CN111158343A - 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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CN111158343A
CN111158343A CN202010025435.7A CN202010025435A CN111158343A CN 111158343 A CN111158343 A CN 111158343A CN 202010025435 A CN202010025435 A CN 202010025435A CN 111158343 A CN111158343 A CN 111158343A
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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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    • 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 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 the actuator and the 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 technology, the engineering system is more and more complex, and the requirement on the safety of the system is 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) e RnIs in a state of,
Figure BDA0002362264050000022
Is the control input, y (t) e RmIs 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 t0＜t1＜…＜tk< …, where tkIs the instant of switching; when σ (t) ═ i, it means that the ith subsystem is activated, where i ∈ N ═ {1,2, … N };
step 2.2 the continuous time switching system (1) in step 2.1 is simplified to:
Figure BDA0002362264050000031
wherein A isi,Bi,Ci,DiIs a matrix of appropriate dimensions, assuming (A)i,Bi) Is controllable (A)i,Ci) 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
fs(t)∈Rmrespectively 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 isσ'(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, with the set Ω ═ t0,t1)∪[t1+Δ1,t2)∪…∪[tk+Δk,tk+1) ∪ … denote matching times, set Ω' ═ t1,t1+Δ1)∪…∪[tk,tk+Δk) ∪ … denotes the time of mismatch tkAnd tk+ΔkK is 0,1, …, and represents the switching instant of σ (t) and σ' (t), respectively;
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 variables η (t), giving the precise 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, μ1＞1,μ2> 1, gamma > 0, if a positive definite matrix P is presenti＞0,Pij> 0, and matrix Qi,QijFor i ≠ j, i, j ∈ N, such that:
Pj≤μ1Pij,Pij≤μ2Pi(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-(t0T) and T+(t0T) is expressed at time [ t0T) total cycles within match and mismatch;
step 4.2 the parameters of the precise state estimation observer (14) are designed as follows:
Figure BDA0002362264050000059
further, suppose B in the system (1)i＝FiThen, the observer-based state feedback controller in step 4 is designed as follows:
Figure BDA0002362264050000061
wherein, KiIs 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 existsi＞0,Rij> 0, and a matrix WiSo 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 gamma1(ii) a Wherein, T-(t0T) and T+(t0And t) are respectively expressed at time [ t ]0T) 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 discloses 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: fault estimates and fault error diagrams for f (t) of simulation example 1, where the solid line is fault and the dashed line is estimate;
FIG. 6: output y (t) vs. with and without adjustment 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 2a(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 2s(t) fault estimation and fault error diagrams thereof, 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 isT,P-1Respectively 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, RnRepresenting a set of n-dimensional real vectors, I and 0The identity matrix and 0 matrix are represented with appropriate dimensions, where the symmetry terms in the symmetry matrix are represented.
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) e RnIs in a state of,
Figure BDA0002362264050000082
Is the control input, y (t) e RmIs the output of the measurement, and,
Figure BDA0002362264050000083
is a perturbation and the perturbation is assumed to be norm-bounded.
Let us assume that the switching signal σ (t) is time-varying, the switching sequence being t0＜t1＜…＜tk< …, where tkIs the instant of switching. When σ (t) ═ i, it means that the ith subsystem is activated, where i ∈ N ═ {1,2, … N }. The original system (1) can be simplified as follows:
Figure BDA0002362264050000084
wherein A isi,Bi,Ci,DiIs 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,Bi) Is controllable (A)i,Ci) 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
fs(t)∈Rmindicating actuator and sensor faults, 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 fa(t) and sensor failure fs(t) defining the following matrix:
Figure BDA0002362264050000091
the dynamic system (3) is rewritten as:
Figure BDA0002362264050000092
wherein the content of the first and second substances,
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 content of the first and second substances,
Figure BDA0002362264050000098
is that
Figure BDA0002362264050000099
Is estimated in the state of (a) of (b),
Figure BDA00023622640500000910
is the output of the observer. L isσ'(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, tkAnd tk+ΔkK is 0,1, and …, and indicates the switching time instant of σ (t) and σ' (t), respectively. That is, the period Ω ═ t0,t1)∪[t1+Δ1,t2)∪…∪[tk+Δk,tk+1) ∪ … denotes the time of matching, and the period Ω' ═ t1,t1+Δ1)∪…∪[tk,tk+Δk) ∪ …, the time of mismatch.
