WO2022012156A1

WO2022012156A1 - Iterative feedback tuning control for rotating inverted pendulum and robust optimisation method therefor

Info

Publication number: WO2022012156A1
Application number: PCT/CN2021/094746
Authority: WO
Inventors: 陶洪峰; 庄志和; 周龙辉; 刘巍; 沈凌志
Original assignee: 江南大学
Priority date: 2020-07-14
Filing date: 2021-05-20
Publication date: 2022-01-20
Also published as: CN111736471A; CN111736471B

Abstract

Disclosed in the present invention are iterative feedback tuning control for a rotating inverted pendulum and a robust optimisation method therefor, relating to the field of robot optimised control, the method comprising: establishing Lagrangian and state-space mathematical models of a rotating inverted pendulum on the basis of an inverted pendulum machine and a hardware structure; designing a rotating inverted pendulum iterative feedback tuning dual closed loop controller; and performing algorithmic convergence analysis for the iterative feedback tuning PD controller; the introduction of an auxiliary factor further optimises the robust iterative feedback tuning angle PD controller, enabling the rotating inverted pendulum system to implement rapid, high-precision tracking of the desired motion trajectory; the control algorithm of the method of the present application is simple and efficient, does not require the acquisition of the parameters of the model itself, and drives the calculation of the unbiased gradient of the indicator function to the controller parameters by means of I/O data; and the algorithm introduces an auxiliary factor, enabling the control system to respond better to changes in the input signal and thereby have better robustness.

Description

An Iterative Feedback Tuning Control of Rotating Inverted Pendulum and Its Robust Optimization Method

technical field

The invention relates to the field of robot optimization control, in particular to an iterative feedback tuning control of a rotating inverted pendulum and a robust optimization method thereof.

Background technique

As a typical underactuated nonlinear system, the rotating inverted pendulum has the characteristics of instability, multi-variable, strong coupling, etc., and integrates the three basic disciplines of mathematics, electricity and mechanics well. Therefore, the control of the inverted pendulum system is not only of great significance, but also extremely challenging, and is highly valued by experts and scholars in control disciplines all over the world. In addition, the rotating inverted pendulum, as the simplest model of many control objects such as robots and rocket flight attitudes, is an ideal experimental platform to verify the correctness of various control theory strategies, and builds a bridge between control theory and practical engineering applications. At the same time, as an experimental device, the structure is simple and the control effect is intuitive, and it is an ideal experimental platform for verifying various control methods. The type in which the pivot is rotationally moved is also called a rotary inverted pendulum. Compared with the linear inverted pendulum, the pendulum is controlled by the movement of the trolley, and the swing arm drives the pendulum to rotate to maintain the upright state, and the nonlinearity is stronger. Taking the rotating inverted pendulum as the controlled object, it is possible to test whether the iterative feedback tuning algorithm has the optimization ability for multi-state, nonlinear and absolutely unstable control systems.

As a typical controlled model, the control research of the rotating inverted pendulum system involves almost most of the control methods, among which the traditional control fields mainly include state feedback control, synovial control and PID control. However, these methods all have some limitations. For example, the state feedback control must have an accurate model of the controlled system, and it is difficult to realize the stable control of the inverted pendulum, especially the high-order inverted pendulum under the condition of insufficient model accuracy. The chattering caused by the switching state of the rotating inverted pendulum restricts its application; PID control is still the most common control method. Among them, the multi-closed-loop PID has a good control effect on the rotary inverted pendulum with a lower order, but the multi-closed-loop PID has a better control effect than the multi-closed-loop PID. Compared with the basic PID, its parameter tuning is more complicated.

SUMMARY OF THE INVENTION

In view of the above problems and technical requirements, the inventor proposes an iterative feedback tuning control of a rotating inverted pendulum and its robust optimization method. First, a rotating inverted pendulum experimental platform based on double closed-loop control is established, and on this basis, iterative The feedback tuning algorithm optimizes the angle PD controller of the rotating inverted pendulum system. Under the PD control strategy framework, based on the basic principle of the IFT algorithm, the parameter optimization theory and the experimental tuning method, according to the performance criterion function of the closed-loop system and the input and output signals automatically Tuning the PID controller parameters, using the Gauss-Newton algorithm to obtain the optimal value of the PID controller parameters, introducing auxiliary factors to continuously iterate the weight factor of the performance criterion function to improve the robustness of the system, and finally realize the stable control of the rotating inverted pendulum .

