A preset damping neural network backstepping control method for a cricket system

Abstract

The application discloses a preset damping neural network backstepping control method for a cricket system and relates to the field of nonlinear system control, and aims at the problem that the existing backstepping method cannot accurately adjust the overshoot of the cricket system. Compared with other adaptive neural network backstepping control methods, the application 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 cricket system is solved.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of nonlinear system control, in particular to a preset damping neural network backstepping control method for a billiard system. BACKGROUND

[0002] The neural network backstepping control method has been widely applied in various practical nonlinear systems, such as billiard system servo control. When designing the tracking control of a nonlinear system, the uncertainty of the nonlinear system must be considered. A commonly used control method is the neural network control method, which is designed based on the backstepping control idea, and combines the neural network to approximate the unknown term function in the system and the derivative of the virtual control function, overcoming the calculation explosion problem caused by the derivative of the virtual control function in the design process of the backstepping control method, greatly reducing the design difficulty of the controller of the uncertain nonlinear system. The adaptive neural network tracking control technology can be referred to in Chinese invention patent CN114019804A, Chinese invention patent CN112192573A and Chinese invention patent CN107577146B. The neural network control method can well solve the stability and disturbance resistance of the system. For transient performance, most research focuses on the convergence speed of the system state, such as finite time, fixed time or predefined time convergence. However, the damping ratio is also a decisive parameter of transient response, and excessive overshoot may cause destructive effects on some devices and components of the system, and zero overshoot may lead to slow convergence speed and long rise time. Therefore, the traditional backstepping method cannot accurately adjust the overshoot of the billiard system. SUMMARY

[0003] The purpose of the present application is to solve the problem that the existing backstepping method cannot accurately adjust the overshoot of the billiard system, and to propose

[0004] The technical scheme adopted by the present application to solve the above technical problems is:

[0005] A preset damping neural network backstepping control method for a billiard system, comprising:

[0006] The control signal u x 、u y of the controller is used as the input of the billiard system to control the billiard system, so as to control the position of the billiard ball;

[0007] The design process of the controller and the control signal u x 、u y includes:

[0008] Step one: determine the state variables x1, x2, x5, x6, and according to the state variables x1, x2, x5, x6, the output signal y x 、y yand input control signal u of the billiards system x 、 y A two-dimensional state space model with unknown nonlinear terms of the system is established, and output signal y x 、 y The target signal x d and y d are tracked, wherein x1 represents the position of the small ball in the x-axis direction, x2 represents the speed of the small ball in the x-axis direction, x5 represents the position of the small ball in the y-axis direction, and x6 represents the speed of the small ball in the y-axis direction.

[0009] Step two: based on the two-dimensional state space model with unknown nonlinear terms of the system, error variables z1, z2, z5 and z6 are defined, which are represented as:

[0010] z1=x1-x d ,

[0011] z2=x2-α1;

[0012] z5=x5-y d ,

[0013] z6=x6-α5;

[0014] Wherein, α1 and α5 are virtual control functions to be designed;

[0015] Step three: the first order derivatives of z1 and z5 are obtained and And the virtual control functions α1 and α5 are designed by and ;

[0016] Step four: introducing command filter approximation and The reconstructed unknown nonlinear terms of the system are obtained;

[0017] Step five: based on the reconstructed unknown nonlinear terms of the system, the virtual control functions α1 and α5 are used, and an adaptive neural network is introduced to design control signals u x and u y ;

[0018] Step six: the Lyapunov function is designed, and the controller parameters are obtained by using the Lyapunov stability criterion, and then the controller is obtained, and the control output of the controller is the control signal u x , u y ;

[0019] The control signal u x , u y is represented as:

[0020]

[0021]

[0022] wherein, denote the neural network fitting the nonlinearity in the x and y directions, respectively, k = bgk f , k denotes a coefficient, b denotes a known constant, k f denotes a gain coefficient, g denotes the gravitational acceleration, m1, m2 > 0 denote design parameters for controlling the damping ratio of the system, m2 = 2a, m1 = a 2 + β 2 a, β denote the angles of rotation of the plate in the x and y directions.

