CN113741469A - Output feedback trajectory tracking control method with preset performance and dead zone input constraint for electromechanical system - Google Patents

Output feedback trajectory tracking control method with preset performance and dead zone input constraint for electromechanical system Download PDF

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CN113741469A
CN113741469A CN202111058058.8A CN202111058058A CN113741469A CN 113741469 A CN113741469 A CN 113741469A CN 202111058058 A CN202111058058 A CN 202111058058A CN 113741469 A CN113741469 A CN 113741469A
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electromechanical system
dead zone
function
tracking
error
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宗广灯
王玉迪
褚晓广
孙海滨
杨东
齐文海
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Qufu Normal University
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0223—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving speed control of the vehicle
    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
    • G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
    • G05D1/02—Control of position or course in two dimensions
    • G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
    • G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
    • G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process

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Abstract

An output feedback trajectory tracking control method for an electromechanical system having predetermined performance and dead band input constraints. Firstly, establishing an electromechanical system model containing dead zone input, carrying out linear processing on the dead zone model, and giving a reference tracking track; secondly, defining a tracking error, introducing a preset performance function, and converting an original system into a system containing preset performance constraints; aiming at the problems that angular positions, angular speeds and motor currents in an electromechanical system are difficult to measure or the measurement precision is low, a neural network state observer is designed to estimate the state of the electromechanical system; finally, a command filter technology is adopted to solve the problem of 'complexity explosion' in the traditional back step method, and under the constraint of preset performance control and dead zone input, the self-adaptive back step method and the neural network state observer are combined to construct self-adaptive output feedback control; the method effectively improves the transient state and the steady state performance of the state unknown system, accurately realizes track tracking, and is suitable for the control of an accurately positioned electromechanical system.

Description

Output feedback trajectory tracking control method with preset performance and dead zone input constraint for electromechanical system
Technical Field
The invention designs an output feedback trajectory tracking control method for an electromechanical system with preset performance and dead zone input constraint, which is mainly applied to output feedback trajectory tracking control of the electromechanical system and belongs to the technical field of automatic control.
Background
In recent years, electromechanical control systems have been widely used in industrial production, particularly in motor control. The industrial electromechanical control generally comprises a current loop, a speed loop, a position loop and three closed-loop control structures. People can set internal parameters of the system according to production requirements, so that position control, speed control and torque control of the electromechanical system are realized, and expected performances are obtained. In the conventional control, the PID algorithm is widely used, but it cannot effectively cope with uncertain information in the system; in recent years, adaptive control is gradually applied to production, however, the control strategy of the adaptive control is too dependent on parameters of a motor model; in consideration of the complexity of the environment where an actual industrial system is located, difficulty in measuring the system state, low measurement accuracy, system performance constraint and the like, people often can only obtain output information and partial state information of the system and cannot master all state information of the system, so that the control of the system becomes more difficult; moreover, during the operation of the electromechanical system, people expect that the actual system reaches a steady state in a short time, and the transition process of the system is prevented from being too long. Therefore, the invention designs an electromechanical system output feedback trajectory tracking control scheme with preset performance control and dead zone input constraint, overcomes the influence of the problems and obtains better performance effect.
Disclosure of Invention
The technical problem to be solved by the invention is as follows: for an electromechanical system containing preset performance control and dead zone input constraint, a self-adaptive output feedback trajectory tracking control strategy is provided, the problems that the system state is not easy to measure and dead zone input factors are solved, the stability of the electromechanical control system is ensured, and the electromechanical control system has good transient performance and steady-state performance.
The invention discloses an output feedback trajectory tracking control method with preset performance and dead zone input constraint for an electromechanical system. Firstly, establishing an electromechanical system model containing dead zone input, carrying out linearization processing on the dead zone model, and giving a reference tracking track; then introducing a preset performance function, and converting the original system into a system with preset performance constraint; aiming at the problems that the angular position, the angular speed and the motor current of a motor in an electromechanical system are difficult to measure and the measurement precision is not high, a neural network state observer is designed to obtain an estimated value of the state of the electromechanical system; and finally, a command filter technology is adopted to solve the problem of complexity explosion in the traditional back step method, and the self-adaptive output feedback learning control method is designed by combining the self-adaptive back step method and the neural network state observer under the constraint of preset performance control and dead zone input. The method effectively solves the problems that the system state is difficult to measure and the transient performance of the system is difficult to measure, accurately realizes the track tracking and is suitable for the output feedback track tracking control of an electromechanical system.
