CN111158343B - Asynchronous fault-tolerant control method for switching system with actuator and sensor faults - Google Patents

Asynchronous fault-tolerant control method for switching system with actuator and sensor faults Download PDF

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CN111158343B
CN111158343B CN202010025435.7A CN202010025435A CN111158343B CN 111158343 B CN111158343 B CN 111158343B CN 202010025435 A CN202010025435 A CN 202010025435A CN 111158343 B CN111158343 B CN 111158343B
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state
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CN111158343A (en
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杜董生
杨月
吴瑜
王业琴
赵环宇
唐中一
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Wang Ping
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Huaiyin Institute of Technology
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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
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    • Y02P—CLIMATE CHANGE MITIGATION TECHNOLOGIES IN THE PRODUCTION OR PROCESSING OF GOODS
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    • Y02P90/02—Total factory control, e.g. smart factories, flexible manufacturing systems [FMS] or integrated manufacturing systems [IMS]

Abstract

The invention relates to an asynchronous fault-tolerant control method for a switching system with faults of an actuator and a sensor, which comprises the following steps of firstly, constructing an augmentation system by taking the faults of the actuator and the sensor as a part of a state through transformation; secondly, a fault estimation observer is provided for the augmented system, and meanwhile, the switching signal of the observer and the original system are supposed to have inevitable lag, so that asynchronous switching between the original system and the observer is caused; furthermore, in order to solve the asynchronous switching problem, the error system is provided to be asymptotically stable and satisfy H ∞ Sufficient conditions for performance indexes; and finally, designing a state feedback controller based on an observer based on the fault estimation information so as to ensure the stability of the closed-loop system. The method not only can accurately estimate the state and the fault of the system, but also can ensure that the closed-loop system is stable under the conditions of faults of an actuator and a sensor and external disturbance.

Description

Asynchronous fault-tolerant control method for switching system with actuator and sensor faults
Technical Field
The invention relates to the technical field of fault estimation, in particular to an asynchronous fault-tolerant control method for a switching system with faults of an actuator and a sensor.
Background
With the rapid development of modern technologies, engineering systems are more and more complex, and the requirements on the safety of the systems are higher and higher. In practice, however, system failure is inevitable and can affect the safe operation of the device, resulting in reduced system performance, economic loss, and even catastrophic results. In this context, fault diagnosis and fault-tolerant control, as an effective fault handling method, play an increasingly important role in the development of recent decades. Fault diagnostics include fault detection, fault isolation, and fault estimation. The fault detection and isolation can judge the running state of the system and obtain the fault occurrence position, and some better research results are obtained. By comparing with the first two parts, the fault estimation can obtain the form of fault information and know the position of the fault, apply the information to the fault-tolerant control, and then use the information to reduce the fault influence or perform the fault-tolerant control. Therefore, fault estimation and fault-tolerant control have been the focus of research.
In the fault estimation technique, the most common method is to design a fault observer (or filter) and a fault estimator. In recent years, a great deal of research has been conducted on possible failures. Such as: the system comprises a sliding mode observer, an unknown input observer, a reduced order observer, an adjustable size dynamic observer and a proportional integral observer. The sliding-mode observer can reconstruct possible faults. A fault estimation method based on an Unknown Input Observer (UIO) is a method for combining a fault estimation observer and a fault-tolerant controller. To reduce the estimation cost, reduced order observers have been implemented to reconstruct the fault, wherein observer-based fault-tolerant control methods have been proposed. In order to make a trade-off between estimation accuracy and estimation cost, a design method of a size-adjustable dynamic observer is provided. Based on the output information, a proportional-integral observer is applied to reconstruct the process (or actuator) fault.
Since many complex dynamic processes can be modeled as a combination of a set of linear systems, such as robotic control systems, chemical reaction processes, automotive industry and switching power converters, etc., conversion systems for both theoretical and practical applications have attracted many researchers. Aiming at the problems of fault diagnosis and fault-tolerant control of the switching system, several innovative methods are provided. Such as fault estimation and fault-tolerant control of actuators and sensors under any switching signal; aiming at a time-lag uncertain switching discrete system with actuator faults and input saturation, the stability problem of robust fault tolerance is researched by utilizing a triangular operator method; the problem of self-adaptive fuzzy finite time fault-tolerant control of a nonlinear switching large-scale system with faults of an actuator and a sensor is researched by combining a common Lyapunov function method and a recursion design method; the problems of fault estimation and compensation of a sensor and an actuator of the hybrid switching system are researched by adopting a data-based projection method respectively; based on a segmented Lyapunov function and an average residence time method, fault estimation and fault tolerance control based on a proportional-integral observer are carried out on a switching fuzzy random system with an actuator fault and a sensor fault.
