CN110412867B - High-precision angular rate control method for magnetically suspended control moment gyroscope frame system - Google Patents

High-precision angular rate control method for magnetically suspended control moment gyroscope frame system Download PDF

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CN110412867B
CN110412867B CN201910414922.XA CN201910414922A CN110412867B CN 110412867 B CN110412867 B CN 110412867B CN 201910414922 A CN201910414922 A CN 201910414922A CN 110412867 B CN110412867 B CN 110412867B
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CN110412867A (en
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李海涛
于江坤
宋鹏
崔馨方
孔令文
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Beihang University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
    • G05B13/04—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators
    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
    • G05B13/045—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance using a perturbation signal
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D13/00—Control of linear speed; Control of angular speed; Control of acceleration or deceleration, e.g. of a prime mover
    • G05D13/62—Control of linear speed; Control of angular speed; Control of acceleration or deceleration, e.g. of a prime mover characterised by the use of electric means, e.g. use of a tachometric dynamo, use of a transducer converting an electric value into a displacement

Abstract

The invention provides a high-precision angular rate control method of a magnetic suspension control moment gyro frame system based on a parameter optimization extended state observer. Firstly, a voltage and torque output equation of a magnetic suspension control moment gyro frame servo system is established, a state space expression is established based on the equation, the state space expression is converted into an integral series connection form to design an extended state observer, and finally lumped interference is eliminated through state feedback. Through analyzing the estimation error in the observer, a cost function for estimating the influence degree of the system lumped interference and the measurement noise is established so as to minimize the comprehensive influence of the system lumped interference and the measurement noise, and further, a self-adaptive dynamic parameter optimization method is designed. The invention provides a parameter dynamic optimization method of an extended state observer, improves the dynamic interference suppression and anti-noise performance of a frame system, and can be used for realizing high-precision angular rate control of a magnetic suspension control moment gyroscope frame system.

Description

High-precision angular rate control method for magnetically suspended control moment gyroscope frame system
Technical Field
The invention belongs to the field of high-precision control of a magnetic suspension control moment gyro frame system based on a parameter optimization observer, and particularly relates to a high-precision angular rate control method of a magnetic suspension control moment gyro frame system based on a parameter optimization extended state observer, which is used for improving the interference suppression capability and the anti-noise capability of the frame system, realizing high-precision angular rate tracking control of a frame servo system in a dynamic process and further realizing high-precision angular rate output of a control moment gyro.
Background
Control Moment Gyros (CMGs) and Inertial Momentum Wheels (IMWs) are the primary inertial actuators. The magnetic suspension control moment gyroscope has the advantages of large output torque, good dynamic performance, high control precision and the like, and becomes a preferred execution mechanism for spacecraft attitude control. The frame servo system is an important component of the magnetic suspension control moment gyro, and the control precision of the angular position and the angular rate of the frame servo system has a serious influence on the precision of the output moment.
The Permanent Magnet Synchronous Motor (PMSM) has the advantages of high power density, high efficiency, large torque, low noise, convenient maintenance and the like, and is an ideal choice for a frame servo motor. However, when the permanent magnet synchronous motor is operated at a low speed, the low-frequency torque ripple may generate a severe rotational speed ripple. Further, the frame servo is an ultra low speed servo, and generally operates in a speed range of 0.01 °/s, so that torque ripple is one of the main factors affecting the frame servo performance.
The periodic torque ripple of the permanent magnet synchronous motor is due to the high harmonics contained in the current. Generally, methods of reducing torque ripple can be classified into two categories. One is the optimized design of the motor body, but is only suitable for newly developed motors, and the optimized design is complex and the cost is high. The other method starts from a motor control strategy, and improves the stator current waveform of the motor through a proper control method so as to reduce the torque ripple. However, the second method is applied on the basis of information that parameters of the motor and torque ripple of the motor must be known in advance. However, since the parameters of the motor change during the operation process, the accuracy of the control method is inevitably affected.
An Extended State Observer (ESO) is used as an effective interference estimation technology to expand lumped interference into a new state of a system, so that the reduction of interference estimation precision caused by inaccurate motor parameters and models can be effectively avoided; however, the traditional extended state observer design method does not generally consider the influence of measurement noise, and the measurement noise is inevitably generated in an actual system; and the parameter configuration method of the extended state observer adopts a bandwidth configuration method mostly, and the parameter configuration is simpler, but the method has certain defects, for example, the change of the parameters in the running process of the motor can cause the reduction of the estimation precision of the lumped interference, the influence of the measurement noise in an actual system is not fully considered, and the like, which can cause the reduction of the angular rate output precision of the magnetic suspension control moment gyro.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method for controlling the angular rate of the magnetically suspended control moment gyroscope frame system based on the parameter optimization extended state observer overcomes the defects of the existing method, the parameter optimization is carried out on the extended state observer through the method, the better estimation effect on the system state is obtained, the interference suppression and noise resistance of the system are improved, and the angular rate output precision of the control moment gyroscope frame system is greatly improved.