[0023] Further, the two-dimensional state space model containing the unknown nonlinear terms of the system is represented as:

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] wherein, f1, f2 denote the nonlinear terms of the system, r denotes the radius of the ball, x, y denote the displacements of the ball in the x and y directions, respectively, I b denotes the moment of inertia of the ball, m denotes the mass of the ball, F x , F y denote the forces acting on the ball in the x and y directions.

[0033] Further, the reconstructed unknown nonlinear terms of the system are represented as:

[0034]

[0035]

[0036] wherein, τ2 and τ6 denote filter parameters.

[0037] Further, the Lyapunov stability criterion is expressed as:

[0038] If there exist positive constants m1, m2, δ > 0 and a Lyapunov function V(z) = z satisfying the following conditions:

[0039]

[0040] then the solution of the system is practically stable with a predetermined system damping ratio ζ,

[0041] In addition,

[0042] (1) If m1 = p 2 -m2k1, m2 = 2p - k1, where p, k1 > 0, then ζ ≈ 1;

[0043] (2) If m1 = α 2 + β 2 -m2k1, m2 = 2α - k1, where α, β, k1 > 0, then

[0044] where δ > 0 represents a bounded constant, z represents an error, and represent the first and second derivatives of the Lyapunov function, respectively, k1 and p are both free parameters.

[0045] Further, the weight update law of the adaptive neural network is expressed as:

[0046]

[0047]

[0048] where μ x , μ y > 0 represent learning rates, and represent weight update rates, S(x) represents a radial basis function, m5, m6 > 0 represent design parameters for controlling the system damping ratio,

[0049] Further, the virtual control functions α1 and α5 in step three are expressed as:

[0050]

[0051]

[0052] where represents the first derivative of x d , represents the first derivative of y d .

[0053] Furthermore, k=7.

[0054] The beneficial effects of the present invention are:

[0055] 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 the cricket system. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 This is a fixed-point tracking response curve diagram of the system in the x-axis direction when ζ≈1, ζ≈0.707, and ζ≈0.6;

[0057] Figure 2 This is a fixed-point tracking response curve diagram of the system in the y-axis direction when ζ≈1, ζ≈0.707, and ζ≈0.6;

[0058] Figure 3 This is a schematic diagram of the motion trajectory of a ball when the system of this application tracks a fixed point when ζ≈1, ζ≈0.707, and ζ≈0.6;

[0059] Figure 4 This is a circular trajectory tracking response curve diagram of the system in the x-axis direction when ζ≈1, ζ≈0.707, and ζ≈0.6;

[0060] Figure 5 This is a circular trajectory tracking response curve diagram of the system in the y-axis direction when ζ≈1, ζ≈0.707, and ζ≈0.6;

[0061] Figure 6 This is a schematic diagram of the motion trajectory of a ball when the system of this application tracks a circular trajectory when ζ≈1, ζ≈0.707, and ζ≈0.6. DETAILED DESCRIPTION

[0062] It should be noted that, unless there is any conflict, the various embodiments disclosed in this application can be combined with each other.

[0063] Specific implementation method 1: refer to Figure 1 Specifically describing this embodiment, a preset damping neural network backstepping control method for a cricket system described in this embodiment includes:

[0064] Using the control signal u of the controller x 、u y As input to the cricket system, it controls the cricket system, thereby controlling the position of the ball;

[0065] The controller and the control signal u x 、u yThe design process comprises:

[0066] Step one: determining state variables x1, x2, x5, x6, and establishing a two-dimensional state space model containing unknown nonlinear terms of the system according to the state variables x1, x2, x5, x6, an output signal y of the billiard system x , y y and an input control signal u of the billiard system x , u y , so that the output signal y x , y y tracks target signals x d and y d , wherein x1 represents the position of the small ball in the x-axis direction, x2 represents the speed of the small ball in the x-axis direction, x5 represents the position of the small ball in the y-axis direction, and x6 represents the speed of the small ball in the y-axis direction;

