CN104834220A

CN104834220A - Adaptive error symbol integration robust repetitive control method for electromechanical servo system

Info

Publication number: CN104834220A
Application number: CN201510261456.8A
Authority: CN
Inventors: 姚建勇; 邓文翔; 刘龙
Original assignee: Nanjing University of Science and Technology
Current assignee: Nanjing University of Science and Technology
Priority date: 2015-05-20
Filing date: 2015-05-20
Publication date: 2015-08-12

Abstract

The invention discloses an adaptive error sign integral robust repetitive control method of an electromechanical servo system, comprising the following steps: establishing a mathematical model of the electromechanical servo system; constructing a system design model; designing an adaptive error sign integral robust repetitive controller; Performance theorems and stability proofs for adaptive error sign-integral robust repetitive controllers. The control method of the present invention can enable the electromechanical servo system to obtain semi-global asymptotic tracking performance, and compared with the traditional linear repetitive control method based on the internal model principle, it can also effectively reduce the sensitivity of the repetitive controller to noise and avoid control high memory requirements of the server.

Description

An Adaptive Error Sign Integral Robust Repetitive Control Method for Electromechanical Servo Systems

技术领域technical field

本发明涉及机电伺服控制技术领域，主要涉及一种机电伺服系统的自适应误差符号积分鲁棒重复控制方法。The invention relates to the technical field of electromechanical servo control, and mainly relates to an adaptive error sign integral robust repetitive control method of an electromechanical servo system.

背景技术Background technique

机电伺服系统具有响应快、传动效率高及能源获取方便等突出优点，广泛应用于众多重要领域。但是随着工业技术水平的提高，对于机电伺服系统的性能要求也越来愈高。机电伺服系统的跟踪性能依赖于控制器的设计，然而系统存在的诸多建模不确定性使得高性能控制器的设计变得困难。在实际工程中，绝大多数机电伺服系统都是在周而复始的执行任务，例如电脑磁盘驱动器、旋转式车床及机械手等。针对执行周期性任务的机电伺服系统的控制问题，传统基于内模原理的线性重复控制是一种易于执行且不依赖于系统动态模型信息的方法，且在只存在周期性建模不确定性的情况下可获得渐近跟踪的性能。传统的重复控制的核心思想是通过基于上一个周期的跟踪误差调整系统控制输入的值以实现逐个周期地提升跟踪性能。但是传统的重复控制存在以下问题：首先，传统的重复控制等价于在一个周期内自动更新周期性建模不确定性的所有值，相当于有无限多个数值需要更新，这就要求控制律的带宽很高，从而也就对实施该控制律的微处理器的内存提出了非常高的要求；其次，由于在传统重复控制律执行的时候，一个周期内的建模不确定性的每个值都是相互独立的，这与离散的随机噪声类似，因此传统的重复控制方法对噪声非常敏感，进而会影响系统的跟踪性能。而误差符号积分鲁棒(RISE)控制方法可以处理任意二阶连续可微的建模不确定性，且基于建模不确定性各阶导数的界已知的假设，可获得渐近跟踪的性能。The electromechanical servo system has outstanding advantages such as fast response, high transmission efficiency and convenient energy acquisition, and is widely used in many important fields. However, with the improvement of the industrial technology level, the performance requirements of the electromechanical servo system are also getting higher and higher. The tracking performance of the electromechanical servo system depends on the design of the controller, however, many modeling uncertainties in the system make it difficult to design a high-performance controller. In actual engineering, most electromechanical servo systems perform tasks repeatedly, such as computer disk drives, rotary lathes, and manipulators. For the control problems of electromechanical servo systems that perform periodic tasks, the traditional linear repetitive control based on the internal model principle is a method that is easy to implement and does not depend on the information of the system dynamic model. Asymptotic tracking performance can be obtained in this case. The core idea of traditional repetitive control is to improve the tracking performance cycle by cycle by adjusting the value of the system control input based on the tracking error of the previous cycle. But the traditional repetitive control has the following problems: First, the traditional repetitive control is equivalent to automatically updating all the values of the periodic modeling uncertainty in one period, which is equivalent to infinitely many values need to be updated, which requires the control law The bandwidth of the control law is very high, so it puts forward very high requirements on the memory of the microprocessor implementing the control law; secondly, when the traditional repetitive control law is executed, each of the modeling uncertainties in one cycle The values are independent of each other, which is similar to discrete random noise, so the traditional repetitive control method is very sensitive to noise, which will affect the tracking performance of the system. The error-sign-integral robust (RISE) control method can deal with any second-order continuous differentiable modeling uncertainty, and based on the assumption that the bounds of each order derivative of the modeling uncertainty are known, the asymptotic tracking performance can be obtained .

发明内容Contents of the invention

本发明的目的在于提供一种噪声敏感性弱、内存需求低及跟踪性能高的机电伺服系统的自适应误差符号积分鲁棒重复控制方法。The object of the present invention is to provide an adaptive error sign integral robust repetitive control method for an electromechanical servo system with weak noise sensitivity, low memory requirement and high tracking performance.

实现本发明目的的技术解决方案为：一种机电伺服系统的自适应误差符号积分鲁棒重复控制方法，包括以下步骤：The technical solution to realize the object of the present invention is: a kind of adaptive error sign integral robust repetitive control method of electromechanical servo system, comprising the following steps:

步骤1，建立机电伺服系统的数学模型；Step 1, establishing a mathematical model of the electromechanical servo system;

步骤2，构建系统设计模型；Step 2, build the system design model;

步骤3，设计自适应误差符号积分鲁棒重复控制器。Step 3, design an adaptive error sign-integral robust repetitive controller.

本发明与现有技术相比，其显著优点是：有效地降低传统基于内模原理的重复控制器对噪声的敏感性以及规避控制器的高内存需求，并获得半全局渐近跟踪的优异性能。仿真结果验证了其有效性。Compared with the prior art, the present invention has the remarkable advantages of effectively reducing the sensitivity of the traditional repetitive controller based on the internal model principle to noise and avoiding the high memory requirement of the controller, and obtaining excellent performance of semi-global asymptotic tracking . Simulation results verify its effectiveness.

