CN107121932B - Motor servo system error symbol integral robust self-adaptive control method - Google Patents

Motor servo system error symbol integral robust self-adaptive control method Download PDF

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CN107121932B
CN107121932B CN201710439765.9A CN201710439765A CN107121932B CN 107121932 B CN107121932 B CN 107121932B CN 201710439765 A CN201710439765 A CN 201710439765A CN 107121932 B CN107121932 B CN 107121932B
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胡健
刘雷
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Nanjing University of Science and Technology
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Abstract

The invention discloses a motor servo system error symbol integral robust self-adaptive control method, which comprises the following steps: establishing a motor position servo system model; designing an error symbol integral robust adaptive controller; according to the error sign integral robust self-adaptive controller, the stability of the motor servo system is proved by utilizing the Lyapunov stability theory, and the global gradual stabilization result of the system is obtained by utilizing the Barbalt theorem. The invention provides a parameter adaptive error sign integral robust adaptive anti-interference controller aiming at parameter uncertainty and unknown nonlinear factors in a motor servo system, the parameter adaptive rate can effectively estimate the unknown parameters in the system, an error sign integral robust term is adopted to overcome other uncertain nonlinear factors in the system, and the control precision in the motor servo system is ensured.

Description

Motor servo system error symbol integral robust self-adaptive control method
Technical Field
The invention relates to a motor servo control technology, in particular to a motor servo system error symbol integral robust self-adaptive control method.
Background
The permanent magnet brushless direct current motor has the characteristics of high response speed, high energy utilization rate, small pollution and the like, and is widely applied to the industrial field. With the rapid development of industrial technologies in recent years, higher requirements are also put on the control technology of the dc motor, and how to improve the motion accuracy of the dc single machine has become a main research direction of the dc motor. In a motor servo system, due to different working conditions and some structural limitations, a real model is difficult to completely reflect in modeling of the system, so when a controller is designed, uncertainty of the model plays a very important role, particularly uncertain nonlinearity seriously deteriorates the control performance of the controller, and accordingly low precision, limit ring oscillation and even instability of the system are caused.
For the non-linear problem existing in the system, the influence of the traditional control method on the control precision of the system is difficult to solve. In recent years, with the development of control theory, various control strategies aiming at uncertainty nonlinearity are proposed in succession, such as sliding mode variable structure control, robust adaptive control, adaptive robustness and the like. However, the control strategy controllers are complex in design and not easy to implement in engineering.
Disclosure of Invention
The invention aims to provide an error sign integral robust self-adaptive control method for a motor servo system, which solves the problem of uncertain nonlinearity in a motor position servo system.
The technical scheme for realizing the purpose of the invention is as follows: a motor servo system error symbol integral robust self-adaptive control method comprises the following steps:
step 1, establishing a motor position servo system model;
step 2, designing an error symbol integral robust self-adaptive controller;
and 3, according to the error sign integral robust self-adaptive controller, carrying out stability verification on the motor servo system by utilizing the Lyapunov stability theory, and obtaining a global gradual stabilization result of the system by utilizing the Barbalt theorem.
Compared with the prior art, the invention has the following remarkable advantages:
the invention provides an error sign integral robust adaptive anti-interference controller based on parameter self-adaptation aiming at parameter uncertainty and unknown nonlinear factors (external disturbance) in a motor servo system, the parameter self-adaptation rate can effectively estimate unknown parameters in the system, an error sign integral robust term is adopted to overcome other uncertain nonlinear factors in the system, and the control precision in the motor servo system is ensured; the results of the simulation verify the validity of the proposed control strategy.
