CN113050493B - Output feedback control method for inverted pendulum system of trolley in networked environment - Google Patents

Output feedback control method for inverted pendulum system of trolley in networked environment Download PDF

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CN113050493B
CN113050493B CN202110296497.6A CN202110296497A CN113050493B CN 113050493 B CN113050493 B CN 113050493B CN 202110296497 A CN202110296497 A CN 202110296497A CN 113050493 B CN113050493 B CN 113050493B
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inverted pendulum
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trolley
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CN113050493A (en
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王晓磊
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Dalian University of Technology
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Abstract

本发明提供了一种网络化环境下的小车倒立摆系统输出反馈控制方法，包括：提出一种新的变增益事件触发输出反馈控制器设计方法；基于所设计的在线迭代算法，在不同触发时刻自动更新事件触发输出反馈控制器增益；构造具有时变李雅普诺夫函数矩阵的非连续李雅普诺夫函数，证明闭环系统的全局一致最终有界性。对所提出方法进行仿真验证，结果表明本发明方法能够有效解决小车倒立摆系统的稳定性问题。与现有的固定增益控制方法相比，本发明方法具有更好的系统性能，同时能够有效的节约网络资源。

Figure 202110296497

The invention provides an output feedback control method of a car inverted pendulum system in a networked environment, including: proposing a new variable gain event triggering output feedback controller design method; based on the designed online iterative algorithm, at different triggering moments Automatically update event-triggered output feedback controller gains; construct discontinuous Lyapunov functions with time-varying Lyapunov function matrices, and prove the globally consistent final boundedness of closed-loop systems. The proposed method is simulated and verified, and the results show that the method of the present invention can effectively solve the stability problem of the inverted pendulum system of the trolley. Compared with the existing fixed gain control method, the method of the present invention has better system performance and can effectively save network resources.

Figure 202110296497

Description

Output feedback control method for inverted pendulum system of trolley in networked environment
Technical Field
The invention relates to the field of control of networked linear systems, in particular to an output feedback control method of an inverted pendulum system of a trolley in a networked environment.
Background
The inverted pendulum system has the characteristics of multivariable, strong coupling, nonlinearity, instability and the like, and becomes an ideal object for checking whether the control method has the capability of rapidly processing instability. In addition, stable control is carried out on the inverted pendulum system, and several key technical indexes in the control field can be effectively checked, wherein the key technical indexes comprise robustness, traceability, rapidity and the like. Therefore, designing different control methods for the inverted pendulum system to be stable has become a hot issue in the current control field.
Due to the fact that digital control technology is greatly improved and applied in the last decades, the inverted pendulum system achieves wireless network connection, and the control signal acquisition is completed based on sampling data of a system model. In the classical control framework based on sampled data, the control signal is updated according to a constant sampling period, which is called time sampling. While time sampling is advantageous for the integration and analysis of the controller, the time sampling approach is not ideal from a resource utilization perspective. In particular in network control systems, communication between sensors and control elements is achieved through limited network bandwidth. In this case, the communication between the sensors and the control components should be as small as possible to increase the utilization of network resources.
In recent years, an event-triggered communication mechanism has been proposed and proved to be a new strategy capable of effectively improving the utilization rate of network resources on the premise of ensuring the performance of a control system. Thus, the event-triggered strategy can be widely applied to the control problem of the inverted pendulum system. The invention patent "a design method of network control system controller based on event trigger" (CN202011118574.0) proposes a design method of network system controller based on event trigger mechanism, and the controller designed in the invention can ensure system stability under the condition of simultaneously considering introducing event trigger communication mechanism and packet loss. The invention patent 'electric power system control method based on event trigger dynamic trigger mechanism' (cn201710982346.x) designs an electric power system control method based on dynamic trigger mechanism, wherein the established event trigger mechanism is not only related to the current output and error, but also related to the output at the previous moment. The invention patent CN201910586233.7 discloses an output feedback controller of a neutral stable saturation system based on event triggering, and an established control mechanism can ensure the stability of the system when output saturation exists in the system. The local controller designed in the invention patent of networking control system and control method based on event trigger mechanism (CN201911245333.X) ensures that the networking control system has limited gain L2 stability and input feed-forward output feedback passivity.
