CN110209148B - Fault estimation method of networked system based on description system observer - Google Patents

Fault estimation method of networked system based on description system observer Download PDF

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CN110209148B
CN110209148B CN201910526147.7A CN201910526147A CN110209148B CN 110209148 B CN110209148 B CN 110209148B CN 201910526147 A CN201910526147 A CN 201910526147A CN 110209148 B CN110209148 B CN 110209148B
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姜顺
张青杭
潘丰
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Anhui Xincheng Youxuan Network Technology Co ltd
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Jiangnan University
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    • 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

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Abstract

A fault estimation method of a networked system based on a description system observer belongs to the field of networked systems. Firstly, establishing a discrete time networking system model under the conditions of sensor saturation, disturbance and faults, and performing state augmentation on an original system by regarding the faults as additional states, so that the system with the faults is equivalently transformed into a description system, and a novel description system observer is designed; obtaining sufficient conditions for asymptotic stability of the mean square of the state estimation error system and solution of a description system observer by using a Lyapunov stability theory and a linear matrix inequality analysis method; and solving the optimization problem by using a Matlab LMI tool kit to obtain parameters describing a system observer, and further obtain the amplitude of the fault and the information of the fault along with the change of the amplitude along with time. The method of the invention considers the packet loss, sensor saturation and fault of the system under the actual condition, and the observer can overcome the influence of the packet loss and disturbance, thus obtaining the ideal estimation effect.

Description

Fault estimation method of networked system based on description system observer
Technical Field
The invention belongs to the field of networked systems, and relates to a fault estimation method of a networked system based on a description system observer.
Background
With the rapid development of network technology, networked control systems are widely used in the control fields of industrial automation and the like. The networked control system is a spatially distributed system in which sensors, actuators, and controllers are connected by a shared digital communication network. Although the networked control system has the characteristics of strong flexibility, simple installation, convenient sharing and the like, the introduction of the network brings new problems. Due to limited spectrum resources, channel interference, network congestion, etc., problems of network-induced delay, packet loss, etc. often occur in the networked control system, which may deteriorate the system performance and become a factor of system instability.
The fault diagnosis method mainly comprises fault detection, fault separation and fault estimation, wherein the fault detection and separation method mainly comprises the steps of judging whether a system has a fault or not by generating a residual error and determining the position of the fault. And the amplitude information of the fault is an important basis of fault-tolerant control, so that the fault diagnosis method based on fault estimation has important significance.
Disclosure of Invention
In view of the above problems in the prior art, the present invention provides a method for estimating a fault of a networked system based on a description system observer. The method considers the conditions of packet loss, disturbance, fault and sensor saturation in the networked system, and performs state augmentation on the original system by regarding the fault as an additional state, so as to equivalently transform the system with the fault into a description system, and design a novel description system observer, so that the networked system can still maintain the progressive stability of mean square under the above conditions and meet a certain H∞The performance index can effectively obtain the fault estimation value of the system.
