CN110531732B - Random fault detection method for nonlinear networked control system - Google Patents

Random fault detection method for nonlinear networked control system Download PDF

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CN110531732B
CN110531732B CN201910283824.7A CN201910283824A CN110531732B CN 110531732 B CN110531732 B CN 110531732B CN 201910283824 A CN201910283824 A CN 201910283824A CN 110531732 B CN110531732 B CN 110531732B
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fault detection
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CN110531732A (en
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潘丰
高敏
王蕾
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Hebei laimeisi power and electrical equipment testing equipment Co.,Ltd.
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Abstract

The invention discloses a random fault detection method of a nonlinear networked control system, and belongs to the field of networked control systems. The method comprises the steps of describing random packet loss and random time delay by using a unified model, establishing a discrete networked control system model, designing a fault detection filter, constructing a filtering error system, and introducing a residual error evaluation mechanism to judge whether a fault occurs; and obtaining sufficient conditions of the mean square index stability of a filtering error system and the existence of the fault detection filter by using a Lyapunov stability theory and a linear matrix inequality method, solving an optimization problem by using a Matlab LMI tool box, and providing parameters of the optimal fault detection filter. The method considers that the system has sensor saturation, random faults, packet loss, time delay and nonlinearity under the actual condition, meets Bernoulli distribution under the occurrence condition, is suitable for general fault detection, and reduces the conservatism.

Description

Random fault detection method for nonlinear networked control system
Technical Field
The invention belongs to the field of networked control systems, and relates to a random fault detection method of a nonlinear networked control system with packet loss and time delay under the saturation constraint of a sensor.
Background
In recent years, with the rapid development of network technology, a Networked Control System (NCS) has begun to receive attention from many scholars. The networked control system has the advantages of convenience in installation and maintenance, high flexibility, easiness in reconstruction and the like, and sensors, actuators, controllers and other system elements in the networked control system are connected through a network. However, the introduction of the network brings new problems, such as data loss, network-induced delay, nonlinearity, etc., which affect the performance and stability of the system and even cause failures, and therefore the failure detection method is a hot spot in recent years.
The key step of fault detection is to design a fault detection filter as a residual error generation mechanism to obtain a residual error signal sensitive to faults, and then judge whether the faults occur or not by utilizing a residual error evaluation mechanism. Due to unpredictability of network changes, many random phenomena such as random nonlinearity, random packet loss, random time delay and the like exist in a networked control system, however, most research results assume faults in the system to be deterministically generated, and many models the time delay and the packet loss separately. Meanwhile, due to physical limitation, the output quantity or the output speed of the sensor cannot be arbitrarily large, the amplitude is limited, and the sensor saturation phenomenon frequently occurs in an actual system.
Disclosure of Invention
Aiming at the problems in the prior art, the invention provides a random fault detection method of a nonlinear networked control system with packet loss and time delay under the saturation constraint of a sensor. Considering the conditions of random faults, random packet loss, random time delay, sensor saturation and random nonlinearity of the networked control system, the fault detection filter is designed, so that the networked control system can still keep the mean square index stable and meet the requirement of H under the conditions∞Performance index and can effectively detect faults.