It should be noted that in the above-mentioned patent of invention relating to the problem of networked system event-triggered control, a fixed gain controller is used throughout the event-triggered time, which makes the controller easy to implement. However, the control performance achieved using the fixed gain control method is not ideal. Therefore, how to design a more relaxed variable gain control method to ensure that the inverted pendulum system obtains better performance in a networked environment is a challenging problem, and related inventions do not exist at present.
Disclosure of Invention
The invention aims to provide an output feedback control method of an inverted pendulum system of a trolley in a networked environment, which aims to solve the problem of stability of the inverted pendulum system of the trolley in the networked environment and obtain better system performance under the condition of saving network resources.
The technical scheme of the invention is as follows:
an output feedback control method of a trolley inverted pendulum system in a networked environment comprises the following steps:
firstly, for the inverted pendulum system, modeling is difficult due to instability of the inverted pendulum system, and after some factors are ignored, a linear primary inverted pendulum model is regarded as a trolley inverted pendulum system.
Then, an event trigger communication mechanism is introduced to save network resources, and a variable gain controller of the inverted pendulum system of the trolley is designed in a networked environment; and establishing a controller design condition expressed by a linear matrix inequality, and solving a controller gain.
And finally, the controller u transmits the control instruction to an actuating mechanism of the inverted pendulum system of the trolley, so that the aim of control is fulfilled.
The invention has the following beneficial effects: the design method of the variable gain event trigger controller based on the output can effectively solve the problem of stability of the inverted pendulum system of the trolley on the premise of ensuring the performance of the system. Creating a non-continuous lyapunov function with a time-varying lyapunov matrix may result in greater design flexibility than a constant lyapunov function. The proposed variable gain controller design method can achieve better system performance than the fixed gain control method.
Drawings
FIG. 1 is a block diagram of the control method steps of the present invention.
FIG. 2 is a schematic diagram of the gain of the controller according to the present invention.
Fig. 3 is a schematic diagram illustrating a data transmission situation of the event trigger device according to the present invention.
Fig. 4 is a diagram illustrating a case where an event trigger device using a fixed gain method transmits data.
Fig. 5 is a schematic diagram of a system state (solid line) when the control method proposed in the present invention is used and a system state (dotted line) when an existing fixed gain control method is used.
Fig. 6 is a schematic diagram of a system state (solid line) when the control method proposed in the present invention is used and a system state (dotted line) when an existing fixed gain control method is used.
Fig. 7 is a schematic diagram of a system state (solid line) when the control method proposed in the present invention is used and a system state (dotted line) when an existing fixed gain control method is used.
Fig. 8 is a schematic diagram of a system state (solid line) when the control method proposed in the present invention is used and a system state (dotted line) when an existing fixed gain control method is used.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
In this embodiment, the output feedback control of the inverted pendulum system of the trolley based on the event trigger mechanism includes the following steps:
step 1: for the inverted pendulum system, due to the instability of the inverted pendulum system, modeling is difficult, and after some factors are ignored, the linear primary inverted pendulum model can be regarded as a trolley inverted pendulum system as follows:
Figure BDA0002984546090000041
Figure BDA0002984546090000042
wherein x and theta are respectively the position coordinate of the trolley and the included angle between the pendulum and the vertical direction,
Figure BDA0002984546090000043
and
Figure BDA0002984546090000044
respectively the speed and the acceleration of the trolley,
Figure BDA0002984546090000045
and
Figure BDA0002984546090000046
angular velocity and angular acceleration, respectively, u being the control input, i.e. the control command, and y being the measurement output.
Modeling the trolley inverted pendulum system into a linear system, specifically as follows:
Figure BDA0002984546090000047
y(t)＝Cx(t)
where x (t), y (t), u (t) are system state, measurement output and control input, respectively. A, B and C are system matrixes. In addition, (A, B) is controllable, and (A, C) is observable.
Step 2: establishing an event trigger communication mechanism, and designing a variable gain controller of the inverted pendulum system of the trolley;
(1) the event-triggered communication mechanism is established as follows:
Figure BDA0002984546090000048
wherein
Figure BDA0002984546090000049
tk-1Which indicates the current moment of the data transmission,
Figure BDA00029845460900000410
which represents the state estimate that is transmitted,
Figure BDA00029845460900000411
representing the state estimate at the current time, e (t) representing the difference between the two.
