CN104793645A

CN104793645A - Magnetic levitation ball position control method

Info

Publication number: CN104793645A
Application number: CN201510180614.7A
Authority: CN
Inventors: 彭辉; 覃业梅; 阮文杰; 高家成
Original assignee: Central South University
Current assignee: Central South University
Priority date: 2015-04-16
Filing date: 2015-04-16
Publication date: 2015-07-22
Anticipated expiration: 2035-04-16
Also published as: CN104793645B

Abstract

The invention discloses a method for controlling the position of a magnetic levitation ball. Aiming at the disadvantage that it is difficult to establish an accurate physical model for the magnetic levitation ball system, a system identification method is used to establish an autoregressive model with a function weight coefficient to describe the difference between the input voltage of the electromagnetic winding and the position of the steel ball. Linear dynamics. In this model, the linear function of the position of the steel ball is used as the weight coefficient of the Gaussian RBF network, and the RBF network is used as the function coefficient of the nonlinear autoregressive model, so that the model can better describe the dynamic characteristics of the maglev ball system. The model has a constant regression coefficient at a certain moment, similar to a linear ARX model. Based on this, the present invention designs a time-varying, locally linear predictive controller, and quickly realizes the optimal position control of the steel ball by solving the quadratic programming online at each time, so as to meet the stable and fast requirements of the magnetic levitation ball system.

Description

A method for controlling the position of a magnetic levitation ball

技术领域technical field

本发明涉及自动控制技术领域，特别是一种磁悬浮球位置控制方法。The invention relates to the technical field of automatic control, in particular to a method for controlling the position of a magnetic levitation ball.

背景技术Background technique

磁悬浮技术是集电磁学、电子技术、控制工程、信号处理、机械学、动力学为一体的、典型的机电一体化技术。磁悬浮技术因其无接触、无摩擦、低噪声等特点已广泛应用于磁悬浮列车、磁悬浮轴承、磁悬浮电机等工程领域。磁悬浮球系统主要是通过对电磁绕组通以一定的电流产生电磁力，使其与钢球重力相平衡，使钢球悬浮在空中而处于平衡状态，达到系统稳定运行的目的。具有单一方向的磁悬浮球系统具有本质的非线性、开环不稳定、快速响应的特点，易受电源及外界环境的影响，某些参数具有较强的不确定性，无法精确测量。而且电磁场磁饱和现象使得电磁场中输入电流与磁感应强度、电磁绕组的磁通链之间不成正比关系，增大了系统的非线性并导致系统的电磁力模型无法用简单的数学方程表达；同时，处于电磁场中的钢球产生电涡流，将反过来影响电磁绕组的电感，使得电磁绕组的电感不为常数，而是关于钢球到电磁铁磁极表面的气隙g的函数，而且与其成非线性关系。因此，建立磁悬浮球系统的精确物理模型是非常困难的，这是磁悬浮球系统实现稳定控制的难点所在。Magnetic levitation technology is a typical mechatronics technology integrating electromagnetics, electronic technology, control engineering, signal processing, mechanics and dynamics. Magnetic levitation technology has been widely used in engineering fields such as magnetic levitation trains, magnetic levitation bearings, and magnetic levitation motors due to its non-contact, frictionless, and low noise characteristics. The magnetic levitation ball system mainly generates electromagnetic force by passing a certain current through the electromagnetic winding, so that it is balanced with the gravity of the steel ball, so that the steel ball is suspended in the air and in a balanced state, so as to achieve the purpose of stable operation of the system. The maglev ball system with a single direction has the characteristics of intrinsic nonlinearity, open-loop instability, and fast response. It is easily affected by the power supply and the external environment. Some parameters have strong uncertainties and cannot be accurately measured. Moreover, the magnetic saturation phenomenon of the electromagnetic field makes the input current in the electromagnetic field, the magnetic induction intensity, and the magnetic flux linkage of the electromagnetic winding not proportional, which increases the nonlinearity of the system and makes the electromagnetic force model of the system unable to be expressed by simple mathematical equations; at the same time, The steel ball in the electromagnetic field produces an eddy current, which will in turn affect the inductance of the electromagnetic winding, so that the inductance of the electromagnetic winding is not a constant, but a function of the air gap g from the steel ball to the surface of the electromagnet pole, and it is nonlinear with it relation. Therefore, it is very difficult to establish an accurate physical model of the magnetic levitation ball system, which is the difficulty in realizing the stable control of the magnetic levitation ball system.

PID控制结构简单，可以调节输入电流/电压使钢球悬浮，不需建立磁悬浮系统的物理模型，但控制参数需要人工整定，自适应性较差，对非线性磁悬浮系统的有效控制范围较小，尤其当钢球位置变化快速时超调较大，钢球抖动较大。用模糊推理自动调节PID的控制参数，可以提高参数随钢球与电磁铁间气隙变化而变化的能力。但该方法依赖于模糊规则库，它的建立受制于设计者的经验。另一方面，通过分析磁悬浮系统工作原理，在一些假设条件的基础上，建立物理模型，然后可对钢球实施自适应控制、滑膜控制、预测控制等。这些方法实现的最大难点在于较难获得能准确描述磁悬浮系统动态特性的精确物理模型，因为，某些假设条件难以在工程应用中得到满足。此外，利用线性化技术对物理模型进行线性化处理，然后设计线性控制策略，能加快在线控制优化速度，但损失了系统非线性特性，弱化了模型对磁悬浮系统的描述能力，从而降低了控制效果。The PID control structure is simple, and the input current/voltage can be adjusted to levitate the steel ball. It is not necessary to establish a physical model of the magnetic levitation system, but the control parameters need to be manually adjusted, and the adaptability is poor. The effective control range of the nonlinear magnetic levitation system is small. Especially when the position of the steel ball changes rapidly, the overshoot is large, and the steel ball shakes greatly. Using fuzzy reasoning to automatically adjust the control parameters of PID can improve the ability of parameters to change with the air gap between the steel ball and the electromagnet. But this method depends on the fuzzy rule base, and its establishment is subject to the designer's experience. On the other hand, by analyzing the working principle of the magnetic levitation system, and on the basis of some assumptions, a physical model is established, and then adaptive control, sliding film control, predictive control, etc. can be implemented on the steel ball. The biggest difficulty in the implementation of these methods is that it is difficult to obtain an accurate physical model that can accurately describe the dynamic characteristics of the maglev system, because some assumptions are difficult to be satisfied in engineering applications. In addition, using linearization technology to linearize the physical model and then designing a linear control strategy can speed up the online control optimization speed, but it loses the nonlinear characteristics of the system and weakens the model's ability to describe the magnetic levitation system, thereby reducing the control effect .