The technical scheme of the present invention is as follows:

An iterative feedback tuning control of a rotating inverted pendulum and its robust optimization method, comprising the following steps:

Step 1: Establish the Lagrangian and state space models of the rotating inverted pendulum;

The rotating inverted pendulum system includes a base, a transmission device, a swing rod and a swing arm. The base is used to ensure the stability of the mechanical structure when the swing rod swings; the end of the swing arm is connected to the swing rod, and the rotation of the DC motor drives the movement of the swing rod through the transmission device; The angle and angular velocity of the arm are obtained through the incremental rotary encoder that comes with the DC motor; the incremental rotary encoder and the pendulum rod are connected through the coupling, and the incremental rotary encoder is driven to rotate to obtain the angle and the angle of the pendulum rod. Angular velocity; in the dynamic model of the rotating inverted pendulum, air resistance, frictional force and tiny items are ignored to simplify the modeling process, the arm and the pendulum rod are regarded as uniform long rods, and the rotating inverted pendulum is set when the pendulum rod is stable and erect The potential energy of the pendulum system is zero;

When the pendulum rod deviates from the upright position angle α, the swing arm drives the pendulum rod to the upright position by rotating β, so the speed v _m of the end of the arm is:

Among them, r ₁ is the distance from the rotation center of the arm to the connection point with the pendulum rod,

is the angular velocity when the arm rotates;

Since the pendulum rod is a uniform long rod, considering the pendulum rod as a mass point, the rotation speed of the pendulum rod v _z is obtained as:

Among them, L is the length of the pendulum rod,

is the angular velocity when the pendulum rod rotates;

Decompose the rotation speed v _{z of the} _{pendulum rod in the vertical direction of the speed v m} at the end of the arm, and take the direction of the rotation plane of the pendulum rod and the horizontal direction speed v _{r of the} ground as the positive direction, we get:

_{Under the combined action of the velocity v m} at the end of the arm _{and the velocity v r} in the horizontal direction of the ground _{, the velocity v b} of the pendulum rod in the horizontal direction is:

The kinetic energy of the pendulum rod includes the rotational kinetic energy generated by rotation and the kinetic energy generated by moving in the horizontal direction. In addition, the overall kinetic energy of the rotating inverted pendulum system also includes the kinetic energy of the arm driven by the DC motor, so the overall kinetic energy V of the rotating inverted pendulum system is obtained. , let J ₁ be the moment of inertia of the pendulum rod, J ₂ be the moment of inertia of the swing arm, m is the mass of the pendulum rod, and bring equations (4) and (5) into:

When the pendulum is upright, it is set as the zero potential energy point, H is the overall potential energy of the rotating inverted pendulum system, E is the Lagrangian function, then the potential energy is reduced to:

The Lagrangian function E is:

It can be seen that the rotation of the swing arm drives the movement of the pendulum rod, and there is no external power input. Let T _output be the output torque of the motor, B _eq is the equivalent viscous friction, and the Lagrangian equation can be obtained as:

Equation (8) is brought into equations (9) and (10) to obtain the nonlinear model of the rotating inverted pendulum:

From the nonlinear model of the rotating inverted pendulum obtained in equation (11), its input is the DC motor torque, but usually the DC motor voltage is used as the control input, so the next step is to model the DC motor, and finally establish a Inverted pendulum nonlinear model with DC motor voltage as input;

So as DC current I _d, E _d is the counter electromotive force, and taking into account the efficiency of the transmission gear ratio, K _T is the motor torque coefficient, K _E of the motor-speed coefficient, K _g is the DC motor with gear arm ratio , η _g is the gear transmission efficiency, η _d is the motor efficiency, U is the DC motor voltage, R is the armature resistance, we get:

T _output = η _d η _g K _g K _T I _d (13)

Putting equations (12) and (13) into equation (11), the nonlinear model of the inverted pendulum with the DC motor voltage as the input can be obtained as:

In order to further establish the state space model of the rotating inverted pendulum, the nonlinear model of the inverted pendulum needs to be linearized. It is noted that the pendulum rod is in an upright state in the stable pendulum control, so the angle of the pendulum rod is small, and sinα≈α, cosα exists at this time ≈1, then the linear model of the rotating inverted pendulum is obtained as:

Next, the state space model of the rotating inverted pendulum is established based on the linear model of the rotating inverted pendulum. In order to simplify the writing settings, the following definitions are made:

b=J ₂ +mr ₁ ² (17)

Substituting equations (16) to (21) into equation (15) to solve

and

for:

select state vector

Where β is the rotation angle of the arm, the input is the DC motor voltage U, and the state space model of the rotating inverted pendulum is obtained as:

Among them, since the pendulum rod and the arm are regarded as uniform long rods, the moment of inertia J ₁ , J ₂ can be obtained as follows:

Among them, r ₂ is the length of the arm, M is the mass of the arm, and ρ is the density of the arm and the pendulum;

Step 2: Design a rotating inverted pendulum iterative feedback tuning double closed-loop controller;

A double closed-loop controller is designed for the state space model of the rotating inverted pendulum, and the iterative feedback tuning algorithm is used to optimize the parameters of the angle PD controller. If C(ρ)=[C _r (ρ) C _y (ρ)], C _r (ρ) , C _y (ρ) is the linear time-invariant transfer function, G is the transfer function of the controlled object, u(t) is the controller output, r(t) is the reference input, and y(t) is the output of the rotating inverted pendulum system , v(t) is an external random disturbance with a mean value of zero, the PID controller parameter is ρ=[K _p K _d ], and the response output under the action of the feedback control system is:

In order to simplify the writing, T ₀ (ρ) and S ₀ (ρ) are abbreviated as T ₀ and S ₀ , and y _d is defined as a given expected input signal, then the tracking error between the expected output and the actual output is:

For a fixed-structure PID controller with controller parameter ρ, by minimizing

In order to improve the tracking control effect of the feedback control system, the performance optimization index function J(ρ) is defined as:

Among them, _Ly and _Lu represent filters based on time series, usually _Ly = _Lu = 1, the number of sampling points is N, and the weight factor of performance measurement is λ; the IFT algorithm optimizes the index function by minimizing the performance J( ρ) directly obtain the PID controller parameter ρ of the system, and then gradually obtain the optimal value of the PID controller parameter ρ through i iterations, ρ _i is the value of ρ in the ith iteration, and in each iteration batch , the partial derivatives of the variables y(ρ _i ) and u(ρ _i ) with respect to the controller parameter ρ _{i are:}

_{The IFT algorithm obtains an estimate of T 0} r, T ₀ (ry) by conducting three experiments in a degree-of-freedom control system, in which the first two are used to estimate the signal T ₀ , first in the ith iteration , in the first experiment, r _i ⁽¹⁾ = r is the input reference signal, y ⁽¹⁾ (ρ _i ) is the output value of the control system obtained by sampling; secondly, the difference between the two signals ry ⁽¹⁾ ( ρ _i) for the second reference input signal experiment r _i ^(2), obtained by sampling ^{_{y (2) (ρ i)}} :

The third test signal used to estimate T ₀ r, to r _i ⁽³⁾ = r as the reference signal input:

According to the controller output value of the three experiments and the output value of the rotating inverted pendulum system, the

unbiased estimate of

You can also get:

The estimated gradient of the performance optimization index function J(ρ _{i ) for the ith iteration based on experimental data is:}

According to the estimated gradient of the performance optimization index function J(ρ _i ) and the PID controller parameters ρ _i of the previous iteration, the Gauss–Newton algorithm is used to calculate the ρ _i+1 for the next iteration update:

Where γ _i > 0 represents the step size, and R _i is a positive definite Hessian matrix to represent the optimization search direction:

Step 3: Convergence analysis of iterative feedback setting angle PD controller;

In order to ensure the convergence of the algorithm, condition 1 is to ensure that the estimated gradient of the performance optimization index function is unbiased, and condition 2 is that the step size sequence γ _{i is} required to converge to zero. )get

for:

IFT experiments set of three experiments algorithm based on _{^{v i (m), m =}} 1,2,3 is the same system independent zero-mean random noise bounded, i.e. _{^{| v i (m) | <}} C, where it is assumed In the three experiments, the limit value C and the mean square value of random noise remain unchanged, then the unbiased estimates of equations (21) and (22) are obtained;

Condition 2 The conditions that need to ensure the convergence of the algorithm usually require that all elements of the _{step sequence γ i satisfy:}

The fourth step: further optimization of the robust iterative feedback setting angle PD controller;

The IFT algorithm relies on experience to select weight values such as λ. However, because the physical meanings of various performance metrics are not the same, and the operating environments are not consistent, the value ranges between them are very different. the same system, relying on experience performance metric selected weighting factors do not have universal λ, considering the range between the various performance metrics, build a cofactor L _i, L _i is a co-factor performance metric taken between Ratio of value ranges:

On this basis, the criterion function J(θ) is modified as:

in,

is the tracking error, u(θ _i ) is the controller output,

and

And the approximate Hessian matrix R _{i is} modified as:

where y _d,max and y _d,min are the maximum and minimum values of the expected output, and u _max and u _min represent the maximum and minimum values of the control signal in all N sampling points during the ith iteration; since these values are given at the end of each iteration, all the sample points will be taken into account, and thus the introduction of co-factors L _i such that the weighting factor λ in different systems are optimized represents the current iteration

