CN110580035B - Motion control system fault identification method under sensor saturation constraint - Google Patents

Motion control system fault identification method under sensor saturation constraint Download PDF

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CN110580035B
CN110580035B CN201910821309.XA CN201910821309A CN110580035B CN 110580035 B CN110580035 B CN 110580035B CN 201910821309 A CN201910821309 A CN 201910821309A CN 110580035 B CN110580035 B CN 110580035B
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matrix
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motion control
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CN110580035A (en
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朱俊威
周巧倩
徐建明
顾曹源
董建伟
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Zhejiang University of Technology ZJUT
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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
    • G05B23/00—Testing or monitoring of control systems or parts thereof
    • G05B23/02—Electric testing or monitoring
    • G05B23/0205—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults
    • G05B23/0218—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults
    • G05B23/0243—Electric testing or monitoring by means of a monitoring system capable of detecting and responding to faults characterised by the fault detection method dealing with either existing or incipient faults model based detection method, e.g. first-principles knowledge model
    • 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
    • G05B2219/00—Program-control systems
    • G05B2219/20—Pc systems
    • G05B2219/24—Pc safety
    • G05B2219/24065—Real time diagnostics

Abstract

A method for identifying faults of a motion control system under saturation constraint of a sensor is used for modeling the motion control system; obtaining a system state space equation and discretizing; constructing an intermediate observer; respectively solving the gain of the intermediate observer considering the saturation constraint of the sensor and the gain of the nominal intermediate observer through a matrix inequality; the fault is estimated using an intermediate observer. The effectiveness of the method is verified by a comparison experiment with a nominal intermediate observer. The method of the invention considers the condition that the actuator of the motion control system has faults, and the invention is not limited to the example, and the estimation effect can meet the requirements of the precision and the real-time performance of the practical application.

Description

Motion control system fault identification method under sensor saturation constraint
Technical Field
The invention belongs to the technical field of industry, and particularly provides a solution of an intermediate observer considering sensor saturation constraint aiming at the problem of motion control system fault identification under the sensor saturation constraint.
Background
With the acceleration of industrial automation processes, motion control systems play an increasingly important role. Due to physical or technical limitations, the sensor cannot provide a signal of infinite amplitude, so that the phenomenon of "sensor saturation" occurs. Under the constraint of sensor saturation, the received output signal is incomplete, so that an observer is difficult to accurately estimate faults, and the application of the traditional estimator design scheme is severely limited.
In the case of sensor saturation, it is clearly important to be able to identify faults. In the prior art, there are a robust observer, a sliding-mode observer, a nominal intermediate observer. These observers are all somewhat robust. However, although the nominal intermediate observer has certain robustness, since the constraint of sensor saturation is not considered, the ideal effect cannot be obtained only by adjusting parameters. Based on H∞The fault estimation method of the performance index ensures the fault estimation performance by inhibiting the influence of disturbance on the output estimation error, however, if the signal loss caused by the saturation constraint of the sensor is too large, the estimation error is correspondingly amplified, so that an ideal estimation effect cannot be obtained. The sliding-mode observer requiring a faultA priori information such as the upper bound of the fault derivative, the upper bound of the fault itself, which however cannot be obtained in practical situations.
Disclosure of Invention
Based on the problems, the invention provides a method for identifying the fault of the motion control system under the saturation constraint of the sensor, which is biased to engineering and is more suitable for actual industrial conditions. Specifically, it constructs an intermediate observer that considers the saturation constraints of the sensors to estimate both state and fault simultaneously by introducing an intermediate variable and taking the fan-bounded condition into the framework. The effectiveness and superiority of the method are verified by comparison experiments with a nominal intermediate observer.
