CN105159077A

CN105159077A - Finite-time continuous sliding mode control method for disturbance compensation of direct drive motor system

Info

Publication number: CN105159077A
Application number: CN201510524462.8A
Authority: CN
Inventors: 刘龙; 姚建勇; 胡健; 邓文翔
Original assignee: Nanjing University of Science and Technology
Current assignee: Nanjing University of Science and Technology
Priority date: 2015-08-24
Filing date: 2015-08-24
Publication date: 2015-12-16
Anticipated expiration: 2035-08-24
Also published as: CN105159077B

Abstract

The invention discloses a finite-time continuous sliding mode control method for interference compensation of a direct-drive motor system. The method includes: establishing a mathematical model of the direct-drive motor system; designing a disturbance observer of the direct-drive motor system and proving the accuracy of the observation; designing Continuous sliding-mode controllers with finite-time convergence based on disturbance observers. The invention eliminates the chattering of sliding mode control, and at the same time enables the system to obtain the steady-state performance that the tracking error is zero within a limited time when there is interference, and enhances the application of the sliding mode control method in the direct drive motor system to resist interference and eliminate slippage. The ability of mode to control chattering and obtain good tracking performance.

Description

Finite-time continuous sliding mode control method for disturbance compensation of direct drive motor system

技术领域technical field

本发明涉及机电伺服控制技术领域，主要涉及一种直驱电机系统的干扰补偿的有限时间连续滑模控制方法。The invention relates to the technical field of electromechanical servo control, and mainly relates to a limited-time continuous sliding mode control method for interference compensation of a direct drive motor system.

背景技术Background technique

在现代工业生产中，直驱电机系统由于消除了与减速齿轮相关的一些机械传动问题如齿隙、强惯性载荷以及结构柔性等非线性问题而在许多机械设备中广泛使用。这些非线性问题都是影响系统性能的主要因素，其存在将会严重恶化系统跟踪性能，因此通过对直驱电机系统进行先进的控制器设计可以获得高精度的控制性能。然而，也正是由于缺少减速齿轮的作用，对直驱电机系统进行控制器设计时需要面临许多外干扰，如参数摄动及外负载干扰等，这些外干扰不再经过减速齿轮而是直接作用于驱动部件，这样同样会严重地恶化控制性能，甚至会使系统降阶、失稳。因此探索先进的控制器设计方法来保证直驱电机系统的高精度控制性能仍是实际工程应用领域的迫切需求。In modern industrial production, direct drive motor systems are widely used in many mechanical equipment due to the elimination of some mechanical transmission problems associated with reduction gears, such as backlash, strong inertial loads, and structural flexibility. These nonlinear problems are the main factors affecting system performance, and their existence will seriously deteriorate system tracking performance. Therefore, high-precision control performance can be obtained through advanced controller design for direct drive motor systems. However, it is precisely because of the lack of the role of the reduction gear that the controller design of the direct drive motor system needs to face many external disturbances, such as parameter perturbation and external load interference, etc. These external disturbances no longer pass through the reduction gear but directly act For the drive components, this will also seriously deteriorate the control performance, and even degrade the system and destabilize it. Therefore, exploring advanced controller design methods to ensure the high-precision control performance of direct drive motor systems is still an urgent need in the field of practical engineering applications.

针对直驱电机系统存在外干扰的问题，许多方法相继被提出。其中滑模控制方法对于处理外干扰的问题是一种非常有效的方法。滑模控制方法的基本思路是针对直驱电机系统的名义模型设计控制器，将真实系统模型与名义模型之间的偏差和干扰统一归类到外干扰中。针对外干扰，传统的滑模控制方法主要是通过增大控制器的鲁棒性来克服外干扰从而迫切系统状态到达滑模面，但是，通过增大不连续项增益的方法来增加控制器的鲁棒性的方法增大了滑模控制的抖振，在实际运用中很可能激发系统高频动态，使系统失稳。因而传统的滑模控制方法具有很大的工程局限性。与此同时，传统的滑模控制方法只能获得渐近跟踪的稳态性能。而在工程实际中，控制器的作用时间不可能趋于无穷大，从而跟踪误差不可能趋于零。Aiming at the problem of external interference in the direct drive motor system, many methods have been proposed one after another. Among them, the sliding mode control method is a very effective method for dealing with external disturbances. The basic idea of the sliding mode control method is to design a controller for the nominal model of the direct drive motor system, and classify the deviation and disturbance between the real system model and the nominal model into external disturbances. For external disturbances, the traditional sliding mode control method mainly overcomes external disturbances by increasing the robustness of the controller so that the state of the system reaches the sliding mode surface, but increases the gain of the discontinuous term to increase the The robustness method increases the chattering of the sliding mode control, which is likely to excite the high-frequency dynamics of the system in practical applications and make the system unstable. Therefore, the traditional sliding mode control method has great engineering limitations. Meanwhile, traditional sliding mode control methods can only obtain steady-state performance of asymptotic tracking. However, in engineering practice, the action time of the controller cannot tend to infinity, so the tracking error cannot tend to zero.

发明内容Contents of the invention

本发明的目的在于提供一种直驱电机系统干扰补偿的有限时间连续滑模控制方法。The purpose of the present invention is to provide a limited-time continuous sliding mode control method for disturbance compensation of a direct drive motor system.