[0067] Step two: defining error variables z1, z2, z5 and z6 based on the two-dimensional state space model containing unknown nonlinear terms of the system, the error variables z1, z2, z5 and z6 being represented as:

[0068] z1 = x1 - x d ,

[0069] z2 = x2 - α1;

[0070] z5 = x5 - y d ,

[0071] z6 = x6 - α5;

[0072] wherein α1 and α5 are virtual control functions to be designed;

[0073] Step three: obtaining first-order derivatives of z1 and z5 and and designing virtual control functions α1 and α5 by using and ;

[0074] Step four: introducing command filter approximations and to obtain reconstructed unknown nonlinear terms of the system;

[0075] Step five: based on the reconstructed unknown nonlinear terms of the system, designing control signals u x and u y by using virtual control functions α1 and α5 and introducing an adaptive neural network;

[0076] Step six: designing a Lyapunov function and obtaining controller parameters by using Lyapunov stability criterion, thereby obtaining a controller, and the control output of the controller being the control signal u x、u y ;

[0077] The control signal u x ,u y is expressed as:

[0078]

[0079]

[0080] wherein, respectively represent the neural network fitting the nonlinearity in the x-axis and y-axis directions, k = bgk f , k represents a coefficient, b represents a known constant, k f represents a gain coefficient, g represents the acceleration of gravity, m1, m2 > 0 represent design parameters for controlling the damping ratio of the system, m2 = 2α, m1 = α 2 + β 2 , α, β represent the angles of rotation of the plate in the x-axis and y-axis directions.

[0081] The two-dimensional state space model of the nonlinear strict feedback system containing uncertainty is expressed as:

[0082]

[0083] wherein, x1, x2 represent state variables of the nonlinear second-order system, represents the first-order derivative of x2, represents the first-order derivative of x1, b represents a known constant, f(x1, x2) represents a known nonlinear continuous function representing the nonlinear dynamics of the system, d(t) represents a disturbance term of the nonlinear second-order system, u represents a control input signal of the nonlinear second-order system, and y represents an output of the nonlinear second-order system.

[0084] The specific form of the state space model of the two-dimensional nonlinear system of the plate-ball system is established as:

[0085] The plate-ball system is described by the following kinetic equations:

[0086]

[0087] wherein, r represents the radius of the ball, x, y respectively represent the displacements of the ball in the x-axis and y-axis directions, and α, β are the angles of rotation of the plate in the x-axis and y-axis directions. b represents the moment of inertia of the ball, I x ,I y represents the moments of inertia of the plate in the x-axis and y-axis directions. m represents the mass of the ball, g represents the acceleration of gravity, F x ,F yrepresents the force on the ball in the x and y directions. τ x ,τ y represents the torque on the plate in the x and y directions.

[0088] Let x [x1; x2; x3; x4; x5; x6; x7; x8] be the state variable of the "plate-ball" system, corresponding to the physical variable of the system respectively Take u x ,u y as the input variable of the system. Take the ball position Y [x1; x5] as the output of the system, the state space model of the "plate-ball" system can be established. The model is as follows:

[0089] The plate-ball system is described by the following equations:

[0090]

[0091] In the formula, the coefficient

[0092] Through analysis, the inclination angle of the plate is generally not greater than 10°, so sin x3, sin x7 can be approximated as x3, x7, therefore, the mathematical model of the system after decoupling and simplification can be decomposed into two direction subsystems of x axis and y axis:

[0093] The plate-ball system is described by the following equations:

[0094]

[0095] The models of two directions are completely symmetrical, so this paper only needs to design the controller for the subsystem of one axis.

[0096] According to the actual system, the models of X axis and Y axis can be simplified as follows:

[0097] The plate-ball system is described by the following equations:

[0098]

[0099] In this formula:

[0100]

[0101] f1 = k r [-m(I y + I b )(gx1cosx3+2x1x2x4+x2x5x8+x1x6x8+m 2x5(gx1x5(cosx3-cosx7)+2x1x5(x6x8-x2x4)+(x1x4-x5x8)(x2x5+x1x6)))],

[0102]

[0103] The control objective is to design a system control input u x ,u y such that the system output ω tracks a given target signal y d .

