CN111459093B - Machine tool spindle precision motion output feedback control method - Google Patents

Machine tool spindle precision motion output feedback control method Download PDF

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CN111459093B
CN111459093B CN202010094778.9A CN202010094778A CN111459093B CN 111459093 B CN111459093 B CN 111459093B CN 202010094778 A CN202010094778 A CN 202010094778A CN 111459093 B CN111459093 B CN 111459093B
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machine tool
tool spindle
parameter
adaptive
sgn
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CN111459093A (en
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杨贵超
洪荣晶
王�华
陈捷
张�浩
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NANJING GONGDA CNC TECHNOLOGY CO LTD
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    • G—PHYSICS
    • G05—CONTROLLING; REGULATING
    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B19/00—Programme-control systems
    • G05B19/02—Programme-control systems electric
    • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
    • G05B19/401—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by control arrangements for measuring, e.g. calibration and initialisation, measuring workpiece for machining purposes

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Abstract

A feedback control method for precision motion output of a machine tool spindle is characterized by comprising the following steps: the method comprises the following specific steps: step (1): establishing a mathematical model of a servo system of a machine tool spindle driving motor; step (2): a parameter self-adaptive law is designed to estimate uncertainty parameters suffered by the system; and (3): designing an adaptive state observer to estimate the speed in the system; and (4): designing a precision motion output feedback controller of a machine tool spindle; and (5): and selecting proper observer bandwidth and diagonal adaptive law matrix and adjusting the value of the parameter to ensure that the position output of the machine tool spindle drive motor servo system accurately tracks the expected position command. The precision motion control method of the machine tool spindle driving motor servo system designed by the invention can ensure that the position output of the motor servo system can track the expected position command asymptotically under the working condition that factors such as measurement noise, strong parameter uncertainty, external interference and the like exist simultaneously and only the angle position measurement value exists.

Description

Machine tool spindle precision motion output feedback control method
Technical Field
The invention relates to a control method, in particular to a feedback control method for precision motion output of a machine tool spindle, and belongs to the field of electromechanical servo control.
Background
Machine tools are mainly used for machining workpieces, and the performance of the machine tool directly determines the final machining performance of the workpiece. At present, a machine tool spindle is mainly driven by a servo motor, and the motor has the advantages of high response speed, convenience in energy acquisition, convenience in maintenance and the like. The motion control of the motor servo system in the fields of industry, engineering and the like is still mainly based on the classical three-loop (current loop, speed loop and position loop) control, however, as the fields are developed towards high performance, the common control method can not meet the system requirements gradually. Therefore, the research of more advanced control methods is urgently required. The motor servo system is inevitably affected by model uncertainty in the operation process, including uncertain influences such as variable load mass/moment inertia, viscous friction, coulomb friction and other friction coefficients which are generally difficult to accurately obtain, external disturbance influences such as sudden external disturbance, cutting force generated when a workpiece is machined and the like, and the system is also affected by measurement noise. The presence of these adverse factors can degrade the desired control performance of the system and even cause the designed closed-loop controller to fail.
At present, aiming at an advanced control strategy considering uncertainty factors of a machine tool spindle driving motor servo system, methods such as adaptive robust control, adaptive sliding mode control, robust adaptive control, active disturbance rejection adaptive control and the like are provided. Typically, an adaptive robust control strategy designs an appropriate online estimation strategy for parameter uncertainty in a system to estimate and feed-forward compensate; and the disturbance such as external interference which possibly occurs is suppressed by improving the nonlinear feedback gain, so that the system performance is improved. The strong non-linear feedback gain often results in design conservatism (i.e., high gain feedback), making it somewhat difficult in engineering applications. It is noted that when disturbance such as external disturbance gradually increases, the designed adaptive robust controller will deteriorate the tracking performance and even cause instability. And the control method can only ensure that the system obtains the bounded tracking performance under the general condition, namely ensure that the tracking error of the system is in a bounded range, and how to obtain the asymptotic tracking with better performance in theory and practical application is worth further research. The self-disturbance-rejection adaptive control strategy estimates unknown parameters and external disturbance by respectively combining adaptive control and an extended state observer on uncertain parameters and time-varying external disturbance in a system, and performs feedforward compensation when designing a controller, thereby resisting the influence of disturbance to a certain extent. However, the above-mentioned control strategies can only ensure that the system achieves bounded tracking performance and that disturbances such as noise in the system are not dealt with. In addition, it is difficult and important to develop a controller that can be designed to work with only output signal measurements.
