CN103019092B

CN103019092B - A kind of forecast Control Algorithm of machine driven system locating platform

Info

Publication number: CN103019092B
Application number: CN201210594298.4A
Authority: CN
Inventors: 董瑞丽; 谭永红; 陈辉
Original assignee: Shanghai Normal University
Current assignee: Shanghai Normal University
Priority date: 2012-12-31
Filing date: 2012-12-31
Publication date: 2016-07-06
Anticipated expiration: 2032-12-31
Also published as: CN103019092A

Abstract

The invention discloses a predictive control method for a positioning platform of a mechanical transmission system. The mechanical transmission system is equipped with a ball screw. The output rotation angle of the servo motor included in the positioning platform of the mechanical transmission system is used as the input signal u(k ), through the gear box speed change and the mechanical transmission of the lead screw, and then drive the worktable to work with load, the displacement y(k) of the worktable is the output of the whole system, and the control error ε is given, According to the system model, the system’s working intervals 4→1→2 and 2→3→4 can be estimated. In step k, the Clarke subgradient of J(k) with respect to u(k) can be calculated at the non-smooth point ,ie, Where j∈J _k , J _k ={1,2,…,t}, |J _k | is the number of elements of J _k , |J _k |≤t ₁ given bounded natural number, t=t +1, if t≤t ₁ , then J _k ={1,…,t}; if t>t ₁ , then J _k =J _k-1 ∪{t}\{tt ₁ }; or in smooth point, The gradient of J(k) with respect to u(k) <maths num="0001"> </maths> where <maths num="0002"> </maths> We make <maths num="0003"> </maths> or <maths num="0004"> </maths> Then we can find the quasi-input u ₁ (k) of the system.

Description

A predictive control method for positioning platform of mechanical transmission system

技术领域technical field

本发明属于机械传动控制技术领域，特别涉及一种机械传动系统定位平台的预测控制方法。The invention belongs to the technical field of mechanical transmission control, in particular to a predictive control method for a positioning platform of a mechanical transmission system.

背景技术Background technique

许多常见的精密加工系统所需的设备，譬如高精度万能铣床、三坐标测量仪等，往往采用含有滚珠丝杠装置的机械传动机构的工作平台，这些机械传动机构由于配合或磨损造成的死区、间隙等非线性特性，而这些特性对系统的控制性能影响是不能忽略的，它会降低系统的控制精度，有时会引起系统的抖动，甚至会出现系统的不稳定。因此，现存的不少技术大都是对这些非线性特性进行补偿，由于这些非线性特性建模的时候会有比较大的误差，并且随着设备的磨损、老化，模型的误差会越来越大，因此会存在着较大的补偿误差，这在超精密加工设备中体现得更为明显，能否找到一种不必对这些非线性特性进行直接补偿，同时又能消除其影响，获得满意控制性能的实时控制方法，成为提高超精密系统控制精度和性能的关键和难点。The equipment required for many common precision machining systems, such as high-precision universal milling machines, three-coordinate measuring instruments, etc., often use the working platform of the mechanical transmission mechanism containing the ball screw device. The dead zone of these mechanical transmission mechanisms due to fit or wear , clearance and other nonlinear characteristics, and the impact of these characteristics on the control performance of the system cannot be ignored, it will reduce the control accuracy of the system, sometimes cause system jitter, and even system instability. Therefore, most of the existing technologies are to compensate these nonlinear characteristics, because there will be relatively large errors when modeling these nonlinear characteristics, and as the equipment wears out and ages, the error of the model will become larger and larger , so there will be a large compensation error, which is more obvious in ultra-precision processing equipment. Can we find a method that does not need to directly compensate these nonlinear characteristics, but can eliminate its influence and obtain satisfactory control performance? The real-time control method has become the key and difficult point to improve the control accuracy and performance of ultra-precision systems.

发明内容Contents of the invention

本发明提出一种不必对这类系统中滚珠丝杠产生的间隙这一类非光滑的非线性特性进行直接补偿、却能消除间隙影响的非光滑预测控制方法。The invention proposes a non-smooth predictive control method that does not need to directly compensate the non-smooth nonlinear characteristics such as the gap produced by the ball screw in this type of system, but can eliminate the effect of the gap.

