CN107121932B - A Robust Adaptive Control Method of Error Symbol Integral for Motor Servo System - Google Patents

Abstract

The invention discloses a motor servo system error symbol integral robust self-adaptive control method, which comprises the following steps: establishing a motor position servo system model; designing an error symbol integral robust adaptive controller; according to the error sign integral robust self-adaptive controller, the stability of the motor servo system is proved by utilizing the Lyapunov stability theory, and the global gradual stabilization result of the system is obtained by utilizing the Barbalt theorem. The invention provides a parameter adaptive error sign integral robust adaptive anti-interference controller aiming at parameter uncertainty and unknown nonlinear factors in a motor servo system, the parameter adaptive rate can effectively estimate the unknown parameters in the system, an error sign integral robust term is adopted to overcome other uncertain nonlinear factors in the system, and the control precision in the motor servo system is ensured.

Description

A Robust Adaptive Control Method of Error Symbol Integral for Motor Servo System

technical field

The invention relates to a motor servo control technology, in particular to a robust self-adaptive control method of error symbol integration of a motor servo system.

Background technique

Permanent magnet brushless DC motor has a wide range of applications in the industrial field due to its own characteristics such as fast response speed, high energy utilization rate, and low pollution. With the rapid development of industrial technology in recent years, higher requirements have also been placed on the control technology of DC motors. How to improve the motion accuracy of DC single machines has become the main research direction of DC motors. In the motor servo system, due to different working conditions and some structural limitations, it is difficult for the system to fully reflect the real model when modeling. Therefore, when designing the controller, the uncertainty of these models plays a very important role, especially Uncertain nonlinearity will seriously deteriorate the control performance of the controller, resulting in low accuracy, limit cycle oscillation, and even system instability.

For the nonlinear problems existing in the system, the traditional control method is difficult to solve its influence on the control accuracy of the system. In recent years, with the development of control theory, various control strategies for uncertain nonlinearity have been proposed, such as sliding mode variable structure control, robust adaptive control, adaptive robust and so on. However, the design of the above control strategy controller is relatively complicated, and it is not easy to implement engineering.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a robust self-adaptive control method of error sign integral of the motor servo system, so as to solve the uncertain nonlinear problem in the motor position servo system.

The technical scheme for realizing the purpose of the present invention is: a method for robust self-adaptive control method of error symbol integration of motor servo system, comprising the following steps:

Step 1, establish a motor position servo system model;

Step 2, designing a robust adaptive controller of error sign integral;

Step 3, according to the error sign integral robust adaptive controller, use Lyapunov stability theory to prove the stability of the motor servo system, and use Barbalat's lemma to obtain the result of the global asymptotic stability of the system.

Compared with the prior art, the significant advantages of the present invention are:

Aiming at the parameter uncertainty and unknown nonlinear factors (external disturbance) in the motor servo system, the present invention proposes an error symbol integral robust adaptive anti-interference controller based on parameter self-adaptation, and the parameter self-adaptation rate can effectively estimate the The unknown parameters of , the error symbol integral robust term is used to overcome other uncertain nonlinear factors in the system, which ensures the control accuracy of the motor servo system. The simulation results verify the effectiveness of the proposed control strategy.

Description of drawings

Figure 1 is a schematic diagram of a motor servo system.

FIG. 2 is a diagram of a robust self-adaptive control strategy diagram of the error symbol integration of the motor servo system of the present invention.

Fig. 3 is the tracking process diagram of the system output of the controller to the given output under the action of disturbance (1).

FIG. 4 is a graph showing the variation of the tracking error of the system with time under the action of disturbance (1).

Figure 5 is a graph of the tracking accuracy of PID control and ARISE control under the action of disturbance (2).

FIG. 6 is a graph of disturbance (2) control input u.

FIG. 7 is a graph of the tracking error of the system under the action of disturbance (3) as a function of time.

FIG. 8 is a graph of the control input v under the action of disturbance (3).

FIG. 9 is a graph of parameter adaptation under the action of disturbance (3).

