CN108536185B - Double-framework magnetic suspension CMG framework system parameter optimization method based on reduced-order cascade extended state observer - Google Patents

Double-framework magnetic suspension CMG framework system parameter optimization method based on reduced-order cascade extended state observer Download PDF

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CN108536185B
CN108536185B CN201810434916.6A CN201810434916A CN108536185B CN 108536185 B CN108536185 B CN 108536185B CN 201810434916 A CN201810434916 A CN 201810434916A CN 108536185 B CN108536185 B CN 108536185B
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CN108536185A (en
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李海涛
侯林
宋鹏
韩邦成
郑世强
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Beihang University
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    • 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

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Abstract

The invention provides a double-frame magnetic suspension CMG frame system parameter optimization method based on a reduced cascade extended state observer. Firstly, a double-frame servo system dynamic model is established, the dynamic model is converted into a system state equation which meets the requirement of establishing a reduced order cascade extended state observer (RCESO) integral chain form through coordinate transformation, the system state equation is substituted into the RCESO state model to obtain a transfer function from system interference to interference estimation, the relation of RCESO parameter co-frequency domain characteristics is analyzed based on the transfer function, and then a parameter configuration method which meets system indexes is designed. The method simplifies the traditional Cascade Extended State Observer (CESO) mathematical model and improves the interference estimation performance, and is suitable for parameter configuration of the cascade extended state observer.

Description

Double-framework magnetic suspension CMG framework system parameter optimization method based on reduced-order cascade extended state observer
Technical Field
The invention belongs to the field of observer-based double-frame magnetic suspension control CMG frame system parameter optimization, and particularly relates to a double-frame magnetic suspension CMG frame system parameter optimization method based on a reduced-order cascade extended state observer, which is used for improving the interference suppression capability of a frame system, realizing high-precision angular rate tracking control of a frame servo system and further realizing high-precision torque output of a control torque gyroscope.
Background
The Control Moment Gyroscope (CMG) has the advantages of large output moment, good dynamic performance and high control precision, and is a preferred actuator for attitude control of large-scale spacecrafts. There are a single frame CMG and a double frame CMG according to the degrees of freedom of the frames. The supporting mode of the high-speed rotor can be divided into a mechanical bearing CMG and a magnetic suspension CMG. The double-frame magnetic suspension CMG has the advantages of large torque output, high precision and two degrees of freedom in output torque, and is very attractive to space station and satellite attitude control systems. The double-frame magnetic suspension CMG mainly comprises a magnetic suspension high-speed rotor system and an inner frame system and an outer frame system, and the working principle is as follows: according to the gyro effect, the rotation of the inner and outer frames forcibly changes the angular momentum direction of the rotor so as to output gyro moment. The angular speed precision of the frame system determines the precision of the output torque of the double-frame magnetic suspension CMG, so that the improvement of the angular speed precision of the frame servo control system has important significance.
Due to the strong gyro effect, an obvious coupling moment can be generated between the inner frame and the outer frame. The coupling torque is non-linear and related to the angular position and velocity of the outer frame, which is one of the main factors affecting the angular velocity of the frame system. In addition, the frame servo system is a very low speed mechanical servo system, and since the gyro coupling moment and the friction moment are non-linear, it is very difficult to construct an accurate system model, the friction moment is another main factor influencing the frame servo performance. In order to realize high-precision speed control of the frame system, the influence of unknown unmatched interference such as coupling torque and nonlinear friction torque on the servo performance of the frame system must be overcome.
In order to solve the problem that the angular rate precision of the frame system is reduced due to the unmatched interference, the simplified feedback linearization control method based on the mode separation method can decouple the double-frame magnetic suspension CMG system, but an accurate mathematical model is needed; the differential geometric decoupling method can decouple the double-frame magnetic suspension CMG system, but cannot completely eliminate the influence of coupling torque; an Extended State Observer (ESO) is used as an effective interference estimation technology to expand lumped interference into a new state of a system, however, if the order of a system state equation is greater than 2, it is difficult to configure ESO parameters meeting the system precision requirement in practical application; cascaded ESO (CESO) simplifies the parameter adjustment of the ESO into the adjustment of two parameters, but the two parameters of the CESO are configured according to a second-order ESO parameter configuration method, and a method for configuring the parameters of the CESO is not provided.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method overcomes the defects of the existing method, and provides the parameter optimization method of the double-frame magnetic suspension CMG frame system based on the reduced cascade extended state observer, which not only improves the angular rate tracking performance of the frame servo system, but also enhances the robustness of the frame system to mismatched interference.