Furthermore, it can be seen that all of the aforementioned work assumes that the observer or filter is synchronized with the subsystem. However, in practical applications, a system identification observer or filter is unavoidable. Thus, in the general case, asynchronous switching between the system and its observer or filter is absolutely present. How to combine asynchronous fault diagnosis with fault-tolerant control becomes a hot spot of current research. Although the asynchronous fault detection problem is a research topic that has received much attention in recent years. In the existing method, by constructing a novel switching strategy based on state and switching delay, the design of an asynchronous fault detection filter of a continuous switching delay system is researched. By utilizing a segmented Lyapunov function and an average residence time method, the design of an asynchronous fault detection observer and the design of a fault-tolerant controller are researched. However, asynchronous fault estimation and fault tolerant control techniques for switching systems have not been considered to our knowledge.
Disclosure of Invention
The purpose of the invention is as follows: the invention relates to an asynchronous fault-tolerant control method for a switching system with faults of an actuator and a sensor, which researches the problems of fault estimation and fault-tolerant control of a continuous time switching system under the conditions of faults of the actuator, faults of the sensor and interference, constructs an augmentation system and provides a fault estimation observer for the augmentation system; based on the fault estimation information, an observer-based state feedback controller is designed, so that the stability of a closed-loop system is ensured.
The technical scheme is as follows: an asynchronous fault tolerant control method for a switching system with actuator and sensor faults, comprising the steps of:
step 1: constructing an augmentation system by transforming actuator and sensor faults as part of a continuous time switching system state;
step 2: a fault estimation observer is provided for the augmentation system in the step 1, and meanwhile, the switching signal of the observer and the continuous time switching system are supposed to have inevitable lag, so that asynchronous switching between the continuous time switching system and the fault estimation observer is caused;
and step 3: giving the error system of the fault estimation observer asymptotically stable and satisfying H ∞ Sufficient conditions for performance indexes;
and 4, step 4: based on the fault estimation information, a state feedback controller based on a fault estimation observer is designed to ensure the stability of a closed-loop system.
Further, the step of constructing an augmentation system in step 1 comprises:
step 2.1 the continuous time switching system is:
Figure BDA0002362264050000021
wherein x (t) ∈ R n Is in a state of,
Figure BDA0002362264050000022
Is a control input, y (t) is ∈ R m Is the output of the measurement, and,
Figure BDA0002362264050000023
is a disturbance and assumes norm-bounded, the switching signal σ (t) is assumed to be time-varying, the switching sequence is t 0 ＜t 1 ＜…＜t k 8230in which t k Is the instant of switching; when σ (t) = i, i ∈ N = {1,2, \8230; N }, indicating that the ith subsystem is activated;
step 2.2 the continuous time switching system (1) in step 2.1 is simplified to:
Figure BDA0002362264050000031
wherein A is i ,B i ,C i ,D i Is a matrix of appropriate dimensions, assuming (A) i ,B i ) Is controllable (A) i ,C i ) Is considerable; the derivative of the actuator fault is norm bounded, i.e.:
Figure BDA0002362264050000032
step 2.3 assuming that both actuator and sensor faults are present in the system (2), the system (2) suffering from the fault can be described as:
Figure BDA0002362264050000033
wherein the content of the first and second substances,
Figure BDA0002362264050000034
f s (t)∈R m respectively representing actuator faults and sensor faults; the following matrix is also defined:
Figure BDA0002362264050000035
the dynamic system (3) is then:
Figure BDA0002362264050000036
wherein the content of the first and second substances,
Figure BDA0002362264050000037
then it is easy to find:
Figure BDA0002362264050000038
namely:
Figure BDA0002362264050000039
step 2.4 System (5) transforms into the augmentation System in step 1:
Figure BDA00023622640500000310
further, the specific design process of the fault estimation observer in the step 2 is as follows:
step 3.1 the fault estimation observer is designed as follows:
Figure BDA0002362264050000041
wherein
Figure BDA0002362264050000042
Is that
Figure BDA0002362264050000043
Is estimated in the state of (a) of (b),
Figure BDA0002362264050000044
is the output of the observer; l is σ'(t) Is the observer gain, σ' (t) is the switching signal of observer equation (8) above; let us assume that there is a delay between the activated subsystem of the augmented system (7) and the observer, set Ω = [ t ] 0 ,t 1 )∪[t 1 +Δ 1 ,t 2 )∪…∪[t k +Δ k ,t k+1 ) U \8230denotesmutually matched time, set omega' = [ t = 1 ,t 1 +Δ 1 )∪…∪[t k ,t k +Δ k ) U \8230, indicating unmatched time; t is t k And t k +Δ k K =0,1, \8230;, respectively representing the switching instant of σ (t), σ' (t);
step 3.2 the state error of the fault estimation observer is as follows:
Figure BDA0002362264050000045
the derivative of the state error e (t) is known from (7) and (8):
Figure BDA0002362264050000046
and:
Figure BDA0002362264050000047
wherein the content of the first and second substances,
Figure BDA0002362264050000048
step 3.3 the error system (9) or (10) is considered stable and satisfies H if it satisfies the following two conditions ∞ Performance index γ:
a: in the absence of disturbances, the error system (9) or (10) is asymptotically stable;
b: when the initial condition is zero, then there are:
Figure BDA0002362264050000049
step 3.4 defines the auxiliary variable η (t) and gives the accurate state estimation observer of the augmented system (7):
Figure BDA0002362264050000051
precision state estimation observer of an augmented system (7):
Figure BDA0002362264050000052
wherein the content of the first and second substances,
Figure BDA0002362264050000053
is the observer gain.