Detailed ways

With reference to Fig. 1 and Fig. 2, a method for robust adaptive control method of error symbol integration of motor servo system includes the following steps:

Step 1. Establish a motor position servo system model;

According to Newton's second law, the dynamic model equation of the motor inertial load is:

代表其他未建模干扰，比如输入饱和，外部时变扰动以及未建模动态；where y is the angular displacement, J _equ is the inertial load, ku is the torque constant, _u is the system control input, B _equ is the viscous friction coefficient, _dn is the constant disturbance to the system,

represents other unmodeled disturbances, such as input saturation, external time-varying disturbances, and unmodeled dynamics;

Write equation (1) in state space form, as follows:

x＝[x₁,x₂]^T表示位置和速度的状态向量；参数集θ＝[θ₁,θ₂,θ₃]^T，其中θ₁＝Je_qu/k_u，θ₂＝B_equ/k_u，θ₃＝d_n/k_u，

表示系统中其他未建模干扰。由于系统参数J_equ，k_u，B_equ，d_n未知，系统参数是不确定的，但系统的大致信息是可以知道的。此外，系统的不确定非线性

也是不能明确建模的，但系统的未建模动态和干扰总是有界的。因而，以下假设总是成立的：in

x=[x ₁ , x ₂ ] ^T represents the state vector of position and velocity; parameter set θ=[θ ₁ , θ ₂ , θ ₃ ] ^T , where θ ₁ =Je _qu /k _u , θ ₂ =B _equ / k _u , θ ₃ =d _n /k _u ,

Represents other unmodeled disturbances in the system. Since the system parameters J _equ , _ku , _Bequ , and d _n are unknown, the system parameters are uncertain, but the general information of the system can be known. Furthermore, the uncertain nonlinearity of the system

Nor can it be explicitly modeled, but the unmodeled dynamics and disturbances of the system are always bounded. Thus, the following assumptions always hold:

Assumption 1: The parameter θ satisfies:

where θ _min = [θ _1min , θ _2min , θ _3min ] ^T , θ _max = [θ _1max , θ _2max , θ _3max ] ^T , they are all known, and θ _1min >0, θ _2min >0, θ _3min >0;

Assumption 2: d(x,t) is bounded and first-order differentiable, i.e.

where δ _d is known.

Step 2. Design the error symbol integral robust adaptive controller, and the specific steps are as follows:

选取x₂为虚拟控制量，使方程

趋于稳定状态；令x_2eq为虚拟控制的期望值，x_2eq与真实状态x₂的误差为z₂＝x₂-x_2eq，对z₁求导得：Step 2-1. Define z ₁ =x ₁ -x _1d as the angular displacement tracking error of the system, x1d is the position command that the system expects to track and the command is second-order continuous differentiable, according to the first equation in equation (2)

Select x ₂ as the virtual control quantity, so that the equation

tends to a stable state; let x _2eq be the expected value of virtual control, the error between x _2eq and the real state x ₂ is z ₂ =x ₂ -x _2eq , and derivation for z ₁ can be obtained:

Design a virtual control law:

In formula (6), k ₁ > 0 is an adjustable gain, then

Since z ₁ (s)=G(s)z ₂ (s), where G(s)=1/(s+k ₁ ) is a stable transfer function, when z ₂ tends to 0, z ₁ also must tend to 0.

Step 2-2. In order to design the controller more conveniently, an auxiliary error signal r(t) is introduced

In formula 8, k ₂ > 0 is an adjustable gain;

According to equations (2), (7) and (8), there are the following expansions of r:

According to equations (2) and (9), there are the following equations:

According to Equation (10), the model-based controller is designed as:

代表θ的估计值，

为估计的误差

β为系统控制增益；k_r为正反馈增益；

为参数自适应率；Γ＞0为可调的正的自调节律增益。Formula (11)

represents the estimated value of θ,

is the estimated error

β is the system control gain; k _r is the positive feedback gain;

is the parameter self-adaptation rate; Γ>0 is the adjustable positive self-adjustment law gain.

