A neural network tracking control method for a double inverted pendulum system with friction compensation

Abstract

The present invention discloses a neural network tracking control method for a two-stage inverted pendulum system with friction compensation, which relates to the field of nonlinear system control and includes the following steps: Step 1, establishing a discrete time model of a two-stage inverted pendulum system; Step 2, describing the tracking control problem of the two-stage inverted pendulum system as a discrete time nonlinear output regulation problem; Step 3, designing a neural network controller based on the discrete time nonlinear output regulation theory without considering friction; Step 4, designing a disturbance observer; Step 5, combining the neural network controller and the disturbance observer to obtain a final controller. The present invention solves the tracking control problem of a two-stage inverted pendulum system under the influence of friction, and has a high-precision tracking performance.

Description

Technical Field

[0001] The invention relates to the field of nonlinear system control, and in particular to a neural network tracking control method of a double inverted pendulum system with friction compensation. Background Art

[0002] The double inverted pendulum is a typical underactuated mechanical system with unstable, multivariable, and strongly coupled nonlinear characteristics. It is often used to verify various advanced control algorithms. The related control methods of the double inverted pendulum have also been widely used in the fields of robotics and aerospace. The related control problems of the double inverted pendulum can be divided into stabilization problems, swing-starting problems, and tracking control problems. Since the tracking control problem not only needs to ensure the stability of the closed-loop system but also requires the system output to track a given reference signal, the tracking control problem of the double inverted pendulum system is challenging. In addition, the nonlinear output regulation theory is widely used in the study of trajectory tracking and interference suppression problems of various nonlinear systems. Considering that most control algorithms are implemented based on digital controllers, it is of great significance to use discrete-time nonlinear output regulation methods to solve the tracking problem of the double inverted pendulum system under time-varying reference signals.

[0003] On the other hand, friction is ubiquitous in mechanical systems and cannot be ignored for control systems that require high-precision tracking performance. Without proper compensation, friction may cause significant tracking errors. Therefore, friction compensation is an important goal in motion control. A common approach is to use additional control inputs to offset the effects of friction. Due to the complexity of friction, which often exhibits unpredictable characteristics, friction compensation is a challenging problem. Summary of the invention

[0004] Based on the technical problems existing in the background technology, the present invention proposes a neural network tracking control method for a two-stage inverted pendulum system with friction compensation. For the two-stage inverted pendulum system, firstly, without considering friction, a neural network controller is designed based on discrete time nonlinear output regulation theory to achieve tracking; then a disturbance observer is designed to overcome the influence of friction on tracking performance.

[0005] The technical solution adopted by the present invention is:

[0006] A neural network tracking control method for a two-stage inverted pendulum system with friction compensation, characterized in that it comprises the following steps:

[0007] Step 1, establishing a discrete time model of a double inverted pendulum system;

[0008] Step 2, describe the tracking control problem of the double inverted pendulum system as a discrete-time nonlinear output regulation problem;

[0009] Step 3, without considering friction, design a neural network controller based on discrete-time nonlinear output regulation theory;

[0010] Step 4, design a disturbance observer;

[0011] Step 5: Combining the neural network controller and the disturbance observer is the final controller.

[0012] Furthermore, the neural network tracking control method of a two-stage inverted pendulum system with friction compensation is characterized in that, in step 1, the discrete time model of the two-stage inverted pendulum system is described as follows:

[0013]

[0014] Among them, κ1=m1l1+m2L1+m3L1, κ2=J1+m1l1 2 +m2L1 2 +m3L1 2 ,κ3=J2+m2l2 2 ,κ4=m2l2,κ5=m2L1l2,T is the sampling time,x1=r, represents the position and speed of the car, x3=θ1∈(-π,π), represents the angle between the pendulum 1 and the vertical line and its angular velocity, x5=θ2∈(-π,π), represents the angle between pendulum 2 and the vertical line and its angular velocity, represents the control input, m c Represents the mass of the car, m1 represents the mass of pendulum 1, m2 represents the mass of pendulum 2, m3 represents the mass of the link mechanism, l1 represents the length from the center of mass of pendulum 1 to the bottom of pendulum 1, l2 represents the length from the center of mass of pendulum 2 to the bottom of pendulum 2, L1 represents the length of pendulum 1, L2 represents the length of pendulum 2, J1 represents the moment of inertia of pendulum 1, J2 represents the moment of inertia of pendulum 2, the total mass of the system M=m1+m2+m3, y represents the output of the system.

