CN106292279A

CN106292279A - Electric machine position servo systems by output feedback control method based on nonlinear observer

Info

Publication number: CN106292279A
Application number: CN201610698040.7A
Authority: CN
Inventors: 李旭东; 姚建勇
Original assignee: Nanjing University of Science and Technology
Current assignee: Nanjing University of Science and Technology
Priority date: 2016-08-20
Filing date: 2016-08-20
Publication date: 2017-01-04
Anticipated expiration: 2036-08-20
Also published as: CN106292279B

Abstract

The invention discloses an output feedback control method of a motor position servo system based on a nonlinear observer, which belongs to the field of motor servo control. The method includes the following steps: establishing a mathematical model of the motor position servo system; designing a nonlinear state observer, Design of a nonlinear observer-based output feedback controller for a motor position servo system. The present invention provides an output feedback control method of a motor position servo system based on a nonlinear observer when only the position state is known but the velocity acceleration is unknown, which reduces hardware costs and is beneficial to application in actual engineering; The controller designed by the invention fully considers the nonlinear friction characteristics and external disturbances of the system, and ensures that the system state tends to a balanced state within a limited time, which improves the tracking performance of the system.

Description

Output Feedback Control Method of Motor Position Servo System Based on Nonlinear Observer

技术领域technical field

本发明涉及电机位置伺服系统领域，特别是一种基于非线性观测器的电机位置伺服系统输出反馈控制方法。The invention relates to the field of a motor position servo system, in particular to an output feedback control method of a motor position servo system based on a nonlinear observer.

背景技术Background technique

电机伺服系统具有响应快、维护方便、传动效率高以及能源获取方便等突出优点，广泛应用于各个重要领域，如机器人、机床、航空航天等。随着这些领域的快速发展，对电机伺服系统跟踪性能的要求也越来越高，系统的性能与控制器的设计密切相关。电机伺服系统是一个典型的非线性系统，在设计控制器的过程中会面临许多建模不确定性包括结构不确定性以及非结构不确定性，这些因素可能会严重恶化期望的控制性能，导致不理想的控制精度，产生极限环振荡甚至使所设计的控制器不稳定，从而使控制器的设计变得困难。The motor servo system has outstanding advantages such as fast response, convenient maintenance, high transmission efficiency and convenient energy acquisition, and is widely used in various important fields, such as robots, machine tools, aerospace, etc. With the rapid development of these fields, the requirements for the tracking performance of the motor servo system are getting higher and higher, and the performance of the system is closely related to the design of the controller. The motor servo system is a typical nonlinear system. In the process of designing the controller, it will face many modeling uncertainties, including structural uncertainties and non-structural uncertainties. These factors may seriously deteriorate the desired control performance, resulting in Unsatisfactory control precision will produce limit cycle oscillation and even make the designed controller unstable, thus making the design of the controller difficult.

目前对于电机伺服系统的控制，基于经典三环控制的方法仍是工业及国防领域的主要方法，其以线性控制理论为基础，由内向外逐层设计电流环，速度环及位置环，各环的控制策略大都采用PID校正及其变型。但是随着工业及国防领域技术水平的不断进步，传统基于线性理论的三环控制方法已逐渐不能满足系统的高性能需求，成为限制电机伺服系统发展的瓶颈因素之一。为了提高电机系统的跟踪性能，许多先进的非线性控制器进行了研究，如鲁棒自适应控制，自适应鲁棒控制，滑模控制等等。然而，所有上述方法中使用的全状态反馈控制方法，也就是说，在运动控制中，除了需要位置信号，还需要速度和加速度信号。但对于许多应用中，由于降低成本的需要，仅位置信息可知。此外，严重的测量噪声通常会污染所测的速度和加速度信号，进而恶化实现性能的全状态反馈控制器。因此，迫切需要设计更为实用的非线性输出反馈控制策略。At present, for the control of the motor servo system, the method based on the classic three-loop control is still the main method in the field of industry and national defense. Based on the linear control theory, the current loop, speed loop and position loop are designed layer by layer from the inside to the outside. Most of the control strategies adopt PID correction and its variants. However, with the continuous improvement of the technical level in the industrial and national defense fields, the traditional three-loop control method based on linear theory has gradually been unable to meet the high performance requirements of the system, and has become one of the bottleneck factors restricting the development of motor servo systems. In order to improve the tracking performance of the motor system, many advanced nonlinear controllers have been studied, such as robust adaptive control, adaptive robust control, sliding mode control and so on. However, the full state feedback control method used in all the above methods, that is, in motion control, requires velocity and acceleration signals in addition to position signals. But for many applications, only the location information is known due to the need to reduce costs. In addition, severe measurement noise often contaminates the measured velocity and acceleration signals, thereby deteriorating the performance of full state feedback controllers. Therefore, it is urgent to design a more practical nonlinear output feedback control strategy.

