A model-based adaptive inverted pendulum angle tracking method

Abstract

The application provides a model adaptive inverted pendulum system angle tracking method, which realizes angle reference control and angular velocity reference control through the proportional integral differential of a reference model error; the core of the method is to solve the uncertainty of the model through the reference model, and to obtain an inverted pendulum angle error damping signal, an angle reference control damping signal, an inverted pendulum angle error differential signal and an inverted pendulum angular velocity error damping signal through the reference model, so as to improve the stability of the system through the damping signals, to ensure the stability of the whole system through the backstepping design, and finally to respectively adaptively estimate the inverted pendulum model angle parameter and the acceleration coefficient parameter through the inverted pendulum angular velocity estimation error signal and the inverted pendulum angle error signal, so that the whole control scheme is completely independent of the model parameters, and the whole scheme has good robustness and adaptive ability.

Description

Technical Field

[0001] This invention relates to the field of industrial inverted pendulum and teaching experimental inverted pendulum control, and more specifically, to a model-adaptive inverted pendulum angle tracking method. Background Technology

[0002] The inverted pendulum system is an open-loop, strongly unstable system, and its study is of particular significance for industrial control and experimental teaching demonstrations. Due to its strong instability, many control methods suffer from insufficient stability margins, often leading to pendulum angle control failure and loss of balance under disturbance conditions. The main reasons for insufficient stability margins are, firstly, the inability to accurately model the parameters, i.e., model uncertainty; and secondly, the difficulty in constructing the damping signal in the control, resulting in insufficient system damping. Model uncertainty and the difficulty in accurately constructing damping are also prevalent in industrial control. Based on these background reasons, this invention proposes a model reference method. Its key feature is the use of the precisely known characteristics of the reference model to accurately and conveniently construct the damping signal required for control, thereby significantly improving the overall stability margin of the control. This makes the invention highly innovative in theory and practical in engineering.

[0003] It should be noted that the information in the background section above is only used to enhance the understanding of the background of the present invention, and therefore may include information that does not constitute prior art known to those skilled in the art. Summary of the Invention

[0004] The purpose of this invention is to provide a model-adaptive inverted pendulum angle tracking method, thereby overcoming the problems of low stability margin and dynamic performance of inverted pendulum control caused by the defects of related technologies.

[0005] According to one aspect of the present invention, a model-adaptive inverted pendulum angle tracking method is provided, comprising the following six steps:

[0006] Step S10: Establish the reference model of the inverted pendulum system as follows: First, select the initial value of the estimated angular velocity of the inverted pendulum as 0, and the initial value of the angle reference control quantity as 0. Calculate the differential estimated value of the inverted pendulum angle based on the estimated angular velocity of the inverted pendulum and the angle reference control quantity, and then integrate to obtain the estimated value of the inverted pendulum angle. Then, set the initial value of the acceleration control signal as 0, set the initial value of the angular velocity reference control quantity as 0, and set the initial values ​​of the angle parameters and acceleration coefficient parameters of the inverted pendulum model as nominal values. Calculate the estimated value of the angular acceleration of the inverted pendulum based on the estimated value of the inverted pendulum angle, the acceleration control signal, and the angular velocity reference control quantity, and then integrate to obtain the estimated value of the inverted pendulum angular velocity.

[0007] Step S20: Install an angle measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angle signal; compare it with the estimated inverted pendulum angle to obtain the inverted pendulum angle estimation error signal; then integrate it to obtain the inverted pendulum angle estimation error integral signal; install an angular velocity measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angular velocity signal; and construct the inverted pendulum angle estimation error damping signal based on the estimated inverted pendulum angular velocity, angle reference control quantity, and inverted pendulum angular velocity signal; then superimpose the inverted pendulum angle estimation error integral signal and the inverted pendulum angle estimation error signal to form the angle reference control quantity.

[0008] Step S30: Compare the estimated angular velocity of the inverted pendulum with the angular velocity signal of the inverted pendulum to obtain the angular velocity estimation error signal of the inverted pendulum; then integrate to obtain the integral signal of the angular velocity estimation error of the inverted pendulum; then, based on the set angle parameters, acceleration coefficient parameters and their nominal values ​​of the inverted pendulum model, as well as the acceleration control signal, angular velocity reference control quantity, and the estimated angular velocity of the inverted pendulum and the angular velocity signal of the inverted pendulum, respectively calculate the damping signal of the angular velocity estimation error of the inverted pendulum and the equivalent control quantity of the angular velocity reference; finally, superimpose the angular velocity estimation error signal of the inverted pendulum and the integral signal of the angular velocity estimation error of the inverted pendulum to form the final acceleration control signal.

[0009] Step S40: Set the desired value of the inverted pendulum angle according to the control task of the inverted pendulum, and compare it with the estimated inverted pendulum angle signal to obtain the inverted pendulum angle error signal; then integrate it to obtain the inverted pendulum angle error integral signal; then calculate the inverted pendulum angle error damping signal according to the estimated inverted pendulum angular velocity signal and the angle reference control quantity; then superimpose the angle reference control quantity, the inverted pendulum angle error signal, and the inverted pendulum angle error integral signal to form the desired inverted pendulum angular velocity signal.