The technical scheme of the invention is as follows:
a method for fault estimation based on a networked system describing a system observer, comprising the steps of:
1) establishing a controlled object model of a networked system with faults and disturbances:
Figure GDA0002998894430000011
wherein:
Figure GDA0002998894430000012
is the state vector of the system and,
Figure GDA0002998894430000013
is the output vector of the system with saturation,
Figure GDA0002998894430000014
is the input of the disturbance of the system,
Figure GDA0002998894430000015
is the fault signal to be estimated, w (k) e l2[0,∞)l2Is a continuous function space of [0, ∞) integrable squared;
Figure GDA0002998894430000016
is a known constant matrix; saturation function σ (·):
Figure GDA0002998894430000017
is defined as
Figure GDA0002998894430000021
Where σ isi(νi)＝sign(νi)·min{νi,max,|νi|}，vi，max> 0 is the known saturation boundary, σ (-) is a multivariate saturation function, σi(. h) is the i-th component of the saturation function σ (-), viIs an unknown scalar quantity, representing a function sigmaiVariation of (·), for a given diagonal matrix R1，R2，R1≥0，R2Not less than 0 and R2＞R1σ (·) satisfies the following inequality:
[σ(y(k))-R1y(k)]T[σ(y(k))-R2y(k)]≤0 (3)
considering the fault signal f (k-1) at the time k-1 as an additional state, the following augmented state vector can be obtained
Figure GDA0002998894430000022
And construct the following augmentation system
Figure GDA0002998894430000023
Wherein,
Figure GDA0002998894430000024
is the augmented state vector of x (k), d (k) is the augmented state vector of w (k),
Figure GDA0002998894430000025
is the augmented state vector of y (k),
Figure GDA0002998894430000026
Inis an n x n dimensional identity matrix, and σ (Cx (k)) can be divided into the sum of a linear portion and a non-linear portion
Figure GDA0002998894430000027
Wherein
Figure GDA0002998894430000028
Is a non-linear vector function, and the saturation function sigma (DEG) satisfies an inequality constraint
Figure GDA0002998894430000029
M1And M2Are all known M x M dimensional symmetric positive definite matrices and M2＞M1Further, the formula (7) shows
Figure GDA00029988944300000210
Wherein
Figure GDA00029988944300000211
Considering the packet loss existing in the system, the measurement output is
Figure GDA00029988944300000212
Wherein: beta is akIs a random sequence satisfying Bernoulli, is used for describing the probability of packet loss occurring in the system, when beta iskWhen 1, no packet loss in the system is indicated, when βkWhen the value is 0, the data packet loss in the system is indicated. The possibility of packet loss is
Figure GDA0002998894430000031
2) Design description system observer:
Figure GDA0002998894430000032
wherein:
Figure GDA0002998894430000033
is the intermediate variable that is the variable between,
Figure GDA0002998894430000034
is in an augmented state
Figure GDA0002998894430000035
The estimated vector of (2);
Figure GDA0002998894430000036
is a parameter matrix to be designed, and T, N can be determined by equation (12).
Figure GDA0002998894430000037
Wherein
Figure GDA0002998894430000038
Is an arbitrarily selectable matrix, In×mIs an n × m dimensional identity matrix.
3) The solvable sufficient conditions for the system mean square asymptotic stability and the description of the system observer parameters are as follows:
Figure GDA0002998894430000039
wherein: w is P1L，
Figure GDA00029988944300000310
Denotes the transpose of the symmetric position matrix, 0 is the zero matrix;
Figure GDA00029988944300000311
Figure GDA00029988944300000312
is a symmetrical positive definite matrix and is characterized in that,
Figure GDA00029988944300000313
is an unknown matrix, gamma > 0 is a given system performance index I is an identity matrix,
Figure GDA00029988944300000314
is a known m x m dimensional symmetric positive definite matrix;
given constant
Figure GDA00029988944300000315
And a system performance index with gamma > 0, solving the equation (13) using the LMI toolkit in MATLAB, if a positive definite matrix P exists1,P2And a non-singular matrix W such that equation (13) holds, the system is asymptotically mean-square stable and satisfies H∞Performance index, can obtain non-optimal describing system observer parameter L ═ P1 -1W, i.e. step 4) can be performed); when the unknown variables have no feasible solution, the system is not mean square asymptotically stable, the parameters of the observer of the non-optimal description system cannot be obtained, and the step 4) cannot be carried out;
4) calculating optimal description system observer parameters
According to
Figure GDA0002998894430000041
The system performance index gamma is calculated, an optimization problem formula (14) is solved by utilizing an LMI tool box in MATLAB, and e (k) is a state estimation error:
Figure GDA0002998894430000042
when equation (14) has a solution, the optimal description system observer parameters can be obtained, and the optimal H∞The performance index is gammaminOf the type using(14) By solving the nonsingular matrix W, the optimal parameter L which describes the system observer can be obtained1 -1W。
When the formula (14) has no solution, the optimal description system observer parameters cannot be obtained;
5) fault estimation of networked systems based on a description system observer
From solving the optimization problem in equation (14), the system-describing observer gain parameter L can be obtained and then calculated from equation (11)
Figure GDA0002998894430000043
Figure GDA0002998894430000044
Thereby obtaining an estimated value I of the faultqIs a q-dimensional unit vector.