In summary, the control technology of the existing machine tool spindle driving motor servo system has the following disadvantages:
1. under the output feedback working condition, the asymptotic tracking performance with better performance is difficult to obtain. For practical systems, due to factors such as economic cost, installation space and weight, it is generally difficult to directly obtain all state measurement values required by designing a controller, and how to design a closed-loop controller with perfect theory and excellent performance under such working conditions is a difficult point of research.
2. The measurement noise disturbance of the system is ignored. In designing a closed loop controller for a servo system of a spindle drive motor of a machine tool, measurement values of system signals are generally utilized, and noise is necessarily introduced into the measurement values. The existence of these measurement noises may cause performance degradation, even instability, etc. of a controller designed by the system, and such adverse factors may cause great influence on the precise movement of the machine tool spindle, even seriously affect the machining precision of the workpiece. How to design the controller while suppressing the influence of measurement noise is worth further intensive study.
Disclosure of Invention
The invention provides a precision motion output feedback control method for a machine tool spindle, aiming at solving the problems that the asymptotic tracking performance with better performance is difficult to obtain under the output feedback working condition in the control of the servo system of the driving motor of the existing machine tool spindle, and the adverse effects brought by measurement noise are neglected.
The technical scheme adopted by the invention for solving the problems is as follows: the method comprises the following specific steps:
a feedback control method for precision motion output of a machine tool spindle specifically comprises the following steps:
step (1): establishing a mathematical model of a servo system of a machine tool spindle driving motor;
step (2): a parameter self-adaptive law is designed to estimate uncertainty parameters suffered by the system;
and (3): designing an adaptive state observer to estimate the speed in the system;
and (4): designing a precision motion output feedback controller of a machine tool spindle;
and (5): and selecting proper observer bandwidth and diagonal adaptive law matrix and adjusting the value of the parameter to ensure that the position output of the machine tool spindle drive motor servo system accurately tracks the expected position command.
The step (1): the method for establishing the mathematical model of the servo system of the spindle driving motor of the machine tool comprises the following steps: according to Newton's second law and simplifying the electrical dynamics of the motor as a proportional link, the motion equation of the machine tool spindle driving motor servo system is as follows:
Figure GDA0002543580950000031
j in the formula (1) is load moment of inertia; y is the angular displacement of the load; k is a radical ofuThe moment amplification factor; u is the control input voltage of the system;
Figure GDA0002543580950000034
and
Figure GDA0002543580950000035
different frictional forces experienced by the load, wherein1、γ2、γ3All are coefficients reflecting different friction shapes; p is a radical of1、p2To reflect the magnitude of the different friction levels; b is a viscous friction coefficient; f (t) is time-varying external interference generated by the workpiece in the machining process;
for the convenience of subsequent controller design, the state vector x is selected as
Figure GDA0002543580950000037
Wherein x1、x2Defining known parameters for angular displacement and angular velocity of the load, respectively
Figure GDA0002543580950000036
Defining a set of unknown parameters theta ═ theta1,θ2,θ3]TWhere the parameter θ is unknown1＝p1/J、θ2＝p2/J、θ3B/J, the kinematic equation of the motor servo system can be converted into the following equation of state form:
Figure GDA0002543580950000032
f in formula (2)d(t)＝f(t)/J，ψ(x1,x2)＝[-Fm1(x2),-Fm2(x2),-x2]T；
A control target: under the working condition that a machine tool spindle driving motor servo system suffers from parameter uncertainty and time-varying disturbance at the same time, an output feedback controller is designed to enable the system output y to be x1Tracking the desired smoothing instruction y as accurately as possibled＝x1d；
Assume that 1: command signal x that the system expects to track1d(t) is three-order continuously derivable and the system expects that the position command, the velocity command and the acceleration command are bounded;
assume 2: the system unknown parameter set theta satisfies:
Figure GDA0002543580950000033
Figure GDA0002543580950000041
theta in the formula (3)max＝[θ1max,θ2max,θ3max]TAnd thetamin＝[θ1min,θ2min,θ3min]TKnown upper and lower bounds for theta, respectively. Furthermore, fd(t)∈C3And satisfies alpha ═ supt≥0L f (t), where α is a known normal number.
