CN109976150B - Centralized Active Disturbance Rejection Control Method for a Class of Underdriven Multiple Input Multiple Output Systems - Google Patents

Abstract

The invention discloses a centralized active disturbance rejection control method for a type of underactuated multiple-input multiple-output system. , design the virtual control variable of the direct drive part; use the unified extended state observer to uniformly estimate and compensate the disturbance and uncertainty part of the underactuated part, and design the virtual control variable of the indirect drive part; The virtual control variables are organically combined to form a comprehensive control variable, which realizes the centralized control of the underactuated system. The Lyapunov method is used to associate the feedback control gain of the control system with the pole configuration of the Hurwitz stability matrix in the system error equation, which ensures the stability of the system. The gain of the control system to be set is also concentrated as a pole to configure a parameter; the entire control system has a compact structure, strong robustness and anti-interference ability, easy parameter setting, and universality.

Description

Centralized Active Disturbance Rejection Control Method for a Class of Underdriven Multiple Input Multiple Output Systems

technical field

The invention relates to a centralized active disturbance rejection control method for a type of underdriven multiple-input multiple-output (MIMO) system, belonging to the field of automatic control.

Background technique

An underactuated system refers to a class of nonlinear systems in which the number of system control inputs is less than the system degrees of freedom. Compared with the full-drive system, the under-actuated system reduces part of the actuators, making it low cost, light and flexible, low energy consumption, and easy to maintain. Wide range of applications. However, due to the lack of input quantity, nonlinearity, parameter perturbation, multi-objective, and susceptibility to interference, the underactuated system is difficult to control, and it is still considered as one of the main open problems in the field of automatic control.

In order to simplify the problem, the underactuated system is often dealt with by the model linearization method. Although it has a certain effect, because it cannot compensate the difference between the nominal model and the actual object, the external disturbance and model uncertainty can easily make the control system unstable. The partial feedback linearization method is used to design the controller of the underactuated system, which is a common method for the stability control of the underactuated system, but this method can only ensure that a part of the state quantity of the underactuated system can be controlled, and whether the control target can be achieved depends on the configuration structure. . By combining the actuated variable and the underactuated variable into a composite variable, the underactuated system is transformed into a full actuated system. The control method of the full actuation system is used to design the controller, which is also widely used in the design of the underactuated system controller. This method has poor robustness when there are uncertainties in the model and serious external interference; the first-order sliding mode surfaces of the driving variables and the under-actuated variables are designed separately, and then combined into a second-level sliding mode surface, and the sliding mode control method is applied. The design and implementation of the underactuated system controller is beneficial to improve the robustness of the underactuated system, but the problem of jitter still exists in practical applications. In order to reduce the influence of parameter perturbation, external disturbance and nonlinear coupling term of the underactuated system, the decoupling algorithm of multi-state feedback is often used, which leads to the complicated control structure and too many parameters to be set. Intelligent control methods, such as neural networks, fuzzy systems and learning algorithms, are also used in the control experiments of underactuated systems, but fuzzy control needs to learn from expert experience and obtain the output of the fuzzy controller through fuzzy reasoning. To obtain internal weights, the learning algorithm requires long-term online learning, which greatly increases the difficulty of controller design and adjustment, making it difficult to apply in engineering. Moreover, the control of the current underactuated system mainly uses the dynamic coupling effect of the direct and indirect drive parts to make a specific underactuated system achieve a specific motion through various control strategies, which is inconvenient to generalize to a general system with more degrees of freedom. underactuated system.

SUMMARY OF THE INVENTION

In view of the above-mentioned problems, the present invention discloses a centralized active disturbance rejection control method for an underdriven MIMO system, and the method is implemented according to the following steps:

Step A, according to the characteristics of the underdriven MIMO model, it is divided into a direct drive part (1) and an indirect drive part (2), and the specific process is as follows:

For an underdriven MIMO system with m inputs, n outputs (1≤m<n):

和间接驱动部分的虚拟控制量

的基础上，再将这两个部分的控制量进行有机组合，形成系统的实际控制量u_i，实现对欠驱动系统所有自由度的控制；考虑到u_i由虚拟控制量

和若干个对u_i有影响的虚拟控制量

组成，为描述方便，定义间接控制部分j对直接控制部分i的影响系数

有影响时，

没有影响时，

为简单起见，将

和对u_i有影响的虚拟控制量

进行线性组合，作为直接驱动部分i的实际控制量，即

The number of independent control variables of the underactuated system shown in formula (1) is less than the system degree of freedom, and it is divided into a direct drive part (2) and an indirect drive part (3). For the part that generates equal output, the corresponding input and output degrees of freedom are equal. The indirect drive part realizes the action of the relevant actuator through nonlinear coupling with the direct drive degree of freedom. Therefore, the virtual control quantity of the direct drive part can be designed.