本发明的技术方案是，一种机械传动系统定位平台的预测控制方法，该机械传动系统带有滚珠丝杠，所述的机械传动系统定位平台包括的饲服电机的输出转角作为系统的输入信号u(k),经过齿轮箱变速及丝杠的机械传动，再带动工作台负载工作，工作台的位移y(k),即为整个系统的输出，其中，The technical solution of the present invention is a predictive control method for a positioning platform of a mechanical transmission system, the mechanical transmission system has a ball screw, and the output rotation angle of the servo motor included in the positioning platform of the mechanical transmission system is used as the input signal of the system u(k), through the gear box speed change and the mechanical transmission of the lead screw, and then drives the workbench to work under load, the displacement y(k) of the workbench is the output of the entire system, where,

所述定位平台的间隙用来描述齿轮箱与丝杠的机械传动的非线性特性，The clearance of the positioning platform is used to describe the nonlinear characteristics of the mechanical transmission of the gearbox and the lead screw,

L₁(·)是一个线性的动态子模型，用来描述工作平台的负载，L ₁ (·) is a linear dynamic sub-model used to describe the load of the working platform,

机械传动系统的非线性特性的输出，即x(k)，且不能直接测量，The output of the nonlinear characteristic of the mechanical transmission system, namely x(k), and cannot be directly measured,

所述定位平台的输入和输出为u(k)和y(k)The input and output of the positioning platform are u(k) and y(k)

所述的间隙的特性描述为，The properties of the gap are described as,

$\overset{^^}{x x} ((k k)) = = \{\begin{matrix} {\overset{^^}{m m}}_{11} ((u u ((k k)) - - {\overset{^^}{D D.}}_{11})),, & u u ((k k)) > > \frac{\overset{^^}{x x} ((k k - - 11))}{{\overset{^^}{m m}}_{11}} + + {\overset{^^}{D D.}}_{11} and and & u u ((k k)) > > u u ((k k - - 11)) \\ \overset{^^}{x x} ((k k - - 11)),, & \frac{\overset{^^}{x x} ((k k - - 11))}{{\overset{^^}{m m}}_{22}} - - {\overset{^^}{D D.}}_{22} \leq \leq u u ((k k)) \leq \leq \frac{\overset{^^}{x x} ((k k - - 11))}{{\overset{^^}{m m}}_{11}} + + {\overset{^^}{D D.}}_{11} \\ {\overset{^^}{m m}}_{22} ((u u ((k k)) + + {\overset{^^}{D D.}}_{22})),, & u u ((k k)) < < \frac{\overset{^^}{x x} ((k k - - 11))}{{\overset{^^}{m m}}_{22}} - - {\overset{^^}{D D.}}_{22} and and & u u ((k k)) < < ((k k - - 11)) \end{matrix} - - - - - - ((11))$

其中：和分别是间隙模型上升和下降的斜率，和分别是模型上升和下降的记忆区的绝对值，并且 $0 < {\hat{m}}_{1} < \infty,$ $0 < {\hat{m}}_{2} < \infty,$ $0 < {\hat{D}}_{1} < \infty$ 和 $0 < {\hat{D}}_{2} < \infty,$ in: and are the rising and falling slopes of the gap model, respectively, and are the absolute values of the model's ascending and descending memory areas, respectively, and $0 < {\hat{m}}_{1} < \infty,$ $0 < {\hat{m}}_{2} < \infty,$ $0 < {\hat{D.}}_{1} < \infty$ and $0 < {\hat{D.}}_{2} < \infty,$

L₁(·)模型可以表述为：The L ₁ (·) model can be expressed as:

$\overset{^^}{y the y} ((k k)) = = - - {Σ Σ}_{i i = = 11}^{{n no}_{a a}} {\overset{^^}{a a}}_{i i} y the y ((k k - - i i)) + + {Σ Σ}_{j j = = 00}^{{n no}_{b b}} {\overset{^^}{b b}}_{j j} \overset{^^}{x x} ((k k - - j j - - d d)) - - - - - - ((22))$

其中n_a和n_b是线性子模型的阶次，d是预测模型的预测步长，和是线性子模型的系数，where n _a and n _b are the order of the linear submodel, d is the prediction step size of the prediction model, and are the coefficients of the linear submodel,

由式（1）和（2）组成机械传动系统的模型，则有线性子系统表示成The model of the mechanical transmission system is composed of formulas (1) and (2), and the linear subsystem is expressed as

