CN111752262B - Actuator fault observer and fault-tolerant controller integrated design method - Google Patents

Actuator fault observer and fault-tolerant controller integrated design method Download PDF

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CN111752262B
CN111752262B CN202010690407.7A CN202010690407A CN111752262B CN 111752262 B CN111752262 B CN 111752262B CN 202010690407 A CN202010690407 A CN 202010690407A CN 111752262 B CN111752262 B CN 111752262B
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沈世焜
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Dragon Totem Technology Hefei Co ltd
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Jiangsu College of Safety Technology
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    • 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
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Abstract

The invention relates to an actuator fault observer and fault-tolerant controller integrated design method, which comprises the following steps: building controlled objects based on state space modelsA fault system model is provided, and a continuous system is discretized; integrating and designing a fault diagnosis observer and a fault-tolerant controller; defining a state estimation error and a fault estimation error to obtain an error system; the generalized interference is simultaneously amplified by the amplification state variable to obtain an amplification system; converting the integration design problem of the observer and the fault-tolerant controller into weighted H by adopting Lyapunov function theory∞Solving the problem by multiple targets of the error augmentation system under the performance index, and setting the performance index of the system; and integrating and solving parameters of the observer and the fault-tolerant controller by adopting a Linear Matrix Inequality (LMI) and a relaxation matrix method. The method is simple and convenient to calculate, can accurately and timely diagnose and control the faults in the system, has a certain inhibiting effect on uncertainty and disturbance of the system, and ensures that the control performance of the system is optimal by the fault-tolerant control system.