[0010] Step S50: Compare the estimated angular velocity signal of the inverted pendulum with the expected angular velocity signal of the inverted pendulum to obtain the angular velocity error signal of the inverted pendulum; then integrate to obtain the integral signal of the angular velocity error of the inverted pendulum; then calculate the angle reference control damping signal based on the inverted pendulum angle estimation error damping signal and the inverted pendulum angle estimation error solution; then superimpose the angle reference control quantity and the inverted pendulum angular velocity estimation signal to form the differential signal of the inverted pendulum angle error; then superimpose the inverted pendulum angle estimation signal, the acceleration control signal, the angular velocity reference control quantity, the angle reference control damping signal, the inverted pendulum angle error damping signal, and the inverted pendulum angle error signal to form the inverted pendulum angular velocity error damping signal.

[0011] Step S60: Adaptively update the angle parameters and acceleration coefficient parameters of the inverted pendulum model based on the inverted pendulum angular velocity estimation error signal and the inverted pendulum angle error signal, respectively; then, based on the inverted pendulum angular velocity error damping signal superimposed with the inverted pendulum angular velocity error signal, the inverted pendulum angular velocity error integral signal, the inverted pendulum angle estimation signal, and the acceleration reference control quantity, form the final acceleration control signal; finally, send the acceleration control signal to the inverted pendulum system, thereby completing the stable tracking of the inverted pendulum angle signal of the inverted pendulum system to the desired inverted pendulum angle signal.

[0012] In one exemplary embodiment of the present invention, the initial value of the estimated angular velocity of the inverted pendulum is selected as 0, the initial value of the angle reference control quantity is selected as 0, the differential estimated value of the inverted pendulum angle is calculated based on the estimated angular velocity of the inverted pendulum and the angle reference control quantity, and then integrated to obtain the estimated value of the inverted pendulum angle; then the initial value of the acceleration control signal is set to 0, the initial value of the angular velocity reference control quantity is set to 0, and the initial values ​​of the angle parameters and acceleration coefficient parameters of the inverted pendulum model are set to nominal values, and the estimated value of the angular acceleration of the inverted pendulum is calculated based on the estimated value of the inverted pendulum angle, the acceleration control signal, and the angular velocity reference control quantity, and then integrated to obtain the estimated value of the inverted pendulum angular velocity, including:

[0013]

[0014]

[0015]

[0016]

[0017] Where ω1 is the estimated angular velocity of the inverted pendulum, φ1 is the estimated angle of the inverted pendulum, and u1 is the angle reference control variable. φ1 is the estimated angle of the inverted pendulum, u is the acceleration control signal, u2 is the angular velocity reference control value, and w is the differential estimate of the inverted pendulum angle. 1g For the angle parameters of the inverted pendulum model, w 2g For acceleration coefficient parameters, ω1 is the estimated value of the inverted pendulum angular acceleration, ω2 is the estimated value of the inverted pendulum angular velocity, and dt represents the integral over the time signal.

[0018] In one exemplary embodiment of the present invention, the inverted pendulum angle estimation error damping signal is constructed based on the estimated inverted pendulum angular velocity, the angle reference control quantity, and the inverted pendulum angular velocity signal; the angle reference control quantity is then formed by superimposing the inverted pendulum angle estimation error integral signal and the inverted pendulum angle estimation error signal.

[0019] e1 = φ1 - φ;

[0020] s1=∫e1dt;

[0021] d1 = ω1 + u1 - ω;

[0022] u1 = -k 11 e1-k 12 s1-k 13 d1;

[0023] Where φ is the inverted pendulum angle signal; e1 is the inverted pendulum angle estimation error signal; s1 is the inverted pendulum angle estimation error integral signal; ω is the inverted pendulum angular velocity signal; d1 is the inverted pendulum angle estimation error damping signal; k 11 k 12 k 13 is a constant control parameter, and u1 is the angle reference control quantity.

[0024] In one exemplary embodiment of the present invention, the estimated angular velocity of the inverted pendulum is compared with the angular velocity signal of the inverted pendulum to obtain an angular velocity estimation error signal; then, integration is performed to obtain an integral signal of the angular velocity estimation error of the inverted pendulum; then, based on the set angle parameters, acceleration coefficient parameters and their nominal values ​​of the inverted pendulum model, as well as the acceleration control signal, the angular velocity reference control quantity, and the estimated angular velocity of the inverted pendulum and the angular velocity signal of the inverted pendulum, the damping signal of the angular velocity estimation error of the inverted pendulum and the equivalent control quantity of the angular velocity reference are calculated respectively; finally, the angular velocity estimation error signal of the inverted pendulum and the integral signal of the angular velocity estimation error of the inverted pendulum are superimposed to form the final angular velocity reference control quantity, including:

[0025] e2 = ω1 - ω;

[0026] s2=∫e2dt;

[0027] d2 = -w 1g φ1+w 2g u+u2-(-w 1a φ+w 2a u);