The technical scheme adopted by the invention for solving the technical problems is as follows: a double-frame magnetic suspension CMG frame system parameter optimization method based on a reduced cascade extended state observer comprises the following steps:
step (1): establishing a state space equation according to a dynamic model of a double-frame servo system
The kinetic and torque balance equations for the inner and outer frame systems can be written as:
Figure BDA0001654372900000021
wherein, thetaxAnd thetayRespectively, the angular position of the inner and outer frames, TxAnd TyOutput torque of the inner and outer frames, KxAnd KyAre respectively the torque coefficient, JxAnd JyIs the equivalent moment of inertia of the inner and outer frames, FxAnd FyIs unmodeled dynamics of the inner and outer frames, fxAnd fyNon-linear friction torque, TxAnd TyIs the output torque of the frame motor, HzIs the angular momentum of the high-speed rotor, uxAnd uyIs the control voltage of the frame motor, IxAnd IyIs the current of a torque motor, RxAnd RyIs the stator resistance of a torque motor, LxAnd LyIs an inductance, CexAnd CeyIs the back emf coefficient.
For the inner frame, the state variables are defined as
Figure BDA0001654372900000022
Control input is ux, coupling moment
Figure BDA0001654372900000023
Unmodeled dynamics FxAnd non-linear friction fxAs the main disturbance of the framework system, it is considered as "lumped disturbance". The state space equation of the inner frame system is expressed as:
Figure BDA0001654372900000024
wherein the content of the first and second substances,
Figure BDA0001654372900000025
g＝[0 0 1/Lx]T,p＝[0 1 0]T,
Figure BDA0001654372900000026
control input uxAnd disturbance d1The total interference is estimated by introducing CESO and the influence of the total interference is compensated in a controller without being in the same channel, but the state space equation does not conform to a CESO integral chain form, so that coordinate transformation is introduced and converted into an integral chain form;
step (2): according to the state space equation in the form of the integral chain of the inner frame system in the step (1), the angular velocity is used as the reference input of the CESO, so that the traditional CESO cascaded by three similar second-order Extended State Observers (ESOs) can be converted into RCESO cascaded by two similar second-order ESOs, and the model of the traditional CESO is simplified. RCESO estimates the lumped interference, and compensates the influence of the lumped interference by combining with a state feedback controller, and finally eliminates the unmatched interference.
And (3): and (3) combining the system state equation and the RCESO state equation in the form of the integral chain in the steps (1) and (2) to obtain a transfer function from the lumped interference to the estimated value of the lumped interference. And the RCESO parameter configuration method meeting the system index requirements is obtained by deducing the mathematical relationship between the RCESO parameters and the frequency characteristics of the transfer function.
Further, the parameter configuration method obtained according to the system state equation in the form of the integral chain and the RCESO model is as follows:
the inner frame state space equation in the form of an integral chain is:
Figure BDA0001654372900000031
y＝[0 1 0]v
wherein v is1＝x1,v2＝x2,
Figure BDA0001654372900000032
The RCESO model is constructed as follows:
Figure BDA0001654372900000033
wherein the content of the first and second substances,
Figure BDA0001654372900000034
is a state variable of the RCESO,
Figure BDA0001654372900000035
for estimating v2，
Figure BDA0001654372900000036
For estimating v3，
Figure BDA0001654372900000037
For estimating f.
Figure BDA0001654372900000038
To estimate the error, define as
Figure BDA0001654372900000039
Figure BDA00016543729000000310
And
Figure BDA00016543729000000311
is a parameter of RCESO. In a framework system, the disturbance f is bounded with an upper frequency of ω0And the derivative is bounded, the steady state error of RCESO can be adjusted
Figure BDA00016543729000000312
And
Figure BDA00016543729000000313
but is limited to very small values.