The step (2): the method for estimating the uncertainty parameters suffered by the system by the design parameter adaptive law comprises the following steps:
defining a discontinuous projection function
Figure GDA0002543580950000042
Comprises the following steps:
Figure GDA0002543580950000043
formula (4) wherein p is 1,2,3 ·pIs the p-th element of the vector-for operations between two vectors "<" or ">" are operations between corresponding elements in the vector;
the parameter self-adaptation law is designed as follows:
Figure GDA0002543580950000044
the step (3): designing an adaptive state observer to estimate the speed in the system comprises the following steps:
based on the system model (2), the adaptive state observer can be designed as follows:
Figure GDA0002543580950000045
in the formula (6) < i >0Is an adjustable normal number, which can be regarded as the bandwidth of the adaptive state observer;
Figure GDA0002543580950000051
ε＝–ξ1/2+3ξ2a/2 wherein
Figure GDA0002543580950000055
And
Figure GDA0002543580950000056
since the system velocity signal is not measurable, to obtain sgn (ε), we introduce the function ω (t) as:
Figure GDA0002543580950000052
thus, we can obtain sgn (ε) by the following equation:
sgn(ε)＝sgn[ω(t)-ω(t-τs)] (8)
in equation (8) < tau >sFor time delay, we can put τ into practical implementationsSelecting sampling time;
due to the fact that
Figure GDA0002543580950000053
(9)
Therefore, only the actual measured value of the position needs to be used in the calculation of sgn (ε).
The step (4): the design of the feedback controller for the precision motion output of the machine tool spindle comprises the following steps:
definition of z1＝x1-x1dFor the tracking error of the system, and defines z2And r is:
Figure GDA0002543580950000054
Figure GDA0002543580950000061
k in formula (10)1And k2Is an adjustable positive gain and
Figure GDA0002543580950000062
based on equations (2) and (10), the expansion of r can be given as:
Figure GDA0002543580950000063
based on equation (11), the actual controller input u can be designed as:
Figure GDA0002543580950000064
k in formula (12)sδ is an adjustable gain and ks＞0、δ＞0，uaFor model compensation terms based on parameter adaptation, us1For the linear robust term, us2Is a non-linear integral robust term. Notably, the controller (12) is designed to perform feed forward compensation based on the desired command, suppressing the effects of measurement noise to some extent. In addition, sgn (z) can be obtained by a method similar to that for sgn (ε)2)；
The adaptive function is designed as
Figure GDA0002543580950000068
Where xi is ═ xi1,ξ2]T，
Figure GDA0002543580950000065
It is noted that the adaptive function τ contains the expression ξ comprising a position measurementTPoFo＝–ξ1/2+3ξ2To obtain
Figure GDA0002543580950000066
Integration on both sides of equation (5) yields:
Figure GDA0002543580950000067
Figure GDA0002543580950000074
in the formula (13)
Figure GDA0002543580950000071
Can be expressed as:
Figure GDA0002543580950000072
as can be seen from the formula (14)
Figure GDA0002543580950000073
Can be obtained by known measurements, when sgn [ (r τ)i]The calculation can be performed by the method of obtaining sgn (. epsilon.).