发明内容Contents of the invention

本发明的目的在于提供一种基于非线性观测器的电机位置伺服系统的输出反馈控制方法。The purpose of the present invention is to provide an output feedback control method of a motor position servo system based on a nonlinear observer.

实现本发明目的的技术方案为：一种基于非线性观测器的电机位置伺服系统的输出反馈控制方法，包括以下步骤：The technical solution for realizing the object of the present invention is: a method for output feedback control of a motor position servo system based on a nonlinear observer, comprising the following steps:

步骤1，建立电机位置伺服系统的数学模型；Step 1, establish the mathematical model of the motor position servo system;

步骤2，设计非线性状态观测器；Step 2, designing a nonlinear state observer;

步骤3，设计电机位置伺服系统的基于非线性观测器的输出反馈控制器。Step 3, design an output feedback controller based on a nonlinear observer for the motor position servo system.

与现有技术相比，本发明的显著优点为：Compared with prior art, remarkable advantage of the present invention is:

(1)本发明在只有位置状态已知，而速度加速度未知的情况下，提供一种基于非线性观测器的电机位置伺服系统的输出反馈控制方法，减少硬件成本，更加有利于在实际工程中应用。(1) The present invention provides an output feedback control method of a motor position servo system based on a nonlinear observer when only the position state is known but the velocity acceleration is unknown, which reduces hardware costs and is more conducive to practical engineering application.

(2)本发明所设计的控制器，充分考虑了系统的非线性摩擦特性以及外干扰等，并且保证系统状态在有限时间内趋于平衡状态，提高了系统的跟踪性能。(2) The controller designed by the present invention fully considers the nonlinear friction characteristics and external disturbances of the system, and ensures that the system state tends to a balanced state within a limited time, which improves the tracking performance of the system.

附图说明Description of drawings

图1为本发明基于非线性观测器的电机位置伺服系统的输出反馈控制方法流程图。Fig. 1 is a flow chart of the output feedback control method of the motor position servo system based on the nonlinear observer in the present invention.

图2为本发明电机伺服系统示意图。Fig. 2 is a schematic diagram of the motor servo system of the present invention.

图3为两种控制器轨迹跟踪指令示意图。Fig. 3 is a schematic diagram of two kinds of controller trajectory tracking instructions.

图4为本发明所设计控制器作用下系统状态x₂的估计值随时间变化的曲线图。Fig. 4 is a graph showing the estimated value of the system state _x2 changing with time under the action of the controller designed in the present invention.

图5为两种控制器跟踪误差随时间变化的曲线图。Figure 5 is a graph showing the tracking error of two controllers as a function of time.

图6为本发明所设计的控制器其控制输入随时间变化的曲线图。Fig. 6 is a graph showing the change of control input with time of the controller designed in the present invention.