Further, the error system given in said step 3 asymptotically stabilizes and satisfies H ∞ The sufficient conditions of the performance indexes are as follows:
step 4.1 for a given constant α > 0, β > 0, μ 1 ＞1,μ 2 > 1, gamma > 0, if a positive definite matrix P is present i ＞0,P ij > 0, and matrix Q i ,Q ij For i ≠ j, i, j ∈ N, such that:
P j ≤μ 1 P ij ,P ij ≤μ 2 P i (15)
Figure BDA0002362264050000054
Figure BDA0002362264050000055
Figure BDA0002362264050000056
wherein the content of the first and second substances,
Figure BDA0002362264050000057
Figure BDA0002362264050000058
if any of the switching signals satisfies the conditions (15) to (17) and the average residence time condition (18), the error system (9) or (10) is asymptotically stable and satisfies H ∞ A performance index γ; wherein, T - (t 0 T) and T + (t 0 T) is expressed at time [ t 0 T) total cycles within match and mismatch;
step 4.2 the parameters of the precise state estimation observer (14) are designed as follows:
Figure BDA0002362264050000059
go toStep by step, assume B in System (1) i ＝F i Then, the observer-based state feedback controller in step 4 is designed as follows:
Figure BDA0002362264050000061
wherein, K i Is the state feedback controller gain; then (20) is substituted into the system (1), and after simplification, the following can be known:
Figure BDA0002362264050000062
Figure BDA0002362264050000063
wherein the content of the first and second substances,
Figure BDA0002362264050000064
further, the state feedback controller asymptotically stabilizes and satisfies H ∞ Sufficient conditions of performance indexes:
step 6.1 for given constants
Figure BDA0002362264050000065
If a positive definite matrix R exists i ＞0,R ij > 0, and a matrix W i So that:
Figure BDA0002362264050000066
Figure BDA0002362264050000067
Figure BDA0002362264050000068
wherein:
Figure BDA0002362264050000069
Figure BDA00023622640500000610
if the following average residence time condition (18) is satisfied for any switching signal, the error system (21) or (22) is gradually stabilized and satisfies H ∞ Performance index gamma 1 (ii) a Wherein, T - (t 0 T) and T + (t 0 And t) are respectively expressed at time [ t ] 0 T) total cycles within match and mismatch;
step 6.2 the parameters in the controller (20) are designed as follows:
Figure BDA00023622640500000611
has the advantages that:
1. different from the prior art that only one fault is considered, the invention simultaneously considers the actuator fault and the sensor fault of the switching system, and provides a novel fault-tolerant controller design method.
2. An asynchronous fault estimation observer is designed, and the state and the fault of a system can be accurately estimated at the same time. In order to reduce the influence of faults on the observer, an auxiliary observer is introduced to improve the observation precision, unknown items in the auxiliary observer are eliminated by using new auxiliary variables, and the accurate observer is obtained.
3. Based on the designed observer, an output feedback fault-tolerant controller is established to ensure the stability of the closed-loop system under the sensor fault, the actuator fault and the external interference.
Drawings
FIG. 1: the invention relates to a flow chart of an asynchronous fault-tolerant control method;
FIG. 2: a continuous time switching system and a fault estimation observer schematic;
FIG. 3: a graph of the relationship of the switching signals σ (t), σ' (t) for asynchronous switching;
FIG. 4: the switching signals sigma (t) and sigma' (t) of simulation example 1 in the embodiment of the invention are shown schematically;
FIG. 5 is a schematic view of: fault estimation and fault error schematic for f (t) of simulation example 1, where the solid line is fault and the dashed line is estimation;
FIG. 6: graphs comparing the output y (t) with and without regulation for simulation example 1;
FIG. 7: a boost converter circuit connection diagram of simulation example 2;
FIG. 8: the switching signals sigma (t) and sigma' (t) of simulation example 2 in the embodiment of the invention are shown schematically;
FIG. 9: f of simulation example 2 a (t) fault estimation and fault error diagrams thereof, wherein the solid line is fault and the dashed line is estimation;
FIG. 10: f of simulation example 2 s (t) fault estimation and fault error schematic of (t), wherein the solid line is fault and the dashed line is estimation;
FIG. 11: the output y (t) with and without regulation of example 2 was simulated.