和其一阶导数是已知的，所以自适应率可以进行积分得到：According to the parameter adaptation rate in equation (11), although r is an unknown quantity,

and its first derivative are known, so the adaptation rate can be integrated to get:

Substitute equation (11) into equation (10) to calculate:

Derive to get:

Step 3. According to the error symbol integral robust adaptive controller, use the Lyapunov stability theory to prove the stability of the motor servo system, and use the Barbalat lemma to obtain the result of the global asymptotic stability of the system, as follows:

Lemma 1:

define helper functions

分别表示z₂(t)、

的初始值。z ₂ (0),

respectively represent z ₂ (t),

the initial value of .

时，则when

time, then

P(t)≥0 (19)

Proof of this lemma:

Integrate both sides of equation (15) at the same time and apply equation (7) to get:

Step-by-step integration of the last two items in equation (20) can be obtained:

therefore

时，有式(17)和(19)成立，引理得证。It can be seen from equation (22) that if the value of β satisfies

When , equations (17) and (19) are established, and the lemma is proved.

According to the above lemma, the Lyapunov function is defined as follows:

为估计的误差，即

is the estimated error, i.e.

Using Lyapunov stability theory to prove the stability, and using Barbalat's lemma to get the result of the global asymptotic stability of the system, so adjust the gains k ₁ , k ₂ , k _r and Γ to make the tracking error of the system infinite in the time domain down to zero. Derivating equation (23) and substituting (7), (8), (14) and (16), we get:

in

definition:

Z=[z ₁ ,z ₂ ,r] (25)

By adjusting the parameters k ₁ , k ₂ and k _r , the symmetric matrix Λ can be made a positive definite matrix,

Then there are:

In formula (27), λ _min (Λ) is the minimum eigenvalue of the symmetric matrix Λ.

From equation (27) and Lyapunov stability theorem, it is concluded that the integral symbol robust adaptive controller designed for the uncertain nonlinearity of the motor servo system can make the system achieve asymptotic stability, and the parameter k of the controller can be adjusted. ₁ , k ₂ , and k _r can make the tracking accuracy approach zero. The schematic diagram of the error symbol integral robust adaptive control principle of the motor servo system is shown in Figure 2.

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

Example

单位rad。Simulation parameters: J _equ =0.00138kg·m ² , _Bequ =0.4N·m/rad, ku = _2.36N ·m/V. Take the controller parameters k ₁ =12, k ₂ =1.5, k _r =1, θ _1n =0.02, θ _2n =0.294, the selected nominal value of è is far from the true value of the parameter, so as to evaluate the performance of the adaptive control law. Effect. The PID controller parameters are k _p =90, k _i =70, k _d =0.3. The given position refers to the input signal

The unit is rad.

干扰(3)常值扰动和其他未建模干扰并存，且输入受限的情况下，d_n＝0.5N·m，

v＝u·0.8。Disturbance (1) Under the condition of only constant disturbance and d _n =0.5N·m. Disturbance (2) When constant disturbance and other unmodeled disturbances coexist, and d _n =0.5N·m,

Disturbance (3) Constant disturbance and other unmodeled disturbances coexist, and when the input is limited, d _n =0.5N·m,

v=u·0.8.

The effect of the control law is shown in Figure 3-Figure 9:

Fig. 3 is the tracking process diagram of the system output of the controller to the given output under the action of disturbance (1). FIG. 4 is a graph showing the variation of the tracking error of the system with time under the action of disturbance (1). Figure 5 is a graph of the tracking accuracy of PID control and ARISE control under the action of disturbance (2). FIG. 6 is a graph of disturbance (2) control input u. FIG. 7 is a graph of the tracking error of the system under the action of disturbance (3) as a function of time. FIG. 8 is a graph of the control input v under the action of disturbance (3). FIG. 9 is a graph of parameter adaptation under the action of disturbance (3).

