CN110007605B  Robust prediction control method of repelling magnetic suspension device  Google Patents
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Abstract
The invention discloses a robust prediction control method of a repulsion type magnetic suspension device, which comprises the steps of collecting historical data N groups of input voltage and output distance of the repulsion type magnetic suspension device, and establishing a nonlinear model; establishing a convex polyhedron state space model of the repelling magnetic suspension device based on the nonlinear model; 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. The invention considers the influence of system modeling error and uncertain interference in the controller design, and is a control method with stable robustness and strong applicability.
Description
Technical Field
The invention relates to the field of automatic control, in particular to a robust prediction control method for a repulsive magnetic suspension device.
Background
The magnetic suspension technology is an electromechanical integration technology, eddy current is formed on the surface of metal by utilizing a highfrequency magnetic field, and then Lorentz magnetic force is generated to suspend metal equipment, so that the contact between the equipment is effectively avoided, the mutual friction is reduced, and the application prospect is wide. The magnetic suspension system is a complex nonlinear system integrating control of electric magnetic force, air gap, current and the like, and an accurate mathematical model of the magnetic suspension system is difficult to obtain in practical application. The identification modeling method based on data driving is a mathematical modeling method independent of system physical mechanism, only uses input and output data of an object to perform modeling, and is widely applied to modeling of a complex nonlinear system.
The PID controller has a simple control algorithm structure and does not depend on an accurate mathematical model of a controlled object, so that the PID controller is widely applied to magnetic suspension control. However, the global control characteristic of the PID controller for a complex nonlinear object is poor, and particularly for a magnetic levitation ball control system with extremely high stability requirement, the situation of instantaneous uncontrolled falling of a levitation ball is very easy to occur in a large range near a boundary. The linear quadratic regulator is a control algorithm based on a controlled object state space model, and is widely applied to control of complex systems. But the robustness and stability of the control still need to be improved because the control is highly dependent on the accuracy of the controlled object model. Model predictive control is an advanced computer control algorithm generated in industrial process control practice and is widely used in the control of complex industrial systems. Through the search of the existing documents, the patent of 'a wind power magnetic suspension yaw motor control method based on model predictive control' (application number: 201810076334.5) provides a predictive control method based on the physical mechanism model design of a magnetic suspension 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 function weight coefficients. However, the two methods do not consider the influence of system modeling errors and uncertain interferences in the design process of the predictive controller, and the stability and robustness of the algorithm cannot be effectively guaranteed. Meanwhile, in the process of establishing a system state space model for subsequent predictive controller design, the patent "201510180614.7" directly uses the state quantity of the system at the current moment to approximate and replace the future state quantity of the system, and performs direct singlepoint linearization approximation processing on the future state space model of the system, and the method itself can greatly affect the accuracy of the model.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a robust prediction control method of a repulsive magnetic suspension device aiming at the defects of the prior art, consider the influence of system modeling errors and uncertain interference, and improve the robustness and applicability of the control method.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: a robust prediction control method of a repulsive magnetic suspension device comprises 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,t}Is an inverse quadratic timevarying coefficient with respect to y (t), and  L  ·   survival of the electrically nonwoven hair_{2}Representing a twonorm operation; relevant parameters of nonlinear modelAre obtained by optimization calculation through an RSNPOM 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.
The specific implementation process of the step 2) comprises 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_{set}The 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 + j1) is the input of the repulsion type magnetic suspension device at the moment t + j1; j is an integer less than or equal to zero;
2) onestep forward prediction polynomial model of repelling magnetic suspension deviceThe structure is as follows:
wherein θ (t) is derivedξ (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 onestep 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, the coefficient matrix A of the onestep forward prediction state vector X (t +1 t) of the repulsion type magnetic suspension device_{t}，B_{t}And 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+gt}，B_{t+gt}And predicting a coefficient matrix of a state vector X (t + g +1 t) for the repelling magnetic suspension device in the future g steps.
The coefficient matrix A_{t+gt}，B_{t+gt}The variation range is within the following convex polyhedron range:
wherein, { lambda ]_{t+gt,μ}1,2,3,4 is the linear coefficient of the convex polyhedron; the 4 vertexes of the convex polyhedron are (A)_{1},B_{1})，(A_{2},B_{2})，(A_{3},B_{3}) And (A)_{4},B_{4}) And, and:
wherein the content of the first and second substances,andare functions relating to y (t), respectivelyMaximum and minimum values of;andare functions relating to y (t), respectivelyMaximum and minimum values of.
In step 3), the optimization objective function is designed as follows:
wherein the content of the first and second substances,I_{2}is 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_{t}the future feedback control rate of the repulsion type magnetic suspension device at the time t.
Solving the optimized objective function by the following set of inequalities:
wherein, the symbol represents the symmetric structure of the matrix; { Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 is an intermediate matrix variable generated by solving the inequality set, namely the convex optimization problem; gamma ray_{0}+ 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, gamma_{0}、{Y,G, Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 , andare all the minimized variable gamma_{0}+ 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 set_{0}+ gamma minimum intermediate variables gamma, gamma_{0}、{Y,G, Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 , andwhen the inequality group has feasible solution, the optimization process is ended, and the obtained result isI.e. the value of the input voltage acting on the windings of the magnetic levitation system at time t.