The step (5) comprises the following steps:
choosing the appropriate observer Bandwidth lo(lo> 0) and a diagonal adaptation law matrix Γ (Γ > 0) and adjust a parameter k1(k1＞0)、k2(k2＞0)、ks(ksThe values of more than 0, alpha (alpha more than 0) and delta (delta more than 0) ensure the position output x of the machine tool spindle driving motor servo system1Accurately tracking desired position instruction x1d。
The invention has the beneficial effects that: the method selects a machine tool spindle driving motor (direct current) servo system as a research object, takes the position output of the servo system capable of accurately tracking an expected position command under the common influence of factors such as angular position measurement values, measurement noise, parameter uncertainty, time-varying external interference and the like as a control target, designs a self-adaptive state observer for estimating unknown state measurement values aiming at the working conditions of the angular position measurement values, and further designs an output feedback controller; a compensation technology based on an expected instruction is adopted for noise suppression aiming at the measured noise, so that the system is ensured to be influenced by the noise as little as possible; estimating the parameter uncertainty through a design self-adaptive law and performing feed-forward compensation; suppressing external interference by designing a nonlinear integral robust term; the method for controlling the precise motion of the servo system of the machine tool spindle driving motor can ensure that the position output of the motor servo system gradually tracks the expected position instruction under the working conditions that factors such as measurement noise, strong parameter uncertainty, external interference and the like exist simultaneously and only angle position measurement values exist, and is more beneficial to realizing the precise motion of the machine tool spindle in complex working conditions in engineering practice. The simulation result verifies the effectiveness of the test paper.
Drawings
FIG. 1 is a schematic diagram of the closed loop control of the position of the motor servo system of the present invention;
FIG. 2 is a schematic diagram and a flow chart of the control principle of the precise motion of a servo system of a spindle driving motor of a machine tool;
FIG. 3 is a graph of the tracking performance of the system and the tracking error over time under the control of the controller designed by the present invention;
FIG. 4 is a graph of parameter estimation performance over time for a controller designed in accordance with the present invention;
FIG. 5 is a plot of angular position estimation performance over time for an adaptive state observer designed in accordance with the present invention;
FIG. 6 is a plot of angular velocity estimation performance of an adaptive state observer over time as contemplated by the present invention;
fig. 7 is a graph of the control input voltage versus time for a controller designed in accordance with the present invention.
Detailed Description
The present embodiment is described with reference to fig. 1 to 7, and a feedback control method for precision motion output of a machine tool spindle includes the following specific steps:
step (1): establishing a mathematical model of a machine tool spindle driving motor servo system, taking a direct current motor as an example, simplifying the electrical dynamics of the motor according to a Newton second law as a proportion link, wherein the motion equation of the machine tool spindle driving motor servo system is as follows:
Figure GDA0002543580950000081
j in the formula (1) is load moment of inertia; y is the angular displacement of the load; k is a radical ofuThe moment amplification factor; u is the control input voltage of the system;
Figure GDA0002543580950000082
and
Figure GDA0002543580950000083
different frictional forces experienced by the load, wherein1、γ2、γ3All are coefficients reflecting different friction shapes; p is a radical of1、p2To reflect the magnitude of the different friction levels; b is a viscous friction coefficient; f (t) is the time-varying external disturbance of the workpiece during machining.
For the convenience of subsequent controller design, the state vector x is selected as
Figure GDA0002543580950000084
Wherein x1、x2The angular displacement and the angular velocity of the load respectively, the kinematic equation of the motor servo system can be converted into the following equation of state form:
Figure GDA0002543580950000085
known parameters in equation (2)
Figure GDA0002543580950000091
Unknown parameter theta1＝p1/J、θ2＝p2/J、θ3＝B/J；fd(t)＝f(t)/J。
Defining a set of unknown parameters theta ═ theta1,θ2,θ3]TThen equation (2) can be further converted to:
Figure GDA0002543580950000092
psi (x) in equation (3)1,x2)＝[-Fm1(x2),-Fm2(x2),-x2]T。
A control target: under the working condition that a machine tool spindle driving motor servo system suffers from parameter uncertainty and time-varying disturbance at the same time, an output feedback controller is designed to enable the system output y to be x1Tracking the desired smoothing instruction y as accurately as possibled＝x1d。
Assume that 1: command signal x that the system expects to track1d(t) is continuously derivable over the third order and the system expects that the position command, the velocity command and the acceleration command are bounded.