发明内容Contents of the invention

本发明所要解决的技术问题是，针对现有技术不足，提供一种磁悬浮球位置控制方法。The technical problem to be solved by the present invention is to provide a method for controlling the position of a magnetic levitation ball in view of the deficiencies in the prior art.

为解决上述技术问题，本发明所采用的技术方案是：一种磁悬浮球位置控制方法，适用于上下移动的单自由度磁悬浮球系统，所述单自由度磁悬浮球系统包括产生电磁场的线圈绕组和检测钢球位置的光电传感器；所述电磁绕组输入电压由控制器控制驱动电路给出；所述光电传感器采集的钢球位置信号通过数据采集卡传送给控制计算机。对磁悬浮球系统建立自回归模型：In order to solve the above technical problems, the technical solution adopted in the present invention is: a method for controlling the position of a magnetic levitation ball, which is suitable for a single-degree-of-freedom magnetic levitation ball system that moves up and down. The single-degree-of-freedom magnetic levitation ball system includes a coil winding for generating an electromagnetic field and A photoelectric sensor for detecting the position of the steel ball; the input voltage of the electromagnetic winding is given by the controller controlling the driving circuit; the position signal of the steel ball collected by the photoelectric sensor is transmitted to the control computer through the data acquisition card. Establish an autoregressive model for the magnetic levitation ball system:

$g g ((t t)) = = {φ φ}_{00} + + {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{g g} g g ((t t - - i i)) + + {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{u u} u u ((t t - - i i)) + + ξ ξ ((t t))$

其中，g(t)为磁悬浮球的位置；u(t)为电磁绕组输入电压；ξ(t)为白噪声； $\{\begin{matrix} φ_{0} = c_{0}^{0} + {V_{1}}^{0} \exp (- 0.03 {| | W (t - 1) - 11.74 | |}^{2}) \\ φ_{i}^{g} = c_{i, 0}^{g} + V_{i, 1}^{g} \exp (- 0.03 {| | W (t - 1) - 11.74 | |}^{2}) \\ φ_{j}^{u} = c_{j, 0}^{u} + V_{j . 1}^{u} \exp (- 1.33 {| | W (t - 1) + 3.82 | |}^{2}) \\ W (t - 1) = g (t - 1) \\ V^{0} = v_{0}^{0} + v_{1}^{0} g (t - 1) \\ {V_{i}}^{g} = v_{i, 0}^{g} + v_{i, 1}^{g} g (t - 1) \\ V_{j}^{u} = v_{j, 0}^{u} + v_{j, 1}^{u} g (t - 1) \\ i = 1,2, . . ., 7; j = 1,2, . . ., 6 \end{matrix};$ V⁰、V_i ^g和V_j ^u为RBF网络的权系数，是钢球位置的一次线性函数；为常数系数，通过SNPOM优化方法辨识，在获得非线性参数的基础上用最小二乘法计算获得。Among them, g(t) is the position of the magnetic levitation ball; u(t) is the input voltage of the electromagnetic winding; ξ(t) is white noise; $\{\begin{matrix} φ_{0} = c_{0}^{0} + {V_{1}}^{0} \exp (- 0.03 {| | W (t - 1) - 11.74 | |}^{2}) \\ φ_{i}^{g} = c_{i, 0}^{g} + V_{i, 1}^{g} \exp (- 0.03 {| | W (t - 1) - 11.74 | |}^{2}) \\ φ_{j}^{u} = c_{j, 0}^{u} + V_{j . 1}^{u} \exp (- 1.33 {| | W (t - 1) + 3.82 | |}^{2}) \\ W (t - 1) = g (t - 1) \\ V^{0} = v_{0}^{0} + v_{1}^{0} g (t - 1) \\ {V_{i}}^{g} = v_{i, 0}^{g} + v_{i, 1}^{g} g (t - 1) \\ V_{j}^{u} = v_{j, 0}^{u} + v_{j, 1}^{u} g (t - 1) \\ i = 1,2, . . ., 7; j = 1,2, . . ., 6 \end{matrix};$ V ⁰ , V _i ^g and V _j ^u are the weight coefficients of the RBF network, which is a linear function of the position of the steel ball; is a constant coefficient, which is identified by the SNPOM optimization method and calculated by the least square method on the basis of obtaining nonlinear parameters.

然后基于所述自回归模型设计预测控制器，优化下列二次规划函数J获得最优控制量，控制磁悬浮球的位置g(t)：Then design a predictive controller based on the autoregressive model, optimize the following quadratic programming function J to obtain the optimal control amount, and control the position g (t) of the magnetic levitation ball:

$\begin{matrix} \underset{\overset{^^}{u u} ((t t))}{min min} J J = = {| | | | \overset{^^}{g g} ((t t)) - - {\overset{^^}{g g}}_{r r} ((t t)) | | | |}_{{1.8 1.8 I I}_{1212}}^{22} + + {| | | | \overset{^^}{u u} ((t t)) | | | |}_{{0.0002 0.0002 I I}_{44}}^{22} + + {| | | | Δ Δ \overset{^^}{u u} ((t t)) | | | |}_{{0.16 0.16 I I}_{44}}^{22} \\ \begin{matrix} s the s . . t t . . & - - 22 twenty two \leq \leq \overset{^^}{g g} ((t t)) \leq \leq 00 \end{matrix} \\ \begin{matrix} 00 \leq \leq \overset{^^}{u u} ((t t)) \leq \leq 1010 \end{matrix} \\ \begin{matrix} - - 33 \leq \leq Δ Δ \overset{^^}{u u} ((t t)) \leq \leq 33 \end{matrix} \end{matrix}$