The technical scheme adopted by the invention for solving the technical problems is as follows:
a high-precision angular rate control method of a magnetic suspension control moment gyro frame system based on a parameter optimization extended state observer is characterized by firstly establishing a voltage and torque output equation of the magnetic suspension control moment gyro frame system, establishing a state space expression based on the equation, converting the state space expression into an integral series connection form to design the extended state observer, and eliminating lumped interference through state feedback; by analyzing the estimation error in the extended state observer, a cost function for evaluating the influence degree of the system lumped interference and the measurement noise is established so as to minimize the comprehensive influence of the system lumped interference and the measurement noise, and then the self-adaptive dynamic parameter optimization method is designed.
The method comprises the following steps:
step (1): establishing a voltage and torque output equation of a magnetic suspension control moment gyroscope frame servo system, which can be written as follows:
Figure RE-GDA0002144424350000021
wherein u isdAnd uqD and q axis stator voltages, respectively; i.e. idAnd iqD and q axis stator currents respectively; l isdAnd LqD, q-axis inductances (L) respectivelyd＝Lq＝L)。λdAnd λqRespectively are d-axis and q-axis magnetic chains of the permanent magnet synchronous motor; r is the stator resistance; ω is the mechanical angular velocity; b is the coefficient of friction; t isLIs the load torque; t iscogIs the cogging torque; t iseIs an electromagnetic torque; p is the number of pole pairs of the motor; j is the moment of inertia.
Taking the q-axis current loop as an example, the design methods of the d-axis current loop and the speed loop are consistent with the q-axis method. For q-axis current, it follows from the voltage and torque output equations above:
Figure RE-GDA0002144424350000022
wherein
Figure RE-GDA0002144424350000023
Is the output of the q-axis current loop;
Figure RE-GDA0002144424350000024
"lumped disturbances" for q-axis stator currents;
Figure RE-GDA0002144424350000025
in known amounts.
Defining a state variable as X ═ X1x2]＝[iqdq(t)]Let us order
Figure RE-GDA0002144424350000026
The state space expression for the q-axis current is then:
Figure RE-GDA0002144424350000031
step (2): outputting a q-axis current loop according to the state space expression of the q-axis current of the framework system in the step (1)
Figure RE-GDA0002144424350000032
As inputs to the extended state observer, the extended state observer can be designed to:
Figure RE-GDA0002144424350000033
wherein z is1For estimating iq；z2For estimating dq(t)；eq＝z1-iq；β1And β2Are parameters of the extended state observer.
According to the state feedback control theory, the control rate is designed as follows:
Figure RE-GDA0002144424350000034
wherein k isqIs a current loop controller parameter;
Figure RE-GDA0002144424350000035
is the reference current for the q-axis input.
By adopting the same method as the steps (1) and (2), the extended state observers and the control rates of the d-axis current loop and the speed loop can be respectively designed as follows:
the d-axis current loop extended state observer is designed as follows:
Figure RE-GDA0002144424350000036
wherein z isd1For estimating id；zd2For estimating lumped disturbance of d-axis βd1And βd2Is a parameter of the extended state observer;
Figure RE-GDA0002144424350000037
ed＝zd1-id。
the d-axis current loop control rate is designed as follows:
Figure RE-GDA0002144424350000038
wherein
Figure RE-GDA0002144424350000039
Is the output of the d-axis current loop, kdAs is the parameter of the current loop controller,
Figure RE-GDA00021444243500000310
is a d-axis reference current.
The extended state observer of the velocity loop is designed to:
Figure RE-GDA00021444243500000311
wherein z isω1For estimating ω;zω2for estimating lumped disturbances of the velocity loop;
Figure RE-GDA0002144424350000041
eω＝zω1-ω；βω1and βω2Are parameters of the extended state observer.
The control rate of the speed loop is designed as follows:
Figure RE-GDA0002144424350000042
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000043
is the output of the velocity loop, kωAs a parameter of the speed loop controller, omega*The reference speed input for the speed loop.
And (3): adding measurement noise which actually exists and affects the system precision into the extended state observer in the steps (1) and (2), so as to obtain the extended state observer containing a noise term, taking q axis as an example, and adding a state space expression of the noise term as follows:
Figure RE-GDA0002144424350000044
where h (t) is measurement noise.
The extended state observer containing the noise term can be designed as:
Figure RE-GDA0002144424350000045
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000046
let e be Z-X, i.e. e1＝z1-x1，e2＝z2-x2(ii) a The differential of the error is then:
Figure RE-GDA0002144424350000047
the differential equation is subjected to laplace transform to obtain:
Figure RE-GDA0002144424350000048
wherein G ═ s + β1)。
Analyzing the above equation, if the estimation accuracy is improved, it is necessary to make the estimation error tend to zero (i.e. e)1→0,e2→ 0), β should be made2>>β1True, but if β2Too large, will introduce more measurement noise, although there is a conflict between the two, it is possible to choose β appropriately2、β1To achieve an optimum value within the acceptable range of the system for both the lumped interference and the measurement noise.