Detailed Description
The present invention will be described in detail with reference to the accompanying drawings.
The invention provides an asynchronous fault-tolerant control method for a switching system with faults of an actuator and a sensor by taking a continuous time switching system model as an implementation object and aiming at faults in the system.
Note that: in the present invention, P is T ,P -1 Respectively representing the transpose of the matrix P and the inverse of the matrix, P > 0 (P < 0) indicating that the P matrix is a positive (negative) definite matrix, R n Representing a set of n-dimensional real vectors, I and 0 representing identity matrices and 0 matrices with appropriate dimensions, where x represents the symmetric terms in the symmetric matrix.
The asynchronous fault-tolerant control method comprises the following steps:
step 1: an augmented system is constructed by transforming actuator and sensor faults as part of the state.
The continuous time switching system comprises:
Figure BDA0002362264050000081
wherein x (t) ∈ R n Is in a state of,
Figure BDA0002362264050000082
Is a control input, y (t) is ∈ R m Is the output of the measurement, and,
Figure BDA0002362264050000083
is a perturbation and is assumed to be norm-bounded.
Let us assume that the switching signal σ (t) is time-varying, the switching sequence being t 0 ＜t 1 ＜…＜t k 8230in which t k Is the instant of switching. When σ (t) = i, i ∈ N = {1,2, \8230; N }, it represents that the ith subsystem is activated. The original system (1) can be simplified as follows:
Figure BDA0002362264050000084
wherein A is i ,B i ,C i ,D i Is a matrix of appropriate dimensions.
For the purposes of the present invention, the following two assumptions need to be made:
the first point is as follows: (A) i ,B i ) Is controllable (A) i ,C i ) Is considerable;
and a second point: the derivative of the actuator fault is norm bounded, i.e.:
Figure BDA0002362264050000085
in the present invention, we assume that both actuator and sensor faults occur in the system (2), and the system (2) suffering from the fault can be described as:
Figure BDA0002362264050000086
wherein the content of the first and second substances,
Figure BDA0002362264050000087
f s (t)∈R m indicating actuator failure and sensor failure, respectively.
Note 1: from the point of view of the prior art, the above assumes that the conditions in the second point mean that the actuator failure is energetically bounded, so it is naturally reasonable to assume the second point in practice.
In order to simultaneously obtain actuator faults f a (t) and sensor failure f s (t) defining the following matrix:
Figure BDA0002362264050000091
the dynamic system (3) is rewritten as:
Figure BDA0002362264050000092
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002362264050000093
this is readily found:
Figure BDA0002362264050000094
namely, it is
Figure BDA0002362264050000095
Thus, the system (5) can be converted into:
Figure BDA0002362264050000096
step 2: a fault estimation observer is provided for an augmentation system (7) in the step 1, and the specific process is as follows:
1) For the augmented system (7) we can construct a fault estimation observer as follows:
Figure BDA0002362264050000097
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002362264050000098
is that
Figure BDA0002362264050000099
Is estimated in the state of (a) of (b),
Figure BDA00023622640500000910
is the output of the observer. L is σ'(t) Is the observer gain determined later, and σ' (t) is the switching signal of the failure estimation observer (8). The switching system and the asynchronous observer are shown in fig. 2.
We assume that a delay between the fault estimation observer (8) and the activated subsystem of the augmented system (7) is unavoidable, the relationship between σ (t), σ' (t) being shown in fig. 3. In the figure, t k And t k +Δ k K =0,1, \ 8230;, respectively, indicates the switching instant of σ (t), σ' (t). That is, the period Ω = [ t = 0 ,t 1 )∪[t 1 +Δ 1 ,t 2 )∪…∪[t k +Δ k ,t k+1 ) U \8230denotesthe time of matching, while period Ω' = [ t = 1 ,t 1 +Δ 1 )∪…∪[t k ,t k +Δ k ) U.S. Pat. No. 8230denotes unmatched time.
Note 2: ideally, the last handover is synchronized with the subsystem, i.e., (σ:)t) = σ' (t), in fact, the period Δ in different environments, since a certain time is required to identify the subsystem to be started and apply the matching observer k Also different, so we assume asynchronous handover maximum delay Δ max Are known a priori and without loss of generality.
2) Defining:
Figure BDA0002362264050000101
the derivative of the state error e (t) is known from (7) and (8):
Figure BDA0002362264050000102
and:
Figure BDA0002362264050000103
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002362264050000104
definition 1: from the viewpoint of the prior art, the error system (9) (or (10)) is considered to be asymptotically stable and satisfy H if it satisfies the following two conditions ∞ Performance index γ:
a: in the absence of disturbances, the error system (9) or (10) is asymptotically stable;
b: when the initial condition is zero, the following conditions are present:
Figure BDA0002362264050000105
it can be seen that there is an unknown item H in the fault estimation observer (8) 2 J 2σ'(t) x (t), in order to improve the estimation accuracy of the fault estimation observer, an auxiliary variable η (t) is defined:
Figure BDA0002362264050000106
at this time, an accurate state estimation observer of the augmented system (7) is given:
Figure BDA0002362264050000107
wherein the content of the first and second substances,
Figure BDA0002362264050000108
is the observer gain that will be determined later.