Establishing an electromechanical system model containing dead zone input, carrying out linearization processing on the dead zone model, and giving a reference tracking track; defining a tracking error, introducing a preset performance function, and converting an original system into a system with preset performance constraint; aiming at the problems that the angular position, the angular speed and the motor current of a motor in an electromechanical system are difficult to measure and the measurement precision is not high, a neural network state observer is designed to obtain an estimated value of the state of the electromechanical system; the command filter technology is adopted to solve the problem of complexity explosion in the traditional back-stepping method; and designing a self-adaptive output feedback learning control scheme based on a self-adaptive backstepping method and a neural network state observer. The detailed process is as follows:
firstly, establishing an electromechanical system model containing dead zone input
Figure BDA0003255451960000021
Wherein the ratio of q,
Figure BDA0003255451960000022
respectively expressed as angular position, angular velocity and angular acceleration of motor, I is motor current, V0Is the input voltage.
Figure BDA0003255451960000023
N＝mG0g/(2Kτ)+W0G0g/Kτ，B＝B0/KτJ is moment of inertia, m is connecting rod mass, G0Is the length of the connecting rod, W0Is the load factor, R0Is the load radius, B0Is a viscous friction coefficient, G is a gravity coefficient, G is an armature inductance, R is an armature resistance, KτAnd KTRespectively conversion coefficient and back emf coefficient.
Constructing an operation dead zone of a converter containing an electromechanical system execution unit, wherein the model is
Figure BDA0003255451960000031
Where upsilon (t) is the input and pir(υ(t))，πlV (t) in the section [ c ]r,+∞)，(-∞,cl]Is continuous and has a normal number kl0,kl1,kr0,kr1Satisfy the following requirements
Figure BDA0003255451960000032
Wherein,
Figure BDA0003255451960000033
and is
Figure BDA00032554519600000310
Is bounded, its supremum satisfies
Figure BDA0003255451960000034
From (2), (3), the model can be converted into
u＝Π(υ(t))＝HT(t)γ(t)υ(t)+d(υ(t))＝F(t)υ(t)+d(υ(t)) (4)
Introducing a state variable x1,x2,x3Let x be1Q is the angular position of the motor,
Figure BDA0003255451960000035
as angular velocity, x, of the motor3Where I is the motor current, the system (1) can be converted into
Figure BDA0003255451960000036
Wherein f is1(x1)＝0,
Figure BDA0003255451960000037
As a function of the uncertainty of the slip in the electromechanical system.
Figure BDA0003255451960000038
y is system input and output, and the expected tracking trajectory is yd。
Second, defining the tracking error as v (t) y-ydIntroducing a predetermined performance function for achieving a desired system overshoot and faster convergence rate
Figure BDA0003255451960000039
Wherein,
Figure BDA00032554519600000311
introducing transformations
Figure BDA00032554519600000312
Can obtain
Figure BDA0003255451960000041
Is derived from (7)
Figure BDA0003255451960000042
Wherein
Figure BDA0003255451960000043
Defining coordinate transformations
Figure BDA00032554519600000416
Derived from (9)
Figure BDA0003255451960000044
Wherein,
Figure BDA0003255451960000045
is the weight error.
Thirdly, converting (5) into a matrix equation of formula (11)
Figure BDA0003255451960000046
Wherein,
Figure BDA0003255451960000047
Figure BDA0003255451960000048
using a radial basis function neural network, a smooth uncertainty function of (11)
Figure BDA0003255451960000049
And the system state variable x may be represented by
Figure BDA00032554519600000410
And
Figure BDA00032554519600000411
to represent
Figure BDA00032554519600000412
Wherein,
Figure BDA00032554519600000413
is composed of
Figure BDA00032554519600000414
Is determined by the estimated value of (c),
Figure BDA00032554519600000415
is an estimate of x, and satisfies
Figure BDA0003255451960000051
Wherein,
Figure BDA0003255451960000052
being an estimate of a state variable, thetaiIs an ideal weight
Figure BDA0003255451960000053
Is estimated by the estimation of (a) a,
Figure BDA0003255451960000054
for the output of the hidden layer, the minimum approximation error εiSum approximation error deltaiAre respectively defined as
Figure BDA0003255451960000055
The requirements are met,
Figure BDA0003255451960000056
defining the error of the state observer as e ═ e1,...,en]T. From (11), (12), it is possible to obtain
Figure BDA0003255451960000057
Wherein δ is [ δ ═ δ1,...,δn]T. Based on the output value of the state observer, a command filter is adopted to design the following coordinate transformation
Figure BDA0003255451960000058
Wherein xiAs virtual error plane, αi-1For virtual control signals, piAs boundary layer error, siIs about the dummy control signal alphai-1And is generated by a command filter, satisfying the following conditions
Figure BDA0003255451960000059
Wherein m isiAre design parameters.
