CN111090945B - Actuator and sensor fault estimation design method for switching system - Google Patents

Actuator and sensor fault estimation design method for switching system Download PDF

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CN111090945B
CN111090945B CN201911326838.9A CN201911326838A CN111090945B CN 111090945 B CN111090945 B CN 111090945B CN 201911326838 A CN201911326838 A CN 201911326838A CN 111090945 B CN111090945 B CN 111090945B
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fault
actuator
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CN111090945A (en
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杨月
杜董生
吴瑜
王业琴
唐中一
周红标
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Huaiyin Institute of Technology
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Abstract

The invention discloses a method for switchingThe fault estimation design method for the actuator and the sensor of the system comprises the following steps: 1) establishing a continuous time switching system model to complete preparation work; 2) designing a fault estimation observer, and designing observer parameters to obtain a gain matrix of the observer; 3) accurately estimating the state of the system on line
Figure DDA0002328595890000011
Actuator failure fa(t) sensor failure fs(t) and a measurement disturbance ω (t). Compared with the prior art, the fault diagnosis observer based on the adaptive observer meets the robustness of the fault diagnosis system to external disturbance, so that the error system is stable and meets the requirement of H∞And the performance index realizes accurate online estimation of faults of an actuator and a sensor of the system.

Description

Actuator and sensor fault estimation design method for switching system
Technical Field
The present invention relates to a design method for an adaptive fault estimation observer for switching systems with actuator and sensor faults.
Background
With the rapid development of science and technology, engineering systems become more and more complex, which makes the requirements for the safety and reliability of the systems higher and higher. In the actual production process, the system is inevitably failed, which can cause the performance reduction, the stability deterioration and the system breakdown of the system, and even bring catastrophic property and personnel loss. An effective method, namely a fault diagnosis technology, is provided for solving the problem of faults in industrial production, and is widely applied. The fault diagnosis technology comprises three parts of fault detection, fault isolation and fault estimation. The fault detection is the primary work of the fault diagnosis technology, namely, whether a system has a fault or not is quickly judged. And determining the fault, and then performing the next fault estimation and other work. Compared with the fault detection technology, the fault estimation technology can obtain more fault information, such as: the form of the failure. At present, observer-based fault diagnosis methods are more popular, such as adaptive observers, sliding mode observers, proportional-integral observers (PIO), Unknown Input Observers (UIO), and the like. Where the output and variance of a proportional-integral observer (PIO) are considered known and can be used, but if there is a measurement disturbance or sensor fault in the system, a conventional observer cannot be applied directly, which may be amplified by the observer gain. Therefore, various observer design methods are proposed, wherein the generalized observer design method is widely applied, including the original system state and the sensor fault. From the above results, it can be seen that the system model plays an important role, and the fault estimation observer can be designed according to the model information. However, in practical applications, it is very difficult to know all the information of the dynamic system, and measurement disturbance may affect the estimation result. Further, it is noted that when the actuator failure is slowly time varying, the PIO can achieve good estimation performance. Conversely, if the fault changes frequently, the estimated performance may be affected by the fault differential.
In an actual engineering system, an electric power system, a chemical system, a mechanical system, an aerospace system and the like can be modeled into a switching system in a mathematical analysis mode, and control theory research based on a model method becomes one of important research directions of a control theory. With a very practical modeling form, the fault diagnosis research of the switching system has become one of the mainstream research directions in the field of control theory, and a great deal of researchers are attracted to research on the fault diagnosis research in recent years. The switching system is a kind of hybrid system, and includes a plurality of subsystems (continuous subsystems or discrete subsystems) and a rule for coordinating switching between the subsystems. The switching signal can be classified into an arbitrary switching and a constrained switching according to the switching characteristics. The average residence time (ADT) technique is a limited switching signal that requires the average running time of the subsystems to be greater than a constant value for a limited time, with less conservation than the classical algorithms. Therefore, the ADT technology is widely used in fault diagnosis and fault-tolerant control.
The present document mainly studies the problem of fault estimation for a class of continuous-time linear switching systems. Firstly, based on average residence time (ADT) and switching Lyapunov function, a fault estimation observer of an augmented switching system is designed, so that an error system is gradually stabilized. On the basis, the fault of the system is accurately estimated. Finally, the estimation effect of the designed observer is illustrated by calculation.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems in the prior art, the invention provides a fault estimation design method for an actuator and a sensor of a switching system, which can accurately estimate the fault form of the system on line, so that an error system is gradually stable, the elimination of external disturbance by a fault diagnosis system is realized, and the on-line fault estimation of the system is satisfied.
