CN111007728B - Motor active-disturbance-rejection self-adaptive control method considering all-state constraint - Google Patents

Motor active-disturbance-rejection self-adaptive control method considering all-state constraint Download PDF

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CN111007728B
CN111007728B CN201911395827.6A CN201911395827A CN111007728B CN 111007728 B CN111007728 B CN 111007728B CN 201911395827 A CN201911395827 A CN 201911395827A CN 111007728 B CN111007728 B CN 111007728B
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CN111007728A (en
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徐张宝
刘庆运
孙船斌
郭永存
韩亮
何旭
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Anhui University of Technology AHUT
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Abstract

The invention discloses a self-adaptive control method for motor active disturbance rejection considering full-state constraint in the technical field of motor control systems, which comprises the following steps: establishing a motor position servo system model; motor active-disturbance-rejection adaptive controller and control considering all-state constraintLaw u and parameter k for adjusting motor active disturbance rejection adaptive control law u considering all-state constraint 1 ，k 2 ，b 1 ，b 2 ，L 1 ，L 2 Omega and delta, so that the system meets the control performance index, the scheme can effectively solve the problems of uncertain nonlinearity and uncertain parameters of a motor servo system, designs a constraint controller based on the barrier Lyapunov function, and finally proves the overall stability of the system through a certificate, the parameters are well converged under the interference condition, and the control precision of the system meets the performance index; meanwhile, the invention simplifies the design of the controller, and the simulation result shows the effectiveness of the controller.

Description

Motor active-disturbance-rejection self-adaptive control method considering all-state constraint
Technical Field
The invention relates to the technical field of motor control systems, in particular to a motor active-disturbance-rejection self-adaptive control method considering all-state constraint.
Background
Due to the wide application in industry, high performance control of motor-driven motion systems has attracted a wide range of attention, including engineers and scientists. However, it is not easy to design a high performance controller for a servo system, and since model uncertainties are widely distributed in the control system, a designer is likely to encounter many model uncertainties, especially unmodeled nonlinearities such as non-structural uncertainties. These uncertainty factors can severely degrade the control performance that can be achieved, resulting in low control accuracy, limit cycle oscillations, and even instability.
The traditional control mode is difficult to meet the requirement of uncertain nonlinear tracking precision, so that a control method which is simple and practical and meets the requirement of system performance needs to be researched. In recent years, various advanced control strategies are applied to a motor servo system, such as robust adaptive control, adaptive robust and the like, and the control achieves good control accuracy. However, the high-precision control result of the control strategy is ensured by large feedback gain, and the high-gain feedback often excites the high-frequency dynamic state of the system to cause instability of the system.
Aiming at the characteristics of uncertain nonlinearity and parameter uncertainty in motor servo, a model of the system is established, an extended state observer and a parameter self-adaptation law of a motor position servo system are respectively designed on the basis, and the system is estimated to estimate the unmodeled interference and unknown parameters and compensate in control input. In addition, in consideration of physical constraint conditions frequently encountered by an actual system, a state constraint controller is designed on the basis of an obstacle Lyapunov function, and effective constraint is carried out on the state of the system.
Disclosure of Invention
The invention aims to provide a motor active-disturbance-rejection adaptive control method considering full-state constraint so as to solve the problems in the background technology.