[0028] u 2a =-w 1g φ1+w 2g u-(-w 1a φ+w 2a u);

[0029] u2 = -u 2a -k 21 e2-k 22 s2-k 23 d2;

[0030] Where e2 is the inverted pendulum angular velocity estimation error signal; s2 is the integrated inverted pendulum angular velocity estimation error signal; w 1aw represents the nominal value of the angle parameter of the inverted pendulum model. 2a The nominal value of the acceleration coefficient parameter, as well as the acceleration control signal, the angular velocity reference control quantity, and the estimated and signal angular velocities of the inverted pendulum are calculated separately. d2 is the damping signal for the inverted pendulum angular velocity estimation error. 2a U1 is the angular velocity reference equivalent control quantity, u2 is the angular velocity reference control quantity, and k is the angular velocity reference equivalent control quantity. 21 k 22 k 23 These are constant control parameters.

[0031] In one exemplary embodiment of the present invention, the desired inverted pendulum angle is set according to the control task of the inverted pendulum, and compared with the estimated inverted pendulum angle signal to obtain the inverted pendulum angle error signal; then, integration is performed to obtain the inverted pendulum angle error integral signal; then, the inverted pendulum angle error damping signal is calculated based on the estimated inverted pendulum angular velocity signal and the angle reference control quantity; finally, the angle reference control quantity, the inverted pendulum angle error signal, and the inverted pendulum angle error integral signal are superimposed to form the desired inverted pendulum angular velocity signal, including:

[0032]

[0033] s3=∫e3dt;

[0034] d3=ω1+u1;

[0035]

[0036] in denoted as , e3 is the expected value of the inverted pendulum angle; s3 is the integral signal of the inverted pendulum angle error; d3 is the damping signal of the inverted pendulum angle error. For the desired angular velocity signal of the inverted pendulum, k 31 k 32 k 33 These are constant control parameters.

[0037] In one exemplary embodiment of the present invention, the inverted pendulum angular velocity error signal is obtained by comparing the estimated inverted pendulum angular velocity signal with the expected inverted pendulum angular velocity signal; then, integration is performed to obtain the inverted pendulum angular velocity error integral signal; then, the inverted pendulum angle estimation error damping signal and the inverted pendulum angle estimation error solution angle reference control damping signal are calculated; then, the angle reference control quantity and the inverted pendulum angular velocity estimation signal are superimposed to form the inverted pendulum angle error differential signal; and then, the inverted pendulum angle estimation signal, acceleration control signal, angular velocity reference control quantity, angle reference control damping signal, inverted pendulum angle error damping signal, and inverted pendulum angle error signal are superimposed to form the inverted pendulum angular velocity error damping signal, including:

[0038] e4=ω1-ω1 d ;

[0039]

[0040] d d3 =ω1+u1+u d1 ;

[0041] d4 = -w 1g φ1+w 2g u+u2-u d1 -k 31 d3-k 32 e3-k 33 d d3 ;

[0042] Where e4 is the inverted pendulum angular velocity error signal; s4 is the inverted pendulum angular velocity error integral signal; u d1 For angle reference control damping signal; d d3 d1 is the differential signal of the inverted pendulum angle error; d2 is the damping signal of the inverted pendulum angular velocity error.

[0043] In one exemplary embodiment of the present invention, the angle parameters and acceleration coefficient parameters of the inverted pendulum model are adaptively updated based on the inverted pendulum angular velocity estimation error signal and the inverted pendulum angular error signal, respectively; then, the final acceleration control signal is formed by superimposing the inverted pendulum angular velocity error signal, the inverted pendulum angular velocity error integral signal, the inverted pendulum angle estimation signal, and the acceleration reference control quantity on the inverted pendulum angular velocity error damping signal.

[0044] w 1g (n+1)=w 1g (n)+k 51 e2T+k 52 e3T;

[0045] w 2g (n+1)=w 2g (n)+k 53 e2T+k 54 e3T;

[0046]

[0047] Where T is a constant integration parameter; k 51 k 52 k 53 k 54 These are constant parameters used to adjust the adaptive convergence speed of the angle and acceleration coefficient parameters of the inverted pendulum model; k 41 k 42 k 43 These are constant control parameters used to adjust the amplitude of the acceleration control signal, thereby controlling the speed and dynamic performance of the entire control system.

[0048] Beneficial effects

[0049] This invention provides a model-adaptive inverted pendulum angle tracking method, with the following four main innovations: First, it proposes a model-adaptive approach to construct a reference model for tracking and controlling the inverted pendulum system, significantly reducing the dependence of the entire control scheme on model parameters. Second, it decomposes the entire control into four loops, achieving decoupled control and making the design of each loop relatively independent and simple. Third, it employs a precisely known reference model to accurately construct the damping signals required for system stability, such as the inverted pendulum angle error damping signal, angle reference control damping signal, inverted pendulum angle error differential signal, and inverted pendulum angular velocity error damping signal. This allows each loop to be matched with reasonable damping to improve the stability of a single loop, thereby improving the overall control stability and stability margin. Fourth, the method provided by this invention is universal and can be easily extended to general systems. That is, it constructs an adaptive reference system, adjusts the reference system parameters through errors, and the method of calculating damping based on the reference system can be applied to many other systems.