实现本发明目的的技术解决方案为：一种直驱电机系统的干扰补偿的有限时间连续滑模控制方法，包括以下步骤：The technical solution to realize the object of the present invention is: a finite-time continuous sliding mode control method for disturbance compensation of a direct drive motor system, comprising the following steps:

步骤1，建立直驱电机系统的数学模型；Step 1, establishing a mathematical model of the direct drive motor system;

步骤2，设计直驱电机系统的干扰观测器；Step 2, design the disturbance observer of the direct drive motor system;

步骤3，设计基于干扰观测器的有限时间收敛的连续滑模控制器。Step 3, design a continuous sliding mode controller based on the finite-time convergence of the disturbance observer.

本发明与现有技术相比，其显著优点为：Compared with the prior art, the present invention has the remarkable advantages of:

(1)本发明将观测器滑模面和控制器滑模面结合，消除了干扰的观测误差，保证了控制器的暂态控制性能；(1) The present invention combines the sliding mode surface of the observer and the sliding mode surface of the controller, eliminates the observation error of interference, and ensures the transient control performance of the controller;

(2)本发明补偿了直驱电机系统的外干扰，同时设计了连续滑模控制器，使得控制器曲线连续化，消除了滑模控制的抖振问题，同时保证了滑模控制策略的鲁棒性；(2) The present invention compensates the external interference of the direct drive motor system, and designs a continuous sliding mode controller at the same time, which makes the controller curve continuous, eliminates the chattering problem of the sliding mode control, and ensures the robustness of the sliding mode control strategy Rod;

(3)本发明不要求系统外干扰的数学表达式存在导数，对于可能存在的导数不存在的外干扰仍可保证良好的控制性能；(3) The present invention does not require that there is a derivative in the mathematical expression of the external interference of the system, and good control performance can still be guaranteed for the external interference that the possible derivative does not exist;

(4)本发明最终可得到跟踪误差有限时间为零的稳态性能，保证了跟踪误差在有限时间内为零。(4) The present invention can finally obtain the steady-state performance that the tracking error is zero in a limited time, which ensures that the tracking error is zero in a limited time.

附图说明Description of drawings

图1为本发明的直驱电机系统的干扰补偿的有限时间连续滑模控制方法流程图。FIG. 1 is a flowchart of a finite-time continuous sliding mode control method for disturbance compensation of a direct drive motor system according to the present invention.

图2为本发明直驱电机系统的原理图。Fig. 2 is a schematic diagram of the direct drive motor system of the present invention.

图3为直驱电机系统的干扰补偿的有限时间连续滑模控制方法(UCFT‐SMC)原理示意图。Figure 3 is a schematic diagram of the principle of the finite-time continuous sliding mode control method (UCFT-SMC) for disturbance compensation of the direct drive motor system.

图4为本发明实施例中UCFT‐SMC控制器作用下系统输出对期望指令的跟踪曲线图。Fig. 4 is a tracking curve of the system output to the expected command under the action of the UCFT-SMC controller in the embodiment of the present invention.

图5为本发明实施例中UCFT‐SMC控制器作用下系统的跟踪误差随时间的变化曲线图。Fig. 5 is a graph showing the variation of the system tracking error with time under the action of the UCFT-SMC controller in the embodiment of the present invention.

图6为本发明实施例中滑模干扰观测器对系统干扰的观测曲线图。FIG. 6 is an observation curve of system disturbance by a sliding mode disturbance observer in an embodiment of the present invention.

图7为本发明实施例中滑模干扰观测器对系统干扰的观测误差随时间的变化曲线图。Fig. 7 is a graph showing the variation of the observation error of the system disturbance with time by the sliding mode disturbance observer in the embodiment of the present invention.

图8为本发明实施例中UFTC‐SMC控制器作用下直驱电机系统控制输入随时间变化的曲线图。Fig. 8 is a graph of the control input of the direct drive motor system changing with time under the action of the UFTC-SMC controller in the embodiment of the present invention.

图9为本发明实施例中SMC控制器作用下直驱电机系统控制输入随时间变化的曲线图。Fig. 9 is a graph showing the control input of the direct drive motor system changing with time under the action of the SMC controller in the embodiment of the present invention.

图10为本发明实施例中UCFT‐SMC、SMC控制器分别作用下系统跟踪误差的对比曲线图。Fig. 10 is a comparative graph of system tracking error under the action of UCFT-SMC and SMC controllers respectively in the embodiment of the present invention.

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，本发明直驱电机系统干扰补偿的有限时间连续滑模控制方法，包括以下步骤：In conjunction with Fig. 1, the finite time continuous sliding mode control method of the interference compensation of the direct drive motor system of the present invention comprises the following steps:

步骤1‐1、本发明所考虑的直驱电机系统是通过配有电气驱动器的永磁直流电机直接驱动惯性负载。结合图2，伺服电机输出端驱动惯性负载，电源通过电气驱动器给伺服电机供电，控制指令通过电器驱动器控制伺服电机运动，光电编码器给控制器反馈电机位置信号，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；Step 1-1. The direct drive motor system considered in the present invention directly drives the inertial load through a permanent magnet DC motor equipped with an electric driver. Combined with Figure 2, the output end of the servo motor drives the inertial load, the power supply supplies power to the servo motor through the electrical driver, the control command controls the movement of the servo motor through the electrical driver, and the photoelectric encoder feeds back the motor position signal to the controller. Considering that the electromagnetic time constant is higher than the mechanical time The constant is much smaller, and the speed of the current loop is much larger than the response speed of the speed loop and the position loop, so the current loop can be approximated as a proportional link;