and the virtual control quantity of the indirect drive part

On the basis of , the control quantities of these two parts are organically combined to form the actual control quantity u _i of the system to realize the control of all degrees of freedom of the underactuated system; considering that _ui is controlled by the virtual control quantity

and several virtual control quantities that have an impact on u _i

composition, for the convenience of description, define the influence coefficient of the indirect control part j to the direct control part i

when it affects

without influence,

For simplicity, the

and the virtual control quantity that affects u _i

Perform a linear combination as the actual control amount of the direct drive part i, that is

Step B, design the virtual control quantity of the direct drive part according to the current state and the target state of the direct drive part;

为使系统(1)收敛于目标点，需要对系统进行状态反馈，直接驱动部分控制量的设计符合分离性原理，采用常规的PD 反馈控制；Assume that the target state of the underdriven MIMO system (1) is

In order to make the system (1) converge to the target point, it is necessary to perform state feedback on the system. The design of the direct drive part of the control variable conforms to the principle of separation, and the conventional PD feedback control is adopted;

The direct drive part of an underdriven MIMO system is expressed as:

Using conventional proportional-derivative (PD) feedback control, the virtual control quantity of _yi is designed:

为直接驱动部分的当前状态值，通过相关传感器测得。Among them, k _i1 , k _i2 are feedback control gain, x _i ,

It is the current state value of the direct drive part, measured by the relevant sensor.

Step C, design a linear expansion state observer, estimate the state, disturbance and uncertainty part of the indirect driving part, and design the virtual control quantity of the indirect driving part, and the specific process is as follows:

The indirect drive part of the underdriven MIMO system described in equation (3) can be expressed as

For the convenience of analysis, formula (6) is further expressed as:

Among them, b _j is the estimated value of b _ji , f _j ( ) is the total disturbance of the state quantity y _j loop, including the internal disturbance, external disturbance and system uncertainty of the y _j loop,

(δ_i,

是正实数)，令x_j1＝y_j，

x_j1＝f_j(·)，

则式(7)可扩张为：Let the unknown perturbation f _j ( ) be bounded and differentiable,

(δ _i ,

is a positive real number), let x _j1 =y _j ,

x _j1 = f _j ( ),

Then formula (7) can be expanded as:

in,

The linear expansion state observer (LESO) is designed according to equation (8):

是y_j的状态估计，L是观测增益向量，采用基于迭代步长h的三阶线性扩张状态观测器的参数序列，

where Z _j =[z _j1 z _j2 z _j3 ] ^T is the state estimate of the vector X _j ,

is the state estimate of y _j , L is the observation gain vector, using the parameter sequence of the third-order linearly expanded state observer based on the iteration step h,

The virtual control quantity of y _j is designed as:

和式(10)代入式(7)，可得：Will

Substitute equation (10) into equation (7), we can get:

Among them, the subscript k=m+1,m+2,...,j-1,j+1,...,n, e _j3 is the error of the summation disturbance f _j ( ) of y _j and its estimated value z _j3 , e _j3 =f _j ( )-z _j3 ;

控制，对

也采用PD控制律设计，即：It can be seen from equation (11) that when e _j3 converges to 0, the final system will not be affected by f _j ( ), but is actually controlled by the variable

control, yes

The PD control law design is also used, namely:

Among them, k _j1 , k _j2 are feedback control amount gains.

In step D, the virtual control quantities of the direct drive part and the indirect drive part are organically combined to form the linear feedback control quantity (LESF) of the underactuated system:

Since the indirect drive part is indirectly controlled by the driver, the control value of the indirect drive part will affect the actual control value of the driver. Therefore, the control quantities of the direct drive part and the indirect drive part need to be synthesized to realize all the underactuated system. For the control of degrees of freedom, for the sake of simplicity, a linear combination method is used as the system control quantity:

For the direct drive part, substitute equation (13) into equation (2) to get:

For the indirect drive part, substitute equation (13) into equation (3) to get:

Define the observation error of the extended state observer as e _j =[e _j1 ,e _j2 ,e _j3 ] ^T , then:

The error of the underdriven MIMO system is defined as ψ(t)=[ψ ₁ , ψ ₂ , ψ ₃ , ψ ₄ ,...,ψ _2n-1 ,ψ _2n ] ^T , then:

According to the above formulas, the system error differential equation is updated as:

和

可扩张为：therefore,

and

Can be expanded to:

Among them, A _ψ is the coefficient matrix of ψ(t), and A _ei is the coefficient matrix of e _i ;

上，即In order to make A _ψ a Hurwitz stable matrix, the eigenvalues of A _ψ are arranged at the point

up, that is

Solving equation (20), we can get:

为方程(20)的解，集体的解析式由实际系统决定，则：in,

is the solution of equation (20), the collective analytical expression is determined by the actual system, then:

That is, for a MIMO system with m inputs and outputs, no matter what the values of m and n are, there is always only one adjustable parameter for the ADRC control quantity described in equation (23).