[0050] It should be understood that the above general description and the following detailed description are exemplary and explanatory only, and are not intended to limit the invention. Attached Figure Description

[0051] The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention. It is obvious that the drawings described below are merely some embodiments of the invention, and those skilled in the art can obtain other drawings based on these drawings without any inventive effort.

[0052] Figure 1 This invention provides a flowchart of a model-adaptive inverted pendulum angle tracking method.

[0053] Figure 2 It is the estimated value of the inverted pendulum angular velocity (unit: degrees per second) provided by the method in the embodiments of the present invention;

[0054] Figure 3 It is the estimated value of the inverted pendulum angle (unit: degrees) provided by the method in the embodiments of the present invention;

[0055] Figure 4 This is the inverted pendulum angle signal (unit: degrees) provided by the method in the embodiments of the present invention;

[0056] Figure 5 This is the inverted pendulum angle estimation error signal (unit: degrees) provided by the method in the embodiments of the present invention;

[0057] Figure 6 It is the inverted pendulum angular velocity estimation error signal (unit: degrees per second) provided by the method in the embodiments of the present invention;

[0058] Figure 7 This is the inverted pendulum angle error signal (unit: degrees) provided by the method in the embodiments of the present invention;

[0059] Figure 8 It is the inverted pendulum angular velocity error signal (unit: degrees per second) provided by the method in the embodiments of the present invention;

[0060] Figure 9 It is the acceleration control signal (unit: meters per second squared) of the method provided in the embodiments of the present invention. Detailed Implementation

[0061] Exemplary embodiments will now be described more fully with reference to the accompanying drawings. However, these exemplary embodiments can be implemented in many forms and should not be construed as limited to the examples set forth herein; rather, these embodiments are provided to make the invention more comprehensive and complete, and to fully convey the concept of the exemplary embodiments to those skilled in the art. The described features, structures, or characteristics can be combined in any suitable manner in one or more embodiments. In the following description, numerous specific details are provided to give a full understanding of embodiments of the invention. However, those skilled in the art will recognize that the technical solutions of the invention may be practiced with one or more of these specific details omitted, or other methods, components, apparatus, steps, etc., may be employed. In other instances, well-known technical solutions are not shown or described in detail to avoid obscuring various aspects of the invention.

[0062] This invention provides a model-adaptive inverted pendulum system angle tracking method. It constructs a reference model and uses the proportional-integral-derivative (PID) function of the reference model error to achieve angle and angular velocity reference control quantities. The core of this method lies in addressing model uncertainties through the reference model. Furthermore, it uses the reference model to obtain the inverted pendulum angle error damping signal, angle reference control damping signal, inverted pendulum angle error derivative signal, and inverted pendulum angular velocity error damping signal. These damping signals improve system stability. Inversion design is then used to ensure overall system stability. Finally, the inverted pendulum angular velocity estimation error signal and inverted pendulum angle error signal are used to adaptively estimate the inverted pendulum model angle parameters and acceleration coefficient parameters, respectively. This achieves a control scheme that is completely independent of model parameters, resulting in excellent robustness and adaptability.

[0063] Below, we will combine the appendix Figure 1This paper further explains and illustrates the model-adaptive inverted pendulum angle tracking method of the present invention. (Reference) Figure 1 As shown, this model-adaptive inverted pendulum angle tracking method includes the following steps:

[0064] Step S10: Establish the reference model of the inverted pendulum system as follows: First, select the initial value of the estimated angular velocity of the inverted pendulum as 0, and the initial value of the angle reference control quantity as 0. Calculate the differential estimated value of the inverted pendulum angle based on the estimated angular velocity of the inverted pendulum and the angle reference control quantity, and then integrate to obtain the estimated value of the inverted pendulum angle. Then, set the initial value of the acceleration control signal as 0, set the initial value of the angular velocity reference control quantity as 0, and set the initial values ​​of the angle parameters and acceleration coefficient parameters of the inverted pendulum model as nominal values. Calculate the estimated value of the angular acceleration of the inverted pendulum based on the estimated value of the inverted pendulum angle, the acceleration control signal, and the angular velocity reference control quantity, and then integrate to obtain the estimated value of the inverted pendulum angular velocity.

[0065] Specifically, this can be broken down into the following two steps. First, select an initial value of 0 for both the estimated angular velocity of the inverted pendulum and the initial value of the angle reference control variable. Then, calculate the differential estimate of the inverted pendulum angle based on the estimated angular velocity and the angle reference control variable, and finally integrate the results to obtain the estimated angle of the inverted pendulum as follows:

[0066]

[0067]

[0068] Where ω1 is the estimated angular velocity of the inverted pendulum, φ1 is the estimated angle of the inverted pendulum, and u1 is the angle reference control variable. φ1 is the estimated value of the angle of the inverted pendulum, which is the differential estimate of the angle of the inverted pendulum.