It can be seen from the above that the algorithm proposed by the present invention can more accurately estimate the interference value in the simulation environment. Compared with the traditional PID control, the controller designed by the present invention can greatly improve the stability of the system with parameter uncertainty and large interference. control precision. The research results show that under the influence of uncertain nonlinearity and parameter uncertainty, the method proposed by the invention can meet the performance index.

Claims (2)

1. A robust self-adaptive anti-interference control method for an error symbol integral of a motor servo system is characterized by comprising the following steps:

step 1, establishing a motor position servo system model; the method specifically comprises the following steps:

according to Newton's second law, the dynamic model equation of the inertia load of the motor is as follows:

wherein y is the angular displacement, J_equIs an inertial load, k_uIs the torque constant, u is the system control input, B_equIs a coefficient of viscous friction, d_nIn order for the system to be subject to constant interference,

for other unmodeled interference;

writing equation (1) into a state space form, as follows:

wherein

x＝[x₁，x₂]^TA state vector representing position and velocity; parameter set theta ═ theta₁，θ₂，θ₃]^TWherein theta₁＝Je_qu/k_u，θ₂＝B_equ/k_u，θ₃＝d_n/k_u，

Representing other unmodeled disturbances in the system; the following assumptions hold:

assume that 1: the parameter theta satisfies:

wherein theta is_min＝[θ_1min，θ_2min，θ_3min]^T，θ_max＝[θ_1max，θ_2max，θ_3max]^TAre all known, and furthermore theta_1min＞0，θ_2min＞0，θ_3min＞0；

Assume 2:

is bounded and differentiable to the first order, i.e.

Wherein delta_dThe method comprises the following steps of (1) knowing;

step 2, designing an error symbol integral robust self-adaptive controller; the method specifically comprises the following steps:

step 2-1, definition of z₁＝x₁-x_1dFor angular displacement tracking error of the system, x_1dIs a position command that the system expects to track and that is continuously differentiable in the second order, according to the first equation in equation (2)

Selecting x₂For virtually controlling the quantities, let equation

Tends to a stable state; let x_2eqFor desired values of virtual control, x_2eqAnd the true state x₂Has an error of z₂＝x₂-x_2eqTo z is to₁And (5) obtaining a derivative:

designing a virtual control law:

in the formula (6), k₁If > 0 is adjustable gain, then

Due to z₁(s)＝G(s)z₂(s) wherein G(s) is 1/(s + k)₁) Is a stable transfer function when z₂When going to 0, z₁Also inevitably tends to 0;

step 2-2, introducing an auxiliary error signal r (t)

In the formula k₂The gain is adjustable when the value is more than 0;

according to equations (2), (7) and (8), there is an expansion of r as follows:

according to the formulae (2) and (9), the following formulae are given:

according to equation (10), the model-based controller is designed as:

in the formula (11), the reaction mixture is,

an estimated value representing the value of theta is,

in order to be able to estimate the error,

β is the system control gain, k_rIn order to gain in the positive feedback, the gain,

the gamma is a parameter self-adaptive rate, and is adjustable positive self-modulation rhythm gain when the gamma is more than 0;

from the parameter adaptation rate in the formula (11), r is an unknown quantity, but r is an unknown quantity

And its first derivative is known, integrating the adaptation rate yields:

calculated by substituting formula (11) for formula (10):

the derivation yields:

and 3, according to the error sign integral robust self-adaptive controller, carrying out stability verification on the motor servo system by utilizing the Lyapunov stability theory, and obtaining a global gradual stabilization result of the system by utilizing the Barbalt theorem.

2. The motor servo system error symbol integral robust adaptive anti-interference control method according to claim 1, wherein the step 3 specifically comprises:

defining auxiliary functions

z₂(0)、

Respectively represents z₂(t)、

An initial value of (1);

is proved to be when

When P (t) ≧ 0, the Lyapunov function is thus defined as follows:

for estimated errors, i.e.

The Lyapunov stability theory is used for stability verification, and the Barbalt theorem is used for obtaining the global gradual stable result of the system, so that the gain k is adjusted₁、k₂、k_rAnd Γ makes the tracking error of the system tend to zero under the condition of infinite time zone.

Images