具体实施方式detailed description

结合图1、图2，本发明的一种基于非线性观测器的电机位置伺服系统的输出反馈控制方法，包括以下步骤：In conjunction with Fig. 1, Fig. 2, a kind of output feedback control method of the motor position servo system based on nonlinear observer of the present invention, comprises the following steps:

步骤一、建立电机位置伺服系统数学模型Step 1. Establish the mathematical model of the motor position servo system

根据牛顿第二定律且简化电机的电气动态为比例环节，电机位置伺服系统的运动方程为：According to Newton's second law and the electrical dynamics of the simplified motor is a proportional link, the motion equation of the motor position servo system is:

$m m \overset{\cdot\cdot \cdot\cdot}{y the y} = = {k k}_{f f} u u - - B B \overset{\cdot &Center Dot;}{y the y} - - {F f}_{f f} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + Δ Δ - - - - - - ((11))$

公式(1)中m为惯性负载参数，y为惯性负载位移，k_f为力矩放大系数，u为系统的控制输入，B为粘性摩擦系数，为可建模的非线性摩擦模型，为速度指令，y_d为位置指令，Δ为外干扰及未建模的摩擦等不确定性项。In the formula (1), m is the inertial load parameter, y is the inertial load displacement, _kf is the torque amplification factor, u is the control input of the system, B is the viscous friction coefficient, is a modelable nonlinear friction model, is the speed command, y _d is the position command, and Δ is the uncertainty items such as external disturbance and unmodeled friction.

选取连续静态摩擦模型为：The continuous static friction model is selected as:

${F f}_{f f n no d d} ((\overset{\cdot &Center Dot;}{y the y})) = = {l l}_{11} tanh tanh (({l l}_{22} {\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + {l l}_{33} [[tanh tanh (({l l}_{44} {\overset{\cdot &Center Dot;}{y the y}}_{d d})) - - tanh tanh (({l l}_{55} {\overset{\cdot &Center Dot;}{y the y}}_{d d}))]] - - - - - - ((22))$

公式(2)中l₁、l₂、l₃、l₄、l₅均为已知常数；此连续静态摩擦模型的特征如下：①此摩擦模型是关于时间连续可微并且关于原点对称的；②库伦摩擦特性可用表达式表征；③静态摩擦系数可用l₁+l₃的值来近似表示；④表达式可以表征Stribeck效应。In formula (2), l ₁ , l ₂ , l ₃ , l ₄ , and l ₅ are all known constants; the characteristics of this continuous static friction model are as follows: ① This friction model is continuously differentiable with respect to time and symmetrical about the origin; ②Coulomb friction characteristic available expression Characterization; ③The static friction coefficient can be approximated by the value of l ₁ +l ₃ ; ④Expression The Stribeck effect can be characterized.

选取状态变量为：x＝[x₁,x₂]^T，则电机位置伺服系统的运动学方程可以转化为状态方程形式：Select the state variable as: x=[x ₁ ,x ₂ ] ^T , then the kinematic equation of the motor position servo system can be transformed into the state equation form:

$\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {x x}_{22} - - \frac{B B}{m m} {x x}_{11} \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{k k}_{f f}}{m m} u u - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot \cdot}{y the y}}_{d d})) + + \frac{Δ Δ}{m m} \end{matrix} - - - - - - ((33))$

公式(3)中x₁表示惯性负载的位移，x₂表示另一个不可知的运动状态。In formula (3), x ₁ represents the displacement of the inertial load, and x ₂ represents another unknowable motion state.