The invention has the beneficial effects that: the invention simultaneously considers the design method of the system observer under the conditions of system faults, sensor saturation constraints and disturbance existing in the networked system, can effectively overcome the influence of packet loss and disturbance in the networked system, and can quickly obtain the estimation of the faults of the actuator.
Drawings
Fig. 1 is a flow chart of a method of fault estimation based on a networked system describing a system observer.
FIG. 2 is
Figure GDA0002998894430000045
The actuator failure estimation map of (1).
FIG. 3 is
Figure GDA0002998894430000046
The actuator failure estimation map of (1).
FIG. 4 is
Figure GDA0002998894430000047
The actuator failure estimation map of (1).
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
Referring to fig. 1, a method for fault estimation based on a networked system describing a system observer includes the following steps:
step 1: modeling networked systems in the presence of system faults and disturbances
The model of the networked system with system faults, disturbances, and sensor saturation constraints is equation (15):
Figure GDA0002998894430000048
wherein:
Figure GDA0002998894430000049
is the state vector of the system and,
Figure GDA00029988944300000410
is the output vector of the system with saturation,
Figure GDA00029988944300000411
is the input of the disturbance of the system,
Figure GDA00029988944300000412
is the fault signal to be estimated, w (k) e l2[0,∞)，l2Is a continuous function space of [0, ∞) integrable squared;
Figure GDA00029988944300000413
is a known constant matrix; saturation function σ (·):
Figure GDA0002998894430000051
is defined as
Figure GDA0002998894430000052
Where σ isi(νi)＝sign(νi)·min{νi,max,|νi|}，vi，max> 0 isThe known saturation boundary, σ (-) is a multivariate saturation function, σi(. h) is the i-th component of the saturation function σ (-), viIs an unknown scalar quantity, representing a function sigmaiVariation of (·), for a given diagonal matrix R1，R2，R1≥0，R2Not less than 0 and R2＞R1σ (·) satisfies the following inequality:
[σ(y(k))-R1y(k)]T[σ(y(k))-R2y(k)]≤0 (17)
considering the fault signal f (k-1) at the time k-1 as an additional state, the following augmented state vector can be obtained
Figure GDA0002998894430000053
And construct the following augmentation system
Figure GDA0002998894430000054
Wherein,
Figure GDA0002998894430000055
is the augmented state vector of x (k), d (k) is the augmented state vector of w (k),
Figure GDA0002998894430000056
is the augmented state vector of y (k),
Figure GDA0002998894430000057
sigma (. beta.) satisfies
Figure GDA0002998894430000058
So σ (Cx (k)) can be divided into a linear part and a non-linear part, InIs an n × n dimensional identity matrix, M1Is a known M x M dimensional symmetric positive definite matrix, M2Is a known m × m dimensional symmetric positive definite matrix:
Figure GDA0002998894430000059
Figure GDA00029988944300000510
wherein
Figure GDA00029988944300000511
Non-linear vector function, M1＞0,M2＞0,M2＞M1,
Figure GDA00029988944300000512
Considering the packet loss existing in the system, the measurement output is
Figure GDA00029988944300000513
Wherein: beta is akIs a random sequence satisfying Bernoulli, is used for describing the probability of packet loss occurring in the system, when beta iskWhen 1, no packet loss in the system is indicated, when βkWhen the value is 0, the data packet loss in the system is indicated. The possibility of packet loss is
Figure GDA0002998894430000061
Step 2: design description system observer:
Figure GDA0002998894430000062
wherein:
Figure GDA0002998894430000063
is the intermediate variable that is the variable between,
Figure GDA0002998894430000064
is in an augmented state
Figure GDA0002998894430000065
The estimated vector of (2);
Figure GDA0002998894430000066
is a parameter matrix to be designed, T, N can be determined by equation (26).
Figure GDA0002998894430000067
Wherein
Figure GDA0002998894430000068
Is an arbitrarily selectable matrix, In×mIs an n × m dimensional identity matrix.