The fourth step, the smooth uncertainty function in (12)
Figure BDA00032554519600000510
And an estimated value of a system state variable x, designing the following adaptive output feedback controller based on the state observer
[1]Design the virtual control signal α1And law of adaptation
Figure BDA00032554519600000511
Is composed of
Figure BDA00032554519600000512
Figure BDA00032554519600000513
Wherein, c1＞0,μ1＞0,γ1Are design parameters.
The sources of the virtual controller (17) and the adaptive law (18) are as follows:
construct the following Lyapunov function
V0＝eTNe (19)
Wherein N is a matrix satisfying N ═ NTIs greater than 0. From (14), (19), one can obtain
Figure BDA0003255451960000061
Wherein M is a matrix satisfying ATN+NA＝-M,λmin(M) is the minimum eigenvalue of matrix M.
The Lyapunov function is designed as follows
Figure BDA0003255451960000062
Consists of (10), (20), (21) and
Figure BDA0003255451960000063
through calculation, the method can obtain
Figure BDA0003255451960000064
Wherein,
Figure BDA0003255451960000065
substituting (17) and (18) into (23) to obtain
Figure BDA0003255451960000066
[2]Design the virtual control signal α2And law of adaptation
Figure BDA0003255451960000067
Is composed of
Figure BDA0003255451960000068
Figure BDA0003255451960000069
Wherein, c2＞0,μ2＞0,γ2Are design parameters.
The sources of the virtual controller (25) and the adaptive law (26) are as follows:
from (12) and (15), it can be seen that,
Figure BDA0003255451960000071
the Lyapunov function is constructed as follows
Figure BDA0003255451960000072
From (15), (16), can be obtained
Figure BDA0003255451960000073
Wherein, Y2(. is) a continuous function and satisfies | Y2(·)|＜Q2,Q2Is a positive scalar quantity. And [1]]The derivation is similar, and through calculation, the method can obtain
Figure BDA0003255451960000074
Substituting (25) and (26) into (30) to obtain
Figure BDA0003255451960000075
[3]Designing the actual controller v,
Figure BDA0003255451960000076
and law of adaptation
Figure BDA0003255451960000077
Is composed of
Figure BDA0003255451960000078
Figure BDA0003255451960000079
Figure BDA00032554519600000710
Wherein, c3＞0,μ3＞0,γ3,l1As a design parameter.
The sources of the actual controllers (32), (33) and the adaptive laws (34) are as follows:
the Lyapunov function is designed as follows
Figure BDA0003255451960000081
Similar to the derivation procedure of [1], it can be derived
Figure BDA0003255451960000082
From (36), it can be derived
Figure BDA0003255451960000083
Wherein,
Figure BDA0003255451960000084
is constant and satisfies
Figure BDA0003255451960000085
That is to say that the first and second electrodes,
Figure BDA0003255451960000086
Λ is a bounded normal number. The actual controllers (32), (33) and the adaptive laws (34) and (37) are substituted into (36) to obtain
Figure BDA0003255451960000087
Wherein,
Figure BDA0003255451960000088
R*is a bounded normal number.
Fifthly, analyzing the stability of the closed loop system
From the above derivation
Figure BDA0003255451960000089
In pair (38)
Figure BDA00032554519600000810
χ2χ3,ρj+1Yj+1Is scaled, can have
Figure BDA00032554519600000811
Wherein K is a bounded positive scalar quantity,
Figure BDA00032554519600000812
selecting appropriate parameters
Figure BDA0003255451960000091
Can obtain the product
Figure BDA0003255451960000092
Further obtain
Figure BDA0003255451960000093
By the definition of (40) and Lyapunov function, we can obtain the Lyapunov function V3And all signals in a closed loop system are bounded. According to (15), (16), and (17), s can be obtained2,
Figure BDA0003255451960000094
α1,x2Is well-defined. Similar to the above analysis, it can be obtained in (6)
Figure BDA0003255451960000095
Therefore, the tracking error always remains within the performance bounds.
The invention has the beneficial effects that:
(1) by constructing a neural network state observer and introducing a preset performance control method, the invention solves the problems of system transient state and steady-state tracking control performance with unknown state. In the operation process of the electromechanical system, the overshoot of the system can be effectively restrained by adjusting the parameters of the preset performance function, and a faster convergence rate is obtained, so that the transition process is completed in a shorter time.
