CN112859613B - High-precision control method of control moment gyro frame system based on harmonic reducer - Google Patents

High-precision control method of control moment gyro frame system based on harmonic reducer Download PDF

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CN112859613B
CN112859613B CN202110085465.1A CN202110085465A CN112859613B CN 112859613 B CN112859613 B CN 112859613B CN 202110085465 A CN202110085465 A CN 202110085465A CN 112859613 B CN112859613 B CN 112859613B
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harmonic reducer
frame system
control
torque motor
frame
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CN112859613A (en
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李海涛
王晓宇
陈祥文
史阳阳
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Beihang University
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    • 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
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Abstract

The invention discloses a high-precision control method for a control moment gyro frame system based on a harmonic reducer. The frame system is modeled from an energy angle by adopting a Lagrange function and a Rayleigh dissipation function, the system is observed by the extended state observer through acquiring control input information of the system and load angular position information of the frame system, an observed estimated value of total disturbance is input into the sliding mode controller, and the sliding mode controller inhibits and compensates the disturbance in the system in real time through control input. Compared with the traditional modeling method, the modeling method provided by the invention has no problem of unmatched interference, and the high-precision frame angular rate servo control can be realized by adopting the composite controller.

Description

High-precision control method of control moment gyro frame system based on harmonic reducer
Technical Field
The invention belongs to the field of control of a control moment gyro frame system, and particularly relates to a high-precision control method of the control moment gyro frame system based on a harmonic reducer, in particular to a single-frame control moment gyro.
Background
The control moment gyroscope is an important spacecraft attitude control actuating mechanism, has the outstanding advantages of large output moment, high precision, no consumption of working media and the like, and is an ideal choice for attitude control actuating mechanisms of agile maneuvering satellites, space stations and large satellite platforms. The single-frame control moment gyro mainly comprises a high-speed rotor system and a frame system, wherein when the high-speed rotor system rotates at a constant speed, the frame forcibly changes the direction of a rotor rotating shaft through rotation to output gyro moment. The accuracy of the gyro moment under a certain angular momentum of the high-speed rotor system depends on the accuracy of the angular rate of the frame system. Therefore, the fact that the high-precision frame angular rate is guaranteed is the key premise that the single-frame control moment gyroscope realizes accurate spacecraft attitude control.
The volume and the weight of the single-frame control moment gyro are limited, so that the volume and the weight of the frame motor are also limited, and therefore when the frame motor is difficult to meet the output requirement of large moment, a harmonic reducer is arranged on an output shaft of the frame motor and serves as a moment amplifying device and a transmission device. The harmonic reducer has nonlinear transmission problems of nonlinear friction, hysteresis, motion error and the like in the transmission process, the problems can reduce the precision of the frame angular rate, wherein the motion error has a particularly prominent influence on the frame angular rate, and the existence of the harmonic reducer can cause the frame angular rate to fluctuate.
In order to solve the problem that the motion error in the transmission process of the harmonic reducer affects the output angular rate servo precision, at present, the motion error of the harmonic reducer is mainly researched from two directions of a mechanical design principle and modeling. Fitting and parameter calibration are two key steps for establishing an accurate model in motion error modeling research. The fitting method is that the motion error components generated by each component of the harmonic reducer which can generate elastic deformation in the transmission process are expressed by different symbols through analysis, then a preliminary model of the motion error is established through analyzing the quantity relation among the components, a polynomial is used for fitting a model curve of the motion error, and finally parameter calibration is carried out through a large amount of experiments and data to obtain a final motion error model. The modeling mode causes the modeling process to be complicated and the parameter configuration to be complex. Gandhi P.S., Ghorbel F.H., closed-loop compensation of kinetic in harmonic drives for precision Control applications [ J ]. IEEE Transactions on Control Systems Technology,2002.10(6):759-768.
The extended state observer 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 the inaccuracy of harmonic reducer parameters and models can be effectively avoided, however, if the order of a state equation is greater than 2, it is difficult to configure extended state observer parameters meeting the requirement of system precision in practical application. The sliding mode controller has a particularly outstanding capability in the aspects of suppressing parameter disturbance and external disturbance as a passive disturbance suppression method, and can greatly enhance the robustness of a controlled system, but the sliding mode controller also has an obvious problem that the discontinuous switching of the controller can cause high-frequency buffeting of the controlled system.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: the method solves the problem that the output angular rate precision of the existing control moment gyro frame servo system based on a harmonic reducer is not high, and provides a modeling method and a control algorithm for improving the output angular rate precision of a frame system.
