CN110794680A - Magnetic levitation ball system prediction tracking control method based on extended state observer - Google Patents

Magnetic levitation ball system prediction tracking control method based on extended state observer Download PDF

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CN110794680A
CN110794680A CN201911112876.4A CN201911112876A CN110794680A CN 110794680 A CN110794680 A CN 110794680A CN 201911112876 A CN201911112876 A CN 201911112876A CN 110794680 A CN110794680 A CN 110794680A
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王军晓
陈林杰
赵磊
俞立
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Zhejiang University of Technology ZJUT
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Abstract

A magnetic levitation ball system prediction tracking control method based on an extended state observer comprises the following steps: 1) establishing a magnetic levitation ball system model; 2) the total disturbance obtained by ESO estimation is eliminated from the adverse effect on the system in a forward compensation mode; 3) obtaining an output prediction value according to a prediction equation, and defining a prediction control performance index; 4) and (4) solving the optimal control input of the system by minimizing the performance index to complete the track tracking task. The invention estimates the unmeasured state variable by an Extended State Observer (ESO), which is more suitable for the practical application of engineering; an estimated value of the total disturbance can be obtained through ESO and compensated in a feedforward control mode, so that the anti-interference capacity of the system is improved; the MPC controller may cause the system to track the desired trajectory.

Description

Magnetic levitation ball system prediction tracking control method based on extended state observer
Technical Field
The invention relates to the field of trajectory tracking of a magnetic levitation ball system, in particular to a magnetic levitation ball system prediction tracking control method based on an extended state observer, which is mainly suitable for the magnetic levitation ball system of which the partial state of the system can not be directly measured by a sensor and is easily influenced by external disturbance.
Background
Magnetic levitation is a technique for suspending objects in a non-contact fashion. This technique eliminates mechanical contact between moving and stationary parts, and therefore has a series of advantages of no friction, no wear, no noise, long life, etc. With the development of subjects such as control theory, novel electromagnetic materials and the like, the magnetic levitation technology has also been developed rapidly. At present, magnetic levitation technology is widely applied in the industrial field, such as magnetic levitation trains, magnetic levitation bearings, magnetic levitation vibration isolators, microelectronic packaging and the like. Therefore, the research result of the track tracking control of the magnetic suspension ball system not only adds a theoretical result to the motion control of the magnetic suspension system, but also lays a solid foundation for the practical application of the magnetic suspension system.
As an advanced control method, Model Predictive Control (MPC) has the advantages of good control effect, strong robustness, capability of effectively overcoming the uncertainty of the system and explicitly processing the system constraint and the like, however, a magnetic levitation ball system is a typical strong nonlinear, open-loop and unstable system, and under the actions of unmodeled dynamics, perturbation of model parameters, external interference and measurement noise, the prediction model often hardly reflects the actual dynamics model of a controlled object, thereby reducing the control performance of the predictive control to a great extent. The Extended State Observer (ESO) can expand the total disturbance influencing the controlled output into a new state variable, so that the system state quantity and the total disturbance obtained by ESO estimation are combined with a model prediction control method, and the track tracking performance and the anti-interference capability of the magnetic levitation ball system can be improved.