Further, the parameter optimization method obtained according to the extended state observer model and the error differential equation is as follows:
firstly, a performance index function is established as follows:
Figure RE-GDA0002144424350000051
wherein P is a symmetric semi-positive constant array, Q (t) is a symmetric semi-positive constant array, R is a symmetric positive constant array, t isfIs the terminal time.
First item in the Performance index M
Figure RE-GDA0002144424350000052
The penalty to the terminal error is represented by a cost function of the terminal error; first term e in integral termT(t) Q (t) e (t) represents the component generated by e (t) in the system operation process, and is used for measuring the error e (t); second term U in integral termT(t) R (t) U (t) represents the component generated by U (t) in the system operation process, and is used for measuring the energy size. Thus, the optimum control sought, U (t), is achieved when M is minimal.
Solving the problem by using a variational method, firstly taking a Hamiltonian as follows:
Figure RE-GDA0002144424350000053
wherein λ isT(t)＝[λ1(t),λ2(t)]Is a lagrange multiplier vector function; and:
Figure RE-GDA0002144424350000054
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000055
the Hamiltonian can be simplified to a known quantity by the following equation:
Figure RE-GDA0002144424350000056
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000057
thus, the requirements for an optimal solution are the collaborative equations:
Figure RE-GDA0002144424350000058
the control equation is:
Figure RE-GDA0002144424350000059
obtaining by solution:
U(t)＝-R-1(t)DT(t)λ(t)
wherein R (t) is positive, so R-1(t) is present.
The conditions for the cross-section were:
Figure RE-GDA00021444243500000510
where φ is the first term in the performance metric M. Since X is 0 and e is Z in ideal output, X is 0 and Z is zero
Figure RE-GDA0002144424350000061
The system is a linear steady system, and the terminal time tf→ infinity, therefore in t removalfAt other times than, λ (t) will also be linear with z (t), i.e.:
λ(t)＝PZ(t)
thus, combining the above conditions, the Riccati equation can be given as follows:
Figure RE-GDA0002144424350000062
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000063
De＝[1 0](ii) a Q and R are semi-positive definite matrixes; and the equation has a unique semi-positive solution matrix P.
The extended state observer parameters are designed as follows:
Figure RE-GDA0002144424350000064
selecting
Figure RE-GDA0002144424350000065
R ═ 1; can be solved to obtain:
Figure RE-GDA0002144424350000066
wherein gamma is greater than 0.
From this solution, the two parameters β of the extended state observer1、β2The selection of (a) can be converted into the selection of a scalar parameter gamma, and the value of the scalar parameter gamma determines the measurement noise sensitivity and the estimation performance of the system lumped interference; and as gamma increases, the system is sensitive to noiseThe degree is improved but the estimation effect of the interference is better.
The observed effect of the extended state observer requires a trade-off between the effects of both the estimation error of the lumped disturbances and the measurement noise. The result of the trade-off is described by the cost function J. The design cost function is:
J(γ)＝χe(t)2+γ
wherein
Figure RE-GDA0002144424350000067
χ is the balance parameter.
e(t)2The term is implicitly dependent on the parameter γ and can be used to evaluate the magnitude of the estimation error affected by the measurement noise; the gamma term is used to evaluate the effect of measurement noise; the function of the coefficient χ is to balance the front and rear parts e (t) of the cost function J2The ratio between the terms and the gamma term reflects the tolerance of the system to noise and collective interference.
By utilizing the cost function, the specific steps of the dynamic adaptive parameter optimization designed by the invention are as follows:
step (1): first order gammal＝α+γm，γr＝β+γm，γm＝γmRespectively representing left, right and middle three selectable values of the parameter gamma, wherein α & gt 0 and β & lt 0 are step sizes in positive and negative directions respectively;
step (2) based on the step (1), after the extended state observer obtains the input u (k) and the output y (k) for the first time, the parameters β thereof1，β2From an initial value gammam＝γ0Determining, calculating γ ═ γl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) And cost function Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptAnd the value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output. For example when Jl＜JrAnd Jl＜JmWhen, get γopt＝γlAnd outputZl(k) As observed values of an extended state observer;
and (3): taking gamma on the basis of the step (2)m＝γoptPreparing to perform a second round operation, namely: after the extended state observer obtains new input u (k) and output y (k), γ ═ γ is calculated respectivelyl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) And cost function Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptThe value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output;
and (4): circularly operating the step (3) in the dynamic operation process of the magnetically suspended control moment gyro frame control system;
according to the steps (1) - (4), the magnetic suspension control moment gyro framework control system can continuously approach and obtain the optimal parameters and the optimal observed values in the circulation of the dynamic self-adaptive parameter optimization program in the dynamic operation process, and the reaction time for obtaining the optimal parameters and the optimal observed values is determined by alpha and beta; with a reasonable choice of the step sizes α, β, a smaller response time can be obtained.