$y the y ((k k)) = = \overset{^^}{y the y} ((k k)) + + ϵ ϵ ((k k))$

其中ε(k)为模型误差，设其为零均值的白噪声，根据Diophantine方程Where ε(k) is the model error, which is assumed to be zero-mean white noise, according to the Diophantine equation

$11 = = F f (({z z}^{- - 11})) \overset{^^}{A A} (({z z}^{- - 11})) + + {z z}^{- - d d} G G (({z z}^{- - 11})) - - - - - - ((33))$

其中 $F (z^{- 1}) = 1 + Σ_{i = 1}^{d} f_{i} z^{- i},$ $\hat{A} (z^{- 1}) = 1 + Σ_{i = 1}^{n_{a}} {\hat{a}}_{i} z^{- i}$ 和 $G (z^{- 1}) = Σ_{j = 0}^{n_{a}} g_{j} z^{- j} .$ 相应的d步超前预测模型为in $f (z^{- 1}) = 1 + Σ_{i = 1}^{d} f_{i} z^{- i},$ $\hat{A} (z^{- 1}) = 1 + Σ_{i = 1}^{{no}_{a}} {\hat{a}}_{i} z^{- i}$ and $G (z^{- 1}) = Σ_{j = 0}^{{no}_{a}} g_{j} z^{- j} .$ The corresponding d-step ahead prediction model is

$\overset{^^}{y the y} ((k k + + d d)) = = G G (({z z}^{- - 11})) y the y ((k k)) + + Q Q (({z z}^{- - 11})) \overset{^^}{x x} ((k k)) - - - - - - ((44))$

其中 $Q (z^{- 1}) = F (z^{- 1}) \hat{B} (z^{- 1}) = Σ_{i = 0}^{n_{b} + d} q_{i} z^{- i}$ 和 $\hat{B} (z^{- 1}) = Σ_{j = 0}^{n_{b}} {\hat{b}}_{j} z^{- j},$ in $Q (z^{- 1}) = f (z^{- 1}) \hat{B} (z^{- 1}) = Σ_{i = 0}^{{no}_{b} + d} q_{i} z^{- i}$ and $\hat{B} (z^{- 1}) = Σ_{j = 0}^{{no}_{b}} {\hat{b}}_{j} z^{- j},$

采用式（5）来作为预测控制的目标函数，Formula (5) is used as the objective function of predictive control,

$J J ((k k)) = = \frac{[[r r ((k k + + d d)) - - \overset{^^}{y the y} ((k k + + d d)) {]]}^{22}}{22} + + \frac{λ λ [[u u ((k k)) - - u u ((k k - - 11)) {]]}^{22}}{22} - - - - - - ((55))$

其中，r(k+d)是参考轨迹，是预测模型的输出，λ是一个非负的加权系数，J(k)是Lipschitz连续的，在该优化函数的非光滑点，用次微分来替代传统意义的梯度，J(k)是伪凸的，在非光滑点存在唯一的最优值，在该点的广义梯度为0，Among them, r(k+d) is the reference trajectory, is the output of the prediction model, λ is a non-negative weighting coefficient, J(k) is Lipschitz continuous, at the non-smooth point of the optimization function, sub-differential is used to replace the traditional gradient, J(k) is pseudo-convex , there is a unique optimal value at the non-smooth point, and the generalized gradient at this point is 0,

在间隙的非光滑点，J(k)关于u(k)的Clarke次梯度可以描述为：At the non-smooth point of the gap, the Clarke subgradient of J(k) with respect to u(k) can be described as:

${&PartialD; &PartialD;}_{u u ((k k))} J J ((k k)) &Element; &Element; conv conv {{{&dtri; &dtri;}_{u u ((k k))} J J ((k k))}} - - - - - - ((66))$

其中是J(k)关于u(k)的在非光滑点附近的光滑点处的梯度，in is the gradient of J(k) with respect to u(k) at smooth points near non-smooth points,

J(k)关于u(k)在光滑点处的梯度为：The gradient of J(k) with respect to u(k) at the smooth point for:

${&dtri; &dtri;}_{u u ((k k))} J J ((k k)) = = \{\begin{matrix} λ λ [[u u ((k k)) - - u u ((k k - - 11))]] - - [[r r ((k k + + d d)) - - \overset{^^}{y the y} ((k k + + d d))]] {q q}_{00} {m m}_{11},, & u u ((k k)) > > \frac{x x ((k k - - 11))}{{m m}_{11}} + + {D D.}_{11} and and & u u ((k k)) > > u u ((k k - - 11)) \\ λ λ [[u u ((k k)) - - u u ((k k - - 11))]] + + 00,, & \frac{x x ((k k - - 11))}{{m m}_{22}} - - {D D.}_{22} < < u u ((k k)) < < \frac{x x ((k k - - 11))}{{m m}_{11}} + + {D D.}_{11} \\ λ λ [[u u ((k k)) - - u u ((k k - - 11))]] - - [[r r ((k k + + d d)) - - \overset{^^}{y the y} ((k k + + d d))]] {q q}_{00} {m m}_{22},, & u u ((k k)) < < \frac{x x ((k k - - 11))}{{m m}_{22}} - - {D D.}_{22} and and & u u ((k k)) < < u u ((k k - - 11)) \end{matrix} - - - - - - ((77))$

则所述的预测控制方法包括步骤，Then described predictive control method comprises steps,

第一步,给出控制误差ε,即： In the first step, the control error ε is given, namely:

第二步,根据系统模型，系统的工作区间4→1→2和2→3→4都能估计出来，在第k步，我们能计算出，在非光滑点，J(k)关于u(k)的Clarke次梯度,i.e.,其中j∈J_k，J_k={1,2,…,t},|J_k|是J_k的元素的个数，|J_k|≤t₁给定的有界的自然数，t＝t+1，如果t≤t₁,那么J_k={1，…,t}；如果t>t₁,那么J_k=J_k-1∪{t}\{t-t₁}；或者在光滑点，J(k)关于u(k)的梯度 ${&dtri;}_{u (k)} J (k);$ In the second step, according to the system model, the working intervals 4→1→2 and 2→3→4 of the system can be estimated. In step k, we can calculate that at the non-smooth point, J(k) is about u( k) Clarke subgradient, ie, Where j∈J _k , J _k ={1,2,…,t}, |J _k | is the number of elements of J _k , |J _k |≤t ₁ given bounded natural number, t=t +1, if t≤t ₁ , then J _k ={1,…,t}; if t>t ₁ , then J _k =J _k-1 ∪{t}\{tt ₁ }; or in smooth point, The gradient of J(k) with respect to u(k) ${&dtri;}_{u (k)} J (k);$

第三步， ${&PartialD;}_{u (k)} J (k) = Σ_{j = 1}^{t} χ_{i} {&PartialD;}_{u (k)}^{i} J (k),$ 其中 $Σ_{j = 1}^{t} χ_{i} = 1 .$ 我们令 ${&PartialD;}_{u (k)} J (k) = 0$ 或者 ${&dtri;}_{u (k)} J (k) = 0,$ 那么我们就能求出系统的准输入u₁(k)；third step, ${&PartialD;}_{u (k)} J (k) = Σ_{j = 1}^{t} χ_{i} {&PartialD;}_{u (k)}^{i} J (k),$ in $Σ_{j = 1}^{t} χ_{i} = 1 .$ we order ${&PartialD;}_{u (k)} J (k) = 0$ or ${&dtri;}_{u (k)} J (k) = 0,$ Then we can find the quasi-input u ₁ (k) of the system;

第四步,如果并且t₁∈(0,1),那么u(k)=u₁(k),我们转入第六步.如果|r(k+d)-y(k+d)|≥t₁ε,那么我们转入第五步；Step four, if And t ₁ ∈ (0,1), then u(k)=u ₁ (k), we turn to the sixth step. If |r(k+d)-y(k+d)|≥t ₁ ε, Then we go to the fifth step;

第五步,根据求出u(k)=u_j(k)，如果那么u(k)=u_j(k)，否则，u(k)=u(k-1)，转到第六步；The fifth step, according to Find u(k)=u _j (k), if Then u(k)=u _j (k), otherwise, u(k)=u(k-1), go to the sixth step;

第六步,k＝k+1,转入第二步。In the sixth step, k=k+1, turn to the second step.