附图说明Description of drawings

图1是本发明机电伺服系统的原理图；Fig. 1 is the schematic diagram of electromechanical servo system of the present invention;

图2是机电伺服系统自适应误差符号积分鲁棒重复控制(ARIPC)方法原理示意图；Fig. 2 is a schematic diagram of the principle of the adaptive error sign integral robust repetitive control (ARIPC) method of the electromechanical servo system;

图3是ARIPC控制器作用下系统实际输出对期望指令的跟踪过程；Figure 3 is the tracking process of the actual output of the system to the expected command under the action of the ARIPC controller;

图4是ARIPC控制器作用下系统的跟踪误差随时间变化的曲线；Fig. 4 is the curve of the system tracking error changing with time under the action of ARIPC controller;

图5是反馈线性化控制器(FLC)控制器、误差符号积分鲁棒控制器(RISE)和自适应误差符号积分鲁棒重复控制器(ARIPC)三种控制器分别作用下系统的跟踪误差对比曲线；Fig. 5 is a comparison of the tracking error of the system under the action of the feedback linearization controller (FLC) controller, error sign integral robust controller (RISE) and adaptive error sign integral robust repetitive controller (ARIPC) respectively curve;

图6是ARIPC控制器作用下参数估计随时间变化的曲线；Fig. 6 is the curve of parameter estimation changing with time under the action of ARIPC controller;

图7是ARIPC控制器作用下系统的控制输入随时间变化的曲线。Figure 7 is the curve of the control input of the system changing with time under the action of the ARIPC controller.

具体实施方式Detailed ways

下面结合附图及具体实施例对本发明作进一步详细说明。The present invention will be described in further detail below in conjunction with the accompanying drawings and specific embodiments.

结合图1～2本发明机电伺服系统的自适应误差符号积分鲁棒重复控制方法，包括以下步骤：In conjunction with Figs. 1-2, the adaptive error sign integral robust repetitive control method of the electromechanical servo system of the present invention comprises the following steps:

(1.1)本发明所考虑的机电伺服系统是通过配有商业电气驱动器的永磁直流电机直接驱动惯性负载，其原理图如图1所示。考虑到电磁时间常数比机械时间常数小得多，且电流环速度远大于速度环和位置环的响应速度，故可将电流环近似为比例环节。(1.1) The electromechanical servo system considered in the present invention directly drives the inertial load through a permanent magnet DC motor equipped with a commercial electric drive, and its schematic diagram is shown in Fig. 1 . Considering that the electromagnetic time constant is much smaller than the mechanical time constant, and the speed of the current loop is much faster than the response speed of the speed loop and the position loop, the current loop can be approximated as a proportional link.

因此，根据牛顿第二定律，机电伺服系统的运动方程为：Therefore, according to Newton's second law, the equation of motion of the electromechanical servo system is:

$M m \overset{\cdot &Center Dot; \cdot \cdot}{y the y} = = u u - - B B \overset{\cdot &Center Dot;}{y the y} - - {A A}_{f f} {S S}_{f f} ((\overset{\cdot \cdot}{y the y})) + + d d ((y the y,, \overset{\cdot &Center Dot;}{y the y},, t t)) - - - - - - ((11))$

式(1)中M为惯性负载参数，B为粘性摩擦系数，A_f为库伦摩擦的幅值，为已知的表征库伦摩擦的形状函数，是系统的建模不确定性，包含其他非线性摩擦、外干扰、未建模动态等，y为惯性负载的位移，u为系统的控制输入，t为时间变量。In formula (1), M is the inertial load parameter, B is the viscous friction coefficient, A _f is the amplitude of Coulomb friction, is a known shape function characterizing Coulomb friction, is the modeling uncertainty of the system, including other nonlinear frictions, external disturbances, unmodeled dynamics, etc., y is the displacement of the inertial load, u is the control input of the system, and t is the time variable.

(1.2)定义状态变量：则式(1)运动方程可写成状态方程：(1.2) Define state variables: Then equation (1) motion equation can be written as state equation:

$\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = {x x}_{22} \\ M m {\overset{\cdot \cdot}{x x}}_{22} = = u u - - B B {x x}_{22} - - {A A}_{f f} {S S}_{f f} (({x x}_{22})) - - {d d}_{11} (({x x}_{11},, {x x}_{22})) - - {d d}_{22} ((t t)) \\ y the y = = {x x}_{11} \end{matrix} - - - - - - ((22))$

式(2)中假设系统总的建模不确定性可分成两部分，即只与系统状态相关的d₁(x₁,x₂)和时变的d₂(t)。对于机械系统来说，建模不确定性绝大部分都只与系统状态相关，因此d₁(x₁,x₂)是主要的建模不确定性。In formula (2), it is assumed that the total modeling uncertainty of the system can be divided into two parts, namely d ₁ (x ₁ , x ₂ ) which is only related to the system state and time-varying d ₂ (t). For mechanical systems, most of the modeling uncertainties are only related to the system state, so d ₁ (x ₁ ,x ₂ ) is the main modeling uncertainty.

系统控制器的设计目标为：给定系统参考信号y_d(t)＝x_1d(t)，设计一个有界的连续的控制输入u使系统输出y＝x₁尽可能地跟踪系统的参考信号。The design goal of the system controller is: given the system reference signal y _d (t) = x _1d (t), design a bounded continuous control input u to make the system output y = x ₁ track the system reference signal as much as possible .

步骤2，构建系统设计模型，步骤如下：Step 2, build the system design model, the steps are as follows:

(2.1)尽管建模不确定性d₁(x₁,x₂)是未知的，但是对于执行周期性任务的机电伺服系统，其建模不确定性在一定时间以后也会呈现出相同的周期性，因此可利用重复控制的方法处理此类周期性建模不确定性，而对于非周期性的建模不确定性d₂(t)可设计鲁棒控制器以抑制其对跟踪性能的影响。因此式(2)可以写成如下形式(2.1) Although the modeling uncertainty d ₁ (x ₁ ,x ₂ ) is unknown, for an electromechanical servo system performing periodic tasks, its modeling uncertainty will show the same period after a certain time Therefore, the repetitive control method can be used to deal with such periodic modeling uncertainties, and for the non-periodic modeling uncertainty d ₂ (t), a robust controller can be designed to suppress its impact on tracking performance . Therefore, formula (2) can be written in the following form

$\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = {x x}_{22} \\ M m {\overset{\cdot \cdot}{x x}}_{22} = = u u - - B B {x x}_{22} - - {A A}_{f f} {S S}_{f f} (({x x}_{22})) + + {d d}_{11} (({x x}_{11 d d},, {\overset{\cdot \cdot}{x x}}_{11 d d})) + + {Δ Δ}_{11} + + {d d}_{22} ((t t)) \end{matrix} - - - - - - ((33))$

式(3)中 $Δ_{1} = d_{1} (x_{1}, x_{2}) - d_{1} (x_{1 d}, {\overset{\cdot}{x}}_{1 d}) .$ In formula (3) $Δ_{1} = d_{1} (x_{1}, x_{2}) - d_{1} (x_{1 d}, {\overset{&Center Dot;}{x}}_{1 d}) .$