Description

Actuator fault observer and fault-tolerant controller integrated design method
The technical field is as follows:
the invention belongs to the field of advanced control of industrial processes, and particularly relates to an actuator fault observer and fault-tolerant controller integrated design method.
Background art:
faults are classified as sensor faults, actuator faults, and other component faults of the system. Of all failures, actuator failures are most common in industrial production. Due to the characteristics of friction, dead zones, saturation, etc., the actuator inevitably experiences some malfunction during its execution, which makes it difficult to reach a specified or desired position. The existence of actuator faults can reduce the operation precision of the system, damage the control performance of the system and even influence the production efficiency. Therefore, the fault-tolerant control technology which is accurate and timely in fault diagnosis and reliable has important significance for guaranteeing stable and efficient operation of the control process and industrial production. At present, a great deal of results are obtained for the research of fault diagnosis and fault-tolerant control of an actuator, a fault-tolerant control system based on the fault diagnosis results is developed rapidly, but most of the known methods firstly solve the parameters of an observer and bring the observation state into a fault-tolerant controller, but when the system is uncertain or transmission is delayed or lost, the fault-tolerant control performance is greatly discounted, and even the system operation is influenced.
The invention content is as follows:
the present invention is directed to solving at least one of the problems of the prior art or the related art. Therefore, an object of the present invention is to provide an integrated design method for an actuator fault observer and a fault-tolerant controller, which is characterized in that: the method comprises the following steps
Step 1, aiming at a closed-loop control system, considering system noise interference and actuator faults, establishing a fault system model of a controlled object based on a state space model, specifically:
step 1.1, selecting a controlled object, and constructing a state equation of the controlled object as follows:
Figure BDA0002588337860000011
wherein x ∈ Rn，u∈Rq，y∈Rm，z∈RhRespectively representing the state, control input, system output and controlled output of the system, f ∈ Rq，d∈RmRespectively representing actuator faults and sensor disturbances, Ac，Bc，Cc，Dc1，D2Is a constant matrix.
Step 1.2, discretizing a controlled object according to a sampling period h of a sensor:
Figure BDA0002588337860000021
in the formula:
Figure BDA00025883378600000212
step 2, the observer integrating fault diagnosis and fault-tolerant control is designed as follows:
Figure BDA0002588337860000023
wherein:
Figure BDA0002588337860000024
the state estimate for x (k) is represented,
Figure BDA0002588337860000025
denotes the fault estimate, L ∈ Rn×m，M∈Rq＝mRepresenting a parameter matrix to be designed.
Step 3, defining state estimation error
Figure BDA0002588337860000026
And fault estimation error
Figure BDA0002588337860000027
And Δ f (k) ═ f (k +1) -f (k), can be obtained
Figure BDA0002588337860000028
Step 4, expanding the state variable:
Figure BDA0002588337860000029
augmented generalized interference: w (k) ═ dT(k) ΔfT(k)]And then an augmentation system is obtained
Figure BDA00025883378600000210
Wherein:
Figure BDA00025883378600000211
and 5, setting the performance of the observer and the fault-tolerant controller
Step 5.1. for making the system robust and stable, the fault estimation error ef(k) Robust to interference w (k), satisfies H∞Performance indexes are as follows:
Figure BDA0002588337860000031
defining the Lyapunov function:
Figure BDA0002588337860000032
by Schur supplement theory, we obtain:
Figure BDA0002588337860000033
after Schur supplement, the Chinese medicine becomes
Figure BDA0002588337860000034
Step 5.2, the system is robust and stable, and the output z (k) has robustness to interference, namely H is satisfied∞Norm constraint conditions are as follows:
Figure BDA0002588337860000035
so the Lyapunov function is chosen:
Figure BDA0002588337860000036
by Schur supplement theory, we obtain:
Figure BDA0002588337860000037
after Schur supplement, the Chinese medicine becomes
Figure BDA0002588337860000038
Step 6, integrating and solving linear matrix inequality of observer and fault-tolerant controller based on LMI
Step 6.1, since G is a positive definite symmetric matrix, G is known to be reversible, and Λ ═ diag { G ═ GTP-1,I,I,I}，Ω1And Ω2Left-hand and right-hand multiplier respectivelyTThe following can be obtained:
Figure BDA0002588337860000039
Figure BDA00025883378600000310
step 6.2, taking P as positive definite symmetric matrix, G as reversible matrix, having (P-G)TP-1(P-G) ≧ 0, the above formula is developed to yield: gTP-1G≥-P+G+GTWhen the left and right ends of the inequality are multiplied by-I at the same time, there is-GTP-1G≤P-G-GTAnd he (X) ═ X is definedT+ X, inequality (1) is changed to
Figure BDA0002588337860000041
The inequality (2) of the same theory becomes
Figure BDA0002588337860000042
Step 6.3 redefining the matrix
Figure BDA0002588337860000043
With Schur supplement, formula (3) and formula (4) become:
Figure BDA0002588337860000044
Figure BDA0002588337860000045
wherein
Figure BDA0002588337860000046
Figure BDA0002588337860000047
Figure BDA0002588337860000051
Figure BDA0002588337860000052
Figure BDA0002588337860000053
Step 6.4. due to the presence of the non-linear term in equation (7)
Figure BDA0002588337860000054
The variable replacement can not be carried out, and an inequality is obtained according to a relaxation matrix method
Figure BDA0002588337860000055
Then there is
Figure BDA0002588337860000056
Step 6.5. definition
Figure BDA0002588337860000057
Calculating state estimation gain by using inequalities (5), (6) and (8) through LMI tool
Figure BDA0002588337860000058
Fault estimation gain
Figure BDA0002588337860000059
Controller gain
Figure BDA00025883378600000510
Drawings
FIG. 1 is a flow chart of an embodiment of the invention
FIG. 2 illustrates actuator fault estimation according to an embodiment of the invention