本发明是针对带有滚珠丝杠一类机械传动装置的定位平台的一种预测控制方法。在机械传动中滚珠丝杠由于配合及磨损所产生的间隙现象不能直接测量，如果不对间隙的影响进行直接补偿的情况下，用一种非光滑的预测控制策略消除间隙对机械传动的不利影响。用非光滑的优化技术来设计该非光滑系统的预测控制器，解决了非光滑点的优化预测控制问题，不必对定位平台中滚珠丝杠产生的间隙进行直接补偿，避免了其他控制方法中需要对间隙求逆的困难。The invention is a predictive control method for a positioning platform with a mechanical transmission device such as a ball screw. In the mechanical transmission, the gap phenomenon caused by the fit and wear of the ball screw cannot be directly measured. If the effect of the gap is not directly compensated, a non-smooth predictive control strategy is used to eliminate the adverse effect of the gap on the mechanical transmission. Using non-smooth optimization technology to design the predictive controller of the non-smooth system solves the problem of optimal predictive control of non-smooth points, and does not need to directly compensate the gap generated by the ball screw in the positioning platform, avoiding the need for other control methods. The difficulty of inverting the gap.

附图说明Description of drawings

图1本发明的含有滚珠丝杠的机械传动机构工作平台的实际物理结构示意图。Fig. 1 is a schematic diagram of the actual physical structure of the working platform of the mechanical transmission mechanism containing the ball screw of the present invention.

图2本发明的采用滚动丝杠的机械传动机构定位平台模型的结构图。Fig. 2 is a structural diagram of a positioning platform model of a mechanical transmission mechanism using a rolling screw according to the present invention.

具体实施方式detailed description

含有机械传动机构工作平台的预测模型，例如，含有滚珠丝杠的机械传动机构工作平台的实际物理结构，可以用如图1所示，The predictive model of the working platform with mechanical transmission mechanism, for example, the actual physical structure of the working platform with mechanical transmission mechanism including ball screw, can be used as shown in Fig. 1,

其中饲服电机的输出转角作为系统的输入信号u(k),经过齿轮箱变速及丝杠的机械传动，再带动工作台（负载）工作，工作台的位移y(k),即为整个系统的输出。The output rotation angle of the feeding motor is used as the input signal u(k) of the system. After the gear box speed change and the mechanical transmission of the lead screw, it drives the workbench (load) to work. The displacement y(k) of the workbench is the whole system Output.

图1所示的系统可以进一步用图2所示的模型结构图表示，其中间隙用来描述齿轮箱与丝杠的机械传动的非线性特性，L₁(·)是一个线性的动态子模型，用来描述工作平台的负载，机械传动的非线性特性的输出，即x(k)不能直接测量，只有整个系统的输入和输出，即：u(k)和y(k)可以直接测量。图2中，间隙的特性可以描述为：The system shown in Fig. 1 can be further represented by the model structure diagram shown in Fig. 2, where the gap is used to describe the nonlinear characteristics of the mechanical transmission between the gearbox and the lead screw, and L ₁ (·) is a linear dynamic sub-model, Used to describe the load of the working platform, the output of the nonlinear characteristics of the mechanical transmission, that is, x(k) cannot be directly measured, only the input and output of the entire system, namely: u(k) and y(k) can be directly measured. In Figure 2, the characteristics of the gap can be described as:

其中：和分别是间隙模型上升和下降的斜率，和分别是模型上升和下降的记忆区的绝对值，并且 $0 < {\hat{m}}_{1} < \infty,$ $0 < {\hat{m}}_{2} < \infty,$ $0 < {\hat{D}}_{1} < \infty$ 和 $0 < {\hat{D}}_{2} < \infty .$ in: and are the rising and falling slopes of the gap model, respectively, and are the absolute values of the model's ascending and descending memory areas, respectively, and $0 < {\hat{m}}_{1} < \infty,$ $0 < {\hat{m}}_{2} < \infty,$ $0 < {\hat{D.}}_{1} < \infty$ and $0 < {\hat{D.}}_{2} < \infty .$

L₁(·)模型可以表述为：The L ₁ (·) model can be expressed as:

其中n_a和n_b是线性子模型的阶次，d是预测模型的预测步长，和是线性子模型的系数。where n _a and n _b are the order of the linear submodel, d is the prediction step size of the prediction model, and are the coefficients of the linear submodel.