其中， $\hat{g} (t) = {[\begin{matrix} \hat{g} (t + 1 | t) & \hat{g} (t + 2 | t) & . . . & \hat{g} (t + 12 | t) \end{matrix}]}^{T}, \hat{g} (t + l | t), l = 1, . . ., 12$ 为t时刻l步向前钢球位置预测变量，根据t时刻所述自回归模型获得l步预测的表达式，即 $\hat{g} (t) = \overset{&OverBar;}{C} {\overset{&OverBar;}{A}}_{t} x (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{B}}_{t} \hat{u} (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{Γ}}_{t} {\overset{&OverBar;}{Φ}}_{t}, {\overset{&OverBar;}{A}}_{t}, {\overset{&OverBar;}{B}}_{t}, \overset{&OverBar;}{C}$ 和为系统预测系数矩阵，由t时刻所述自回归模型及多步预测变量获得； ${\hat{g}}_{r} (t) = {[\begin{matrix} g_{r} (t + 1) & g_{r} (t + 2) & . . . & g_{r} (t + 12) \end{matrix}]}^{T},$ g_r(t+l)为t时刻给定的l步向前参考位置； $\hat{u} (t) = {[\begin{matrix} u (t) & u (t + 1) & u (t + 2) & u (t + 3) \end{matrix}]}^{T},$ u(t+p),p＝0,1,2,3为t时刻要优化的电磁绕组输入电压，仅取第一项u(t)作用于被控磁悬浮球； $Δ \hat{u} (t) = {[\begin{matrix} Δu (t) & Δu (t + 1) & Δu (t + 2) & Δu (t + 3) \end{matrix}]}^{T},$ △u(t)＝u(t)-u(t-1)为输入电压增量； $\{\begin{matrix} x (t) = {[\begin{matrix} x_{1, t} & x_{2, t} & . . . & x_{7, t} \end{matrix}]}^{T} \\ x_{1, t} = g (t) \\ x_{k, t} = Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{g} g (t - i) + Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{u} u (t - i) \\ k = 2,3, . . ., 7 \end{matrix}$ 为状态向量； ${\overset{&OverBar;}{Φ}}_{t} = {[\begin{matrix} Φ_{t}^{T} & Φ_{t + 1}^{T} & . . . & Φ_{t + 11}^{T} \end{matrix}]}^{T},$ $Φ_{t} = [\begin{matrix} φ_{0} \\ 0 \\ . \\ . \\ . \\ 0 \end{matrix}];$ φ₀、φ_i ^g和φ_j ^u为自回归模型的回归函数系数，φ₇ ^u＝0；I为单位矩阵。in, $\hat{g} (t) = {[\begin{matrix} \hat{g} (t + 1 | t) & \hat{g} (t + 2 | t) & . . . & \hat{g} (t + 12 | t) \end{matrix}]}^{T}, \hat{g} (t + l | t), l = 1, . . ., 12$ is the l-step forward steel ball position predictor variable at time t, and the expression of l-step prediction is obtained according to the autoregressive model at time t, namely $\hat{g} (t) = \overset{&OverBar;}{C} {\overset{&OverBar;}{A}}_{t} x (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{B}}_{t} \hat{u} (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{Γ}}_{t} {\overset{&OverBar;}{Φ}}_{t}, {\overset{&OverBar;}{A}}_{t}, {\overset{&OverBar;}{B}}_{t}, \overset{&OverBar;}{C}$ and is the system prediction coefficient matrix, obtained by the autoregressive model and multi-step predictor variables described at time t; ${\hat{g}}_{r} (t) = {[\begin{matrix} g_{r} (t + 1) & g_{r} (t + 2) & . . . & g_{r} (t + 12) \end{matrix}]}^{T},$ g _r (t+l) is the l-step forward reference position given at time t; $\hat{u} (t) = {[\begin{matrix} u (t) & u (t + 1) & u (t + 2) & u (t + 3) \end{matrix}]}^{T},$ u(t+p),p=0,1,2,3 is the electromagnetic winding input voltage to be optimized at time t, and only the first item u(t) is used to act on the controlled magnetic levitation ball; $Δ \hat{u} (t) = {[\begin{matrix} Δu (t) & Δu (t + 1) & Δ u (t + 2) & Δu (t + 3) \end{matrix}]}^{T},$ △u(t)=u(t)-u(t-1) is the input voltage increment; $\{\begin{matrix} x (t) = {[\begin{matrix} x_{1, t} & x_{2, t} & . . . & x_{7, t} \end{matrix}]}^{T} \\ x_{1, t} = g (t) \\ x_{k, t} = Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{g} g (t - i) + Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{u} u (t - i) \\ k = 2,3, . . ., 7 \end{matrix}$ is the state vector; ${\overset{&OverBar;}{Φ}}_{t} = {[\begin{matrix} Φ_{t}^{T} & Φ_{t + 1}^{T} & . . . & Φ_{t + 11}^{T} \end{matrix}]}^{T},$ $Φ_{t} = [\begin{matrix} φ_{0} \\ 0 \\ . \\ . \\ . \\ 0 \end{matrix}];$ φ ₀ , φ _i ^g and φ _j ^u are the regression function coefficients of the autoregressive model, φ ₇ ^u = 0; I is the identity matrix.