[0015] Furthermore, the neural network tracking control method for a two-stage inverted pendulum system with friction compensation is characterized in that, in step 2, the tracking control problem of the two-stage inverted pendulum system is described as a discrete-time nonlinear output regulation problem, and the process is as follows:

[0016] 2.1, assuming that the position reference trajectory of the double inverted pendulum system is a sinusoidal signal y d (k) = AsinωkT, this position reference trajectory can be generated by an external system in the following form:

[0017]

[0018] Its output is y d (k) = v1(k).

[0019] 2.2, define tracking error as:

[0020] e(k)=x1(k)-v1(k)(3)

[0021] 2.3, the double inverted pendulum system, the external system and the tracking error (3) can be written in the following compact form:

[0022]

[0023] At this point, the tracking control problem of the double inverted pendulum system has been described as a discrete-time nonlinear output regulation problem, and its control goal is to make the steady-state tracking error small enough under the premise of ensuring the stability of the closed-loop system.

[0024] where x(v) = [x1(k), x2(k), …, x6(k)] T H(x(k),u(k),v(k))=x1(k)-v1(k)

[0025]

[0026]

[0027] B1=κ1κ3cos x3(k)-κ4κ5cos x5(k)cos(x3(k)-x5(k))

[0028]

[0029] B2=κ2κ4cos x5(k)-κ1κ5cos x3(k)cos(x3(k)-x5(k))

[0030]

[0031] Furthermore, the neural network tracking control method of the two-stage inverted pendulum system with friction compensation is characterized in that, in step 3, a neural network controller is designed based on discrete-time nonlinear output regulation theory without considering friction, and the process is as follows:

[0032] 3.1, the Jacobian matrix of the system is as follows:

[0033]

[0034]

[0035] Where Δ1=κ2κ3-κ5 2 .

[0036] It can be shown that (A,B) is stabilizable. Therefore there exists a matrix K such that the eigenvalues ​​of A+BK are all inside the unit circle.

[0037] 3.2, solve the discrete regulator equation, the process is as follows:

[0038] The discrete regulator equation has the following form:

[0039]

[0040] where x(v)=[x1(v),x2(v),…,x6(v)] T represents the steady-state state, and u(v) represents the steady-state input.

[0041] Through calculation, the partial solution of the discrete regulator equation can be obtained as follows:

[0042]

[0043] Where E is the identity matrix. Let χ(v) = col(x3(v), x4(v), x5(v), x6(v)), then χ(v) satisfies the following equation:

[0044]

[0045] in

[0046] χ=col(x3,x4,x5,x6)

[0047]

[0048]

[0049]

[0050]

[0051]

[0052] 3.3, use the feedforward neural network function to find the approximate solution of χ(v), the process is as follows:

[0053] Since the exact solution of χ(v) is not available, the approximate solution of χ(v) can be found through the following feedforward neural network function:

[0054]

[0055] Where φ(λ)=(1-e -λ ) / (1+e -λ ), the weight vector W is composed of real numbers Composition, where n=1,...,N, m=1,2.

[0056] Let χ(W,v)=col(x3(W,v),x4(W,v),x5(W,v),x6(W,v)). According to the general approximation theorem, given ε>0 and a compact set containing the origin Since χ(v) is sufficiently smooth, there exist N,W that satisfy:

[0057]

[0058] Next, we use the parameter optimization method to find a set of suitable weights W.