[0069] The second step involves setting the initial value of the acceleration control signal to 0, the initial value of the angular velocity reference control quantity to 0, and the initial values ​​of the angle parameters and acceleration coefficient parameters of the inverted pendulum model to their nominal values. Then, based on the estimated angle value of the inverted pendulum, the acceleration control signal, and the angular velocity reference control quantity, the estimated angular acceleration value of the inverted pendulum is calculated. Finally, integration is performed to obtain the estimated angular velocity value of the inverted pendulum as follows:

[0070]

[0071]

[0072] Where u is the acceleration control signal; u2 is the angular velocity reference control value, and w 1g For the angle parameters of the inverted pendulum model, w 2g For acceleration coefficient parameters, ω1 is the estimated value of the inverted pendulum angular acceleration, ω2 is the estimated value of the inverted pendulum angular velocity, and dt represents the integral over the time signal.

[0073] Step S20: Install an angle measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angle signal; compare it with the estimated inverted pendulum angle to obtain the inverted pendulum angle estimation error signal; then integrate it to obtain the inverted pendulum angle estimation error integral signal; install an angular velocity measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angular velocity signal; and construct the inverted pendulum angle estimation error damping signal based on the estimated inverted pendulum angular velocity, angle reference control quantity, and inverted pendulum angular velocity signal; then superimpose the inverted pendulum angle estimation error integral signal and the inverted pendulum angle estimation error signal to form the angle reference control quantity.

[0074] Specifically, this can be broken down into the following three steps. First, install an angle-measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angle signal; compare this signal with the estimated inverted pendulum angle to obtain the inverted pendulum angle estimation error signal; then integrate the signals to obtain the integrated inverted pendulum angle estimation error signal as follows:

[0075] e1 = φ1 - φ;

[0076] s1=∫e1dt;

[0077] Where φ is the inverted pendulum angle signal; e1 is the inverted pendulum angle estimation error signal; and s1 is the inverted pendulum angle estimation error integral signal.

[0078] The second step involves installing an angular velocity measuring gyroscope on the inverted pendulum system to measure the angular velocity signal of the inverted pendulum; and constructing the inverted pendulum angle estimation error damping signal based on the estimated angular velocity value, angle reference control quantity, and inverted pendulum angular velocity signal as follows:

[0079] d1 = ω1 + u1 - ω;

[0080] Where ω is the inverted pendulum angular velocity signal; d1 is the inverted pendulum angle estimation error damping signal.

[0081] The third step involves superimposing the damping signal of the inverted pendulum angle estimation error, the integral signal of the inverted pendulum angle estimation error, and the inverted pendulum angle estimation error signal to form the angle reference control quantity as follows:

[0082] u1 = -k 11 e1-k 12 s1-k 13 d1;

[0083] Where k 11 k 12 k 13 is a constant control parameter, and u1 is the angle reference control quantity.

[0084] Step S30: Compare the estimated angular velocity of the inverted pendulum with the angular velocity signal of the inverted pendulum to obtain the angular velocity estimation error signal of the inverted pendulum; then integrate to obtain the integral signal of the angular velocity estimation error of the inverted pendulum; then, based on the set angle parameters, acceleration coefficient parameters and their nominal values ​​of the inverted pendulum model, as well as the acceleration control signal, angular velocity reference control quantity, and the estimated angular velocity of the inverted pendulum and the angular velocity signal of the inverted pendulum, respectively calculate the damping signal of the angular velocity estimation error of the inverted pendulum and the equivalent control quantity of the angular velocity reference; finally, superimpose the angular velocity estimation error signal of the inverted pendulum and the integral signal of the angular velocity estimation error of the inverted pendulum to form the final acceleration control signal.

[0085] Specifically, this can be broken down into the following three steps. First, compare the estimated inverted pendulum angular velocity with the inverted pendulum angular velocity signal to obtain the inverted pendulum angular velocity estimation error signal; then integrate the results to obtain the integrated inverted pendulum angular velocity estimation error signal as follows:

[0086] e2 = ω1 - ω;

[0087] s2=∫e2dt;

[0088] Where e2 is the inverted pendulum angular velocity estimation error signal; s2 is the inverted pendulum angular velocity estimation error integral signal.

[0089] The second step involves calculating the inverted pendulum angular velocity estimation error damping signal and the equivalent angular velocity reference control quantity based on the set angle parameters, acceleration coefficient parameters, and nominal values ​​of the inverted pendulum model, as well as the acceleration control signal, angular velocity reference control quantity, and the estimated and signal angular velocity of the inverted pendulum, as follows:

[0090] d2 = -w 1g φ1+w 2g u+u2-(-w 1a φ+w 2a u);

[0091] u 2a =-w 1g φ1+w 2g u-(-w 1a φ+w 2a u);

[0092] Where w 1a w represents the nominal value of the angle parameter of the inverted pendulum model. 2a The nominal value of the acceleration coefficient parameter, as well as the acceleration control signal, the angular velocity reference control quantity, and the estimated and signal angular velocities of the inverted pendulum are calculated separately. d2 is the damping signal for the inverted pendulum angular velocity estimation error. 2a This is the angular velocity reference equivalent control quantity.