Disclosure of Invention
In order to solve the problems that a speed signal in a magnetic levitation ball system cannot be directly measured by a sensor and the system cannot be interfered by external disturbance, the invention provides a magnetic levitation ball system prediction tracking control method based on an extended state observer. And then, performing optimization control on nominal systems except the total disturbance by adopting a model prediction control method to realize track tracking.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a magnetic levitation ball system prediction tracking control method based on an extended state observer comprises the following steps:
1) establishing a magnetic levitation ball system model;
defining m as the mass of the magnetic suspension small ball, g as the gravity acceleration, h as the instantaneous gap between the surface of the electromagnet and the steel ball, i as the instantaneous current passing through the electromagnetic coil, F as the electromagnetic force, u as the controlled voltage applied to the electromagnetic coil, R as the equivalent resistance of the electromagnetic coil, L as the self-inductance of the electromagnetic coil, and K as the mutual inductance of the electromagnetic coil, then modeling the magnetic suspension ball system according to the physical law as follows:
Figure BDA0002273247060000021
for the electromagnetic force F (i, h) at the equilibrium point (i)0,h0) Expansion of Taylor series as follows
F(i,h)＝F(i0,h0)+ki(i-i0)+kh(h-h0)+O(i,h) (2)
Wherein F (i)0,h0)＝mg，
Figure BDA0002273247060000022
O (i, h) is a higher order term of F (i, h);
then, the system (1) is rewritten as
Let h be x1，
Figure BDA0002273247060000024
Defining state variable x ═ x1x2]TThe state space equation of the magnetic levitation ball system is expressed as
Wherein
Figure BDA0002273247060000026
C＝[1 0]And y is the measurement output, y is,
Figure BDA0002273247060000027
this term is considered to be the total disturbance of the system, including unmodeled dynamics of the system, linearization error, and external disturbances;
2) designing an extended state observer;
for the system (4), the total disturbance d is expanded into a new state variable, x3The resulting expanded state system is denoted as d
Wherein
Figure BDA0002273247060000032
Let the sampling period be TsThen the discrete-time model of the system (4) is represented as
Figure BDA0002273247060000034
Wherein
Figure BDA0002273247060000035
Cd＝C；
The discrete-time model of the extended state system (5) is represented as
Figure BDA0002273247060000036
Wherein
Figure BDA0002273247060000037
The extended state observer is designed for the system (7) as follows
Figure BDA0002273247060000038
Wherein
Figure BDA0002273247060000039
In order to expand the state quantity of the state observer,for observer estimation output, L is an observation gain matrix to be designed;
3) designing a closed-loop controller;
the estimated value of the system state obtained by ESO estimation is used
Figure BDA00022732470600000311
And total disturbance estimate
Figure BDA00022732470600000312
Integrated into the system controller design;
combining the formula (6) and the formula (8), ignoring the total disturbance d, and obtaining an incremental discrete time prediction model of a nominal system
Figure BDA00022732470600000313
Wherein
Figure BDA00022732470600000314
For the state estimate increment at step k +1,
Figure BDA0002273247060000041
incremental state estimates for k steps, Δ uC(k)＝uC(k)-uC(k-1) control increments for step k;
setting a prediction time domain to NpControl time domain as NcAnd N isc≤NpTo derive the predictive equations of the system, let it controlThe control quantity outside the field being constant, i.e. Δ uC(k+i)＝0,i＝Nc,Nc+1,...,Np-1；
The prediction equation of the output k + i step of the system is derived from equation (9) as
Figure BDA0002273247060000042
Definition of NpStep output prediction vector sum NcThe step input increment vector is as follows
Figure BDA0002273247060000043
According to the formulae (10), (11), system NpThe step output prediction is calculated by the following prediction equation
Figure BDA0002273247060000044
Wherein
Figure BDA0002273247060000045
Figure BDA0002273247060000046
Defining an objective function as
J＝||Γy(Yp(k+1|k)-R(k+1))||2+||ΓuΔUC(k)||2(14)
Wherein gamma isyAnd ΓuWeight matrices for tracking error and control increment, respectively, R (k +1) is the desired trajectory;
substituting formula (12) for formula (14), for Δ UC(k) Make a derivative of
Figure BDA0002273247060000051
Then the optimal control sequence
Figure BDA0002273247060000052
Is shown as
Figure BDA0002273247060000053
Wherein
Figure BDA0002273247060000054
Taking the first element of the optimal control sequence to act on the system, the control increment is calculated by the following formula
ΔuC(k)＝KmpcEp(k+1|k) (17)
Wherein KmpcFor predicting the control gain, is expressed as
Figure BDA0002273247060000055
The system optimal control input from equation (17) is
uC(k)＝uC(k-1)+ΔuC(k) (19)
The closed-loop control law taking into account the interference compensation is
Figure BDA0002273247060000056
The magnetic levitation ball system prediction tracking control process based on the extended state observer obtained by the analysis is as follows:
s1: calculating S from equation (13)x，SuAnd I, then K is calculated by the formula (18)mpc；
S2: at time k, the state quantity of ESO estimation is obtainedAnd total disturbance
Figure BDA0002273247060000058
Computing
Figure BDA0002273247060000059
Error E is calculated from equation (16)p(k+1|k)；
S3: the control increment Δ u is calculated from equation (17)C(k) Thereby obtaining a predictive control optimum control input u by the calculation of the formula (19)C(k) Calculating a system closed-loop control law u (k) by the formula (20);
s4: at time k +1, k is made k +1, and the process returns to step S1.