Optimal range of weight ratios to u(θ _{i ).}

The beneficial technical effects of the present invention are:

To optimize the parameters of the angle PD controller of the rotating inverted pendulum experimental platform, the iterative feedback tuning algorithm obtains the Gauss–Newton gradient through three closed-loop experiments to update the parameters, which enables the control system to respond quickly to changes in the input signal and has better robustness sex. The invention combines the idea of iterative design and numerical optimization, links the performance index function with the I/O data, does not need to obtain the parameters of the model itself, and avoids the accuracy of the controlled object and the disturbance characteristic model in the optimization process. estimated requirements. At the same time, a model-free method for calculating the unbiased gradient of the indicator function to the controller parameters (that is, the unbiased signal of the system output differential) is presented, which improves the applicability of the algorithm in the control of complex systems.

Description of drawings

Figure 1 is a schematic diagram of the swing model of the rotating inverted pendulum.

Figure 2 is a block diagram of the double closed-loop control structure for iterative feedback tuning of the rotating inverted pendulum.

Figure 3 is the mechanical structure diagram of the rotating inverted pendulum experimental platform.

Figure 4 is the hardware structure diagram of the rotating inverted pendulum experimental platform.

Figure 5 is the overall program design diagram of DSPACE.

FIG. 6 is a schematic diagram of the trajectory of the rotating inverted pendulum pendulum rod, the criterion function and the change of the controller parameters in the iterative process.

FIG 7 is a factor L _i before and after the introduction of the auxiliary rotation inverted pendulum schematic tracking error.

detailed description

The specific embodiments of the present invention will be further described below with reference to the accompanying drawings.

The present application discloses an iterative feedback tuning control of a rotating inverted pendulum and a robust optimization method thereof,

Step 1: Establish the Lagrangian and state space model of the rotating inverted pendulum based on the mechanical and hardware structure of the inverted pendulum;

Figure 1 is a schematic diagram of the swing model of the rotating inverted pendulum. On this basis, the mathematical model of the rotating inverted pendulum is constructed. The rotating inverted pendulum system includes a base, a transmission device, a swing rod and a swing arm. The base is used to ensure the mechanical structure of the swing rod. Stable; the end of the arm is connected to the pendulum rod, and the rotation of the DC motor drives the movement of the pendulum rod through the transmission device; the angle and angular velocity of the arm are obtained through the incremental rotary encoder that comes with the DC motor; the incremental rotation is connected through the coupling The rotary encoder and the pendulum rod are used to drive the incremental rotary encoder to rotate to obtain the angle and angular velocity of the pendulum rod; in the dynamic model of the rotating inverted pendulum, air resistance, friction force and tiny items are ignored to simplify the modeling process , the swing arm and the pendulum rod are regarded as a uniform long rod, and the potential energy of the rotating inverted pendulum system is set to zero when the pendulum rod is in a stable erection. Combined with Table 1, the meanings of the physical quantities of the rotating inverted pendulum are:

Table 1 Significance of each physical quantity of rotating inverted pendulum

is the angular velocity when the arm rotates;

Among them, L is the length of the pendulum rod,

is the angular velocity when the pendulum rod rotates;

_{Under the combined action of the velocity v m} at the end of the boom _{and the velocity v r} in the horizontal direction of the ground, the velocity vb of the pendulum in the horizontal direction is:

The Lagrangian function E is:

T _output = η _d η _g K _g K _T I _d (13) Putting equations (12) and (13) into equation (11), the nonlinear model of the inverted pendulum with the DC motor voltage as input is:

b=J ₂ +mr ₁ ² (17)

Substituting equations (16) to (21) into equation (15) to solve

and

for:

select state vector

where r ₂ is the length of the arm, M is the mass of the arm, and ρ is the density of the arm and the pendulum.

Substituting the actual parameter values of the inverted pendulum shown in Table 2 into equations (16) to (25), the specific state space model of the rotating inverted pendulum is obtained as follows:

According to the state space model, it can be obtained from the Lyapunov criterion and the rank criterion that the rotating inverted pendulum is an unstable but completely controllable and observable system. Therefore, the angle, angular velocity and The angle and angular velocity of the pendulum are controlled, and these parameters are completely observable.