The beneficial effects of the present invention are: using a unified extended state observer to uniformly estimate and compensate the disturbance and uncertain parts of the underactuated system, thereby improving the robustness and auto-disturbance rejection capability of the system; The synthetic control variables formed by the organic combination of virtual control variables realize the centralized control of the degrees of freedom of the system; the Lyapunov method is used to correlate the feedback control gain of the control system with the pole configuration of the Hurwitz stability matrix in the system error equation to ensure the stability of the system , the gain of the control system to be set is also concentrated as a pole and a parameter is configured, and a centralized active disturbance rejection controller for an underactuated system is formed comprehensively; the control system designed by the invention has a centralized structure, small parameters to be set, easy design, and universal sex.

Description of drawings

Fig. 1 is the flow chart of the centralized active disturbance rejection control of the underactuated system;

Fig. 2 is the simulation result of the secondary inverted pendulum controlled by the present invention. The three curves in the figure represent the trolley displacement x(t), the primary pendulum swing angle θ ₁ (t), the secondary pendulum pendulum angle from top to bottom, respectively. θ ₂ (t).

Detailed ways

In order to make the purpose, technical solutions and advantages of the present invention clearer, the present invention will be described in detail below by taking the linear secondary inverted pendulum produced by Shenzhen Yuanchuangxing Technology Co., Ltd. as an example.

The system model of the linear second-order inverted pendulum can be expressed as:

in:

θ₁,

θ₂,

分别代表小车位移、小车速度、摆杆1角度、摆杆1角速度、摆杆2角度、摆杆2角速度，其它各符号的含义及具体数值见表1。The six state quantities x in the state equation,

θ ₁ ,

θ ₂ ,

They represent the cart displacement, cart speed, pendulum rod 1 angle, pendulum rod 1 angular velocity, pendulum rod 2 angle, and pendulum rod 2 angular velocity, respectively. The meanings and specific values of other symbols are shown in Table 1.

Table 1 Physical parameters of Yuanchuangxing linear two-stage inverted pendulum system

Substitute the physical parameters in Table 1 into the above equations, and obtain: c ₁₁ =52.797, c ₁₂ =-16.100, c ₁₃ =3.7407, c ₂₁ =-44.5267, c ₂₂ =-43.9288, c ₂₃ =0.597.

The control goal of the secondary inverted pendulum is that when the trolley reaches the target position, the pendulum rod is in a vertically stable equilibrium position, so the target state of the inverted pendulum is

According to formula (5), the direct drive part is designed, that is, the virtual feedback control quantity of the trolley displacement is:

According to formula (12), the indirect drive part is designed, that is, the virtual control quantities of the swing angle θ 1 of the swing rod ₁ and the swing angle θ ₂ of the swing rod 2 are:

Substituting equations (25)-(27) into equation (13), the feedback control amount of the linear second-order inverted pendulum is obtained:

According to formula (22), b ₁ and b ₂ are estimated values of c ₁₃ and c ₂₃ respectively, and take the actual values, namely: b ₁ =c ₁₃ , b ₂ =c ₂₃ .

According to the stable point coordinates of the second-order inverted pendulum of the straight line, the system error is defined as:

According to the above formulas, the error differential equation of the linear second-order inverted pendulum system is obtained:

Expanding the error differential equation of the second-order inverted pendulum, we get:

in:

According to formula (21), we have

In order to simplify the operation process, after substituting the system parameters in Table 1, and solving equation (30), we can get:

优选为10，将以上数据代入式(28)，得出二级倒立摆的控制量：According to b ₁ =c ₁₃ , b ₂ =c ₂₃ , we can get b ₁ =3.7407, b ₂ =0.597, the numerical calculation iteration step size is h=0.001s according to the system hardware, after sufficient tuning,

It is preferably 10. Substitute the above data into formula (28) to obtain the control amount of the secondary inverted pendulum:

Assuming that the target position of the car is 0.2m, a step disturbance of 10% is applied to the car at 5s, and a step disturbance of 10% is applied to the secondary pendulum at 8s. Control, the experimental results are shown in Figure 2, the results show that after the car reaches the target position, its motion range is less than 0.002m without disturbance, and the up and down swing angles are all close to 0°, and the control effect is very good; at 5s After applying a 10% step disturbance to the trolley, the swing amplitudes of the upper and lower swing rods are both less than 1.2° and tend to be stable soon. Less than 0.02m, the swing amplitudes of the upper and lower swing rods are also less than 0.8°, and the time to stabilize is less than 2s, indicating that the method of the present invention has good stability and anti-interference performance for the underactuated system.