The technical conception of the invention is as follows: firstly, model uncertainty and external disturbance are regarded as total disturbance, an Extended State Observer (ESO) is adopted to estimate the total disturbance, the influence of the disturbance is eliminated in a forward compensation mode, and the anti-interference capability of the system is improved. And then, performing optimization control on nominal systems except the total disturbance by adopting a model prediction control method to realize track tracking.
The invention has the following beneficial effects: the unmeasured state variable is estimated by an Extended State Observer (ESO), so that the method is more suitable for the practical application of engineering; an estimated value of the total disturbance can be obtained through ESO and compensated in a feedforward control mode, so that the anti-interference capacity of the system is improved; the MPC controller may cause the system to track the desired trajectory.
Drawings
Fig. 1 is a schematic view of a mechanical body of a magnetic levitation ball system, wherein 1 is an external support, 2 is an excitation electromagnet fixing threaded rod, 3 is an excitation electromagnet, 4 is a small iron ball, 5 is a laser displacement sensor, and 6 is a laser displacement sensor fixing support;
FIG. 2 is a block diagram of a magnetic levitation ball system predictive tracking control structure based on an extended state observer;
FIG. 3 is a diagram of magnetic levitation ball system predictive tracking control simulation based on an extended state observer.
Detailed Description
The invention is further described below with reference to the accompanying drawings.
Referring to fig. 1 to 3, a magnetic levitation ball system prediction tracking control method based on an extended state observer includes the following steps:
1) establishing a magnetic levitation ball system model;
defining m as the mass of the magnetic suspension small ball, g as the gravity acceleration, h as the instantaneous gap between the surface of the electromagnet and the steel ball, i as the instantaneous current passing through the electromagnetic coil, F as the electromagnetic force, u as the controlled voltage applied to the electromagnetic coil, R as the equivalent resistance of the electromagnetic coil, L as the self-inductance of the electromagnetic coil, and K as the mutual inductance of the electromagnetic coil, then the magnetic suspension ball system can be modeled as follows according to the physical law:
Figure BDA0002273247060000061
for the electromagnetic force F (i, h) at the equilibrium point (i)0,h0) Expansion of Taylor series as follows
F(i,h)＝F(i0,h0)+ki(i-i0)+kh(h-h0)+O(i,h) (2)
Wherein F (i)0,h0)＝mg，
Figure BDA0002273247060000062
O (i, h) is a higher order term of F (i, h);
then, the system (1) is rewritten as
Figure BDA0002273247060000063
Let h be x1，
Figure BDA0002273247060000064
Defining state variable x ═ x1x2]TThe state space equation of the magnetic levitation ball system is expressed as
Figure BDA0002273247060000071
WhereinC＝[1 0]And y is the measurement output, y is,
Figure BDA0002273247060000073
this term is considered as the total disturbance of the system, including unmodeled dynamics of the system, linearization errors, and external disturbances; .