Therefore, the dynamic adaptive parameter optimization process can enable the system to almost always operate under the condition of obtaining the optimal observed value, and the interference suppression capability, the anti-noise capability and the control precision of the whole system are greatly improved.
The basic principle of the invention is as follows: according to the invention, a state space expression is established according to a voltage and torque output equation of a magnetic suspension control moment gyroscope frame servo system, and is converted into an integral series connection form, so that an extended state observer is designed, and lumped interference is eliminated in a state feedback mode. By analyzing the estimation error in the observation equation, a cost function for evaluating the influence degree of the system lumped interference and the measurement noise is established so as to minimize the comprehensive influence of the system lumped interference and the measurement noise; and then an optimal parameter optimization method for adaptively and dynamically searching system control precision is designed. An optimal balance point is found between the accuracy of lumped interference estimation and the anti-noise capability of the extended state observer under the action of optimal parameters, the comprehensive performance of dynamic interference suppression and anti-noise of the whole system is improved, and therefore high-precision angular rate output of the magnetic suspension control moment gyroscope is achieved.
Compared with the prior art, the invention has the advantages that:
1. the design of the extended state observer takes the measurement noise item into consideration, so that the risk of introducing unnecessary measurement noise into the traditional extended state observer is avoided, the anti-noise capability of the system is improved, and the extended state observer has more practical engineering application value.
2. By the dynamic adaptive parameter optimization configuration method in the motor operation process, the reduction of lumped interference estimation precision caused by fixed parameters such as a bandwidth configuration method can be effectively avoided, the expansion state observer can almost be ensured to obtain optimal parameter values and optimal observed values all the time, the interference suppression capability and the noise resistance capability of the whole system are greatly improved, the moment pulsation problem is greatly improved, and high-precision angular rate output can be realized.
Drawings
FIG. 1 is a flow chart of a control method of the present invention;
FIG. 2 is a control block diagram of the extended state observer of the whole system;
FIG. 3 is a q-axis current loop control block diagram;
FIG. 4 is a block diagram of dynamic adaptive parameter optimization of an extended state observer;
Detailed Description
The invention is further described with reference to the following figures and detailed description.
Fig. 1 is a flow chart of the control method of the present invention. A high-precision angular rate control method of a magnetic suspension control moment gyro frame system based on a parameter optimization extended state observer is characterized by firstly establishing a voltage and torque output equation of a magnetic suspension control moment gyro frame servo system, establishing a state space expression based on the equation, converting the state space expression into an integral series connection form to design an extended state observer, and finally eliminating lumped interference through state feedback. By analyzing estimation errors in an equation, a cost function for evaluating the influence degree of system lumped interference and measurement noise is established so as to minimize the comprehensive influence of the system lumped interference and the measurement noise, and then a self-adaptive dynamic parameter optimization method is designed, so that the dynamic interference suppression and anti-noise performance of a frame system is improved, and the high-precision angular rate control of a magnetic suspension control moment gyro frame system is realized.
The specific implementation steps are as follows:
(1) designing state space expression of integral series form according to frame servo system voltage and torque output equation
The voltage and torque output equation of the magnetic suspension control moment gyro frame servo system can be written as follows:
Figure RE-GDA0002144424350000081
wherein u isdAnd uqD and q axis stator voltages, respectively; i.e. idAnd iqD and q axis stator currents respectively; l isdAnd LqD, q-axis inductances (L) respectivelyd＝Lq＝L)。λdAnd λqRespectively are d-axis and q-axis magnetic chains of the permanent magnet synchronous motor; r is the stator resistance; ω is the mechanical angular velocity; b is the coefficient of friction; t isLIs the load torque; t iscogIs the cogging torque; t iseIs an electromagnetic torque; p is the number of pole pairs of the motor; j is the moment of inertia.
As shown in fig. 2, the entire system control block diagram can be divided into a current loop and a speed loop, wherein the current loop is further divided into a q-axis current loop and a d-axis current loop. Taking the q-axis current loop as an example, the state space equation design method of the d-axis current loop and the speed loop is consistent with the q-axis method. For q-axis current, it can be obtained from equation (1):
Figure RE-GDA0002144424350000091
wherein
Figure RE-GDA0002144424350000092
Is the output of the q-axis current loop;
Figure RE-GDA0002144424350000093
"lumped disturbances" for q-axis stator currents;
Figure RE-GDA0002144424350000094
in known amounts.
Defining a state variable as x ═ x1x2]＝[iqdq(t)]Let us order
Figure RE-GDA0002144424350000095
The state space expression for the q-axis current is then:
Figure RE-GDA0002144424350000096
(2) extended state observer and control law design
According to the above formula (3), the q-axis current is outputted
Figure RE-GDA0002144424350000097
As inputs to the extended state observer, the extended state observer can be designed to:
Figure RE-GDA0002144424350000098
wherein z is1For estimating iq；z2For estimating dq(t)；eq＝z1-iq；β1And β2Are parameters of the extended state observer.