Defining state estimation errors
Figure GDA0002998894430000069
So that the error equation is
Figure GDA00029988944300000610
By substituting the formula (19) and the formula (25)
Figure GDA00029988944300000611
Wherein
Figure GDA00029988944300000612
By designing the state estimation error system as described above, the description system observer design with sensor saturation constraints can be converted to H∞The problem of fault estimation and the following requirements are fulfilled:
(1) the state estimation error system (29) is asymptotically mean square stable.
(2) At zero initial conditions, H of the system∞The performance index γ satisfies the following inequality.
Figure GDA0002998894430000071
And the performance index γ is required to be as small as possible.
And step 3: state estimation error system mean square asymptotic stabilization and sufficient condition for describing system observer solution
Constructing a Lyapunov function:
Figure GDA0002998894430000072
by utilizing a Lyapunov stability theory and a linear matrix inequality analysis method, the conditions of state estimation error system formula (29) mean square progressive stability and sufficient condition for describing the solution of a system observer are obtained. The method comprises the following steps:
step 3.1: and the state estimation error system is a sufficient condition for gradual stabilization of mean square.
Assuming that equation (32) holds:
Figure GDA0002998894430000073
wherein
Figure GDA0002998894430000074
The Lyapunov function (31) is biased along the track of the system
Figure GDA0002998894430000075
From equation (22), one can obtain:
Figure GDA0002998894430000076
definition of
Figure GDA0002998894430000077
The combined formulae (34) to (35) can give
E{ΔV(k)}≤ξT(k)Φ1ξ(k) (36)
According to Lyapunov stability theory, constants are given
Figure GDA0002998894430000081
If a positive definite matrix P exists1＞0，P2> 0, the non-singular matrix W is such that phi1If < 0, equation (36) holds and the system becomes increasingly stable as a mean square. When the sufficient condition of the step 3.1 is met, the step 3.2 is executed again; if the sufficiency of step 3.1 is not met, the state estimation error system (18) is not asymptotically mean square stable and step 3.2 cannot be performed.
Step 3.2: h of the system∞Performance analysis and characterization of sufficient conditions for System observer Presence
First, H is carried out∞Analysis of the performance index assumes that equation (37) holds:
Figure GDA0002998894430000082
wherein
Figure GDA0002998894430000083
To satisfy H of the system∞Performance indexes are as follows:
Figure GDA0002998894430000084
definition eta (k) ═ xiT(k) θT(k)]TThen canTo obtain
E{V(k+1)}-E{V(k)}+E{eT(k)e(k)}-γ2dT(k)d(k)＝ηT(k)Φη(k)＜0 (40)
The k is added from 0 to ∞ at the same time on both sides of equation (40):
Figure GDA0002998894430000085
assuming that the initial state of the system is η (0) ═ 0 and the system is asymptotically stable in the mean square, it is known that both values of V (∞) and V (0) are 0, and therefore the requirement of the performance index in equation (39) can be satisfied.
In order to satisfy a sufficient condition for describing the existence of a solution in the system observer, it is necessary that the expression (6) is satisfied. From equation (37), the original matrix can be written as follows:
Figure GDA0002998894430000091
applying Schur's complement theorem, equation (42) can be converted into:
Figure GDA0002998894430000092
for both sides of formula (43), left-and right-multiplying diag { I, I, I, I, P1,P1< CHEM > may have the formula (13).
Solving with LMI toolkit in MATLAB, given constants
Figure GDA0002998894430000093
And an index gamma > 0, when a positive definite matrix P is present1,P2And a non-singular matrix W such that equation (13) holds, the system is asymptotically mean-square stable and satisfies H∞Performance index, can obtain non-optimal describing system observer parameter L ═ P1 -1W, i.e. step 4) can be performed); when the above unknown variables have no feasible solution, the system is not mean square asymptotically stable and non-optimal can not be obtainedThe system observer parameters are well described, step 4) cannot be performed;
and 4, step 4: calculating optimal description system observer parameters
For the state estimation error system (29), an optimization problem formula (7) is solved by utilizing an LMI tool box in MATLAB, and if the formula (7) has a solution, the optimal H is obtained∞Performance index is λminObtaining optimal description system observer parameters; if equation (7) has no solution, optimal system-describing observer parameters cannot be obtained.