步骤二、设计非线性状态观测器Step 2. Design a nonlinear state observer

对未知状态x₂进行估计，首先引入坐标转换体系，引入新状态ξ：To estimate the unknown state x ₂ , first introduce the coordinate transformation system and introduce a new state ξ:

ξ＝x₂-k₁x₁ (4)ξ=x ₂ -k ₁ x ₁ (4)

公式(4)中k₁为设计参数。然后对公式(4)左右两边同时微分，并联合公式(3)可得ξ的动态为：In formula (4), k ₁ is a design parameter. Then differentiate the left and right sides of formula (4) and combine formula (3) to obtain the dynamics of ξ as follows:

$\overset{\cdot &Center Dot;}{ξ ξ} = = - - {k k}_{11} ξ ξ + + \frac{{k k}_{f f}}{m m} u u + + {k k}_{11} \frac{B B}{m m} {x x}_{11} - - {k k}_{11}^{22} {x x}_{11} - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + \frac{Δ Δ}{m m} - - - - - - ((55))$

根据方程(5)，设计出状态观测器为：According to equation (5), the state observer is designed as:

$\overset{\cdot &Center Dot;}{\overset{^^}{ξ ξ}} = = - - {k k}_{11} \overset{^^}{ξ ξ} + + \frac{{k k}_{f f}}{m m} u u + + {k k}_{11} \frac{B B}{m m} {x x}_{11} - - {k k}_{11}^{22} {x x}_{11} - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) - - - - - - ((66))$

公式(6)中是状态ξ的估计值。In formula (6) is the estimated value of state ξ.

定义为状态观测器的估计误差，由公式(5)、(6)可得估计误差的动态方程为：definition is the estimation error of the state observer, and the dynamic equation of the estimation error can be obtained from formulas (5) and (6):

$\overset{\cdot &Center Dot;}{\overset{~ ~}{ξ ξ}} = = - - {k k}_{11} \overset{~ ~}{ξ ξ} - - \frac{Δ Δ}{m m} - - - - - - ((77))$

根据公式(7)可得：According to the formula (7), we can get:

$ξ ξ ((t t)) \leq \leq {e e}^{- - {k k}_{11} t t} ξ ξ ((00)) + + \frac{{δ δ}_{d d}}{k k} [[11 - - {e e}^{- - {k k}_{11} t t}]] - - - - - - ((88))$

公式(8)中δ_d为一未知常数。通过调整设计参数k₁可以使估计误差在有限时间内趋于很小的值，因此状态观测器有良好的稳态观测性能。δ _d in formula (8) is an unknown constant. By adjusting the design parameter k ₁ , the estimation error tends to a small value within a limited time, so the state observer has good steady-state observation performance.

步骤三、设计电机位置伺服系统的基于非线性观测器的输出反馈控制器，其具体步骤如下：Step 3. Design the output feedback controller based on the nonlinear observer of the motor position servo system. The specific steps are as follows:

控制器设计的目标是使电机位置伺服系统的位置输出x₁尽可能准确地跟踪期望跟踪的位置指令x_1d；The goal of the controller design is to make the position output x ₁ of the motor position servo system track the desired tracked position command x _1d as accurately as possible;

设计电机位置伺服系统的基于非线性观测器的输出反馈控制器如下：Design the nonlinear observer based output feedback controller of the motor position servo system as follows:

定义变量如下：Define variables as follows:

$\begin{matrix} {z z}_{11} = = {x x}_{11} - - {x x}_{11 d d} \\ {z z}_{22} = = \overset{^^}{ξ ξ} - - {α α}_{11} \end{matrix} - - - - - - ((99))$

其中α₁为虚拟控制量，设计如下：Among them, _α1 is the virtual control quantity, which is designed as follows:

$\begin{matrix} {α α}_{11} = = {α α}_{11 a a} + + {α α}_{11 s the s} \\ {α α}_{11 a a} = = {\overset{\cdot \cdot}{x x}}_{11 d d} + + ((\frac{B B}{m m} - - {k k}_{11})) {x x}_{11} \\ {α α}_{11 s the s} = = {α α}_{11 s the s 11} + + {α α}_{11 s the s 22} \\ {α α}_{11 s the s 11} = = - - {k k}_{s the s 11} {z z}_{11} \end{matrix} - - - - - - ((1010))$

公式(10)中k_s1为设计参数，为速度指令，α_1s2满足如下条件：k _s1 in formula (10) is the design parameter, As the speed command, α _1s2 satisfies the following conditions:

$\begin{matrix} {z z}_{11} (({α α}_{11 s the s 22} - - \overset{~ ~}{ξ ξ})) \leq \leq {ϵ ϵ}_{11} \\ {z z}_{11} {α α}_{11 s the s 22} \leq \leq 00 \end{matrix} - - - - - - ((1111))$

其中ε₁＞0是一个设计参数，在此给出满足(11)的α_1s2的一个形式where ε ₁ >0 is a design parameter, here a form of α _1s2 satisfying (11) is given

$\begin{matrix} {h h}_{11} &GreaterEqual; &Greater Equal; {δ δ}_{ξ ξ}^{22} \\ {α α}_{22 s the s 22} = = - - \frac{{h h}_{11}}{22 {ϵ ϵ}_{11}} {z z}_{11} \end{matrix} - - - - - - ((1212))$

其中δξ是的上界where δξ is upper bound of

基于非线性观测器的输出反馈控制器设计如下：The output feedback controller based on nonlinear observer is designed as follows:

$\begin{matrix} u u = = {u u}_{a a} + + {u u}_{s the s} \\ {u u}_{a a} = = - - \frac{m m}{{k k}_{f f}} {z z}_{11} + + \frac{m m}{{k k}_{f f}} {k k}_{11} \overset{^^}{ξ ξ} - - \frac{B B}{{k k}_{f f}} {k k}_{11} {x x}_{11} + + \frac{m m}{{k k}_{f f}} {k k}_{11}^{22} {x x}_{11} + + \frac{11}{{k k}_{f f}} {F f}_{f f n no d d} (({\overset{\cdot \cdot}{y the y}}_{d d})) + + \frac{m m}{{k k}_{f f}} {\overset{\cdot &Center Dot;}{α α}}_{11 c c} \\ {u u}_{s the s} = = {u u}_{s the s 11} + + {u u}_{s the s 22} \\ {u u}_{s the s 11} = = - - \frac{m m}{{k k}_{f f}} {k k}_{s the s 22} {z z}_{22} \end{matrix} - - - - - - ((1313))$

其中k_s2为设计参数，为α₁时间导数中可计算部分。u_s2满足如下条件where k _s2 is the design parameter, is the computable part _of the time derivative of α1. u _s2 satisfies the following conditions

$\begin{matrix} {z z}_{22} (({u u}_{s the s 22} + + \frac{\partial \partial {α α}_{11}}{\partial \partial {x x}_{11}} \overset{~ ~}{ξ ξ})) \leq \leq {ϵ ϵ}_{22} \\ {z z}_{22} {u u}_{s the s 22} \leq \leq 00 \end{matrix} - - - - - - ((1414))$

其中ε₂＞0是一个设计参数。在此给出满足(14)的α_1s2的一个形式where ε ₂ >0 is a design parameter. Here a form of α _1s2 satisfying (14) is given

$\begin{matrix} {h h}_{22} &GreaterEqual; &Greater Equal; | | \frac{\partial \partial {α α}_{11}}{\partial \partial {x x}_{11}} | | {δ δ}_{ξ ξ}^{22} \\ {u u}_{s the s 22} = = - - \frac{{h h}_{22}}{22 {ϵ ϵ}_{22}} {z z}_{22} \end{matrix} - - - - - - ((1515))$

分析电机位置伺服系统的稳定性：Analyze the stability of the motor position servo system:

根据控制理论中系统的稳定性分析，选取李亚普诺夫方程为：According to the stability analysis of the system in control theory, the Lyapunov equation is selected as:

${V V}_{11} = = \frac{11}{22} {z z}_{11}^{22} + + \frac{11}{22} {z z}_{22}^{22} - - - - - - ((1616))$

运用李亚普诺夫稳定性理论进行稳定性证明，对(16)式求导，并将公式(10)、(12)、(13)、(15)带入可得：Using Lyapunov's stability theory to prove the stability, deriving the formula (16), and bringing the formulas (10), (12), (13), and (15) into it, we can get:

${V V}_{22} ((t t)) \leq \leq {e e}^{- - λ λ t t} {V V}_{22} ((00)) + + \frac{{ϵ ϵ}_{11} + + {ϵ ϵ}_{22}}{λ λ} [[11 - - {e e}^{- - λ λ t t}]] - - - - - - ((1717))$

其中λ是一个正实数，从而可以使系统达到渐进稳定。Where λ is a positive real number, so that the system can be asymptotically stable.