(2.2)由于所考虑的机电伺服系统执行的是周期性的任务，因此期望跟踪的位置指令x_1d(t)是周期性的，即(2.2) Since the considered electromechanical servo system performs periodic tasks, it is expected that the tracked position command x _1d (t) is periodic, namely

x_1d(t-T)＝x_1d(t) (4)x _1d (tT) = x _1d (t) (4)

其中T是已知的周期。注意到只与x_1d和有关，因此也是周期性的，在如下的设计过程中令因此有where T is a known period. noticed only with x _1d and Related, and therefore periodic, in the following design process let Therefore there are

D_d(t-T)＝D_d(t) (5)D _d (tT) = D _d (t) (5)

对D_d(t)利用傅立叶级数展开得Using Fourier series expansion for D _d (t), we get

${D D.}_{d d} ((t t)) = = \frac{{a a}_{00}}{22} + + {Σ Σ}_{n no = = 11}^{\infty \infty} (({a a}_{n no} cos cos nωt nωt + + {b b}_{n no} sin sin nωt nωt)) - - - - - - ((66))$

式(6)中ω＝2π/T。考虑到机械系统的传递函数在物理意义上等价为一个具有有限频宽的低通滤波器，因此D_d(t)可以用式(6)中的有限项进行近似，即In formula (6), ω=2π/T. Considering that the transfer function of the mechanical system is physically equivalent to a low-pass filter with finite bandwidth, so D _d (t) can be approximated by the finite term in formula (6), namely

${D D.}_{d d} ((t t)) = = \frac{{a a}_{00}}{22} + + {Σ Σ}_{n no = = 11}^{m m} (({a a}_{n no} cos cos nωt nωt + + {b b}_{n no} sin sin nωt nωt)),, m m < < \infty \infty - - - - - - ((77))$

(2.3)为简化系统方程，定义未知常值参数矢量(2.3) To simplify the system equation, define the unknown constant parameter vector

θ＝[a₁,b₁,...,a_m,b_m]^T (8)θ=[a ₁ ,b ₁ ,...,a _m ,b _m ] ^T (8)

根据式(7)和(8)，系统的模型(3)可写成According to equations (7) and (8), the system model (3) can be written as

式(9)中D₂(t)＝a₀/2+d₂(t)。且M、B和A_f为系统物理参数已知的名义值，可用于控制器的设计，参数名义值与其真值之间的偏差可归并到系统建模不确定性D₂(t)中。In formula (9) D ₂ (t)=a ₀ /2+d ₂ (t). And M, B and A _f are the known nominal values of the physical parameters of the system, which can be used in the design of the controller, and the deviation between the nominal values of the parameters and their true values can be included in the system modeling uncertainty D ₂ (t).

为便于控制器设计，假设如下：For the convenience of controller design, the assumptions are as follows:

假设1：系统参考指令信号x_1d(t)是二阶连续可微的，且其各阶导数有界。Assumption 1: The system reference command signal x _1d (t) is second-order continuous differentiable, and its derivatives of each order are bounded.

假设2：建模不确定性d₁(x₁,x₂)和D₂(t)都是二阶连续可微的，且满足如下条件：Assumption 2: The modeling uncertainties d ₁ (x ₁ ,x ₂ ) and D ₂ (t) are both second-order continuous differentiable and satisfy the following conditions:

$| | {φ φ}_{11} (({x x}_{11},, {x x}_{22})) | | = = | | \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{11}} | | \leq \leq {ϵ ϵ}_{11},, | | {φ φ}_{22} (({x x}_{11},, {x x}_{22})) | | = = | | \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{22}} | | \leq \leq {ϵ ϵ}_{22} - - - - - - ((1010))$

$| | {D D.}_{22} ((t t)) | | \leq \leq {δ δ}_{00},, | | {\overset{\cdot &Center Dot;}{D D.}}_{22} ((t t)) | | \leq \leq {δ δ}_{11},, | | {\overset{\cdot &Center Dot; \cdot &Center Dot;}{D D.}}_{22} ((t t)) | | \leq \leq {δ δ}_{11} - - - - - - ((1111))$

式(10)和(11)中ε₁,ε₂,δ₀是未知的正数，δ₁,δ₂是已知正数。In formulas (10) and (11), ε ₁ , ε ₂ , δ ₀ are unknown positive numbers, and δ ₁ , δ ₂ are known positive numbers.

步骤3，设计自适应误差符号积分鲁棒重复控制器，步骤如下：Step 3, design an adaptive error sign-integral robust repetitive controller, the steps are as follows:

(3.1)定义z₁＝x₁-x_1d为系统的跟踪误差，根据式(9)中的第一个方程选取x₂为虚拟控制，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求导可得：(3.1) Define z ₁ = x ₁ -x _1d as the tracking error of the system, according to the first equation in formula (9) Picking _x2 as the dummy control makes the equation tends to a stable state; let x _2eq be the expected value of the virtual control, the error between x _2eq and the real state x ₂ is z ₂ = x ₂ -x _2eq , and the derivative of z ₁ can be obtained:

${\overset{\cdot &Center Dot;}{z z}}_{11} = = {x x}_{22} - - {\overset{\cdot &Center Dot;}{x x}}_{11 d d} = = {z z}_{22} + + {x x}_{22 eq eq} - - {\overset{\cdot &Center Dot;}{x x}}_{11 d d} - - - - - - ((1212))$

设计虚拟控制律：Design a virtual control law:

${x x}_{22 eq eq} = = {\overset{\cdot &Center Dot;}{x x}}_{11 d d} - - {k k}_{11} {z z}_{11} - - - - - - ((1313))$

式中k₁＞0为可调增益，则Where k ₁ >0 is the adjustable gain, then

${\overset{\cdot \cdot}{z z}}_{11} = = {z z}_{22} - - {k k}_{11} {z z}_{11} - - - - - - ((1414))$

由于z₁(s)＝G(s)z₂(s),式中G(s)＝1/(s+k₁)是一个稳定的传递函数，当z₂趋于0时，z₁也必然趋于0。所以在接下来的设计中，将以使z₂趋于0为主要设计目标。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. Therefore, in the following design, the main design goal will be to make z ₂ tend to 0.

