CN110007605A - A robust predictive control method for a repulsive magnetic levitation device - Google Patents

Abstract

The invention discloses a robust predictive control method for a repelling magnetic levitation device, which collects N groups of historical data of the input voltage and output distance of the repulsive magnetic levitation device, and establishes a nonlinear model; Convex polyhedron state space model; based on the convex polyhedron state space model, obtain the optimal objective function of the robust control of the repulsive magnetic levitation device, solve the objective function, and obtain the input voltage value acting on the winding of the repulsive magnetic levitation device at time t. The present invention considers the influence of system modeling error and uncertain disturbance in the controller design, and is a robust, stable and highly applicable control method.

Description

A robust predictive control method for a repulsive magnetic levitation device

technical field

The invention relates to the field of automatic control, in particular to a robust predictive control method for a repelling magnetic levitation device.

Background technique

Magnetic levitation technology is an electromechanical integration technology. It uses a high-frequency magnetic field to form eddy currents on the metal surface, thereby generating Loren magnetic force to suspend metal equipment, effectively avoiding contact between equipment and reducing mutual friction. application prospects. The magnetic levitation system is a complex nonlinear system that integrates the control of electromagnetic force, air gap, and current. It is difficult to obtain its accurate mathematical model in practical applications. The data-driven identification modeling method is a kind of mathematical modeling method that does not depend on the physical mechanism of the system. It only uses the input and output data of the object for modeling, and is widely used in the modeling of complex nonlinear systems.

PID controller has been widely used in magnetic levitation control because of its simple control algorithm structure and no dependence on the precise mathematical model of the controlled object. However, the PID controller has poor global control characteristics for complex nonlinear objects, especially for the magnetic levitation ball control system with extremely high stability requirements. The linear quadratic regulator is a control algorithm based on the state space model of the controlled object, and is also widely used in the control of complex systems. However, because it is highly dependent on the accuracy of the controlled object model, the robustness and stability of its control still need to be improved. Model predictive control is an advanced computer control algorithm produced in the practice of industrial process control, and is widely used in the control of complex industrial systems. Through the retrieval of existing literature, it is found that the patent "A Wind Electromagnetic Suspension Yaw Motor Control Method Based on Model Predictive Control" (application number: 201810076334.5) proposes a predictive control method based on the design of the physical mechanism model of the magnetic levitation system. The patent "A Magnetic Levitation Ball Position Control Method" (application number: 201510180614.7) proposes a predictive control method based on an autoregressive model with a function weight coefficient. However, the above two methods do not consider the influence of system modeling errors and uncertain disturbances in the process of predictive controller design, and the stability and robustness of the algorithms cannot be effectively guaranteed. At the same time, the patent "201510180614.7" in the process of establishing the system state space model for the subsequent predictive controller design, directly uses the state quantity of the system at the current moment to approximate the future state quantity of the system, and directly analyzes the future state space model of the system. Single-point linearization approximation, the method itself will have a greater impact on the accuracy of the model.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is to provide a robust predictive control method for a repelling magnetic levitation device in view of the deficiencies of the prior art, which improves the robustness and applicability of the control method by considering the influence of system modeling errors and uncertain disturbances. sex.

In order to solve the above-mentioned technical problems, the technical scheme adopted in the present invention is: a robust predictive control method of a repelling magnetic levitation device, comprising the following steps:

1) Collect N groups of historical data of the input voltage and output distance of the repelling magnetic levitation device, and establish the following nonlinear model:

Among them, y(t) is the distance between the bottom of the globe of the repulsive magnetic levitation device and the infrared reflection position sensor at time t, that is, the output of the repulsive magnetic levitation device; u(t) is the time t The control board of the repulsive magnetic levitation device applies to the windings , that is, the input of the repelling magnetic levitation device; ξ(t+1) is a term that includes modeling errors and uncertain disturbances, and |ξ(t+1)|≤1; {a _0,t ,a _1,t ,b _1,t ,a _2,t ,b _2,t } are time-varying coefficients of inverse quadratic function with respect to y(t), and ||·|| ₂ represents two-norm operation; nonlinear The relevant parameters of the model All are optimized and calculated by the R-SNPOM optimization method;