因此，根据牛顿第二定律，直驱电机系统的运动方程为：Therefore, according to Newton's second law, the motion equation of the direct drive motor system is:

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

式(1)中m为惯性负载参数，k_i为力矩放大系数，B为粘性摩擦系数，是建模误差，包括m、k_i、B的名义值与真实值之间的偏差以及外负载干扰；y为惯性负载的位移，为惯性负载的速度，u为系统的控制输入，t为时间变量；In formula (1), m is the inertial load parameter, k _i is the moment amplification factor, B is the viscous friction coefficient, is the modeling error, including the deviation between the nominal value and the real value of m, k _i , B and the external load disturbance; y is the displacement of the inertial load, is the speed of the inertial load, u is the control input of the system, and t is the time variable;

步骤1‐2、定义状态变量：则式(1)运动方程转化为状态方程：Step 1‐2, define state variables: Then the equation of motion (1) is transformed into the equation of state:

$\begin{matrix} {\overset{\cdot \cdot}{x x}}_{11} = = {x x}_{22} \\ {\overset{\cdot \cdot}{x x}}_{22} = = {θ θ}_{11} u u - - {θ θ}_{22} {x x}_{22} + + d d ((x x,, t t)) \\ y the y = = {x x}_{11} \end{matrix} - - - - - - ((22))$

式(2)中，均为名义值且已知。可认为是系统总干扰，包括外负载干扰、未建模摩擦、未建模动态、系统实际参数与建模参数的偏离造成的不确定性。f(t,x₁,x₂)即为上述x₁表示惯性负载的位移，x₂表示惯性负载的速度；In formula (2), Both are nominal and known. It can be considered as the total disturbance of the system, including the uncertainty caused by external load disturbance, unmodeled friction, unmodeled dynamics, and the deviation between the actual system parameters and the modeled parameters. f(t,x ₁ ,x ₂ ) is the above x ₁ represents the displacement of the inertial load, x ₂ represents the velocity of the inertial load;

因为在直驱电机系统中，系统的状态和参数都是有界的，故系统总干扰量d(x,t)满足：Because in the direct drive motor system, the state and parameters of the system are bounded, so the total disturbance d(x,t) of the system satisfies:

|d(x,t)|≤D(3)|d(x,t)|≤D(3)

式(3)中D为已知正常数，即d(x,t)具有已知的上界。In formula (3), D is a known constant, that is, d(x, t) has a known upper bound.

步骤2，设计干扰观测器并证明观测的准确性:Step 2, design the disturbance observer and prove the accuracy of the observation:

步骤2‐1、设计干扰观测器：Step 2-1. Design disturbance observer:

定义观测器滑模面s₁为：Define the observer sliding mode surface _s1 as:

s₁＝z₁-x₂(4)s ₁ =z ₁ -x ₂ (4)

其中，z₁为观测器内动态；Among them, z ₁ is the internal dynamics of the observer;

${\overset{\cdot &Center Dot;}{z z}}_{11} = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{11} u u - - - - - - ((55))$

式(5)中，k₁、β₁、ε₁、p₁和q₁均为干扰观测器系数；p₁<q₁，且均为正奇数，k₁、β₁、ε₁均为正数，β₁≥D；In formula (5), k ₁ , β ₁ , ε ₁ , p ₁ and q ₁ are all interference observer coefficients; p ₁ <q ₁ , and they are all positive odd numbers; k ₁ , β ₁ , and ε ₁ are all positive number, β ₁ ≥ D;

sign(0)∈[-1,1]sign(0)∈[-1,1]

则d(x,t)的估计为：Then the estimate of d(x,t) for:

$\overset{^^}{d d} ((x x,, t t)) = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} - - - - - - ((77))$

由式(2)、(4)、(5)有：From formula (2), (4), (5) have:

$\begin{matrix} {\overset{\cdot \cdot}{s the s}}_{11} = = {\overset{\cdot \cdot}{z z}}_{11} - - {\overset{\cdot \cdot}{x x}}_{22} \\ = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} - - d d ((x x,, t t)) \end{matrix} - - - - - - ((88))$

步骤2‐2、定义干扰观测器李雅普诺夫方程：Step 2‐2, define the disturbance observer Lyapunov equation:

${V V}_{11} ((t t)) = = \frac{11}{22} {s the s}_{11}^{22} - - - - - - ((99))$

又因β₁≥D，则：And because β ₁ ≥ D, then:

$\begin{matrix} {\overset{\cdot &Center Dot;}{V V}}_{11} ((t t)) = = {s the s}_{11} {\overset{\cdot &Center Dot;}{s the s}}_{11} \\ = = {s the s}_{11} [[- - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} - - d d ((x x,, t t))]] \\ = = - - {k k}_{11} {s the s}_{11}^{22} - - {β β}_{11} {s the s}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{(({p p}_{11} / / {q q}_{11})) {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | | | {s the s}_{11} | | + + {θ θ}_{22} {x x}_{22} {s the s}_{22} - - d d ((x x,, t t)) {s the s}_{11} \\ \leq \leq - - {k k}_{11} {s the s}_{11}^{22} - - {β β}_{11} | | {s the s}_{11} | | - - {ϵ ϵ}_{11} {s the s}_{11}^{(({p p}_{11} / / {q q}_{11})) {q q}_{11}} + + d d ((x x,, t t)) {s the s}_{11} \\ \leq \leq - - {k k}_{11} {s the s}_{11}^{22} - - {ϵ ϵ}_{11} {s the s}_{11}^{(({p p}_{11} / / {q q}_{11})) {q q}_{11}} = = - - 22 {k k}_{11} {V V}_{11} ((t t)) - - 22^{(({p p}_{11} / / {q q}_{11})) / / 22 {q q}_{11}} {ϵ ϵ}_{11} {V V}_{11}^{(({p p}_{11} / / {q q}_{11})) / / 22 {q q}_{11}} ((t t)) \end{matrix} - - - - - - ((1010))$

若存在一正定函数V₀(t)满足以下不等式：If there is a positive definite function V ₀ (t) satisfying the following inequality:

${\overset{\cdot \cdot}{V V}}_{00} ((t t)) + + {αV αV}_{00} ((t t)) + + {λV λV}_{00}^{γ γ} ((t t)) \leq \leq 00,, &ForAll; &ForAll; t t > > {t t}_{00} - - - - - - ((1111))$

则，V₀(t)在时间t_s内收敛到平衡点，其中，Then, V ₀ (t) converges to the equilibrium point within time t _s , where,

${t t}_{s the s} \leq \leq {t t}_{00} + + \frac{11}{α α ((11 + + γ γ))} l l n no \frac{{αV αV}_{00}^{11 - - γ γ} (({t t}_{00})) + + λ λ}{λ λ} - - - - - - ((1212))$

其中，α>0，λ>0，0<γ<1；Among them, α>0, λ>0, 0<γ<1;

故，V₁(t)将在有限时间内收敛到平衡点，即存在一个时间t₂点，在t₂之后，V₁(t)恒为零，由V₁(t)的表达式(9)可知，V₁(t)为零后，s₁也为零，此时也将收敛到零，又因d(x,t)估计误差 Therefore, V ₁ (t) will converge to the equilibrium point within a finite time, that is, there is a time t ₂ point, after t ₂ , V ₁ ( _t ) is always zero, and the expression (9 ) we can see that after V ₁ (t) is zero, s ₁ is also zero, at this time It will also converge to zero, and because of the estimation error of d(x,t)

$\begin{matrix} \overset{~ ~}{d d} ((x x,, t t)) = = \overset{^^}{d d} ((x x,, t t)) - - d d ((x x,, t t)) \\ = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} - - d d ((x x,, t t)) \\ = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} - - {\overset{\cdot &Center Dot;}{x x}}_{22} + + {θ θ}_{11} u u - - {θ θ}_{22} {x x}_{22} \\ = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) - - {\overset{\cdot &Center Dot;}{x x}}_{22} + + {θ θ}_{11} u u \\ = = {\overset{\cdot &Center Dot;}{z z}}_{11} - - {\overset{\cdot &Center Dot;}{x x}}_{22} \\ = = {\overset{\cdot &Center Dot;}{s the s}}_{11} \end{matrix} - - - - - - ((1313))$

则干扰的估计误差也将在有限时间t₂内为0；即在t₂后有 The estimation error of the disturbance will also be 0 for a finite time t ₂ ; that is, after t ₂ there is

得到干扰观测器：Get the disturbance observer:

$\overset{^^}{d d} ((x x,, t t)) = = - - {k k}_{11} {s the s}_{11} - - {β β}_{11} s the s i i g g n no (({s the s}_{11})) - - {ϵ ϵ}_{11} {s the s}_{11}^{{p p}_{11} / / {q q}_{11}} - - | | {θ θ}_{22} {x x}_{22} | | s the s i i g g n no (({s the s}_{11})) + + {θ θ}_{22} {x x}_{22} . .$

步骤3，设计基于干扰观测器的有限时间收敛的连续滑模控制器:Step 3, design the continuous sliding mode controller based on the finite-time convergence of the disturbance observer:

定义直驱电机系统位置跟踪误差e₀(t)、误差变量e₁(t)：Define the position tracking error e ₀ (t) and error variable e ₁ (t) of the direct drive motor system:

e₀(t)＝x₁-x_d(t)(14)e ₀ (t) = x ₁ -x _d (t) (14)

${e e}_{11} ((t t)) = = {x x}_{22} - - {\overset{\cdot &Center Dot;}{x x}}_{d d} ((t t)) + + {s the s}_{11} - - - - - - ((1515))$

其中，x_d(t)为系统参考位置信号，x_d(t)是二阶连续的，且系统参考位置信号x_d(t)、系统参考速度信号系统参考加速度信号都是有界的；Among them, x _d (t) is the system reference position signal, x _d (t) is second-order continuous, and the system reference position signal x _d (t), system reference speed signal System Reference Acceleration Signal are bounded;

定义滑模控制器滑模面s:Define the sliding mode controller sliding surface s:

$s the s = = {e e}_{11} ((t t)) + + &Integral; &Integral; {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} d d t t - - - - - - ((1616))$

其中λ₀、λ₁、α₁、α₂均为滑模控制器参数，且均大于零，并且λ₀、λ₁满足表达式z²+λ₁z+λ₀是Hurwitz的，其中_z为微分算子，α₁、α₂满足α₂∈(0,1)；则，Among them, λ ₀ , λ ₁ , α ₁ , and α ₂ are sliding mode controller parameters, and they are all greater than zero, and λ ₀ and λ ₁ satisfy the expression z ² +λ ₁ z+λ ₀ is Hurwitz, where _z is Differential operator, α ₁ , α ₂ satisfy α ₂ ∈ (0,1); then,

$\begin{matrix} \overset{\cdot &Center Dot;}{s the s} = = s the s = = {\overset{\cdot \cdot}{e e}}_{11} ((t t)) + + {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} \\ = = {θ θ}_{11} u u - - {θ θ}_{22} {x x}_{22} + + d d ((x x,, t t)) - - {\overset{\cdot\cdot \cdot\cdot}{x x}}_{d d} + + {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} \\ + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} + + {\overset{\cdot &Center Dot;}{s the s}}_{11} \end{matrix} - - - - - - ((1717))$

得到滑模控制器u为：The sliding mode controller u is obtained as:

$\begin{matrix} u u = = - - \frac{11}{{θ θ}_{11}} [[- - {θ θ}_{22} {x x}_{22} + + d d ((x x,, t t)) - - {\overset{\cdot\cdot \cdot\cdot}{x x}}_{d d} + + {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} \\ + + {λ λ}_{22} s the s + + {λ λ}_{33} s the s i i g g n no ((s the s)) | | s the s {| |}^{{α α}_{00}}]] \end{matrix} - - - - - - ((1818))$

其中λ₂、λ₃、α₀为控制器参数，且λ₂>0、λ₃>0、0<α₀<1。Where λ ₂ , λ ₃ , and α ₀ are controller parameters, and λ ₂ >0, λ ₃ >0, 0<α ₀ <1.

步骤4，系统全局稳定及误差有限时间内为零测试:Step 4, the global stability of the system and the zero error test within a limited time:

将式(18)代入式(17)有：Substituting formula (18) into formula (17) has:

$\begin{matrix} \overset{\cdot &Center Dot;}{s the s} = = d d ((t t)) - - \overset{^^}{d d} ((t t)) + + {\overset{\cdot &Center Dot;}{s the s}}_{11} - - {λ λ}_{22} s the s + + {λ λ}_{33} s the s i i g g n no ((s the s)) | | s the s {| |}^{{α α}_{00}} \\ = = - - {λ λ}_{22} s the s - - {λ λ}_{33} s the s i i g g n no ((s the s)) | | s the s {| |}^{{α α}_{00}} \end{matrix} - - - - - - ((1919))$

定义滑模控制器李雅普诺夫方程：Define the Lyapunov equation for the sliding mode controller:

$V V ((t t)) = = \frac{11}{22} {s the s}^{22} - - - - - - ((2020))$

则有：Then there are:

$\begin{matrix} \overset{\cdot &Center Dot;}{V V} ((t t)) = = s the s \overset{\cdot &Center Dot;}{s the s} \\ = = s the s ((- - {λ λ}_{22} s the s - - {λ λ}_{33} s the s i i g g n no ((s the s)) | | s the s {| |}^{{α α}_{00}})) \\ = = - - 22 {λ λ}_{22} V V ((t t)) - - 22^{(({α α}_{00} + + 11)) / / 22} {λ λ}_{33} {V V}^{(({α α}_{00} + + 11)) / / 22} ((t t)) \end{matrix} - - - - - - ((21 twenty one))$

则滑模控制器滑模面s将在有限时间内为零，即存在一个时间点t₁，在t₁之后有s＝0，由式(16)可知，此时有：Then the sliding mode surface s of the sliding mode controller will be zero within a finite time, that is, there is a time point t ₁ , and s=0 after t ₁ , it can be seen from formula (16), at this time:

${\overset{\cdot &Center Dot;}{e e}}_{11} ((t t)) = = {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} - - - - - - ((22 twenty two))$

则有：Then there are:

$\begin{matrix} {\overset{\cdot &Center Dot;}{e e}}_{00} ((t t)) = = {e e}_{11} ((t t)) - - {s the s}_{11} \\ {\overset{\cdot &Center Dot;}{e e}}_{11} ((t t)) = = {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} \end{matrix} - - - - - - ((23 twenty three))$

又因s₁、s₂也是有限时间内为0，设t₁为s为零的时刻，t₂为s₁为零的时刻，则存在t₃＝max{t₁,t₂}，经过t₃时刻后有：And because s ₁ and s ₂ are also 0 within a limited time, let t ₁ be the time when s is zero, and t ₂ be the time when s ₁ is zero, then there exists t ₃ =max{t ₁ ,t ₂ }, after t After ₃ moments there is:

$\begin{matrix} {\overset{\cdot &Center Dot;}{e e}}_{00} ((t t)) = = {e e}_{11} ((t t)) \\ {\overset{\cdot &Center Dot;}{e e}}_{11} ((t t)) = = {λ λ}_{00} s the s i i g g n no (({e e}_{00} ((t t)))) | | {e e}_{00} ((t t)) {| |}^{{α α}_{11}} + + {λ λ}_{11} s the s i i g g n no (({e e}_{11} ((t t)))) | | {e e}_{11} ((t t)) {| |}^{{α α}_{22}} \end{matrix} - - - - - - ((24 twenty four))$

又因若存在一系统如式(24)所示，且各参数λ₀、λ₁、α₁、α₂均大于零，并且λ₀、λ₁满足表达式z²+λ₁z+λ₀是Hurwitz的，其中_z为微分算子，α₁、α₂满足α₂∈(0,1)，则该系统状态e₀(t)、e₁(t)将在有限时间内稳定到平衡点，即存在一时间点t₄，e₀(t)、e₁(t)将在t₄之后收敛到零，即系统的跟踪误差有限时间内为零。And because if there exists a system as shown in formula (24), and all parameters λ ₀ , λ ₁ , α ₁ , α ₂ are greater than zero, and λ ₀ , λ ₁ satisfy the expression z ² +λ ₁ z+λ ₀ is Hurwitz's, where _z is a differential operator, and α ₁ and α ₂ satisfy α ₂ ∈ (0,1), then the system state e ₀ (t), e ₁ (t) will stabilize to the equilibrium point within a finite time, that is, there is a time point t ₄ , e ₀ (t), e ₁ (t) will converge to zero after _t4 , that is, the tracking error of the system is zero in a finite time.

综上可知，针对直驱电机系统(2)设计的干扰补偿的有限时间连续滑模控制方法使系统得到有限时间内跟踪误差为零的结果，调节观测器系数k₁、β₁、ε₁、p₁、q₁、可以使观测器的跟踪误差在有限时间内趋于零，调节控制器参数λ₀、λ₁、α₁、α₂、λ₂、λ₃、α₀可以使系统的跟踪误差在有限时间内趋于零。直驱电机系统干扰补偿的有限时间连续滑模控制器原理示意图如图3所示。通过获取的系统状态x₁、x₂，期望跟踪指令x_d构造控制器滑模面s和观察器滑模面s₁，通过滑模干扰观测器观测直驱电机系统的外干扰，将观测到的外干扰传递给干扰补偿的有限时间连续滑模控制器，控制器计算出控制量后作用到电机驱动器中，从而控制直驱电机跟踪期望指令。In summary, the finite-time continuous sliding mode control method of disturbance compensation designed for the direct drive motor system (2) enables the system to obtain the result that the tracking error is zero within a finite time, and adjust the observer coefficients k ₁ , β ₁ , ε ₁ , p ₁ , q ₁ , can make the tracking error of the observer tend to zero within a limited time, and adjust the controller parameters λ ₀ , λ ₁ , α ₁ , α ₂ , λ ₂ , λ ₃ , α ₀ to make the tracking of the system The error tends to zero in a finite time. The schematic diagram of the finite-time continuous sliding mode controller for disturbance compensation of the direct drive motor system is shown in Fig. 3 . Through the obtained system state x ₁ , x ₂ , the expected tracking command x _d constructs the controller sliding mode surface s and the observer sliding mode surface s ₁ , and observes the external disturbance of the direct drive motor system through the sliding mode disturbance observer, it will be observed The external disturbance is transmitted to the finite-time continuous sliding mode controller for disturbance compensation. The controller calculates the control quantity and acts on the motor driver, so as to control the direct drive motor to track the desired command.

实施例Example

为考核所设计的控制器性能，在仿真中取如下参数对直驱电机系统进行建模：In order to assess the performance of the designed controller, the following parameters are taken in the simulation to model the direct drive motor system:

惯性负载参数m＝0.00026kg·m²；粘性摩擦系数B＝0.00143·m·s/rad；力矩放大系数k_u＝1.11N·m/V；Inertial load parameter m=0.00026kg·m ² ; viscous friction coefficient B=0.00143·m·s/rad; torque amplification factor k _u =1.11N·m/V;

给定系统的期望指令为：x_d＝20sin(t)[1-exp(-0.01t³)]o；The desired command of a given system is: x _d =20sin(t)[1-exp(-0.01t ³ )]o;

干扰水平：d(x,t)＝(0.1/0.00026)sin(0.5πt)[1-exp(-0.01t³)]N·m。Interference level: d(x,t)=(0.1/0.00026) sin(0.5πt)[1-exp(-0.01t ³ )]N·m.

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

干扰补偿的有限时间稳定连续滑模控制(UCFT‐SMC)控制器：取干扰观测器参数k₁＝5000、β₁＝500、ε₁＝0.05、p₁＝3和q₁＝5；控制器参数λ₀＝32、λ₁＝36、α₁＝0.25、α₂＝0.4、λ₂＝56、λ₃＝60、α₀＝0.3。Finite-time stable continuous sliding mode control (UCFT-SMC) controller for disturbance compensation: take disturbance observer parameters k ₁ =5000, β ₁ =500, ε ₁ =0.05, p ₁ =3 and q ₁ =5; the controller Parameters λ ₀ =32, λ ₁ =36, α ₁ =0.25, α ₂ =0.4, λ ₂ =56, λ ₃ =60, α ₀ =0.3.