The present invention provides the following solutions to solve the above technical problems:
a method for identifying faults of a motion control system under the saturation constraint of a sensor comprises the following steps:
step 1), determining a transfer function of a motion control system;
through system identification, the transfer function of the motion control system is determined as shown in the formula (1):
Figure RE-GDA0002264158200000011
where G(s) is the transfer function of the motion control system, s is a variable of the transfer function, K, TsIs the identified parameter;
step 2), establishing a state space equation of the motion control system and discretizing, wherein the process is as follows:
2.1) converting the transfer function into a state space equation and discretizing the state space equation, and considering the condition that an actuator fault exists in the system:
Figure RE-GDA0002264158200000021
wherein, A is the state matrix of the system, B is the input matrix, x represents the state quantity of the system, k represents the time k, y represents the system outputAmount, u is the system input, auIndicating actuator failure, EaRepresenting a fault gain matrix, and C is an output matrix of the system;
2.2) define the saturation function:
σ(v)＝sign(v)min{1，|v|} (3)
2.3) estimator side received signal:
s(k)＝σ(Cx(k)) (4)
2.4) System rewrite of sensor saturation plus actuator failure as:
Figure RE-GDA0002264158200000022
step 3), constructing an intermediate observer, wherein the process is as follows:
3.1) introduction of intermediate variables
τ(k)＝au(k)-wE′ax(k) (6)
Wherein, the superscript "'" represents the transposition of the matrix, τ represents the intermediate variable, and w is the tuning parameter;
3.2) designing an intermediate observer based on the intermediate variables as shown in (7):
Figure RE-GDA0002264158200000023
wherein the content of the first and second substances,
Figure RE-GDA0002264158200000024
an estimate of the system state quantity x is represented,
Figure RE-GDA0002264158200000025
an estimated value of the intermediate variable tau is represented,
Figure RE-GDA0002264158200000026
indicating a failure to an actuator auL represents the intermediate observer gain to be designed;
step 4), solving the gain of the intermediate observer considering the saturation constraint of the sensor by using a matrix inequality, wherein the process is as follows:
4.1) constructing a matrix as shown in the formula (8):
Figure RE-GDA0002264158200000027
Figure RE-GDA0002264158200000031
∑11＝Aa′P1Aa-ε1C′AC-P1
∑12＝Aa′P1Ab+C′H′Ba
∑13＝Aa′P1B+C′H′B
∑22＝A′bP1Ab+B′aP2Ba-C′H′Ba-B′aHC+C′bP3Cb+εC′bCb-P2
∑23＝A′bP1B+B′aP2B-C′H′B+CbP3Ca
∑33＝B′P1B+B′P2B+C′aP3Ca+εC′aCa-P3
Aa＝A-Bk，Ab＝Bk+wBE′a
wherein, represents a symmetric element, P1、P2Representing the positive definite matrix to be solved, H representing the matrix to be solved, P3Representing the scalar to be solved, I representing the unit matrix, n1、∏2、∑11、∑12、∑13、∑22、∑23、∑33Representing the intermediate matrix, w being tuning parameters, ε1For a given scalar;
4.2) solving the matrix inequality pi < 0 to obtain P1、P2、P3H, the intermediate observer gain L is as shown in equation (10):
L＝P2 -1H (10)
wherein the superscript "-1" represents the inverse of the matrix, so that an accurate estimation of the actuator failure is achieved by the intermediate observer (7).
Step 5), solving the nominal intermediate observer gain by a matrix inequality, wherein the process is as follows:
5.1) constructing a matrix as shown in the formula (11):
Figure RE-GDA0002264158200000032
∑11＝A′cP1Ac-A′cHC-C′H′Ac-C′bP2Cb+εC′bCb-P1
∑12＝A′cP1Ea-C′H′Ea+C′bP2Ca
∑22＝E′aP1Ea+C′aP2Ca+εC′aCa-P2
Ac＝A+wEaE′a，Ca＝I-wE′aEa，Cb＝wE′a(I-wEaE′a-A)
wherein, represents a symmetric element, P1Representing the positive definite matrix to be solved, H representing the matrix to be solved, P2Representing the scalar to be solved, I representing the unit array, sigma11、∑12、∑22Representing a middle matrix, w is a tuning parameter, and epsilon is a given scalar;
5.2) solving the inequality pi of the matrix to obtain P1、P2H, the intermediate observer gain L is as shown in equation (12):
L＝P1 -1H (12)
wherein the superscript "-1" represents the inverse of the matrix, so that the estimation of the actuator failure is effected by the intermediate observer (7).