[0059]

[0060]

[0061] In order to make sup v∈Γ J(W,v) is small enough, so a new objective function is constructed:

[0062]

[0063] in is a dense finite set in Γ. The appropriate weight vector W can be obtained by the following weight update law:

[0064]

[0065] Among them, ρ j is the training step length, and the initial weight W0 is a set of randomly generated vectors. Based on the weights obtained from offline training, the approximate values ​​of x3(v), x4(v), x5(v), and x6(v) can be calculated.

[0066] 3.4, design the neural network controller as follows:

[0067]

[0068] in

[0069]

[0070]

[0071] Furthermore, the neural network tracking control method for a two-stage inverted pendulum system with friction compensation is characterized in that, in step 4, a disturbance observer is designed, specifically:

[0072] 4.1, the above design only considers the ideal situation, that is, no friction. In practice, small cars are often affected by friction, usually including static friction, Coulomb friction and viscous friction. The model is as follows:

[0073] F f (k) = f c (k)sign(v c (k))+bv c (k) (17)

[0074] where f c (k) ≥ 0 is the time-varying friction parameter, which represents the sum of static friction and Coulomb friction, b ≥ 0 is the viscous friction parameter, v c =x2 is the speed.

[0075] 4.2, considering that the control input of the system in this method is acceleration, the equivalent friction force is defined as follows:

[0076]

[0077] In order to eliminate the influence of friction, the disturbance observer is designed as follows:

[0078]

[0079] in is the estimated value of the equivalent friction force n(k), is the estimated value of the velocity x2(k), a1,a2,w1,w2∈R + Tunable parameters defined by the user.

[0080] Furthermore, the neural network tracking control method for a two-stage inverted pendulum system with friction compensation is characterized in that the final controller in step 5 is:

[0081]

[0082] The advantages of the present invention are:

[0083] The invention proposes a neural network tracking control method with friction compensation, which solves the tracking control of a two-stage inverted pendulum in the presence of friction and has a high-precision tracking performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0084] Figure 1 This is the block diagram of the neural network control method with friction compensation.

[0085] Figure 2 Schematic diagram of a double inverted pendulum.

[0086] Figure 3 This is the tracking simulation effect diagram of the double inverted pendulum system.

[0087] Figure 4 is the tracking error graph.

[0088] Figure 5 is the friction compensation diagram.

[0089] Figure 6 is the control input diagram. DETAILED DESCRIPTION

[0090] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the accompanying drawings 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.

[0091] Example 1.

[0092] A neural network tracking control method for a two-stage inverted pendulum system with friction compensation comprises the following steps:

[0093] Step 1, establishing a discrete time model of a double inverted pendulum system;

[0094] Step 2, describe the tracking control problem of the double inverted pendulum system as a discrete-time nonlinear output regulation problem;

[0095] Step 3, without considering friction, design a neural network controller based on discrete-time nonlinear output regulation theory;

[0096] Step 4, design a disturbance observer;

[0097] Step 5: Combining the neural network controller and the disturbance observer is the final controller.

[0098] In step 1, the discrete time model of the double inverted pendulum system is described as follows:

[0099]

[0100] Among them, κ1=m1l1+m2L1+m3L1, κ2=J1+m1l1 2 +m2L1 2 +m3L1 2 ,κ3=J2+m2l2 2 ,κ4=m2l2,κ5=m2L1l2,T is the sampling time,x1=r, represents the position and speed of the car, x3=θ1∈(-π,π), represents the angle between the pendulum 1 and the vertical line and its angular velocity, x5=θ2∈(-π,π), represents the angle between pendulum 2 and the vertical line and its angular velocity, represents the control input, m c Represents the mass of the car, m1 represents the mass of pendulum 1, m2 represents the mass of pendulum 2, m3 represents the mass of the link mechanism, l1 represents the length from the center of mass of pendulum 1 to the bottom of pendulum 1, l2 represents the length from the center of mass of pendulum 2 to the bottom of pendulum 2, L1 represents the length of pendulum 1, L2 represents the length of pendulum 2, J1 represents the moment of inertia of pendulum 1, J2 represents the moment of inertia of pendulum 2, the total mass of the system M=m1+m2+m3, y represents the output of the system.