Drawings
Fig. 1 is a schematic diagram of a motor servo system.
FIG. 2 is a diagram of an error symbol integral robust adaptive control strategy for a motor servo system according to the present invention.
Fig. 3 is a graph of the tracking process of the system output of the controller to a given output under the influence of disturbance (1).
Fig. 4 is a graph of the tracking error of the system over time under the influence of disturbance (1).
FIG. 5 is a graph of PID control and ARISE control tracking accuracy under the effect of disturbance (2).
Fig. 6 is a graph of the disturbance (2) control input u.
Fig. 7 is a graph of the tracking error of the system over time under the influence of disturbance (3).
Fig. 8 is a graph of the control input v under the influence of disturbance (3).
Fig. 9 is a graph of parameter adaptation under the influence of interference (3).
Detailed Description
With reference to fig. 1 and fig. 2, a method for adaptively controlling the error sign integral robustness of a motor servo system includes the following steps:
step 1, 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 BDA0001319615310000021
wherein y represents an angular displacement, JequRepresenting the inertial load, kuRepresenting the torque constant, u being the system control input, BequRepresents a viscous friction coefficient, dnRepresenting the constant interference experienced by the system,
Figure BDA0001319615310000024
represent other unmodeled disturbances, such as input saturation, external time-varying disturbances, and unmodeled dynamics;
writing equation (1) into a state space form, as follows:
Figure BDA0001319615310000022
wherein
Figure BDA0001319615310000025
x＝[x1,x2]TA state vector representing position and velocity; parameter set theta ═ theta1,θ2,θ3]TWherein theta1＝Jequ/ku，θ2＝Bequ/ku，θ3＝dn/ku，
Figure BDA0001319615310000023
Representing other unmodeled disturbances in the system. Due to the system parameter Jequ，ku，Bequ，dnUnknown, system parameters are uncertain, but general information of the system is known. Furthermore, the uncertainty nonlinearity of the system
Figure BDA0001319615310000031
Nor can it be modeled explicitly, but the unmodeled dynamics and disturbances of the system are always bounded. Thus, the following assumptions always hold:
assume that 1: the parameter theta satisfies:
Figure BDA0001319615310000032
wherein theta ismin＝[θ1min,θ2min,θ3min]T，θmax＝[θ1max,θ2max,θ3max]TAll of which are known, and furthermore theta1min＞0，θ2min＞0，θ3min＞0；
Assume 2: d (x, t) is bounded and differentiable to the first order, i.e.
Figure BDA0001319615310000033
Wherein deltadAre known.
Step 2, designing an error symbol integral robust self-adaptive controller, which comprises the following specific steps:
step 2-1, definition of z1＝x1-x1dFor angular displacement tracking error of the system, x1d is a position command that the system expects to track and that command is continuously differentiable in the second order, according to the first equation in equation (2)
Figure BDA0001319615310000034
Selecting x2For virtually controlling the quantities, let equation
Figure BDA0001319615310000035
Tends to a stable state; let x2eqFor desired values of virtual control, x2eqAnd the true state x2Has an error of z2＝x2-x2eqTo z is to1And (5) obtaining a derivative:
Figure BDA0001319615310000036
designing a virtual control law:
Figure BDA0001319615310000037
in the formula (6), k1If > 0 is adjustable gain, then
Figure BDA0001319615310000038
Due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1And necessarily tends to 0.
Step 2-2, in order to design the controller more conveniently, an auxiliary error signal r (t) is introduced
Figure BDA0001319615310000041
In formula 8, k2The gain is adjustable when the value is more than 0;
according to equations (2), (7) and (8), there is an expansion of r as follows:
Figure BDA0001319615310000042
according to the formulae (2) and (9), the following formulae are given:
Figure BDA0001319615310000043
according to equation (10), the model-based controller is designed as:
Figure BDA0001319615310000044
formula (11)
Figure BDA0001319615310000045
An estimated value representing the value of theta is,
Figure BDA0001319615310000046
as an estimated error
Figure BDA0001319615310000047
β is the system control gain, krA positive feedback gain;
Figure BDA0001319615310000048
is the parameter adaptation rate; gamma > 0 is a tunable positive autorhythmic gain.
From the parameter adaptation rate in the formula (11), r is an unknown quantity, but
Figure BDA0001319615310000049
And its first derivative are known, so the adaptation rate can be integrated to give:
Figure BDA00013196153100000410
calculated by substituting formula (11) for formula (10):