The technical scheme is as follows: the invention provides a fault estimation design method for an actuator and a sensor of a switching system, which comprises the following steps:
step 1: constructing a continuous time switching system model, and completing related preparation work, wherein the related preparation work comprises introducing a replacement variable and reconstructing an augmentation system;
the continuous time switching system model is as follows:
Figure GDA0002540072810000021
wherein, x (t) ∈ Rn，
Figure GDA0002540072810000022
d(t)∈RdRespectively representing a state vector, a control input vector, a measurable output vector and an external disturbance vector of the system; the system has actuator fault, sensor fault and measurement disturbance caused by
Figure GDA0002540072810000023
Is represented by RnRepresenting a set of n-dimensional real vectors, Ai，Bi，Ci，Di，Fi，Wi，GiRepresenting a known constant matrix of appropriate dimensions; sigmai(t) is a switching signal, which is required to satisfy:
Figure GDA0002540072810000024
step 2: aiming at the continuous time switching system model in the step 1, designing a fault estimation observer, and designing observer parameters to obtain a gain matrix of the observer;
the fault estimation observer is designed as follows:
Figure GDA0002540072810000031
wherein the content of the first and second substances,
Figure GDA0002540072810000032
and
Figure GDA0002540072810000033
which represents the state of the observer,
Figure GDA0002540072810000034
respectively represent z (t), x (t),
Figure GDA0002540072810000035
fa(t) and y (t) is an adjustable parameter, and the gain of the fault estimation observer is
Figure GDA0002540072810000036
And step 3: estimating the state of the system on line
Figure GDA0002540072810000037
Actuator failure fa(t) sensor failure fs(t) and a measurement disturbance ω (t). Further, the process of introducing the replacement variable and reconstructing the augmented system in step 1 is as follows:
2.1) introducing the variable η (t):
let η (t) be [ omega ]T(t)，fs T(t)]T，
Figure GDA0002540072810000038
The output equation can be written as follows:
Figure GDA0002540072810000039
wherein the content of the first and second substances,
Figure GDA00025400728100000310
2.2) introduction of variables
Figure GDA00025400728100000311
Order to
Figure GDA00025400728100000312
Based on the output equation (3) rewritten in 3.1), we have:
Figure GDA00025400728100000313
wherein the content of the first and second substances,
Figure GDA00025400728100000314
Figure GDA00025400728100000315
from the output equation (4), we get:
Figure GDA0002540072810000041
therefore, the temperature of the molten metal is controlled,
Figure GDA0002540072810000042
adding at the same time on both sides of formula (5)
Figure GDA0002540072810000043
Obtaining:
Figure GDA0002540072810000044
wherein the content of the first and second substances,
Figure GDA0002540072810000045
because of the fact that
Figure GDA0002540072810000046
Is of full rank, with its left inverse present
Figure GDA0002540072810000047
To express, then can know
Figure GDA0002540072810000048
Thus exist
Figure GDA0002540072810000049
So that
Figure GDA00025400728100000410
The following can be obtained:
Figure GDA00025400728100000411
wherein the content of the first and second substances,
Figure GDA00025400728100000412
2.3) the output y (t) is measurable, and
Figure GDA00025400728100000413
is difficult to measure, in the above formula (7), the right side of the equal sign contains
Figure GDA00025400728100000414
To this end, a new variable z (t) is introduced:
Figure GDA00025400728100000415
wherein:
Figure GDA00025400728100000416
from (8) can be obtained:
Figure GDA0002540072810000051
further, the system state error and parameters of the fault estimation observer in the step 2 are designed as follows:
3.1) systematic state error:
defining:
Figure GDA0002540072810000052
then there is e (t) ═ ez(t), it is possible to obtain:
Figure GDA0002540072810000053
the following can be obtained:
Figure GDA0002540072810000054
3.2) definition:
Figure GDA0002540072810000055
then there are:
Figure GDA0002540072810000056
wherein the content of the first and second substances,
Figure GDA0002540072810000057
Figure GDA0002540072810000058
definition of
Figure GDA0002540072810000061