Figure BDA00029845460900000412
The event threshold value is exponentially decreased, and the parameters respectively satisfy the following conditions: epsilon is more than 1, 0 is more than or equal to sigma and less1 and e0Is more than or equal to 0. Furthermore, 0. ltoreq. t0≤t1≤…tk≦ … is defined as the sequence of event-triggered times.
(2) Assuming the sensor has computational capability, the state estimate can be derived from a state observer through processing of the measured output, the observer being established as follows:
Figure BDA0002984546090000051
Figure BDA0002984546090000052
wherein
Figure BDA0002984546090000053
And
Figure BDA0002984546090000054
respectively an estimate of the state of the system and an estimate of the output, L being the observer gain matrix to be designed.
(3) Designing a controller with variable gain based on the established event trigger mechanism and the state observer, wherein the structure is as follows:
Figure BDA0002984546090000055
wherein KkIs the controller gain to be designed, which is updated at the kth trigger time,
Figure BDA0002984546090000056
representing a positive integer. In addition, the data is sampled
Figure BDA0002984546090000057
Is transmitted to the controller side and the value is kept unchanged until the next time the sampled data arrives.
Defining new variables
Figure BDA0002984546090000058
And
Figure BDA0002984546090000059
an augmentation system of the following form is then obtained:
Figure BDA00029845460900000510
wherein
Figure BDA00029845460900000511
(4) The stability of the lyapunov function analysis system is established in the form:
V(t)＝ξT(t)Pk(t)ξ(t)
wherein the Lyapunov function V (t) is discontinuous for t ≧ 0, a Lyapunov matrix Pk(t) is time-varying positive and satisfies the following inequality relationship:
Figure BDA00029845460900000512
wherein, λ (P)k(t)) represents Pk-1The singular value of (a) is,λ(Pk-1) And
Figure BDA00029845460900000513
respectively represent Pk-1Minimum and maximum singular values of, the symbols max, min and RNRespectively, a maximum value, a minimum value, and a positive integer. In addition, Pk(t) is a continuous linear function, as follows:
Figure BDA0002984546090000061
wherein P isk-1> 0 and Pk+1> 0 denotes the time tk-1And tkIs given constant, δ ≧ 1 serves as a feasible solution for adjusting the linear matrix inequality.
In the embodiments of the present invention, for a given parameter e > 1, 0 ≦ σ < 1, e0Greater than or equal to 0 and delta greater than or equal to 1. If matrix P is presentk-1Observer gain matrix L and controller gain matrix KkAnd k is 1,2, …, satisfying the following inequality:
Figure BDA0002984546090000062
and
Figure BDA0002984546090000063
wherein, denotes the symmetric term of the matrix, I is the identity matrix, and phi is a positive constant.
The inverted pendulum system of the vehicle can achieve globally consistent and ultimately bounded stability, and all signals are contained within the following ranges:
Figure BDA0002984546090000064
(5) an event trigger mechanism gain, an observer gain, and a controller gain are determined.
1) Setting k to 1 and t to t0
By using the pole placement method, the following inequality is solved:
Figure BDA0002984546090000065
obtaining an initial parameter P0，K1And L. Further according to Pk(t) definition, assuming P0＝P1。
2) Setting k 2 and t1
According to the parameter P obtained in step 1)1And L, by solvingThe following inequality:
Figure BDA0002984546090000071
to obtain K2。
3)t∈(t0,t1)
According to the parameters P obtained in 1) and 2)0，K1And L, solving the following inequality:
Figure BDA0002984546090000072
Figure BDA0002984546090000073
to obtain P2。
4) Setting k to 3 and t to t2
P obtained according to 3)2By solving the following inequality:
Figure BDA0002984546090000074
to obtain K3。
5)t∈(t1,t2)
According to the parameters P obtained in 1) and 2)1And K2By solving the following inequality:
Figure BDA0002984546090000075
Figure BDA0002984546090000076
to obtain P3。
Repeating the steps for N times.
And (6) ending.
According to 1) -N) in step 2, the event trigger mechanism gain, the observer gain and the controller gain matrix can be calculated.
And step 3: and (3) transmitting the control signal u established in the step (2) to an actuating mechanism of the inverted pendulum system of the trolley by the controller, thereby achieving the purpose of control.