The RCESO state is setObtained by Cheng La transformation
Figure BDA00016543729000000314
And v2、uxThe relation of (A) is as follows:
Figure BDA00016543729000000315
combining the state equation of the RCESO and the state space equation of the framework system, the above equation can be:
Figure BDA00016543729000000316
thus from f to
Figure BDA00016543729000000317
The transfer function can be described as:
Figure BDA0001654372900000041
the amplitude-frequency characteristic and the phase-frequency characteristic of the interference transfer function are as follows:
Figure BDA0001654372900000042
in order to make RCESO at a given frequency bandwidth ω ∈ [0, ω0]The method has good observation capability and provides the following frequency domain performance indexes:
Figure BDA0001654372900000043
wherein A is0And
Figure BDA0001654372900000044
in the system bandwidth range ω ∈ [0, ω respectively0]Maximum amplitude-frequency error and maximum phase-frequency error.
In order to meet the requirements of amplitude-frequency error and phase-frequency error and combine the stable conditions of RCESO
Figure BDA0001654372900000045
It can be found that A (ω) is at ω ∈ [0, ω [ ]0]Inner monotonic decrease, increasing lag phase angle with increasing ω, amplitude frequency error and phase frequency error simultaneously at ω ═ ω0The time reaches the maximum value in the frequency domain index. Once A (ω)0) And
Figure BDA00016543729000000411
meet the performance index of frequency domain, A (omega) and
Figure BDA0001654372900000046
at ω ∈ (0, ω)0) And is certainly satisfied.
The following equation is established:
Figure BDA0001654372900000047
Figure BDA0001654372900000048
by taking the intersection of the two formulas, the RCESO parameter meeting the frequency domain performance index can be obtained
Figure BDA0001654372900000049
And
Figure BDA00016543729000000410
to the final range of (c). The RCESO subjected to parameter optimization is combined with a state feedback controller to compensate the influence of lumped interference, and finally mismatching interference is eliminated.
The basic principle of the invention is as follows: according to the invention, an integral chain type state equation is established according to a frame servo control system dynamic model, a reduced-order cascading extended state observer model is established, and the traditional cascading extended state observer model is simplified. The system state equation in the form of an integral chain is combined with the RCESO model to obtain a transfer function from lumped interference to an estimated value of the lumped interference. And the RCESO parameter configuration method meeting the system index requirements is obtained by deducing the mathematical relationship between the RCESO parameters and the frequency characteristics of the transfer function. The RCESO optimized by the parameters can estimate the lumped interference more accurately, and the influence of the state feedback controller on the lumped interference is compensated, so that the system disturbance is suppressed, and the high-precision frame angular rate output is realized.
Compared with the prior art, the invention has the advantages that:
1. three second-order ESOs with the same parameter are cascaded and reduced into two second-order ESOs with the same parameter, so that a CESO model is simplified. The problems of high order and complex mathematical model of the traditional CESO caused by high order of the system can be well solved.
2. By two parameters for RCESO
Figure BDA0001654372900000051
And
Figure BDA0001654372900000052
the optimization design is carried out, the problem of poor interference estimation performance caused by the fact that the traditional CESO parameters are configured according to a second-order ESO parameter configuration method is solved, and the interference estimation performance of RCESO is improved. And by compensating the interference estimation item in the composite controller, the overall anti-interference capability of the system is improved, and the accuracy of the output angular rate of the system is improved.
Drawings
FIG. 1 is a flow chart of a control algorithm for a frame angular rate servo system;
FIG. 2 is a schematic diagram of a conventional CESO;
FIG. 3 is a diagram of a conventional CESO structure;
FIG. 4 is a schematic representation of RCESO according to the present invention;
FIG. 5 is a diagram of the RCESO structure of the present invention;
FIG. 6 is a block diagram of the overall control algorithm of the RCESO-based frame angular rate servo system.
Detailed Description
The invention is further described with reference to the following figures and detailed description.