(2) Aiming at the situation that the converter has a dead zone in the operation process, the design does not use the median theorem to construct a dead zone inverse, and does not need the known dead zone parameter limit and a linear function except the dead zone to eliminate the influence of the dead zone on the system stability, so that an input dead zone model is further optimized;
(3) the designed adaptive output to the feedback learning controller can overcome the problem of complexity explosion generated by the traditional backstepping method, and the burden of online calculation is reduced; the design can effectively solve the problems of unknown system state and transient and steady-state performance of the system, and accurately realize the track tracking of the electromechanical control system.
Drawings
Fig. 1 is a design block diagram of the present invention.
FIG. 2 is a design flow chart of the present invention.
FIG. 3 is a graph of the output curve of the present invention and a reference curve.
FIG. 4 is a graph of the output of an electromechanical system based on a PID algorithm versus a reference graph.
FIG. 5 is a graph of tracking error versus predetermined performance bounds for the present invention.
FIG. 6 is a graph of electromechanical system tracking error versus predetermined performance bounds based on a PID algorithm.
FIG. 7 is a graph of control input curves for the present invention.
FIG. 8 is a graph of electromechanical system control input based on a PID algorithm.
FIG. 9 is a graph showing the actual and estimated angular positions of the motor according to the present invention.
Fig. 10 is a graph showing an actual angular velocity curve and an estimated angular velocity curve of the motor according to the present invention.
Fig. 11 is a graph showing an actual current curve and an estimated current curve of the motor according to the present invention.
FIG. 12 is a graph of adaptive law according to the present invention.
Detailed Description
As shown in FIG. 1, the method comprises the following specific steps:
firstly, establishing an electromechanical system physical model containing dead zone input, carrying out linearization processing on the dead zone model, and giving a reference tracking track.
Figure BDA0003255451960000101
Wherein the ratio of q,
Figure BDA0003255451960000102
respectively expressed as angular position, angular velocity, angular acceleration of the motor. I represents the motor current, V0Representing the input voltage.
Figure BDA0003255451960000103
N＝mG0g/(2Kτ)+W0G0g/Kτ，B＝B0/Kτ. The parameters in the electromechanical system model are shown in table 1. Wherein,
Figure BDA0003255451960000104
and is
Figure BDA0003255451960000105
Is bounded, its supremum satisfies
Figure BDA0003255451960000106
From (2), (3), the model can be converted into
u＝Π(υ(t))＝HT(t) gamma (t) upsilon (t) + d (upsilon (t)) ═ f (t) upsilon (t) + d (upsilon (t)) (4) introduces a state variable x1,x2,x3Let x be1Q is the angular position of the motor,
Figure BDA0003255451960000111
as angular velocity, x, of the motor3Where I is the motor current, the system (1) can be converted into
Figure BDA0003255451960000112
Wherein f is1(x1)＝0,
Figure BDA0003255451960000113
As a function of the uncertainty of the slip in the electromechanical system.
Figure BDA0003255451960000114
y is system input and output, and the initial state is x (0) [0.01,0.13,0.01 ]]TThe expected tracking trajectory is yd＝0.25cos(0.2t)。。
Second, defining the tracking error as v (t) y-ydTo achieve the desired system overshoot and faster convergence rate, a predetermined performance function is introduced as follows
Figure BDA0003255451960000115
Wherein,
Figure BDA00032554519600001111
s＝1,kmax＝1.4,kmin＝0.9。
introducing transformations
Figure BDA00032554519600001112
Can obtain
Figure BDA0003255451960000116
Is derived from (7)
Figure BDA0003255451960000117
Wherein
Figure BDA0003255451960000118
Defining coordinate transformations
Figure BDA0003255451960000119
Derived from (9)
Figure BDA00032554519600001110
Wherein,
Figure BDA0003255451960000121
is the weight error.
Thirdly, converting (5) into a matrix equation of formula (11)
Figure BDA0003255451960000122
Wherein,
Figure BDA0003255451960000123
Figure BDA0003255451960000124
using a radial basis function neural network, a smooth uncertainty function of (11)
Figure BDA0003255451960000125
And the system state variable x may be represented by
Figure BDA0003255451960000126
And
Figure BDA0003255451960000127
to represent
Figure BDA0003255451960000128
Wherein,
Figure BDA0003255451960000129
is composed of
Figure BDA00032554519600001210
Is determined by the estimated value of (c),
Figure BDA00032554519600001211
is an estimate of x, and satisfies
Figure BDA00032554519600001212
Wherein,
Figure BDA00032554519600001213
is changed into a stateEstimate of quantity, thetaiIs an ideal weight
Figure BDA00032554519600001214
Is estimated by the estimation of (a) a,
Figure BDA00032554519600001215
for the output of the hidden layer, A, Bi,BnAnd C is already given.