2) Based on the above nonlinear model, establish a convex polyhedron state space model of the repelling magnetic levitation device;

3) Based on the convex polyhedron state space model, obtain the optimal objective function of the robust control of the repulsive magnetic levitation device, solve the objective function, and obtain the input voltage value acting on the winding of the repulsive magnetic levitation device at time t.

The specific implementation process of step 2) includes:

1) Define the input increment and output increment of the repelling magnetic levitation device as follows:

Among them, y(t+j) is the output of the repulsive magnetic levitation device at time t+j; y _set is the expected value of the repulsive magnetic levitation device at time t; u(t+j) is the input of the repulsive magnetic levitation device at time t+j, u(t+j-1) is the input of the repelling magnetic levitation device at time t+j-1; j is an integer less than or equal to zero;

2) One-step forward prediction polynomial model of repulsive magnetic levitation device The structure is as follows:

where θ(t) is the derivation The intermediate quantity generated in the process; ξ(t+1|t) is the quantity including the system modeling error and uncertain disturbance, and |ξ(t+1|t)|≤1;

3) Based on the above-mentioned one-step forward prediction polynomial model and the definition of the input increment and output increment of the system, the state space model corresponding to the polynomial model of the repulsive magnetic levitation device is derived as follows:

Among them, the coefficient matrix A _t , B _t and X(t|t) of the one-step forward prediction state vector X(t+1|t) of the repelling magnetic levitation device are the parameters and states calculated by the nonlinear model at time t, respectively; is the input increment of the repelling magnetic levitation device at time t; the variation range of Ξ(t) is in the vector and between; A _t+g|t , B _t+g|t are the coefficient matrix of the future g-step forward prediction state vector X(t+g+1|t) of the repulsive magnetic levitation device.

The coefficient matrices A _t+g|t and B _t+g|t vary in the range of the following convex polyhedron:

Among them, {λ _{t+g|t, μ} |μ=1, 2, 3, 4} is the linear coefficient of the convex polyhedron; the four vertices of the convex polyhedron are (A ₁ , B ₁ ), (A ₂ , B ₂ ), (A ₃ , B ₃ ) and (A ₄ , B ₄ ), and:

in, and respectively as a function of y(t) the maximum and minimum values; and respectively as a function of y(t) the maximum and minimum values.

In step 3), the optimization objective function is designed as follows:

in, I ₂ is the unit matrix; X(t+g|t) is the t+g step system state quantity predicted by the model at time t; Input control increment for the t+g step repelling magnetic levitation device predicted at time t; g≥1, F _t is the future feedback control rate of the repelling magnetic levitation device at time t.

The optimization objective function is solved by the following set of inequalities:

Among them, the symbol * represents the symmetric structure of the matrix; {Q _μ |μ=1, 2, 3, 4} is the intermediate matrix variable generated by solving the above inequality group, that is, the convex optimization problem; γ ₀ +γ is the above convex optimization problem The optimization objective value of {(A _μ ,B _μ )|μ=1,2,3,4} is the vertex of the convex polyhedron model; γ, γ ₀ , {Y,G,Q _μ |μ=1,2,3 ,4} and are intermediate variables obtained during the solution process of minimizing variables γ ₀ +γ; in solving the minimization problem When , the optimization function automatically finds the intermediate variables γ, γ ₀ , {Y,G,Q _μ |μ=1,2,3,4} that satisfy the minimum γ ₀ +γ When the above inequality group has a feasible solution, the optimization process ends, and the obtained That is, the input voltage value acting on the winding of the magnetic suspension system at time t.