Figure BDA00013196153100000411
the derivation yields:
Figure BDA00013196153100000412
step 3, according to the error sign integral robust self-adaptive controller, utilizing the Lyapunov stability theory to perform stability verification on the motor servo system, and utilizing the Barbalt theorem to obtain the global gradual stabilization result of the system, which is specifically as follows:
introduction 1:
defining auxiliary functions
Figure BDA0001319615310000051
Figure BDA0001319615310000052
z2(0)、
Figure BDA0001319615310000053
Respectively represents z2(t)、
Figure BDA0001319615310000054
Is started.
When in use
Figure BDA0001319615310000055
When it is, then
Figure BDA0001319615310000056
Figure BDA0001319615310000057
P(t)≥0 (19)
Proof of this lemma:
integrating the two sides of the equation (15) simultaneously and applying the equation (7) to obtain:
Figure BDA0001319615310000058
the latter two terms in equation (20) are integrated in steps to yield:
Figure BDA0001319615310000059
thus, it is possible to provide
Figure BDA00013196153100000510
As can be seen from the equation (22), if the value of β satisfies
Figure BDA0001319615310000061
In this case, the following equations (17) and (19) hold.
According to the above reasoning, the lyapunov function is defined as follows:
Figure BDA0001319615310000062
Figure BDA0001319615310000063
for estimated errors, i.e.
Figure BDA0001319615310000064
The Lyapunov stability theory is used for stability verification, and the Barbalt theorem is used for obtaining the global gradual stable result of the system, so that the gain k is adjusted1、k2、krAnd Γ makes the tracking error of the system tend to zero under the condition of infinite time zone. The derivation of equation (23) and substitution of (7), (8), (14), and (16) yields:
Figure BDA0001319615310000065
wherein
Figure BDA0001319615310000066
Defining:
Z＝[z1,z2,r](25)
Figure BDA0001319615310000067
by adjusting the parameter k1、k2、krThe symmetric matrix Λ may be made positive definite,
then there are:
Figure BDA0001319615310000068
in formula (27) < lambda >, < lambda >minAnd (Λ) is the minimum eigenvalue of the symmetric matrix Λ.
The conclusion is drawn from equation (27) and the lyapunov stability theorem: the integral symbol robust self-adaptive controller designed aiming at uncertain nonlinearity existing in a motor servo system can enable the system to achieve the effect of gradual stabilization, and the parameter k of the controller is adjusted1、k2、krThe tracking accuracy can be continuously approached to zero. The schematic diagram of the error sign integral robust adaptive control principle of the motor servo system is shown in fig. 2.
The invention is further illustrated by the following examples and figures.
Examples
Simulation parameters: j. the design is a squareequ＝0.00138kg·m2，Bequ＝0.4N·m/rad，ku2.36N · m/V. Taking a controller parameter k1＝12，k2＝1.5，kr＝1，θ1n＝0.02，θ2nThe nominal value of [ theta ] is selected to be 0.294, which is far from the true value of the parameter, to evaluate the effect of the adaptive control law. PID controller parameter is kp＝90，ki＝70，kd0.3. Given position reference input signal
Figure BDA0001319615310000071
Unit rad.
Disturbance (1) in a regime where only constant disturbances are present and dn0.5N · m. Interference (2) when constant perturbations coexist with other unmodeled ones, and dn＝0.5N·m，
Figure BDA0001319615310000072
Interference (3) constant disturbance and other unmodeled disturbances coexist, and in the case of limited input, dn＝0.5N·m，
Figure BDA0001319615310000073
v＝u·0.8。
The control law effects are shown in fig. 3-9:
fig. 3 is a graph of the tracking process of the system output of the controller to a given output under the influence of disturbance (1). Fig. 4 is a graph of the tracking error of the system over time under the influence of disturbance (1). FIG. 5 is a graph of PID control and ARISE control tracking accuracy under the effect of disturbance (2). Fig. 6 is a graph of the disturbance (2) control input u. Fig. 7 is a graph of the tracking error of the system over time under the influence of disturbance (3). Fig. 8 is a graph of the control input v under the influence of disturbance (3). Fig. 9 is a graph of parameter adaptation under the influence of interference (3).
Therefore, the interference value can be accurately estimated by the algorithm provided by the invention under the simulation environment, and compared with the traditional PID control, the controller designed by the invention can greatly improve the parameter uncertainty and the control precision of a large interference system. Research results show that under the influence of uncertain nonlinearity and parameter uncertainty, the method provided by the invention can meet performance indexes.