The invention relates to a method for identifying faults of a motion control system under the saturation constraint of a sensor, which constructs an intermediate observer considering the saturation constraint of the sensor by introducing an intermediate variable to simultaneously estimate states and faults. Meanwhile, a comparison experiment is carried out with a nominal middle observer, and the effectiveness and superiority of the method are verified.
Compared with the prior nominal intermediate observer technology, the invention has the beneficial effects that: under the influence of sensor saturation, the fault identification effect is good, and the performance can be continuously improved by adjusting specific adjusting parameters.
Drawings
FIG. 1a illustrates actuator failure a under an intermediate observer considering sensor saturation constraintsuThe estimated effect map of (2);
FIG. 1b illustrates a nominal middle observer vs. actuator failure auThe estimated effect map of (2);
FIG. 2a is a graph of the effect of estimation on state 1 in an intermediate observer taking into account sensor saturation constraints;
FIG. 2b is a diagram of the estimated effect on state 1 under a nominal intermediate observer;
FIG. 3a is a graph of the estimated effect on State 2 under an intermediate observer taking into account sensor saturation constraints;
FIG. 3b is a diagram of the effect of the estimation of state 2 under a nominal intermediate observer;
FIG. 4 is a response curve of output 1 under sensor constraints;
fig. 5 is a response curve of output 2 under sensor constraints.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention clearer, the technical solutions of the present invention are further described below with reference to the accompanying drawings and practical experiments.
Referring to fig. 1a to 5, a method for identifying a fault of a motion control system under a sensor saturation constraint includes the following steps:
1) determining a motion control system transfer function;
2) establishing a state space equation of the motion control system and discretizing;
3) constructing an intermediate observer;
4) solving the gain of an intermediate observer considering the saturation constraint of the sensor through a matrix inequality;
5) the nominal intermediate observer gain is solved by a matrix inequality.
Further, in the step 1), determining a transfer function of the motion control system:
through system identification, the transfer function of the motion control system is determined as shown in the formula (1):
Figure RE-GDA0002264158200000041
where g(s) is the transfer function of the motion control system, s is a variable of the transfer function, K-0.08373, Ts0.02433 is the identified parameter.
Further, in the step 2), a state space equation of the motion control system is established and discretized, and the process is as follows:
2.1) converting the transfer function into a state space equation and discretizing the state space equation, and considering the conditions that an actuator fault and a sensor are saturated in the system:
Figure RE-GDA0002264158200000051
wherein the state matrix
Figure RE-GDA0002264158200000052
Input matrix
Figure RE-GDA0002264158200000053
Output matrix
Figure RE-GDA0002264158200000054
x represents the system stateState quantity, k represents k time, u is system input, actuator fault au50sin (0.2t) +3, fault gain matrix
Figure RE-GDA0002264158200000055
Further, in the step 3), an intermediate observer is constructed, and the process is as follows:
3.1) defining intermediate variables as shown in equation (3):
τ(k)＝au(k)-wE′ax(k) (3)
where the superscript "'" denotes the transpose of the matrix, τ denotes the intermediate variable, auIndicating actuator failure, EaIndicating fault gain, w is a tuning parameter;
3.2) based on the intermediate variables, designing an intermediate observer as shown in equation (4):
Figure RE-GDA0002264158200000056
wherein the content of the first and second substances,
Figure RE-GDA0002264158200000057
an estimate of the system state quantity x is represented,
Figure RE-GDA0002264158200000058
an estimated value of the intermediate variable tau is represented,
Figure RE-GDA0002264158200000059
indicating a failure to an actuator auL represents the intermediate observer gain to be designed;
step 4), solving the gain of the intermediate observer considering the saturation constraint of the sensor by using a matrix inequality, wherein the process is as follows:
4.1) constructing a matrix as shown in the formula (5):
Figure RE-GDA00022641582000000510
Figure RE-GDA00022641582000000511
∑11＝Aa′P1Aa-ε1C′AC-P1
∑12＝Aa′P1Ab+C′H′Ba
∑13＝Aa′P1B+C′H′B
∑22＝A′bP1Ab+B′aP2Ba-C′H′Ba-B′aHC+C′bP3Cb+εC′bCb-P2
∑23＝A′bP1B+B′aP2B-C′H′B+CbP3Ca
∑33＝B′P1B+B′P2B+C′aP3Ca+εC′aCa-P3
Aa＝A-Bk，Ab＝Bk+wBE′a
wherein, represents a symmetric element, P1、P2Representing the positive definite matrix to be solved, H representing the matrix to be solved, P3Representing the scalar to be solved, I representing the unit matrix, n1、∏2、∑11、∑12、∑13、∑22、∑23、∑33Representing the intermediate matrix, w being tuning parameters, ε1For a given scalar;
solving the matrix inequality pi < 0, and taking w as 1000 to obtain
Figure RE-GDA0002264158200000061
Figure RE-GDA0002264158200000062
The intermediate observer gain L is shown as equation (7):
L＝P2 -1H (7)
obtaining the gain of the intermediate observer
Figure RE-GDA0002264158200000063
Thus, the intermediate observer (5) can accurately estimate the actuator fault.
Step 5), solving the nominal intermediate observer gain by a matrix inequality, wherein the process is as follows:
5.1) constructing a matrix as shown in the formula (8):
Figure RE-GDA0002264158200000064
∑11＝Ac′P1Ac-Ac′HC-C′bP2Cb+εC′bCb-P1
∑12＝A′cP2Ea-C′H′Ea+C′bP2Ca
∑22＝E′aP1Ea+C′aP2Ca+εC′aCa-P2
Ac＝A+wEaE′a，Ca＝I-wE′aEa，Cb＝wE′a(I-wEaE′a-A)
wherein, denotes symmetric elements, P1 denotes positive definite matrix to be solved, H denotes matrix to be solved, P denotes2Representing the scalar to be solved, I representing the unit array, sigma11、∑12、∑22Representing a middle matrix, w is a tuning parameter, and epsilon is a given scalar;
4.2) solving the matrix inequality pi < 0, and taking w as 50 to obtain
Figure RE-GDA0002264158200000071
Figure RE-GDA0002264158200000072
The intermediate observer gain L is shown as equation (9):
L＝P1 -1H (9)
obtaining the gain of the intermediate observer
Figure RE-GDA0002264158200000073
Thus, the intermediate observer (4) can estimate the actuator fault.
The experimental result shows that the method provided by the invention can accurately estimate the fault of the actuator in real time under the condition that the sensor is saturated, and the operation result can meet the requirements of precision and real-time performance of practical application.
The embodiments of the present invention have been described and illustrated in detail above with reference to the accompanying drawings, but are not limited thereto. Many variations and modifications are possible which remain within the knowledge of a person skilled in the art, given the concept underlying the invention.