Figure BDA00032554519600001216
Is an initial value of the state observer, K ═ 85,400,3610]TFor the controller gain matrix, the minimum approximation error εiSum approximation error deltaiAre respectively defined as
Figure BDA00032554519600001217
Figure BDA00032554519600001218
The requirements are met,
Figure BDA00032554519600001219
defining the error of the state observer as e ═ e1,...,en]TFrom (11), (12), it is possible to obtain
Figure BDA0003255451960000131
Wherein δ is [ δ ═ δ1,...,δn]T. Based on the output value of the state observer, a command filter is adopted to design the following coordinate transformation
Figure BDA0003255451960000132
Wherein xiAs virtual error plane, αi-1For virtual control signals, piAs boundary layer error, siIs about the dummy control signal alphai-1And is generated by a command filter, satisfying the following conditions
Figure BDA0003255451960000133
Wherein s (0) ═ 0.1,0.1]TAs an initial value of the command filter, m2＝m30.01 is a design parameter.
The fourth step, the smooth uncertainty function in (12)
Figure BDA0003255451960000134
And an estimated value of a system state variable x, designing the following adaptive output feedback controller based on the state observer
[1]Design the virtual control signal α1And law of adaptation
Figure BDA0003255451960000135
Is composed of
Figure BDA0003255451960000136
Figure BDA0003255451960000137
Wherein, c1＝68,μ1＝0.1,γ1＝0.001。
The sources of the virtual controller (17) and the adaptive law (18) are as follows:
construct the following Lyapunov function
V0＝eTNe (19)
Wherein N is a matrix satisfying N ═ NTIs greater than 0. From (14), (19), one can obtain
Figure BDA0003255451960000141
Wherein M is a matrix satisfying ATN+NA＝-M,λmin(M) is the minimum eigenvalue of matrix M.
The Lyapunov function is designed as follows
Figure BDA0003255451960000142
For (21) to be derived
Figure BDA0003255451960000143
Through calculation, the method can obtain
Figure BDA0003255451960000144
Wherein,
Figure BDA0003255451960000145
substituting (17) and (18) into (23) to obtain
Figure BDA0003255451960000146
[2]Design the virtual control signal α2And law of adaptation
Figure BDA0003255451960000147
Is composed of
Figure BDA0003255451960000148
Figure BDA0003255451960000149
Wherein, c2＝26,μ2＝1,γ2＝0.008。
The sources of the virtual controller (25) and the adaptive law (26) are as follows:
as can be seen from (12) and (15),
Figure BDA00032554519600001410
the Lyapunov function is constructed as follows
Figure BDA0003255451960000151
From (15), (16), can be obtained
Figure BDA0003255451960000152
Wherein, Y2(. is) a continuous function and satisfies | Y2(·)|＜Q2,Q2Is a positive scalar quantity. And [1]]Similarly, through calculation, can obtain
Figure BDA0003255451960000153
Substituting (25) and (26) into (30) to obtain
Figure BDA0003255451960000154
[3]Designing the actual controller v,
Figure BDA0003255451960000155
and law of adaptation
Figure BDA0003255451960000156
Is composed of
Figure BDA0003255451960000157
Figure BDA0003255451960000158
Figure BDA0003255451960000159
Wherein, c3＝38,μ3＝1,γ3＝0.01,l1＝0.01。
The sources of the actual controllers (32), (33) and the adaptive laws (34) are as follows:
the following Lyapunov function was designed,
Figure BDA00032554519600001510
similar to the derivation procedure of [1], it can be derived
Figure BDA0003255451960000161
From (36), it can be derived
Figure BDA0003255451960000162
Wherein,
Figure BDA0003255451960000163
is constant and satisfies
Figure BDA0003255451960000164
That is to say that the first and second electrodes,
Figure BDA0003255451960000165
Λ is a bounded normal number. The actual controllers (32), (33) and the adaptive laws (34) and (37) are substituted into (36) to obtain
Figure BDA0003255451960000166
Wherein,
Figure BDA0003255451960000167
R*is a bounded normal number.