Compared with the prior art, the present invention has the following beneficial effects: the present invention takes into account that the actual magnetic levitation system is unavoidably affected by external interference, and the present invention utilizes a data-driven time-varying coefficient quasi-linear regression structure model. The magnetic levitation system is modeled, and the modeling error of the system and the influence of external uncertain interference are considered in the modeling process. In order to overcome the shortcomings that the general predictive control method is difficult to guarantee the stability and robustness of the algorithm, the present invention proposes a robust predictive control method that can be realized by solving the linear matrix inequality group based on the established time-varying coefficient regression model of the magnetic levitation system. The influence of system modeling errors and uncertain disturbances is considered in the design of the controller, which is a robust, stable and applicable control method.

Description of drawings

FIG. 1 is a structural diagram of a repelling magnetic levitation device aimed at by the present invention.

Detailed ways

The structure of the repelling magnetic levitation device targeted by the present invention is shown in Figure 1, wherein: 1 is the globe shell (radius is 20 cm), 3 is a square cylindrical magnet, and 2 is a bracket between the globe shell 1 and the square cylindrical magnet 3 , 4 is the infrared reflection position sensor (model ST178H), 5 is the iron core (section radius is 2.5 cm), 6 is the winding (the number of turns is 3500), and 7 is the control circuit board based on the microcontroller. The system controls the distance between the bottom of the globe 2 and the infrared reflection position sensor 4 by adjusting the voltage applied to the winding 6 by the control board 7 (the output voltage range is 0V-20V) (the control distance is 5cm-25cm). A specific embodiment of the robust control method for a repelling magnetic levitation device according to the present invention includes the following steps:

Step 1: For the repulsive magnetic levitation device shown in Figure 1, collect 2500 sets of historical data of the input voltage and output distance of the system, and establish the following nonlinear model of the system:

In the above formula, y(t) is the distance between the bottom of the globe 2 and the infrared reflection position sensor 4 at time t, that is, the output of the system; u(t) is the voltage applied by the control board 7 to the winding 6 at time t, that is, The input quantity of the system; ξ(t+1) is the term including modeling error and uncertain disturbance, and |ξ(t+1)|≤1; {a _0,t ,a _1,t ,b _1,t ,a _2,t ,b _2,t } is the inverse quadratic function time-varying coefficient about y(t), and ||·|| ₂ represents the two-norm operation; the relevant parameters of the model All are calculated by the R-SNPOM optimization method (see the literature for the R-SNPOM optimization method: Zeng X., Peng H., Zhou F., 2018, A regularized SNPOM for stableparameter estimation of RBF-AR(X)model, IEEE Transactions on Neural Networksand Learning Systems,29,No.4,779-791.), the specific parameter values obtained by optimization are:

Step 2: Based on the inverse quadratic function time-varying coefficient regression model (1) of the magnetic levitation system established in Step 1, a convex polyhedron state space model of the system is established. The specific process is as follows:

The input increment and output increment of the maglev system are defined as follows:

In the above formula, y(t+j) is the output of the system at time t+j; y _set ∈ [5,25] is the expected value of the system at time t; u(t+j) is the input of the system at time t+j, u (t+j-1) is the input of the system at time t+j-1; j=0,-1,-2,.... From the above definition, the one-step forward prediction of the model can be derived The structure is as follows:

In the above formula, θ(t) is the derivation The intermediate quantity generated in the process can be calculated from the output expectation and historical data of the system; ξ(t+1|t) is the quantity including the system modeling error and uncertain disturbance, and |ξ(t+1|t)| ≤1.