滑模控制器(SMC)：为了迫使系统状态到达滑模面，选取的控制器参数为λ₀＝32、λ₁＝36、k＝40。Sliding mode controller (SMC): In order to force the system state to reach the sliding mode surface, the selected controller parameters are λ ₀ =32, λ ₁ =36, k=40.

在系统存在模型不确定性d(x,t)＝(0.1/0.00026)sin(0.5πt)[1-exp(-0.01t³)]N·m时，UC‐SMC控制器作用下系统输出对期望指令的跟踪、跟踪误差曲线如图4，图5所示；图4中期望指令和UCFT‐SMC控制器作用下系统输出曲线几乎重合，同时，结合图5，可知，在UCFT‐SMC控制器作用下系统具有良好的跟踪性能，稳态跟踪误差在0.08°内；When the system has model uncertainty d(x,t)=(0.1/0.00026)sin(0.5πt)[1-exp(-0.01t ³ )]N m, the system output under the action of UC-SMC controller The tracking and tracking error curves of the expected command are shown in Figure 4 and Figure 5; in Figure 4, the expected command and the system output curve under the action of the UCFT-SMC controller almost coincide. At the same time, combined with Figure 5, it can be seen that the UCFT-SMC controller Under the action, the system has good tracking performance, and the steady-state tracking error is within 0.08°;

图6、图7为UCFT‐SMC控制器作用下干扰观测曲线和观测误差随时间变化曲线，图6中干扰观测曲线与系统中实际干扰曲线基本重合；从图6、图7中可以看出，所设计的干扰观测器对系统存在的模型不确定观测非常准确，并且由图7可知，在经过很短的时间后观测误差迅速收敛到较小的界内，约为0.15Nm。Fig. 6 and Fig. 7 are the interference observation curve and observation error curve with time under the action of UCFT-SMC controller. The interference observation curve in Fig. 6 basically coincides with the actual interference curve in the system; it can be seen from Fig. 6 and Fig. 7 that The designed disturbance observer is very accurate in observing the model uncertainty existing in the system, and it can be seen from Fig. 7 that the observation error quickly converges to a small boundary, about 0.15Nm, after a short period of time.

图8、图9为UCFT‐SMC控制器作用下和传统滑模控制器(SMC)作用下系统的控制输入随时间的变化曲线。传统的滑模控制方法主要是通过增大控制器的鲁棒性来克服模型不确定性从而迫切系统状态到达滑模面，由于不连续项增益取值较大，从图9中可以看出，系统的控制输入出现了大幅度的抖振，这在实际工程运用中将可能激发高频动态，严重时将使系统发散。而具有干扰补偿的有限时间连续滑模控制方法(UCFT‐SMC)由于补偿了系统存在的干扰，同时使控制输入连续化，从而从理论上消除了滑模控制器的不连续项，消除了滑模控制的抖振。从图8中亦可看出，系统的控制输入为一低频连续曲线，不存在高频抖振，便于在工程实际中运用，从而验证了UCFT‐SMC可以消除滑模控制的抖振问题。Fig. 8 and Fig. 9 are the control input curves of the system under the action of UCFT-SMC controller and traditional sliding mode controller (SMC) over time. The traditional sliding mode control method mainly overcomes the model uncertainty by increasing the robustness of the controller so that the state of the system reaches the sliding mode surface urgently. Since the gain of the discontinuity term is large, it can be seen from Figure 9 that The control input of the system has a large amount of chattering, which may stimulate high-frequency dynamics in actual engineering applications, and will cause the system to diverge in severe cases. The finite-time continuous sliding mode control method with disturbance compensation (UCFT-SMC) compensates the disturbance in the system and makes the control input continuous at the same time, thus theoretically eliminating the discontinuous term of the sliding mode controller and eliminating the slippage. mode controlled chattering. It can also be seen from Figure 8 that the control input of the system is a low-frequency continuous curve, and there is no high-frequency chattering, which is convenient for application in engineering practice, thus verifying that UCFT-SMC can eliminate the chattering problem of sliding mode control.

图10为UCFT‐SMC控制器作用下和传统滑模控制器(SMC)作用下系统的跟踪误差随时间的变化曲线。从图10中两种控制器的跟踪误差对比可以看出本发明所提出的UCFT‐SMC控制器的跟踪误差相较于SMC控制器要小，UCFT‐SMC控制器的跟踪误差的幅值约为0.08°，SMC控制器的稳态跟踪误差的幅值约为0.1°。Fig. 10 is the tracking error curve of the system under the action of UCFT-SMC controller and traditional sliding mode controller (SMC) over time. From the comparison of the tracking errors of the two controllers in Figure 10, it can be seen that the tracking error of the UCFT-SMC controller proposed by the present invention is smaller than that of the SMC controller, and the magnitude of the tracking error of the UCFT-SMC controller is about 0.08°, the magnitude of the steady-state tracking error of the SMC controller is about 0.1°.

Claims

1. a finite time continuous sliding mode control method for direct drive motor system disturbance compensation, is characterized in that, comprises the following steps:

Step 1, establishing a mathematical model of the direct drive motor system;

Step 2, design the disturbance observer of the direct drive motor system;

Step 3, design a continuous sliding mode controller based on the finite-time convergence of the disturbance observer.