In order to achieve the purpose, the invention provides the following technical scheme: a motor active-disturbance-rejection self-adaptive control method considering full-state constraint comprises the following steps:
the method comprises the following steps: establishing a motor position servo system model; according to Newton's second law, the dynamic model equation of the inertia load of the motor is as follows:
Figure BDA0002346267710000021
in the formula: y is angular displacement, m is inertial load, k f Is the torque constant, u is the system control input, b is the viscous friction coefficient,
Figure BDA0002346267710000022
for other unmodeled disturbances, non-linear friction, external disturbances, and unmodeled dynamics are referred to;
defining state variables
Figure BDA0002346267710000023
Writing equation (1) as a state space form：
Figure BDA0002346267710000024
In the formula: x = [ x = 1 ,x 2 ] T For the state vector of position and velocity, assume that the output states of the system are constrained in a set Ω, Ω = { x = i :|x i |≤c i ,i＝1,2},c i Constant > 0, define unknown parameter set θ = [ ] 1 ,θ 2 ,] T Wherein θ 1 ＝k f /m，θ 2 ＝b/m，
Figure BDA0002346267710000025
Representing concentrated interference;
assuming d (x, t) is sufficiently smooth, i.e. | d (x, t) | ≦ δ 1 ，
Figure BDA0002346267710000026
In the formula: delta. For the preparation of a coating 1 ，δ 2 The method comprises the following steps of (1) knowing;
assume again that instruction x is expected 1d (t) and the time i-th derivative thereof
Figure BDA0002346267710000027
i =1,2 satisfies x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure BDA0002346267710000028
υ i > 0 is a constant, L 1 >0 is a design parameter.
Step two: designing a motor active-disturbance-rejection self-adaptive controller and a control law u considering all-state constraint;
step three: parameter k of motor active-disturbance-rejection adaptive control law u considering all-state constraint is adjusted 1 ，k 2 ，b 1 ，b 2 ，L 1 ，L 2 ω and δ, to make the system meet the control performance criteria.
Further, the second step includes the following steps:
s1: constructing an extended state observer of the motor system;
s2: designing a motor active-disturbance-rejection adaptive controller system considering full-state constraint;
s3: and verifying the stability of the system.
Further, the step S1 includes the following steps:
s101: equation (2) is converted to the following form:
Figure BDA0002346267710000031
in the formula:
Figure BDA0002346267710000032
s102: designing an extended state observer according to formula (4) in step S101, wherein the extended state observer has the following structure:
Figure BDA0002346267710000033
in the formula:
Figure BDA0002346267710000034
is x i I =1,2,3; omega>0 is a parameter of the extended state observer, let
Figure BDA0002346267710000035
i＝1,2,3；
If expanded state x 3 = d (x, t), h (t) being defined as x 3 The estimated error dynamics of the observer can be obtained as:
Figure BDA0002346267710000036
definition of
Figure BDA0002346267710000037
i =1,2,3, then
Figure BDA0002346267710000038
In the formula: ε = [ ε ] 1 ,ε 2 ,ε 3 ] T ,
Figure BDA0002346267710000039
B＝[0,0,1] T ,C＝[0,1,0] T ；
If in an expanded state
Figure BDA00023462677100000310
Then it can be obtained:
Figure BDA00023462677100000311
since matrix a satisfies the hervetz criterion, there is a positive definite symmetry of matrix P satisfying the following equation: a. The T P+PA＝-2I (9)
In the formula: the matrix I is an identity matrix;
further, the step S2 includes the steps of: first of all, z is defined 1 ＝x 1 -x 1d ，
z 2 ＝x 2 -α 1 ，α 1 For the virtual control law, the barrier Lyapunov function is defined again:
Figure BDA0002346267710000041
in the formula: b 1 >0，L 1 >0 is a constant, V 1 The time derivative of (a) is:
Figure BDA0002346267710000042
let the virtual control law α 1 The design is as follows:
Figure BDA0002346267710000043
in the formula: k is a radical of 1 >0 is the feedback gain of the feedback signal,by substituting equation (12) into equation (11), the following can be obtained:
Figure BDA0002346267710000044
if z is 2 If not =0, then
Figure BDA0002346267710000045
Then define the barrier lyapunov function:
Figure BDA0002346267710000046
in the formula: b 2 >0，L 2 >0 is a constant; v 2 The time derivative of (a) is:
Figure BDA0002346267710000047
combining equation (2) can obtain:
Figure BDA0002346267710000048
finally, based on the disturbance estimation of the extended state observer, the control input u is set as follows:
Figure BDA0002346267710000051
in the formula: u. u a As a model compensation term, u s For the robust term, k 2 >When 0 is a feedback gain, equation (17) is substituted into equation (16), the following can be obtained:
Figure BDA0002346267710000052
further, the step S3 specifically includes: if (1) a suitable parameter can be selected to satisfy the following equation: c. C 2 ≥|α 1 | max +L 2 (19) (2) the system initial value z (0) can satisfy the following conditions:
Figure BDA0002346267710000053
then, as can be seen from equation (17):
when the interference is time invariant, i.e., h (t) =0, the adaptive law is designed to:
Figure BDA0002346267710000054
the system is asymptotically convergent, i.e., when t → ∞ z 1 → 0, all signals are bounded, the system state can be effectively constrained;
when system disturbances are time-varying, all signals in a closed-loop control system are bounded, and the system state can be effectively constrained, as defined by the positive lyapunov function:
Figure BDA0002346267710000055
it satisfies:
Figure BDA0002346267710000061
further, when the interference is time invariant, the lyapunov function is defined as follows:
Figure BDA0002346267710000062
the derivation of equation (24) and the substitution of equations (7), (18) and (21) can be obtained:
Figure BDA0002346267710000063
from x 1 ＝z 1 +x 1d (t) and the desired instruction x 1d (t) and the time i-th derivative thereof
Figure BDA0002346267710000064
i =1,2 satisfies x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure BDA0002346267710000065
υ i > 0 is a constant, L 1 >0 is a design parameter, and | x can be obtained 1 |≤c 1 Thus x 1 Is bounded; and alpha is 1 Is z 1 And
Figure BDA0002346267710000066
function of z 1 And
Figure BDA0002346267710000067
is bounded, and a can be obtained 1 Is bounded; from | x 2 |≤|α 1 | max +|z 2 I and I z 2 |≤L 2 Let x know 2 ≤c 2 Is bounded; thus it can be verified that all signals of the system are bounded, the system state is restrainable;
when the system interference is time-varying, derivation is performed on equation (22), and equations (8), (21) are substituted to obtain:
Figure BDA0002346267710000068
in the formula:
Figure BDA0002346267710000071
integration of equation (26) yields:
Figure BDA0002346267710000072
from equation (27), it can be demonstrated that all signals in a closed loop system are bounded, the system state is constrained, and therefore the controller is converged and the system is stable.
Compared with the prior art, the invention has the beneficial effects that: the invention establishes a motor position servo system model for the characteristics of a motor position servo system; the motor active disturbance rejection adaptive controller considering the full-state constraint is designed to estimate unmodeled disturbance and carry out feedforward compensation, and meanwhile, the adaptive controller is used for estimating system position parameters, so that the problems of uncertain nonlinearity and uncertain parameters of a motor servo system can be effectively solved, the constraint controller is designed based on the barrier Lyapunov function, the overall stability of the system is finally proved through a certificate, the parameter convergence is good under the interference condition, and the system control precision meets the performance index; meanwhile, the invention simplifies the design of the controller, and the simulation result shows the effectiveness of the controller.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art that other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a schematic diagram of a motor actuator of the present invention;
FIG. 2 is a schematic diagram of the system control strategy of the present invention;
fig. 3 is a graph of interference estimation and interference estimation error in accordance with the present invention;
FIG. 4 is a graph of system parameter estimation according to the present invention;
FIG. 5 is a graph of output state curves for the design control method of the present invention;