[0093] The third step involves superimposing the inverted pendulum angular velocity estimation error signal and the inverted pendulum angular velocity estimation error integral signal based on the damped signal of the inverted pendulum angular velocity estimation error and the equivalent control quantity of the angular velocity reference, to form the final angular velocity reference control quantity as follows:

[0094] u2 = -u 2a -k 21 e2-k 22 s2-k 23 d2;

[0095] Where u2 is the angular velocity reference control quantity, k 21 k 22 k 23 These are constant control parameters.

[0096] Step S40: Set the desired value of the inverted pendulum angle according to the control task of the inverted pendulum, and compare it with the estimated inverted pendulum angle signal to obtain the inverted pendulum angle error signal; then integrate it to obtain the inverted pendulum angle error integral signal; then calculate the inverted pendulum angle error damping signal according to the estimated inverted pendulum angular velocity signal and the angle reference control quantity; then superimpose the angle reference control quantity, the inverted pendulum angle error signal, and the inverted pendulum angle error integral signal to form the desired inverted pendulum angular velocity signal.

[0097] Specifically, this can be broken down into the following three steps. First, set the desired inverted pendulum angle based on the control task of the inverted pendulum, and compare it with the estimated inverted pendulum angle signal to obtain the inverted pendulum angle error signal; then integrate the values ​​to obtain the integral inverted pendulum angle error signal as follows:

[0098]

[0099] s3=∫e3dt;

[0100] in s3 represents the expected value of the inverted pendulum angle, e3 represents the error signal of the inverted pendulum angle, and s3 represents the integral signal of the error of the inverted pendulum angle.

[0101] The second step involves calculating the inverted pendulum angle error damping signal based on the estimated inverted pendulum angular velocity signal and the angle reference control quantity, as follows:

[0102] d3=ω1+u1;

[0103] Where d3 is the inverted pendulum angle error damping signal.

[0104] The third step involves superimposing the inverted pendulum angle error damping signal, the angle reference control quantity, the inverted pendulum angle error signal, and the inverted pendulum angle error integral signal to form the desired inverted pendulum angle velocity signal as follows:

[0105]

[0106] in For the desired angular velocity signal of the inverted pendulum, k 31 k 32 k 33 These are constant control parameters.

[0107] Step S50: Compare the estimated angular velocity signal of the inverted pendulum with the expected angular velocity signal of the inverted pendulum to obtain the angular velocity error signal of the inverted pendulum; then integrate to obtain the integral signal of the angular velocity error of the inverted pendulum; then calculate the angle reference control damping signal based on the inverted pendulum angle estimation error damping signal and the inverted pendulum angle estimation error solution; then superimpose the angle reference control quantity and the inverted pendulum angular velocity estimation signal to form the differential signal of the inverted pendulum angle error; then superimpose the inverted pendulum angle estimation signal, the acceleration control signal, the angular velocity reference control quantity, the angle reference control damping signal, the inverted pendulum angle error damping signal, and the inverted pendulum angle error signal to form the inverted pendulum angular velocity error damping signal.

[0108] Specifically, it can be broken down into the following four steps. First, compare the estimated inverted pendulum angular velocity signal with the expected inverted pendulum angular velocity signal to obtain the inverted pendulum angular velocity error signal; then integrate the signals to obtain the integrated inverted pendulum angular velocity error signal as follows:

[0109]

[0110] s4=∫e4dt;

[0111] Where e4 is the inverted pendulum angular velocity error signal; s4 is the inverted pendulum angular velocity error integral signal.

[0112] The second step involves calculating the reference control damping signal for the inverted pendulum angle estimation error based on the inverted pendulum angle estimation error, as follows:

[0113]

[0114] Where u d1 This is the angle reference control damping signal.

[0115] The third step involves superimposing the angle reference control quantity and the estimated angular velocity of the inverted pendulum onto the angle reference control damping signal to form the differential signal of the inverted pendulum angle error, as follows:

[0116] d d3 =ω1+u1+u d1 ;

[0117] Where d d3 This is the differential signal of the inverted pendulum angle error.

[0118] The fourth step involves superimposing the inverted pendulum angle estimation signal, acceleration control signal, angular velocity reference control signal, angle reference control damping signal, inverted pendulum angle error damping signal, and inverted pendulum angle error signal onto the differential signal of the inverted pendulum angle error to form the inverted pendulum angular velocity error damping signal as follows:

[0119] d4 = -w 1g φ1+w 2g u+u2-u d1 -k 31 d3-k 32 e3-k 33 d d3 ;

[0120] Where d4 is the inverted pendulum angular velocity error damping signal.