Table 2: The actual parameters of the rotating inverted pendulum

Combined with Fig. 2, a double closed-loop controller is designed for the state space model of the rotating inverted pendulum, and the iterative feedback tuning algorithm is used to optimize the parameters of the angle PD controller, if C(ρ)=[C _r (ρ) C _y (ρ)] , C _r (ρ), C _y (ρ) are linear time-invariant transfer functions, G is the transfer function of the controlled object, u(t) is the controller output, r(t) is the reference input, y(t) is the output of the rotating inverted pendulum system, v(t) is the external random disturbance with zero mean, the PID controller parameter is ρ=[K _p K _d ], and the response output under the action of the feedback control system is:

Among them, _Ly and _Lu represent filters based on time series, usually _Ly = _Lu = 1, the number of sampling points is N, and the weight factor of performance measurement is λ; the IFT algorithm optimizes the index function by minimizing the performance J ( ρ) directly obtain the PID controller parameter ρ of the system, and then gradually obtain the optimal value of the PID controller parameter ρ through i iterations, ρ _i is the value of ρ in the ith iteration, and in each iteration batch , the partial derivatives of the variables y(ρ _i ) and u(ρ _i ) with respect to the controller parameter ρ _{i are:}

unbiased estimate of

You can also get:

In order to design the stable pendulum controller to keep the pendulum rod upright and stable, first design a closed loop for the pendulum rod angle, the upright state is used as a dynamic balance, and integral control is not required, but the differential control is added to improve the control speed when the angle change rate is large, and finally the pendulum is reversed. The rod angle adopts PD controller; when the pendulum rod is stable and upright, the arm should remain stationary, so a closed loop is added to control the position of the arm, based on the integral of the speed, the speed of the arm adopts a PI controller; due to the closed loop of the speed It is an interference quantity of angle control, and the influence of PI controller on angle control needs to be reduced. Therefore, iterative feedback tuning algorithm is used for the tuning optimization of the PD controller parameters of the rotating inverted pendulum angle during the stable pendulum process.

Taking the specific state space model of the rotary inverted pendulum derived from equation (26) as the unknown unknown controlled object of the control system, and by obtaining the tracking error between the actual angle of the pendulum rod collected by the incremental rotary encoder and the given angle, the design The angle IFT-PD controller is used to output the DC motor voltage.

In the present application, the swing control of the rotary inverted pendulum is realized by the position feedback of the swing arm and the speed feedback of the swing rod. The position feedback of the swing arm restricts the swing rod from swinging around the desired position, and the speed feedback of the swing rod makes the swing rod The swing angle is gradually increased.

Step 3: Convergence analysis of iterative feedback setting angle PD controller;

In order to ensure the convergence of the algorithm, condition 1 is to ensure that the estimated gradient of the performance optimization index function is unbiased, and condition 2 is that the step size sequence γ _{i is} required to converge to zero. ) get

for:

The basic requirement of these convergence conditions is that the reference input signal r(t) remains bounded throughout the optimization iteration. _{Although the matrix R i} that determines the update direction does not affect the convergence ability of IFT, the ideal choice is to speed up the convergence speed through the Gauss–Newton direction. Therefore, using the Gauss-Newton optimization algorithm can ensure the convergence of the algorithm, so that the designed IFT algorithm can quickly cover a fixed optimization point. This conclusion does not make any assumptions about the nature of the system except for the time-invariant condition, so the conclusion is applicable to simple PID controllers or more complex controllers.

Step 4: Introduce auxiliary factors to further optimize the robust iterative feedback setting angle PD controller;

On this basis, the criterion function J(θ) is modified as:

in,

is the tracking error, u(θ _i ) is the controller output,

and

And the approximate Hessian matrix R _{i is} modified as:

Optimal range of weight ratios to u(θ _{i ).}

Figure 3 is the mechanical structure of the rotating inverted pendulum experimental platform, including the base, transmission device, DC motor, pendulum and arm and other parts, Figure 4 is the hardware structure diagram of the rotating inverted pendulum experimental platform, the hardware structure of the rotating inverted pendulum can be obtained by DSPACE It consists of programming controller, IR2104 DC motor driver board, ETS25 absolute rotary encoder, 50V/4.9A DC motor and STM32 board for SPI communication with ETS25. DSPACE sends a PWM signal to the IR2104 DC motor driver board, and the IR2104 DC motor driver board controls the voltage and direction of the DC motor according to PWM, and further DSPACE reads the position and speed of the motor, and the motor drives the rotating arm to rotate through the transmission belt. The shaft drives the rotary encoder to rotate, and finally the position and speed of the pendulum rod are read by the STM32 minimum system through SPI, and sent to DSPACE through the serial port.