由此，式（1）和（2）就组成了机械传动机构的模型。则线性子系统可以表示成Thus, formulas (1) and (2) constitute the model of the mechanical transmission mechanism. Then the linear subsystem can be expressed as

$y the y ((k k)) = = \overset{^^}{y the y} ((k k)) + + ϵ ϵ ((k k))$

其中ε(k)为模型误差，设其为零均值的白噪声。根据Diophantine方程Where ε(k) is the model error, which is assumed to be zero-mean white noise. According to the Diophantine equation

其中 $F (z^{- 1}) = 1 + Σ_{i = 1}^{d} f_{i} z^{- i},$ $\hat{A} (z^{- 1}) = 1 + Σ_{i = 1}^{n_{a}} {\hat{a}}_{i} z^{- i}$ 和 $G (z^{- 1}) = Σ_{j = 0}^{n_{a}} g_{j} z^{- j} .$ 相应的d步超前预测模型为in $f (z^{- 1}) = 1 + Σ_{i = 1}^{d} f_{i} z^{- i},$ $\hat{A} (z^{- 1}) = 1 + Σ_{i = 1}^{{no}_{a}} {\hat{a}}_{i} z^{- i}$ and $G (z^{- 1}) = Σ_{j = 0}^{{no}_{a}} g_{j} z^{- j} .$ The corresponding d-step ahead forecasting model is

其中 $Q (z^{- 1}) = F (z^{- 1}) \hat{B} (z^{- 1}) = Σ_{i = 0}^{n_{b} + d} q_{i} z^{- i}$ 和 $\hat{B} (z^{- 1}) = Σ_{j = 0}^{n_{b}} {\hat{b}}_{j} z^{- j} .$ in $Q (z^{- 1}) = f (z^{- 1}) \hat{B} (z^{- 1}) = Σ_{i = 0}^{{no}_{b} + d} q_{i} z^{- i}$ and $\hat{B} (z^{- 1}) = Σ_{j = 0}^{{no}_{b}} {\hat{b}}_{j} z^{- j} .$

2.采用丝杠机械传动机构工作平台的预测控制2. Predictive control of the working platform using the screw mechanical transmission mechanism

我们用式（5）来作为预测控制的目标函数。We use formula (5) as the objective function of predictive control.

其中，r(k+d)是参考轨迹，是预测模型的输出，λ是一个非负的加权系数。Among them, r(k+d) is the reference trajectory, is the output of the prediction model, and λ is a non-negative weighting coefficient.

J(k)是Lipschitz连续的，因此，在该优化函数的非光滑点，我们可以用次微分来替代传统意义的梯度，又因为J(k)是伪凸的，所以在非光滑点存在唯一的最优值，并且在该点的广义梯度为0.J(k) is Lipschitz continuous, therefore, at the non-smooth point of the optimization function, we can replace the gradient in the traditional sense with sub-differentiation, and because J(k) is pseudo-convex, there is a unique The optimal value of , and the generalized gradient at this point is 0.

因此，在间隙的非光滑点，J(k)关于u(k)的Clarke次梯度可以描述为：Therefore, at the non-smooth point of the gap, the Clarke subgradient of J(k) with respect to u(k) can be described as:

其中是J(k)关于u(k)的在非光滑点附近的光滑点处的梯度.in is the gradient of J(k) with respect to u(k) at smooth points near non-smooth points.

该预测的算法为：The algorithm for this prediction is:

第二步,根据系统模型，系统的工作区间4→1→2和2→3→4都能估计出来，在第k步，我们能计算出，在非光滑点，J(k)关于u(k)的Clarke次梯度,i.e.,其中j∈J_k，J_k={1,2,…,t},|J_k|是J_k的元素的个数，|J_k|≤t₁给定的有界的自然数，t＝t+1，如果t≤t₁,那么J_k={1，…,t}；如果t>t₁,那么J_k=J_k-1∪{t}\{t-t₁}；或者在光滑点，J(k)关于u(k)的梯度 ${&dtri;}_{u (k)} J (k) .$ In the second step, according to the system model, the working intervals 4→1→2 and 2→3→4 of the system can be estimated. In step k, we can calculate that at the non-smooth point, J(k) is about u( k) Clarke subgradient, ie, Where j∈J _k , J _k ={1,2,…,t}, |J _k | is the number of elements of J _k , |J _k |≤t ₁ given bounded natural number, t=t +1, if t≤t ₁ , then J _k ={1,…,t}; if t>t ₁ , then J _k =J _k-1 ∪{t}\{tt ₁ }; or in smooth point, The gradient of J(k) with respect to u(k) ${&dtri;}_{u (k)} J (k) .$

第三步， ${&PartialD;}_{u (k)} J (k) = Σ_{j = 1}^{t} χ_{i} {&PartialD;}_{u (k)}^{i} J (k),$ 其中 $Σ_{j = 1}^{t} χ_{i} = 1 .$ 我们令 ${&PartialD;}_{u (k)} J (k) = 0$ 或者 ${&dtri;}_{u (k)} J (k) = 0,$ 那么我们就能求出系统的准输入u₁(k)。third step, ${&PartialD;}_{u (k)} J (k) = Σ_{j = 1}^{t} χ_{i} {&PartialD;}_{u (k)}^{i} J (k),$ in $Σ_{j = 1}^{t} χ_{i} = 1 .$ we order ${&PartialD;}_{u (k)} J (k) = 0$ or ${&dtri;}_{u (k)} J (k) = 0,$ Then we can find out the quasi-input u ₁ (k) of the system.