与现有技术相比，本发明所具有的有益效果为：本发明采用系统辨识建模的思想，利用系统实际运行的输入/输出数据包含系统动态特性的相关信息建立磁悬浮球系统的动态模型，可以最大可能地描述出系统的动态特性，而不用考虑磁饱和现象及涡流效应对模型的影响(这些影响均已包含在辨识数据中)。该方法适用于这类强非线性、强不确定性的复杂系统，可推广至其他类似系统；本发明建立的带函数权RBF-ARX模型的RBF网络系数依存于钢球在电磁场中的位置，提高了RBF网络的函数逼近能力，使获得的一组伪线性ARX模型能更好地描述非线性磁悬浮球系统的动态特性。系统的一步预测输出建模误差在±0.4％以内；本发明基于带函数权RBF-ARX模型设计时变的、局部线性的预测控制器，能快速优化计算最优控制量，减少在线优化时间，有利于在磁悬浮球系统采样周期(5毫秒)内完成优化计算，实现对钢球位置的快速、稳定控制。Compared with the prior art, the beneficial effects of the present invention are: the present invention adopts the idea of system identification and modeling, uses the input/output data of the actual operation of the system to include the relevant information of the system dynamic characteristics to establish the dynamic model of the magnetic levitation ball system, The dynamic characteristics of the system can be described to the greatest possible extent, without considering the influence of magnetic saturation and eddy current effects on the model (these influences have been included in the identification data). This method is suitable for such complex systems with strong nonlinearity and strong uncertainty, and can be extended to other similar systems; the RBF network coefficient of the weighted RBF-ARX model established by the present invention depends on the position of the steel ball in the electromagnetic field, The function approximation ability of the RBF network is improved, so that a set of pseudo-linear ARX models obtained can better describe the dynamic characteristics of the nonlinear magnetic levitation ball system. The modeling error of the one-step prediction output of the system is within ±0.4%; the present invention designs a time-varying and locally linear predictive controller based on the RBF-ARX model with function weight, which can quickly optimize and calculate the optimal control amount, and reduce the online optimization time. It is beneficial to complete the optimization calculation within the sampling period (5 milliseconds) of the magnetic levitation ball system, and realize fast and stable control of the position of the steel ball.

附图说明Description of drawings

图1为本发明磁悬浮球系统结构图。Fig. 1 is a structural diagram of the magnetic levitation ball system of the present invention.

具体实施方式Detailed ways

本发明磁悬浮球系统的系统结构如图1所示，是一个仅能控制钢球上下方向移动的单自由度系统。PC机9通过控制器输出控制电压，经D/A转换器8传输给电磁绕组驱动电路6，电磁绕组2在通以相应电流的情况下产生电磁感应，在绕组下方形成电磁场，对处于场中的钢球1施加电磁感应力F，使钢球上/下移动，调整电磁铁与钢球间的气隙g(即钢球位置)，直至电磁力F与钢球重力G平衡；同时，LED光源3与光电板4构成的光电传感器用来检测钢球位置，相应的电压信号经处理电路5及A/D转换器7传回PC机输出。图1所示系统中，钢球1的半径为12.5毫米、质量为22克，电磁绕组2的匝数为2450、等效电阻为13.8欧姆。The system structure of the magnetic levitation ball system of the present invention is shown in Figure 1, which is a single-degree-of-freedom system that can only control the movement of the steel ball in the up and down direction. The PC 9 outputs the control voltage through the controller, and transmits it to the electromagnetic winding drive circuit 6 through the D/A converter 8, and the electromagnetic winding 2 generates electromagnetic induction when the corresponding current is passed through, forming an electromagnetic field under the winding, and the pair in the field The steel ball 1 applies an electromagnetic induction force F to make the steel ball move up/down, and adjust the air gap g between the electromagnet and the steel ball (that is, the position of the steel ball) until the electromagnetic force F is balanced with the gravity G of the steel ball; at the same time, the LED The photoelectric sensor composed of the light source 3 and the photoelectric board 4 is used to detect the position of the steel ball, and the corresponding voltage signal is transmitted back to the PC through the processing circuit 5 and the A/D converter 7 for output. In the system shown in Fig. 1, the radius of the steel ball 1 is 12.5 mm, the mass is 22 grams, the number of turns of the electromagnetic winding 2 is 2450, and the equivalent resistance is 13.8 ohms.

本发明所述磁悬浮球系统受到磁饱和及涡流效应影响，还受到电源及外界干扰的影响。为此，采用基于带函数权RBF-ARX模型的建模方法，构建电磁绕组输入电压与电磁场中钢球位置间关系的动态模型。在本发明中，利用数据辨识技术，采用钢球位置的线性函数作权系数的、高斯核的RBF网络作为非线性ARX模型中的函数系数。该模型是一种具有线性ARX模型结构的非线性时变模型，它的自变量是电磁绕组输入电压、钢球位置的回归量，钢球位置为表征系统状态的信号量，采用与钢球位置线性相关的函数逼近RBF神经网络的常数权，然后用该RBF结构对模型参数进行实时在线调整。带函数权RBF-ARX模型在局部的线性区间内与线性ARX模型非常近似，另外它的参数能随着系统非线性状态而自动更新、自动调整，具有良好的全局适应特性。The magnetic levitation ball system of the present invention is affected by magnetic saturation and eddy current effect, and is also affected by power supply and external interference. To this end, the modeling method based on the weighted RBF-ARX model with function is used to construct a dynamic model of the relationship between the input voltage of the electromagnetic winding and the position of the steel ball in the electromagnetic field. In the present invention, the data identification technology is used, and the linear function of the position of the steel ball is used as the weight coefficient and the RBF network of the Gaussian kernel is used as the function coefficient in the nonlinear ARX model. This model is a nonlinear time-varying model with a linear ARX model structure. Its independent variables are the regression quantities of the input voltage of the electromagnetic winding and the position of the steel ball. The position of the steel ball is a semaphore representing the state of the system. The linearly related function approximates the constant weight of the RBF neural network, and then uses the RBF structure to adjust the model parameters online in real time. The RBF-ARX model with function weight is very similar to the linear ARX model in the local linear interval, and its parameters can be automatically updated and adjusted with the nonlinear state of the system, and has good global adaptation characteristics.