2. the finite time continuous sliding mode control method of direct drive motor system disturbance compensation according to claim 1, is characterized in that, the mathematical model of setting up direct drive motor system described in step 1 is specifically as follows:

Step 1-1. The direct drive motor system directly drives the inertial load through the permanent magnet DC motor equipped with an electric driver; according to Newton's second law, the motion equation of the direct drive motor system is:

In formula (1), m is the inertial load parameter, k _i is the moment amplification factor, B is the viscous friction coefficient, is the modeling error, y is the displacement of the inertial load, is the speed of the inertial load, u is the control input of the system, and t is the time variable;

Step 1-2, define state variables: Then the equation of motion (1) is transformed into the equation of state:

In formula (2), are nominal and known; is the total interference of the system, f(t,x ₁ ,x ₂ ) is the above x ₁ represents the displacement of the inertial load, x ₂ represents the velocity of the inertial load;

Because in the direct drive motor system, the state and parameters of the system are bounded, so the total disturbance d(x,t) of the system satisfies:

|d(x,t)|≤D(3)

In formula (3), D is a known constant, that is, d(x, t) has a known upper bound.

3. the finite time continuous sliding mode control method of direct drive motor system disturbance compensation according to claim 1, is characterized in that, the described disturbance observer of design direct drive motor system in step 2, steps are as follows:

Define the observer sliding mode surface _s1 as:

s ₁ =z ₁ -x ₂ (4)

Among them, z ₁ is the internal dynamics of the observer;

In formula (5), k ₁ , β ₁ , ε ₁ , p ₁ and q ₁ are all interference observer coefficients; p ₁ <q ₁ , and they are all positive odd numbers; k ₁ , β ₁ , and ε ₁ are all positive number, β ₁ ≥ D;

Then design the observer of the disturbance d(x,t) of the direct drive motor system as:

From formula (2), (4), (5) have:

.

4. the finite time continuous sliding mode control method of direct drive motor system disturbance compensation according to claim 3, is characterized in that, the observation accuracy test process of described disturbance observer is:

Define the disturbance observer Lyapunov equation:

And because β ₁ ≥ D, then:

If there is a positive definite function V ₀ (t) satisfying the following inequality:

Then, V ₀ (t) converges to the equilibrium point within time t _s , where,

Among them, α>0, λ>0, 0<γ<1;

Therefore, V ₁ (t) will converge to the equilibrium point within a finite time, that is, there is a time t ₂ point, after t ₂ , V ₁ ( _t ) is always zero, and the expression (9 ) we can see that after V ₁ (t) is zero, s ₁ is also zero, at this time It will also converge to zero, and because of the estimation error of d(x,t)

The estimation error of the disturbance will also be 0 for a finite time t ₂ ; that is, after t ₂ there is

Get the disturbance observer:

5. the finite-time continuous sliding mode control method of direct-drive motor system disturbance compensation according to claim 1, is characterized in that, the design described in step 3 is based on the continuous sliding mode controller of the finite-time convergence of disturbance observer, specifically as follows:

Define the position tracking error e ₀ (t) and error variable e ₁ (t) of the direct drive motor system:

e ₀ (t) = x ₁ -x _d (t) (14)

Among them, x _d (t) is the system reference position signal, x _d (t) is second-order continuous, and the system reference position signal x _d (t), system reference speed signal System Reference Acceleration Signal are bounded;

Define the sliding mode controller sliding surface s:

Among them, λ ₀ , λ ₁ , α ₁ , and α ₂ are sliding mode controller parameters, and they are all greater than zero, and λ ₀ and λ ₁ satisfy the expression z ² +λ ₁ z+λ ₀ is Hurwitz, where z is Differential operator, α ₁ , α ₂ satisfy α ₂ ∈ (0,1); then,

The sliding mode controller u is obtained as:

Where λ ₂ , λ ₃ , and α ₀ are controller parameters, and λ ₂ >0, λ ₃ >0, 0<α ₀ <1.

6. the finite-time continuous sliding mode control method of direct-drive motor system disturbance compensation according to claim 5, is characterized in that, described direct-drive motor system overall stability and tracking error are zero tests in a limited time, specifically as follows:

Substituting formula (18) into formula (17) has:

Define the Lyapunov equation for the sliding mode controller:

Then there are:

Then the sliding mode surface s of the sliding mode controller will be zero within a finite time, that is, there is a time point t ₁ , and s=0 after t ₁ , it can be seen from formula (16), at this time:

Then there are:

And because s ₁ and s ₂ are also 0 in a finite time, t ₁ is the moment when s is zero, and t ₂ is the moment when s ₁ is zero, then there exists t ₃ =max{t ₁ ,t ₂ }, after t ₃ After the moment there is:

And because if there exists a system as shown in formula (24), and all parameters λ ₀ , λ ₁ , α ₁ , α ₂ are greater than zero, and λ ₀ , λ ₁ satisfy the expression z ² +λ ₁ z+λ ₀ is Hurwitz's, where z is a differential operator, and α ₁ and α ₂ satisfy α ₂ ∈ (0,1), then the system state e ₀ (t), e ₁ (t) will stabilize to the equilibrium point within a finite time, that is, there is a time point t ₄ , e ₀ (t), e ₁ (t) will converge to zero after _t4 , that is, the tracking error of the system is zero in a finite time.