(3.2)为获得一个额外的控制器设计自由度，定义一个辅助的误差信号r(t)：(3.2) In order to obtain an additional degree of freedom in controller design, define an auxiliary error signal r(t):

$r r = = {\overset{\cdot &Center Dot;}{z z}}_{22} + + {k k}_{22} {z z}_{22} - - - - - - ((1515))$

式(15)中k₂＞0为可调的增益。由于r(t)中含有位置的加速度信号，因此在实际中认为是不可测量的，即r(t)仅为辅助设计所用，并不具体出现在所设计的控制器中。In formula (15), k ₂ >0 is an adjustable gain. Because r(t) contains the acceleration signal of the position, it is considered unmeasurable in practice, that is, r(t) is only used for auxiliary design and does not specifically appear in the designed controller.

根据式(9)和(15)可得According to formulas (9) and (15), we can get

基于式(16)，可设计控制器如下Based on formula (16), the controller can be designed as follows

式(17)中k_r为正的反馈增益。u_a为用于提高系统跟踪精度的基于模型的补偿项，u_s为鲁棒控制律且其中u_s1为用于稳定系统名义模型的线性鲁棒反馈项，u_s2为非线性鲁棒项用于克服建模不确定性对系统性能的影响。为参数θ的估计值，定义为参数估计误差。In formula (17), k _r is a positive feedback gain. u _a is a model-based compensation item used to improve system tracking accuracy, u _s is a robust control law and where u _s1 is a linear robust feedback item for stabilizing the system nominal model, u _s2 is a nonlinear robust item for To overcome the impact of modeling uncertainty on system performance. is the estimated value of the parameter θ, define is the parameter estimation error.

(3.3)设计参数自适应律为：(3.3) The design parameter adaptive law is:

式(18)中Γ为对角正定矩阵，表征参数自适应率。由于式(18)中的参数自适应律含有不可测的信号r(t)，因此对其采用分部积分展开如下In formula (18), Γ is a diagonal positive definite matrix, which represents the parameter adaptive rate. Since the parameter adaptive law in formula (18) contains an unmeasurable signal r(t), it is expanded by integral by parts as follows

将式(17)代入式(16)中得：Substitute formula (17) into formula (16):

根据误差符号积分鲁棒控制器设计方法，积分鲁棒项u_s2可设计为：According to the error sign integral robust controller design method, the integral robust term u _s2 can be designed as:

${u u}_{s the s 22} = = - - {&Integral; &Integral;}_{00}^{t t} [[{k k}_{r r} {k k}_{22} {z z}_{22} + + βsign β sign (({z z}_{22}))]] dτ dτ - - - - - - ((21 twenty one))$

对式(20)求导并结合式(18)和(21)可得Deriving formula (20) and combining formulas (18) and (21), we can get

本例中，还对前述设计的控制器进行性能及稳定性证明，具体如下：In this example, the performance and stability of the previously designed controller are also verified, as follows:

控制器性能：使用参数自适应律(19)，控制器反馈增益k₁,k₂,k_r取得足够大以使如下定义的矩阵Λ为正定矩阵：Controller performance: Using the parameter adaptive law (19), the controller feedback gains k ₁ , k ₂ , k _r are made sufficiently large so that the matrix Λ defined as follows is a positive definite matrix:

$Λ Λ = = [\begin{matrix} {k k}_{11} & - - 11 / / 22 & 00 \\ - - 11 / / 22 & {k k}_{22} & - - 11 / / 22 \\ 00 & - - 11 / / 22 & {k k}_{33} \end{matrix}] - - - - - - ((23 twenty three))$

式(23)中则设计的自适应误差符号积分鲁棒重复控制器可使闭环系统中所有信号均有界，且系统获得半全局渐近输出跟踪性能，即当t→∞时，z₁→0。In formula (23) Then the designed adaptive error sign-integral robust repetitive controller can make all signals in the closed-loop system bounded, and the system obtains semi-global asymptotic output tracking performance, that is, when t→∞, z ₁ →0.

稳定性证明：Proof of Stability:

在稳定性证明之前，先给出如下两个引理：Before the stability proof, the following two lemmas are given:

引理1：定义辅助函数Lemma 1: Defining helper functions

$L L ((t t)) = = r r [[{\overset{\cdot &Center Dot;}{D D.}}_{22} ((t t)) - - βsign β sign (({z z}_{22}))]] - - - - - - ((24 twenty four))$

如果控制器增益β的选取满足如下条件，即If the selection of the controller gain β satisfies the following conditions, namely

$β β &GreaterEqual; &Greater Equal; {δ δ}_{11} + + \frac{11}{{k k}_{22}} {δ δ}_{22} - - - - - - ((2525))$

则如下定义的函数P(t)恒为非负Then the function P(t) defined as follows is always non-negative

$P P ((t t)) = = β β | | {z z}_{22} ((00)) | | - - {z z}_{22} ((00)) {\overset{\cdot &Center Dot;}{D D.}}_{22} ((00)) - - {&Integral; &Integral;}_{00}^{t t} L L ((τ τ)) dτ dτ - - - - - - ((2626))$

引理1的证明：Proof of Lemma 1:

对式(24)两边积分并运用式(15)得：Integrate both sides of formula (24) and use formula (15) to get:

${&Integral; &Integral;}_{00}^{t t} L L ((τ τ)) dτ dτ = = {&Integral; &Integral;}_{00}^{t t} {k k}_{22} {z z}_{22} (({\overset{\cdot \cdot}{D D.}}_{22} - - βsgn βsgn (({z z}_{22})))) dτ dτ + + {&Integral; &Integral;}_{00}^{t t} {\overset{\cdot \cdot}{z z}}_{22} {\overset{\cdot \cdot}{D D.}}_{22} dτ dτ - - {&Integral; &Integral;}_{00}^{t t} {\overset{\cdot \cdot}{z z}}_{22} βsgn βsgn (({z z}_{22})) dτ dτ - - - - - - ((2727))$

对式(27)中后两项进行分部积分可得：Integrating the last two terms in formula (27) by parts can get:

$\begin{matrix} {&Integral; &Integral;}_{00}^{t t} L L ((τ τ)) dτ dτ = = {&Integral; &Integral;}_{00}^{t t} {k k}_{22} {z z}_{22} (({\overset{\cdot &Center Dot;}{D D.}}_{22} - - βsgn βsgn (({z z}_{22})))) dτ dτ + + {z z}_{22} {\overset{\cdot &Center Dot;}{D D.}}_{22} {| |}_{00}^{t t} - - {&Integral; &Integral;}_{00}^{t t} {z z}_{22} {\overset{\cdot &Center Dot; \cdot \cdot}{D D.}}_{22} dτ dτ - - β β | | {z z}_{22} | | {| |}_{00}^{t t} \\ = = {&Integral; &Integral;}_{00}^{t t} {k k}_{22} {z z}_{22} (({\overset{\cdot &Center Dot;}{D D.}}_{22} - - \frac{11}{{k k}_{22}} {\overset{\cdot \cdot \cdot \cdot}{D D.}}_{22} - - βsgn βsgn (({z z}_{22})))) dτ dτ + + {z z}_{22} {\overset{\cdot &Center Dot;}{D D.}}_{22} - - {z z}_{22} ((00)) {\overset{\cdot &Center Dot;}{D D.}}_{22} ((00)) - - β β | | {z z}_{22} | | + + β β | | {z z}_{22} ((00)) | | \end{matrix} - - - - - - ((2828))$

故so

${&Integral; &Integral;}_{00}^{t t} L L ((τ τ)) dτ dτ \leq \leq {&Integral; &Integral;}_{00}^{t t} {k k}_{22} | | {z z}_{22} | | ((| | {\overset{\cdot &Center Dot;}{D D.}}_{22} | | + + \frac{11}{{k k}_{22}} | | {\overset{\cdot &Center Dot; \cdot \cdot}{D D.}}_{22} | | - - β β)) dτ dτ + + | | {z z}_{22} | | (({\overset{\cdot &Center Dot;}{D D.}}_{22} - - β β)) + + β β | | {z z}_{22} ((00)) | | - - {z z}_{22} ((00)) {\overset{\cdot &Center Dot;}{D D.}}_{22} ((00)) - - - - - - ((2929))$

从式(29)可以看出，若β的选取满足式(25)所示的条件时，易推断引理1成立。It can be seen from formula (29) that if the selection of β satisfies the conditions shown in formula (25), it is easy to deduce that Lemma 1 holds.

引理2：定义状态空间中的域其中κ∈R是正的常数。并令连续可微的函数V(t,ξ):R₊×D→R₊满足如下条件：Lemma 2: Defining domains in state space where κ∈R is a positive constant. And let the continuously differentiable function V(t,ξ):R ₊ ×D→R ₊ satisfy the following conditions:

$\begin{matrix} {W W}_{11} ((ξ ξ)) \leq \leq V V ((t t,, ξ ξ)) \leq \leq {W W}_{22} ((ξ ξ)) \\ \overset{\cdot \cdot}{V V} ((t t,, ξ ξ)) \leq \leq - - W W ((ξ ξ)) \end{matrix} - - - - - - ((3030))$

式(30)中，W₁(ξ),W₂(ξ)∈R为对于任意t≥0及任意ξ∈D都为连续正定的函数，W(ξ)∈R为一致连续正半定函数。In formula (30), W ₁ (ξ), W ₂ (ξ)∈R are continuous positive definite functions for any t≥0 and any ξ∈D, and W(ξ)∈R is a uniform continuous positive semidefinite function .

若条件(30)满足且ξ(0)∈S，则If condition (30) is satisfied and ξ(0)∈S, then

$\underset{t t &RightArrow; &Right Arrow; \infty \infty}{lim lim} W W ((ξ ξ)) = = 00 - - - - - - ((3131))$

域S的定义为The domain S is defined as

$S S \overset{Δ Δ}{= =} {{ξ ξ &Element; &Element; D D. | | {W W}_{22} ((ξ ξ)) \leq \leq δ δ}},, δ δ < < \underset{| | | | ξ ξ | | | | = = κ κ}{min min} {W W}_{11} ((ξ ξ)) - - - - - - ((3232))$

式中δ∈R是正的常数。where δ∈R is a positive constant.

定义矢量z＝[z₁,z₂,r]^T, Define vector z=[z ₁ ,z ₂ ,r] ^T ,

选取Lyapunov函数如下Select the Lyapunov function as follows

$V V = = \frac{11}{22} {z z}_{11}^{22} + + \frac{11}{22} {z z}_{22}^{22} + + \frac{11}{22} {Mr Mr.}^{22} + + \frac{11}{22} {\overset{~ ~}{θ θ}}^{T T} {Γ Γ}^{- - 11} \overset{~ ~}{θ θ} + + P P - - - - - - ((3333))$

则函数V满足如下性质：Then the function V satisfies the following properties:

$\begin{matrix} {W W}_{11} ((ξ ξ)) \leq \leq V V \leq \leq {W W}_{22} ((ξ ξ)),, \\ {W W}_{11} \overset{Δ Δ}{= =} {v v}_{11} {| | | | ξ ξ | | | |}^{22},, {W W}_{22} \overset{Δ Δ}{= =} {v v}_{22} {| | | | ξ ξ | | | |}^{22} \end{matrix} - - - - - - ((3434))$

式(34)中 $v_{1} \overset{Δ}{=} \frac{1}{2} \min {1, M, λ_{\min} (Γ^{- 1})}, v_{2} \overset{Δ}{=} \frac{1}{2} \max {1, M, λ_{\min} (Γ^{- 1})} .$ In formula (34) $v_{1} \overset{Δ}{=} \frac{1}{2} \min {1, m, λ_{\min} (Γ^{- 1})}, v_{2} \overset{Δ}{=} \frac{1}{2} \max {1, m, λ_{\min} (Γ^{- 1})} .$

求函数V对时间的微分，并结合式(14)、(15)、(22)和(26)可得Calculate the differential of the function V with respect to time, and combine formulas (14), (15), (22) and (26) to get

式中λ_min(Λ)为式(23)中定义的矩阵Λ的最小特征值。where λ _min (Λ) is the minimum eigenvalue of the matrix Λ defined in formula (23).