2) Designing an extended state observer;
for the system (4), the total disturbance d is expanded into a new state variable, x3The resulting expanded state system is denoted as d
Figure BDA0002273247060000074
Wherein
Figure BDA0002273247060000075
Let the sampling period be TsThen the discrete-time model of the system (4) is represented as
Figure BDA0002273247060000076
WhereinCd＝C；
The discrete-time model of the extended state system (5) is represented as
Figure BDA0002273247060000078
Wherein
Figure BDA0002273247060000079
The extended state observer is designed for the system (7) as follows
Figure BDA00022732470600000710
WhereinIn order to expand the state quantity of the state observer,
Figure BDA00022732470600000712
for observer estimation output, L is an observation gain matrix to be designed;
3) designing a closed-loop controller;
the estimated value of the system state obtained by ESO estimation is usedAnd total disturbance estimate
Figure BDA0002273247060000082
Integrated into the system controller design;
combining the formula (6) and the formula (8), ignoring the total disturbance d, and obtaining an incremental discrete time prediction model of a nominal system
Figure BDA0002273247060000083
Wherein
Figure BDA0002273247060000084
For the state estimate increment at step k +1,
Figure BDA0002273247060000085
incremental state estimates for k steps, Δ uC(k)＝uC(k)-uC(k-1) control increments for step k;
setting a prediction time domain to NpControl time domain as NcAnd N isc≤NpTo derive the prediction equation for the system, the control quantity outside the control time domain is made constant, i.e. Δ uC(k+i)＝0,i＝Nc,Nc+1,...,Np-1；
The output prediction equation of the k + i step of the system is derived from the formula (9) as
Figure BDA0002273247060000086
Definition of NpStep output prediction vector sum NcThe step input increment vector is as follows
Figure BDA0002273247060000087
According to the formulae (10), (11), system NpThe step output prediction is calculated by the following prediction equation
Figure BDA0002273247060000088
Wherein
Figure BDA0002273247060000089
Defining an objective function as
J＝||Γy(Yp(k+1|k)-R(k+1))||2+||ΓuΔUC(k)||2(14)
Wherein gamma isyAnd ΓuWeight matrices for tracking error and control increment, respectively, R (k +1) is the desired trajectory;
substituting formula (12) for formula (14), for Δ UC(k) Make a derivative ofThen the optimal control sequence
Figure BDA0002273247060000093
Is shown as
Figure BDA0002273247060000094
Wherein
Taking the first element of the optimal control sequence to act on the system, the control increment is calculated by the following formula
ΔuC(k)＝KmpcEp(k+1|k) (17)
Wherein KmpcFor predicting the control gain, is expressed as
The system optimal control input available from equation (17) is
uC(k)＝uC(k-1)+ΔuC(k) (19)
The closed-loop control law taking into account the interference compensation is
Figure BDA0002273247060000097
The magnetic levitation ball system prediction tracking control process based on the extended state observer obtained by the analysis is as follows:
s1: calculating S from equation (13)x，SuAnd I, then K is calculated by the formula (18)mpc；
S2: at time k, the state quantity of ESO estimation is obtained
Figure BDA0002273247060000098
And total disturbance
Figure BDA0002273247060000099
Computing
Figure BDA00022732470600000910
Error E is calculated from equation (16)p(k+1|k)；
S3: the control increment Δ u is calculated from equation (17)C(k) Thereby obtaining a predictive control optimum control input u by the calculation of the formula (19)C(k) Calculating a system closed-loop control law u (k) by the formula (20);
s4: at time k +1, k is made k +1, and the process returns to step S1.
Setting Γ in conjunction with fig. 3y＝[1 1 1 3 3]，Γu＝[0.1 0.1]From the initial state x ═ 00 in the simulation experiment]TAnd starting, adding disturbance when t is 10s, and tracking the magnetic suspension small ball to a specified track h of 0.0425m by using a magnetic suspension ball system prediction tracking control method based on the extended state observer, and effectively inhibiting the influence of the disturbance.