The technical scheme of the invention is as follows:
a random fault detection method of a nonlinear networked control system is characterized by comprising the following steps:
1) establishing a mathematical model of a nonlinear networked control system with random faults, random packet loss, random time delay and sensor saturation:
Figure GDA0002988410480000011
Figure GDA0002988410480000012
wherein: k is a discrete time index and k e [ -d [ - ]q,N-1]N is a natural number set;
Figure GDA0002988410480000013
a state vector for the networked control system;
Figure GDA0002988410480000014
is the initial value of the state vector;
Figure GDA0002988410480000015
is a control input vector;
Figure GDA0002988410480000021
unknown input vector of finite energy, belonging to2[0, ∞) space,/[2[0, ∞) is the space of square multiplicative vectors;
Figure GDA0002988410480000022
a fault signal vector to be detected;
Figure GDA0002988410480000023
is a nonlinear vector value function in a networked control system and satisfies the following conditions of [ g (x (k)) -R1x(k)]T[g(x(k))-R2x(k)]≤0，
Figure GDA0002988410480000024
R2-R1Is a symmetric positive definite matrix;
Figure GDA0002988410480000025
outputting a vector for measuring the system under the conditions of considering sensor saturation, random packet loss and time delay;
Figure GDA0002988410480000026
is a constant time delay, j is 1, …, q, d1＜d2＜…＜dqQ is the maximum delay dqThe subscript of (a) is,
Figure GDA0002988410480000027
is a set of positive integers;
Figure GDA0002988410480000028
is a non-linear part of the sensor saturation, and satisfies
Figure GDA0002988410480000029
And
Figure GDA00029884104800000210
Figure GDA00029884104800000211
and
Figure GDA00029884104800000212
is a diagonal matrix of the angles,
Figure GDA00029884104800000230
is a symmetric positive definite matrix; tau (k) is a random variable of the time delay of k time and the data packet loss condition;
H{τ(k)＝0}and
Figure GDA00029884104800000213
is a random variable of the occurrence conditions of packet loss and time delay and meets the requirements
Figure GDA00029884104800000214
Figure GDA00029884104800000215
Figure GDA00029884104800000216
And
Figure GDA00029884104800000217
is a constant matrix of the system; α (k) is a random variable for the occurrence of a fault; δ (k) is a random variable that occurs non-linearly; α (k) and δ (k) satisfy Bernoulli distribution:
Figure GDA00029884104800000218
Figure GDA00029884104800000219
wherein: prob {. represents the event occurrence probability, Var {. represents the variance, E {. represents the mathematical expectation; e { α (k) } represents the probability that α (k) ═ 1 occurs,
Figure GDA00029884104800000220
is a specific probability value，
Figure GDA00029884104800000221
Is the variance of α (k); e { δ (k) } represents the probability that δ (k) ═ 1 occurs,
Figure GDA00029884104800000222
is a specific numerical value of the probability that,
Figure GDA00029884104800000223
is the variance of δ (k);
Figure GDA00029884104800000224
and
Figure GDA00029884104800000225
is a known constant;
2) designing a fault detection filter:
Figure GDA00029884104800000226
wherein:
Figure GDA00029884104800000227
detecting a state vector of the filter for the fault;
Figure GDA00029884104800000228
a residual vector output for the fault detection filter;
Figure GDA00029884104800000229
is the parameter of the fault detection filter that needs to be determined;
3) constructing a filtering error system model:
Figure GDA0002988410480000031
wherein:
Figure GDA0002988410480000032
θ(k)＝[uT(k) wT(k) fT(k)]T，e(k)＝r(k)-f(k)，
Figure GDA0002988410480000033
Figure GDA0002988410480000034
Figure GDA0002988410480000035
Figure GDA0002988410480000036
Figure GDA0002988410480000037
Figure GDA0002988410480000038
Figure GDA0002988410480000039
Figure GDA00029884104800000310
Figure GDA00029884104800000311
0 and I represent a zero matrix and an identity matrix of appropriate dimensions, respectively;
detecting nets with residual evaluation mechanismWhether the fault of the coordination control system occurs or not, a residual evaluation function J (k) and a threshold value JthFormula (4) and formula (5), respectively:
Figure GDA00029884104800000312
Figure GDA00029884104800000313
wherein: l is a finite evaluation time duration, threshold JthRepresenting the supremum of the residual error evaluation function J (k) when no fault occurs, and sup represents the supremum for determining a certain function;
whether the networked control system fails is detected by the formula (6):
Figure GDA0002988410480000041