Note 2: ideally, the last switch is synchronized with the subsystem, i.e., σ (t) — σ' (t), in effect, due to the identification of the subsystem to be started and the application of the matched observationThe device needs a certain time, the period delta in different environmentskAlso different, so we assume asynchronous handover maximum delay ΔmaxAre 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 content of the first and second substances,
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, then there are:
Figure BDA0002362264050000105
it can be seen that there is an unknown item H in the fault estimation observer (8)2J2σ'(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 σi(t) at an arbitrary time t2＞t1Greater than 0, let Nσi(t)(t1,t2) Representing the switching signal sigmai(t) at (t)1,t2) The number of handovers in. For a given N0≥0,τa> 0 if the following holds:
Figure BDA0002362264050000111
then constant τaReferred to as average residence time, N0Called 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, mu1＞1,μ2> 1, gamma > 0, if a positive definite matrix P is presenti＞0,Pij> 0, and matrix Qi,QijFor i ≠ j, i, j ∈ N, such that:
Pj≤μ1Pij,Pij≤μ2Pi(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-(t0T) and T+(t0T) is expressed at time [ t0T) 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:
Vi(t)＝eT(t)Pie(t) (20)
from the error system equation (9), it can be seen that:
Figure BDA0002362264050000121
definition of Qi＝PiLiThen 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, Vi(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:
Vij(t)＝eT(t)Pije(t) (24)
according to the error system (10), it is known that:
Figure BDA0002362264050000125
definition of Qij＝PijLiThen (25) can be converted into:
Figure BDA0002362264050000126
from the condition (17), it can be obtained:
Figure BDA0002362264050000127
thus, during the mismatch period, Vij(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 zero initial condition e (t) 0, and nonzero d (t) e L2[0, ∞), when t ∈ Ω is considered from the lyapunov function in equation (20), there is:
Figure BDA0002362264050000133
wherein the content of the first and second substances,
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,tk+Δk) When k is 1,2, …, we consider the lyapunov function in equation (25), similar to the proof when t e Ω, we have:
Figure BDA0002362264050000138
we can conclude as follows:
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, which ultimately gives the gain in the precision state estimation observer (14), is given by the auxiliary variable η (t) defined in equation (13):
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 the system (1)In, we assume Bi＝FiThe form of controller chosen at this time is as follows:
Figure BDA0002362264050000144
wherein, KiIs 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 existsi＞0,Rij> 0, and a matrix WiSo 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 gamma1。
Wherein, T-(t0T) and T+(t0T) is expressed at time [ t0T) 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:
Vi(t)＝xT(t)Xix(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 { RiMultiplying theta by I, I, I }1iThe following can be obtained:
Figure BDA0002362264050000165
wherein the content of the first and second substances,
Figure BDA0002362264050000166
from (47), theta can be known2i< 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:
Vij(t)＝xT(t)Xijx(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 { RiMultiplying theta by I, I, I }1ijThe following can be obtained:
Figure BDA0002362264050000175
wherein the content of the first and second substances,
Figure BDA0002362264050000176
from (48), theta can be known2ij< 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
When 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 choose toKnown parameter mu1＝μ2By solving the condition of theorem 1, 1.2, α, the following observer gain can be obtained:
Figure BDA0002362264050000189
further, assuming that the interference is band-limited white noise, the simulation is the same in both cases within 200s, and subsystems 1,2, and 3 are activated when σ (t) is 1, σ (t) is 2, and σ (t) is 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 α1＝0.7,β1＝1.05,ε1＝0.05,ε2＝0.05,γ1＝0.5,
Figure BDA00023622640500001811
By solving theorem 2, the gain in the controller can be found:
K1＝[13.5645 -5.0943],K2＝[8.2348 2.9745],K3＝[-0.3245 -1.2345]
y (t) output results with and without regulation are shown in fig. 6, and simulation results show that the designed controller has good fault tolerance.
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 Rin＝25Ω,Lin＝20mH,C＝15μF,rs＝10Ω,Lout＝30mH,R＝25Ω,υsIf mu is 18V, if1＝μ2When the average residence time condition in equation (18) is applied to equation (15) to equation (17), equation (1.2) and equation α is 0.9, the observer gain can be obtained:
Figure BDA0002362264050000193
the simulation is the same for 200s in both cases, and when σ (t) is 1 and σ (t) is 2, subsystems 1 and 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 faults of the actuator and the corresponding fault estimation errors; 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 α1＝0.7,β1＝1.05,ε1＝0.05,ε2＝0.05,γ1＝0.5,
Figure BDA0002362264050000196
By solving theorem 2, the gain in the controller can be found:
K1＝[-0.2343 16.2365 3.8344],K2＝[3.2389 11.4566 0.3479],
K3＝[0.4523 -8.4235 0.0013]
y (t) output results with and without regulation are shown in fig. 11, and simulation results show that the designed controller has good fault tolerance.
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;
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.