As shown in fig. 1, a parameter optimization method for a double-frame magnetic suspension CMG frame system based on a reduced-order cascade extended state observer includes the steps of firstly, performing dynamic modeling on a frame servo control system, and establishing a state equation in an integral chain form according to the dynamic model; according to the system state equation and the RCESO state equation in the form of integral chain, the system state equation and the RCESO state equation are combined to obtain a transfer function from lumped interference to an estimated value of the lumped interference. And the RCESO parameter configuration method meeting the system index requirements is obtained by deducing the mathematical relationship between the RCESO parameters and the frequency characteristics of the transfer function. RCESO estimates the lumped interference and compensates the influence of the lumped interference by combining with a state feedback controller, so that the system disturbance is restrained and the high-precision frame angular rate output is realized.
The specific embodiment of the invention is as follows:
(1) establishing an integral chain type state space equation according to a double-frame servo system dynamic model
The kinetic and torque balance equations for the inner and outer frame systems can be written as:
Figure BDA0001654372900000061
wherein, thetaxAnd thetayRespectively, the angular position of the inner and outer frames, TxAnd TyOutput torque of the inner and outer frames, KxAnd KyAre respectively the torque coefficient, JxAnd JyIs the equivalent moment of inertia of the inner and outer frames, FxAnd FyIs unmodeled dynamics of the inner and outer frames, fxAnd fyNon-linear friction torque, TxAnd TyIs the output torque of the frame motor, HzIs the angular momentum of the high-speed rotor, uxAnd uyIs the control voltage of the frame motor, IxAnd IyIs the current of a torque motor, RxAnd RyIs the stator resistance of a torque motor, LxAnd LyIs an inductance, CexAnd CeyIs the back emf coefficient.
For the inner frame, the state variables are defined as
Figure BDA0001654372900000062
Control input is uxMoment of coupling
Figure BDA0001654372900000063
Unmodeled dynamics FxAnd non-linear friction fxAs the main disturbance of the framework system, it is considered as "lumped disturbance". The state space equation of the inner frame system is expressed as:
Figure BDA0001654372900000064
wherein the content of the first and second substances,
Figure BDA0001654372900000065
g＝[0 0 1/Lx]T,p＝[0 1 0]T,
Figure BDA0001654372900000066
as can be seen from (2), control input uxAnd disturbance d1Not in the same channel, resulting in a mismatch problem. The problem of mismatch can be solved by introducing CESO to estimate the total interference and compensate its effect in the controller. However, (2) does not conform to the traditional CESO integral chain form, and therefore requires the introduction of coordinate transformations. The new coordinate is defined as v ═ v1,v2,v3]ΤWherein v is1＝x1,v2＝x2,
Figure BDA0001654372900000067
The inner frame state space equation in the form of an integral chain is:
Figure BDA0001654372900000068
wherein the content of the first and second substances,
Figure BDA0001654372900000069
(2) construction of RCESO model
For a third order system, if we look at the angular position v1As a reference input for CESO, then linear CESO would be cascaded by three similar second-order ESOs with the same parameters, the structure of CESO being shown in fig. 2. The state variable of CESO is defined as z ═ z1,z2,z3,z4,z5,z6]ΤWherein z is1For estimating v1，z2For estimating v2，z4For estimating v3，z6For estimating f, and z3And z5Is an intermediate variable. The estimation error is defined as e1＝z1-v1，e2＝z2-v2，e3＝z3-z2，e4＝z4-v3，e5＝z5-z4，e6＝z6-f。β1And β2Are two parameters of CESO. As shown in fig. 3, the equation of state for CESO is described as follows:
Figure BDA0001654372900000071
since the control target of the frame system is to realize accurate tracking control of the angular velocity, the angular velocity v can be adjusted2As a reference input in CESO design. This shows that we can construct a reduced order CESO, only needing to estimate v2，v3And f. Thus, a conventional CESO can be converted to an RCESO cascaded by two similar second-order ESOs, which simplifies the model of conventional CESO. The structure of RCESO is shown in fig. 4.
