0) and δ are constants. Therefore, all functions on the right-hand side of inequalities (4) and (5) are local Lipschitz functions with respect to η1. Therefore, the comparison principle can be used to analyze the upper and lower bounds of the solutions of (4) and (5) by using the following equations:

[0088]

[0089]

[0090] in also, and is the solution of the following equation:

[0091]

[0092]

[0093] Applying the comparison principle to (6) and (4) and (7) and (5), we can obtain:

[0094]

[0095] Integrating (10) yields:

[0096]

[0097] Bundle and Substituting (11) into (8) gives

[0098]

[0099] The characteristic equation of differential equations (8) and (9) is given by

[0100] s 2 + ps + q = 0. (13)

[0101] Two cases are discussed:

[0102] 1) The case of double roots:

[0103] Choose a constant k > 0, then

[0104] (s + k) 2 = s 2 + ps + q, (14)

[0105] where p = 2k and q = k 2 The solutions of equations (8) and (9) can be defined as

[0106]

[0107]

[0108] where a1, a2, a4 and a5 are constants. According to the approximation solution of (2), we have

[0109]

[0110] where a7 = (a1 + a4) / 2, a8 = (a2 + a5) / 2, and It can be seen that is asymptotically stable. According to (17), the trajectories of e1(t) and have the approximate characteristic roots, which indicates that e1(t) is practically stable, and the overshoot and are approximately. Taking the derivative of we have

[0111]

[0112] Using the initial condition Equation (18) can be rewritten as

[0113]

[0114] Let ​To find the p t > 0 p If a8=0, according to the initial conditions, a7=0, so This means The overshoot is σ=0%. If a8≠0, when t>0, No solution, is always positive or negative, so As t changes monotonically, The overshoot is also σ=0%. Since e1(t) and It has an approximate overshoot, so the overshoot of the system output y is σ≈0%.

[0115] 2) When the characteristic roots are different negative real numbers

[0116] If we choose two constants α, β>0, then:

[0117] [s+(α+βi)][s+(α-βi)]=s 2 +ps+q, (20)

[0118] Where p = 2α, q = α 2 +β 2 Therefore, the solutions of (8) and (9) can be given by the following equations:

[0119]

[0120]

[0121] Where b1, b2, b4 and b5 are constants. The approximate solution of (2) can be given by the following formula:

[0122]

[0123] where b7 = (b1 + b4) / 2, b8 = (b2 + b5) / 2 and It can be seen that is asymptotically stable. It can be obtained that e1(t) is actually stable and is consistent with With approximately the same overshoot.

[0124] In order to get To calculate the overshoot, we introduce the auxiliary angle θ and rewrite (23) as follows:

[0125]

[0126] in

[0127] The time derivative of is:

[0128]

[0129] because We can get tan(θ)=β / α. Let We can get tan(βt p +θ)=β / α. Then, we can get:

[0130]

[0131] Due to t p Indicates the first peak time, and t p =π / β. Based on the initial conditions Put t p =π / βSubstitute into (24):

[0132]

[0133] Substituting (27) into (2), according to The overshoot of the system output y is obtained:

[0134]

[0135] The proof is complete.

[0136] The specific form of the state space model of the two-dimensional nonlinear system of the brushless DC motor is:

[0137] The brushless DC motor system is described by the following equation:

[0138]

[0139] Where J represents the moment of inertia, ω represents the speed, T e =Ei / ω is the electromagnetic torque, i is the armature current, E=2k e nω represents the sum of the back electromotive force of the two-phase windings, k e is the back electromotive force coefficient, n represents the number of pole pairs, T L Represents the load moment, k f represents the damping coefficient, u is the voltage input of the two-phase winding, R, L, and M refer to the stator winding resistance, self-inductance, and mutual inductance of each phase winding, respectively.

[0140] Assumption: Load torque T L and is bounded and locally Lipschitz continuous.