The following error system can be derived from equation (14) above:
Figure GDA0002540072810000062
3.3) observer parameter design:
case 1: when in use
Figure GDA0002540072810000063
The error system equation (15) is asymptotically stable and satisfies H∞The performance index γ, that is:
Figure GDA0002540072810000064
case 2: when in use
Figure GDA0002540072810000065
If there is a positive definite matrix Pi＝Pi T> 0, and matrix QiSo that:
Figure GDA0002540072810000066
wherein the content of the first and second substances,
Figure GDA0002540072810000067
the ADT constraint is satisfied for any switching signal:
Figure GDA0002540072810000068
the error system (15) is stable and satisfies H∞The performance index γ. By solving the linear matrix inequality (17), the gain matrix in the fault estimation observer in 4.1) can be obtained as:
Figure GDA0002540072810000069
further, the state of the system is estimated online in step 3
Figure GDA00025400728100000610
Actuator failure fa(t) sensor failure fs(t) and a measured disturbance ω (t), which can be done as follows:
Figure GDA00025400728100000611
has the advantages that:
1. based on the adaptive observer technique, a new adaptive dynamic proportional-integral observer (PIO) is designed to estimate system states, actuator and sensor faults, and measure disturbances. In a dynamic proportional-integral observer (PIO), the output and output differential information of an original system are used, which is different from the existing method, the fault form of the system can be accurately estimated on line, an error system is enabled to be gradually stable, the elimination of external disturbance by a fault diagnosis system is realized, and the online fault estimation of the system is satisfied.
2. The designed fault estimation observer satisfies H∞The performance is gradually stable;
3. the fault estimation design method provided by the invention has general adaptability.
Drawings
FIG. 1: a flow diagram of the present invention;
FIG. 2: switching signal diagram sigma in the inventioni(t)；
FIG. 3: external disturbances in the system: white noise d (t);
FIG. 4: actuator fault signal fa(t) and estimation thereof
Figure GDA0002540072810000071
FIG. 5: sensor fault signal fs(t) and estimation thereof
Figure GDA0002540072810000072
FIG. 6: measuring disturbance signal omega (t) and its estimation
Figure GDA0002540072810000073
Detailed Description
The present invention will be described in detail with reference to the accompanying drawings.
The invention provides a design method of a self-adaptive fault estimation observer 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 occurring in the system.
And (4) note marking: p involved in the algorithm of the inventionT，P-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, RnRepresenting 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.
In the present invention, some of the quotations and related definitions are used, and a brief description is given first:
1) definition 1: for arbitrary switching signal sigmai(t) and an arbitrary time t2＞t1Is greater than 0, order
Figure GDA0002540072810000079
Is shown in the time period (t)1，t2) Inner number of handovers, for a given N0≥0，τa> 0 if the following holds:
Figure GDA0002540072810000074
then constant τaReferred to as average residence time, N0Called buffeting limit value.
2) Note 1: the switching signal involved in the invention is a slow switching signal, and has a residence time longer than the classical residence timeThe switching signal is less conservative and more general, and for simplicity, the invention makes the buffeting boundary N0＝0。
3) Introduction 1: considering continuous time switching system model
Figure GDA0002540072810000075
Let α > 0, μ > 1 all be given constants
Figure GDA0002540072810000076
Function(s)
Figure GDA0002540072810000077
And two types
Figure GDA0002540072810000078
Function k1，k2Such that:
Figure GDA0002540072810000081
and is
Figure GDA0002540072810000082
Vi(x(t))≤μVi(x (t)), then the switching system
Figure GDA0002540072810000083
ADT constraints are satisfied for any switching signal:
Figure GDA0002540072810000084
the system is stable.
4) Assume that 1: matrix Wi，GiIs of full rank.
The fault estimation method comprises the following steps:
step 1: and establishing a continuous time switching system model and carrying out preparation work.