[0104] The above system can be simplified as:

[0105]

[0106] The proposed system state space expression for cricket is:

[0107]

[0108] where k = bgk f , is a Lipschitz continuous function; y x and denotes the system output, and the controller is designed such that it can track the signal x d and y d .

[0109] The Lyapunov stability criterion is expressed as:

[0110] Second-order Lyapunov stability criterion: if there exist positive numbers m1, m2, δ > 0, and a Lyapunov function V(z) = z = y - y d satisfying the following conditions:

[0111]

[0112] then the solution of system (1) is practically stable at a predetermined system damping ratio ζ.

[0113] In addition,

[0114] (1) If m1 = p 2 -m2k1, m2 = 2p - k1, where p, k1 > 0, then ζ ≈ 1;

[0115] (2) If m1 = α 2 + β 2 -m2k1, m2 = 2α - k1, where α, β, k1 > 0, then

[0116] It will be shown that criterion 1 can ensure the system tracking error is practically stable in the case of a predetermined overshoot, the proof is as follows:

[0117] Inequality (8) is a second-order differential inequality, removing the absolute value symbol of inequality (8), we get

[0118]

[0119] This means that inequality (8) can be divided into two differential inequalities, because f is a Lipschitz continuous function, so the comparison principle can be used to analyze the upper and lower bounds of the solution of (8). The boundary conditions are as follows:

[0120]

[0121]

[0122] Where And Respectively represent the solutions of equations (10) and (11). According to the comparison principle, the solution of inequality (8) satisfies In this way, we will change the research object from a second-order differential inequality to two second-order ordinary differential equations.

[0123] The characteristic equation of two differential equations (10) and (11) is:

[0124] s 2 +m2s+m1=0 (12)

[0125] This application can select a normal number k, so that:

[0126] (s+p) 2 =s 2 +m2s+m1 (13) Where m2=2p,q=p 2 Then the solutions of (4) and (5) can be defined as:

[0127]

[0128]

[0129] Where a1, a2,..., a6 are constants related to the initial conditions of the system. Since The approximate solution of inequality (8) is:

[0130] z(t)≈(a7+a8t)e -pt (16)

[0131] Where the time derivative of z(t) is given by:

[0132]

[0133] where a7, a8 are constants related to the initial conditions of the system. Since the initial conditions and Thus, a8 - pa7 = 0, and we have:

[0134]

[0135] So, z(t) is actually stable, and the damping ratio ζ of the system is ζ ~ 1.

[0136] If we choose two positive constants a, β > 0, then:

[0137] [s + (a + βi)][s + (a - βi)] = s 2 + m2s + m1 (19)

[0138] where m2 = 2a, m1 = a 2 + β 2 Thus, the solutions of (4) and (5) can be defined as:

[0139]

[0140]

[0141] where b1, b2,..., b6 are constants related to the initial conditions of the system. According to the comparison principle, we have Thus, the approximate solution of inequality (8) can be expressed as:

[0142] z(t) ~ (b7 cos βt + b8 sin βt)e -αt ,(22)

[0143] where b7, b8 are constants related to the initial conditions of the system. It follows that the solution of (22) is asymptotically stable.

[0144] The damping ratio is defined as:

[0145]

[0146] This shows that the solution z(t) of the system has an underdamped response.

[0147] Define error variables z1 = x1 - x d , z2 = x2 - a1; z5 = x5 - y d , z6 = x6 - a5; where a1 and a5 represent virtual control functions.

[0148] Using To design α1, α5. The specific process is:

[0149] Calculate by taking the derivative

[0150]

[0151] Therefore, α1 and α5 are designed as follows:

[0152]

[0153] Introducing command filters to approximate To reconstruct the nonlinear terms of the system. The specific process is:

[0154] Introducing instruction filtering: We can use the following instruction filter to obtain Estimated value of:

[0155]

[0156] Where τ1, τ2 are filter parameters. So we have:

[0157]

[0158] where ξ1 is and The estimation error between and The estimation error between .