构建钢球位置与电磁绕组输入电压间动态特性模型的函数权系数型RBF-ARX模型，采用列维布格马奎尔特方法(Levenberg-Marquardt Method,LMM)和线性最小二乘法(LeastSquare Method,LSM)相结合的SNPOM优化方法(详见：Peng H,Ozaki T,Haggan-Ozaki V,Toyoda Y.2003，A parameter optimization method for the radial basis function type models)辨识该模型参数，获得如下结构：Construct the function weight type RBF-ARX model of the dynamic characteristic model between the position of the steel ball and the input voltage of the electromagnetic winding, using the Levenberg-Marquardt Method (LMM) and the linear least square method (LeastSquare Method, LSM) combined with the SNPOM optimization method (see: Peng H, Ozaki T, Haggan-Ozaki V, Toyoda Y.2003, A parameter optimization method for the radial basis function type models) to identify the model parameters and obtain the following structure:

$g g ((t t)) = = {φ φ}_{00} + + {φ φ}_{11}^{g g} g g ((t t - - 11)) + + {φ φ}_{22}^{g g} g g ((t t - - 22)) + + . . . . . . + + {φ φ}_{77}^{g g} g g ((t t - - 77)) + + {φ φ}_{11}^{u u} u u ((t t - - 11)) + + {φ φ}_{22}^{u u} u u ((t t - - 22)) + + . . . . . . + + {φ φ}_{66}^{u u} u u ((t t - - 66)) + + ξ ξ ((t t)) - - - - - - ((11))$

其中，in,

$\{\begin{matrix} {φ φ}_{00} = = {c c}_{00}^{00} + + {V V}_{11}^{00} exp exp ((- - 0.03 0.03 {| | | | W W ((t t - - 11)) - - 11.74 11.74 | | | |}^{22})) \\ {φ φ}_{i i}^{g g} = = {c c}_{i i,, 00}^{g g} + + {V V}_{i i,, 11}^{g g} exp exp ((- - 0.03 0.03 {| | | | W W ((t t - - 11)) - - 11.74 11.74 | | | |}^{22})) \\ {φ φ}_{j j}^{u u} = = {c c}_{j j,, 00}^{u u} + + {V V}_{j j . . 11}^{u u} exp exp ((- - 1.33 1.33 {| | | | W W ((t t - - 11)) + + 3.82 3.82 | | | |}^{22})) \\ W W ((t t - - 11)) = = g g ((t t - - 11)) \\ {V V}^{00} = = {v v}_{00}^{00} + + {v v}_{11}^{00} g g ((t t - - 11)) \\ {V V}_{i i}^{g g} = = {v v}_{i i,, 00}^{g g} + + {v v}_{i i,, 11}^{g g} g g ((t t - - 11)) \\ {V V}_{j j}^{u u} = = {v v}_{j j,, 00}^{u u} + + {v v}_{j j,, 11}^{u u} g g ((t t - - 11)) \\ i i = = 1,2 1,2,, . . . . . .,, 77;; j j = = 1,2 1,2,, . . . . . .,, 66 \end{matrix} - - - - - - ((22))$

g(t)为钢球的位置，也是钢球与电磁铁间的气隙；u(t)为电磁绕组的输入电压；φ₀、φ_i ^g和φ_j ^u分别为自回归模型的回归函数系数；W(t-1)为系统工作点状态，用钢球位置g(t-1)来表征系统工作点状态；V⁰、和为钢球位置的一次线性函数，作为RBF神经网络的函数权系数；||·||为2范数；ξ(t)为白噪声；为线性常数系数，通过SNPOM优化方法辨识，在获得非线性参数的基础上用LSM计算获得。令 $θ_{L} = {c_{0}^{0}, c_{i, 0}^{g}, c_{j, 0}^{u}, v_{0}^{0}, v_{i, 0}^{g}, v_{j, o}^{u} v_{1}^{0}, v_{i, 1}^{g}, v_{j, 1}^{u} | i = 1, . . ., 7, j = 1, . . ., 6},$ $θ_{N} = {0.03,1.33,11.74, - 3.82},$ 式(1)改写为则g(t) is the position of the steel ball, which is also the air gap between the steel ball and the electromagnet; u(t) is the input voltage of the electromagnetic winding; φ ₀ , φ _i ^g and φ _j ^u are the regression functions of the autoregressive model coefficient; W(t-1) is the state of the system operating point, and the position of the steel ball g(t-1) is used to characterize the state of the system operating point; V ⁰ , and is a linear function of the position of the steel ball, and is used as the function weight coefficient of the RBF neural network; ||·|| is a 2-norm; ξ(t) is white noise; It is a linear constant coefficient, which is identified by SNPOM optimization method and calculated by LSM on the basis of obtaining nonlinear parameters. make $θ_{L} = {c_{0}^{0}, c_{i, 0}^{g}, c_{j, 0}^{u}, v_{0}^{0}, v_{i, 0}^{g}, v_{j, o}^{u} v_{1}^{0}, v_{i, 1}^{g}, v_{j, 1}^{u} | i = 1, . . ., 7, j = 1, . . ., 6},$ $θ_{N} = {0.03, 1.33, 11.74, - 3.82},$ Formula (1) is rewritten as but

其中，为钢球位置的观测数据，SNPOM进行模型辨识时共使用4000个观测数据。in, It is the observation data of the steel ball position, and SNPOM uses a total of 4000 observation data for model identification.