We define the state variables of RCESO as
Figure BDA0001654372900000072
Wherein
Figure BDA0001654372900000073
For estimating v2，
Figure BDA0001654372900000074
For estimating v3，
Figure BDA0001654372900000075
For estimating f, and
Figure BDA0001654372900000076
is an intermediate variable. The estimation error is defined as
Figure BDA0001654372900000077
Figure BDA0001654372900000078
As shown in fig. 5, the state equation for RCESO is described as follows:
Figure BDA0001654372900000079
wherein the content of the first and second substances,
Figure BDA00016543729000000710
and
Figure BDA00016543729000000711
is a parameter of RCESO. Subtracting (3) from (5), the error equation for RCESO is as follows:
Figure BDA00016543729000000712
the steady state error of RCESO is:
Figure BDA00016543729000000713
in a framework system, the disturbance f is bounded with an upper frequency of ω0And its derivative is bounded. Therefore, the steady state error of RCESO can be adjusted
Figure BDA00016543729000000714
And
Figure BDA00016543729000000715
but is limited to very small values.
(3) Obtaining an interference transfer function
And (3) converting the RCESO state equation in Laplace to obtain:
Figure BDA0001654372900000081
1)
Figure BDA0001654372900000082
and v2、uxThe relation of (A) is as follows:
Figure BDA0001654372900000083
2)
Figure BDA0001654372900000084
and v2、uxThe relation of (A) is as follows:
Figure BDA0001654372900000085
combining the state equation of the RCESO and the state space equation of the framework system, the above equation can be:
Figure BDA0001654372900000086
thus from f to
Figure BDA0001654372900000087
The transfer function can be described as:
Figure BDA0001654372900000088
(4) RCESO parameter configuration method design
The amplitude-frequency characteristic and the phase-frequency characteristic of the interference transfer function are as follows:
Figure BDA0001654372900000089
in order to make RCESO at a given frequency bandwidth ω ∈ [0, ω0]The method has good observation capability and provides the following frequency domain performance indexes:
Figure BDA0001654372900000091
wherein A is0And
Figure BDA0001654372900000092
in the system bandwidth range ω ∈ [0, ω respectively0]Maximum amplitude-frequency error and maximum phase-frequency error.
In order to satisfy the requirements of amplitude frequency error and phase frequency error given by formula (14) and combine the stable condition of RCESO
Figure BDA0001654372900000093
It can be found that A (ω) is at ω ∈ [0, ω [ ]0]Inner monotone decreasing. Further, as can be seen from (13), the hysteresis phase angle increases as ω increases. Therefore, the amplitude frequency error and the phase frequency error are simultaneously set to ω0The time reaches a maximum value in the frequency domain index (14). Once A (ω)0) And
Figure BDA0001654372900000094
meet the frequency domain performance index (14), A (omega) and
Figure BDA0001654372900000095
at ω ∈ (0, ω)0) And is certainly satisfied.
The following equation is established:
Figure BDA0001654372900000096
Figure BDA0001654372900000097
by taking (1)5) The intersection of (16) and (16) can obtain the RCESO parameter satisfying the frequency domain performance index (14)
Figure BDA0001654372900000098
And
Figure BDA0001654372900000099
to the final range of (c). In actual engineering the phase angle error will not exceed 90 deg., which means that the phase angle error is in the first quadrant. This can be expressed as
Figure BDA00016543729000000910
Stability conditions combining (15), (16) and RCESO
Figure BDA00016543729000000911
The results are shown below:
① when
Figure BDA00016543729000000912
When the temperature of the water is higher than the set temperature,
Figure BDA00016543729000000913
wherein the content of the first and second substances,
Figure BDA00016543729000000914
② when b1When the ratio is less than 1, the reaction solution is,
Figure BDA0001654372900000101
in the above two equations, the stable condition of the reduced-order cascaded extended state observer
Figure BDA0001654372900000102
Can be expressed as
Figure BDA0001654372900000103
Wherein
Figure BDA0001654372900000104
Thus, for any ω0＞0，0＜A0＜1，
Figure BDA0001654372900000105
The estimate of the total interference f by the RCESO satisfies the frequency domain performance indicator (14) if the parameter of the RCESO satisfies any one of the conditions (17) and (18).