[0159] Design control input u x ,u y ; The specific process is:

[0160] 1) The time derivative of z2 is derived as follows:

[0161]

[0162] The time derivative of z6 is derived as follows:

[0163]

[0164] 2) Control input u x ,u y The design is as follows:

[0165]

[0166]

[0167] Among them, m1, m2, m5, m6>0 are the design parameters for controlling the damping ratio of the system. The neural network is denoted by W S(x). Substitute (27) into (29) to get

[0168]

[0169]

[0170] The update rate of the neural network is designed; the specific process is as follows:

[0171] 1) Define a neural network The input vector of the neural network is denoted by x, and Ω is a compact set; S(x) = [s1(x), s2(x),..., sN(x)]Tis the basis function vector of the neural network. N (x) T The weight vector is denoted by W = [ω1, ω2,..., ωN]T; the number of nodes of the neural network is denoted by N. N T s i (x) is denoted by:

[0172]

[0173] where m i = [m 1i ,m 2i ,...,m vi ] T is the center vector, and d i is the width of the basis function.

[0174] 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:

[0175]

[0176] then

[0177]

[0178] 3) Use the defined neural network W T S(x) to approximate unknown terms f1, f5; according to the above lemma, the neural network weight update law is designed as follows:

[0179]

[0180]

[0181] where μ x , μ y > 0 represent the learning rate.

[0182] ​With the help of the criterion proposed in step 1, the effectiveness of the overshoot controllable neural network backstepping controller (28)(29) for the cricket system will be proved below. The proof process is as follows:

[0183] According to the instruction filter we can get and The relationship between:

[0184]

[0185]

[0186] Where ξ2,ξ6 represent the approximation error, represents the upper bound of ξ2,ξ6. Substitute equation (33) into equation (31):

[0187]

[0188]

[0189] According to the conclusion in the lemma, we can get:

[0190]

[0191]

[0192] Substitute equation (35) into equation (27):

[0193]

[0194]

[0195] Select the Lyapunov function as V1=z1, then:

[0196]

[0197] in and is bounded, so is bounded, that is So we get:

[0198]

[0199] in

[0200] Similarly, this application selects the Lyapunov function as V2=z5, so:

[0201]

[0202] wherein and is bounded, thus ξ1 is bounded, i.e. Thus we get:

[0203]

[0204] wherein

[0205] According to the criterion, it can be obtained that the system tracking error is practically stable under the predetermined overshoot, and the proof is completed.

[0206] The beneficial effects of the present application are verified by the following examples:

[0207] Example 1:

[0208] For the system, the constant k = 7, for the fixed point tracking, the system target signal is X d1 (t) = 100, Y d1 (t) = 100, for the circular trajectory tracking, the system target signal is X d2 (t) = 100sin t + 100, Y d2 (t) = 100cos t) + 100. The command filter parameters τ2 = τ6 = 100, The neural network parameters γ = 0.00001, d = 0.5, m = 0 ω(0) = 0. The damping ratio of the system is different, and different parameters are generated as follows:

[0209]

[0210] The fixed point tracking response curve graph in the x-axis direction when ζ ≈ 1, ζ ≈ 0.707, ζ ≈ 0.6 is as shown in Figure 1 ;

[0211] The fixed point tracking response curve graph in the y-axis direction when ζ ≈ 1, ζ ≈ 0.707, ζ ≈ 0.6 is as shown in Figure 2 ;

[0212] The movement trajectory of the small ball when tracking the fixed point when ζ ≈ 1, ζ ≈ 0.707, ζ ≈ 0.6 is as shown in Figure 3 ;

[0213] The circular trajectory tracking response curve graph in the x-axis direction when ζ ≈ 1, ζ ≈ 0.707, ζ ≈ 0.6 is as shown in Figure 4 ;

[0214] The circular trajectory tracking response curve graph in the y-axis direction when ζ ≈ 1, ζ ≈ 0.707, ζ ≈ 0.6 is as shown in Figure 5 ;