The purpose of this embodiment is to ensure that the designed variable gain controller can make the inverted pendulum system of the trolley realize the global consistent bounded stability. In addition, the design method of the variable gain controller provided by the invention can obtain better system performance than the existing design method of the fixed gain controller.
The method is applied to an inverted pendulum system of a trolley to verify the effectiveness of the inverted pendulum system.
The inverted pendulum model of the vehicle is as follows:
Figure BDA0002984546090000081
Figure BDA0002984546090000082
wherein M is the trolley mass, M is the pendulum mass, l is the pendulum length, b is the friction of the trolley,
Figure BDA0002984546090000083
is the pendulum inertia. In addition, the selection of the respective parameters is as follows: m0.5 kg, b 0.1N/M/s, l 0.3M and I0.006 kg M2。
Let e, σ 0.034 and e00.01 is a parameter in the event trigger condition, and when the event trigger condition is established, the current state estimation value
Figure BDA0002984546090000084
Will be transmitted to the controller. Further, the definition ∈ ═ e, σ ═ 0.934, and ∈ e00.0001 as a parameter in the non-trigger condition. Thus, it is possible to provideWith this method, a smaller number of triggers can be obtained. Without loss of generality, let t be00, the first trigger occurs at time t 00. By performing 1) in (5), the following initial gain matrix K can be obtained1And L
K1＝[17.0386 13.0877 -50.0520 -9.8150]
Figure BDA0002984546090000091
Then, based on the initial gain matrix K1And L, obtaining an event-triggered controller gain by performing 1) -N) in (5), as shown in fig. 2. The time and transmission interval of the data transmitted by the event trigger mechanism are shown in fig. 3. As can be seen from fig. 2-3, the gain of the event-triggered controller is varied and the event-triggered controller gain will switch when the event unit is triggered. In addition, fig. 4 plots the release data time and release interval for the event trigger mechanism obtained when using the fixed gain approach. As can be seen from fig. 3-4, the number of events triggered using the method of the present invention is 241, and the number of events triggered using the fixed gain method is 320. Fig. 5-8 show the system states obtained using the method of the present invention and the fixed gain method. Fig. 5-8 show that better control performance can be achieved with the method of the present invention with fewer triggers. Therefore, it can be seen from the simulation results that the design method disclosed in the present invention is effective.
Finally, the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting, although the present invention has been described in detail with reference to the preferred embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered in the claims of the present invention.

Claims (1)

1.一种网络化环境下的小车倒立摆系统输出反馈控制方法，其特征在于，步骤如下：1. a trolley inverted pendulum system output feedback control method under a networked environment, is characterized in that, step is as follows: 步骤1：将直线一级倒立摆模型视为小车倒立摆系统，如下所示：Step 1: Consider the linear first-order inverted pendulum model as a trolley inverted pendulum system, as shown below:
Figure FDA0003361405150000011
Figure FDA0003361405150000011
Figure FDA0003361405150000012
Figure FDA0003361405150000012
 其中x和θ分别为小车位置坐标和钟摆与垂直方向的夹角，
Figure FDA0003361405150000013
和
Figure FDA0003361405150000014
分别为小车速度和小车加速度，
Figure FDA0003361405150000015
和
Figure FDA0003361405150000016
分别是角速度和角加速度，u为控制输入，即控制指令，y为测量输出；where x and θ are the position coordinates of the trolley and the angle between the pendulum and the vertical direction, respectively,