利用磁悬浮球系统局部线性的特性设计时变、线性的预测控制器。将式(1)改写为多项式A time-varying, linear predictive controller is designed using the local linearity of the magnetic levitation ball system. Rewrite equation (1) as a polynomial

$g g ((t t)) = = {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{g g} g g ((t t - - i i)) + + {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{u u} u u ((t t - - i i)) + + {φ φ}_{00} + + ξ ξ ((t t)) - - - - - - ((44))$

其中， $φ_{7}^{u} = 0 .$ in, $φ_{7}^{u} = 0 .$

定义系统的状态变量：Define the state variables of the system:

$\{\begin{matrix} x x ((t t)) = = {[\begin{matrix} {x x}_{11,, t t} & {x x}_{22,, t t} & . . . . . . & {x x}_{77,, t t} \end{matrix}]}^{T T} \\ {x x}_{11,, t t} = = g g ((t t)) \\ {x x}_{k k,, t t} = = {Σ Σ}_{i i = = 11}^{77 - - k k + + 11} {φ φ}_{i i + + k k - - 11}^{g g} g g ((t t - - i i)) + + {Σ Σ}_{i i = = 11}^{77 - - k k + + 11} {φ φ}_{i i + + k k - - 11}^{u u} u u ((t t - - i i)) \\ k k = = 2,3 2,3,, . . . . . .,, 77 \end{matrix} - - - - - - ((55))$

则式(1)的状态空间模型为：Then the state space model of formula (1) is:

$\{\begin{matrix} x x ((t t + + 11)) = = {A A}_{t t} x x ((t t)) + + {B B}_{t t} u u ((t t)) + + {Φ Φ}_{t t} + + Ξ ξ ((t t + + 11)) \\ g g ((t t)) = = Cx Cx ((t t)) \end{matrix} - - - - - - ((66))$

这里here

$\{\begin{matrix} {A A}_{t t} = = [\begin{matrix} {φ φ}_{11}^{g g} & 11 & 00 & 00 & 00 \\ {φ φ}_{22}^{g g} & 00 & 11 & 00 & 00 \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ {φ φ}_{66}^{g g} & 00 & 00 & 00 & 11 \\ {φ φ}_{77}^{g g} & 00 & 00 & 00 & 00 \end{matrix}],, {B B}_{t t} = = [\begin{matrix} {φ φ}_{11}^{u u} \\ {φ φ}_{22}^{u u} \\ . . \\ . . \\ . . \\ {φ φ}_{66}^{u u} \\ 00 \end{matrix}] \\ {Φ Φ}_{t t} = = [\begin{matrix} {φ φ}_{00} \\ 00 \\ . . \\ . . \\ . . \\ 00 \end{matrix}],, Ξ ξ ((t t + + 11)) = = [\begin{matrix} ξ ξ ((t t + + 11)) \\ 00 \\ . . \\ . . \\ . . \\ 00 \end{matrix}],, C C = = {[\begin{matrix} 11 \\ 00 \\ . . \\ . . \\ . . \\ 00 \end{matrix}]}^{T T} \end{matrix} - - - - - - ((77))$

上式中φ₀、φ_i ^g和φ_j ^u不是常数，而是随着钢球位置g(t)而变化。定义相关的预测变量：In the above formula, φ ₀ , φ _i ^g and φ _j ^u are not constant, but change with the position of steel ball g(t). Define the associated predictor variables:

$\{\begin{matrix} \overset{^^}{x x} ((t t)) = = {[\begin{matrix} \overset{^^}{x x} {((t t + + 11 | | t t))}^{T T} & \overset{^^}{x x} {((t t + + 22 | | t t))}^{T T} & . . . . . . & \overset{^^}{x x} {((t t + + 1212 | | t t))}^{T T} \end{matrix}]}^{T T} \\ \overset{^^}{g g} ((t t)) = = {[\begin{matrix} \overset{^^}{g g} ((t t + + 11 | | t t)) & \overset{^^}{g g} ((t t + + 22 | | t t)) & . . . . . . & \overset{^^}{g g} ((t t + + 1212 | | t t)) \end{matrix}]}^{T T} \\ \overset{^^}{u u} ((t t)) = = {[\begin{matrix} u u ((t t)) & t t ((t t + + 11)) & t t ((t t + + 22)) & t t ((t t + + 33)) \end{matrix}]}^{T T} \\ {\overset{&OverBar; &OverBar;}{Φ Φ}}_{t t} = = {[\begin{matrix} {Φ Φ}_{t t}^{T T} & {Φ Φ}_{t t + + 11}^{T T} & . . . . . . & {Φ Φ}_{t t + + 1111}^{T T} \end{matrix}]}^{T T} \end{matrix} - - - - - - ((88))$

其中，是多步向前预测状态向量，是钢球位置多步向前预测向量，是绕组输入电压多步向前预测向量。假设u(t+j)＝u(t+3)(j≥4)，从(5)～(7)，可以得到：in, is the multi-step forward prediction state vector, is the multi-step forward prediction vector of the steel ball position, is the winding input voltage multi-step forward prediction vector. Suppose u(t+j)=u(t+3)(j≥4), from (5)～(7), we can get:

$\{\begin{matrix} \overset{^^}{x x} ((t t)) = = {\overset{&OverBar; &OverBar;}{A A}}_{t t} x x ((t t)) + + {\overset{&OverBar; &OverBar;}{B B}}_{t t} \overset{^^}{u u} ((t t)) + + {\overset{&OverBar; &OverBar;}{Γ Γ}}_{t t} {\overset{&OverBar; &OverBar;}{Φ Φ}}_{t t} \\ \overset{^^}{g g} ((t t)) = = \overset{&OverBar; &OverBar;}{C C} \overset{^^}{x x} ((t t)) = = \overset{&OverBar; &OverBar;}{C C} {\overset{&OverBar; &OverBar;}{A A}}_{t t} x x ((t t)) + + \overset{&OverBar; &OverBar;}{C C} {\overset{&OverBar; &OverBar;}{B B}}_{t t} \overset{^^}{u u} ((t t)) + + \overset{&OverBar; &OverBar;}{C C} {\overset{&OverBar; &OverBar;}{Γ Γ}}_{t t} {\overset{&OverBar; &OverBar;}{Φ Φ}}_{t t} \end{matrix} - - - - - - ((99))$

这里，和为系统预测系数矩阵，可由式(5)～(8)获得，依赖于t时刻电磁场中钢球的位置，均为随钢球位置g(t)变化的值。here, and is the system prediction coefficient matrix, which can be obtained from formulas (5) to (8), depends on the position of the steel ball in the electromagnetic field at time t, and is a value that varies with the position of the steel ball g(t).