4) the mean square index of the filtering error system is stable and meets H∞The performance index and the sufficient conditions of the fault detection filter are as follows:
Figure GDA0002988410480000042
wherein:
denotes the transpose of the symmetric position matrix,
Figure GDA0002988410480000043
Figure GDA0002988410480000044
Ψ3＝diag{-I,-I,-I,P-G-GT,P-G-GT,P-G-GT,P-G-GT,P-G-GT}，
Figure GDA0002988410480000045
Figure GDA0002988410480000046
Figure GDA0002988410480000047
Figure GDA0002988410480000048
Figure GDA0002988410480000049
Figure GDA00029884104800000410
Figure GDA00029884104800000411
Figure GDA0002988410480000051
Figure GDA0002988410480000052
Figure GDA0002988410480000053
Figure GDA0002988410480000054
Figure GDA0002988410480000055
Figure GDA0002988410480000056
Figure GDA0002988410480000057
Figure GDA0002988410480000058
wherein:
Figure GDA0002988410480000059
is an unknown matrix, λ1> 0 is a variable that is not known,
Figure GDA00029884104800000510
βjj is 0, …, q is a given constant, γ > 0 is a given index;
given a positive scalar quantity
Figure GDA00029884104800000511
βjJ is 0, …, q, an index with gamma > 0, solving inequality (7) by using Matlab LMI toolbox; when inequality (7) has a solution, a positive definite matrix P, Q existsjJ-1, …, q, matrix G,
Figure GDA00029884104800000512
and a positive scalar λ1The mean square index of the filtering error system (3) is stable and meets the requirement of H∞Performance indexes, namely, parameters of the fault detection filter can be obtained, and the step 5) is carried out; when the inequality (7) is not solved, the filtering error system (3) is not stable in mean square index, and can not obtain the parameters of the fault detection filter, and the process is finished;
5) calculating optimal fault detection filter parameters
According to
Figure GDA00029884104800000513
And (3) solving a performance index gamma, and solving an optimization problem by using a Matlab LMI tool box:
Figure GDA00029884104800000514
when formula (8) has a solution, the optimum H∞The performance index is gammaminThe parameters of the optimal fault detection filter are obtained as follows:
Figure GDA00029884104800000515
wherein:
Figure GDA00029884104800000516
is a non-singular matrix; turning to step 6);
when the formula (8) is not solved, the optimal fault detection filter cannot be obtained, and the process is finished;
6) networked control system random fault detection
According to the input of the fault detection filter obtained when the networked control system actually operates
Figure GDA00029884104800000619
Obtaining residual signals r (k) output by the fault detection filter according to a fault detection filter formula (2), and then obtaining a residual evaluation function J (k) and a threshold value J through calculation according to formulas (4) and (5)thAnd finally, judging whether the random fault occurs or not according to the formula (6).
The invention has the beneficial effects that: the invention simultaneously considers the design method of the fault detection filter under the conditions of random time delay, random packet loss, sensor saturation, random nonlinearity and random faults in the networked control system, only considers deterministic faults and uses less unified models to describe the limitations of random packet loss and random time delay when compared with the traditional fault detection filter design modeling, and the method has more practical significance and reduces the conservatism.
Drawings
Fig. 1 is a flowchart of a random fault detection method of a nonlinear networked control system with packet loss and time delay under the constraint of sensor saturation.
Fig. 2 is a structural diagram of a nonlinear networked control system with packet loss and time delay under the constraint of sensor saturation. In the figure:
Figure GDA0002988410480000061
is a control input vector;
Figure GDA0002988410480000062
an unknown input vector of finite energy;
Figure GDA0002988410480000063
a fault signal vector to be detected;
Figure GDA0002988410480000064
the input vector of the fault detection filter is the measurement output vector of the system under the conditions of considering sensor saturation, random packet loss and time delay;
Figure GDA0002988410480000065
a residual signal vector output for the fault detection filter;
Figure GDA0002988410480000066
is the residual error vector.
FIG. 3 is w (k) ≠ 0,
Figure GDA0002988410480000067
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA0002988410480000068
residual signal plot of time.