The basic principle of the invention is as follows: the method comprises the steps of constructing a real-time relation between a load end angular rate of a frame system and a motor end angular rate by analyzing and deducing harmonic speed reducer motion errors, and then establishing a frame system model based on a Lagrangian function and a Rayleigh dissipation function; the disturbance in the frame system is estimated by adopting the extended state observer with low requirement on the accuracy of the model, and then the estimated disturbance is compensated to the control input of the frame system through the sliding mode controller, so that the disturbance is restrained and compensated, and the constructed system is favorable for reducing the influence on the angular rate output accuracy of the frame system caused by the motion error of the harmonic reducer.
The technical scheme adopted by the invention for solving the problems is as follows: a high-precision control method for a control moment gyro frame system based on a harmonic reducer comprises the following steps: the system comprises a sliding mode controller, a power amplifier, a torque motor, an encoder, a harmonic reducer, a wave generator of the harmonic reducer, a flexible gear of the harmonic reducer, a frame system load, a rotary transformer and an extended state observer, wherein the sliding mode controller comprises the following components: wherein, the sliding mode controller restrains and compensates disturbance in the system in real time through control input to generate a control signal, actual control current is output through a power amplifier to drive the torque motor, an encoder fixedly connected on a torque motor shaft is used for measuring the angular position of the torque motor, the output end of the torque motor is rigidly connected with the input end of a harmonic reducer and has the same rotating speed, a wave generator of the harmonic reducer is in transmission connection with a flexible gear of the harmonic reducer, reverse transmission with a reduction ratio of N is arranged between the wave generator of the harmonic reducer and the flexible gear of the harmonic reducer, the flexible gear of the harmonic reducer is also rigidly connected with a frame system load and has the same rotating speed, a rotary transformer fixedly connected with the frame system load is used for obtaining the angular position of the frame system load, and the angular position of the frame system load and the control output of the sliding mode controller are used as input information of an expansion state observer together, the extended state observer generates an estimator, and the estimator and an angular rate command of a control moment gyro frame angular rate servo system, a torque motor angular position and related variables, a frame system load angular position and related variables given by an attitude control computer are used as input information of a sliding mode controller, so that the method for realizing high-precision control is characterized by comprising the following steps of:
step (1): establishing a state space equation based on a Lagrangian function and a Rayleigh dissipation function from the energy perspective;
step (2): designing an extended state observer;
and (3): designing a sliding mode controller;
and (4): designing parameters of a composite control algorithm;
further, the step (1) specifically includes:
the motion error generated by the harmonic reducer (5) is as follows:
Figure BDA0002910637670000031
wherein the content of the first and second substances,
Figure BDA0002910637670000032
is the motion error, thetapIs due to manufacture and installationPure motion error, theta, caused by mounting error, backlash, or the liketIs the torsion angle theta caused by the elastic deformation of a flexible gear (5.2) of the harmonic reducer under the condition of loadmIs the angular position of the torque motor, thetalIs the frame system load angular position, N is the reduction ratio of the harmonic reducer;
the above equation is derived over time t:
Figure BDA0002910637670000033
wherein the content of the first and second substances,
Figure BDA0002910637670000034
instantaneous speed ratio which is the input and output angular rate of the harmonic reducer;
the kinetic and dissipation energies of the frame system are respectively:
Figure BDA0002910637670000035
wherein, JmAnd JlMoment of inertia of the load of the torque motor and frame system, respectively, BmAnd BlThe damping coefficients of the load of the torque motor and the load of the frame system are respectively;
the lagrangian equation of motion for the framework system is:
Figure BDA0002910637670000036
wherein, L is T-V, V is the elastic potential energy of the flexible gear in the transmission process of the harmonic reducer; τ is a control input to the frame system;
substituting the formulas (2) and (3) into the formula (4) to obtain the frame system load rotating speed
Figure BDA0002910637670000038
System equation for basic state variables:
Figure BDA0002910637670000037
wherein the content of the first and second substances,
Figure BDA0002910637670000041
Tkthe torque is required by the torque motor to drive the harmonic reducer;
defining a state variable as
Figure BDA0002910637670000042
The control output of the composite controller is u-tau, namely the control output of the frame system is y-x based on the control input of the frame system of the harmonic reducer2B is the control input gain, a is the indirect variable, f (x) is the main interference of the frame system, which is regarded as the 'lumped disturbance', and the state space equation of the frame system can be obtained as follows:
Figure BDA0002910637670000043
wherein the content of the first and second substances,
Figure BDA0002910637670000044