下面结合具体实施例对本发明作进一步说明。The present invention will be further described below in conjunction with specific examples.

实施例Example

电机伺服系统的参数取值如下：The parameter values of the motor servo system are as follows:

m＝0.02kg·m²,k_f＝5N·m·V^-1,B＝10N·m·rad-1·s^-1,l₁＝0.1N·m,l₂＝700s·rad^-1,l₃＝0.06N·m,l₄＝15s·rad^-1,l₅＝1.5s·rad^-1 m=0.02kg·m ² , k _f =5N·m·V ^-1 , B=10N·m·rad-1·s ^-1 , l ₁ =0.1N·m, l ₂ =700s·rad ^-1 , l ₃ =0.06N·m, l ₄ =15s·rad ^-1 , l ₅ =1.5s·rad ^-1

本发明的控制器参数k₁＝800,k_s1＝2000,k_s2＝500。The controller parameters of the present invention are k ₁ =800, k _s1 =2000, and k _s2 =500.

PID控制器参数为k_p＝1000,k_i＝50,k_d＝0.1。The parameters of the PID controller are k _p =1000, _ki =50, and k _d =0.1.

位置角度输入信号 Position angle input signal

图3是两种控制器轨迹跟踪指令示意图。Fig. 3 is a schematic diagram of two controller trajectory tracking instructions.

图4是本发明所设计控制器作用下系统状态x₂的估计值随时间变化的曲线，从图中可以看出其估计值渐渐接近于名义值，并在名义值附近一定范围内波动，从而能够准确地将系统的状态估计出来。Fig. 4 is the curve of the estimated value of the system state _x2 changing with time under the action of the designed controller of the present invention, as can be seen from the figure, its estimated value gradually approaches the nominal value, and fluctuates within a certain range near the nominal value, thereby It can accurately estimate the state of the system.

图5是两种控制器跟踪误差随时间变化的曲线，可以看出本发明所设计控制器明显优于PID控制器。Fig. 5 is the curve of the tracking error of the two controllers changing with time, it can be seen that the controller designed by the present invention is obviously better than the PID controller.

图6是本发明所设计的控制器其控制输入随时间变化的曲线，从图中可以看出，本发明所得到的控制输入信号连续，利于在工程实际中应用。Fig. 6 is the time-varying curve of the control input of the controller designed by the present invention. It can be seen from the figure that the control input signal obtained by the present invention is continuous, which is beneficial to the application in engineering practice.

Claims

1. a kind of output feedback control method based on the motor position servo system of nonlinear observer, is characterized in that, comprises the following steps:

Step 1, establish the mathematical model of the motor position servo system;

Step 2, designing a nonlinear state observer;

Step 3, design an output feedback controller based on a nonlinear observer for the motor position servo system.

2. according to the output feedback control method of the motor position servo system based on nonlinear observer according to claim 1, it is characterized in that, step 1 is specifically:

According to Newton's second law and the electrical dynamics of the simplified motor is a proportional link, the motion equation of the motor position servo system is:

m m \overset{\cdot\cdot \cdot\cdot}{y the y} = = {k k}_{f f} u u - - B B \overset{\cdot &Center Dot;}{y the y} - - {F f}_{f f} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + Δ Δ - - - - - - ((11))