The control law is as follows:
Figure BDA0001654372900000106
wherein r isvFor the angular rate command, rpFor reference rotor position information, v, obtained after integration of angular rate commands1Is the angular position of the frame, v2In order to be the angular rate of the frame,
Figure BDA0001654372900000107
an additional control quantity generated for the cascaded extended state observer; k is a radical of1、k2、k3For 3 design parameters of the controller, bxAs a framework system parameter, uxControlling the output for the controller.
The RCESO subjected to parameter optimization estimates the lumped interference, and the influence of the upper-type state feedback controller on the lumped interference is compensated, so that system disturbance is suppressed, and high-precision frame angular rate output is realized. The relationship diagram of the overall control algorithm of the frame angular rate servo system is shown in FIG. 6.
(5) Specific parameter configuration
The whole control algorithm needs 5 parameters to be configured, namely: controller parameter k1、k2、k3And two parameters of a cascaded extended state observer
Figure BDA0001654372900000108
Wherein the controller parameter k1、k2、k3According to the pole configuration mode, two parameters of RCESO are configured
Figure BDA0001654372900000109
Configured as described above. By adopting the control algorithm, the speed servo precision of the frame system can be improved.
Taking the angular rate servo system of the magnetic suspension control moment gyro frame with the angular momentum of 20Nms as an example, the angular rate bandwidth is 5 Hz. The system parameters are shown in table 1.
TABLE 1 System parameters
Figure BDA0001654372900000111
Controller parameter k1、k2、k3The pole arrangement mode is as follows:
k1＝1.1×106，k2＝1.2×105，k3＝1.1×103。
the frequency domain performance index of the frame system is selected as A0＝0.1,
Figure BDA0001654372900000112
Then according to the parameter configuration method herein, since
Figure BDA0001654372900000113
Therefore (18) is a sufficient condition for satisfying the frequency domain index. Two parameters of a cascaded extended state observer
Figure BDA0001654372900000114
Is configured to:
Figure BDA0001654372900000115
so take a set of parameters as
Figure BDA0001654372900000116
Simulation verifies that when the angular rate instruction is 5 DEG/s, the fluctuation of the system output angular rate is only 0.013 DEG/s, and compared with a high-precision friction compensation control method (patent number: ZL201510561163.1) of a double-frame magnetic suspension CMG frame system, the system output angular rate precision is improved by 9.1%.

Claims (1)

1. A double-frame magnetic suspension CMG frame system parameter optimization method based on a reduced cascade extended state observer is characterized by comprising the following steps:
step (1): establishing a state space equation according to a dynamic model of a double-frame servo system
The kinetic and torque balance equations for the inner and outer frame systems can be written as:
Figure FDA0002524082660000011
wherein, thetaxAnd thetayRespectively, the angular position of the inner and outer frames, TxAnd TyOutput torque of the inner and outer frames, KxAnd KyAre respectively the torque coefficient, JxAnd JyIs the equivalent moment of inertia of the inner and outer frames, FxAnd FyIs unmodeled dynamics of the inner and outer frames, fxAnd fyNon-linear friction torque, TxAnd TyIs the output torque of the frame motor, HzIs the angular momentum of the high-speed rotor, uxAnd uyIs the control voltage of the frame motor, IxAnd IyIs the current of a torque motor, RxAnd RyIs the stator resistance of a torque motor, LxAnd LyIs an inductance, CexAnd CeyIs the back electromotive force coefficient;
for the inner frame, the state variables are defined as
Figure FDA0002524082660000012
Control input is uxMoment of coupling
Figure FDA0002524082660000013
Unmodeled dynamics FxAnd is notLinear friction fxAs the main disturbance of the framework system, it is considered as "lumped disturbance", and the state space equation of the inner framework system is expressed as:
Figure FDA0002524082660000014
wherein the content of the first and second substances,
Figure FDA0002524082660000015
g＝[0 0 1/Lx]T，p＝[0 1 0]T，
Figure FDA0002524082660000016
control input uxAnd disturbance d1The total interference is estimated by introducing CESO and the influence of the total interference is compensated in a controller without being in the same channel, but the state space equation does not conform to a CESO integral chain form, so that coordinate transformation is introduced and converted into an integral chain form;