FIG. 3 is a state responsive angular velocity of an embodiment of the invention
FIG. 4 is a state responsive armature current for an embodiment of the invention
The specific implementation mode is as follows:
the invention is further explained below with reference to the figures and the examples.
Example 1
Referring to fig. 1, according to step 1.1, the controlled object is selected to be a direct current motor, and a mathematical model of the controlled object comprises the following two differential equations
Figure BDA0002588337860000061
Wherein iaω and vaRepresenting armature current, angular velocity and armature voltage, respectively. RaIs armature resistance, LaIs an inductance. K and KbIs the voltage and motor constant, JmIs moment of inertia, BmCoefficient of friction.
Further, the controlled object mathematical model is constructed as a controlled object state equation as follows:
Figure BDA0002588337860000062
let x1＝Ia，x2＝ωm，
Figure BDA0002588337860000063
C＝[0 1]Then the above formula is rewritten as
Figure BDA0002588337860000064
And considering system noise interference and actuator faults, establishing a fault system model of the controlled object based on a state space model:
Figure BDA0002588337860000065
wherein x ∈ Rn，u∈Rq，y∈Rm，z∈RhRespectively representing the state, control input, system output and controlled output of the system, f ∈ Rq，d∈RmRespectively representing actuator faults and sensor disturbances, Ac，Bc，Cc，Dc1，D2Is a constant matrix.
According to step 1.2, let Ra＝1.2，La＝0.05，K＝0.6，Kb＝0.6，Jm＝0.1352，BmWhen discretization is performed at a sampling period h of 0.01s of 0.3, the system matrix parameters of the linear discretization model can be obtained as follows
Figure BDA0002588337860000066
CZ＝C＝[1 0]
In order to verify the design of the fault estimation observer and the fault-tolerant controller, a fault signal is set as follows:
Figure BDA0002588337860000071
and the unknown disturbance input signal d (k) is a random signal with an amplitude smaller than 0.1.
Step 2, the observer integrating fault diagnosis and fault-tolerant control is designed as follows:
Figure BDA0002588337860000072
wherein:
Figure BDA0002588337860000073
the state estimate for x (k) is represented,
Figure BDA0002588337860000074
denotes the fault estimate, L ∈ Rn×m，M∈Rq＝mRepresenting a parameter matrix to be designed.
Step 3, defining state estimation error
Figure BDA0002588337860000075
And fault estimation error
Figure BDA0002588337860000076
And Δ f (k)) F (k +1) -f (k), available as
Figure BDA0002588337860000077
Step 4, expanding the state variable:
Figure BDA0002588337860000078
augmented generalized interference: w (k) ═ dT(k) ΔfT(k)]And then an augmentation system is obtained
Figure BDA0002588337860000079
Wherein:
Figure BDA00025883378600000710
and 5, setting the performance of the observer and the fault-tolerant controller
Step 5.1. for making the system robust and stable, the fault estimation error ef(k) Robust to interference w (k), satisfies H∞Performance indexes are as follows:
Figure BDA0002588337860000081
defining the Lyapunov function:
Figure BDA0002588337860000082
by Schur supplement theory, we obtain:
Figure BDA0002588337860000083
after Schur supplement, the Chinese medicine becomes
Figure BDA0002588337860000084
Step 5.2, the system is robust and stable, and the output z (k) has robustness to interference, namely the system meets the requirementH∞Norm constraint conditions are as follows:
Figure BDA0002588337860000085
so the Lyapunov function is chosen:
Figure BDA0002588337860000086
by Schur supplement theory, we obtain:
Figure BDA0002588337860000087
after Schur supplement, the Chinese medicine becomes
Figure BDA0002588337860000088
Step 6, integrating and solving linear matrix inequality of observer and fault-tolerant controller based on LMI
Step 6.1, since G is a positive definite symmetric matrix, G is known to be reversible, and Λ ═ diag { G ═ GTP-1,I,I,I}，Ω1And Ω2Left-hand and right-hand multiplier respectivelyTThe following can be obtained:
Figure BDA0002588337860000089
Figure BDA00025883378600000810
step 6.2, taking P as positive definite symmetric matrix, G as reversible matrix, having (P-G)TP-1(P-G) ≧ 0, the above formula is developed to yield: gTP-1G≥-P+G+GTWhen the left and right ends of the inequality are multiplied by-I at the same time, there is-GTP-1G≤P-G-GTAnd he (X) ═ X is definedT+ X, inequality (1) is changed to
Figure BDA0002588337860000091
The inequality (2) of the same theory becomes
Figure BDA0002588337860000092
Step 6.3 redefining the matrix
Figure BDA0002588337860000093
With Schur supplement, formula (3) and formula (4) become:
Figure BDA0002588337860000094
Figure BDA0002588337860000095
wherein
Figure BDA0002588337860000096
Figure BDA0002588337860000097
Figure BDA0002588337860000101
Figure BDA0002588337860000102
Figure BDA0002588337860000103
Step 6.4. due to the presence of non-threads in formula (7)Sexual item
Figure BDA0002588337860000104
The variable replacement can not be carried out, and an inequality is obtained according to a relaxation matrix method
Figure BDA0002588337860000105
Then there is
Figure BDA0002588337860000106
Step 6.5. definition
Figure BDA0002588337860000107
Calculating state estimation gain by using inequalities (5), (6) and (8) through LMI tool
Figure BDA0002588337860000108
Fault estimation gain
Figure BDA0002588337860000109
Controller gain
Figure BDA00025883378600001010
The specific parameters are solved as follows: when M is equal to 1.1111, M is,
Figure BDA00025883378600001011
K＝[-0.8724 0]。
referring to fig. 2, the system initial state is x (k) ═ 11]TWhen the system has a fault, the method provided by the invention can accurately and rapidly estimate the size of the fault. Referring to fig. 3 and 4, the fault-tolerant controller designed by the present invention can maintain good performance of the system when a fault occurs, and it can be seen by comparison that the system has a large fluctuation amplitude and a large oscillation frequency without the fault-tolerant controller provided by the present invention. On the contrary, if the fault-tolerant control method provided by the invention is adopted, the armature current and the angular speed in the system state have small fluctuation and tend to be stable in a short time, thereby achieving the fault-tolerant controlThe purpose is.