Assume 2: the system unknown parameter set theta satisfies:
Figure GDA0002543580950000093
theta in the formula (4)max＝[θ1max,θ2max,θ3max]TAnd thetamin＝[θ1min,θ2min,θ3min]TKnown upper and lower bounds for theta, respectively. Furthermore, fd(t)∈C3And satisfies alpha ═ supt≥0L f (t), where α is a known normal number.
Further, the invention claims
Figure GDA0002543580950000094
An estimated value of the representative value,
Figure GDA0002543580950000097
error of estimation of expression ·minAnd · amaxRespectively, the minimum and maximum values of · and sgn (·) represents a standard sign function with respect to ·.
Step (2): the design parameter adaptation law estimates the uncertainty parameters suffered by the system.
Defining a discontinuous projection function
Figure GDA0002543580950000096
Comprises the following steps:
Figure GDA0002543580950000101
formula (5) wherein p is 1,2,3 ·pFor the p-th element of the vector, for an operation between two vectors "<" or ">" is an operation between corresponding elements in the vector.
The parameter self-adaptation law is designed as follows:
Figure GDA0002543580950000102
in the formula (6)
Figure GDA0002543580950000103
Γ is a constant diagonal positive definite adaptive law matrix and τ is an adaptive function. For any adaptive function τ, the use of the projection function (6) can ensure that:
Figure GDA0002543580950000104
and (3): an adaptive state observer is designed to estimate the velocity in the system.
Based on the system model (3), the adaptive state observer can be designed as follows:
Figure GDA0002543580950000111
in formula (8) < i >0Is an adjustable normal number, which can be regarded as the bandwidth of the adaptive state observer;
Figure GDA0002543580950000112
ε＝–ξ1/2+3ξ2a/2 wherein
Figure GDA0002543580950000113
And
Figure GDA0002543580950000114
since the system velocity signal is not measurable, to obtain sgn (ε), we introduce the function ω (t) as:
Figure GDA0002543580950000115
thus, we can obtain sgn (ε) by the following equation:
sgn(ε)＝sgn[ω(t)-ω(t-τs)] (10)
in equation (10) < tau >sFor time delay, we can put τ into practical implementationsAnd selecting sampling time.
Due to the fact that
Figure GDA0002543580950000116
Therefore, only the actual measured value of the position needs to be used in the calculation of sgn (ε).
The dynamic equation of the observer estimation error obtained from the formulas (3) and (8) is:
Figure GDA0002543580950000117
definitions xi ═ xi1,ξ2]TThen equation (12) can be further converted to:
Figure GDA0002543580950000121
in the formula (13)
Figure GDA0002543580950000122
From matrix AoThe definition of (A) is such that it satisfies the Hurwitz criterion, and thus there must be a positive and symmetric matrix
Figure GDA0002543580950000123
So that
Figure GDA0002543580950000128
This is true.