Claims (1)

1. A method for identifying faults of a motion control system under the saturation constraint of a sensor is characterized by comprising the following steps:
step 1), determining a transfer function of a motion control system;
through system identification, the transfer function of the motion control system is determined as shown in the formula (1):
Figure FDA0002786906450000011
where G(s) is the transfer function of the motion control system, s is a variable of the transfer function, K, TsIs the identified parameter;
step 2), establishing a state space equation of the motion control system and discretizing, wherein the process is as follows:
2.1) converting the transfer function into a state space equation and discretizing the state space equation, and considering the condition that an actuator fault exists in the system:
Figure FDA0002786906450000012
wherein, A is the state matrix of the system, B is the input matrix, x represents the state quantity of the system, k represents the time k, y represents the output quantity of the system, u is the input of the system, auIndicating actuator failure, EaRepresenting a fault gain matrix, and C is an output matrix of the system;
2.2) define the saturation function:
σ(v)＝sign(v)min{1，|v|} (3)
2.3) estimator side received signal:
s(k)＝σ(Cx(k)) (4)
2.4) System rewrite of sensor saturation plus actuator failure as:
Figure FDA0002786906450000013
step 3), constructing an intermediate observer, wherein the process is as follows:
3.1) introduction of intermediate variables
τ(k)＝au(k)-wE′ax(k) (6)
Wherein, the superscript "'" represents the transposition of the matrix, τ represents the intermediate variable, and w is the tuning parameter;
3.2) designing an intermediate observer based on the intermediate variables as shown in (7):
Figure FDA0002786906450000014
wherein the content of the first and second substances,
Figure FDA0002786906450000015
an estimate of the system state quantity x is represented,
Figure FDA0002786906450000016
an estimated value of the intermediate variable tau is represented,
Figure FDA0002786906450000017
indicating a failure to an actuator auL represents the intermediate observer gain to be designed;
step 4), solving the gain of the intermediate observer considering the saturation constraint of the sensor by using a matrix inequality, wherein the process is as follows:
4.1) constructing a matrix as shown in the formula (8):
Figure FDA0002786906450000021
Figure FDA0002786906450000022
∑11＝Aa′P1Aa-ε1C′ΛC-P1
∑12＝Aa′P1Ab+C′H′Ba
∑13＝Aa′P1B+C′H′B
∑22＝A′bP1Ab+B′aP2Ba-C′H′Ba+B′aHC+C′bP3Cb+εC′bCb-P2
∑23＝A′bP1B+B′aP2B-C′H′B+CbP3Ca
∑33＝B′P1B+B′P2B+C′aP3Ca+εC′aCa-P3
Aa＝A-Bk，Ab＝Bk+wBE′a
wherein, represents a symmetric element, P1、P2Representing the positive definite matrix to be solved, H representing the matrix to be solved, P3Representing a scalar to be solved, I representing a unit array, Π1、Π2、∑11、∑12、∑13、∑22、∑23、∑33Representing the intermediate matrix, w being tuning parameters, ε1For a given scalar;
4.2) solving the inequality pi of the matrix to be less than 0 to obtain P1、P2、P3H, the intermediate observer gain L is as shown in equation (10):
L＝P2 -1H (10)
wherein the superscript "-1" represents the inverse of the matrix, so that an accurate estimation of the actuator fault is achieved by the intermediate observer (7);
step 5), solving the nominal intermediate observer gain by a matrix inequality, wherein the process is as follows:
5.1) constructing a matrix as shown in the formula (11):
Figure FDA0002786906450000023
Φ11＝A′cP1Ac-A′cHC-C′H′Ac-C′bP2Cb+εC′bCb-P1
Φ12＝A′cP1Ea-C′H′Ea+C′bP2Ca
Φ22＝E′aP1Ea+C′aP2Ca+εC′aCa-P2
Ac＝A+wEaE′a，Ca＝I-wE′aEa,Cb＝wE′a(I-wEaE′a-A)
wherein, represents a symmetric element, P1Representing the positive definite matrix to be solved, H representing the matrix to be solved, P2Representing a scalar to be solved, I representing a unit matrix, phi11、Φ12、Φ22Representing a middle matrix, w is a tuning parameter, and epsilon is a given scalar;
5.2) solving the inequality pi of the matrix to obtain P1、P2H, the intermediate observer gain L is as shown in equation (12):
L＝P1 -1H (12)
wherein the superscript "-1" represents the inverse of the matrix, so that the estimation of the actuator failure is effected by the intermediate observer (7).
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