[0121] Step S60: Adaptively update the angle parameters and acceleration coefficient parameters of the inverted pendulum model based on the inverted pendulum angular velocity estimation error signal and the inverted pendulum angle error signal, respectively; then, based on the inverted pendulum angular velocity error damping signal superimposed with the inverted pendulum angular velocity error signal, the inverted pendulum angular velocity error integral signal, the inverted pendulum angle estimation signal, and the acceleration reference control quantity, form the final acceleration control signal; finally, send the acceleration control signal to the inverted pendulum system, thereby completing the stable tracking of the inverted pendulum angle signal of the inverted pendulum system to the desired inverted pendulum angle signal.

[0122] Specifically, this can be broken down into the following two steps. The first step involves adaptively updating the angle parameters and acceleration coefficient parameters of the inverted pendulum model based on the aforementioned inverted pendulum angular velocity estimation error signal and inverted pendulum angle error signal, as follows:

[0123] w 1g (n+1)=w 1g (n)+k 51 e2T+k 52 e3T;

[0124] w 2g (n+1)=w 2g (n)+k 53 e2T+k 54 e3T;

[0125] Where T is a constant integration parameter; k 51 k 52 k 53 k 54 These are constant parameters used to adjust the adaptive convergence speed of the angle and acceleration coefficient parameters of the inverted pendulum model.

[0126] The second step involves superimposing the inverted pendulum angular velocity error damping signal, the inverted pendulum angular velocity error integral signal, the inverted pendulum angle estimation signal, and the acceleration reference control quantity to form the final acceleration control signal as follows:

[0127]

[0128] Where k 41 k 42 k 43 is a constant control parameter used to adjust the amplitude of the acceleration control signal, thereby controlling the speed and dynamic performance of the entire control system. u is the final acceleration control signal, which is sent to the inverted pendulum system to ensure stable tracking of the desired pendulum angle signal by the inverted pendulum system.

[0129] Case Implementation and Analysis of Computer Solution Results

[0130] In step S10, a reference model of the inverted pendulum system is established according to the aforementioned method, and the estimated angular velocity of the inverted pendulum is obtained as follows: Figure 2 As shown, the estimated value of the inverted pendulum angle is obtained as follows: Figure 3 As shown.

[0131] In step S20, an angle measuring gyroscope is installed on the inverted pendulum system to measure the inverted pendulum angle signal, such as... Figure 4 As shown; the inverted pendulum angle estimation error signal is obtained as follows: Figure 5 As shown. Select k. 11 =30, k 12 =5.2, k 13 =0.01 is a constant control parameter.

[0132] In step S30, k is set 21 =28, k 22 =0.8, k 23 =0.04, the estimated error signal of the inverted pendulum angular velocity is obtained as follows: Figure 6 As shown.

[0133] In step S40, the desired inverted pendulum angle is set to 5 degrees; k is selected. 31 =15.5, k 32 =2.1, k 33 =0.2, the inverted pendulum angle error signal is obtained as follows Figure 7 As shown.

[0134] In step S50, the inverted pendulum angular velocity error signal is obtained as follows: Figure 8 As shown. In step S60, k is selected. 41 =10, k 42 =1.5, k 43=0.15, the acceleration control signal is obtained as follows Figure 9 As shown.

[0135] Depend on Figure 3 and Figure 4 It can be seen that the reference system can track the state changes of the original system well; moreover, the states of both the reference system and the original system can stably converge to the desired swing angle of 5 degrees; while the... Figure 5 and Figure 6 , Figure 7 as well as Figure 8 It can be seen that the errors of all four loops converge to zero quickly, therefore the closed-loop stability of the entire system is excellent. Figure 9 As can be seen, the total control quantity is less than 3, which meets the actual needs of engineering and demonstrates that the method provided by this invention has great practical engineering value.