And (4): designing a feedback controller for the precision motion output of a machine tool spindle, which comprises the following specific steps:
definition of z1＝x1-x1dFor the tracking error of the system, and defines z2And r is:
Figure GDA0002543580950000124
k in formula (14)1And k2Is an adjustable positive gain and
Figure GDA0002543580950000125
based on equations (3) and (14), the expansion of r can be given as:
Figure GDA0002543580950000126
based on equation (15), the actual controller input u can be designed as:
Figure GDA0002543580950000127
k in formula (16)sδ is an adjustable gain and ks＞0、δ＞0，uaFor model compensation terms based on parameter adaptation, us1For the linear robust term, us2Is a non-linear integral robust term. Notably, the controller (16) is designed to perform feed forward compensation based on the desired command, suppressing the effects of measurement noise to some extent. In addition, sgn (z) can be obtained by a method similar to that for sgn (ε)2)。
Substituting equation (16) into equation (15) yields:
Figure GDA0002543580950000131
from equations (14) and (17), the derivative of r with respect to time can be found as:
Figure GDA0002543580950000132
the helper function Δ (t) in equation (18) is
Figure GDA0002543580950000133
And is
Figure GDA0002543580950000134
Based on assumptions 1 and 2 and using the median theorem, one can obtain:
Figure GDA0002543580950000135
in the formula (20), z ═ z1,z2]T，e＝[z1,z2,ξ1,ξ2]TAnd ρi(. 1,2,3 represents a positive increasing function,. mu.iAnd i ═ 1,2,3,4 represent some normal numbers.
And (5): analyzing the stability of a closed loop system of the position of a servo system of a spindle driving motor of a machine tool:
before analyzing the stability, a theory is first introduced:
introduction 1: if an auxiliary function L (t) is defined as:
Figure GDA0002543580950000141
in the formula (21), β and η are normal numbers. If the selected control parameters delta, beta and eta meet the following conditions:
Figure GDA0002543580950000142
ζ in equation (22)1＝supt≥0I Delta (t) I and
Figure GDA0002543580950000143
known as normal. The following inequality is always true:
Figure GDA0002543580950000144
theory 1: if the adaptive function is designed as
Figure GDA0002543580950000145
And selecting an appropriate gain k1、k2、ksδ and loThe controller is then designed to ensure that all signals of the closed loop system are bounded and asymptotic estimation and asymptotic tracking performance can be achieved, i.e., ξ → 0 and z when t → ∞1→0。
It is noted that the adaptive function τ contains the expression ξ comprising a position measurementTPoFo＝–ξ1/2+3ξ2To obtain
Figure GDA0002543580950000146
Integration on both sides of equation (6) yields:
Figure GDA0002543580950000147
in the formula (24)
Figure GDA0002543580950000151
Can be expressed as:
Figure GDA0002543580950000152
as can be seen from the formula (25)
Figure GDA0002543580950000153
Can be obtained by known measurements, when sgn [ (r τ)i]The calculation can be performed by the method of obtaining sgn (. epsilon.).
Based on the lemma 1, we define a non-negative auxiliary function P (t) as
Figure GDA0002543580950000154
According to the stability analysis of a system in a control theory, selecting a Lyapunov candidate function V (χ, t) as follows:
Figure GDA0002543580950000155
equation (27) wherein χ (t) represents
Figure GDA0002543580950000156
Wherein
Figure GDA0002543580950000157
The derivation of equation (27) can be:
Figure GDA0002543580950000158
in the formula (28)
Figure GDA0002543580950000159
And satisfy
Figure GDA00025435809500001510
Where ρ is4(. and ρ)5(. cndot.) is a positive, non-decreasing function.
With some simplification, we can get:
Figure GDA00025435809500001511
phi ═ P in formula (29)oFo‖/lo，
Figure GDA0002543580950000161
Satisfied by design control parameters
Figure GDA0002543580950000162
Can obtain
Figure GDA0002543580950000163
(wherein c is a normal number). Therefore, the results in theory 1 can be obtained.
And (6): choosing the appropriate observer Bandwidth loAnd a diagonal adaptive law matrix Γ and adjust parameter k1、k2、ksAlpha and delta values ensure the position output x of a machine tool spindle drive motor servo system1Accurately tracking desired position instruction x1d. The parameters are selected in the relevant portions of the detailed description.