And 5: fault estimation of networked systems based on a description system observer
According to the actuator fault occurring when the networked system actually operates, the parameter L of the descriptive system observer is obtained by the formula (25), and then the parameter L is obtained by the calculation of the formula (11)
Figure GDA0002998894430000094
Figure GDA0002998894430000095
Thus obtaining the amplitude of the fault and its time-varying information.
Example (b):
by adopting the fault estimation method of the networked system based on the description system observer, provided by the invention, the filtering error system (29) is stable in mean square progression under the condition of considering external disturbance and fault. The specific implementation method comprises the following steps:
the model of a certain uninterruptible power supply networked system is formula (15), and the system parameters are given as follows:
Figure GDA0002998894430000101
Figure GDA0002998894430000102
the saturation function is taken here as:
Figure GDA0002998894430000103
wherein
Figure GDA0002998894430000104
To demonstrate the role of the description system observer, assume that the fault signal f (k) is:
Figure GDA0002998894430000105
meanwhile, in the system (1), a disturbance input is given, and in a real system, the disturbance input is always present, assuming that the disturbance input is as follows:
Figure GDA0002998894430000106
using the conditions given above, making the matrix T non-singular by selecting the appropriate matrix S, solving equation (13) through the LMI toolbox in MATLAB, using the LMI method, when
Figure GDA0002998894430000107
The minimum performance index γ of 42.19 can be derived, describing the system observer parameters as shown below
Figure GDA0002998894430000108
Figure GDA0002998894430000109
When in use
Figure GDA0002998894430000111
The minimum performance indicator γ can be derived as 52.29, describing the system observer parameters as follows:
Figure GDA0002998894430000112
Figure GDA0002998894430000113
when in use
Figure GDA0002998894430000114
The minimum performance indicator γ can be found to be 22.35, describing the system observer parameters as follows:
Figure GDA0002998894430000115
Figure GDA0002998894430000116
assume the initial state of the system
Figure GDA0002998894430000117
Initial state of observer
Figure GDA0002998894430000118
By simulating by MATLAB software, the parameters of the fault estimator can be obtained
Figure GDA0002998894430000119
And
Figure GDA00029988944300001110
the fault estimates for time are shown in fig. 2, 3 and 4.
In short, from the simulation result, the designed description system observer is effective, and in the networked system, even if the packet loss rate is continuously increased, the effect of the observer is not deteriorated, and an ideal estimation result can still be obtained.

Claims (1)

1. A method for fault estimation based on a networked system describing a system observer, characterized by comprising the following steps:
1) establishing a controlled object model of a networked system with faults and disturbances:
Figure FDA0002998894420000011
wherein:
Figure FDA0002998894420000012
is the state vector of the system and,
Figure FDA0002998894420000013
is the output vector of the system with saturation,
Figure FDA0002998894420000014
is the input of the disturbance of the system,
Figure FDA0002998894420000015
is the fault signal to be estimated, w (k) e l2[0,∞)，l2Is [0, ∞) square-summable discrete function space;
Figure FDA0002998894420000016
is a known constant matrix; saturation function
Figure FDA0002998894420000017
Is defined as
Figure FDA0002998894420000018
Where σ isi(νi)＝sign(νi)·min{νi,max,|νi|}，vi，max> 0 is the known saturation boundary, σ (-) is a multivariate saturation function, σi(. h) is the ith of the saturation function σ (-)Component, viIs an unknown scalar quantity which represents the function sigmaiVariation of (·), for a given diagonal matrix R1，R2，R1≥0，R2Not less than 0 and R2＞R1σ (·) satisfies the following inequality:
[σ(y(k))-R1y(k)]T[σ(y(k))-R2y(k)]≤0 (3)
regarding the fault signal f (k-1) at the time k-1 as an additional state, the following augmented state vector is obtained
Figure FDA0002998894420000019
And construct the following augmentation system
Figure FDA00029988944200000110