Note 3: constructing an augmented vector by integrating actuator and sensor failures
Figure BDA0002362264050000109
If the observer (14) is successfully solved, states, actuator faults and sensor faults in the system (1) can be simultaneously obtained.
Definition 2: from the point of view of the prior art, for any switching signal σ i (t) at an arbitrary time t 2 ＞t 1 Greater than 0, let N σi(t) (t 1 ,t 2 ) Representing the switching signal sigma i (t) at (t) 1 ,t 2 ) The number of handovers in. For a given N 0 ≥0,τ a > 0 if the following holds:
Figure BDA0002362264050000111
then constant τ a Referred to as average residence time, N 0 Called buffeting limit value.
And step 3: in order to solve the problem of asynchronous switching, the error system is provided to be asymptotically stable and meet the requirement of H ∞ The specific process of the sufficient conditions of the performance indexes is as follows:
theorem 1: for a given constant α > 0, β > 0, μ 1 ＞1,μ 2 > 1, gamma > 0, if a positive definite matrix P is present i ＞0,P ij > 0, and matrix Q i ,Q ij For i ≠ j, i, j ∈ N, such that:
P j ≤μ 1 P ij ,P ij ≤μ 2 P i (15)
Figure BDA0002362264050000112
Figure BDA0002362264050000113
Figure BDA0002362264050000114
wherein the content of the first and second substances,
Figure BDA0002362264050000115
Figure BDA0002362264050000116
if any of the switching signals satisfies the conditions (15) to (17) and the average residence time condition (18), the error system (9) or (10) is asymptotically stable and satisfies H ∞ The performance index γ.
Wherein, T - (t 0 T) and T + (t 0 T) is expressed at time [ t 0 T) total period of match and mismatch, while parameters in the precision state estimation observer (14) are given:
Figure BDA0002362264050000117
proof of theorem 1:
first, the stability of the error systems (9) and (10) is established:
case 1: when t ∈ Ω, consider the following lyapunov function:
V i (t)＝e T (t)P i e(t) (20)
from the error system equation (9), it can be seen that:
Figure BDA0002362264050000121
definition of Q i ＝P i L i Then the above formula (21) can be converted into:
Figure BDA0002362264050000122
from the condition (16), it can be obtained:
Figure BDA0002362264050000123
thus, during the matching period, V i (t) satisfies:
Figure BDA0002362264050000124
case 2: when t e Ω', the error system is represented by equation (10), in which case we consider the following piecewise lyapunov function:
V ij (t)＝e T (t)P ij e(t) (24)
according to the error system (10), it is known that:
Figure BDA0002362264050000125
definition of Q ij ＝P ij L i Then (25) can be converted into:
Figure BDA0002362264050000126
from the condition (17), it can be obtained:
Figure BDA0002362264050000127
thus, during the mismatch period, V ij (t) satisfies:
Figure BDA0002362264050000128
consider the following piecewise Lyapunov function:
Figure BDA0002362264050000129
when t ∈ Ω', from the conditions (15) and (28), there are:
Figure BDA0002362264050000131
similar to the stability demonstration in the literature, we obtained:
Figure BDA0002362264050000132
thus, if the average residence time conditional expression (18) is satisfied, we conclude that: when t → ∞, V (t) converges to 0.