FIG. 6 is a graph of tracking error for the desired command and for both controllers according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Referring to fig. 1 and fig. 2, the present invention provides a technical solution: a motor active-disturbance-rejection self-adaptive control method considering full-state constraint comprises the following steps:
the method comprises the following steps: establishing a motor position servo system model; according to Newton's second law, the dynamic model equation of the inertia load of the motor is as follows:
Figure BDA0002346267710000081
in the formula: y is angular displacement, m is inertial load, k f Is the torque constant, u is the system control input, b is the viscous friction coefficient,
Figure BDA0002346267710000082
for other unmodeled disturbances, non-linear friction, external disturbances, and unmodeled dynamics are referred to;
defining state variables
Figure BDA0002346267710000083
Equation (1) is written as a state space form:
Figure BDA0002346267710000084
in the formula: x = [ x = 1 ,x 2 ] T For the state vector of position and velocity, assume that the output states of the system are constrained in a set Ω, Ω = { x = i :|x i |≤c i ,i＝1,2},c i Constant > 0, define unknown parameter set θ = [ ] 1 ,θ 2 ,] T Wherein theta 1 ＝k f /m，θ 2 ＝b/m，
Figure BDA0002346267710000085
Representing concentrated interference;
suppose d (x, t) is sufficiently smoothThat is | d (x, t) | is less than or equal to δ 1 ，
Figure BDA0002346267710000086
In the formula: delta. For the preparation of a coating 1 ，δ 2 The method comprises the following steps of (1) knowing;
assume again that instruction x is expected 1d (t) and the time i-th derivative thereof
Figure BDA0002346267710000087
i =1,2 satisfying x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure BDA0002346267710000088
υ i > 0 is a constant, L 1 >0 is a design parameter.
Step two: designing a motor active-disturbance-rejection adaptive controller and a control law u considering all-state constraint;
step three: parameter k of motor active-disturbance-rejection adaptive control law u considering all-state constraint 1 ，k 2 ，b 1 ，b 2 ，L 1 ，L 2 ω and δ, to make the system meet the control performance criteria.
The second step comprises the following steps:
s1: constructing an extended state observer of the motor system;
s2: designing a motor active-disturbance-rejection adaptive controller system considering full-state constraint;
s3: and verifying the stability of the system.
The step S1 includes the steps of:
s101: equation (2) is converted to the following form:
Figure BDA0002346267710000091
in the formula:
Figure BDA0002346267710000092
s102: designing an extended state observer according to formula (4) in step S101, wherein the extended state observer has the following structure:
Figure BDA0002346267710000093
in the formula:
Figure BDA0002346267710000094
is x i I =1,2,3; omega>0 is a parameter of the extended state observer,
order to
Figure BDA0002346267710000095
i＝1,2,3；
If expanded state x 3 = d (x, t), define h (t) as x 3 The time derivative of (2) can be obtained
The estimation error dynamics is:
Figure BDA0002346267710000096
definition of
Figure BDA0002346267710000097
i =1,2,3, then
Figure BDA0002346267710000098
In the formula: ε = [ ε ] 1 ,ε 2 ,ε 3 ] T ,
Figure BDA0002346267710000099
B＝[0,0,1] T ,C＝[0,1,0] T ；
If in an expanded state
Figure BDA0002346267710000101
Then it is possible to obtain:
Figure BDA0002346267710000102
since the matrix A satisfies HelvelzFor the criterion, there is a positive definite symmetry matrix P satisfying the following equation: a. The T P+PA＝-2I (9)
In the formula: the matrix I is an identity matrix;
the step S2 includes the steps of: first of all, z is defined 1 ＝x 1 -x 1d ，z 2 ＝x 2 -α 1 ，α 1 For virtual control
Law making, then defining an obstacle Lyapunov function:
Figure BDA0002346267710000103
in the formula: b 1 >0，L 1 >0 is a constant, V 1 The time derivative of (a) is:
Figure BDA0002346267710000104
let the virtual control law α 1 The design is as follows:
Figure BDA0002346267710000105
in the formula: k is a radical of 1 >When 0 is a feedback gain, equation (12) is substituted into equation (11), and:
Figure BDA0002346267710000106
if z is 2 If not =0, then
Figure BDA0002346267710000107
Then the barrier lyapunov function is defined:
Figure BDA0002346267710000108