Compared with the prior art, the invention has the beneficial effects that: the invention takes the influence of external interference which cannot be avoided by an actual magnetic suspension system into consideration, the invention utilizes a timevarying coefficient quasilinear regression structure model based on data driving to model the magnetic suspension system, and the influence of modeling error and external uncertain interference of the system is taken into consideration in the modeling process. In order to overcome the defect that the stability and robustness of an algorithm are difficult to ensure by a common prediction control method, the invention provides a robust prediction control method which can be realized by solving a linear matrix inequality set based on an established timevarying coefficient regression model of a magnetic suspension system, and the influence of system modeling errors and uncertain interference is considered in the design of a controller, so that the method is a control method with stable robustness and strong applicability.
Drawings
Fig. 1 is a structural view of a repulsive magnetic levitation apparatus to which the present invention is directed.
Detailed Description
The structure of the repulsive magnetic suspension device to which the invention is directed is shown in fig. 1, wherein: the globe comprises a globe shell (the radius is 20 cm), a square cylindrical magnet 3, a support between the globe shell 1 and the square cylindrical magnet 3, an infrared reflection position sensor 4 (the model is ST178H), an iron core 5 (the section radius is 2.5 cm), a winding 6 (the number of turns is 3500), and a control circuit board 7 based on a single chip microcomputer. The system controls the distance (the control distance range is 5 cm25 cm) 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 0V20V). The specific embodiment of the robust control method of the repulsive magnetic suspension device comprises the following steps of:
step 1: for the repulsive magnetic levitation device shown in fig. 1, a historical data 2500 group of input voltage and output distance of the system is acquired, and the following nonlinear model of the system is established:
in the above formula, y (t) is the distance between the bottom of the globe 2 and the infrared reflection position sensor 4 at the time t, namely the output quantity of the system, u (t) is the voltage applied to the winding 6 by the control board 7 at the time t, namely the input quantity of the system, ξ (t +1) is a term containing modeling error and uncertain disturbance, and  ξ (t +1)  is less than or equal to 1, and { a +1 ≦ is_{0,t},a_{1,t},b_{1,t},a_{2,t},b_{2,t}Is an inverse quadratic timevarying coefficient with respect to y (t), and  L  ·   survival of the electrically nonwoven hair_{2}Representing a twonorm operation; relevant parameters of the modelAre obtained by optimization calculation through an RSNPOM optimization method (the RSNPOM optimization method is detailed in the documents of Zeng X, Peng H, Zhou F, 2018, A customized SNPOM for standing parameter estimation of RBFAR (X) model, IEEE Transactions on Neural networks and Learning Systems,29, No.4, 779:
step 2: based on the magnetic suspension system inverse quadratic function type timevarying coefficient regression model (1) established in the step 1, a convex polyhedron state space model of the system is established, and the specific process is as follows:
the input increment and the output increment of the magnetic levitation system are defined as follows:
in the above formula, y (t + j) is the output of the system at the time t + j; y is_{set}∈[5,25]The expected value of the system at the moment t; u (t + j) is the input of the system at the moment t + j, and u (t + j1) is the input of the system at the moment t + j1; j is 0, 1, 2. From the above definitions, one step forward prediction of the model can be derivedThe structure is as follows:
in the above formula, θ (t) is derivedThe intermediate quantities that are generated in the process,ξ (t +1 t) is the amount of uncertainty interference and system modeling error involved, and  ξ (t +1 t)  ≦ 1.
The state vector of the magnetic levitation system is defined as follows:
based on the above definition of the onestep forward prediction polynomial model and the input/output deviation amount of the system, the state space model corresponding to the polynomial model of the system can be derived as follows:
in the above equation, the coefficient matrix A of the onestep forward prediction state vector X (t +1 t) of the system_{t}，B_{t}And X (t  t) are the parameters and states respectively calculated by the model identified in step S1 at time t;the xi (t) cannot be accurately obtained at the moment t because the unknown interference term ξ (t +1 t) of the system is contained in the xi (t), but the change range of the xi (t) is less than or equal to 1 in the vector quantity because ξ (t +1 t) Andto (c) to (d); coefficient matrix A of forward predicted state vector X (t + g +1 t) for g steps in the future of the system_{t+gt}，B_{t+gt}The variation range of the method cannot be directly calculated at the time t, but is within the following convex polyhedron range:
in the above formula, { λ_{ } _{ } _{ } _{ } _{t+gt,μ}1,2,3,4 is a linear coefficient of the polyhedron; the 4 vertexes of the polyhedron are (A)_{1},B_{1})，(A_{2},B_{2})，(A_{3},B_{3}) And (A)_{4},B_{4}) And is and
wherein the content of the first and second substances, because y (t) e [5,25]Then, then Thus, 4 vertices (A) of the convex polyhedron set (9) can be calculated_{1},B_{1})，(A_{2},B_{2})，(A_{3},B_{3}) And (A)_{4},B_{4})。
And step 3: based on the system convex polyhedron state space model (78) established in the step 2, the optimized objective function design of the robust control method for the repulsive magnetic suspension device is as follows:
in the above formula, the first and second carbon atoms are,i is 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;and inputting a control increment for the system at the step t + g predicted at the moment t. The future control rate structure design of the robust predictive controller is as follows:g≥1，F_{t}the future feedback control rate of the system at the time t.