According to the state feedback control theory, the control law is designed as follows:
Figure RE-GDA0002144424350000099
wherein k isqIs a current loop controller parameter;
Figure RE-GDA00021444243500000910
is the reference current for the q-axis input.
Based on the design of the control law, a q-axis current loop control block diagram is shown in fig. 3.
By adopting the same method as the q-axis current loop, the extended state observers and the control laws of the d-axis current loop and the speed loop can be respectively designed as follows:
the d-axis current loop extended state observer is designed as follows:
Figure RE-GDA00021444243500000911
wherein z isd1For estimating id；zd2For estimating lumped disturbance of d-axis βd1And βd2Is a parameter of the extended state observer;
Figure RE-GDA00021444243500000912
ed＝zd1-id。
the d-axis current loop control law is designed as follows:
Figure RE-GDA0002144424350000101
wherein
Figure RE-GDA0002144424350000102
Is the output of the d-axis current loop, kdAs is the parameter of the current loop controller,
Figure RE-GDA0002144424350000103
is a d-axis reference current.
The extended state observer of the velocity loop is designed to:
Figure RE-GDA0002144424350000104
wherein z isω1For estimating ω; z is a radical ofω2For estimating lumped disturbances of the velocity loop;
Figure RE-GDA0002144424350000105
eω＝zω1-ω；βω1and βω2Are parameters of the extended state observer.
The control law of the speed ring is designed as follows:
Figure RE-GDA0002144424350000106
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000107
is the output of the velocity loop, kωAs a parameter of the speed loop controller, omega*The reference speed input for the speed loop.
According to the designed extended state observer and control law of each ring, a control block diagram of the extended state observer of the whole system is shown in fig. 2.
(3) Adding noise terms to extended state observer controller and analyzing estimation error
The measurement noise which actually exists and affects the system precision is added into the extended state observer, so that the extended state observer containing a noise term can be obtained, and taking the q axis as an example, a state space expression added with the noise term is as follows:
Figure RE-GDA0002144424350000108
where h (t) is measurement noise.
The extended state observer containing the noise term can be designed as:
Figure RE-GDA0002144424350000109
wherein the content of the first and second substances,
Figure RE-GDA00021444243500001010
let e be Z-X, i.e. e1＝z1-x1，e2＝z2-x2(ii) a The differential of the error is then:
Figure RE-GDA00021444243500001011
laplace transform of equation (12) yields:
Figure RE-GDA0002144424350000111
wherein G ═ s + β1)。
As can be seen from the analysis (13), to improve the estimation accuracy, it is necessary to make the estimation error tend to zero (i.e. e)1→0,e2→ 0), β should be made2>>β1True, but if β2Too large, will introduce more measurement noise, although there is a conflict between the two, it is possible to choose β appropriately2、β1To achieve an optimum value within the acceptable range of the system for both the lumped interference and the measurement noise.
(4) Constructing cost function and searching dynamic self-adapting optimum parameter according to the cost function
For the established extended state observer containing the noise term, to seek the optimal control, a performance index function is firstly established as follows:
Figure RE-GDA0002144424350000112
wherein P is a symmetric semi-positive constant array, Q (t) is a symmetric semi-positive constant array, R is a symmetric positive constant array, t isfIs the terminal time.
First item in the Performance index M
Figure RE-GDA0002144424350000113
The penalty to the terminal error is represented by a cost function of the terminal error; first term e in integral termT(t) Q (t) e (t) represents the component of the system during operation resulting from e (t) and is used to measure the error e(t) size; second term U in integral termT(t) R (t) U (t) represents the component generated by U (t) in the system operation process, and is used for measuring the energy size. Thus, the optimum control sought, U (t), is achieved when M is minimal.
Solving the problem by using a variational method, firstly taking a Hamiltonian as follows:
Figure RE-GDA0002144424350000114
wherein λ isT(t)＝[λ1(t),λ2(t)]Is a lagrange multiplier vector function; and:
Figure RE-GDA0002144424350000115
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000116
the Hamiltonian can be simplified to a known quantity by the following equation:
Figure RE-GDA0002144424350000117
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000118
thus, the requirements for an optimal solution are the collaborative equations:
Figure RE-GDA0002144424350000121
the control equation is:
Figure RE-GDA0002144424350000122
obtaining by solution:
U(t)＝-R-1(t)DT(t)λ(t)
wherein, due to R (t)Positive, so R-1(t) is present.