in the actual situation where the device is,
Figure BDA0002910637670000045
thus can be used
Figure BDA0002910637670000046
Instead of Y, the b term can be rewritten as
Figure BDA0002910637670000047
So that the b term becomes a constant term;
further, the step (2) specifically includes:
defining expanded state variables as x3＝f(x1,x2)，
Figure BDA0002910637670000048
The expansion state space equation of the framework system is:
Figure BDA0002910637670000049
defining a state variable of the extended state observer as z ═ z1,z2,z3]TWherein z is1Is thetalEstimate of z2Is that
Figure BDA00029106376700000416
Estimate of z3Is an estimate of h;
the state equation of the extended state observer of the framework system is:
Figure BDA00029106376700000410
wherein, beta1、β2、β3Three design parameters of the extended state observer;
further, the step (3) specifically includes:
designing a sliding mode surface function s as follows:
Figure BDA00029106376700000411
wherein the content of the first and second substances,
Figure BDA00029106376700000412
Figure BDA00029106376700000413
and
Figure BDA00029106376700000414
are each thetalExpected value of and
Figure BDA00029106376700000415
c is to be setCalculating a sliding mode surface function parameter;
the following control law is adopted in the controller:
Figure BDA0002910637670000051
wherein ε >0 is the switching gain, k >0 is the state feedback gain;
further, the step (4) specifically includes:
the parameters k, c and epsilon of the controller are carried out according to a traditional pole allocation mode; extended state observer parameter beta1、β2、β3Are all configured at the observer bandwidth omega0And, the following equation is satisfied:
Δ(λ)＝λ3+β1λ2+β2λ+β3＝(λ+ω0)3 (ω0＞0) (11)
wherein λ isiI is 1,2,3 is the root of the observer characteristic equation; omega0The selection principle is the controller bandwidth omegac2-5 times of the total weight of the powder.
Compared with the prior art, the invention has the advantages that:
1. according to the invention, the frame system is modeled from the energy perspective by adopting the Lagrange function and the Rayleigh dissipation function, and the modeling thought of the frame system can reduce the influence of unmodeled dynamics in the system on the frame system, reduce the estimation burden of the extended state observer, and improve the estimation precision of the extended state observer, thereby improving the control precision of the control method.
2. According to the method, the extended state observer with low requirement on model accuracy is adopted to estimate the disturbance in the frame system, the problem that parameters are difficult to configure when the order of the extended state observer is high is solved, and the estimation accuracy of the extended state observer is improved; the sliding mode controller can ensure that the controlled state variable of the system can keep changing back and forth on a balance point as long as the switch gain is larger than the disturbance of the system after the controlled state variable of the system reaches the sliding mode surface, so that the system can still keep stable when the extended state observer has a certain estimation error, and the buffeting problem is relieved.
Drawings
FIG. 1 is a control diagram of a frame angular rate servo system;
FIG. 2 is a schematic diagram of a harmonic reducer-based frame system;
FIG. 3 is a schematic view of a frame system energy transfer component;
FIG. 4 is a flow chart of a control algorithm for the frame angular rate servo system.
The reference numbers in the figures mean: the system comprises a sliding mode controller 1, a power amplifier 2, a torque motor 3, an encoder 4, a harmonic reducer 5, a wave generator of the harmonic reducer 5.1, a flexible gear of the harmonic reducer 5.2, a frame system load 6, a rotary transformer 7 and a cascade expansion state observer 8.
Detailed Description
The technical solutions in the embodiments of the present invention will be described below with reference to the drawings in the embodiments of the present invention.
As shown in fig. 1, a control moment gyro frame system based on a harmonic reducer includes: the method comprises the following steps of (1) a sliding mode controller, a power amplifier 2, a torque motor 3, an encoder 4, a harmonic reducer 5, a frame system load 6, a rotary transformer 7 and an extended state observer 8: the sliding mode controller 1 inhibits and compensates disturbance in a system in real time through control input to generate a control signal, actual control current is output through a power amplifier 2 to drive a torque motor 3, an encoder 4 fixedly connected to a torque motor shaft is used for measuring the angular position of the torque motor 3, the output end of the torque motor 3 is rigidly connected with the input end of a harmonic reducer 5 at the same rotating speed, the harmonic reducer 5 is used as a torque amplifying device and a transmission device, output torque acts on a frame system load 6, a rotary transformer 7 fixedly connected with the frame system load 6 is used for obtaining the angular position of the frame system load 6, the angular position of the frame system load 6 and the control output of the sliding mode controller 1 are used as input information of an expansion state observer 8, the expansion state observer 8 generates an estimation quantity, and the angular speed instruction, the control torque gyroscope frame angular speed servo system, the control torque servo system and the angular speed servo system are given by an attitude control computer, The angular position of the torque motor 3 and the related variables, the angular position of the frame system load 6 and the related variables together serve as input information for the sliding mode controller 1.