[0141] The specific form of establishing a two-dimensional state space model of a nonlinear strict feedback system with uncertainty is:

[0142] Define x1=ω,x2=i,u is the control input,y is the speed output, then:

[0143]

[0144] Where c1 = 2k e n / J, c2=1 / 2(LM), f1(x1)=(-T L -k f x1) / J

[0145] f2(x1,x2)=-(k e nx1+Rx2) / (LM). The motor system parameters are: k e =6.6v / krpm, n=2, J=173g·cm 2 ,LM=0.8mH,k f = 0. Note that f1 and f2 are unknown because T L and R is unknown.

[0146] The control purpose is to design the system control input u so that the output speed of the brushless DC motor system tracks the given target signal y d .

[0147] Define error variables z1, z2:

[0148] Define error variable z1 = x1 - y d , z2=x2-α1; where α1 represents the virtual control function.

[0149] Introducing command filters to approximate To reconstruct the nonlinear terms of the system. The specific process is:

[0150] 1) Take the time derivative of z1 and get:

[0151]

[0152] 2) Separate the uncertain terms: If is known and available, then we can get:

[0153]

[0154] Equation (26) implies that the nonlinear You can use and the known state c1x2 to reconstruct. However, The parsed expression for contains an unknown term.

[0155] 3) Introducing command filtering: This application can use the following command filter to obtain the estimated value of

[0156]

[0157] where τ1is a filter parameter. Then we have

[0158]

[0159] where ξ1is and the estimation error between

[0160] 4) Reconstruct the uncertainty term: Substitute (28) into (26) to get

[0161]

[0162] This means that the uncertainty can be approximated by with an error of ξ1.

[0163] Substitute (29) into (25) to get

[0164]

[0165] Use the found in step four to design α1. α1is designed as

[0166]

[0167] Substitute (31) into (30) to get

[0168]

[0169] Design the control input u; the specific process is as follows:

[0170] 1) The time derivative of z2is derived as follows:

[0171]

[0172] 2) The design of the control input uis as follows:

[0173]

[0174] where p, q > 0 are design parameters for the control system overshoot, W T S(x) represents the neural network. Substitute (34) into (33) to get

[0175]

[0176] Design the update rate of the neural network; the specific process is as follows:

[0177] 1) Define a neural network Represents the input vector of the neural network, Ω is a compact set; S(x)=[s1(x),s2(x)...,s N (x)] T represents the basis function vector, W=[ω1,ω2,...,ω N ] T Represents the weight vector; the number of nodes in the neural network is represented by N. i (x) means:

[0178]

[0179] where m i =[m 1i ,m 2i ,…,m vi ] T represents the center vector, d i Indicates the width of the basis function.

[0180] 2) Lemma: Assume is a continuous function. If there exists γ>0, and a radial basis function neural network W T S(x) satisfies the following conditions:

[0181]

[0182] but:

[0183]

[0184] 3) Using the defined neural network W T S(x) approximates the unknown According to the above lemma, the weight update law of the neural network is designed:

[0185]

[0186] Where γ>0 represents the learning rate, express The approximate value of is defined as follows:

[0187]

[0188] Where τ2>0 is a filtering parameter.

[0189] With the help of the criterion proposed in step 1, the effectiveness of the overshoot controllable neural network backstepping controller (34) for the DC brushless motor will be proved below. The proof process is as follows:

[0190] According to the instruction filter we can get and The relationship between:

[0191]

[0192] Where ξ2 represents the approximation error, represents the upper bound of ξ2. Substitute equation (41) into equation (39):

[0193]

[0194] According to the conclusion in the lemma, we can get:

[0195]

[0196] Substitute equation (43) into equation (35):

[0197]

[0198] Select the Lyapunov function as V=z1, then we have:

[0199]

[0200] in and is bounded, so is bounded, that is So we get:

[0201]

[0202] in According to the proposed criterion, it can be concluded that the system tracking error is actually stable under the condition of predetermined overshoot. The proof is complete.