FIG. 4 shows w (k) ≠ 0,
Figure GDA0002988410480000069
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA00029884104800000610
time-dependent residual evaluation function graph.
FIG. 5 shows w (k) ≠ 0,
Figure GDA00029884104800000611
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA00029884104800000612
time-dependent residual evaluation function graph.
FIG. 6 shows w (k) ≠ 0,
Figure GDA00029884104800000613
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA00029884104800000614
time-dependent residual evaluation function graph.
FIG. 7 shows w (k) ≠ 0,
Figure GDA00029884104800000615
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA00029884104800000616
time-dependent residual evaluation function graph.
FIG. 8 shows w (k) ≠ 0,
Figure GDA00029884104800000617
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA00029884104800000618
time-dependent residual evaluation function graph.
Detailed Description
The following further describes the embodiments of the present invention with reference to the drawings.
Referring to fig. 1, a random fault detection method for a nonlinear networked control system with packet loss and time delay under the constraint of sensor saturation includes the following steps:
step 1: establishing a mathematical model of a nonlinear networked control system with random faults, random packet loss, random time delay and sensor saturation
The mathematical model of the nonlinear networked control system with random faults, random packet loss, random time delay and sensor saturation is established as formula (10):
Figure GDA0002988410480000071
Figure GDA0002988410480000072
H{τ(k)＝0}and
Figure GDA0002988410480000073
the random variable of the occurrence conditions of packet loss and time delay meets the following requirements:
Figure GDA0002988410480000074
wherein: beta is aj> 0 is a known scalar, j equals 0, …, q and
Figure GDA0002988410480000075
if it is not
Figure GDA0002988410480000076
Representing the measured output regardless of whether or not time delay is present
Figure GDA0002988410480000077
By probability
Figure GDA0002988410480000078
Arrive at the fault detection filter at a certain moment and the probability of packet loss is
Figure GDA0002988410480000079
If it is not
Figure GDA00029884104800000710
Representing no packet loss.
Assuming that the nonlinear vector value function g (x (k)) satisfies
[g(x(k))-R1x(k)]T[g(x(k))-R2x(k)]0 ≦ 0 (11) wherein:
Figure GDA00029884104800000711
and R is2-R1Is a symmetric positive definite matrix.
Considering that in a networked control system, the sensor is saturated, the saturation function σ (·):
Figure GDA00029884104800000712
is of [ L1，L2]，L1And L2Is a diagonal matrix, and L2-L1Is a symmetric positive definite matrix, σ (·) satisfies:
[σ(C0x(k))-L1C0x(k)]T[σ(C0x(k))-L2C0x(k)]≤0 (12)
[σ(Cjx(k-dj))-L1Cjx(k-dj)]T[σ(Cjx(k-dj))-L2Cjx(k-dj)]≤0,j＝1,…,q (13)
for convenience of processing, let σ (C)0x (k)) and σ (C)jx(k-dj) J ═ 1, …, q is divided into linear and non-linear portions:
σ(C0x(k))＝φ(C0x(k))+L1C0x(k) (14)
σ(Cjx(k-dj))＝φ(Cjx(k-dj))+L1Cjx(k-dj),j＝1,…,q (15)
so that:
Figure GDA00029884104800000713
Figure GDA00029884104800000714
wherein: phi (C)0x (k)) and phi (C)jx(k-dj) J ═ 1,2, …, q is a non-linear vector function,
Figure GDA00029884104800000715
step 2: designing fault detection filter
Designing a fault detection filter formula (2), selecting alpha (k) to represent the probability of fault occurrence, wherein alpha (k) is a random variable satisfying Bernoulli 0-1 sequence distribution, and for the moment k, when alpha (k) is 0, the system is indicated to be free of fault and works normally, and the probability is
Figure GDA00029884104800000716
When alpha (k) is 1, a fault is indicated, and the probability is
Figure GDA00029884104800000717
The larger the probability of a failure in the system. The probability of occurrence of random nonlinearity is represented by δ (k), which is a random variable satisfying the Bernoulli 0-1 sequence distribution, and when δ (k) is 0 for the time k, it indicates that the system has no occurrence of nonlinearity, and the probability is
Figure GDA0002988410480000081
When δ (k) ═ 1, it indicates that nonlinearity occurs with a probability of
Figure GDA0002988410480000082
The larger the probability that non-linearity occurs in the system.