in the formula: b 2 >0，L 2 >0 is a constant; v 2 The time derivative of (a) is:
Figure BDA0002346267710000109
in conjunction with equation (2) one can obtain:
Figure BDA00023462677100001010
finally, based on the disturbance estimation of the extended state observer, the control input u is set as follows:
Figure BDA0002346267710000111
in the formula: u. of a As a model compensation term, u s For the robust term, k 2 >When 0 is a feedback gain, equation (17) is substituted into equation (16), and:
Figure BDA0002346267710000112
the step S3 specifically includes: if (1) a suitable parameter can be selected to satisfy the following equation:
c 2 ≥|α 1 | max +L 2 (19)
(2) The system initial value z (0) can satisfy the following condition:
Figure BDA0002346267710000113
then, as can be seen from equation (17):
when the interference is time-invariant, i.e., h (t) =0, the adaptive law is designed to:
Figure BDA0002346267710000114
the system is asymptotically convergent, i.e., when t → ∞ z 1 → 0, all signals are bounded, the system state can be effectively constrained;
when system disturbances are time-varying, all signals in a closed-loop control system are bounded, and the system state can be effectively constrained, as defined by the positive lyapunov function:
Figure BDA0002346267710000115
it satisfies:
Figure BDA0002346267710000121
when the interference is time invariant, the lyapunov function is defined as follows:
Figure BDA0002346267710000122
the derivation of equation (24) and the substitution of equations (7), (18) and (21) can be obtained:
Figure BDA0002346267710000123
from x 1 ＝z 1 +x 1d (t) and expected instruction x 1d (t) and the time i-th derivative thereof
Figure BDA0002346267710000124
i =1,2 satisfying x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure BDA0002346267710000125
υ i > 0 is a constant, L 1 >0 is a design parameter, and | x can be obtained 1 |≤c 1 Thus x 1 Is bounded; and alpha is 1 Is z 1 And
Figure BDA0002346267710000126
function of z 1 And
Figure BDA0002346267710000127
is bounded, canGet alpha 1 Is bounded; from | x 2 |≤|α 1 | max +|z 2 I and I z 2 |≤L 2 Knowing x 2 ≤c 2 Is bounded; thus, it can be verified that all signals of the system are bounded, and the system state is restrainable;
when the system interference is time-varying, derivation is performed on equation (22), and equations (8), (21) are substituted to obtain:
Figure BDA0002346267710000128
in the formula:
Figure BDA0002346267710000131
integration of equation (26) yields:
Figure BDA0002346267710000132
it is demonstrated by equation (27) that all signals in a closed loop system are bounded and the system state is constrained, so the controller is convergent and the system is stable.
For example: the initial state of the system is x 1 (0)＝0.02,x 2 (0) =0, design controller herein models the system in simulation taking the following parameters: m =0.01,kg · m 2 ，k f ＝5，b＝1.25N·s/m，θ 1n ＝600,θ 2n ＝80，θ 3n ＝0，k 1 ＝70,k 2 ＝0.001，b 1 ＝2，b 2 ＝0.01，L 1 ＝2，L 2 =10, ω =800, pid controller parameter k p ＝110，k i ＝70，k d =0.3, position angle input signal y d (t)＝0.2sin(πt)[1-exp(-0.01t 3 )]rad, d (x, t) =10sin (2 π t) N · m, the effect of the control law is shown in fig. 3-6, the algorithm proposed by the present invention can accurately estimate the interference value and the system state under the simulation environment, compared with the traditional PID control, the control designed by the present inventionThe controller can greatly improve the control precision of the system under the condition of large interference and has better state constraint performance; research results show that the method provided by the scheme can meet performance indexes under uncertain nonlinear influence.
In the description herein, references to the description of "one embodiment," "an example," "a specific example" or the like are intended to mean that a particular feature, structure, material, or characteristic described in connection with the embodiment or example is included in at least one embodiment or example of the invention. In this specification, the schematic representations of the terms used above do not necessarily refer to the same embodiment or example. Furthermore, the particular features, structures, materials, or characteristics described may be combined in any suitable manner in any one or more embodiments or examples.