Fig. 2 is a schematic diagram of the mechanical connection of the torque motor 3, the encoder 4, the harmonic reducer 5, the frame system load 6, and the rotary transformer 7 in fig. 1. The encoder 4 is installed on one side of a torque motor shaft, the torque motor 3 is fixedly connected with the input end of the harmonic speed reducer 5 through a rigid shaft, the harmonic speed reducer 5 is connected with a frame system load 6, and a rotary transformer 7 is installed at the frame system load end.
As shown in fig. 3, the energy transfer component of the harmonic reducer-based control moment gyro frame system includes: a torque motor 3, a wave generator 5.1 of a harmonic reducer, a flexible gear 5.2 of the harmonic reducer, and a frame system load 6: wherein, the torque motor 3 is rigidly connected with the wave generator 5.1 of the harmonic reducer at the same rotating speed, and the kinetic energy and the dissipation energy at the torque motor side are respectively TmAnd Fm(ii) a A wave generator 5.1 of the harmonic reducer is in transmission connection with a flexible gear 5.2 of the harmonic reducer, reverse transmission with a reduction ratio of N exists between the wave generator 5.1 of the harmonic reducer and the flexible gear 5.2 of the harmonic reducer, and the flexible gear 5.2 has elastic potential energy of V in the transmission process of the harmonic reducer 5; the flexible gear 5.2 of the harmonic reducer is also rigidly connected with the frame system load 6 and has the same rotating speed, and the kinetic energy and the dissipation energy of the frame system load side are respectively TlAnd Fl(ii) a The torque motor 3 is part of the energy input in the frame system, and the existence of the damping causes the energy dissipation of the frame system, and the frame system is a non-conservative system.
As shown in fig. 4, the process of establishing the control algorithm of the control moment gyro frame system based on the harmonic reducer comprises the following steps:
step (1): establishing a state space equation based on Lagrangian function and Rayleigh dissipation function from the energy perspective
The motion error generated by the harmonic reducer (5) is as follows:
Figure BDA0002910637670000061
wherein the content of the first and second substances,
Figure BDA0002910637670000062
is the motion error, thetapDue to pure movement errors, theta, caused by manufacturing, mounting errors, backlash, or the liketIs the torsion angle theta caused by the elastic deformation of a flexible gear (5.2) of the harmonic reducer under the condition of loadmIs the angular position of the torque motor, thetalIs the frame system load angular position, N is the reduction ratio of the harmonic reducer;
the above equation is derived over time t:
Figure BDA0002910637670000071
wherein the content of the first and second substances,
Figure BDA0002910637670000072
instantaneous speed ratio which is the input and output angular rate of the harmonic reducer;
the kinetic and dissipation energies of the frame system are respectively:
Figure BDA0002910637670000073
wherein, JmAnd JlMoment of inertia of the load of the torque motor and frame system, respectively, BmAnd BlThe damping coefficients of the load of the torque motor and the load of the frame system are respectively;
the lagrangian equation of motion for the framework system is:
Figure BDA0002910637670000074
wherein, L is T-V, V is the elastic potential energy of the flexible gear in the transmission process of the harmonic reducer; τ is a control input to the frame system;
substituting the formulas (2) and (3) into the formula (4) to obtain the frame system load rotating speed
Figure BDA00029106376700000711
System equation for basic state variables:
Figure BDA0002910637670000075
wherein the content of the first and second substances,
Figure BDA0002910637670000076
Tkthe torque is required by the torque motor to drive the harmonic reducer;
defining a state variable as
Figure BDA0002910637670000077
The control output of the composite controller is u-tau, namely the control output of the frame system is y-x based on the control input of the frame system of the harmonic reducer2B is the control input gain, a is the indirect variable, f (x) is the main interference of the frame system, which is regarded as the 'lumped disturbance', and the state space equation of the frame system can be obtained as follows:
Figure BDA0002910637670000078
wherein the content of the first and second substances,
Figure BDA0002910637670000079
in the actual situation where the device is,
Figure BDA00029106376700000710
thus can be used
Figure BDA0002910637670000081
Instead of Y, the b term can be rewritten as
Figure BDA0002910637670000082
So that the b term becomes a constant term;
step (2): extended state observer design
Defining expanded state variables as x3＝f(x1,x2)，
Figure BDA0002910637670000083