Claims (1)

1. An actuator fault observer and fault-tolerant controller integrated design method is characterized in that: the method comprises the following steps
Step 1, aiming at a closed-loop control system, considering system noise interference and actuator faults, establishing a fault system model of a controlled object based on a state space model, specifically:
step 1.1, selecting a controlled object, and constructing a state equation of the controlled object as follows:
Figure FDA0003029370930000011
wherein x ∈ Rn，u∈Rq，y∈Rm，z∈RhRespectively representing the state, control input, system output and controlled output of the system, f ∈ Rq，d∈RmRespectively, representing actuator faults and sensor disturbances, Ac，Bc，C，Cz，Dc1，D2Is a constant matrix;
step 1.2, discretizing a controlled object according to a sampling period h of a sensor:
Figure FDA0003029370930000012
in the formula:
Figure FDA0003029370930000013
step 2, the observer integrating fault diagnosis and fault-tolerant control is designed as follows:
Figure FDA0003029370930000014
wherein:
Figure FDA0003029370930000015
the state estimate for x (k) is represented,
Figure FDA0003029370930000016
representing a fault estimate, the controller gain K ∈ Rq×nThe state estimation gain L is equal to Rn×mThe fault estimation gain M is equal to Rq×mIs a parameter matrix to be designed;
step 3, defining state estimation error
Figure FDA0003029370930000017
And fault estimation error
Figure FDA0003029370930000018
And Δ f (k) ═ f (k +1) -f (k), can be obtained
Figure FDA0003029370930000019
Step 4, expanding the state variable: ζ (k) ═ xT(k) ex T(k) ef T(k)]TThe generalized interference is amplified: w (k) ═ dT(k) ΔfT(k)]And then an augmentation system is obtained
Figure FDA0003029370930000021
Wherein:
Figure FDA0003029370930000022
wherein IqIs a q-order identity matrix;
and 5, setting the performance of the observer and the fault-tolerant controller
Step 5.1. for making the system robust and stable, the fault estimation error ef(k) Robust to interference w (k), satisfies H∞Performance indexes are as follows:
Figure FDA0003029370930000023
wherein gamma iseIs constant and has a value of gammae> 0, defining the Lyapunov function:
Figure FDA0003029370930000028
by Schur supplement theory, we obtain:
Figure FDA0003029370930000024
after Schur supplement, the Chinese medicine becomes
Figure FDA0003029370930000025
Wherein I is an identity matrix;
step 5.2, the system is robust and stable, and the output z (k) has robustness to interference, namely H is satisfied∞Norm constraint conditions are as follows:
Figure FDA0003029370930000026
wherein gamma iszIs constant and has a value of gammaz> 0, so the Lyapunov function was chosen:
Figure FDA0003029370930000029
by Schur supplement theory, we obtain:
Figure FDA0003029370930000027
after Schur supplement, the Chinese medicine becomes
Figure FDA0003029370930000031
Step 6, integrating and solving linear matrix inequality of observer and fault-tolerant controller based on LMI
Step 6.1, since G is a positive definite symmetric matrix, G is known to be reversible, and Λ ═ diag { G ═ GTP-1,I,I,I}，Ω1And Ω2Left-hand and right-hand multiplier respectivelyTThe following can be obtained:
Figure FDA0003029370930000032
Figure FDA0003029370930000033
step 6.2, taking P as positive definite symmetric matrix, G as reversible matrix, having (P-G)TP-1(P-G) ≧ 0, the above formula is developed to yield: gTP-1G≥-P+G+GTWhen the left and right ends of the inequality are multiplied by-I at the same time, there is-GTP-1G≤P-G-GTAnd he (X) ═ X is definedT+ X, inequality (1) is changed to
Figure FDA0003029370930000034
The inequality (2) of the same theory becomes
Figure FDA0003029370930000035
Step 6.3 redefining the matrix
Figure FDA0003029370930000041
With Schur supplement, formula (3) and formula (4) become:
Figure FDA0003029370930000042
Figure FDA0003029370930000043
wherein
Figure FDA0003029370930000044
And IpIs an identity matrix of order p, IhIs an h-order identity matrix
Step 6.4. due to the presence of the non-linear term in equation (7)
Figure FDA0003029370930000051
The variable replacement can not be carried out, and an inequality is obtained according to a relaxation matrix method
Figure FDA0003029370930000052
Wherein eta is constant and has eta > 0, InIs an n-order identity matrix, IrIs an r-order identity matrix, then
Figure FDA0003029370930000053
Step 6.5. definition
Figure FDA0003029370930000054
Calculating state estimation gain by using inequalities (5), (6) and (8) through LMI tool
Figure FDA0003029370930000055
Fault estimation gain
Figure FDA0003029370930000056
Controller gain
Figure FDA0003029370930000057
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