And step 3: constructing a filtering error system model
Defining a residual error signal vector:
e(k)＝r(k)-f(k) (18)
according to the equations (2), (11) and (18), the filtering error system equation (3) is obtained by a state augmentation method:
definition 1: when θ (k) is 0, if a constant α > 0 exists and κ ∈ (0,1) makes equation (19) true under any initial condition, the filter error system (3) exponentially stabilizes in the mean square sense.
Figure GDA0002988410480000083
Theorem 1: given matrix S1，S2And S3Wherein
Figure GDA0002988410480000084
If and only if
Figure GDA0002988410480000085
Or
Figure GDA0002988410480000086
When true, then S1+S3 TS2 -1S3< 0 is established
Theorem 2: for matrix a, Q ═ QTAnd P > 0, A if matrix G is present such that equation (22) holdsTPA-Q < 0 holds true.
Figure GDA0002988410480000087
Theorem 3: let T0(·),T1(·),…TpIs about a variable
Figure GDA0002988410480000088
Quadratic function of (1), Tj(x)＝xTΦjx is not less than 0(j is 0,1, …, p), wherein
Figure GDA0002988410480000089
Then
Figure GDA00029884104800000810
Sufficient conditions are established: presence of lambda1≥0,…,λpNot less than 0 such that
Figure GDA00029884104800000811
And constructing a residual evaluation function J (k) and a threshold value J (th) as an equation (4) and an equation (5), respectively, wherein the equation (6) can be used for judging whether the fault occurs. When the value in the residual evaluation function is larger than the threshold value, a fault occurs and an alarm is given, otherwise, no fault occurs.
And 4, step 4: the mean square index of the filtering error system is stable and meets H∞Performance index and sufficiency condition for fault detection filter presence
Constructing a Lyapunov function:
Figure GDA0002988410480000091
and (3) obtaining sufficient conditions for the stability of the mean square index and the existence of the fault detection filter of the filtering error system (3) by utilizing a Lyapunov stability theory and a linear matrix inequality method. The method comprises the following steps:
step 4.1: firstly, the stability of a filtering error system is judged, and a sufficient condition for the mean square index of the filtering error is obtained.
Assuming that the inequality (25) holds:
Figure GDA0002988410480000092
wherein:
Figure GDA0002988410480000093
Figure GDA0002988410480000094
Figure GDA0002988410480000095
Figure GDA0002988410480000096
Figure GDA0002988410480000097
Figure GDA0002988410480000098
Figure GDA0002988410480000099
Figure GDA00029884104800000910
Figure GDA00029884104800000911
Figure GDA00029884104800000912
defining:
Θ(k)＝[ηT(k)ηT(k-1)…ηT(0)]T，ΔV(k)＝Ε{V(k+1)}-V(k)
note that when i is 0, …, q,
Figure GDA00029884104800000913
when θ (k) is 0,
Figure GDA0002988410480000101
equation (11) can be converted to:
Figure GDA0002988410480000102
according to the equations (16) and (17), it is possible to obtain:
Figure GDA0002988410480000103
Figure GDA0002988410480000104
combining formulas (26), (27), (28) and (29), according to theorem 3, we can obtain:
Figure GDA0002988410480000105
wherein:
Figure GDA0002988410480000106
ηl＝[ηT(k-d1)ηT(k-d2)…ηT(k-dq)]T，
Figure GDA0002988410480000111
from theorem 1, if the linear matrix inequality (25) holds, it is known that the non-zero
Figure GDA0002988410480000112
E { Δ V (k) } < 0, the positive scalar χ > 0 can always be found so that equation (31) holds.
