CN104793645B - A method for controlling the position of a magnetic levitation ball - Google Patents

Abstract

The invention discloses a kind of magnetic levitation ball position control method, the shortcoming of precise physical model is difficult to set up for maglev ball system, sets up tape function weight coefficient type autoregression model to describe the non-linear dynamic characteristic between electromagnetism winding input voltage and steel ball position using system identifying method.The model weight coefficient of the once linear function of steel ball position as Gauss RBF networks, and with function type coefficient of the RBF networks as non linear autoregressive model, the model is preferably portrayed the dynamic characteristic of maglev ball system.At a time regression coefficient is constant to the model, similar to a linear ARX model.Based on this, the present invention one time-varying of design, local linear predictive controller, by being controlled in each moment line solver quadratic programming come the quick optimal location for realizing steel ball, meets maglev ball system stabilization, quickly required.

Description

A method for controlling the position of a magnetic levitation ball

technical field

The invention relates to the technical field of automatic control, in particular to a method for controlling the position of a magnetic levitation ball.

Background technique

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.

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 the PID can improve the ability of the 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 .

Contents of the invention

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.

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:

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; 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.

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:

in, 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 with is the system prediction coefficient matrix, which is obtained by the autoregressive model and multi-step predictor variables described at time t; g _r (t+l) is the l-step forward reference position given at time 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; △u(t)=u(t)-u(t-1) is the input voltage increment; is the state vector; 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.

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.

Description of drawings

Fig. 1 is a structural diagram of the magnetic levitation ball system of the present invention.

detailed description

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.

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.

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 (Least Square 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:

in,

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 working point state of the system, and the steel ball position g(t-1) is used to characterize the working point state of the system; V ⁰ , with 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 Formula (1) is rewritten as but

in, It is the observation data of the steel ball position, and SNPOM uses a total of 4000 observation data for model identification.

A time-varying and linear predictive controller is designed by utilizing the local linearity of the maglev ball system. Rewrite equation (1) as a polynomial

in,

Define the state variables of the system:

Then the state space model of formula (1) is:

here

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:

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:

here, with 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).

Therefore, a linear predictive controller that changes with t can be designed for the local linear characteristics of the RBF-ARX model with functional weights of the magnetically levitated ball system at time t.

Define Control Increments and the desired output variable

Then there is the quadratic programming optimization function of local linear predictive control

in, I is the identity matrix.

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 (2)

1. A magnetic levitation ball position control method is characterized by comprising the following steps:

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

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

is a linear function of the position of the steel ball;the constant coefficient is identified by an SNPOM optimization method; g (t-1) is the position of the magnetic levitation ball at the time of t-1;

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

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wherein,g_r(t + l) is the given forward reference position for l steps at time t, l being 1,2, …, 12;u (t + p) is the input voltage of the electromagnetic winding to be optimized at the moment t, only the first term u (t) is taken to act on the controlled magnetic levitation ball, and p is 0,1,2 and 3;Δ u (t) u (t-1) is the input voltage increment; andpredicting a coefficient matrix for the system;is a state vector;φ₀、andare the coefficients of the regression function of the autoregressive model,i is an identity matrix; i is₁₂Is an identity matrix of 12 orders; i is₄Is an identity matrix of order 4.

2. The magnetic levitation ball position control method as recited in claim 1,

<mrow> <msub> <mover> <mi>B</mi> <mo>&amp;OverBar;</mo> </mover> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>B</mi> <mi>t</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>4</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mn>4</mn> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mrow> <mn>4</mn> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>11</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>10</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mn>9</mn> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> <mtd> <mrow> <munderover> <mi>&amp;Sigma;</mi> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> </mrow> <mn>11</mn> </munderover> <msup> <mrow> <mo>(</mo> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mrow> <mn>11</mn> <mo>-</mo> <mi>i</mi> </mrow> </msup> <msub> <mi>B</mi> <mi>t</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>;</mo> </mrow>

wherein:

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>A</mi> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>1</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>2</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>6</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>7</mn> <mi>g</mi> </msubsup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <msub> <mi>B</mi> <mi>t</mi> </msub> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>1</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>2</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&amp;phi;</mi> <mn>6</mn> <mi>u</mi> </msubsup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>C</mi> <mo>=</mo> <msup> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mtable> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> <mtr> <mtd> <mo>.</mo> </mtd> </mtr> </mtable> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mfenced> <mi>T</mi> </msup> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>2