第四步,如果并且t₁∈(0,1),那么u(k)=u₁(k),我们转入第六步.如果|r(k+d)-y(k+d)|≥t₁ε,那么我们转入第五步。Step four, if And t ₁ ∈ (0,1), then u(k)=u ₁ (k), we turn to the sixth step. If |r(k+d)-y(k+d)|≥t ₁ ε, Then we go to the fifth step.

第五步,根据求出u(k)=u_j(k)，如果那么u(k)=u_j(k)，否则，u(k)=u(k-1)，转到第六步.The fifth step, according to Find u(k)=u _j (k), if Then u(k)=u _j (k), otherwise, u(k)=u(k-1), go to the sixth step.

可以采用角编码器作为传感器，对其测量值进行解码和差分后获得含有机械平台的速度，并传送到数字信号处理器（DSP）中。DSP计算控制算法输出。经过放大器放大后驱动饲服电机。An angle encoder can be used as a sensor, and its measured value is decoded and differentiated to obtain the speed of the mechanical platform and sent to the digital signal processor (DSP). DSP calculation control algorithm output. After being amplified by the amplifier, it drives the feeding motor.

Claims

1. the forecast Control Algorithm of a machine driven system locating platform, this machine driven system is with ball-screw, it is characterized in that, the output corner of the servomotor that described machine driven system locating platform includes is as input signal u (k) of system, through the machine driving of gear-box speed change and leading screw, then drives workbench loaded work piece, displacement y (k) of workbench, it is the output of whole system, wherein

The gap of described locating platform is used for the mechanically operated nonlinear characteristic describing gear-box with leading screw,

L₁() is a linear dynamic sub-model, is used for describing the load of work platforms,

The output of the nonlinear characteristic of machine driven system, i.e. x (k), and can not directly measure,

The input of described locating platform and be output as u (k) and y (k)

The characteristic in described gap is described as,

\hat{x} (k) = \{\begin{matrix} {\hat{m}}_{1} (u (k) - {\hat{D}}_{1}), & \begin{matrix} u (k) > \frac{\hat{x} (k - 1)}{{\hat{m}}_{1}} + {\hat{D}}_{1} a n d & u (k) > u (k - 1) \end{matrix} \\ \hat{x} (k - 1), & \frac{\hat{x} (k - 1)}{{\hat{m}}_{2}} - {\hat{D}}_{2} \leq u (k) \leq \frac{x (k - 1)}{{\hat{m}}_{1}} + {\hat{D}}_{1} \\ {\hat{m}}_{2} (u (k) + {\hat{D}}_{2}), & \begin{matrix} u (k) < \frac{\hat{x} (k - 1)}{{\hat{m}}_{2}} - {\hat{D}}_{2} a n d & u (k) < u (k - 1) \end{matrix} \end{matrix} - - - (1)

Wherein:WithIt is the slope of gap former raising and lowering respectively,WithIt is the absolute value in the memory district of model raising and lowering respectively, and

0 < {\hat{m}}_{1} < \infty,

0 < {\hat{m}}_{2} < \infty,

0 < {\hat{D}}_{1} < \infty

With

0 < {\hat{D}}_{2} < \infty,

L₁() model formulation is:

\hat{y} (k) = - Σ_{i = 1}^{n_{a}} {\hat{a}}_{i} y (k - i) + Σ_{j = 0}^{n_{b}} {\hat{b}}_{j} \hat{x} (k - j - d) - - - (2)

Wherein n_aAnd n_bBeing the order of linear submodel, d is the prediction step of forecast model,WithIt is the coefficient of linear submodel,

Be made up of the model of machine driven system formula (1) and (2), then linear subsystem is expressed as

y (k) = \hat{y} (k) + ϵ (k)

Wherein ε (k) is model error, if it is the white noise of zero-mean, according to Diophantine equation

1 = F (z^{- 1}) \hat{A} (z^{- 1}) + z^{- d} G (z^{- 1}) - - - (3)

Wherein

F (z^{- 1}) = 1 + Σ_{i = 1}^{d} f_{i} z^{- i},

\hat{A} (z^{- 1}) = 1 + Σ_{i = 1}^{n_{a}} {\hat{a}}_{i} z^{- i}

With

G (z^{- 1}) = Σ_{j = 0}^{n} g_{j} z^{- j} .