In the formula (1), m is the inertial load parameter, y is the inertial load displacement, _kf is the torque amplification factor, u is the control input of the system, B is the viscous friction coefficient, is a modelable nonlinear friction model, is the speed command, y _d is the position command, Δ is the uncertainty items such as external disturbance and unmodeled friction;

The continuous static friction model is selected as:

{F f}_{f f n no d d} ((\overset{\cdot &Center Dot;}{y the y})) = = {l l}_{11} tanh tanh (({l l}_{22} {\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + {l l}_{33} [[tanh tanh (({l l}_{44} {\overset{\cdot &Center Dot;}{y the y}}_{d d})) - - tanh tanh (({l l}_{55} {\overset{\cdot &Center Dot;}{y the y}}_{d d}))]] - - - - - - ((22))

In formula (2), l ₁ , l ₂ , l ₃ , l ₄ , and l ₅ are all known constants. The characteristics of this continuous static friction model are as follows: ① This friction model is continuously differentiable with respect to time and symmetrical about the origin; ② Coulomb friction characteristics through the expression Characterization; ③The static friction coefficient is approximated by the value of l ₁ +l ₃ ; ④Expression Can characterize the Stribeck effect;

Select the state variable as: x=[x ₁ ,x ₂ ] ^T , then the kinematic equation of the motor position servo system can be transformed into the state equation form:

\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {x x}_{22} - - \frac{B B}{m m} {x x}_{11} \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{{k k}_{f f}}{m m} u u - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + \frac{Δ Δ}{m m} \end{matrix} - - - - - - ((33))

In formula (3), x ₁ represents the displacement of the inertial load, and x ₂ represents another unknowable motion state.

3. according to the output feedback control method of the motor position servo system based on nonlinear observer according to claim 1, it is characterized in that, step 2 is specially:

Design a nonlinear state observer to estimate the unknown state x ₂ , firstly introduce the coordinate transformation system and introduce a new state ξ:

ξ=x ₂ -k ₁ x ₁ (4)

In the formula (4), k ₁ is the design parameter, and then differentiate the left and right sides of the formula (4) simultaneously, and combine the formula (3) to obtain the dynamics of ξ as follows:

\overset{\cdot &Center Dot;}{ξ ξ} = = - - {k k}_{11} ξ ξ + + \frac{{k k}_{f f}}{m m} u u + + {k k}_{11} \frac{B B}{m m} {x x}_{11} - - {k k}_{11}^{22} {x x}_{11} - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + \frac{Δ Δ}{m m} - - - - - - ((55))

According to equation (5), the state observer is designed as:

\overset{\cdot &Center Dot;}{\overset{^^}{ξ ξ}} = = - - {k k}_{11} \overset{^^}{ξ ξ} + + \frac{{k k}_{f f}}{m m} u u + + {k k}_{11} \frac{B B}{m m} {x x}_{11} - - {k k}_{11}^{22} {x x}_{11} - - \frac{11}{m m} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) - - - - - - ((66))

In formula (6) is the estimated value of state ξ;

definition is the estimation error of the state observer, and the dynamic equation of the estimation error can be obtained from formulas (5) and (6):

\overset{\cdot &Center Dot;}{\overset{~ ~}{ξ ξ}} = = - - {k k}_{11} \overset{~ ~}{ξ ξ} - - \frac{Δ Δ}{m m} - - - - - - ((77))

According to the formula (7), we can get:

ξ ξ ((t t)) \leq \leq {e e}^{- - {k k}_{11} t t} ξ ξ ((00)) + + \frac{{δ δ}_{d d}}{k k} [[11 - - {e e}^{- - {k k}_{11} t t}]] - - - - - - ((88))

δ _d in formula (8) is an unknown constant; by adjusting the design parameter k ₁ , the estimation error can tend to a small value within a limited time, so the state observer has good steady-state observation performance.