Figure GDA0002988410480000113
Namely:
E{ΔV(k)}＜-χ||η(k)||2 (32)
according to definition 1, a filter error system (3) mean square index stabilization can be obtained.
According to the Lyapunov stability theory, when θ (k) is 0, a positive scalar is given
Figure GDA0002988410480000114
βj,j＝0,…,q，
Figure GDA0002988410480000115
Gamma and fault detection filter parameter Af,Bf,Cf,DfThe presence of a positive definite matrix P > 0, Qj> 0, j-1, …, q and a scalar λ1> 0 so that equation (25) holds. When the sufficient condition of the step 4.1 is met, the step 4.2 is executed again; if the sufficiency of step 4.1 is not established, then the filtering error system (3) is not mean square index stable and step 4.2 cannot be performed.
Step 4.2: adequate condition for fault detection filter existence
When θ (k) ≠ 0,
Figure GDA0002988410480000116
wherein: ξ (k) ═ ζT(k)θT(k)]T。
Xi is obtained from the formula (25)T(k) Ω ξ (k) < 0, resulting in:
Figure GDA0002988410480000117
considering the zero initial condition and the stable mean square index of the filtering error system (3), further comprising:
Figure GDA0002988410480000118
satisfy H∞Performance index.
Equation (25) can be written in the form of equation (36) according to theorem 1:
Figure GDA0002988410480000121
equation (36) may be converted to:
Figure GDA0002988410480000122
wherein:
Figure GDA0002988410480000123
Ξ3＝diag{-I,-I,-I,-P-1,-P-1,-P-1,-P-1,-P-1}
according to theorem 2, if matrix G exists such that inequality (38) holds, equation (37) holds.
Figure GDA0002988410480000124
Wherein:
Figure GDA0002988410480000125
decomposition of P and G gives:
Figure GDA0002988410480000126
order:
Figure GDA0002988410480000131
equation (38) and equation (7) are equivalent to each other through conventional calculation.
Solving by using a Matlab LMI tool box, and giving a positive scalar quantity when theta (k) is not equal to 0
Figure GDA0002988410480000132
βj,j＝0,…,q,
Figure GDA0002988410480000133
An index with gamma > 0 and a fault detection filter parameter Af,Bf,Cf,DfThe presence of a positive definite matrix P > 0, Qj> 0, j-1, …, q and a scalar λ1> 0 such that equation (25) holds; the filtering error system (3) is mean square index stable and satisfies H∞Performance index, obtaining the parameters of the fault detection filter, and then executing the step 5; if equation (25) does not hold, then the filter error system (3) is not mean square stable and the fault detection filter parameters cannot be solved, and step 5 cannot be performed.
And 5: calculating optimal fault detection filter parameters
For the filter error system (3), the optimization problem equation (8) is solved using the Matlab LMI toolkit. If formula (8) has a solution, the optimal H is obtained∞The performance index is gammaminThe optimum fault detection filter parameter is equation (9). If equation (8) has no solution, an optimal fault detection filter cannot be obtained.
Step 6: networked control system random fault detection
According to the network control systemInput to a fault detection filter obtained at run-time
Figure GDA00029884104800001312
Obtaining residual signals r (k) output by the fault detection filter according to a fault detection filter formula (2), and then obtaining a residual evaluation function J (k) and a threshold value J through calculation according to formulas (4) and (5)thAnd finally, judging whether the random fault occurs or not according to the formula (6).