[0101] In step 2, the tracking control problem of the double inverted pendulum system is described as a discrete-time nonlinear output regulation problem, and the process is as follows:

[0102] 2.1, assuming that the position reference trajectory of the double inverted pendulum system is a sinusoidal signal y d (k) = AsinωkT, this position reference trajectory can be generated by an external system in the following form:

[0103]

[0104] Its output is y d (k) = v1(k).

[0105] 2.2, define tracking error as:

[0106] e(k)=x1(k)-v1(k) (3)

[0107] 2.3, the double inverted pendulum system, the external system and the tracking error (3) can be written in the following compact form:

[0108]

[0109] At this point, the tracking control problem of the double inverted pendulum system has been described as a discrete-time nonlinear output regulation problem, and its control goal is to make the steady-state tracking error small enough under the premise of ensuring the stability of the closed-loop system.

[0110] where x(v) = [x1(k), x2(k), …, x6(k)] T

[0111] H(x(k),u(k),v(k))=x1(k)-v1(k)

[0112]

[0113]

[0114] B1=κ1κ3cos x3(k)-κ4κ5cos x5(k)cos(x3(k)-x5(k))

[0115]

[0116] B2=κ2κ4cos x5(k)-κ1κ5cos x3(k)cos(x3(k)-x5(k))

[0117]

[0118] In step 3, without considering friction, the neural network controller is designed based on discrete-time nonlinear output regulation theory. The process is as follows:

[0119] 3.1, the Jacobian matrix of the system is as follows:

[0120]

[0121]

[0122] Where Δ1=κ2κ3-κ5 2 .

[0123] It can be proved is stabilizable. Therefore there exists a matrix K such that All eigenvalues ​​of are inside the unit circle.

[0124] 3.2, solve the discrete regulator equation, the process is as follows:

[0125] The discrete regulator equation has the following form:

[0126]

[0127] where x(v)=[x1(v),x2(v),…,x6(v)] T represents the steady-state state, and u(v) represents the steady-state input.

[0128] Through calculation, the partial solution of the discrete regulator equation can be obtained as follows:

[0129]

[0130] Where E is the identity matrix. Let χ(v) = col(x3(v), x4(v), x5(v), x6(v)), then χ(v) satisfies the following equation:

[0131]

[0132] in

[0133] χ=col(x3,x4,x5,x6)

[0134]

[0135]

[0136]

[0137]

[0138]

[0139] 3.3, use the feedforward neural network function to find the approximate solution of χ(v), the process is as follows:

[0140] Since the exact solution of χ(v) is not available, the approximate solution of χ(v) can be found through the following feedforward neural network function:

[0141]

[0142] Where φ(λ)=(1-e -λ ) / (1+e -λ ), the weight vector W is composed of real numbers Composition, where n=1,...,N, m=1,2.

[0143] Let χ(W,v)=col(x3(W,v),x4(W,v),x5(W,v),x6(W,v)). According to the general approximation theorem, given ε>0 and a compact set containing the origin Since χ(v) is sufficiently smooth, there exist N,W that satisfy:

[0144]

[0145] Next, we use the parameter optimization method to find a set of suitable weights W.

[0146]

[0147]

[0148] In order to make sup v∈Γ J(W,v) is small enough, so a new objective function is constructed:

[0149]

[0150] in is a dense finite set in Γ. The appropriate weight vector W can be obtained by the following weight update law:

[0151]

[0152] Among them, ρ jis the training step length, and the initial weight W0 is a set of randomly generated vectors. Based on the weights obtained from offline training, the approximate values ​​of x3(v), x4(v), x5(v), and x6(v) can be calculated.

[0153] 3.4, design the neural network controller as follows:

[0154]

[0155] in

[0156]

[0157]

[0158] In step 4, the disturbance observer is designed, specifically:

[0159] 4.1, the above design only considers the ideal situation, that is, no friction. In practice, small cars are often affected by friction, usually including static friction, Coulomb friction and viscous friction. The model is as follows:

[0160] F f (k) = f c (k)sign(v c (k))+bv c (k) (17)

[0161] where f c (k) ≥ 0 is the time-varying friction parameter, which represents the sum of static friction and Coulomb friction, b ≥ 0 is the viscous friction parameter, v c =x2 is the speed.