The conditions for the cross-section were:
Figure RE-GDA0002144424350000123
where φ is the first term in the performance metric M. Since X is 0 and e is Z in ideal output, X is 0 and Z is zero
Figure RE-GDA0002144424350000124
The system is a linear steady system, and the terminal time tf→ infinity, therefore in t removalfAt other times than, λ (t) will also be linear with z (t), i.e.:
λ(t)＝PZ(t)
thus, combining the above conditions, the Riccati equation can be given as follows:
Figure RE-GDA0002144424350000125
wherein the content of the first and second substances,
Figure RE-GDA0002144424350000126
De＝[1 0](ii) a Q and R are semi-positive definite matrixes; and the equation has a unique semi-positive solution matrix P.
The extended state observer parameters are designed as follows:
Figure RE-GDA0002144424350000127
selecting
Figure RE-GDA0002144424350000128
R ═ 1; can be solved to obtain:
Figure RE-GDA0002144424350000129
wherein gamma is greater than 0.
Expansion according to the formula (16)Two parameters β of the state observer1、β2The selection of (a) can be converted into the selection of a scalar parameter gamma, and the value of the scalar parameter gamma determines the measurement noise sensitivity and the estimation performance of the system lumped interference; and as gamma increases, the noise sensitivity of the system increases, but the interference estimation is more effective.
The observed effect of the extended state observer requires a trade-off between the effects of both the estimation error of the lumped disturbances and the measurement noise. The result of the trade-off is described by the cost function J. The design cost function is:
J(γ)＝χe(t)2+γ (17)
wherein
Figure RE-GDA0002144424350000131
χ is the balance parameter.
e(t)2The term is implicitly dependent on the parameter γ and can be used to evaluate the magnitude of the estimation error affected by the measurement noise; the gamma term is used to evaluate the effect of measurement noise; the function of the coefficient χ is to balance the front and rear parts e (t) of the cost function J2The ratio between the terms and the gamma term reflects the tolerance of the system to noise and collective interference.
As shown in fig. 4, a block diagram is optimized for the dynamic adaptive parameters of the extended state observer. By utilizing the cost function (17), the specific steps of the dynamic adaptive parameter optimization designed by the invention are as follows:
step (1): first order gammal＝α+γm，γr＝β+γm，γm＝γmRespectively representing left, right and middle three selectable values of the parameter gamma, wherein α & gt 0 and β & lt 0 are step sizes in positive and negative directions respectively;
step (2) based on the step (1), after the extended state observer obtains the input u (k) and the output y (k) for the first time, the parameters β thereof1，β2From an initial value gammam＝γ0Determining, calculating γ ═ γl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) Sum cost letterNumber Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptAnd the value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output. For example when Jl＜JrAnd Jl＜JmWhen, get γopt＝γlAnd output Zl(k) As observed values of an extended state observer;
and (3): taking gamma on the basis of the step (2)m＝γoptPreparing to perform a second round operation, namely: after the extended state observer obtains new input u (k) and output y (k), γ ═ γ is calculated respectivelyl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) And cost function Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptThe value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output;
and (4): circularly operating the step (3) in the dynamic operation process of the magnetically suspended control moment gyro frame control system;
in the dynamic operation process of the magnetic suspension control moment gyroscope frame control system, the optimal parameters and the optimal observed values are continuously approached and obtained in the circulation of the dynamic adaptive parameter optimization program, and the reaction time for obtaining the optimal parameters and the optimal observed values is determined by alpha and beta; with a reasonable choice of the step sizes α, β, a smaller response time can be obtained.
Therefore, the dynamic adaptive parameter optimization process of the extended state observer can enable the system to operate almost always under the condition of obtaining the optimal observation value, and the interference suppression capability, the anti-noise capability and the control precision of the whole system are greatly improved.
(5) Specific parameter configuration
The whole control algorithm needs 6 parameters to be configured, namely: controller parameter kq、kd、kωAnd positive and negative direction steps α, β and a balance parameter χ in the cost function, wherein,the positive and negative direction step sizes α, β and the balance parameter χ need to be selected according to actual system conditions and control effects.
Controller parameter kq、kd、kωConfigured as data shown in table 1:
TABLE 1 controller parameter configuration
Figure RE-GDA0002144424350000141
The positive and negative direction step lengths alpha and beta are respectively designed as 1 and-1; the balance parameter χ is adjusted appropriately according to the accuracy effect of the system output and can be set to 0.00000001.
In order to verify the superiority of the high-precision angular rate control method of the magnetic suspension control moment gyro frame system based on the parameter optimization extended state observer, simulation setting is carried out according to the frame servo system parameters of a magnetic suspension force control moment gyro prototype so as to compare the method with the existing method. Simulation verifies that when the angular rate instruction is 5 degrees/s, the fluctuation of the system output angular rate is only 0.0109 degrees/s, and compared with a parameter optimization method (patent number: 201810420964.X) of a double-frame magnetic suspension CMG frame system based on a reduced cascade extended state observer in Chinese patents, the precision of the system output angular rate is improved by 16.15%.