The continuous time switching system model is as follows:
Figure GDA0002540072810000085
wherein, x (t) ∈ Rn，
Figure GDA0002540072810000086
d(t)∈RdRespectively representing a state vector, a control input vector, a measurable output vector and an external disturbance vector of the system. The actuator fault, sensor fault and measurement disturbance of the system can be respectively composed of
Figure GDA0002540072810000087
Is represented by RnRepresenting a set of n-dimensional real vectors, Ai，Bi，Ci，Di，Fi，Wi，GiRepresenting a known constant matrix of appropriate dimensions; sigmai(t) is a switching signal, which is required to satisfy:
Figure GDA0002540072810000088
the parameters of each constant real matrix of the system are expressed as follows:
Figure GDA0002540072810000089
Figure GDA00025400728100000810
Figure GDA00025400728100000811
Figure GDA00025400728100000812
Figure GDA00025400728100000813
for simplicity, we first make some variable substitutions, the process is as follows:
1) introducing variable η (t):
let η (t) be [ omega ]T(t)，fs T(t)]T，
Figure GDA0002540072810000091
The output equation can be written as follows:
Figure GDA0002540072810000092
wherein the content of the first and second substances,
Figure GDA0002540072810000093
2) introducing variables
Figure GDA0002540072810000094
Order to
Figure GDA0002540072810000095
The output equation rewritten in equation (3), we have:
Figure GDA0002540072810000096
wherein the content of the first and second substances,
Figure GDA0002540072810000097
Figure GDA0002540072810000098
at this time, from the output equation (4), we find:
Figure GDA0002540072810000099
thus:
Figure GDA00025400728100000910
we add to both sides of formula (5) at the same time
Figure GDA00025400728100000911
Thereby obtaining:
Figure GDA00025400728100000912
wherein the content of the first and second substances,
Figure GDA00025400728100000913
because of the fact that
Figure GDA0002540072810000101
Is full rank, so its left inverse exists
Figure GDA0002540072810000102
To express, then can know
Figure GDA0002540072810000103
Thus exist
Figure GDA0002540072810000104
So that
Figure GDA0002540072810000105
Based on the above analysis it can be found that:
Figure GDA0002540072810000106
wherein the content of the first and second substances,
Figure GDA0002540072810000107
3) in general, we know that the output y (t) is measurable, and
Figure GDA0002540072810000108
is difficult to measure, in the above formula (7), the right side of the equal sign contains
Figure GDA0002540072810000109
To avoid this problem, we introduce a new variable z (t):
order to
Figure GDA00025400728100001010
Is easy to obtain
Figure GDA00025400728100001011
Thus, from (8):
Figure GDA00025400728100001012
step 2: the following fault estimation observer is designed, and the specific content is as follows:
1) based on the transformation of the above equation (9), the following fault observer is designed, and the process is as follows:
Figure GDA0002540072810000111
in the formula (10), the compound represented by the formula (10),
Figure GDA0002540072810000112
and
Figure GDA0002540072810000113
which represents the state of the observer,
Figure GDA0002540072810000114
respectively represent z (t), x (t),
Figure GDA0002540072810000115
fa(t) and y (t), are adjustable parameters,
Figure GDA0002540072810000116
representing the observer gain.
2) Error of system state:
defining:
Figure GDA0002540072810000117
we have e (t) ═ ez(t), then, it can be known that:
Figure GDA0002540072810000118
then, it can be known that:
Figure GDA0002540072810000119
3) if it is determined that
Figure GDA00025400728100001110
Then there are:
Figure GDA0002540072810000121
wherein the content of the first and second substances,
Figure GDA0002540072810000122
Figure GDA0002540072810000123
if we define
Figure GDA0002540072810000124
The following error system can be derived from (14):
Figure GDA0002540072810000125
4) the observer parameter design process is carried out as follows:
theorem: for a given parameter γ > 0, μ > 1, α > 0:
case 1: when in use
Figure GDA0002540072810000126
The error system (15) is asymptotically stable and satisfies H∞The performance index γ, that is:
Figure GDA0002540072810000127
case 2: when in use
Figure GDA0002540072810000128
If there is a positive definite matrix Pi＝Pi T> 0, and matrix QiSo that:
Figure GDA0002540072810000129
wherein the content of the first and second substances,
Figure GDA00025400728100001210
the ADT constraint is satisfied for any switching signal:
Figure GDA00025400728100001211
the error system (15) is stable and satisfies H∞The performance index γ. By solving the linear matrix inequality (17), the gain matrix in the fault estimation observer in step 4.1) can be obtained as:
Figure GDA00025400728100001212
the proof process of theorem is as follows: for a given parameter γ > 0, μ > 1, α > 0:
(1) when in use
Figure GDA00025400728100001213
Then, we build the stability of the stand (15): the following switching Lyapunov function is defined:
Figure GDA00025400728100001214
wherein, Pi> 0, one can get:
Figure GDA0002540072810000131
we define:
Figure GDA0002540072810000132
the formula (16) is also satisfied by the following formula:
Figure GDA0002540072810000133
that is, when t → ∞,
Figure GDA0002540072810000134
then J < 0 and the error system of equation (15) is asymptotically stable.