Example (b):
by adopting the random fault detection method of the nonlinear networked control system with packet loss and time delay under the sensor saturation constraint, the filtering error system (3) is stable in mean square index under the condition of no external disturbance and fault, namely when theta (k) is 0. When theta (k) ≠ 0, the filtering error system (3) is mean square exponential stable and satisfies H∞Performance index. The specific implementation method comprises the following steps:
the mathematical model of a certain three-tank system is formula (10), and the system parameters are given as:
Figure GDA0002988410480000134
Figure GDA0002988410480000135
q＝2，d1＝1，d2＝2，
Figure GDA0002988410480000136
Figure GDA0002988410480000137
u(k)＝Kx(k)，
Figure GDA0002988410480000138
Figure GDA0002988410480000139
let beta0＝0.1，β1＝0.2，β20.3, different random failure probabilities are obtained
Figure GDA00029884104800001310
And nonlinear probability
Figure GDA00029884104800001311
H∞Performance index gammaminAs shown in table 1.
TABLE 1 different random failure probabilities
Figure GDA0002988410480000141
And random non-linear probability
Figure GDA0002988410480000142
H under circumstances∞Performance index gammamin
Figure GDA0002988410480000143
As can be seen from Table 1, as the random failure probability or the random non-linear probability of the networked control system increases, the corresponding H∞Performance index gammaminWith a consequent increase, i.e. a deterioration of the disturbance suppression performance.
Get
Figure GDA0002988410480000144
β0＝0.1，β1＝0.2，β2＝0.3，
Figure GDA0002988410480000145
Applying MATLAB LMI toolbox to obtain H for filtering error system (3)∞Performance index gammamin1.4993, the optimal parameters for the fault detection filter are:
Figure GDA0002988410480000146
Cf＝[-0.0010 -0.0023 -0.0015]，Df＝1×10-4×[0.3150 -0.0492]
the nonlinear function g (x (k)) is 0.4sin (x (k)), and the nonlinear part of the saturation function is:
Figure GDA0002988410480000147
Figure GDA0002988410480000148
Figure GDA0002988410480000149
the fault signal and the unknown disturbance are respectively:
Figure GDA00029884104800001410
w(k)＝e-0.02k sin(0.2k)
Figure GDA00029884104800001411
graphs of residual errors r (k) and residual error evaluation functions j (k) of the networked control system are shown in fig. 3 and fig. 4, and when the evaluation time length L obtained by the residual error evaluation machine adopted in the present invention is 400, the calculation formula of the threshold value is:
Figure GDA00029884104800001412
after 200 times of simulation operation without fault, the average value J is obtainedth＝4.1388×10-4For the final threshold, 4.0704 × 10 can be found by comparing with the residual evaluation function (4)-4＝J(86)＜Jth＜J(87)＝4.2024×10-4The fault detection filter can detect the fault within 17 time steps after the fault occurs at the moment k is 70And (4) occurrence of failure.
At beta0＝0.1，β1＝0.2，β2＝0.5，
Figure GDA0002988410480000151
In the case of (2), different random failure probabilities
Figure GDA0002988410480000152
The corresponding thresholds and the time length for discriminating the fault are shown in table 2, and the corresponding residual evaluation function graphs are shown in fig. 5, fig. 6, fig. 7 and fig. 8.
TABLE 2 different random failure probabilities
Figure GDA0002988410480000154
J in case ofthAnd length of time to discriminate failure
Figure GDA0002988410480000153
It can be seen that the occurrence of random faults can be effectively detected, and the greater the probability of the occurrence of the faults in the networked control system, the shorter the time required for detecting the faults.