[0162] 4.2, considering that the control input of the system in this method is acceleration, the equivalent friction force is defined as follows:

[0163]

[0164] In order to eliminate the influence of friction, the disturbance observer is designed as follows:

[0165]

[0166] in is the estimated value of the equivalent friction force n(k), is the estimated value of the velocity x2(k), a1,a2,w1,w2∈R + Tunable parameters defined by the user.

[0167] In step 5, the final controller is:

[0168]

[0169] Figure 1 FIG. 1 is a schematic diagram of a neural network control method with friction compensation. In order to verify the effectiveness of the proposed method, the present invention simulates and verifies the control effect of a neural network controller (20) with friction compensation. Figure 2 As shown. The mass of the trolley is m c =1.32Kg, pendulum 1 mass m1 = 0.05kg, pendulum 2 mass m2 = 0.13kg, link mechanism mass m3 = 0.236kg, l1 = 0.0775m, l2 = 0.25m, moment of inertia of pendulum 1 Moment of inertia of pendulum 2 Sampling time T=0.005s.

[0170] The controller parameters are set as:

[0171] a1=5,a2=4,w1=2,w2=2,

[0172] K=[-59.7109 -42.8161 -113.6927 1.8385 286.4569 48.5711]

[0173] Obtained through computer training, the training parameters are set as:

[0174] N=20,Γ={v∈R 2 |||v||≤0.5},ω=0.4π, Γ d ={v1=Asinθ, v2=Acosθ|A=0.02i, i=1,2,...25, j=1,2,...,20}.

[0175] The tracking trajectory of this embodiment is a 0.5sin(0.4πkT)m sinusoidal reference signal. The friction force is set to

[0176] Figure 3 This is the tracking simulation effect diagram of the two-stage inverted pendulum system. Figure 4 is the tracking error graph, Figure 5 is the friction compensation diagram, Figure 6 is the output graph of the controller (20). Figure 3-5 It can be seen from the simulation results that under the action of the neural network controller with friction compensation, the interference of friction can be effectively suppressed and the position reference signal can be quickly tracked.

[0177] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any technician familiar with the technical field can make equivalent replacements or changes according to the technical scheme and inventive concept of the present invention within the technical scope disclosed by the present invention, which should be covered by the protection scope of the present invention.