Example (b):
the motor servo system parameters are as follows: load moment of inertia J is 0.02kg m2(ii) a Coefficient of moment amplification ku＝5N·m/V；B＝1.5N·m·s/rad，p1＝0.2N·m·s/rad，p2＝0.1N·m·s/rad，γ1＝705，γ2＝16，γ31.6; added time-varying external interference fd(t) ═ 2sin (π t) N · m; the position command that the system expects to track is a curve x1d(t)＝0.2sin(πt)[1-exp(-0.01t3)]rad。
Designing control parameters:
the control parameter is selected as k1＝200,k2＝100,ks＝200，l0＝400，α1＝1，δ＝5，θmin＝[1,1,10]T,θmax＝[50,45,150]T，
Figure GDA0002543580950000164
Г＝diag{1.5×10-3,1.8×10-3,2.8×10-2}，。
The controller has the following effects: fig. 3 is a graph of the tracking performance of the system under the action of the controller designed by the invention and the change of the tracking error along with time, and it can be seen that the tracking error shows a gradually decreasing trend, thereby verifying the effectiveness of the accurate motion tracking performance of the designed controller. FIG. 4 is a graph of the parameter estimation performance of the controller designed according to the present invention, which shows that the estimated value of the parameter gradually approaches a certain value and tends to be stable; FIG. 5 is a plot of angular position estimation performance over time for an adaptive state observer designed in accordance with the present invention, from which it can be seen that the angular position estimate closely tracks the actual angular position; fig. 6 is a curve of the angular velocity estimation performance of the adaptive state observer, which changes with time, and it can be seen from the graph that the estimation error of the angular velocity gradually approaches 0, thereby verifying the high-precision estimation performance of the angular velocity; FIG. 7 is a control input voltage versus time curve for a controller designed according to the present invention, from which it can be seen that the resulting control input signal is regular and continuously derivable and bounded, which is advantageous for implementation in engineering practice.

Claims (2)

1. A feedback control method for precision motion output of a machine tool spindle is characterized by comprising the following steps: the method comprises the following specific steps:
step (1): establishing a mathematical model of a servo system of a machine tool spindle driving motor;
step (2): a parameter self-adaptive law is designed to estimate uncertainty parameters suffered by the system;
and (3): designing an adaptive state observer to estimate the speed in the system;
and (4): designing a precision motion output feedback controller of a machine tool spindle;
and (5): selecting proper observer bandwidth and diagonal adaptive law matrix and adjusting parameter values to ensure that the position output of a machine tool spindle drive motor servo system accurately tracks an expected position instruction;
the step (1): the method for establishing the mathematical model of the servo system of the spindle driving motor of the machine tool comprises the following steps:
according to Newton's second law and simplifying the electrical dynamics of the motor as a proportional link, the motion equation of the machine tool spindle driving motor servo system is as follows:
Figure FDA0003314108520000011
j in the formula (1) is load moment of inertia; y is the angular displacement of the load; k is a radical ofuThe moment amplification factor; u is the control input voltage of the system;
Figure FDA0003314108520000012
and
Figure FDA0003314108520000013
different frictional forces experienced by the load, wherein1、γ2、γ3All are coefficients reflecting different friction shapes; p is a radical of1、p2To reflect the magnitude of the different friction levels; b is a viscous friction coefficient; f (t) is time-varying external interference generated by the workpiece in the machining process;
for the convenience of subsequent controller design, the state vector x is selected as
Figure FDA0003314108520000014
Wherein x1、x2Defining known parameters for angular displacement and angular velocity of the load, respectively
Figure FDA0003314108520000015
Defining a set of unknown parameters theta ═ theta1,θ2,θ3]TWhere the parameter θ is unknown1＝p1/J、θ2＝p2/J、θ3B/J, the kinematic equation of the motor servo system is converted into the following state equationThe program form:
Figure FDA0003314108520000016
f in formula (2)d(t)＝f(t)/J，ψ(x1,x2)＝[-Fm1(x2),-Fm2(x2),-x2]T；