The state vector that defines the magnetic levitation system is as follows:

Based on the above-mentioned definition of the one-step forward prediction polynomial model and the input and output deviation of the system, the state space model corresponding to the polynomial model of the system can be deduced as follows:

In the above formula, the coefficient matrix A _t , B _t and X(t|t) of the one-step forward prediction state vector X(t+1|t) of the system are calculated by the model identified by step S1 at time t, respectively. parameters and status; is the input increment of the system at time t, and is the amount to be optimized by the algorithm; Ξ(t) cannot be accurately obtained at time t, because Ξ(t) contains the unknown interference term ξ(t+1|t) of the system, but due to ξ(t+1|t)|≤1, it can be seen that the variation range of Ξ(t) is in the vector and between; the coefficient matrix A _t+g|t and B _t+g|t of the system’s future g step forward prediction state vector X(t+g+1|t) cannot be directly calculated at time t, but its variation range In the range of the following convex polyhedra:

In the above formula, {λ _{t+g|t, μ} |μ=1,2,3,4} is the linear coefficient of the polyhedron; the four vertices of the polyhedron are (A ₁ , B ₁ ), (A ₂ , B ₂ ), (A ₃ , B ₃ ) and (A ₄ , B ₄ ), and

in, Because y(t)∈[5,25], then Therefore, the 4 vertices (A ₁ , B ₁ ), (A ₂ , B ₂ ), (A ₃ , B ₃ ) and (A ₄ , B ₄ ) of the convex polyhedron set (9) can be calculated.

Step 3: Based on the system convex polyhedron state space model (7-8) established in step 2, the optimization objective function of the robust control method for the repelling magnetic levitation device according to the present invention is designed as follows:

In the above formula, I is the unit matrix; X(t+g|t) is the state quantity of the t+g step system predicted by the model at time t; Enter the control increment for the t+g step system predicted at time t. The future control rate structure design of the robust predictive controller of the present invention is as follows: g≥1, F _t is the future feedback control rate of the system at time t.

Based on the controller optimization objective function of the above design, the optimal control rate of the robust control method for the repelling magnetic levitation device according to the present invention is obtained by solving the following linear matrix inequalities:

In the above formula, the symbol * represents the symmetric structure of the matrix; F _t = YG ^-1 is the future feedback control rate of the system at time t; {Q _μ | μ = 1, 2, 3, 4} is generated to solve the above convex optimization problem The _{intermediate matrix variable of} X(t|t) is the known parameter matrix at time t; {(A _μ ,B _μ )|μ=1,2,3,4} is the vertex of the system polyhedron model (8) in step 2. γ, γ ₀ , {Y, G, Q _μ | μ = 1, 2, 3, 4} and Both are intermediate variables obtained during the solution process of minimizing the variable γ ₀ +γ. When solving the minimization problem (12), the optimization function will automatically find the intermediate variables γ, γ ₀ , {Y,G,Q _μ |μ that satisfy the minimum γ ₀ +γ according to the above inequality constraints (13-16) =1,2,3,4} and When the above optimization problem (12-16) has a feasible solution, the optimization process ends. At this time, the obtained That is, the input voltage value acting on the winding of the magnetic suspension system at time t.

Claims (5)

1. A robust prediction control method of a repulsive magnetic suspension device is characterized by comprising the following steps:

1) acquiring historical data N groups of input voltage and output distance of the repulsion type magnetic suspension device, and establishing a nonlinear model as follows:

wherein y (t) is the distance between the bottom of a globe of the repulsion type magnetic suspension device and the infrared reflection position sensor at the time t, namely the output quantity of the repulsion type magnetic suspension device, u (t) is the voltage applied to a winding by a control board of the repulsion type magnetic suspension device at the time t, namely the input quantity of the repulsion type magnetic suspension device, ξ (t +1) is a term containing modeling error and uncertain disturbance, and | ξ (t +1) | is less than or equal to 1; { a%_0,t,a_1,t,b_1,t,a_2,t,b_2,tIs an inverse quadratic time-varying coefficient with respect to y (t), and | L | · | | survival of the electrically non-woven hair₂Representing a two-norm operation; relevant parameters of non-linear modelAre obtained by optimization calculation through an R-SNPOM optimization method;

2) establishing a convex polyhedron state space model of the repelling magnetic suspension device based on the nonlinear model;

3) and obtaining an optimized objective function for the robust control of the repelling magnetic suspension device based on the convex polyhedron state space model, and solving the objective function to obtain an input voltage value acting on a winding of the repelling magnetic suspension device at the moment t.