Fifthly, analyzing the stability of the closed loop system
From the above derivation
Figure BDA0003255451960000168
In pair (38)
Figure BDA0003255451960000169
χ2χ3,ρj+1Yj+1Is scaled, can have
Figure BDA00032554519600001610
Wherein K is a bounded positive scalar quantity,
Figure BDA00032554519600001611
selecting appropriate parameters
Figure BDA00032554519600001612
Can obtain the product
Figure BDA00032554519600001613
Further, in the present invention,
Figure BDA00032554519600001614
by the definition of (40) and Lyapunov function, we can obtain the Lyapunov function V3And all messages in a closed loop systemThe numbers are bounded. According to (15), (16), and (17), s can be obtained2,
Figure BDA0003255451960000172
α1,x2Is well-defined. Similar to the above analysis, it can be obtained in (6)
Figure BDA0003255451960000173
Therefore, the tracking error always remains within the performance bounds.
Taking the specific implementation of the electromechanical system as an example, the detailed structural parameters of the system are shown in table 1, the effectiveness of the control method of the invention is further verified, and the performance comparison is respectively performed from reference curve tracking, error tracking, predetermined performance boundary, control input voltage and electromechanical system state estimation through comparison with the traditional PID control strategy trajectory tracking performance, and the detailed analysis is as follows:
TABLE 1 electromechanical System parameters
Figure BDA0003255451960000171
Fig. 3 shows an experimental graph of an output curve and a reference curve of the electromechanical system, and it can be seen from the results that the maximum steady-state tracking error is 0.0024m, and the tracking effect of the system at the initial moment is good.
An experimental graph of an output curve and a reference curve of the electromechanical system based on the PID is shown in FIG. 4, and it can be seen from the results that the maximum steady-state tracking error is 0.03m, and the system has a large tracking error in 0-10 s, and the expected tracking effect cannot be achieved.
FIG. 5 shows an experimental graph of a tracking error curve and a predetermined performance boundary curve of an electromechanical system. It can be seen that the maximum tracking error is 0.32m, the steady state error fluctuates at 0.0015m when the system is operating steadily, and the tracking error always remains within the bounds of the predetermined performance function.
FIG. 6 shows an experimental graph of a tracking error curve and a predetermined performance boundary curve based on a PID electromechanical system. It can be seen that the maximum tracking error is 0.25m, the steady state error fluctuates at 0.018m when the system is operating steadily, and the tracking error has exceeded the constraints of the predetermined performance function within 0-15 s.
Fig. 7 shows an experimental graph of the control input curve of the electromechanical system. It can be obtained that the maximum control input of the system is 268V, when the time exceeds 1s, the system tends to be stable, and the control input is 2.5V.
FIG. 8 shows an experimental graph of a control input curve based on a PID electromechanical system. It can be obtained that the maximum control input of the system is 27V, when the time exceeds 1s, the system tends to be stable, and the control input is 3V.
Fig. 9 shows an experimental graph of an actual curve and an estimated curve of the angular position of the motor. It can be obtained that the maximum error between the actual value and the estimated value of the motor angular position is 0.35m, and the error between the actual value and the estimated value of the motor angular position is 0.002m after the time exceeds 1 s.
Fig. 10 shows an experimental diagram of an actual angular velocity curve and an estimated angular velocity curve of the motor. It can be obtained that the maximum error between the actual value and the estimated value of the motor angular velocity is 1.2m/s, and the error between the actual value and the estimated value of the motor angular velocity is 0.2m/s after the time exceeds 1 s.
Fig. 11 shows an experimental diagram of an actual curve and an estimated curve of the motor current. It can be obtained that the maximum error between the actual value and the estimated value of the motor current is 2.3A, and when the time exceeds 1s, the error between the actual value and the estimated value of the motor current is 0.17A.
FIG. 12 is a diagram of an adaptive law curve experiment of an electromechanical system. It can be seen that the experimental results for motor angular position, motor angular velocity, motor current, and adaptive law in an electromechanical system are 0.4, 2.5, and 34, respectively, and remain bounded.
The invention provides an output feedback trajectory tracking control method with preset performance and dead zone input constraint for an electromechanical system. According to the designed preset performance function, the invention solves the transient and steady-state performance tracking problem of the electromechanical system, optimizes the situation that the converter has dead zones in the operation period, and further reduces the burden and cost of online calculation based on the designed command filter. By comparing with the traditional PID control strategy trajectory tracking performance, the effectiveness of the invention for the electromechanical system is further verified respectively from the aspects of reference curve tracking, error tracking, predetermined performance boundary constraint, control input voltage and electromechanical system state estimation.