Secondly, consider H ∞ Performance index, for a zero initial condition e (t) =0, and nonzero d (t) ∈ L 2 [0, ∞), when t ∈ Ω is considered from the lyapunov function in equation (20), there is:
Figure BDA0002362264050000133
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002362264050000134
if the conditional expression (16) is established, it is possible to obtain:
Figure BDA0002362264050000135
that is:
Figure BDA0002362264050000136
from the formula (34):
Figure BDA0002362264050000137
when t ∈ [ t ] k ,t k +Δ k ) K =1,2, \8230, when we consider the lyapunov function in equation (25), similar to the proof when t e Ω, we have:
Figure BDA0002362264050000138
we can conclude that:
Figure BDA0002362264050000139
from (37):
Figure BDA00023622640500001310
from (35) (38), analogously to H ∞ Designing a technical proof method, we can deduce:
Figure BDA0002362264050000141
from the above definition 2 we have: the error system formula (9) or (10) is asymptotically stable and satisfies H ∞ The performance index γ, finally gives the gain in the precision state estimation observer (14). By the auxiliary variable η (t) defined in equation (13), it can be derived:
Figure BDA0002362264050000142
by comparison with the (14) form, we have:
Figure BDA0002362264050000143
and 4, step 4: based on the fault estimation information, a state feedback controller based on a fault estimation observer is designed, so that the stability of a closed-loop system is ensured, and the specific process is as follows:
in system (1), we assume B i ＝F i The form of controller chosen at this time is as follows:
Figure BDA0002362264050000144
wherein, K i Is the controller gain to be designed, substituting (42) into the system (1), we have:
Figure BDA0002362264050000151
wherein the content of the first and second substances,
Figure BDA0002362264050000152
then there are:
Figure BDA0002362264050000153
and:
Figure BDA0002362264050000154
theorem 2:for a given constant
Figure BDA0002362264050000155
If a positive definite matrix R exists i ＞0,R ij > 0, and a matrix W i So that:
Figure BDA0002362264050000156
Figure BDA0002362264050000157
Figure BDA0002362264050000158
Figure BDA0002362264050000159
wherein:
Figure BDA00023622640500001510
Figure BDA00023622640500001511
the error system (44) or (45) is asymptotically stable and satisfies H for any switching signal satisfying the conditions (46) to (48) and the average dwell time condition (49) ∞ Performance index gamma 1 。
Wherein, T - (t 0 T) and T + (t 0 T) is expressed at time [ t 0 T) total period of match and mismatch, while the parameters in the controller (42) are designed as follows:
Figure BDA0002362264050000161
proof of theorem 2:
case 1: when t ∈ Ω, consider the following lyapunov function:
V i (t)＝x T (t)X i x(t) (51)
according to the system (44) there are:
Figure BDA0002362264050000162
wherein the content of the first and second substances,
Figure BDA0002362264050000163
definition of
Figure BDA0002362264050000164
Diagonal matrix diag { R i Multiplying theta by I, I, I } 1i The following can be obtained:
Figure BDA0002362264050000165
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002362264050000166
from (47), theta can be known 2i < 0, we have:
Figure BDA0002362264050000167
as can be seen from (34):
Figure BDA0002362264050000171
case 2: when t ∈ Ω', consider the piecewise lyapunov function as follows:
V ij (t)＝x T (t)X ij x(t) (57)
similar to the method of attestation at t e Ω, we have, according to the system (44):
Figure BDA0002362264050000172
wherein the content of the first and second substances,
Figure BDA0002362264050000173
definition of
Figure BDA0002362264050000174
Diagonal matrix diag { R i Multiplying theta by I, I, I } 1ij The following can be obtained:
Figure BDA0002362264050000175
wherein the content of the first and second substances,
Figure BDA0002362264050000176
from (48), theta can be known 2ij < 0, we have:
Figure BDA0002362264050000177
from (61):
Figure BDA0002362264050000178
from formulae (55) and (60), analogously to H ∞ Proof of design technique, we can conclude that:
Figure BDA0002362264050000181
wherein the content of the first and second substances,
Figure BDA0002362264050000182
in addition, when
Figure BDA0002362264050000183
In this case, the condition (47) or (48) can be ensured
Figure BDA0002362264050000184
This means when
Figure BDA0002362264050000185
The closed loop system (44) or (45) is asymptotically stable.
The proposed inventive method is verified by two examples below:
simulation example 1: considering the system (3), we present the following parameters:
Figure BDA0002362264050000186
Figure BDA0002362264050000187
Figure BDA0002362264050000188
we select the known parameter μ 1 ＝μ 2 =1.2, α =1.2, by solving the condition of theorem 1, the following observer gain can be obtained:
Figure BDA0002362264050000189
further, assuming that the interference is band-limited white noise, the simulation in both cases is the same within 200s, and subsystems 1,2, and 3 are activated when σ (t) =1, σ (t) =2, and σ (t) =3, respectively. The switching signals σ (t), σ' (t) are shown in fig. 4, assuming the actuator and sensor failures are of the form:
Figure BDA00023622640500001810
fig. 5 is (solid line is fault, dashed line is estimate) the actuator fault estimate and its corresponding fault estimate error.
Let alpha 1 ＝0.7,β 1 ＝1.05,ε 1 ＝0.05,ε 2 ＝0.05,γ 1 ＝0.5,
Figure BDA00023622640500001811
By solving theorem 2, the gain in the controller can be found:
K 1 ＝[13.5645 -5.0943],K 2 ＝[8.2348 2.9745],K 3 ＝[-0.3245 -1.2345]
the output results of y (t) with and without regulation are shown in fig. 6, and the simulation results show that the designed controller has good fault-tolerant performance.
Simulation example 2: the proposed design is illustrated with a boost converter circuit, see fig. 7.