(2) When in use
Figure GDA0002540072810000135
In time, order:
Figure GDA0002540072810000136
in the formula (22), the reaction mixture is,
Figure GDA0002540072810000137
therefore, if (17) is true, φ < 0 can be obtained. Given below is H∞Proof of performance index, for the system (15), defines:
Figure GDA0002540072810000138
then there are:
Figure GDA0002540072810000139
the formula (22) also represents:
Figure GDA00025400728100001310
then we have J < 0, that is when
Figure GDA00025400728100001311
The certification is complete.
And step 3: for accurately estimating the state of the system on line
Figure GDA00025400728100001312
Actuator failure fa(t) sensor failure fs(t) and measuring the disturbance ω (t), and performing the following work by a specific method:
the alternative known sensor fault estimation and measurement disturbance estimation from the variables described above can be represented by the following equations:
Figure GDA0002540072810000141
in summary, the algorithm of the present invention comprises the following steps:
the first step is as follows: an augmentation system (7) is constructed.
The second step is that: calculating the gain matrix K of the observer according to (17)1i，K2iAnd an unknown matrix Pi，Qi。
The third step: based on the fault estimation observer (equation (10)), a state estimate can be obtained
Figure GDA0002540072810000142
And actuator fault estimation
Figure GDA0002540072810000143
The fourth step: since the state estimate has already been obtained in the third step, we can obtain a sensor fault estimate by (25)
Figure GDA0002540072810000144
And measurement disturbance estimation
Figure GDA0002540072810000145
When α is equal to 0.5, mu is equal to 1.01, and gamma is equal to 0.2, the product can be obtained
Figure GDA0002540072810000146
By using a linear matrix inequality tool in MATLAB, a gain matrix K can be obtained1i，K2i：
Figure GDA0002540072810000147
Figure GDA0002540072810000148
K21＝1.0*e8[-1.3358 0.0474]，K22＝1.0*e7[-1.8078 8.4465]，
K23＝1.0*e8[-2.0972 1.5217]
Assuming that the switching system actuator fails, the failure model is as follows:
Figure GDA0002540072810000149
assuming a ramp signal when a sensor of the system fails, the failure model is as follows:
Figure GDA00025400728100001410
we consider the measured disturbance as a time-varying signal, modeled as follows:
Figure GDA0002540072810000151
for the simulation, the switching signal of the system is shown in FIG. 3, FIG. 4 is the external disturbance white noise and the actuator fault signal f in the systema(t) and estimation thereof
Figure GDA0002540072810000152
As shown in fig. 4, sensor failure signal fs(t) and estimation thereof
Figure GDA0002540072810000153
FIG. 6 shows the measured disturbance signal ω (t) and its estimation, as shown in FIG. 5
Figure GDA0002540072810000154
From the simulation result, when the system has a fault, the fault noble observer designed by the invention can estimate the fault of the actuator, the fault of the sensor and the measurement disturbance on line, and 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 (4)

1. A design method for fault estimation of actuators and sensors of a switching system, comprising the steps of:
step 1: constructing a continuous time switching system model, and completing related preparation work, wherein the related preparation work comprises introducing a replacement variable and reconstructing an augmentation system;
the continuous time switching system model is as follows:
Figure FDA0002540072800000011
wherein, x (t) ∈ Rn，
Figure FDA0002540072800000012
d(t)∈RdRespectively representing a state vector, a control input vector, a measurable output vector and an external disturbance vector of the system; actuator failure, sensor failure of a systemAnd measuring the disturbance respectively by
Figure FDA0002540072800000013
Is represented by RnRepresenting a set of n-dimensional real vectors, Ai，Bi，Ci，Di，Fi，Wi，GiRepresenting a known constant matrix of appropriate dimensions; sigmai(t) is a switching signal, which is required to satisfy: sigmai(t)：[0，∞)→{0，1}，
Figure FDA0002540072800000014
Step 2: aiming at the continuous time switching system model in the step 1, designing a fault estimation observer, and designing observer parameters to obtain a gain matrix of the observer;
the fault estimation observer is designed as follows:
Figure FDA0002540072800000015
wherein the content of the first and second substances,
Figure FDA0002540072800000016
and
Figure FDA0002540072800000017
which represents the state of the observer,
Figure FDA0002540072800000018
respectively represent z (t), x (t),
Figure FDA0002540072800000019
fa(t) and y (t) is an adjustable parameter, and the gain of the fault estimation observer is
Figure FDA00025400728000000110
And step 3: on-lineEstimating the state x (t) of the system, the actuator fault fa(t) sensor failure fs(t) and a measurement disturbance ω (t).