Claims (1)

1. An output feedback trajectory tracking control method for an electromechanical system with preset performance and dead zone input constraints is characterized in that an electromechanical system model with dead zone input is established, the dead zone model of the electromechanical system is subjected to linearization processing, a reference tracking trajectory is given, a tracking error is defined, a preset performance function is introduced, and an original system is converted into a system with the preset performance constraints; aiming at the problems that three state variables of a motor angular position, an angular velocity and a motor current in an electromechanical system are difficult to measure or low in measurement precision, a neural network state observer is designed to obtain an estimated value of the states; the command filter technology is adopted to solve the problem of complexity explosion in the traditional back-stepping method; designing a self-adaptive output feedback learning control scheme based on a self-adaptive backstepping method and a neural network state observer; the method comprises the following steps:
firstly, establishing an electromechanical system model containing dead zone input
Figure FDA0003255451950000011
Wherein,
Figure FDA0003255451950000012
respectively expressed as angular position, angular velocity and angular acceleration of motor, I is motor current, V0In order to input the voltage, the voltage is,
Figure FDA0003255451950000013
N＝mG0g/(2Kτ)+W0G0g/Kτ，B＝B0/Kτj is moment of inertia, m is connecting rod mass, G0Is the length of the connecting rod, W0Is the load factor, R0Is the load radius, B0Is a viscous friction coefficient, G is a gravity coefficient, G is an armature inductance, R is an armature resistance, KτAnd KTRespectively a conversion coefficient and a back electromotive force coefficient,
constructing an operation dead zone of a converter containing an electromechanical system execution unit, wherein the model is
Figure FDA0003255451950000014
Where upsilon (t) is the input and pir(υ(t))，πlV (t) in the section [ c ]r,+∞)，(-∞,cl]Is continuous and has a normal number kl0,kl1,kr0,kr1Satisfy the following requirements
Figure FDA0003255451950000015
Wherein,
Figure FDA0003255451950000016
and is
Figure FDA0003255451950000017
Is bounded, its supremum satisfies
Figure FDA00032554519500000213
From (2), (3), the model can be converted into
u＝∏(υ(t))＝HT(t)Υ(t)υ(t)+d(υ(t))＝F(t)υ(t)+d(υ(t)) (4)
Introducing a state variable x1,x2,x3Let x be1Q is the angular position of the motor,
Figure FDA0003255451950000021
as angular velocity, x, of the motor3Where I is the motor current, the system (1) can be converted into
Figure FDA0003255451950000022
Wherein,
Figure FDA0003255451950000023
as a function of the uncertainty in the slip in the electromechanical system,
Figure FDA0003255451950000024
y is system input and output, and the expected tracking trajectory is yd；
Second, defining the tracking error as v (t) y-ydIntroducing a predetermined performance function for achieving a desired system overshoot and faster convergence rate
Figure FDA0003255451950000025
Wherein,
Figure FDA0003255451950000026
introducing transformations
Figure FDA0003255451950000027
Can obtain
Figure FDA0003255451950000028
Is derived from (7)
Figure FDA0003255451950000029
Wherein
Figure FDA00032554519500000210
Defining coordinate transformations
Figure FDA00032554519500000211
Derived from (9)
Figure FDA00032554519500000212
Wherein,
Figure FDA0003255451950000031
is the weight error;
thirdly, converting (5) into a matrix equation of formula (11)
Figure FDA0003255451950000032
Wherein,
Figure FDA0003255451950000033
Figure FDA0003255451950000034
using a radial basis function neural network, a smooth uncertainty function of (11)
Figure FDA0003255451950000035
And the system state variable x may be represented by
Figure FDA0003255451950000036
And
Figure FDA0003255451950000037
to represent
Figure FDA0003255451950000038
Wherein,
Figure FDA0003255451950000039
is composed of
Figure FDA00032554519500000310
Is determined by the estimated value of (c),
Figure FDA00032554519500000311
is an estimate of x, and satisfies
Figure FDA00032554519500000312
Wherein,
Figure FDA00032554519500000313
being an estimate of a state variable, thetaiIs an ideal weight
Figure FDA00032554519500000314
Is estimated by the estimation of (a) a,
Figure FDA00032554519500000315
for the output of the hidden layer, the minimum approximation error εiSum approximation error deltaiAre respectively defined as
Figure FDA00032554519500000316
Figure FDA00032554519500000317
The requirements are met,
Figure FDA00032554519500000318
defining the error of the state observer as e ═ e1,...,en]TFrom (11), (12), it is possible to obtain