[0215] The movement trajectory of the small ball when tracking the circular trajectory is as shown in the figure when ζ≈1, ζ≈0.707, ζ≈0.6. Figure 6

[0216] Conclusion: From the experiment, it can be known that the control signal u designed by the preset damping neural network backstepping control method of the cricket system can not only stabilize the closed-loop system, but also can determine the damping ratio of the system in advance according to the proposed parameter setting rule. x y

[0217] It should be noted that the specific embodiments are only an explanation and illustration of the technical solutions of the present application, and cannot limit the protection scope. Any partial change made according to the claims and the specification of the present application shall still fall within the protection scope of the present application.​​​

Claims

1. A preset damping neural network backstepping control method for a cricket system, characterized in that: include: Using the control signal of the controller 、 As input to the cricket system, it controls the cricket system, thereby controlling the position of the ball; The controller and the control signal 、 The design process includes: Step 1: Determine the state variables 、 、 、 , and according to the state variables 、 、 、 , the output signal of the cricket system 、 and the input control signal of the cricket system 、 Establish a two-dimensional state space model containing unknown nonlinear terms of the system so that the output signal 、 Tracking target signal and ,in, Represents a small ball Axis position, Represents a small ball Direction speed, Represents a small ball Axis position, Represents a small ball Speed ​​of direction; Step 2: Define the error variable based on the two-dimensional state space model containing the unknown nonlinear terms of the system 、 、 as well as , the error variable 、 、 as well as Expressed as: ， ； ， ； in, and is the virtual control function to be designed; Step 3: Get and The first derivative of and , and use and Design virtual control function and ; Step 4: Introducing command filter approximation and , get the unknown nonlinear terms of the reconstructed system; Step 5: Based on the unknown nonlinear terms of the reconstructed system, use the virtual control function and , and introduce adaptive neural network to design control signal and ; Step 6: Design the Lyapunov function and use the Lyapunov stability criterion to obtain the controller parameters, and then obtain the controller. The control output of the controller is the control signal 、 ; The control signal Expressed as: in, Respectively Axis and Axis direction fitting nonlinear neural network, , represents the coefficient, represents a known constant, represents the gain coefficient, represents the acceleration due to gravity, represents the design parameter for controlling the system damping ratio, , Indicates that the tablet is Axis and The angle of rotation about the axis.

2. The preset damping neural network backstepping control method for a cricket system according to claim 1, characterized in that: The two-dimensional state space model containing unknown nonlinear terms of the system is expressed as: in, 、 represents the nonlinear term of the system, represents the radius of the ball, Respectively represent the ball in Axis direction and The displacement in the axial direction, represents the moment of inertia of the ball, represents the mass of the ball, Indicates that the ball is Axis and The force in the axial direction.

3. The preset damping neural network backstepping control method for a cricket system according to claim 2, characterized in that: The unknown nonlinear term of the reconstructed system is expressed as: in, and Represents the filter parameters.

4. The preset damping neural network backstepping control method for a cricket system according to claim 3, characterized in that: The Lyapunov stability criterion is expressed as: If there is a positive constant , and a Lyapunov function The following conditions are met: Then the solution of the system is determined in advance by the damping ratio of the system The following is actually stable, also, (1) If , ,in ,So ; (2) If , ,in ,So ; in, represents a bounded constant, Indicates error, and denote the first and second order derivatives of the Lyapunov function, respectively. and are free parameters.

5. The preset damping neural network backstepping control method for a cricket system according to claim 4, characterized in that: The weight update law of the adaptive neural network is expressed as: in, represents the learning rate, and represents the weight update rate, represents the radial basis function, represents the design parameter used to control the damping ratio of the system.

6. The preset damping neural network backstepping control method for a cricket system according to claim 5, characterized in that: The virtual control function in step 3 and Expressed as: in, express The first derivative of express The first derivative of .

7. The preset damping neural network backstepping control method for a cricket system according to claim 6, characterized in that: described =7.

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