The preferred embodiments of the invention disclosed above are intended to be illustrative only. The preferred embodiments are not intended to be exhaustive or to limit the invention to the precise embodiments disclosed. Obviously, many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, to thereby enable others skilled in the art to best understand the invention for and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims (2)

1. A motor active-disturbance-rejection self-adaptive control method considering full-state constraint is characterized in that: the method comprises the following steps:
the method comprises the following steps: establishing a motor position servo system model; according to Newton's second law, the dynamic model equation of the inertia load of the motor is as follows:
Figure FDA0003924949690000011
in the formula: y is angular displacement, m is inertial load, k f Is the torque constant, u is the system control input, b is the viscous friction coefficient,
Figure FDA0003924949690000012
for other unmodeled disturbances, non-linear friction, external disturbances, and unmodeled dynamics are referred to;
defining state variables
Figure FDA0003924949690000013
Equation (1) is written as a state space form:
Figure FDA0003924949690000014
in the formula: x = [ x = 1 ,x 2 ] T For the state vector of position and velocity, assume that the output states of the system are constrained in a set Ω, Ω = { x = i :|x i |≤c i ,i＝1,2},c i Constant > 0, define unknown parameter set θ = [ ] 1 ,θ 2 ,] T Wherein θ 1 ＝k f /m，θ 2 ＝b/m，
Figure FDA0003924949690000015
Representing concentrated interference;
assuming d (x, t) is sufficiently smooth, i.e. | d (x, t) | ≦ δ 1 ，
Figure FDA0003924949690000016
In the formula: delta 1 ，δ 2 The method comprises the following steps of (1) knowing;
assume again that the desired instruction x 1d (t) and the time i-th derivative thereof
Figure FDA0003924949690000017
Satisfy x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure FDA0003924949690000018
υ i > 0 is a constant, L 1 >0 is a design parameter;
υ 0 is x 1d (t) an upper bound value; c. C 1 Is a constant;
step two: a motor active disturbance rejection adaptive controller and a motor active disturbance rejection adaptive control law u with full-state constraint considered are designed, and the method comprises the following steps:
s1: constructing an extended state observer of the motor system;
the step S1 includes the steps of:
s101: equation (2) is converted to the following form:
Figure FDA0003924949690000019
in the formula:
Figure FDA0003924949690000021
Figure FDA0003924949690000022
are each theta 1 、θ 2 Is determined by the estimated value of (c),
Figure FDA0003924949690000023
are respectively theta 1 、θ 2 The estimated error of (2);
s102: designing an extended state observer according to formula (4) in step S101, wherein the extended state observer has the following structure:
Figure FDA0003924949690000024
in the formula:
Figure FDA0003924949690000025
is x i I =1,2,3; omega>0 is a parameter of the extended state observer, let
Figure FDA0003924949690000026
If expanded state x 3 = d (x, t), h (t) being defined as x 3 The estimated error dynamics of the observer can be obtained as:
Figure FDA0003924949690000027
definition of
Figure FDA0003924949690000028
Then
Figure FDA0003924949690000029
In the formula: ε = [ ε ] 1 ,ε 2 ,ε 3 ] T ,
Figure FDA00039249496900000210
B＝[0,0,1] T ,C＝[0,1,0] T ；
If in an expanded state
Figure FDA00039249496900000211
Then it is possible to obtain:
Figure FDA00039249496900000212
Figure FDA00039249496900000213
since matrix a satisfies the hervetz criterion, there is a positive definite symmetry of matrix P satisfying the following equation: a. The T P+PA＝-2I (9)
In the formula: the matrix I is an identity matrix;
s2: designing a motor active-disturbance-rejection adaptive controller system considering full-state constraint;