4. the output feedback control method of the motor position servo system based on nonlinear observer according to claim 1, is characterized in that, step 3 is specifically:

Define variables as follows:

\begin{matrix} {z z}_{11} = = {x x}_{11} - - {x x}_{11 d d} \\ {z z}_{22} = = \overset{^^}{ξ ξ} - - {α α}_{11} \end{matrix} - - - - - - ((99))

Among them, x _1d is the desired tracking position command, α ₁ is the virtual control value, and the design is as follows:

\begin{matrix} {α α}_{11} = = {α α}_{11 a a} + + {α α}_{11 s the s} \\ {α α}_{11 a a} = = {\overset{\cdot \cdot}{x x}}_{11 d d} + + ((\frac{B B}{m m} - - {k k}_{11})) {x x}_{11} \\ {α α}_{11 s the s} = = {α α}_{11 s the s 11} + + {α α}_{11 s the s 22} \\ {α α}_{11 s the s 11} = = - - {k k}_{s the s 11} {z z}_{11} \end{matrix} - - - - - - ((1010))

k _s1 in formula (10) is the design parameter, As the speed command, α _1s2 satisfies the following conditions:

\begin{matrix} {z z}_{11} (({α α}_{11 s the s 22} - - \overset{~ ~}{ξ ξ})) \leq \leq {ϵ ϵ}_{11} \\ {z z}_{11} {α α}_{11 s the s 22} \leq \leq 00 \end{matrix} - - - - - - ((1111))

where ε ₁ >0 is a design parameter, here a form of α _1s2 satisfying (11) is given

\begin{matrix} {h h}_{11} &GreaterEqual; &Greater Equal; {δ δ}_{ξ ξ}^{22} \\ {α α}_{22 s the s 22} = = - - \frac{{h h}_{11}}{22 {ϵ ϵ}_{11}} {z z}_{11} \end{matrix} - - - - - - ((1212))

where δ _ξ is upper bound of

The output feedback controller based on nonlinear observer is designed as follows:

\begin{matrix} u u = = {u u}_{a a} + + {u u}_{s the s} \\ {u u}_{a a} = = - - \frac{m m}{{k k}_{f f}} {z z}_{11} + + \frac{m m}{{k k}_{f f}} {k k}_{11} \overset{^^}{ξ ξ} - - \frac{B B}{{k k}_{f f}} {k k}_{11} {x x}_{11} + + \frac{m m}{{k k}_{f f}} {k k}_{11}^{22} {x x}_{11} + + \frac{11}{{k k}_{f f}} {F f}_{f f n no d d} (({\overset{\cdot &Center Dot;}{y the y}}_{d d})) + + \frac{m m}{{k k}_{f f}} {\overset{\cdot &Center Dot;}{α α}}_{11 c c} \\ {u u}_{s the s} = = {u u}_{s the s 11} + + {u u}_{s the s 22} \\ {u u}_{s the s 11} = = - - \frac{m m}{{k k}_{f f}} {k k}_{s the s 22} {z z}_{22} \end{matrix} - - - - - - ((1313))

where k _s2 is the design parameter, is the computable part of the time derivative of α ₁ ; u _s2 satisfies the following conditions

\begin{matrix} {z z}_{22} (({u u}_{s the s 22} + + \frac{\partial \partial {α α}_{11}}{\partial \partial {x x}_{11}} \overset{~ ~}{ξ ξ})) \leq \leq {ϵ ϵ}_{22} \\ {z z}_{22} {u u}_{s the s 22} \leq \leq 00 \end{matrix} - - - - - - ((1414))

where ε ₂ >0 is a design parameter; here a form of α _1s2 satisfying (14) is given

\begin{matrix} {h h}_{22} &GreaterEqual; &Greater Equal; | | \frac{\partial \partial {α α}_{11}}{\partial \partial {x x}_{11}} | | {δ δ}_{ξ ξ}^{22} \\ {u u}_{s the s 22} = = - - \frac{{h h}_{22}}{22 {ϵ ϵ}_{22}} {z z}_{22} \end{matrix} . . - - - - - - ((1515))