Wherein the state vector is defined as:
Figure BDA0002362264050000191
and the model parameters are as follows:
Figure BDA0002362264050000192
selecting a parameter value of R in ＝25Ω,L in ＝20mH,C＝15μF,r s ＝10Ω,L out ＝30mH,R＝25Ω,υ s =18V, if let μ 1 ＝μ 2 By applying the average residence time condition in equation (18) to solve equations (15) to (17) =1.2, α =0.9, the observer gain can be obtained:
Figure BDA0002362264050000193
the simulation in both cases is the same within 200s, and when σ (t) =1, σ (t) =2, the subsystems 1,2 are activated, respectively. The switching signals σ (t), σ' (t) are shown in fig. 8, and further, the actuator failure and the sensor failure are in the following forms:
Figure BDA0002362264050000194
Figure BDA0002362264050000195
the simulation results are shown in fig. 9 and 10, wherein fig. 8 shows the fault estimation results of the actuator fault and the corresponding fault estimation error; FIG. 9 depicts a fault estimation result and a corresponding fault estimation error for a sensor fault. From the simulation results of fig. 9 and 10, it can be concluded that the method can successfully achieve the goal of fault estimation.
Let alpha 1 ＝0.7,β 1 ＝1.05,ε 1 ＝0.05,ε 2 ＝0.05,γ 1 ＝0.5,
Figure BDA0002362264050000196
By solving theorem 2, the gain in the controller can be found:
K 1 ＝[-0.2343 16.2365 3.8344],K 2 ＝[3.2389 11.4566 0.3479],
K 3 ＝[0.4523 -8.4235 0.0013]
the output results of y (t) with and without regulation are shown in fig. 11, and the simulation results show that the designed controller has good fault-tolerant performance.
The invention researches the problem of fault-tolerant control of output feedback faults of an asynchronous observer of a switching system with faults of an actuator and a sensor. The asynchronous fault estimation observer is used for estimating the state and the fault of the system, firstly, the virtual observer is adopted to reduce the fault influence, and then, the real observer is obtained from the virtual observer. Based on the designed observer, H based on the observer is provided ∞ And a new criterion is designed, so that the fault closed-loop system is ensured to be asymptotically stable under a given interference attenuation level, and the method has practical value.
The above embodiments are merely illustrative of the technical concepts and features of the present invention, and the purpose of the embodiments is to enable those skilled in the art to understand the contents of the present invention and implement the present invention, and not to limit the protection scope of the present invention. All equivalent changes and modifications made according to the spirit of the present invention should be covered within the protection scope of the present invention.

Claims (6)

1. An asynchronous fault tolerant control method for switching systems with actuator and sensor faults, comprising the steps of:
step 1: constructing an augmentation system by transforming actuator and sensor faults as part of a continuous time switching system state;
the augmentation system is as follows:
Figure FDA0003912557630000011
wherein x (t) ∈ R n Is in a state of,
Figure FDA00039125576300000113
Is a control input, y (t) is ∈ R m Is the output of the measurement, and,
Figure FDA00039125576300000114
is a disturbance, A i ,B i ,C i ,D i Is a matrix of appropriate dimensions, assuming (A) i ,B i ) Is controllable (A) i ,C i ) Is considerable; the derivative of the actuator fault is norm bounded, i.e.:
Figure FDA0003912557630000012
Figure FDA0003912557630000013
Figure FDA0003912557630000014
Figure FDA0003912557630000015
indicating an actuator failure and a sensor failure respectively,
Figure FDA0003912557630000016
namely that
Figure FDA0003912557630000017
And 2, step: a fault estimation observer is provided for the augmentation system in the step 1, and meanwhile, the switching signal of the fault estimation observer and the continuous time switching system are supposed to have inevitable lag, so that asynchronous switching between the continuous time switching system and the fault estimation observer is caused;
the fault estimation observer is:
Figure FDA0003912557630000018
wherein the content of the first and second substances,
Figure FDA0003912557630000019
in order to gain the observer,
Figure FDA00039125576300000110
is that
Figure FDA00039125576300000111
Is estimated in the state of (a) of (b),
Figure FDA00039125576300000112
is the output of the observer, L σ'(t) Is the observer gain, σ' (t) is the switching signal, assuming the activated subsystem of the augmentation system (7)Delay with observer, set Ω = [ t = [) 0 ,t 1 )∪[t 1 +Δ 1 ,t 2 )∪…∪[t k +Δ k ,t k+1 ) U (8230) ("omega" = [ t ") representing mutually matched time 1 ,t 1 +Δ 1 )∪…∪[t k ,t k +Δ k ) U \8230denotesunmatched time; t is t k And t k +Δ k K =0,1, \ 8230, representing the switching instants of σ (t), σ' (t), respectively;
Figure FDA0003912557630000021
and step 3: giving the error system of the fault estimation observer asymptotically stable and satisfying H ∞ The sufficient conditions of the performance indexes are as follows:
for a given constant α > 0, β > 0, μ 1 ＞1,μ 2 > 1, gamma > 0, if a positive definite matrix P is present i ＞0,P ij > 0, and matrix Q i ,Q ij For i ≠ j, i, j ∈ N, such that:
P j ≤μ 1 P ij ,P ij ≤μ 2 P i (15)
Figure FDA0003912557630000022
Figure FDA0003912557630000023
Figure FDA0003912557630000024
wherein the content of the first and second substances,
Figure FDA0003912557630000025
Figure FDA0003912557630000026
if any of the switching signals satisfies the conditions (15) - (17) and the average dwell time condition (18), the error system is asymptotically stable and satisfies H ∞ A performance index γ; wherein, T - (t 0 T) and T + (t 0 T) is expressed at time [ t 0 T) total cycles within match and mismatch;
and 4, step 4: based on the fault estimation information, a state feedback controller based on a fault estimation observer is designed to ensure the stability of a closed-loop system;
the state feedback controller based on the fault estimation observer is as follows:
Figure FDA0003912557630000027
wherein, K i Is the state feedback controller gain, let B i ＝F i ；
The error system giving the state feedback controller is asymptotically stable and satisfies H ∞ Sufficient conditions of performance indexes:
for a given constant
Figure FDA0003912557630000031
If a positive definite matrix R exists i ＞0,R ij > 0, and a matrix W i So that:
Figure FDA0003912557630000032
Figure FDA0003912557630000033
Figure FDA0003912557630000034
wherein:
Figure FDA0003912557630000035
Figure FDA0003912557630000036
if the following average dwell time condition (18) is satisfied for any switching signal, the error system is asymptotically stable and satisfies H ∞ Performance index gamma 1 (ii) a Wherein, T - (t 0 T) and T + (t 0 And t) are respectively expressed at time [ t ] 0 And t) total cycles within match and mismatch.
2. The asynchronous fault-tolerant control method for switching systems with actuator and sensor faults according to claim 1, characterized in that the step of constructing an augmented system in step 1 comprises:
step 2.1 the continuous time switching system is:
Figure FDA0003912557630000037
assuming that the perturbation is norm-bounded, assuming that the switching signal σ (t) is time-varying, the switching sequence is t 0 ＜t 1 ＜…＜t k 8230in which t k Is the instant of switching; when σ (t) = i, i ∈ N = {1,2, \8230; N }, indicating that the ith subsystem is activated;
step 2.2 the continuous time switching system (1) in step 2.1 is simplified to:
Figure FDA0003912557630000041
step 2.3 assuming that both actuator and sensor faults are present in the system (2), the system (2) suffering from the fault can be described as:
Figure FDA0003912557630000042
the following matrix is also defined:
Figure FDA0003912557630000043
the dynamic system (3) is then:
Figure FDA0003912557630000044
wherein the content of the first and second substances,
Figure FDA0003912557630000045
then the following results are obtained:
Figure FDA0003912557630000049
namely:
Figure FDA0003912557630000046
step 2.4 system (5) is converted to the augmentation system described in step 1.
3. The asynchronous fault-tolerant control method for a switching system with an actuator and a sensor fault according to claim 2, characterized in that the fault estimation observer in step 2 is specifically designed as follows:
step 3.1 the fault estimation observer is represented as follows:
Figure FDA0003912557630000047
wherein σ' (t) is a switching signal of the observer (8);
step 3.2 the state error of the fault estimation observer is as follows:
Figure FDA0003912557630000048
the derivative of the state error e (t) is known from (7) and (8):
Figure FDA0003912557630000051
and:
Figure FDA0003912557630000052
wherein the content of the first and second substances,
Figure FDA0003912557630000053
step 3.3 the error system (9) or (10) is considered stable and satisfies H if it satisfies the following two conditions ∞ Performance index γ:
a: in the absence of disturbances, the error system (9) or (10) is asymptotically stable;
b: when the initial condition is zero, then there are:
Figure FDA0003912557630000054
step 3.4 defines the auxiliary variable η (t) and gives the accurate state estimation observer of the augmented system (7):
Figure FDA0003912557630000055
the precise state estimation observer of the augmented system (7) is the fault estimation observer (14) in step 2.
4. Asynchronous fault-tolerant control method for switching systems with actuator and sensor faults according to claim 3, characterized in that the parameters of the step 3 fault estimation observer (14) are designed as follows:
Figure FDA0003912557630000056
5. an asynchronous fault-tolerant control method for a switching system with actuator and sensor faults according to claim 4, characterized in that the state feedback controller (20) is substituted into the continuous-time switching system in step 4, and after simplification:
Figure FDA0003912557630000057
Figure FDA0003912557630000058
wherein the content of the first and second substances,
Figure FDA0003912557630000061
6. asynchronous fault-tolerant control method for switching systems with actuator and sensor failure according to claim 5, characterized in that the parameters in the state feedback controller (20) are designed as follows:
Figure FDA0003912557630000062
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