The DSPACE real-time simulation system is a development and testing work platform for a control system based on MATLAB/Simulink in a real-time environment developed by the German DSPACE company. It can be seamlessly connected with MATLAB/Simulink. In this application, the model of DSPACE used is DS1104, which is a real-time control system based on PowerPC603 floating-point processor, and the operating frequency can reach 250MHz. In order to meet the demand for some advanced I/O ports, this model includes a slave DSP subsystem based on the TMS320F204DSP microcontroller. For Rapid Control Prototyping (RCP), specific interface connectors and connector panels provide easy access to all DSPACE input and output signals.

This application also provides the specific design of software and hardware based on DSPACE rotating inverted pendulum:

The specific robust optimization scheme of rotating inverted pendulum double closed-loop control based on iterative feedback tuning is as follows:

1) For the rotating inverted pendulum linear model (15), set the initial angle θ _{0 of} the pendulum rod, the initial control signal Δu ₀ , the desired trajectory y _d , and the sampling period ΔT.

2) Select the initial parameter ρ _{1 of the} angle PD controller and design the performance optimization index J(ρ _i ) according to formula (17), and give a threshold value J _max .

3) times of rotation inverted pendulum swings After three experiments _{^{were: r i (1) = y}} d, r i (2) = y (1) (ρ i), r i (3) = y d, ^{The y (1)} (ρ _i ) obtained in the first experiment is used as the reference input for the second experiment, and the second and third experiments obtain the controller input values u ⁽²⁾ (ρ _i ), u ⁽³⁾ (ρ _i ) and the system output values y ⁽²⁾ (ρ _i ), y ⁽³⁾ (ρ _i ) to calculate the gradient of the controller parameters.

4) Calculate the estimated gradient according to equations (21) and (22) using the results of the second and third experiments

The factor K _{i is} introduced to obtain the value of the weight factor λ and the Hessian matrix R _i is obtained on the basis of formula (30).

5) Determine whether the system performance optimization index J(ρ _i ) is smaller than J _max , if it is smaller, go to 6) to end, otherwise perform step 3).

6) end;

In order to realize the speed regulation and forward and reverse rotation of the DC motor, this patent adopts the typical control circuit of the DC motor, the H-bridge drive circuit. By controlling the on and off of the MOS tube, the magnitude of the motor voltage and the direction of the current are changed to realize the control of the DC motor.

The angle and speed of the pendulum rod in the rotary inverted pendulum system are collected through a rotary encoder ETS25 that transmits data through an SPI signal. In this design, a STM32 minimum system is used as an intermediary, that is, it communicates with ETS25 through the STM32 board, and then sends the encoder signal to DSPACE through RS232 serial communication. There is only one SPI signal data line used by the rotary encoder ETS25, and it needs 5V to be pulled up to provide a high level. For the sensor, it is a slave device, so it is connected to the MOSI of the STM32 microcontroller. The acquisition of the arm angle and speed is detected by the Hall sensor that comes with the DC motor. Since the reading program of this type of sensor is integrated on DSPACE, the reading of the angle and speed of the arm is relatively simple.

Figure 5 is the overall program design diagram of DSPACE. In this application, the initial angle θ ₀ of the pendulum of the inverted pendulum is 0.1rad, and the gain of the PD controller is θ=[150 45], which is taken as θ ₁ and imported into the rotating inverted pendulum based on DSPACE In the double closed-loop control system, the trajectory sampling data of the pendulum rod is obtained, and these data are processed offline in MATLAB to update the PD controller. As the number of iterations increases gradually, the optimization effect of the IFT algorithm is tested on this basis. The trajectories of the rotating inverted pendulum rod with iteration numbers i=1, i=3 and i=20 are shown in Fig. 6(a), and the criterion functions J(θ _i ), k _P and k _{D under these iteration numbers} The changes of , are shown in Figure 6(b), (c), (d). It can be seen that with the continuous iteration of the IFT algorithm, the control effect of the pendulum angle has been significantly improved, and the corresponding criterion function J(θ _i ) shows that the input and output errors gradually decrease as the iteration progresses, and the PD controller parameter θ _i finally converges to θ ₂₀ =[325 45.7]. We were selected for this further [lambda] ₁ = ^10-4 and λ ₂ = 10 ^-5, and with the introduction of co-factors L _i batch change, before and after the introduction of co-factors L _i rotary inverted pendulum tracking error is shown in Figure 7, rotation inverted as the pendulum tracking error iterative process steadily decreased, and after the introduction of cofactors L _i, to further reduce the tracking error, the control shows that the overall performance of the system is further improved.

The above descriptions are only preferred embodiments of the present application, and the present invention is not limited to the above embodiments. It can be understood that other improvements and changes directly derived or thought of by those skilled in the art without departing from the spirit and concept of the present invention should be considered to be included within the protection scope of the present invention.