Based on the designed controller optimization objective function, the optimal control rate of the robust control method for the repulsive magnetic levitation device is obtained by solving the following linear matrix inequality set:
in the above formula, the symbol represents a symmetric structure of the matrix; f_{t}＝YG^{1}The future feedback control rate of the system at the moment t; { Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 is an intermediate matrix variable generated by solving the convex optimization problem; gamma ray_{0}+ gamma is the optimized target value of the convex optimization problem (12), and gamma are simultaneously_{0}Also an intermediate quantity generated in the above optimization process; coefficient matrix A_{t}、B_{t}、X (t  t) is a parameter matrix known at time t; { (A)_{μ},B_{μ}) And . mu. ═ 1,2,3,4} is a vertex of the system polyhedron model (8) in the step 2. Gamma, gamma_{0}、{Y,G, Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 , andare all the minimized variable gamma_{0}The intermediate variables obtained in the + gamma solution process. When solving the minimization problem (12), the optimization function automatically searches for gamma which meets the requirement according to the inequality constraint conditions (1316)_{0}+ gamma minimum intermediate variables gamma, gamma_{0}、{Y,G, Q_{ } _{ } _{ } _{ } _{μ}1,2,3,4 , andwhen the above is mentionedWhen the optimization problem (1216) has a feasible solution, the optimization process is ended. At this time, obtainedI.e. the value of the input voltage acting on the windings of the magnetic levitation system at time t.
Claims (1)
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,t}Is an inverse quadratic timevarying coefficient with respect to y (t), and  L  ·   survival of the electrically nonwoven hair_{2}Representing a twonorm operation; relevant parameters of nonlinear modelAre obtained by optimization calculation through an RSNPOM optimization method;
2) establishing a convex polyhedron state space model of the repelling magnetic suspension device based on the nonlinear model;
3) based on the convex polyhedron state space model, obtaining an optimized objective function for robust control of the repelling magnetic suspension device, and solving the optimized objective function to obtain an input voltage value acting on a winding of the repelling magnetic suspension device at the moment t;
the specific implementation process of the step 2) comprises 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_{set}The 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 + j1) is the input of the repulsion type magnetic suspension device at the moment t + j1; j is an integer less than or equal to zero;
(2) onestep forward prediction polynomial model of repelling magnetic suspension deviceThe structure is as follows:
wherein the content of the first and second substances,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 onestep 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, the coefficient matrix A of the onestep forward prediction state vector X (t +1 t) of the repulsion type magnetic suspension device_{t}，B_{t}And 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+gt}，B_{t+gt}Predicting a coefficient matrix of a state vector X (t + g +1 t) for the repelling magnetic suspension device in the future g steps;
the coefficient matrix A_{t+gt}，B_{t+gt}The variation range is within the following convex polyhedron range:
wherein, { lambda ]_{t+gt,μ}1,2,3,4 is the linear coefficient of the convex polyhedron; the 4 vertexes of the convex polyhedron are (A)_{1},B_{1})，(A_{2},B_{2})，(A_{3},B_{3}) And (A)_{4},B_{4}) And, and:
wherein the content of the first and second substances,andare functions relating to y (t), respectivelyMaximum and minimum values of;andare functions relating to y (t), respectivelyMaximum and minimum values of;
in step 3), the optimization objective function is designed as follows:
wherein the content of the first and second substances,I_{2}is 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;t + g step rejection for prediction at time tInputting control increment by a magnetic suspension device;F_{t}for the future feedback control rate of the repulsion type magnetic suspension device at the time t,
solving the optimized objective function by the following set of inequalities:
wherein, the symbol represents the symmetric structure of the matrix; { Q_{μ}1,2,3,4 is an intermediate matrix variable generated by solving the inequality set, namely the convex optimization problem; gamma ray_{0}+ 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, gamma_{0}、{Y,G,Q_{μ}1,2,3,4 , andare all the minimized variable gamma_{0}+ gamma solving intermediate variables obtained in the process; in solving the minimization problemThe optimization function is based on the above inequalityFormula group constraint condition automatic search for gamma satisfying_{0}+ gamma minimum intermediate variables gamma, gamma_{0}、{Y,G,Q_{μ}1,2,3,4 , andwhen the inequality group has feasible solution, the optimization process is ended, and the obtained result isI.e. the value of the input voltage acting on the windings of the magnetic levitation system at time t.
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