Claims (1)

1. A magnetic levitation ball system prediction tracking control method based on an extended state observer is characterized by comprising the following steps:
1) establishing a magnetic levitation ball system model;
defining m as the mass of the magnetic suspension small ball, g as the gravity acceleration, h as the instantaneous gap between the surface of the electromagnet and the steel ball, i as the instantaneous current passing through the electromagnetic coil, F as the electromagnetic force, u as the controlled voltage applied to the electromagnetic coil, R as the equivalent resistance of the electromagnetic coil, L as the self-inductance of the electromagnetic coil, and K as the mutual inductance of the electromagnetic coil, then modeling the magnetic suspension ball system according to the physical law as follows:
Figure FDA0002273247050000011
for the electromagnetic force F (i, h) at the equilibrium point (i)0,h0) Expansion of Taylor series as follows
F(i,h)＝F(i0,h0)+ki(i-i0)+kh(h-h0)+O(i,h) (2)
Wherein F (i)0,h0)＝mg，
Figure FDA0002273247050000012
O (i, h) is a higher order term of F (i, h);
then, the system (1) is rewritten as
Figure FDA0002273247050000013
Let h be x1，Defining state variable x ═ x1x2]TThe state space equation of the magnetic levitation ball system is expressed as
Figure FDA0002273247050000015
Wherein
Figure FDA0002273247050000016
C＝[1 0]And y is the measurement output, y is,
Figure FDA0002273247050000017
this term is considered to be the total disturbance of the system, including unmodeled dynamics of the system, linearization error, and external disturbances;
2) designing an extended state observer;
for the system (4), the total disturbance d is expanded into a new state variable, x3The resulting expanded state system is denoted as d
Wherein
Figure FDA0002273247050000021
Let the sampling period be TsThen the discrete-time model of the system (4) is represented as
Figure FDA0002273247050000022
WhereinCd＝C；
The discrete-time model of the extended state system (5) is represented as
Wherein
Figure FDA0002273247050000025
The extended state observer is designed for the system (7) as follows
Figure FDA0002273247050000026
WhereinIn order to expand the state quantity of the state observer,
Figure FDA0002273247050000028
for observer estimation output, L is an observation gain matrix to be designed;
3) designing a closed-loop controller;
the estimated value of the system state obtained by ESO estimation is used
Figure FDA0002273247050000029
And total disturbance estimateIntegrated into the system controller design;
combining the formula (6) and the formula (8), ignoring the total disturbance d, and obtaining an incremental discrete time prediction model of a nominal system
Figure FDA00022732470500000211
WhereinIn the shape of k +1 stepThe increment of the state estimation value is increased,
Figure FDA00022732470500000213
incremental state estimates for k steps, Δ uC(k)＝uC(k)-uC(k-1) control increments for step k;
setting a prediction time domain to NpControl time domain as NcAnd N isc≤NpTo derive the prediction equation for the system, the control quantity outside the control time domain is made constant, i.e. Δ uC(k+i)＝0,i＝Nc,Nc+1,...,Np-1；
The prediction equation of the output k + i step of the system is derived from equation (9) as
Figure FDA00022732470500000214
Definition of NpStep output prediction vector sum NcThe step input increment vector is as follows
Figure FDA00022732470500000215
According to the formulae (10), (11), system NpThe step output prediction is calculated by the following prediction equation
Figure FDA0002273247050000031
Wherein
Figure FDA0002273247050000033
Defining an objective function as
J＝||Γy(Yp(k+1|k)-R(k+1))||2+||ΓuΔUC(k)||2(14)
Wherein gamma isyAnd ΓuWeight matrices for tracking error and control increment, respectively, R (k +1) is the desired trajectory;
substituting formula (12) for formula (14), for Δ UC(k) Make a derivative of
Figure FDA0002273247050000034
Then the optimal control sequence
Figure FDA0002273247050000035
Is shown as
Wherein
Figure FDA0002273247050000037
Taking the first element of the optimal control sequence to act on the system, the control increment is calculated by the following formula
ΔuC(k)＝KmpcEp(k+1|k) (17)
Wherein KmpcFor predicting the control gain, is expressed as
Figure FDA0002273247050000038
The system optimal control input from equation (17) is
uC(k)＝uC(k-1)+ΔuC(k) (19)
The closed-loop control law taking into account the interference compensation is
Figure FDA0002273247050000039
The magnetic levitation ball system prediction tracking control process based on the extended state observer obtained by the analysis is as follows:
s1: calculating S from equation (13)x，SuAnd I, then K is calculated by the formula (18)mpc；
S2: at time k, the state quantity of ESO estimation is obtained
Figure FDA00022732470500000310
And total disturbance
Figure FDA00022732470500000311
Computing
Figure FDA00022732470500000312
Error E is calculated from equation (16)p(k+1|k)；
S3: the control increment Δ u is calculated from equation (17)C(k) Thereby obtaining a predictive control optimum control input u by the calculation of the formula (19)C(k) Calculating a system closed-loop control law u (k) by the formula (20);
s4: at time k +1, k is made k +1, and the process returns to step S1.
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