Claims

1. A neural network tracking control method for a double inverted pendulum system with friction compensation, characterized in that: The following steps are involved: Step 1, establishing a discrete time model of a double inverted pendulum system; Step 2, describe the tracking control problem of the double inverted pendulum system as a discrete-time nonlinear output regulation problem; Step 3, without considering friction, design a neural network controller based on discrete time nonlinear output regulation theory; Step 4, design a disturbance observer; Step 5, combining the neural network controller and the disturbance observer to obtain the final controller; In step 1, the discrete time model of the double inverted pendulum system is described as follows: Among them, κ1=m1l1+m2L1+m3L1, κ2=J1+m1l1 2 +m2L1 2 +m3L1 2 ,κ3=J2+m2l2 2 ,κ4=m2l2,κ5=m2L1l2,T is the sampling time,x1=r, represents the position and speed of the car, x3=θ1∈(-π,π), represents the angle between the pendulum 1 and the vertical line and its angular velocity, x5=θ2∈(-π,π), represents the angle between pendulum 2 and the vertical line and its angular velocity, represents the control input, m c represents the mass of the car, m1 represents the mass of pendulum 1, m2 represents the mass of pendulum 2, m3 represents the mass of the link mechanism, l1 represents the length from the center of mass of pendulum 1 to the bottom of pendulum 1, l2 represents the length from the center of mass of pendulum 2 to the bottom of pendulum 2, L1 represents the length of pendulum 1, L2 represents the length of pendulum 2, J1 represents the moment of inertia of pendulum 1, J2 represents the moment of inertia of pendulum 2, the total mass of the system M=m1+m2+m3, y represents the output of the system; In step 2, the tracking control problem of the double inverted pendulum system is described as a discrete-time nonlinear output regulation problem, and the process is as follows: 2.1, assuming that the position reference trajectory of the double inverted pendulum system is a sinusoidal signal y d (k) = AsinωkT, this position reference trajectory can be generated by an external system in the following form: Its output is y d (k) = v1(k); 2.2, define tracking error as: e(k)=x1(k)-v1(k) (3) 2.3, the double inverted pendulum system (1), the external system (2) and the tracking error (3) can be written in the following compact form: x(k+1)=F(x(k),u(k),v(k)) v(k+1)=Sv(k) (4) e(k)=H(x(k),u(k),v(k)) At this point, the tracking control problem of the double inverted pendulum system has been described as a discrete-time nonlinear output regulation problem, and its control goal is to make the steady-state tracking error small enough under the premise of ensuring the stability of the closed-loop system; in, x(v)=[x1(k),x2(k),…,x6(k)] T H(x(k),u(k),v(k))=x1(k)-v1(k) Step 3, without considering friction, design a neural network controller based on discrete time nonlinear output regulation theory. The process is as follows: 3.1, the Jacobian matrix of the system is as follows: in, Δ1=κ2κ3-κ5 2 ; prove is stabilizable; therefore there exists a matrix K such that The eigenvalues ​​of are all inside the unit circle; 3.2, solve the discrete regulator equation, the process is as follows: The discrete regulator equation has the following form: where x(v)=[x1(v),x2(v),…,x6(v)] T represents the steady state, and u(v) represents the steady-state input; by calculation, the partial solution of the discrete regulator equation can be obtained as follows: Where E is the identity matrix; let χ(v) = col(x3(v), x4(v), x5(v), x6(v)), then χ(v) satisfies the following equation: in χ=col(x3,x4,x5,x6) 3.3, use the feedforward neural network function to find the approximate solution of χ(v), the process is as follows: Since the exact solution of χ(v) is not available, the approximate solution of χ(v) can be found through the following feedforward neural network function: Where φ(λ)=(1-e -λ ) / (1+e -λ ), the weight vector W is composed of real numbers Composition, where n=1,...,N, m=1,2; Let χ(W,v)=col(x3(W,v),x4(W,v),x5(W,v),x6(W,v)); According to the general approximation theorem, given ε>0 and a compact set containing the origin Since χ(v) is sufficiently smooth, there exist N,W that satisfy: Next, we use the parameter optimization method to find a set of suitable weight vectors W; let In order to make sup v∈Γ J(W,v) is small enough, so a new objective function is constructed: in is a dense finite set in Γ; the appropriate weight vector W can be obtained by the following weight update law: Among them, ρ j is the training step length, and the initial weight W0 is a set of randomly generated vectors; based on the weights obtained from offline training, the approximate values ​​of x3(v), x4(v), x5(v), and x6(v) can be calculated; 3.4, design the neural network controller as follows: in In step 4, the disturbance observer is designed, specifically: 4.1, the design in step 3 considers the ideal situation that there is no friction effect. In practice, small cars are often affected by friction, including static friction, Coulomb friction and viscous friction. The model is as follows: F f (k) = f c (k)sign(v c (k))+bv c (k) (17) where f c (k) ≥ 0 is the time-varying friction parameter, which represents the sum of static friction and Coulomb friction, b ≥ 0 is the viscous friction parameter, v c =x2 is the speed; 4.2, considering that the control input of the system in this method is acceleration, the equivalent friction force is defined as follows: In order to eliminate the influence of friction, the disturbance observer is designed as follows: in is the estimated value of the equivalent friction force n(k), is the estimated value of the velocity x2(k), a1, a2, w1, Tunable parameters defined by the user.

2. The neural network tracking control method of a double inverted pendulum system with friction compensation according to claim 1 is characterized in that: The final controller described in step 5 is:

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