Claims (3)

1. A high-precision angular rate control method of a magnetic suspension control moment gyro frame system based on a parameter optimization extended state observer is characterized by firstly establishing a voltage and torque output equation of the magnetic suspension control moment gyro frame system, establishing a state space expression based on the equation, converting the state space expression into an integral series connection form to design the extended state observer, and eliminating lumped interference through state feedback; establishing a cost function for evaluating the influence degree of the system lumped interference and the measurement noise by analyzing the estimation error in the extended state observer so as to minimize the comprehensive influence of the system lumped interference and the measurement noise, and further designing a self-adaptive dynamic parameter optimization method;
the method comprises the following steps:
step (1): establishing a voltage and torque output equation of a magnetic suspension control moment gyroscope frame servo system, which comprises the following steps:
Figure FDA0002540895460000011
wherein u isdAnd uqD and q axis stator voltages, respectively; i.e. idAnd iqD and q axis stator currents respectively; l isdAnd LqD, q-axis inductances, L, respectivelyd＝Lq＝L；λdAnd λqRespectively are d-axis and q-axis magnetic chains of the permanent magnet synchronous motor; r is the stator resistance; ω is the mechanical angular velocity; b is the coefficient of friction; t isLIs the load torque; t iscogIs the cogging torque; t iseIs an electromagnetic torque; p is the number of pole pairs of the motor; j is the moment of inertia;
for q-axis current, it can be derived from the voltage equation above:
Figure FDA0002540895460000012
wherein
Figure FDA0002540895460000013
Is the output of the q-axis current loop;
Figure FDA0002540895460000014
"lumped disturbances" for q-axis stator currents;
Figure FDA0002540895460000015
is a known amount;
defining a state variable as X ═ X1x2]＝[iqdq(t)]Let us order
Figure FDA0002540895460000016
The state space expression for the q-axis current is then:
Figure FDA0002540895460000021
step (2): outputting a q-axis current loop according to the state space expression of the q-axis current of the framework system in the step (1)
Figure FDA0002540895460000022
As input to the extended state observer, the extended state observer is designed to:
Figure FDA0002540895460000023
wherein z is1For estimating iq；z2For estimating dq(t)；eq＝z1-iq；β1And β2Is a parameter of the extended state observer;
according to the state feedback control theory, the control rate is designed as follows:
Figure FDA0002540895460000024
wherein k isqIs a current loop controller parameter;
Figure FDA0002540895460000025
is the reference current of the q-axis input;
respectively designing the extended state observers and the control rates of the d-axis current loop and the speed loop by adopting the same method as the step (1) and the step (2), wherein the method comprises the following steps:
the d-axis current loop extended state observer is designed as follows:
Figure FDA0002540895460000026
wherein z isd1For estimating id；zd2For estimating lumped disturbance of d-axis βd1And βd2Is a parameter of an extended state observer；
Figure FDA0002540895460000027
ed＝zd1-id；
The d-axis current loop control rate is designed as follows:
Figure FDA0002540895460000031
wherein
Figure FDA0002540895460000032
Is the output of the d-axis current loop, kdAs is the parameter of the current loop controller,
Figure FDA0002540895460000033
is a d-axis reference current;
the extended state observer of the velocity loop is designed to:
Figure FDA0002540895460000034
wherein z ω 1 is used to estimate ω; z is a radical ofω2For estimating lumped disturbances of the velocity loop;
Figure FDA0002540895460000035
eω＝zω1-ω；βω1and βω2Is a parameter of the extended state observer;
the control rate of the speed loop is designed as follows:
Figure FDA0002540895460000036
wherein the content of the first and second substances,
Figure FDA0002540895460000037
is the speed loop output, kωAs a parameter of the speed loop controller, omega*A reference speed input for a speed loop;
and (3): adding measurement noise which actually exists and affects the system precision into the designed extended state observer in the step (2), so as to obtain the extended state observer containing a noise term, taking the q axis as an example, and adding a state space expression of the noise term as follows:
Figure FDA0002540895460000038
wherein h (t) is measurement noise;
the extended state observer containing the noise term can be designed as:
Figure FDA0002540895460000039
wherein the content of the first and second substances,
Figure FDA0002540895460000041
let e be Z-X, i.e. e1＝z1-x1，e2＝z2-x2(ii) a The differential of the error is then:
Figure FDA0002540895460000042
the differential equation is subjected to laplace transform to obtain:
Figure FDA0002540895460000043
wherein G ═ s + β1)；
Analyzing the above equation, if the estimation accuracy is to be improved, the estimation error needs to be driven to zero, i.e. e1→0,e2→ 0, β should be made2>>β1True, but if β2Too large will introduce more measurement noise, and although there is a conflict between the two, it is possible to choose β appropriately2、β1Size of (1), order setThe influence of both the total interference and the measurement noise reaches an optimum value within the acceptable range of the system.