由于 $Δ = d_{1} (x_{1}, x_{2}) - d_{1} (x_{1 d}, {\overset{\cdot}{x}}_{1 d}),$ 因此because $Δ = d_{1} (x_{1}, x_{2}) - d_{1} (x_{1 d}, {\overset{\cdot}{x}}_{1 d}),$ therefore

$\overset{\cdot &Center Dot;}{Δ Δ} = = \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{11}} {x x}_{22} - - \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11 d d},, {\overset{\cdot \cdot}{x x}}_{11 d d}))}{&PartialD; &PartialD; {x x}_{11 d d}} {\overset{\cdot \cdot}{x x}}_{11 d d} + + \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{22}} {\overset{\cdot \cdot}{x x}}_{22} - - \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11 d d},, {\overset{\cdot \cdot}{x x}}_{11 d d}))}{&PartialD; &PartialD; {\overset{\cdot \cdot}{x x}}_{11 d d}} {\overset{\cdot \cdot \cdot \cdot}{x x}}_{11 d d} - - - - - - ((3636))$

又因为also because

$\begin{matrix} {x x}_{22} = = {\overset{\cdot &Center Dot;}{x x}}_{11 d d} + + {z z}_{22} - - {k k}_{11} {z z}_{11} \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = {\overset{\cdot &Center Dot; \cdot \cdot}{x x}}_{11 d d} + + r r - - (({k k}_{11} + + {k k}_{22})) {z z}_{22} + + {k k}_{11}^{22} {z z}_{11} \end{matrix} - - - - - - ((3737))$

所以式(36)可写成So equation (36) can be written as

$\begin{matrix} \overset{\cdot \cdot}{Δ Δ} = = {\overset{\cdot &Center Dot;}{x x}}_{11 d d} [[\frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{11}} - - \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11 d d},, {\overset{\cdot \cdot}{x x}}_{11 d d}))}{&PartialD; &PartialD; {x x}_{11 d d}}]] + + \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{11}} (({z z}_{22} - - {k k}_{11} {z z}_{11})) \\ + + {\overset{\cdot &Center Dot; \cdot &Center Dot;}{x x}}_{11 d d} [[\frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{22}} - - \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11 d d},, {\overset{\cdot &Center Dot;}{x x}}_{11 d d}))}{&PartialD; &PartialD; {\overset{\cdot \cdot}{x x}}_{11 d d}}]] + + \frac{&PartialD; &PartialD; {d d}_{11} (({x x}_{11},, {x x}_{22}))}{&PartialD; &PartialD; {x x}_{22}} [[r r - - (({k k}_{11} + + {k k}_{22})) {z z}_{22} + + {k k}_{11}^{22} {z z}_{11}]] \end{matrix} - - - - - - ((3838))$

由假设2中式(10)可知From Assumption 2, Equation (10), we can know

$\begin{matrix} \overset{\cdot \cdot}{Δ Δ} = = {\overset{\cdot \cdot}{x x}}_{11 d d} [[{φ φ}_{11} (({x x}_{11},, {x x}_{22})) - - {φ φ}_{11} (({x x}_{11 d d},, {\overset{\cdot &Center Dot;}{x x}}_{11 d d}))]] + + {φ φ}_{11} (({x x}_{11},, {x x}_{22})) (({z z}_{22} - - {k k}_{11} {z z}_{11})) \\ + + {\overset{\cdot \cdot \cdot \cdot}{x x}}_{11 d d} [[{φ φ}_{22} (({x x}_{11},, {x x}_{22})) - - {φ φ}_{22} (({x x}_{11 d d},, {\overset{\cdot \cdot}{x x}}_{11 d d}))]] + + {φ φ}_{22} (({x x}_{11},, {x x}_{22})) [[r r - - (({k k}_{11} + + {k k}_{22})) {z z}_{22} + + {k k}_{11}^{22} {z z}_{11}]] \end{matrix} - - - - - - ((3939))$

因此有下式成立Therefore, the following formula holds

$\begin{matrix} | | \overset{\cdot \cdot}{Δ Δ} | | \leq \leq | | {\overset{\cdot &Center Dot;}{x x}}_{11 d d} | | {ρ ρ}_{11} ((| | | | z z | | | |)) | | | | z z | | | | + + {ϵ ϵ}_{11} ((| | {z z}_{22} | | + + {k k}_{11} | | {z z}_{11} | |)) + + | | {\overset{\cdot &Center Dot; \cdot &Center Dot;}{x x}}_{11 d d} | | {ρ ρ}_{22} ((| | | | z z | | | |)) | | | | z z | | | | + + {ϵ ϵ}_{22} [[| | r r | | + + (({k k}_{11} + + {k k}_{22})) | | {z z}_{22} | | + + {k k}_{11}^{22} | | {z z}_{11} | |]] \\ \leq \leq ρ ρ ((| | | | z z | | | |)) | | | | z z | | | | \end{matrix} - - - - - - ((4040))$

式(40)中，ρ₁(||z||),ρ₂(||z||)和ρ(||z||)都是正的不减函数，且In formula (40), ρ ₁ (||z||), ρ ₂ (||z||) and ρ(||z||) are all positive non-decreasing functions, and

$ρ ρ ((| | | | z z | | | |)) = = | | {\overset{\cdot &Center Dot;}{x x}}_{11 d d} | | {ρ ρ}_{11} ((| | | | z z | | | |)) + + | | {\overset{\cdot &Center Dot; \cdot &Center Dot;}{x x}}_{11 d d} | | {ρ ρ}_{22} ((| | | | z z | | | |)) + + (({k k}_{11} + + 11)) {ϵ ϵ}_{11} + + (({k k}_{11}^{22} + + {k k}_{11} + + {k k}_{22} + + 11)) {ϵ ϵ}_{22} - - - - - - ((4141))$

利用式(41)并结合以下不等式性质Using equation (41) combined with the following inequality properties

$ρ ρ ((| | | | z z | | | |)) | | | | z z | | | | | | r r | | \leq \leq \frac{11}{22} {ρ ρ}^{22} ((| | | | z z | | | |)) {| | | | z z | | | |}^{22} + + \frac{11}{22} {r r}^{22} - - - - - - ((4242))$

则式(35)可以写成Then equation (35) can be written as

$\overset{\cdot \cdot}{V V} \leq \leq - - [[{λ λ}_{min min} ((Λ Λ)) - - \frac{11}{22} {ρ ρ}^{22} ((| | | | z z | | | |))]] {| | | | z z | | | |}^{22} - - - - - - ((4343))$

因此基于式(43)可以得到，当时，Therefore, based on formula (43), it can be obtained that when hour,

$\overset{\cdot &Center Dot;}{V V} \leq \leq - - γ γ {| | | | z z | | | |}^{22} = = W W ((ξ ξ)) - - - - - - ((4444))$

式中γ为正的常数。根据式(44)和(34)可知，函数V定义在如下域内where γ is a positive constant. According to formulas (44) and (34), the function V is defined in the following domain

$D D. \overset{Δ Δ}{= =} {{ξ ξ &Element; &Element; {R R}^{22 m m + + 33} | | | | | | ξ ξ | | | | \leq \leq {ρ ρ}^{- - 11} ((\sqrt{{λ λ}_{min min} ((Λ Λ))}))}} - - - - - - ((4545))$