Figure FDA0003361405150000013
and
Figure FDA0003361405150000014
are the speed of the car and the acceleration of the car, respectively,
Figure FDA0003361405150000015
and
Figure FDA0003361405150000016
are the angular velocity and angular acceleration respectively, u is the control input, that is, the control command, and y is the measurement output; 将上述小车倒立摆模型建模为线性系统，具体如下：The above inverted pendulum model is modeled as a linear system, as follows:
Figure FDA0003361405150000017
Figure FDA0003361405150000017
 y(t)＝Cx(t)y(t)=Cx(t) 其中x(t)，y(t)，u(t)分别是系统状态，测量输出和控制输入；A，B，C是系统矩阵；(A,B)可控，(A,C)可观测；where x(t), y(t), u(t) are the system state, measurement output and control input, respectively; A, B, C are system matrices; (A, B) are controllable, (A, C) are observable ; 步骤2：建立事件触发通信机制，设计小车倒立摆系统的变增益控制器；Step 2: Establish an event-triggered communication mechanism and design a variable gain controller for the inverted pendulum system of the trolley; (1)事件触发通信机制建立如下：(1) The event-triggered communication mechanism is established as follows:
Figure FDA0003361405150000018
Figure FDA0003361405150000018
 其中
Figure FDA0003361405150000019
tk-1表示当前数据传输时刻，
Figure FDA00033614051500000110
代表被传输的状态估计，
Figure FDA00033614051500000111
表示当前时刻的状态估计，e(t)表示二者差值；
Figure FDA00033614051500000112
为指数递减的事件阈值，其中的参数分别满足：∈＞1，0≤σ＜1和∈0≥0；0≤t0≤t1≤…tk≤…被定义为事件触发时刻序列；in
Figure FDA0003361405150000019
t k-1 represents the current data transmission moment,
Figure FDA00033614051500000110
represents the transmitted state estimate,
Figure FDA00033614051500000111
Represents the state estimate at the current moment, and e(t) represents the difference between the two;
Figure FDA00033614051500000112
is an exponentially decreasing event threshold, where the parameters satisfy: ∈>1, 0≤σ<1 and ∈ 0 ≥0; 0≤t 0 ≤t 1 ≤…t k ≤… is defined as the event trigger time sequence; (2)假定传感器具有计算能力，通过对测量输出的处理，可从状态观测器得到状态估计值，观测器被建立如下：(2) Assuming that the sensor has computing power, the state estimation value can be obtained from the state observer by processing the measurement output, and the observer is established as follows:
Figure FDA0003361405150000021
Figure FDA0003361405150000021
Figure FDA0003361405150000022
Figure FDA0003361405150000022
 其中
Figure FDA0003361405150000023
和
Figure FDA0003361405150000024
分别是系统状态的估计和输出的估计，L是将要被设计的观测器增益矩阵；in
Figure FDA0003361405150000023
and
Figure FDA0003361405150000024
are the estimation of the system state and the output, respectively, and L is the observer gain matrix to be designed; (3)基于所建立的事件触发机制和状态观测器，设计具有变增益的控制器，结构如下：(3) Based on the established event trigger mechanism and state observer, a controller with variable gain is designed. The structure is as follows:
Figure FDA0003361405150000025
Figure FDA0003361405150000025
 其中Kk是将要被设计的控制器增益，它在第kth个触发时刻进行更新，
Figure FDA0003361405150000026
表示正整数；另外，采样数据
Figure FDA0003361405150000027
被传输到控制器一侧，并保持数值不变直到下一次采样数据到来；where K k is the controller gain to be designed, which is updated at the kth trigger time,
Figure FDA0003361405150000026
Represents a positive integer; in addition, the sampled data
Figure FDA0003361405150000027
It is transmitted to the controller side, and keeps the value unchanged until the next sampling data arrives; 定义新变量
Figure FDA0003361405150000028
和
Figure FDA0003361405150000029
然后得到如下形式的增广系统：define new variable
Figure FDA0003361405150000028
and
Figure FDA0003361405150000029
Then an augmented system of the form:
Figure FDA00033614051500000210
Figure FDA00033614051500000210
 其中in
Figure FDA00033614051500000211
Figure FDA00033614051500000211
 (4)建立如下形式的李雅普诺夫函数分析系统的稳定性：(4) Establish the following form of Lyapunov function to analyze the stability of the system: V(t)＝ξT(t)Pk(t)ξ(t)V(t)=ξ T (t)P k (t)ξ(t) 其中李雅普诺夫函数V(t)对于t≥0是非连续的，李雅普诺夫矩阵Pk(t)是时变正定的，并满足如下不等式关系：where the Lyapunov function V(t) is discontinuous for t≥0, and the Lyapunov matrix Pk (t) is time-varying positive definite and satisfies the following inequality:
Figure FDA00033614051500000212
Figure FDA00033614051500000212
 其中，λ(Pk(t))表示Pk-1的奇异值，λ(Pk-1)和
Figure FDA00033614051500000213
分别表示Pk-1的最小和最大奇异值，符号max，min和RN分别表示最大值，最小值和正整数；另外，Pk(t)是连续的线性函数，如下所示：where λ(P k (t)) represents the singular value of P k- 1 , λ (P k-1 ) and
Figure FDA00033614051500000213
represent the minimum and maximum singular values of P k-1 , respectively, and the symbols max, min, and R N represent the maximum, minimum, and positive integers, respectively; in addition, P k (t) is a continuous linear function, as follows:
Figure FDA0003361405150000031
Figure FDA0003361405150000031
 其中Pk-1＞0和Pk+1＞0分别表示在时刻tk-1和tk的常值矩阵，δ≥1是给定的常值，其作用是调节线性矩阵不等式的可行解；where P k-1 > 0 and P k+1 > 0 represent constant value matrices at time t k-1 and t k respectively, δ≥1 is a given constant value, and its function is to adjust the feasible solution of the linear matrix inequality ; 如果存在矩阵Pk-1，观测器增益矩阵L和控制器增益矩阵Kk，k＝1,2,…，满足如下不等式：If there is a matrix P k-1 , the observer gain matrix L and the controller gain matrix K k , k=1, 2, . . . satisfy the following inequalities:
Figure FDA0003361405150000032
Figure FDA0003361405150000032
 和and
Figure FDA0003361405150000033
Figure FDA0003361405150000033
 其中，★表示矩阵的对称项，I为单位矩阵，φ是正的常值；Among them, ★ represents the symmetric term of the matrix, I is the identity matrix, and φ is a positive constant value; 则小车倒立摆系统能够实现全局一致最终有界稳定，并且所有的信号都包含在如下范围内：Then the car inverted pendulum system can achieve globally consistent and eventually bounded stability, and all signals are included in the following ranges:
Figure FDA0003361405150000034
Figure FDA0003361405150000034
 (5)确定事件触发机制增益，观测器增益以及控制器增益；(5) Determine event trigger mechanism gain, observer gain and controller gain; 1)设定k＝1和t＝t0 1) Set k=1 and t=t 0 通过使用极点配置法，求解如下不等式：By using the pole placement method, solve the following inequalities:
Figure FDA0003361405150000035
Figure FDA0003361405150000035
 得到初始参数P0，K1和L；此外根据Pk(t)的定义，假定P0＝P1；Obtain the initial parameters P 0 , K 1 and L; in addition, according to the definition of P k (t), it is assumed that P 0 =P 1 ; 2)设定k＝2和t＝t1 2) Set k=2 and t=t 1 根据步骤1)中获得的参数P1和L，通过求解以下不等式：According to the parameters P 1 and L obtained in step 1), by solving the following inequalities:
Figure FDA0003361405150000041
Figure FDA0003361405150000041
 得到K2；get K 2 ; 3)t∈(t0,t1)3) t∈(t 0 ,t 1 ) 根据1)和2)中得到的参数P0，K1和L，求解如下不等式：According to the parameters P 0 , K 1 and L obtained in 1) and 2), the following inequalities are solved:
Figure FDA0003361405150000042
Figure FDA0003361405150000042
Figure FDA0003361405150000043
Figure FDA0003361405150000043
 得到P2；get P 2 ; 4)设定k＝3和t＝t2 4) Set k=3 and t=t 2 根据3)中获得的P2，通过求解如下不等式：According to P 2 obtained in 3), by solving the following inequality:
Figure FDA0003361405150000044
Figure FDA0003361405150000044
 得到K3；get K 3 ; 5)t∈(t1,t2)5) t∈(t 1 ,t 2 ) 根据1)和2)中得到的参数P1和K2，通过求解如下不等式：According to the parameters P 1 and K 2 obtained in 1) and 2), by solving the following inequalities:
Figure FDA0003361405150000045
Figure FDA0003361405150000045
Figure FDA0003361405150000046
Figure FDA0003361405150000046
 得到P3；get P 3 ; 重复步骤1)-5)N次；Repeat steps 1)-5) N times; 结束；Finish; 根据步骤(5)中的1)-5)计算出事件触发机制增益，观测器增益和控制器增益矩阵；Calculate event trigger mechanism gain, observer gain and controller gain matrix according to 1)-5) in step (5); 步骤3：控制器将其在步骤2中建立的控制指令u传递给小车倒立摆系统的执行机构，从而实现控制的目的。Step 3: The controller transmits the control instruction u established in step 2 to the actuator of the trolley inverted pendulum system, thereby realizing the purpose of control.
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