2. The method for controlling the high-precision angular rate of the magnetically suspended control moment gyroscope frame system based on the parameter optimization extended state observer as claimed in claim 1, wherein the parameter optimization method obtained according to the extended state observer model and the error differential equation is as follows:
for the established extended state observer containing the noise term, to seek the optimal control, a performance index function is firstly established as follows:
Figure FDA0002540895460000044
wherein P is a symmetric semi-positive constant array, Q (t) is a symmetric semi-positive constant array, R is a symmetric positive constant array, t isfIs the terminal time;
first item in the Performance index M
Figure FDA0002540895460000045
The penalty to the terminal error is represented by a cost function of the terminal error; first term e in integral termT(t) Q (t) e (t) represents the component generated by e (t) in the system operation process, and is used for measuring the error e (t); second term U in integral termT(t) R (t) U (t) represents the component generated by U (t) in the system operation process, and is used for measuring the energy; therefore, the found optimal control U (t) is obtained when M is minimum;
solving the above function by using a variational method, firstly, taking a Hamiltonian as:
Figure FDA0002540895460000051
wherein λ isT(t)＝[λ1(t),λ2(t)]Is a lagrange multiplier vector function; and:
Figure FDA0002540895460000052
wherein the content of the first and second substances,
Figure FDA0002540895460000053
the Hamiltonian can be reduced to a known quantity by the following equation:
Figure FDA0002540895460000054
wherein the content of the first and second substances,
Figure FDA0002540895460000055
thus, the requirements for an optimal solution are the collaborative equations:
Figure FDA0002540895460000056
the control equation is:
Figure FDA0002540895460000057
obtaining by solution:
U(t)＝-R-1(t)DT(t)λ(t)
wherein R (t) is positive, so R-1(t) present;
the conditions for the cross-section were:
Figure FDA0002540895460000058
wherein phi is a first term in the performance index M; since X is 0 and e is Z in ideal output, X is 0 and Z is zero
Figure FDA0002540895460000061
The system is a linear steady system, and the terminal time tf→ infinity, therefore in t removalfAt other times than, λ (t) will also be linear with z (t), i.e.:
λ(t)＝PZ(t)
thus, combining the above conditions, the Riccati equation can be given as follows:
Figure FDA0002540895460000062
wherein the content of the first and second substances,
Figure FDA0002540895460000063
De＝[1 0](ii) a Q and R are semi-positive definite matrixes; and the equation has a unique semi-positive definite solution matrix P;
the extended state observer parameters are designed as follows:
Figure FDA0002540895460000064
selecting
Figure FDA0002540895460000065
R ═ 1; can be solved to obtain:
Figure FDA0002540895460000066
wherein gamma is greater than 0;
from this solution, the two parameters β of the extended state observer1、β2The selection of (a) can be converted into the selection of a scalar parameter gamma, and the value of the scalar parameter gamma determines the measurement noise sensitivity and the estimation performance of the system lumped interference; along with the increase of gamma, the noise sensitivity of the system is improved, but the estimation effect of interference is better;
the observation effect of the extended state observer requires a compromise between the effects of both the estimation error of the lumped disturbances and the measurement noise, the result of which is described by the cost function J; the design cost function is:
J(γ)＝χe(t)2+γ
wherein
Figure FDA0002540895460000067
χ is a balance parameter;
e(t)2the term is implicitly dependent on the parameter γ and can be used to evaluate the magnitude of the estimation error affected by the measurement noise; the gamma term is used to evaluate the effect of measurement noise; the function of the coefficient χ is to balance the front and rear parts e (t) of the cost function J2The ratio between the terms and the gamma term reflects the tolerance of the system to noise and collective interference.
3. The method for controlling the high-precision angular rate of the magnetically suspended control moment gyroscope frame system based on the parameter optimization extended state observer as claimed in claim 2, wherein the cost function designed in step (3) is used, and the specific steps of the designed dynamic adaptive parameter optimization are as follows:
step (1): first order gammal＝α+γm，γr＝β+γm，γm＝γmRespectively, left, right, and middle three selectable values of the parameter gamma, wherein α > 0, β<0 is the step length in the positive and negative directions respectively;
step (2) based on the step (1), after the extended state observer obtains the input u (k) and the output y (k) for the first time, the parameters β thereof1，β2From an initial value gammam＝γ0Determining, calculating γ ═ γl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) And cost function Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptThe value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output; for example when Jl<JrAnd Jl<JmWhen, get γopt＝γlAnd output Zl(k) As observed values of an extended state observer;
and (3): taking gamma on the basis of the step (2)m＝γoptPreparing to perform a second round operation, namely: after the extended state observer obtains new input u (k) and output y (k), γ ═ γ is calculated respectivelyl，γ＝γrAnd γ ═ γmObserved value Z of under-action extended state observerl(k)，Zr(k)，Zm(k) And cost function Jl，Jr，Jm(ii) a Optimizing the parameter gammaoptThe value is taken as a gamma value which enables the cost function to obtain the minimum value, and the observed value of the extended state observer under the action of the corresponding parameter is output;
and (4): and (3) circularly operating in the dynamic operation process of the magnetically suspended control moment gyro frame control system.
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