[0203] The following examples are used to verify the beneficial effects of the present invention:

[0204] Example 1:

[0205] For the system (Eq. 24), constants c1 = 1526, c2 = 625; the system target signal y d (t) = 100. Command filter parameter τ1 = τ2 = 30, Neural network parameters

[0206] N=100,γ=0.01,d j =0.5,m j =0,ω j (0)=0,j=1,…,100. According to the parameter rule proposed in criterion 1, when the system

[0207] Controller parameters when the system overshoot is about 5%:

[0208] α=38.00, β=40.00, p=2α=76.00, q=(α 2 +β 2 ) / c1=1.99. When the overshoot is about 0%, the controller parameters are: k=40.00, p=2α=80.00, q=k 2 / c1=2.10.

[0209] Conclusion: Figure 1 It can be seen that the control signal u designed by the overshoot controllable neural network backstepping control method of the DC brushless motor can not only stabilize the closed-loop system, but also predetermine the overshoot of the system according to the proposed parameter setting rules.

[0210] It should be noted that the specific embodiments are merely explanations and illustrations of the technical solutions of the present invention and cannot be used to limit the scope of protection. Any minor changes made based on the claims and description of the present invention shall still fall within the scope of protection of the present invention.

Claims

1. A method for overshoot controllable neural network backstepping control of a brushless DC motor, characterized in that: The method comprises: Using the control signal of the controller As the input of the brushless DC motor system, it controls the brushless DC motor system and obtains the actual speed output; The controller and the control signal The design process is as follows: Step 1: Determine the state variables , state variables , and according to the state variables , state variables , the control signal of the brushless DC motor system and the output signal of the brushless DC motor system, establish a two-dimensional state space model containing unknown nonlinear terms of the system, so that the system speed output Tracking target signal ,in, Indicates the speed, Indicates current; Step 2: Define the error variable and , the error variable and Expressed as: in, Represents the virtual control function to be designed; Step 3: Get The first derivative of , and use the instruction filter to approximate , reconstruct the unknown nonlinear terms of the system; Step 4: Design a virtual control function using the unknown nonlinear terms of the reconstructed system ; Step 5: According to the virtual control function ,get , and then get The first derivative of , then based on , and introduce adaptive neural network, design neural network weight update law and control signal , control signal Expressed as: in, is a radial basis function neural network, , and is the design parameter for controlling the overshoot of the system, , and are the self-inductance of each phase winding and the mutual inductance between the two phase windings; Step 6: Design the Lyapunov stability criterion and use the Lyapunov stability criterion to obtain the controller parameters, and then obtain the controller. The output of the controller is the control signal ; The virtual control function in step 4 Expressed as: ； also, (1) If , ,in ,but ; (2) If , ,in ,but in, is a bounded constant, 、 、 is the design parameter, is the tracking error at time 0, is the system overshoot, for The estimated value of , is the moment of inertia, is the back electromotive force coefficient, is the pole pair number.

2. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 1, characterized in that: The two-dimensional state space model containing the unknown nonlinear terms of the system in step 1 is expressed as: in, for The first derivative of for The first derivative of , , is the moment of inertia, is the back electromotive force coefficient, is the pole pair number, and are the self-inductance of each phase winding and the mutual inductance between the two phase windings, , , is the load torque, is the damping coefficient, is the stator winding resistance for each phase winding.

3. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 2, characterized in that: The unknown nonlinear term of the reconstructed system is expressed as: in, for The estimated value of for The estimated value of are the filter parameters, for and The estimation error between .

4. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 3, characterized in that: described The first derivative of Expressed as: in, for The first derivative of Represents the uncertainty of the system.

5. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 4, characterized in that: The neural network weight update law is expressed as: in, , is a filtering parameter, , is the learning rate, for The estimated value of for The estimated value of represents the basis function vector.

6. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 5, characterized in that: The Lyapunov stability criterion is specifically: If there is a positive constant , and a Lyapunov function The following conditions are met: Then the solution of the system is to predetermine the system overshoot as The lower one is actually stable; in, is the error, and are the first and second order derivatives of the Lyapunov function, respectively.

7. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 6, characterized in that: described , .

8. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 7, characterized in that: described , .

9. The overshoot controllable neural network backstepping control method of a brushless DC motor according to claim 8, characterized in that: described .

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