As mentioned above, similar technical solutions can be derived in conjunction with the content of the solutions given in the drawings and descriptions. However, any simple modifications, equivalent changes and modifications made according to the technical essence of the present invention still fall within the scope of the technical solutions of the present invention.

Claims (1)

1. A centralized active disturbance rejection control method of an under-actuated multiple-input multiple-output (MIMO) system is characterized by comprising the following steps:

step A, splitting an under-actuated MIMO system into a direct drive part and an indirect drive part:

for an under-actuated MIMO system with m input and n output, m is more than or equal to 1 and less than n:

the method comprises the steps of splitting the driving mode into a direct driving part (2) and an indirect driving part (3), and designing virtual control quantity of the direct driving part

i is 1,2, …, m, and indirect drive section virtual control amount

The virtual control quantities of the two parts are organically combined to form the actual control quantity u of the system_iTo realize the control of all the degrees of freedom of the under-actuated system; defining the influence coefficient of the indirect control part j to the direct control part i

If the indirect control section j has an effect on the direct control section i, then

If there is no influence, then

For simplicity, will

And pair of u_iInfluential virtual control volume

Linearly combined as an actual control quantity of the direct drive section i, i.e.

And B, designing a virtual control quantity of the direct drive part according to the current state and the target state of the direct drive part:

assuming a target state v for an under-actuated MIMO system (1)₁,

In order to make the system (1) converge on a target point, state feedback needs to be carried out on the system, and a direct-drive part of the under-driven MIMO system is represented as follows:

design y using conventional proportional-derivative (PD) feedback control_iVirtual control amount of (2):

wherein k is_i1,k_i2For feedback control gain, x_i,

The current state value of the direct driving part i is measured by a relevant sensor;

and step C, designing a linear extended state observer, estimating the state, disturbance and uncertain parts of the indirect driving part, and designing the virtual control quantity of the indirect driving part, wherein the specific flow is as follows:

the indirect driving part of the under-driven MIMO system described in equation (3) can be expressed as:

wherein f is_j(. is a state quantity y_jThe sum of (a) and (b) is disturbed,

b_jis b is_jiAn estimated value of (d);

let unknown disturbance f_j(. a) bounded and differentiable, i.e.

_j,

Is a positive real number, let x_j1＝y_j，

x_j3＝f_j(·)，

Formula (6) can be expanded as:

wherein,

a Linear Extended State Observer (LESO) is designed according to equation (7):

wherein Z is_j＝[z_j1,z_j2,z_j2]^TIs a vector X_jIs estimated in the state of (a) of (b),

is y_jL is an observation gain vector, adopts a parameter sequence of a third-order linear extended state observer based on numerical calculation iteration step length h,

will y_jThe virtual control amount of (2) is designed as follows:

to pair

And (3) designing by adopting a conventional PD control law:

wherein k is_j1,k_j2Is a feedback control quantity gain;

and D, organically synthesizing the virtual control quantities of the direct drive part and the indirect drive part to form a linear feedback control quantity of the under-actuated system:

for simplicity, the feedback control quantity of the under-actuated system is designed in a linear combination mode:

for the direct drive portion, formula (11) can be substituted for formula (2):

for the indirect drive section, formula (11) may be substituted for formula (3):

wherein, subscript k ═ m +1, m +2, …, j-1, j +1, …, n;

defining the observation error of the extended state observer as e_j＝[e_j1,e_j2,e_j2]^TAnd then:

defining the error of the under-actuated MIMO system as psi (t) [. psi₁,ψ₂,ψ₃,ψ₄,…,ψ_2n-1,ψ_2n]^TAnd then:

wherein, subscript q ═ 1,2, …, n;

from the above equations, the system error differential equation can be updated as:

the under-actuated system error differential equation is expanded to:

wherein A is_ψCoefficient matrix of ψ (t), A_eiIs e_iThe matrix of coefficients of (a) is,

to make A_ψIs a Hurwitz stabilization matrix, A_ψAre all arranged at points

To do so, i.e.

Wherein λ is A_ψE is an identity matrix,

representing the bandwidth of the extended state observer;

by solving equation (19), it is possible to obtain:

wherein,

the specific analytic expression is determined by the actual system and is the solution of the equation (20);

by substituting formula (21) for formula (12), the control amount of the under-actuated system can be obtained:

that is, for the MIMO system with m inputs and n outputs, the centralized auto-disturbance-rejection controller described in equation (22) always has only one adjustable parameter no matter what the values of m and n are

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