2. The asynchronous fault-tolerant control method for switching systems with actuator and sensor faults as recited in claim 1, wherein the constructing an augmentation system step in step 1 comprises:
step 2.1 the continuous time switching system is:
Figure FDA0002362264040000011
wherein x (t) e RnIs in a state of,
Figure FDA0002362264040000012
Is the control input, y (t) e RmIs the output of the measurement, and,
Figure FDA0002362264040000013
is a disturbance and assumes norm-bounded, the switching signal σ (t) is assumed to be time-varying, the switching sequence is t0＜t1＜…＜tk< …, where tkIs the instant of switching; when σ (t) ═ i, it means that the ith subsystem is activated, where i ∈ N ═ {1,2, … N };
step 2.2 the continuous time switching system (1) in step 2.1 is simplified to:
Figure FDA0002362264040000014
wherein A isi,Bi,Ci,DiIs a matrix of appropriate dimensions, assuming (A)i,Bi) Is controllable (A)i,Ci) Is considerable; the derivative of the actuator fault is norm bounded, i.e.:
Figure FDA0002362264040000015
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 FDA0002362264040000021
wherein the content of the first and second substances,
Figure FDA0002362264040000022
fs(t)∈Rmrespectively representing actuator faults and sensor faults; the following matrix is also defined:
Figure FDA0002362264040000023
the dynamic system (3) is then:
Figure FDA0002362264040000024
wherein the content of the first and second substances,
Figure FDA0002362264040000025
then the following results are obtained:
Figure FDA0002362264040000026
namely:
Figure FDA0002362264040000027
step 2.4 System (5) transforms into the augmentation System in step 1:
Figure FDA0002362264040000028
3. the asynchronous fault-tolerant control method for switching systems with actuator and sensor faults 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 FDA0002362264040000029
wherein
Figure FDA00023622640400000210
Is that
Figure FDA00023622640400000211
Is estimated in the state of (a) of (b),
Figure FDA00023622640400000212
is the output of the observer, Lσ'(t)Is the observer gain, σ' (t) is the switching signal of the observer (8), assuming that there is a delay between the observer and the activated subsystem of the augmented system (7), the set Ω ═ t0,t1)∪[t1+Δ1,t2)∪…∪[tk+Δk,tk+1) ∪ … denote matching times, set Ω' ═ t1,t1+Δ1)∪…∪[tk,tk+Δk) ∪ … denotes the time of mismatch tkAnd tk+ΔkK is 0,1, …, and represents the switching instant of σ (t) and σ' (t), respectively;
step 3.2 the state error of the fault estimation observer is as follows:
Figure FDA0002362264040000031
the derivative of the state error e (t) is known from (7) and (8):
Figure FDA0002362264040000032
and:
Figure FDA0002362264040000033
wherein the content of the first and second substances,
Figure FDA0002362264040000034
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 FDA0002362264040000035
step 3.4 defines the auxiliary variables η (t), giving the precise state estimation observer of the augmented system (7):
Figure FDA0002362264040000036
the precise state estimation observer of the augmented system (7) is then:
Figure FDA0002362264040000037
wherein the content of the first and second substances,
Figure FDA0002362264040000038
is the observer gain.
4. The asynchronous fault-tolerant control method for switching systems with actuator and sensor faults according to claim 3, characterized in that the error system given in 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, μ1＞1,μ2> 1, gamma > 0, if a positive definite matrix P is presenti＞0,Pij> 0, and matrix Qi,QijFor i ≠ j, i, j ∈ N, such that：
Pj≤μ1Pij,Pij≤μ2Pi(15)
Figure FDA0002362264040000041
Figure FDA0002362264040000042
Figure FDA0002362264040000043
Wherein the content of the first and second substances,
Figure FDA0002362264040000044
Figure FDA0002362264040000045
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-(t0T) and T+(t0T) is expressed at time [ t0T) total cycles within match and mismatch;
step 4.2 the parameters of the precise state estimation observer (14) are designed as follows:
Figure FDA0002362264040000046
5. asynchronous fault-tolerant control method for switching systems with actuator and sensor faults according to claim 4, characterized in that B in the system (1) is assumedi＝FiThen, the observer-based state feedback controller in step 4 is designed as follows:
Figure FDA0002362264040000047
wherein, KiIs the state feedback controller gain; then (20) is substituted into the system (1), and after simplification, the following can be known:
Figure FDA0002362264040000048
Figure FDA0002362264040000051
wherein d is1(t)＝Kσ'(t)J1e(t),d2(t)＝J3e(t),
Figure FDA0002362264040000052
6. The asynchronous fault-tolerant control method for switching systems with actuator and sensor faults as claimed in claim 5, characterized in that the state feedback controller asymptotically stabilizes and satisfies H∞Sufficient conditions of performance indexes:
step 6.1 for a given constant α1＞0,β1＞0,ε1＞0,ε2＞0,γ1＞0，
Figure FDA0002362264040000053
If a positive definite matrix R existsi＞0,Rij> 0, and a matrix WiSo that:
Figure FDA0002362264040000054
Figure FDA0002362264040000055
Figure FDA0002362264040000056
wherein:
Figure FDA0002362264040000057
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 gamma1(ii) a Wherein, T-(t0T) and T+(t0And t) are respectively expressed at time [ t ]0T) total cycles within match and mismatch;
step 6.2 the parameters in the state feedback controller (20) are designed as follows:
Figure FDA0002362264040000058
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