Claims

1. A model-adaptive inverted pendulum angle tracking method, characterized in that, Includes the following steps: Step S10: Establish the reference model of the inverted pendulum system as follows: First, select an initial value of 0 for the estimated angular velocity of the inverted pendulum and an initial value of 0 for the angle reference control quantity. Calculate the differential estimated value of the inverted pendulum angle based on the estimated angular velocity and the angle reference control quantity, and then integrate to obtain the estimated angle value of the inverted pendulum. Then, set the initial value of the acceleration control signal to 0, the initial value of the angular velocity reference control quantity to 0, and the initial values ​​of the angle parameters and acceleration coefficient parameters of the inverted pendulum model to nominal values. Calculate the estimated angular acceleration of the inverted pendulum based on the estimated angle value, the acceleration control signal, and the angular velocity reference control quantity, and then integrate to obtain the estimated angular velocity value of the inverted pendulum as follows: ； ； ； ； in This is an estimate of the angular velocity of the inverted pendulum. This is an estimated value for the angle of the inverted pendulum. For angle reference control quantity, This is the estimated value of the differential angle of the inverted pendulum. This is an estimated value for the angle of the inverted pendulum; For acceleration control signals; This is the angular velocity reference control quantity. For the angle parameters of the inverted pendulum model, For acceleration coefficient parameters, This is an estimate of the inverted pendulum angular acceleration. This is an estimate of the angular velocity of the inverted pendulum. Indicates the integration over a time signal; Step S20: Install an angle measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angle signal; compare it with the estimated inverted pendulum angle to obtain the inverted pendulum angle estimation error signal; then integrate the signals to obtain the inverted pendulum angle estimation error integral signal; install an angular velocity measuring gyroscope on the inverted pendulum system to measure the inverted pendulum angular velocity signal; and construct the inverted pendulum angle estimation error damping signal based on the estimated inverted pendulum angular velocity, angle reference control quantity, and inverted pendulum angular velocity signal; then superimpose the inverted pendulum angle estimation error integral signal and the inverted pendulum angle estimation error signal to form the angle reference control quantity as follows: ； ； ； ； in This is the inverted pendulum angle signal; This is the error signal for estimating the inverted pendulum angle; The integral signal is the estimation error of the inverted pendulum angle. This is the angular velocity signal of the inverted pendulum; This is the damping signal for estimating the inverted pendulum angle error; , , These are constant control parameters; Step S30: Compare the estimated value of the inverted pendulum angular velocity with the inverted pendulum angular velocity signal to obtain the inverted pendulum angular velocity estimation error signal; Then, integration is performed to obtain the integral signal of the inverted pendulum angular velocity estimation error. Next, based on the set angle parameters, acceleration coefficient parameters and their nominal values ​​of the inverted pendulum model, as well as the acceleration control signal, angular velocity reference control quantity, and the estimated and signaled angular velocity of the inverted pendulum, the damping signal of the inverted pendulum angular velocity estimation error and the equivalent control quantity of the angular velocity reference are calculated respectively. Finally, the inverted pendulum angular velocity estimation error signal and the integral signal of the inverted pendulum angular velocity estimation error are superimposed to form the final angular velocity reference control quantity as follows: ； ； ； ； ； in This is the error signal for estimating the angular velocity of the inverted pendulum; The integral signal is used to estimate the angular velocity error of the inverted pendulum. These are the nominal values ​​of the angle parameters for the inverted pendulum model. The nominal value of the acceleration coefficient parameter, as well as the acceleration control signal, the angular velocity reference control quantity, and the estimated value and signal of the inverted pendulum angular velocity are calculated separately. The damping signal for estimating the angular velocity of the inverted pendulum. For angular velocity, the reference equivalent control quantity, , , These are constant control parameters; Step S40: Set the expected value of the inverted pendulum angle according to the control task of the inverted pendulum, and compare it with the estimated inverted pendulum angle signal to obtain the inverted pendulum angle error signal; Then, integration is performed to obtain the inverted pendulum angle error integral signal; next, the inverted pendulum angle error damping signal is calculated based on the inverted pendulum angle velocity estimation signal and the angle reference control quantity; finally, the angle reference control quantity, the inverted pendulum angle error signal, and the inverted pendulum angle error integral signal are superimposed to form the desired inverted pendulum angle velocity signal as follows: ； ； ； ； in Let the expected value of the inverted pendulum angle be , This is the inverted pendulum angle error signal; This is the integral signal of the inverted pendulum angle error; This is the damping signal for the inverted pendulum angle error; The desired signal for the inverted pendulum's angular velocity. , , These are constant control parameters; Step S50: Compare the estimated inverted pendulum angular velocity signal with the expected inverted pendulum angular velocity signal to obtain the inverted pendulum angular velocity error signal; then integrate to obtain the inverted pendulum angular velocity error integral signal; then calculate the inverted pendulum angle estimation error damping signal and the angle reference control damping signal based on the inverted pendulum angle estimation error; then superimpose the angle reference control quantity and the inverted pendulum angular velocity estimation signal to form the inverted pendulum angle error differential signal; finally, superimpose the inverted pendulum angle estimation signal, acceleration control signal, angular velocity reference control quantity, angle reference control damping signal, inverted pendulum angle error damping signal, and inverted pendulum angle error signal to form the inverted pendulum angular velocity error damping signal as follows: ； ； ； ； ； in This is the angular velocity error signal of the inverted pendulum; This is the integral signal of the inverted pendulum's angular velocity error. The angle reference control damping signal; This is the differential signal of the inverted pendulum angle error; This is the damping signal for the angular velocity error of the inverted pendulum; Step S60: Adaptively update the angle parameters and acceleration coefficient parameters of the inverted pendulum model based on the inverted pendulum angular velocity estimation error signal and the inverted pendulum angle error signal, respectively; then, superimpose the inverted pendulum angular velocity error signal, the inverted pendulum angular velocity error integral signal, the inverted pendulum angle estimation signal, and the acceleration reference control quantity to form the final acceleration control signal; finally, send the acceleration control signal to the inverted pendulum system, thereby completing the stable tracking of the inverted pendulum angle signal to the desired inverted pendulum angle signal by the inverted pendulum system as follows: ； ； ； in The parameter is a constant integral. , , , These are constant parameters used to adjust the adaptive convergence speed of the angle and acceleration coefficient parameters of the inverted pendulum model. , , These are constant control parameters used to adjust the amplitude of the acceleration control signal, thereby controlling the speed and dynamic performance of the entire control system.

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