Claims (2)

1. A robust self-adaptive anti-interference control method for an error symbol integral of a motor servo system is characterized by comprising the following steps:
step 1, establishing a motor position servo system model; the method specifically comprises the following steps:
according to Newton's second law, the dynamic model equation of the inertia load of the motor is as follows:
Figure FDA0002449862270000011
wherein y is the angular displacement, JequIs an inertial load, kuIs the torque constant, u is the system control input, BequIs a coefficient of viscous friction, dnIn order for the system to be subject to constant interference,
Figure FDA00024498622700000110
for other unmodeled interference;
writing equation (1) into a state space form, as follows:
Figure FDA0002449862270000012
wherein
Figure FDA0002449862270000019
x＝[x1，x2]TA state vector representing position and velocity; parameter set theta ═ theta1，θ2，θ3]TWherein theta1＝Jequ/ku，θ2＝Bequ/ku，θ3＝dn/ku，
Figure FDA0002449862270000013
Representing other unmodeled disturbances in the system; the following assumptions hold:
assume that 1: the parameter theta satisfies:
Figure FDA0002449862270000014
wherein theta ismin＝[θ1min，θ2min，θ3min]T，θmax＝[θ1max，θ2max，θ3max]TAre all known, and furthermore theta1min＞0，θ2min＞0，θ3min＞0；
Assume 2:
Figure FDA0002449862270000015
is bounded and differentiable to the first order, i.e.
Figure FDA0002449862270000016
Wherein deltadThe method comprises the following steps of (1) knowing;
step 2, designing an error symbol integral robust self-adaptive controller; the method specifically comprises the following steps:
step 2-1, definition of z1＝x1-x1dFor angular displacement tracking error of the system, x1dIs a position command that the system expects to track and that is continuously differentiable in the second order, according to the first equation in equation (2)
Figure FDA0002449862270000017
Selecting x2For virtually controlling the quantities, let equation
Figure FDA0002449862270000018
Tends to a stable state; let x2eqFor desired values of virtual control, x2eqAnd the true state x2Has an error of z2＝x2-x2eqTo z is to1And (5) obtaining a derivative:
Figure FDA0002449862270000021
designing a virtual control law:
Figure FDA0002449862270000022
in the formula (6), k1If > 0 is adjustable gain, then
Figure FDA0002449862270000023
Due to z1(s)＝G(s)z2(s) wherein G(s) is 1/(s + k)1) Is a stable transfer function when z2When going to 0, z1Also inevitably tends to 0;
step 2-2, introducing an auxiliary error signal r (t)
Figure FDA00024498622700000212
In the formula k2The gain is adjustable when the value is more than 0;
according to equations (2), (7) and (8), there is an expansion of r as follows:
Figure FDA0002449862270000024
according to the formulae (2) and (9), the following formulae are given:
Figure FDA0002449862270000025
according to equation (10), the model-based controller is designed as:
Figure FDA0002449862270000026
in the formula (11), the reaction mixture is,
Figure FDA0002449862270000027
an estimated value representing the value of theta is,
Figure FDA0002449862270000028
in order to be able to estimate the error,
Figure FDA0002449862270000029
β is the system control gain, krIn order to gain in the positive feedback, the gain,
Figure FDA00024498622700000210
the gamma is a parameter self-adaptive rate, and is adjustable positive self-modulation rhythm gain when the gamma is more than 0;
from the parameter adaptation rate in the formula (11), r is an unknown quantity, but r is an unknown quantity
Figure FDA00024498622700000211
And its first derivative is known, integrating the adaptation rate yields:
Figure FDA0002449862270000031
calculated by substituting formula (11) for formula (10):
Figure FDA0002449862270000032
the derivation yields:
Figure FDA0002449862270000033
and 3, according to the error sign integral robust self-adaptive controller, carrying out stability verification on the motor servo system by utilizing the Lyapunov stability theory, and obtaining a global gradual stabilization result of the system by utilizing the Barbalt theorem.
2. The motor servo system error symbol integral robust adaptive anti-interference control method according to claim 1, wherein the step 3 specifically comprises:
defining auxiliary functions
Figure FDA0002449862270000034
Figure FDA0002449862270000035
z2(0)、
Figure FDA0002449862270000036
Respectively represents z2(t)、
Figure FDA0002449862270000037
An initial value of (1);
is proved to be when
Figure FDA0002449862270000038
When P (t) ≧ 0, the Lyapunov function is thus defined as follows:
Figure FDA0002449862270000039
Figure FDA00024498622700000310
for estimated errors, i.e.
Figure FDA00024498622700000311
The Lyapunov stability theory is used for stability verification, and the Barbalt theorem is used for obtaining the global gradual stable result of the system, so that the gain k is adjusted1、k2、krAnd Γ makes the tracking error of the system tend to zero under the condition of infinite time zone.
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