${\overset{&OverBar;}{A}}_{t} = [\begin{matrix} A_{t} \\ {(A_{t})}^{2} \\ . \\ . \\ . \\ {(A_{t})}^{12} \end{matrix}],$ ${\overset{&OverBar;}{A}}_{t} = [\begin{matrix} A_{t} \\ {(A_{t})}^{2} \\ . \\ . \\ . \\ {(A_{t})}^{12} \end{matrix}],$

${\overset{&OverBar; &OverBar;}{B B}}_{t t} = = [\begin{matrix} {B B}_{t t} & 00 & 00 & 00 \\ {A A}_{t t} {B B}_{t t} & {B B}_{t t} & 00 & 00 \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ {(({A A}_{t t}))}^{33} {B B}_{t t} & {(({A A}_{t t}))}^{22} {B B}_{t t} & {A A}_{t t} {B B}_{t t} & {B B}_{t t} \\ {(({A A}_{t t}))}^{44} {B B}_{t t} & {(({A A}_{t t}))}^{33} {B B}_{t t} & {(({A A}_{t t}))}^{22} {B B}_{t t} & {Σ Σ}_{i i = = 33}^{44} {(({A A}_{t t}))}^{44 - - i i} {B B}_{t t} \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ {(({A A}_{t t}))}^{1111} {B B}_{t t} & {(({A A}_{t t}))}^{1010} {B B}_{t t} & {(({A A}_{t t}))}^{99} {B B}_{t t} & {Σ Σ}_{i i = = 33}^{1111} {(({A A}_{t t}))}^{1111 - - i i} {B B}_{t t} \end{matrix}] - - - - - - ((1111))$

故而可针对磁悬浮球系统在t时刻的带函数权RBF-ARX模型的局部线性特性设计随t而变化的线性预测控制器。Therefore, a linear predictive controller that changes with t can be designed according to the local linear characteristics of the RBF-ARX model with functional weights of the maglev ball system at time t.

定义控制增量及期望输出变量 Define Control Increments and the desired output variable

$\{\begin{matrix} Δ Δ \overset{^^}{u u} ((t t)) = = {[[Δu Δu ((t t)) Δu Δu ((t t + + 11)) Δu Δu ((t t + + 22)) Δu Δu ((t t + + 33))]]}^{T T} \\ {\overset{^^}{g g}}_{r r} ((t t)) = = {[[{g g}_{r r} ((t t + + 11)) {g g}_{r r} ((t t + + 22)) . . . . . . {g g}_{r r} ((r r + + 1212))]]}^{T T} \\ Δu Δu ((t t)) = = u u ((t t)) - - u u ((t t - - 11)) \end{matrix} - - - - - - ((1212))$

则有局部线性预测控制的二次规划优化函数Then there is the quadratic programming optimization function of local linear predictive control

$\begin{matrix} \underset{\overset{^^}{u u} ((t t))}{min min} J J = = {| | | | \overset{^^}{g g} ((t t)) - - {\overset{^^}{g g}}_{r r} ((t t)) | | | |}_{{1.8 1.8 I I}_{1212}}^{22} + + {| | | | \overset{^^}{u u} ((t t)) | | | |}_{{0.0002 0.0002 I I}_{44}}^{22} + + {| | | | Δ Δ \overset{^^}{u u} ((t t)) | | | |}_{{0.16 0.16 I I}_{44}}^{22} \\ \begin{matrix} s the s . . t t . . & - - 22 twenty two \leq \leq \overset{^^}{g g} ((t t)) \leq \leq 00 \end{matrix} \\ \begin{matrix} 00 \leq \leq \overset{^^}{u u} ((t t)) \leq \leq 1010 \end{matrix} \\ \begin{matrix} - - 33 \leq \leq Δ Δ \overset{^^}{u u} ((t t)) \leq \leq 33 \end{matrix} \end{matrix} - - - - - - ((1313))$

其中，I为单位矩阵。in, I is the identity matrix.

对磁悬浮球系统，式(13)为一个二次规划的优化问题，通过在线优化即可获得最优控制量。从而将非线性磁悬浮球系统的预测控制简化为随钢球位置状态变化的、线性的预测控制，能大大节约最优控制量的在线优化时间，使钢球快速地达到稳定状态。For the maglev ball system, formula (13) is a quadratic programming optimization problem, and the optimal control amount can be obtained through online optimization. Therefore, the predictive control of the nonlinear magnetic levitation ball system is simplified to a linear predictive control that changes with the position and state of the steel ball, which can greatly save the online optimization time of the optimal control amount, and make the steel ball quickly reach a stable state.

Claims

1. A magnetic levitation ball position control method, is characterized in that, comprises the following steps:

1) Establish an autoregressive model for the magnetic levitation ball system:

g g ((t t)) = = {φ φ}_{00} + + {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{g g} g g ((t t - - i i)) + + {Σ Σ}_{i i = = 11}^{77} {φ φ}_{i i}^{u u} u u ((t t - - i i)) + + ξ ξ ((t t))

Among them, g(t) is the position of the magnetic levitation ball at time t; u(t) is the input voltage of the electromagnetic winding; ξ(t) is white noise;

\{\begin{matrix} φ_{0} = c_{0}^{0} + V_{1}^{0} \exp (- 0.03 {| | W (t - 1) - 11.74 | |}^{2}) \\ φ_{i}^{g} = c_{i, 0}^{g} + V_{i, l}^{g} \exp (- 0.03 | | W (t - 1) - 11.74 {| |}^{2}) \\ φ_{j}^{u} = c_{j, 0}^{u} + V_{j, 1}^{u} \exp (- 1.33 | | {W (t - 1) + 3.82 | |}^{2}) \\ W (t - 1) = g (t - 1) \\ V^{0} = v_{0}^{0} + v_{1}^{0} g (t - 1) \\ V_{i}^{g} = v_{i, 0}^{g} + v_{i, 1}^{g} g (t - 1) \\ V_{j}^{u} = v_{j, 0}^{g} + v_{j, 1}^{u} g (t - 1) \\ i = 1,2, . . ., 7; j = 1,2, . . ., 6 \end{matrix};