step (2): according to the state space equation in the form of the integral chain of the inner frame system in the step (1), by taking the angular velocity as the reference input of the CESO, the traditional CESO cascaded by three similar second-order Extended State Observers (ESOs) can be converted into RCESO cascaded by two similar second-order ESOs, so that the model of the traditional CESO is simplified;
and (3): combining the system state equation in the form of the integral chain and the RCESO state equation in the steps (1) and (2) to obtain a transfer function from lumped interference to an estimated value of the lumped interference; obtaining an RCESO parameter configuration method meeting the system index requirements by deducing the mathematical relationship between the RCESO parameters and the frequency characteristics of the transfer function; RCESO estimates the lumped interference, and compensates the influence of the lumped interference by combining with a state feedback controller, and finally eliminates the unmatched interference;
the parameter configuration method is obtained according to the system state equation in the form of the integral chain and the RCESO model as follows:
the inner frame state space equation in the form of an integral chain is:
Figure FDA0002524082660000021
y＝[0 1 0]v
wherein v is1＝x1,v2＝x2,
Figure FDA0002524082660000022
The RCESO model is constructed as follows:
Figure FDA0002524082660000023
wherein the content of the first and second substances,
Figure FDA0002524082660000024
is a state variable of the RCESO,
Figure FDA0002524082660000025
for estimating v2，
Figure FDA0002524082660000026
For estimating v3，
Figure FDA0002524082660000027
For the purpose of estimating the value of f,
Figure FDA0002524082660000028
to estimate the error, define as
Figure FDA0002524082660000029
Figure FDA00025240826600000210
And
Figure FDA00025240826600000211
is a parameter of RCESO; in frame systems, the noiseThe motion f is bounded and has an upper limit frequency of ω0And the derivative is bounded, the steady state error of RCESO can be adjusted
Figure FDA00025240826600000212
And
Figure FDA00025240826600000213
but is limited to very small values;
the RCESO state equation is transformed by Ralstonian transformation to obtain
Figure FDA00025240826600000214
And v2、uxThe relation of (A) is as follows:
Figure FDA00025240826600000215
combining the state equation of the RCESO and the state space equation of the framework system, the above equation can be:
Figure FDA00025240826600000216
thus from f to
Figure FDA00025240826600000217
The transfer function can be described as:
Figure FDA00025240826600000218
the amplitude-frequency characteristic and the phase-frequency characteristic of the interference transfer function are as follows:
Figure FDA0002524082660000031
in order to make RCESO at a given frequency bandwidth ω ∈ [0, ω0]The method has good observation capability and provides the following frequency domain performance indexes:
Figure FDA0002524082660000032
wherein A is0And
Figure FDA0002524082660000033
in the system bandwidth range ω ∈ [0, ω respectively0]Maximum amplitude-frequency error and maximum phase-frequency error;
in order to meet the requirements of amplitude-frequency error and phase-frequency error and combine the stable conditions of RCESO
Figure FDA0002524082660000034
It can be found that A (ω) is at ω ∈ [0, ω [ ]0]Inner monotonic decrease, increasing lag phase angle with increasing ω, amplitude frequency error and phase frequency error simultaneously at ω ═ ω0The time reaches the maximum value in the frequency domain index; once A (ω)0) And
Figure FDA0002524082660000035
meet the performance index of frequency domain, A (omega) and
Figure FDA0002524082660000036
at ω ∈ (0, ω)0) Is certainly satisfied;
the following equation is established:
Figure FDA0002524082660000037
Figure FDA0002524082660000038
by taking the intersection of the two formulas, the RCESO parameter meeting the frequency domain performance index can be obtained
Figure FDA0002524082660000039
And
Figure FDA00025240826600000310
and in the final range, the RCESO subjected to parameter optimization compensates the influence of the lumped interference by combining with the state feedback controller, and finally the unmatched interference is eliminated.
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