可以推断，在域D内z，是有界的；由假设1可知系统状态变量x₁和x₂在域D内是有界的；因为参数θ有界，因此也有界，根据式(17)和(21)可知控制输入u是有界的；基于式(14)、(15)和(22)可知即因此可知W(ξ)是一致连续的函数，则根据引理2可得，若系统初始条件满足ξ(0)∈S时，且S定义为It can be deduced that within the domain D z, is bounded; from assumption 1 we know that the system state variables x ₁ and x ₂ are bounded in the domain D; because the parameter θ is bounded, so is also bounded, according to formulas (17) and (21), it can be known that the control input u is bounded; based on formulas (14), (15) and (22), it can be seen that Right now Therefore, it can be seen that W(ξ) is a consistent and continuous function. According to Lemma 2, if the initial condition of the system satisfies ξ(0)∈S, and S is defined as

$S S \overset{Δ Δ}{= =} {{ξ ξ ((t t)) &Element; &Element; D D. | | {W W}_{11} ((ξ ξ)) < < {v v}_{11} {ρ ρ}^{- - 11} ((\sqrt{22 {λ λ}_{min min} ((Λ Λ))}))}} - - - - - - ((4646))$

有结论当t→∞，||z||→0成立，即系统获得半全局渐近跟踪的性能。机电伺服系统的自适应误差符号积分鲁棒重复(ARRPC)控制原理示意图如图2所示。It is concluded that when t→∞, ||z||→0 holds true, the system obtains the performance of semi-global asymptotic tracking. The schematic diagram of the adaptive error sign integral robust repetition (ARRPC) control principle of the electromechanical servo system is shown in Fig. 2 .

为验证所设计的控制器性能，在仿真中取如下参数对机电伺服系统进行建模：In order to verify the performance of the designed controller, the following parameters are taken in the simulation to model the electromechanical servo system:

惯性负载参数m＝0.01kg·m²；粘性摩擦系数B＝0.2N·m·s/rad；库伦摩擦的幅值A_f＝0.1N·m·s/rad；系统建模不确定性d₁(x₁,x₂)＝x₁+x₂(N·m),d₂(t)＝0.2sint(N·m)。Inertial load parameter m=0.01kg·m ² ; viscous friction coefficient B=0.2N·m·s/rad; Coulomb friction amplitude A _f =0.1N·m·s/rad; system modeling uncertainty d ₁ (x ₁ , x ₂ )=x ₁ +x ₂ (N·m), d ₂ (t)=0.2 sint(N·m).

给定系统的期望指令为x_1d＝0.5sin(πt)[1-exp(-0.01t³)](rad)。The desired command for a given system is x _1d =0.5 sin(πt)[1-exp(-0.01t ³ )](rad).

取如下的控制器以作对比：Take the following controller for comparison:

自适应误差符号积分鲁棒重复控制器(ARIPC)：取控制器参数k₁＝250，k₂＝20，k_r＝2，β＝2，自适应增益Г＝diag{100,50,50,50}。Adaptive error sign-integrated robust repetitive controller (ARIPC): take controller parameters k ₁ =250, k ₂ =20, k _r =2, β=2, adaptive gain Г=diag{100,50,50, 50}.

误差符号积分鲁棒控制器(RISE)：即所设计的ARIPC控制器中不含自适应模型补偿部分，对比RISE控制器是为了验证ARIPC控制器中自适应模型补偿部分对周期性的建模不确定性的抑制能力。为保证对比公平性，其控制器参数与ARIPC控制器中对应的参数相同。Error Sign Integral Robust Controller (RISE): That is, the designed ARIPC controller does not contain the adaptive model compensation part. The purpose of comparing the RISE controller is to verify that the adaptive model compensation part of the ARIPC controller does not model the periodicity. Deterministic suppression capabilities. To ensure the fairness of the comparison, its controller parameters are the same as those in the ARIPC controller.

反馈线性化控制器(FLC)：控制器设计如下Feedback Linearized Controller (FLC): The controller is designed as follows

$u u = = M m {\overset{\cdot \cdot}{x x}}_{22 eq eq} + + B B {x x}_{22} + + {A A}_{f f} {S S}_{f f} (({x x}_{22})) - - {k k}_{22} {z z}_{22}$

用以验证ARIPC控制器中自适应模型补偿项和非线性鲁棒项分别对周期性建模不确定性和非周期性建模不确定性的抑制能力。其控制器参数与ARIPC中对应参数相同。It is used to verify the ability of the adaptive model compensation term and the nonlinear robust term in the ARIPC controller to suppress the periodic modeling uncertainty and the non-periodic modeling uncertainty respectively. Its controller parameters are the same as the corresponding parameters in ARIPC.

ARIPC控制器作用下系统输出对期望指令的跟踪、ARIPC控制器跟踪误差、三种控制器分别作用下的跟踪误差对比分别如图3，图4和图5所示。由图4和图5可知，所设计的ARIPC控制器的暂态和稳态跟踪性能都要优于相对比的RISE控制器和FLC控制器，RISE控制器由于缺少自适应补偿，获得较差的跟踪性能，而FLC控制器既没有自适应模型补偿也没有非线性鲁棒反馈则作用，获得最差的跟踪性能。The tracking of the system output to the expected command under the action of the ARIPC controller, the tracking error of the ARIPC controller, and the comparison of the tracking errors under the action of the three controllers are shown in Figure 3, Figure 4 and Figure 5, respectively. It can be seen from Figure 4 and Figure 5 that the transient and steady-state tracking performance of the designed ARIPC controller is better than that of the comparative RISE controller and FLC controller. tracking performance, while the FLC controller has neither adaptive model compensation nor nonlinear robust feedback function, the worst tracking performance is obtained.

图6是ARIPC控制器作用下周期性不确定性按Fourier级数展开后各基函数前常值参数估计随时间变化的曲线。从图中可以看出，尽管存在非周期性建模不确定性，各参数估计仍能很好地收敛真值Fig. 6 is the curve of the constant parameter estimation of each basis function changing with time after the periodic uncertainty is expanded according to the Fourier series under the action of the ARIPC controller. It can be seen from the figure that despite the non-periodic modeling uncertainty, the parameter estimates converge well to the true value

图7是系统在ARIPC控制器作用下系统控制输入随时间变化的曲线图。从图中可以看出，所获得的控制输入是低频连续的信号，更利于在实际应用中的执行。Fig. 7 is a curve diagram of system control input changing with time under the action of ARIPC controller. It can be seen from the figure that the obtained control input is a low-frequency continuous signal, which is more conducive to the implementation in practical applications.