Claims

An iterative feedback tuning control of a rotating inverted pendulum and a robust optimization method thereof, characterized in that the method comprises:

Step 1: Establish the Lagrangian and state space models of the rotating inverted pendulum;

The rotating inverted pendulum system includes a base, a transmission device, a swing rod and a swing arm. The base is used to ensure the stability of the mechanical structure when the swing rod swings; the end of the swing arm is connected to the swing rod, and the rotation of the DC motor passes through the transmission. The device drives the movement of the pendulum rod; the angle and angular velocity of the arm are obtained through the incremental rotary encoder that comes with the DC motor; the incremental rotary encoder is connected with the The pendulum rod drives the incremental rotary encoder to rotate to obtain the angle and angular velocity of the pendulum rod; in constructing the dynamic model of the rotating inverted pendulum, air resistance, friction force and tiny items are ignored to simplify the modeling In the process, the swing arm and the pendulum rod are regarded as a uniform long rod, and the potential energy of the rotating inverted pendulum system is set to zero when the pendulum rod is in a stable erection;

When the pendulum rod deviates from the upright position angle α, the swing arm drives the pendulum rod to the upright position by rotating β, so the speed v m of the end of the arm is:

Among them, r 1 is the distance from the rotation center of the arm to the connection point with the pendulum rod,
is the angular velocity when the arm rotates;

Since the pendulum rod is a uniform long rod, considering the pendulum rod as a mass point, the rotation speed of the pendulum rod v z is obtained as:

Among them, L is the length of the pendulum rod,
is the angular velocity when the pendulum rod rotates;

Decompose the rotation speed v z of the swing rod in the vertical direction of the speed v m at the end of the arm, and take the direction of the rotation plane of the swing rod and the speed v r in the horizontal direction of the ground as the positive direction to obtain:

Under the combined action of the speed v m at the end of the boom and the speed v r in the horizontal direction of the ground , the speed v b of the pendulum rod in the horizontal direction is:

The kinetic energy of the pendulum rod includes the rotational kinetic energy generated by rotation and the kinetic energy generated by moving in the horizontal direction. In addition, the overall kinetic energy of the rotating inverted pendulum system also includes the kinetic energy of the arm driven by the DC motor. Therefore, we obtain The overall kinetic energy V of the rotating inverted pendulum system, let J 1 be the moment of inertia of the pendulum rod, J 2 be the moment of inertia of the swing arm, m is the mass of the pendulum rod, and the equations (4) and (5) are brought in to obtain:

The pendulum rod is set as the zero potential energy point when it is upright, H is the overall potential energy of the rotating inverted pendulum system, E is the Lagrangian function, then the potential energy after the deflection angle α is reduced to:

The Lagrangian function E is:

It can be seen that the rotation of the swing arm drives the movement of the pendulum rod, and there is no external power input, let T output be the output torque of the motor, and B eq be the equivalent viscous friction, and the Lagrangian equation can be obtained as:

Equation (8) is brought into equations (9) and (10) to obtain the nonlinear model of the rotating inverted pendulum:

In the nonlinear model of the rotating inverted pendulum obtained from equation (11), its input is the DC motor torque, but usually the DC motor voltage is used as the control input, so the DC motor is modeled next , and finally establish an inverted pendulum nonlinear model with the DC motor voltage as the input;

So as DC current I d, E d is the counter electromotive force, and taking into account the efficiency of the transmission gear ratio, K T is the motor torque coefficient, K E of the motor-speed coefficient, K g is the arm with the DC motor Gear ratio, η g is the gear transmission efficiency, η d is the motor efficiency, U is the voltage of the DC motor, R is the armature resistance, obtain:

T output = η d η g K g K T I d (13)

Putting equations (12) and (13) into equation (11), the nonlinear model of the inverted pendulum with the DC motor voltage as the input is obtained as:

In order to further establish the state space model of the rotating inverted pendulum, the nonlinear model of the inverted pendulum needs to be linearized. It is noted that the pendulum rod is in an upright state in the stable pendulum control, so the angle of the pendulum rod is small, and sinα exists at this time. ≈α, cosα≈1, then the linear model of the rotating inverted pendulum is obtained as:

Next, the state space model of the rotating inverted pendulum is established based on the linear model of the rotating inverted pendulum. In order to simplify the writing settings, the following definitions are made:

Substituting equations (16) to (21) into equation (15) to solve
and
for:

select state vector
Where β is the rotation angle of the arm, the input is the DC motor voltage U, and the state space model of the rotating inverted pendulum is obtained as:

Since the pendulum rod and the swing arm are regarded as uniform long rods, the moment of inertia J 1 and J 2 can be obtained as follows:

Among them, r 2 is the length of the arm, M is the mass of the arm, and ρ is the density of the arm and the pendulum;

Step 2: Design a rotating inverted pendulum iterative feedback tuning double closed-loop controller;

The double closed-loop controller is designed according to the state space model of the rotating inverted pendulum, and the parameters of the angle PD controller are optimized using an iterative feedback tuning algorithm. If C(ρ)=[C r (ρ) C y (ρ)], C r (ρ), C y (ρ) are linear time-invariant transfer functions, G is the transfer function of the controlled object, u(t) is the controller output, r(t) is the reference input, and y(t) is the The output of the rotating inverted pendulum system, v(t) is an external random disturbance with a mean value of zero, and the PID controller parameter is ρ=[K p K d ]. On this basis, the response output under the action of the feedback control system is:

In order to simplify the writing, T 0 (ρ) and S 0 (ρ) are abbreviated as T 0 and S 0 , and y d is defined as a given expected input signal, then the tracking error between the expected output and the actual output is:

For a fixed-structure PID controller with controller parameter ρ, by minimizing
In order to improve the tracking control effect of the feedback control system, the performance optimization index function J(ρ) is defined as:

Wherein Ly and Lu represent filters based on time series, usually Ly = Lu =1, the number of sampling points is N, and the weight factor of the performance measurement is λ; the IFT algorithm is to optimize the index function by minimizing the performance J(ρ) directly obtains the PID controller parameter ρ of the system, and then gradually obtains the optimal value of the PID controller parameter ρ through i iterations, where ρ i is the value of ρ in the ith iteration, where In each iteration batch, the partial derivatives of the variables y(ρ i ) and u(ρ i ) with respect to the controller parameter ρ i are:

The IFT algorithm is obtained by conducting three experiments in the DOF control system to obtain the estimated value of T 0 r, T 0 (ry), in the three experiments, the first two are used to estimate the signal T 0 , first in the ith time In the iteration, the first experiment takes r i (1) = r as the input reference signal, y (1) (ρ i ) is the output value of the control system obtained by sampling; secondly, the difference between the two signals ry (1) (ρ i) for the second reference input signal experiment r i (2), obtained by sampling y (2) (ρ i) :

The third test signal used to estimate T 0 r, to r i (3) = r as the reference signal input:

According to the controller output value of the three experiments and the output value of the rotating inverted pendulum system, the
unbiased estimate of
You can also get:

The estimated gradient of the performance optimization index function J(ρ i ) based on the ith iteration of the experimental data is:

According to the estimated gradient of the performance optimization index function J(ρ i ) and the PID controller parameters ρ i of the previous iteration, the Gauss-Newton algorithm is used to calculate the updated ρ i+1 of the next iteration:

where γ i > 0 represents the step size, and R i is a positive definite Hessian matrix to represent the optimization search direction:

Step 3: Convergence analysis of iterative feedback setting angle PD controller;

In order to ensure the convergence of the algorithm, Condition 1 is to ensure that the estimated gradient of the performance optimization index function is unbiased, and Condition 2 is that the step sequence γ i is required to converge to zero. In order to ensure Condition 1, from equation (18) to (20) get
for:

The set of three experiments IFT experiments based algorithm v i (m), m = 1,2,3 is the same system independent zero-mean random noise bounded, i.e. | v i (m) | < C, Assuming that the limit value C and the mean square value of random noise in the three experiments remain unchanged, the unbiased estimates of equations (21) and (22) are obtained;

Condition 2 The conditions that need to ensure the convergence of the algorithm usually require that all elements of the step sequence γ i satisfy:

The fourth step: further optimization of the robust iterative feedback setting angle PD controller;

The IFT algorithm relies on experience to select weight values such as λ. However, because the physical meanings of the performance metrics are not the same, and the operating environments are not consistent, the value ranges between them vary greatly. Therefore, if simultaneously controlling a plurality of the same system, the right to rely on the experience of the selected performance metric is not weighting factor λ is universal, considering the range between each of the performance metric to build a cofactor L i, the auxiliary factor L i is the ratio between the range of performance metrics:

On this basis, the criterion function J(θ) is modified as:

in,
is the tracking error, u(θ i ) is the controller output,

and
And the approximate Hessian matrix R i is modified as:

where y d,max and y d,min are the maximum and minimum values of the expected output, and u max and u min represent the maximum and minimum values of the control signal in all N sampling points during the ith iteration; since these values are given at the end of each iteration, all the sample points will be taken into account, and thus the introduction of the co-factor L i such that the weighting factor λ in different systems are optimized represents the current iteration
Optimal range of weight ratios to u(θ i ).