The expansion state space equation of the framework system is:
Figure BDA0002910637670000084
defining a state variable of the extended state observer as z ═ z1,z2,z3]TWherein z is1Is thetalEstimate of z2Is that
Figure BDA0002910637670000085
Estimate of z3Is an estimate of h;
the state equation of the extended state observer of the framework system is:
Figure BDA0002910637670000086
wherein, beta1、β2、β3Three design parameters of the extended state observer;
and (3): sliding mode controller design
Designing a sliding mode surface function s as follows:
Figure BDA0002910637670000087
wherein the content of the first and second substances,
Figure BDA0002910637670000088
and
Figure BDA0002910637670000089
are each thetalExpected value of and
Figure BDA00029106376700000810
is expected toThe value c is the sliding mode surface function parameter to be designed;
the following control law is adopted in the controller:
Figure BDA00029106376700000811
wherein ε >0 is the switching gain, k >0 is the state feedback gain;
and (4): parameter design of composite control algorithm
The parameters k, c and epsilon of the controller are carried out according to a traditional pole allocation mode; extended state observer parameter beta1、β2、β3Are all configured at the observer bandwidth omega0Here, the following equation is satisfied:
Δ(λ)＝λ3+β1λ2+β2λ+β3＝(λ+ω0)3 (ω0＞0) (11)
wherein λ isiI is 1,2,3 is the root of the observer characteristic equation; omega0The selection principle is the controller bandwidth omegac2-5 times of the total weight of the powder.
Taking a single-frame control moment gyro frame control system based on a harmonic reducer with angular momentum of 200Nms as an example, the angular rate bandwidth is 5Hz, and the specific parameter configuration is shown in table 1.
TABLE 1 Single frame control moment gyro frame system parameters
Figure BDA0002910637670000091
The controller parameters k, c, epsilon are configured in a conventional pole configuration:
k＝3、c＝5、ε＝3；
cascaded extended state observer parameter beta1、β2、β3Can be according to beta1＝3ω0，
Figure BDA0002910637670000092
Is configured to:
β1＝300、β2＝30000、β3＝100000；
through simulation verification, a 5 DEG/s step reference signal acts on a frame system when t is 0.2s, the tracking error of the composite control method is between-0.05 DEG/s and 0.05 DEG/s, the tracking error is smaller by one order of magnitude compared with the tracking error of a state feedback method based on an extended state observer, the tracking error is smaller by 75% compared with the tracking error of a proportional-integral-derivative (PID) control method, and compared with the two control methods, the system output angular rate precision is improved by 50%.
Portions of the invention not disclosed in detail are well within the skill of the art.

Claims (1)

1. A high-precision control method for a control moment gyro frame system based on a harmonic reducer comprises the following steps: the device comprises a sliding mode controller (1), a power amplifier (2), a torque motor (3), an encoder (4), a harmonic reducer (5), a wave generator (5.1) of the harmonic reducer, a flexible gear (5.2) of the harmonic reducer, a frame system load (6), a rotary transformer (7) and an expansion state observer (8): wherein, the sliding mode controller (1) restrains and compensates disturbance in the system in real time by controlling input to generate a control signal, the actual control current is output by the power amplifier (2) to drive the torque motor (3), the encoder (4) fixedly connected on the shaft of the torque motor is used for measuring the angular position of the torque motor (3), the output end of the torque motor (3) is rigidly connected with the input end of the harmonic reducer (5) and has the same rotating speed, the wave generator (5.1) of the harmonic reducer is in transmission connection with the flexible wheel (5.2) of the harmonic reducer, reverse transmission with the reduction ratio of N is arranged between the wave generator (5.1) of the harmonic reducer and the flexible wheel (5.2) of the harmonic reducer, the flexible wheel (5.2) of the harmonic reducer is also in rigid connection with the frame system load (6) and has the same rotating speed, the rotary transformer (7) fixedly connected with the frame system load (6) is used for obtaining the angular position of the frame system load (6), the angular position of the frame system load (6) and the control output of the sliding mode controller (1) are used as the input information of an extended state observer (8), the extended state observer (8) generates an estimation quantity, and the estimation quantity and the angular speed command of a control moment gyro frame angular speed servo system, the angular position and the related variable of a torque motor (3), the angular position and the related variable of the frame system load (6) which are given by an attitude control computer are used as the input information of the sliding mode controller (1), so that the high-precision control method is realized, and the method is characterized by comprising the following steps:
step (1): establishing a state space equation based on a Lagrangian function and a Rayleigh dissipation function from the energy perspective;
step (2): designing an extended state observer;
and (3): designing a sliding mode controller;
and (4): designing parameters of a composite control algorithm;
wherein, the step (1) specifically comprises:
the motion error generated by the harmonic reducer (5) is as follows:
Figure FDA0003524220070000011
wherein the content of the first and second substances,
Figure FDA0003524220070000012
is the motion error, thetapDue to pure movement errors, theta, caused by manufacturing, mounting errors, backlash, or the liketIs the torsion angle theta caused by the elastic deformation of a flexible gear (5.2) of the harmonic reducer under the condition of loadmIs the angular position, theta, of the torque motor (3)lIs the angular position of the frame system load (6), N is the reduction ratio of the harmonic reducer (5);
the above equation is derived over time t:
Figure FDA0003524220070000013
wherein the content of the first and second substances,
Figure FDA0003524220070000021
for the input and output of the harmonic reducer (5)Instantaneous rotational speed ratio of angular rate;
the kinetic and dissipation energies of the frame system are respectively:
Figure FDA0003524220070000022
wherein, JmAnd JlThe moment of inertia of the torque motor (3) and the frame system load (6), BmAnd BlDamping coefficients of the torque motor (3) and the frame system load (6) are respectively set;
the lagrangian equation of motion for the framework system is:
Figure FDA0003524220070000023
wherein, L is T-V, and V is the elastic potential energy of the flexible gear (5.2) in the transmission process of the harmonic reducer (5); τ is a control input to the frame system;
substituting the formulas (2) and (3) into the formula (4) to obtain the rotating speed of the frame system load (6)
Figure FDA0003524220070000024
System equation for basic state variables:
Figure FDA0003524220070000025
wherein the content of the first and second substances,
Figure FDA0003524220070000026
Tkthe torque is required by the torque motor (3) to drive the harmonic reducer (5);
defining a state variable as
Figure FDA0003524220070000027
The control output of the composite controller is u-tau, namely based on a harmonic reducer (5) frameThe control input of the system and the control output of the frame system are y ═ x2B is the control input gain, a is the indirect variable, f (x) is the main disturbance of the frame system, which is regarded as the 'lumped disturbance', and the state space equation of the frame system can be obtained as follows:
Figure FDA0003524220070000028
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003524220070000029
in the actual situation where the device is,
Figure FDA00035242200700000210
thus can be used
Figure FDA00035242200700000211
Instead of Y, the b term can be rewritten as
Figure FDA00035242200700000212
So that the b term becomes a constant term;
the step (2) specifically comprises:
defining expanded state variables as x3＝f(x1,x2)，
Figure FDA0003524220070000031
The expansion state space equation of the framework system is:
Figure FDA0003524220070000032
defining the state variable of the extended state observer (8) as z ═ z1,z2,z3]TWherein z is1Is thetalEstimate of z2Is that
Figure FDA0003524220070000033
Estimate of z3Is an estimate of h;
the state equation of the extended state observer (8) of the framework system is then:
Figure FDA0003524220070000034
wherein, beta1、β2、β3Is three design parameters of the extended state observer (8);
the step (3) specifically comprises:
designing a sliding mode surface function s as follows:
Figure FDA0003524220070000035
wherein the content of the first and second substances,
Figure FDA0003524220070000036
Figure FDA0003524220070000037
and
Figure FDA0003524220070000038
are each thetalExpected value of and
Figure FDA0003524220070000039
c is a sliding mode surface function parameter to be designed;
the following control law is adopted in the controller:
Figure FDA00035242200700000310
wherein ε >0 is the switching gain, k >0 is the state feedback gain;
the step (4) specifically comprises:
the parameters k, c and epsilon of the controller are carried out according to a traditional pole allocation mode; extended state observer (8) parameter beta1、β2、β3Are all configured at the observer bandwidth omega0And, the following equation is satisfied:
Δ (λ)＝λ3+β1λ2+β2λ+β3＝(λ+ω0)3(ω0>0) (11)
wherein λ isiI is 1,2,3 is the root of the observer characteristic equation; omega0The selection principle is the controller bandwidth omegac2-5 times of the total weight of the powder.
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