Wherein,
Figure FDA00029988944200000111
is the augmented state vector of x (k), d (k) is the augmented state vector of w (k),
Figure FDA00029988944200000112
is the augmented state vector of y (k),
Figure FDA00029988944200000113
Inis an n x n dimensional identity matrix, and σ (Cx (k)) can be divided into the sum of a linear portion and a non-linear portion
Figure FDA00029988944200000114
Wherein phi: (
Figure FDA0002998894420000021
x (k is a non-linear vector function, and the saturation function σ () satisfies an inequality constraint
Figure FDA0002998894420000022
M1And M2Are all known M x M dimensional symmetric positive definite matrices and M2＞M1Further, the formula (7) shows
Figure FDA0002998894420000023
Wherein
Figure FDA0002998894420000024
Considering the packet loss existing in the system, the measurement output is
Figure FDA0002998894420000025
Wherein: beta is akIs a random sequence satisfying Bernoulli, is used for describing the probability of packet loss occurring in the system, when beta iskWhen 1, no packet loss in the system is indicated, when βkWhen the value is 0, the data packet in the system is completely lost; the possibility of packet loss is
Figure FDA0002998894420000026
2) Design description system observer:
Figure FDA0002998894420000027
wherein:
Figure FDA0002998894420000028
is the intermediate variable that is the variable between,
Figure FDA0002998894420000029
is in an augmented state
Figure FDA00029988944200000210
(ii) an estimate of (d);
Figure FDA00029988944200000211
is a parameter matrix to be designed, and T, N can be determined by formula (12);
Figure FDA00029988944200000212
wherein
Figure FDA00029988944200000213
Is an arbitrarily selectable matrix, In×mIs an n x m dimensional identity matrix
3) The solvable sufficient conditions for the system mean square asymptotic stability and the description of the system observer parameters are as follows:
Figure FDA0002998894420000031
wherein: w is P1L，
Figure FDA0002998894420000032
Denotes the transpose of the symmetric position matrix, 0 is the zero matrix;
Figure FDA0002998894420000033
Figure FDA0002998894420000034
is a symmetrical positive definite matrix and is characterized in that,
Figure FDA0002998894420000035
is an unknown nonsingular matrix, gamma > 0 is a given system performance index, I is an identity matrix,
Figure FDA0002998894420000036
is a known m x m dimensional symmetric positive definite matrix;
given constant
Figure FDA0002998894420000037
And a system performance index with gamma > 0, solving the formula (13) by using an LMI toolbox in MATLAB when a positive definite matrix P exists1,P2And a non-singular matrix W such that equation (13) holds, the system is asymptotically mean-square stable and satisfies H∞Performance index, can obtain non-optimal describing system observer parameter L ═ P1 -1W, i.e. step 4) can be performed); when the unknown variables have no feasible solution, the system is not mean square asymptotically stable, the parameters of the observer of the non-optimal description system cannot be obtained, and the step 4) cannot be carried out;
4) calculating optimal description system observer parameters
According to
Figure FDA0002998894420000038
The system performance index gamma is calculated, an optimization problem formula (14) is solved by utilizing an LMI tool box in MATLAB, and e (k) is a state estimation error:
Figure FDA0002998894420000039
when equation (14) has a solution, the optimal description system observer parameters can be obtained, and the optimal H∞The performance index is gammaminBy calculating the nonsingular matrix W using equation (14), the optimum parameter L ═ P describing the system observer can be obtained1 -1W；
When the formula (14) has no solution, the optimal description system observer parameters cannot be obtained;
5) fault estimation of networked systems based on a description system observer
According to the actuator fault occurring in the actual operation of the networked system, the parameter L of the system observer is obtained by the formula (14), and then the parameter L is obtained by the calculation of the formula (11)
Figure FDA00029988944200000310
Figure FDA00029988944200000311
Thus obtaining an estimate of the fault, IqIs a q-dimensional unit vector.
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