2. The actuator and sensor fault estimation design method for switching system as claimed in claim 1, characterized in that the process of introducing replacement variables and reconstructing the augmented system in step 1 is as follows:
2.1) introducing the variable η (t):
let η (t) be [ omega ]T(t)，fs T(t)]T，
Figure FDA0002540072800000021
The output equation can be written as follows:
Figure FDA0002540072800000022
wherein the content of the first and second substances,
Figure FDA0002540072800000023
2.2) introduction of variables
Figure FDA0002540072800000024
Order to
Figure FDA0002540072800000025
Based on the output equation (3) rewritten in 3.1), we have:
Figure FDA0002540072800000026
wherein the content of the first and second substances,
Figure FDA0002540072800000027
Figure FDA0002540072800000028
from the output equation (4), we get:
Figure FDA0002540072800000029
therefore, the temperature of the molten metal is controlled,
Figure FDA00025400728000000210
adding at the same time on both sides of formula (5)
Figure FDA00025400728000000211
Obtaining:
Figure FDA0002540072800000031
wherein the content of the first and second substances,
Figure FDA0002540072800000032
because of the fact that
Figure FDA0002540072800000033
Is of full rank, with its left inverse present
Figure FDA0002540072800000034
To express, then can know
Figure FDA0002540072800000035
Thus exist
Figure FDA0002540072800000036
So that
Figure FDA0002540072800000037
The following can be obtained:
Figure FDA0002540072800000038
wherein the content of the first and second substances,
Figure FDA00025400728000000315
2.3) the output y (t) is measurable, and
Figure FDA00025400728000000310
is difficult to measure, in the above formula (7), the right side of the equal sign contains
Figure FDA00025400728000000311
To this end, a new variable z (t) is introduced:
Figure FDA00025400728000000312
wherein:
Figure FDA00025400728000000313
from (8) can be obtained:
Figure FDA00025400728000000314
3. the actuator and sensor fault estimation design method for switching systems of claim 2, wherein the system state errors and parameters of the step 2 fault estimation observer are designed as follows:
3.1) systematic state error:
defining:
Figure FDA0002540072800000041
then there is e (t) ═ ez(t), it is possible to obtain:
Figure FDA0002540072800000042
the following can be obtained:
Figure FDA0002540072800000043
3.2) definition:
Figure FDA0002540072800000044
then there are:
Figure FDA0002540072800000045
wherein the content of the first and second substances,
Figure FDA0002540072800000046
Figure FDA0002540072800000047
definition of
Figure FDA0002540072800000048
The following error system can be derived from equation (14) above:
Figure FDA0002540072800000049
3.3) observer parameter design:
case 1: when in use
Figure FDA00025400728000000410
The error system equation (15) is asymptotically stable and satisfies H∞The performance index γ, that is:
Figure FDA0002540072800000051
case 2: when in use
Figure FDA0002540072800000052
If there is a positive definite matrix Pi＝Pi T> 0, and matrix QiSo that:
Figure FDA0002540072800000053
wherein the content of the first and second substances,
Figure FDA0002540072800000059
the ADT constraint is satisfied for any switching signal:
Figure FDA0002540072800000055
the error system (15) is stable and satisfies H∞The performance index γ is obtained by solving a linear matrix inequality (17), and the gain matrix in the fault estimation observer in 4.1) is:
Figure FDA0002540072800000056
4. the actuator and sensor fault estimation design method for switching system as claimed in claim 3, characterized in that the state of the system is estimated online in step 3
Figure FDA0002540072800000057
Actuator failure fa(t) sensor failure fs(t) and a measured disturbance ω (t), which can be done as follows:
Figure FDA0002540072800000058
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