由上可知，本发明提出的机电伺服系统的自适应误差符号积分鲁棒重复控制方法有效地降低传统基于内模原理的重复控制器对噪声的敏感性以及规避控制器的高内存需求，并获得半全局渐近跟踪的优异性能。It can be seen from the above that the adaptive error sign integral robust repetitive control method of the electromechanical servo system proposed by the present invention effectively reduces the sensitivity of the traditional repetitive controller based on the internal model principle to noise and avoids the high memory requirement of the controller, and obtains Excellent performance for semi-global asymptotic tracking.

虽然本发明已以较佳实施例揭露如上，然其并非用以限定本发明。本发明所属技术领域中具有通常知识者，在不脱离本发明的精神和范围内，当可作各种的更动与润饰。因此，本发明的保护范围当视权利要求书所界定者为准。Although the present invention has been disclosed above with preferred embodiments, it is not intended to limit the present invention. Those skilled in the art of the present invention can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, the scope of protection of the present invention should be defined by the claims.

Claims

1. an adaptive error symbolic integration robust repetitive control method for electromechanical servo system, is characterized in that, comprise the following steps:

Step 1, set up the mathematical model of electromechanical servo system;

Step 2, constructing system design a model;

Step 3, design adaptive error symbolic integration Robust Repetitive Controller.

2. the adaptive error symbolic integration robust repetitive control method of electromechanical servo system according to claim 1, is characterized in that, set up the mathematical model of electromechanical servo system described in step 1, specific as follows:

Step 1.1, according to Newton second law, the equation of motion of electromechanical servo system is:

In formula (1), M is inertia load parameter, and B is viscosity friction coefficient, A _ffor the amplitude of Coulomb friction, for the shape function of known sign Coulomb friction, the modeling being system is uncertain, and comprise other non-linear frictions, outer interference, Unmarried pregnancy, y is the displacement of inertia load, and u is the control inputs of system, and t is time variable;

Step 1.2, definition status variable: then formula (1) equation of motion can be write as state equation:

The modeling uncertainty that in formula (2), supposing the system is total can be divided into two parts, namely only relevant to system state d ₁(x ₁, x ₂) and time become d ₂t (), for mechanical system, the uncertain overwhelming majority of modeling is all only relevant to system state, therefore d ₁(x ₁, x ₂) be main modeling uncertainty,

The design object of system controller is: given system reference signal y _d(t)=x _1dt (), the continuous print control inputs u of a design bounded makes system export y=x ₁the reference signal of tracker as much as possible.

3. the adaptive error symbolic integration robust repetitive control method of electromechanical servo system according to claim 2, it is characterized in that, described in step 2, constructing system designs a model, and step is as follows:

Although the uncertain d of step 2.1 modeling ₁(x ₁, x ₂) be unknown, but for the electromechanical servo system of performance period property task, its modeling uncertainty also can present identical periodicity after a certain time, therefore this type of periodicity modeling of the method process of Repetitive controller can be utilized uncertain, and for the uncertain d of acyclic modeling ₂t () can design robust controller to suppress it on the impact of tracking performance, therefore formula (2) can be write as following form

In formula (3)

Step 2.2, due to considered electromechanical servo system perform be periodic task, therefore expect follow the tracks of position command x _1dt () is periodic, namely

x _1d(t-T)＝x _1d(t) (4)

Wherein T is the known cycle, notices only and x _1dwith relevant, be therefore also periodic, make in following design process therefore have

D _d(t-T)＝D _d(t) (5)

To D _dt () utilizes fourier progression expanding method to obtain

ω=2 π/T in formula (6), considers that the transport function of mechanical system is equivalent to the low-pass filter that has limited frequency range in the physical sense, therefore D _dt () is similar to the finite term in formula (6), namely

Step 2.3, define unknown constant parameter vector

θ＝[a ₁,b ₁,...,a _m,b _m] ^T(8)

According to formula (7) and (8), the model (3) of system can be write as

In formula (9) d ₂(t)=a ₀/ 2+d ₂(t), and M, B and A _ffor the nominal value that system physical parameter is known, the deviation between parameter nominal value and its true value can be integrated into the uncertain D of system modelling ₂in (t);

Suppose as follows:

Suppose 1: system reference command signal x _1dt () is that Second Order Continuous can be micro-, and its all-order derivative bounded;

Suppose 2: the uncertain d of modeling ₁(x ₁, x ₂) and D ₂t () is all that Second Order Continuous can be micro-, and meet following condition:

Formula (10) and (11) middle ε ₁, ε ₂, δ ₀unknown positive number, δ ₁, δ ₂it is known positive number.

4. the adaptive error symbolic integration robust repetitive control method of electromechanical servo system according to claim 3, it is characterized in that, design adaptive error symbolic integration Robust Repetitive Controller described in step 3, step is as follows:

Step 3.1, definition z ₁=x ₁-x _1dfor the tracking error of system, according to first equation in formula (9) choose x ₂for virtual controlling, make equation tend towards stability state; Make x _2eqfor the expectation value of virtual controlling, x _2eqwith time of day x ₂error be z ₂=x ₂-x _2eq, to z ₁differentiate can obtain:

Design virtual controlling rule:

K in formula ₁> 0 is adjustable gain, then

Due to z ₁(s)=G (s) z ₂(s), G (s)=1/ (s+k in formula ₁) be a stable transport function, work as z ₂when being tending towards 0, z ₁also 0 must be tending towards.So in ensuing design, will to make z ₂be tending towards 0 for main design goal;

Step 3.2, be the extra Controller gain variations degree of freedom of acquisition one, define an auxiliary error signal r (t):

K in formula (15) ₂> 0 is adjustable gain, due in r (t) containing the acceleration signal of position, think immeasurablel in practice, to be namely only Computer Aided Design used for r (t), specifically do not appear in designed controller;

Can obtain according to formula (9) and (15)

Based on formula (16), can CONTROLLER DESIGN as follows

K in formula (17) _rfor positive feedback gain.U _afor the compensation term based on model for improving systematic tracking accuracy, u _sfor Robust Control Law and wherein u _s1for the linear robust feedback term for systems stabilisation nominal plant model, u _s2for non linear robust item is for overcoming the uncertain impact on system performance of modeling, for the estimated value of parameter θ, definition for parameter estimating error;

Step 3.3, design parameter adaptive law are:

In formula (18), Γ is diagonal angle positive definite matrix, characterization parameter adaptive rate; Because the parameter update law in formula (18) contains immesurable signal r (t), therefore integration by parts is adopted to be unfolded as follows to it

Formula (17) is substituted in formula (16) and obtains:

According to error symbol integration robust Controller Design method, integration robust item u _s2can be designed to:

To formula (20) differentiate and convolution (18) and (21) can obtain

。