A control target: under the working condition that a machine tool spindle driving motor servo system suffers from parameter uncertainty and time-varying disturbance at the same time, an output feedback controller is designed to enable the system output y to be x1Tracking the desired smoothing instruction y as accurately as possibled＝x1d；
Assume that 1: command signal x that the system expects to track1d(t) is three-order continuously derivable and the system expects that the position command, the velocity command and the acceleration command are bounded;
assume 2: the system unknown parameter set theta satisfies:
Figure FDA0003314108520000021
theta in the formula (3)max＝[θ1max,θ2max,θ3max]TAnd thetamin＝[θ1min,θ2min,θ3min]TA known upper and lower bound of θ, respectively; furthermore, fd(t)∈C3And satisfies alpha ═ supt≥0L f (t), where α is a known normal number;
the step (2): the method for estimating the uncertainty parameters suffered by the system by the design parameter adaptive law comprises the following steps:
defining a discontinuous projection function
Figure FDA0003314108520000022
Comprises the following steps:
Figure FDA0003314108520000023
formula (4) wherein p is 1,2,3 ·pThe p-th element being the vector; for an operation between two vectors < or > is an operation between corresponding elements in the vectors;
the parameter self-adaptation law is designed as follows:
Figure FDA0003314108520000024
the step (3): designing an adaptive state observer to estimate the speed in the system comprises the following steps:
based on the system model (2), the adaptive state observer can be designed as follows:
Figure FDA0003314108520000025
in the formula (6) < i >0Is an adjustable normal number, which is considered as the bandwidth of the adaptive state observer;
Figure FDA0003314108520000026
ε＝–ξ1/2+3ξ2a/2 wherein
Figure FDA0003314108520000027
And
Figure FDA0003314108520000028
since the system velocity signal is not measurable, to obtain sgn (ε), we introduce the function ω (t) as:
Figure FDA0003314108520000029
therefore, sgn (∈) is obtained by the following equation:
sgn(ε)＝sgn[ω(t)-ω(t-τs)] (8)
in equation (8) < tau >sFor time delay, τ is used in practical implementationsSelecting sampling time;
due to the fact that
Figure FDA0003314108520000031
Therefore, only the actual measurement of the position need be utilized in calculating sgn (ε);
the step (4): the design of the feedback controller for the precision motion output of the machine tool spindle comprises the following steps:
definition of z1＝x1-x1dFor the tracking error of the system, and defines z2And r is:
Figure FDA0003314108520000032
k in formula (10)1And k2Is an adjustable positive gain and
Figure FDA0003314108520000033
based on equations (2) and (10), the expansion of r can be given as:
Figure FDA0003314108520000034
based on equation (11), the actual controller input u can be designed as:
Figure FDA0003314108520000035
k in formula (12)sδ is an adjustable gain and ks＞0、δ＞0，uaFor model compensation based on parameter adaptationFree item of compensation, us1For the linear robust term, us2A non-linear integral robust term; notably, the controller (12) is designed to perform feed-forward compensation based on the desired command, suppressing the effect of measurement noise to some extent; further, sgn (z) is obtained by the method for obtaining sgn (. epsilon.)2)；
The adaptive function is designed as
Figure FDA0003314108520000036
Where xi is ═ xi1,ξ2]T，
Figure FDA0003314108520000037
It is noted that the adaptive function τ contains the expression ξ comprising a position measurementTPoFo＝–ξ1/2+3ξ2To obtain
Figure FDA0003314108520000041
Integration on both sides of equation (5) yields:
Figure FDA0003314108520000042
in the formula (13)
Figure FDA0003314108520000043
Expressed as:
Figure FDA0003314108520000044
as can be seen from the formula (14)
Figure FDA0003314108520000045
Obtained by known measurements, when sgn [ (T)i]The calculation was performed by the method for obtaining sgn (. epsilon.).
2. The machine tool spindle precision motion output feedback control method according to claim 1, characterized in that: the step (5) comprises the following steps:
choosing the appropriate observer Bandwidth loAnd a diagonal adaptive law matrix Γ and adjust parameter k1、k2、ksAlpha and delta values ensure the position output x of a machine tool spindle drive motor servo system1Accurately tracking desired position instruction x1d(ii) a Wherein lo＞0；Γ＞0；k1＞0；k2＞0；ks＞0；α＞0；δ＞0。
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