V ⁰ , and is the weight coefficient of the RBF network, which is a linear function of the position of the steel ball; is a constant coefficient, identified by the SNPOM optimization method; g(t-1) is the position of the maglev ball at time t-1;

2) Design a predictive controller based on the autoregressive model, optimize the following quadratic programming function J to obtain the optimal control quantity, and control the position g (t) of the magnetic levitation ball:

\{\begin{matrix} \underset{\overset{^^}{u u} ((t t))}{min min} & J J = = {| | | | \overset{^^}{g g} ((t t)) - - {\overset{^^}{g g}}_{r r} ((t t)) | | | |}_{{1.8 1.8 I I}_{1212}}^{22} + + {| | | | \overset{^^}{u u} ((t t)) | | | |}_{0.0002 0.0002 {I I}_{44}}^{22} + + {| | | | Δ Δ \overset{^^}{u u} ((t t)) | | | |}_{{0.16 0.16 I I}_{44}}^{22} \\ s the s . . t t . . & - - 22 twenty two \leq \leq \overset{^^}{g g} ((t t)) \leq \leq 00 \\ 00 \leq \leq \overset{^^}{u u} ((t t)) \leq \leq 1010 \\ - - 33 \leq \leq Δ Δ \overset{^^}{u u} ((t t)) \leq \leq 33 \end{matrix}

in,

{\hat{g}}_{r} (t) = {[\begin{matrix} g_{r} (t + 1) & g_{r} (t + 2) & &Center Dot; &Center Dot; &Center Dot; & g_{r} (t + 12) \end{matrix}]}^{T},

g _r (t+l) is the l-step forward reference position given at time t, l=1,2,...,12;

\hat{u} (t) = {[\begin{matrix} u (t) & u (t + 1) & u (t + 2) & u (t + 3) \end{matrix}]}^{T},

u(t+p) is the electromagnetic winding input voltage to be optimized at time t, and only the first item u(t) is used to act on the controlled magnetic levitation ball, p=0,1,2,3;

Δ u (t) = {[\begin{matrix} Δu (t) & Δ u (t + 1) & Δ u (t + 2) & Δu (t + 3 \end{matrix}]}^{T},

Δu(t)=u(t)-u(t-1) is the input voltage increment;

\hat{g} (t) = \overset{&OverBar;}{C} {\overset{&OverBar;}{A}}_{t} x (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{B}}_{t} \hat{u} (t) + \overset{&OverBar;}{C} {\overset{&OverBar;}{Γ}}_{t} {\overset{&OverBar;}{Φ}}_{t},

and is the system prediction coefficient matrix;

\{\begin{matrix} x (t) =^{{[\begin{matrix} x_{1, t} & x_{2, t} & \cdot &Center Dot; \cdot & x_{7, t} \end{matrix}]}^{T}} \\ x_{1, t} = g (t) \\ x_{k, t} = Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{g} g (t - i) + Σ_{i = 1}^{7 - k + 1} φ_{i + k - 1}^{u} u (t - i) \\ k = 2,3, . . ., 7 \end{matrix}

is the state vector;

{\overset{&OverBar;}{Φ}}_{t} = {[\begin{matrix} Φ_{t}^{T} & Φ_{t + 1}^{T} & \cdot \cdot \cdot & Φ_{t + 11}^{T} \end{matrix}]}^{T},

Φ_{t} = [\begin{matrix} φ_{0} \\ 0 \\ \cdot \\ &Center Dot; \\ &Center Dot; \\ 0 \end{matrix}];

φ ₀ , and is the regression function coefficient of the autoregressive model,

φ_{7}^{u} = 0;

I is the identity matrix.

2. The method for controlling the position of the magnetic levitation ball according to claim 1, wherein:

{\overset{&OverBar;}{A}}_{t} = [\begin{matrix} A_{t} \\ {(A_{t})}^{2} \\ . \\ . \\ . \\ {(A_{t})}^{12} \end{matrix}],

{\overset{&OverBar; &OverBar;}{B B}}_{t t} = = [\begin{matrix} {B B}_{t t} & 00 & 00 & 00 \\ {A A}_{t t} {B B}_{t t} & {B B}_{t t} & 00 & 00 \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ {(({A A}_{t t}))}^{33} {B B}_{t t} & {(({A A}_{t t}))}^{22} {B B}_{t t} & {A A}_{t t} {B B}_{t t} & {B B}_{t t} \\ {(({A A}_{t t}))}^{44} {B B}_{t t} & {(({A A}_{t t}))}^{33} {B B}_{t t} & {(({A A}_{t t}))}^{22} {B B}_{t t} & {Σ Σ}_{i i = = 33}^{44} {(({A A}_{t t}))}^{44 - - i i} {B B}_{t t} \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ . . & . . & . . & . . \\ {(({A A}_{t t}))}^{1111} {B B}_{t t} & {(({A A}_{t t}))}^{1010} {B B}_{t t} & {(({A A}_{t t}))}^{99} {B B}_{t t} & {Σ Σ}_{i i = = 33}^{1111} {(({A A}_{t t}))}^{1111 - - i i} {B B}_{t t} \end{matrix}];;

in:

\{\begin{matrix} {A A}_{t t} = = [\begin{matrix} {φ φ}_{11}^{g g} & 11 & 00 & 00 & 00 \\ {φ φ}_{22}^{g g} & 00 & 11 & 00 & 00 \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ . . & . . & . . & . . & . . \\ {φ φ}_{66}^{g g} & 00 & 00 & 00 & 11 \\ {φ φ}_{77}^{g g} & 00 & 00 & 00 & 00 \end{matrix}],, {B B}_{t t} [\begin{matrix} {φ φ}_{11}^{u u} \\ {φ φ}_{22}^{u u} \\ . . \\ . . \\ . . \\ {φ φ}_{66}^{u u} \\ 00 \end{matrix}] \\ {[\begin{matrix} 11 \\ 00 \\ . . \\ . . \\ . . \\ 00 \end{matrix}]}^{T T} \end{matrix} . .