Figure FDA00032554519500000319
Wherein δ is [ δ ═ δ1,...,δn]TDesigning the following coordinate transformation based on the output value of the state observer and using a command filter
Figure FDA0003255451950000041
Wherein, χiAs virtual error plane, αi-1For virtual control signals, piAs boundary layer error, siIs about the dummy control signal alphai-1And is generated by a command filter, satisfying the following conditions
Figure FDA0003255451950000042
Wherein m isiIs a design parameter;
the fourth step, the smooth uncertainty function in (12)
Figure FDA0003255451950000043
And an estimated value of a system state variable x, and designing the following state observer-based adaptive output feedback controller:
[1]designing a virtual control signal alpha1And law of adaptation
Figure FDA0003255451950000044
Is composed of
Figure FDA0003255451950000045
Figure FDA0003255451950000046
Wherein, c1＞0,μ1＞0,γ1In order to design the parameters of the device,
the sources of the virtual controller (17) and the adaptive law (18) are as follows:
construct the following Lyapunov function
V0＝eTNe (19)
Wherein N is a matrix satisfying N ═ NT> 0, from (14), (19), can be obtained
Figure FDA0003255451950000047
Wherein M is a matrix satisfying ATN+NA＝-M,λmin(M) is the minimum eigenvalue of matrix M,
the Lyapunov function is designed as follows
Figure FDA0003255451950000048
Consists of (10), (20) and (21) a
Figure FDA0003255451950000051
Through calculation, the method can obtain
Figure FDA0003255451950000052
Wherein,
Figure FDA0003255451950000053
substituting (17) and (18) into (23) to obtain
Figure FDA0003255451950000054
[2]Designing a virtual control signal alpha2And law of adaptation
Figure FDA0003255451950000055
Is composed of
Figure FDA0003255451950000056
Figure FDA0003255451950000057
Wherein, c2＞0,μ2＞0,γ2In order to design the parameters of the device,
the sources of the virtual controller (25) and the adaptive law (26) are as follows:
from (12) and (15) can be obtained
Figure FDA0003255451950000058
The Lyapunov function is constructed as follows
Figure FDA0003255451950000059
From (15), (16) can be obtained
Figure FDA00032554519500000510
Wherein, Y2(. is) a continuous function and satisfies | Y2(·)|＜Q2,Q2Is a positive scalar quantity, and [1]The derivation is similar, and through calculation, the method can obtain
Figure FDA0003255451950000061
Substituting (25) and (26) into (30) to obtain
Figure FDA0003255451950000062
[3]The control signal is designed as follows
Figure FDA0003255451950000063
And law of adaptation
Figure FDA0003255451950000064
Figure FDA0003255451950000065
Figure FDA0003255451950000066
Figure FDA0003255451950000067
Wherein, c3＞0,μ3＞0,γ3,l1In order to design the parameters,
the sources of the actual controllers (32), (33) and the adaptive laws (34) are as follows:
the Lyapunov function is designed as follows
Figure FDA0003255451950000068
Similar to the derivation procedure of [1], it can be derived
Figure FDA0003255451950000069
From (36) can be derived
Figure FDA00032554519500000610
Wherein,
Figure FDA0003255451950000071
is constant and satisfies
Figure FDA0003255451950000072
That is to say that the first and second electrodes,
Figure FDA0003255451950000073
lambda is a bounded normal number, and the actual controllers (32), (33), adaptive laws (34) and (37) are substituted into (36) to obtain
Figure FDA0003255451950000074
Wherein,
Figure FDA0003255451950000075
R*is a bounded normal number;
fifthly, analyzing the stability of the closed loop system
From the above derivation
Figure FDA0003255451950000076
In pair (38)
Figure FDA0003255451950000077
Is scaled, can have
Figure FDA0003255451950000078
Wherein K is a bounded positive scalar quantity,
Figure FDA0003255451950000079
selecting appropriate parameters
Figure FDA00032554519500000710
Can obtain the product
Figure FDA00032554519500000711
Further, in the present invention,
Figure FDA00032554519500000712
by the definition of (40) and Lyapunov function, we can obtain the Lyapunov function V3And all signals in the closed loop system are bounded, according to (15), (16) and (17), it is possible to obtain
Figure FDA00032554519500000713
Is similar to the above analysis, can be obtained in (6)
Figure FDA00032554519500000714
Therefore, the tracking error always remains within the performance bounds.
CN202111058058.8A 2021-09-09 2021-09-09 Output feedback trajectory tracking control method with preset performance and dead zone input constraint for electromechanical system Withdrawn CN113741469A (en)

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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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