Corresponding d walks advanced prediction model

\hat{y} (k + d) = G (z^{- 1}) y (k) + Q (z^{- 1}) \hat{x} (k) - - - (4)

Wherein

Q (z^{- 1}) = F (z^{- 1}) \hat{B} (z^{- 1}) = Σ_{i = 0}^{n_{b} + d} q_{i} z^{- i}

With

\hat{B} (z^{- 1}) = Σ_{j = 0}^{n_{b}} {\hat{b}}_{j} z^{- j},

Employing formula (5) is used as the object function of PREDICTIVE CONTROL,

J (k) = \frac{{[r (k + d) - \hat{y} (k + d)]}^{2}}{2} + \frac{λ {[u (k) - u (k - 1)]}^{2}}{2} - - - (5)

Wherein, r (k+d) is reference locus,It is the output of forecast model, λ is the weight coefficient of a non-negative, J (k) is Lipschitz continuous print, at the Non-smooth surface point of this majorized function, substituting the gradient of traditional sense by subdifferential, J (k) is pseudo-convex, optimal value at Non-smooth surface point existence anduniquess, generalized gradient at this point is 0, the Non-smooth surface point in gap, and J (k) can be described as about the Clarke subgradient of u (k):

\partial_{u (k)} J (k) &Element; c o n v {{&dtri;}_{u (k)} J (k)} - - - (6)

WhereinIt is the J (k) gradient about the slick spot place near Non-smooth surface point of u (k),

J (k) is about the u (k) gradient at slick spot placeFor:

{&dtri;}_{u (k)} J (k) = \{\begin{matrix} λ [u (k) - u (k - 1)] - [r (k + d) - \hat{y} (k + d)] q_{0} m_{1}, & \begin{matrix} u (k) > \frac{x (k - 1)}{m_{1}} + D_{1} a n d & u (k) > u (k - 1) \end{matrix} \\ λ [u (k) - u (k - 1)] + 0, & \frac{x (k - 1)}{m_{2}} - D_{2} < u (k) < \frac{x (k - 1)}{m_{1}} + D_{1} \\ λ [u (k) - u (k - 1)] - [r (k + d) - \hat{y} (k + d)] q_{0} m_{2}, & \begin{matrix} u (k) < \frac{x (k - 1)}{m_{2}} - D_{2} a n d & u (k) < u (k - 1) \end{matrix} \end{matrix} - - - (7)

Then described forecast Control Algorithm includes step,

The first step, provides control error ε, that is:

Second step, according to formula (1), the operation interval 4 → 1 → 2 and 2 → 3 → 4 of system can estimate, and walks in kth, calculates, at Non-smooth surface point, J (k) about the Clarke subgradient of u (k), i.e.,Wherein j ∈ J_k, J_k=1,2 ..., t}, | J_k| it is J_kThe number of element, | J_k|≤t₁The natural number of given bounded, t=t+1, if t≤t₁, so J_k=1 ..., t}；If t is > t₁, so J_k=J_k-1∪{t}\{t-t₁}；Or at slick spot, J (k) is about the gradient of u (k)

3rd step,

\partial_{u (k)} J (k) = Σ_{j = 1}^{t} χ_{i} \partial_{u (k)}^{j} J (k),

Wherein

Σ_{j = 1}^{t} χ_{i} = 1,

Order

\partial_{u (k)} J (k) = 0

Or

{&dtri;}_{u (k)} J (k) = 0,

Obtain the quasi-input u of system₁(k)；

4th step, ifAnd t₁∈ (0,1), so u (k)=u₁K (), proceeds to the 6th step, if | r (k+d)-y (k+d) | is >=t₁ε, proceeds to the 5th step；

5th step, according toObtain u (k)=u_j(k), ifSo u (k)=u_j(k), otherwise, u (k)=u (k-1), forward the 6th step to；

6th step, k=k+1, proceed to second step.