2. The robust predictive control method for a repulsive magnetic levitation device as recited in claim 1, wherein the step 2) is implemented by the following steps:

1) the input increment and the output increment of the repulsive magnetic levitation device are defined as follows:

wherein y (t + j) is the output of the repulsive magnetic suspension device at the moment t + j; y is_setThe expected value of the repulsive magnetic suspension device at the time t is shown; u (t + j) is the input of the repulsion type magnetic suspension device at the moment t + j, and u (t + j-1) is the input of the repulsion type magnetic suspension device at the moment t + j-1; j is an integer less than or equal to zero;

2) one-step forward prediction polynomial model of repelling magnetic suspension deviceThe structure is as follows:

wherein,to deriveξ (t +1| t) is the quantity containing system modeling error and uncertain disturbance, and | ξ (t +1| t) | is less than or equal to 1;

3) based on the one-step forward prediction polynomial model and the definition of the input increment and the output increment of the system, the state space model corresponding to the polynomial model of the repulsive magnetic suspension device is deduced as follows:

wherein, one of the repulsion type magnetic suspension deviceCoefficient matrix A of forward-step prediction state vector X (t +1| t)_t，B_tAnd X (t | t) are respectively a parameter and a state calculated by the nonlinear model at the time t;is the input increment of the repulsion type magnetic suspension device at the time t; xi (t) in a vectorAndto (c) to (d); a. the_t+g|t，B_t+g|tAnd predicting a coefficient matrix of a state vector X (t + g +1| t) for the repelling magnetic suspension device in the future g steps.

3. The robust predictive control method for repelling magnetic levitation devices according to claim 2, wherein the coefficient matrix a_t+g|t，B_t+g|tThe variation range is within the following convex polyhedron range:

wherein, { lambda ]_t+g|t,μ1,2,3,4 is the linear coefficient of the convex polyhedron; the 4 vertexes of the convex polyhedron are (A)₁,B₁)，(A₂,B₂)，(A₃,B₃) And (A)₄,B₄) And, and:

wherein,andare functions relating to y (t), respectivelyMaximum and minimum values of;andare functions relating to y (t), respectivelyMaximum and minimum values of.

4. The robust predictive control method for repelling magnetic levitation devices according to claim 3, wherein in step 3), the optimization objective function is designed as follows:

wherein,I₂is a unit array; x (t + g | t) is the system state quantity of the step t + g predicted by the model at the moment t;inputting a control increment for the t + g step repulsion type magnetic suspension device predicted at the t moment;g≥1，F_tthe future feedback control rate of the repulsion type magnetic suspension device at the time t.

5. The robust predictive control method for repelling magnetic levitation devices as recited in claim 4, wherein the optimization objective function is solved by the following set of inequalities:

wherein, the symbol represents the symmetric structure of the matrix; { Q_μ1,2,3,4 is an intermediate matrix variable generated by solving the inequality set, namely the convex optimization problem; gamma ray₀+ gamma is the optimized target value { (A) of the convex optimization problem_μ,B_μ) 1,2,3,4 is the vertex of the convex polyhedron model; gamma, gamma₀、{Y,G,Q_μ1,2,3,4 |, andare all the minimized variable gamma₀+ gamma solving intermediate variables obtained in the process; in solving the minimization problemThen, the optimization function automatically searches for the gamma satisfying the constraint conditions of the inequality set₀+ gamma minimum intermediate variables gamma, gamma₀、{Y,G,Q_μ1,2,3,4 |, andwhen the above is mentionedWhen the inequality group has feasible solution, the optimization process is ended, and the obtained solution is obtainedI.e. the value of the input voltage acting on the windings of the magnetic levitation system at time t.