the step S2 includes the steps of: first of all, z is defined 1 ＝x 1 -x 1d ，z 2 ＝x 2 -α 1 ，α 1 For the virtual control law, the barrier Lyapunov function is defined again:
Figure FDA0003924949690000031
Figure FDA0003924949690000032
in the formula: b 1 >0，L 1 >0 is a constant, V 1 The time derivative of (a) is:
Figure FDA0003924949690000033
let the virtual control law alpha 1 The design is as follows:
Figure FDA0003924949690000034
in the formula: k is a radical of 1 >When 0 is a feedback gain, equation (12) is substituted into equation (11), and:
Figure FDA0003924949690000035
if z is 2 =0, then
Figure FDA0003924949690000036
Then define the barrier lyapunov function:
Figure FDA0003924949690000037
in the formula: b 2 >0，L 2 >0 is a constant; v 2 The time derivative of (a) is:
Figure FDA0003924949690000038
in conjunction with equation (2) one can obtain:
Figure FDA0003924949690000039
finally, based on the disturbance estimation of the extended state observer, the control inputs u are set as follows:
Figure FDA00039249496900000310
in the formula: u. of a As a model compensation term, u s For the robust term, k 2 >When 0 is a feedback gain, equation (17) is substituted into equation (16), the following can be obtained:
Figure FDA0003924949690000041
s3: verifying the stability of the system;
the step S3 specifically comprises the following steps: if (1) a suitable parameter can be selected to satisfy the following equation: c. C 2 ≥|α 1 | max +L 2 (19)；
(2) The system initial value z (0) can satisfy the following condition:
Figure FDA0003924949690000042
then, as can be seen from equation (17):
when the interference is time invariant, i.e., h (t) =0, the adaptive law is designed to:
Figure FDA0003924949690000043
the system is asymptotically convergent, i.e., when t → infinity, z 1 → 0, all signals are bounded, the system state can be effectively constrained;
when system disturbances are time-varying, all signals in a closed-loop control system are bounded, and the system state can be effectively constrained as follows defining a positive lyapunov function:
Figure FDA0003924949690000044
it satisfies:
Figure FDA0003924949690000045
wherein, delta and lambda are transition parameters;
step three: parameter k of motor active-disturbance-rejection adaptive control law u considering all-state constraint 1 ，k 2 ，b 1 ，b 2 ，L 1 ，L 2 ω and δ, to make the system meet the control performance criteria.
2. The adaptive control method for auto-disturbance-rejection of the motor considering the full-state constraint as claimed in claim 1, wherein: when the interference is time invariant, the lyapunov function is defined as follows:
Figure FDA0003924949690000051
the derivation of equation (24) and substitution of equations (7), (18), and (21) yields:
Figure FDA0003924949690000052
from x 1 ＝z 1 +x 1d (t) and expected instruction x 1d (t) and the time i-th derivative thereof
Figure FDA0003924949690000053
Satisfy x 1d (t)≤υ 0 ≤c 1 -L 1 ，
Figure FDA0003924949690000054
υ i > 0 is a constant, L 1 >0 is a design parameter, and | x can be obtained 1 |≤c 1 Thus x 1 Is bounded; and alpha is 1 Is z 1 And
Figure FDA0003924949690000055
function of z 1 And
Figure FDA0003924949690000056
is bounded, and can be given a 1 Is bounded; from | x 2 |≤|α 1 | max +|z 2 I and I z 2 |≤L 2 Let x know 2 ≤c 2 Is bounded; thus, it can be verified that all signals of the system are bounded, and the system state is restrainable; when the system interference is time-varying, derivation of equation (22) and substitution of equations (8), (21) can be obtained:
Figure FDA0003924949690000057
in the formula:
Figure FDA0003924949690000061
integration of equation (26) yields:
Figure FDA0003924949690000062
it is demonstrated by equation (27) that all signals in a closed loop system are bounded and the system state is constrained, so the controller is convergent and the system is stable.
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