Claims (1)

1. A random fault detection method of a nonlinear networked control system is characterized by comprising the following steps:
1) establishing a mathematical model of a nonlinear networked control system with random faults, random packet loss, random time delay and sensor saturation:
Figure FDA0002988410470000011
Figure FDA0002988410470000012
wherein: k is an index of the discrete time,and k e [ -d ]q,N-1]N is a natural number set;
Figure FDA0002988410470000013
a state vector for the networked control system;
Figure FDA0002988410470000014
is the initial value of the state vector;
Figure FDA0002988410470000015
is a control input vector;
Figure FDA0002988410470000016
unknown input vector of finite energy, belonging to2[0, ∞) space,/[2[0, ∞) is the space of square multiplicative vectors;
Figure FDA0002988410470000017
a fault signal vector to be detected;
Figure FDA0002988410470000018
is a nonlinear vector value function in a networked control system and meets the requirement
Figure FDA0002988410470000019
R2-R1Is a symmetric positive definite matrix;
Figure FDA00029884104700000110
outputting a vector for measuring the system under the conditions of considering sensor saturation, random packet loss and time delay;
Figure FDA00029884104700000111
is a constant time delay, j is 1, …, q, d1＜d2＜…＜dqQ is the maximum delay dqThe subscript of (a) is,
Figure FDA00029884104700000112
is a set of positive integers;
Figure FDA00029884104700000113
and
Figure FDA00029884104700000114
is a non-linear part of the sensor saturation, and satisfies
Figure FDA00029884104700000115
And
Figure FDA00029884104700000116
Figure FDA00029884104700000117
and
Figure FDA00029884104700000118
is a diagonal matrix of the angles,
Figure FDA00029884104700000119
is a symmetric positive definite matrix;
tau (k) is a random variable of the time delay of k time and the data packet loss condition;
H{τ(k)＝0}and
Figure FDA00029884104700000120
is a random variable of the occurrence conditions of packet loss and time delay and meets the requirements
Figure FDA00029884104700000121
Figure FDA00029884104700000122
Figure FDA00029884104700000123
And
Figure FDA00029884104700000124
is a constant matrix of the system;
α (k) is a random variable for the occurrence of a fault; δ (k) is a random variable that occurs non-linearly; α (k) and δ (k) satisfy Bernoulli distribution:
Figure FDA00029884104700000125
Figure FDA00029884104700000126
wherein: prob {. represents the event occurrence probability, Var {. represents the variance, E {. represents the mathematical expectation; e { α (k) } represents the probability that α (k) ═ 1 occurs,
Figure FDA0002988410470000021
is a specific numerical value of the probability that,
Figure FDA0002988410470000022
Figure FDA0002988410470000023
is the variance of α (k); e { δ (k) } represents the probability that δ (k) ═ 1 occurs,
Figure FDA0002988410470000024
is a specific numerical value of the probability that,
Figure FDA0002988410470000025
Figure FDA0002988410470000026
is the variance of δ (k);
Figure FDA0002988410470000027
and
Figure FDA0002988410470000028
is a known constant;
2) designing a fault detection filter:
Figure FDA0002988410470000029
wherein:
Figure FDA00029884104700000210
detecting a state vector of the filter for the fault;
Figure FDA00029884104700000211
a residual vector output for the fault detection filter;
Figure FDA00029884104700000212
is the parameter of the fault detection filter that needs to be determined;
3) constructing a filtering error system model:
Figure FDA00029884104700000213
wherein:
Figure FDA00029884104700000214
θ(k)＝[uT(k) wT(k) fT(k)]T，e(k)＝r(k)-f(k)，
Figure FDA00029884104700000215
Figure FDA00029884104700000216
Figure FDA00029884104700000217
Figure FDA00029884104700000218
Figure FDA00029884104700000219
Figure FDA00029884104700000220
Figure FDA00029884104700000221
Figure FDA00029884104700000222
Figure FDA00029884104700000223
0 and I represent a zero matrix and an identity matrix of appropriate dimensions, respectively;
detecting whether a fault of a networked control system occurs by using a residual error evaluation mechanism, and evaluating a function J (k) and a threshold value J by using a residual errorthFormula (4) and formula (5), respectively:
Figure FDA0002988410470000031
Figure FDA0002988410470000032
wherein: l is a finite evaluation time duration, threshold JthRepresenting the supremum of the residual error evaluation function J (k) when no fault occurs, and sup represents the supremum for determining a certain function;
whether the networked control system fails is detected by the formula (6):
Figure FDA0002988410470000033
4) the mean square index of the filtering error system is stable and meets H∞The performance index and the sufficient conditions of the fault detection filter are as follows:
Figure FDA0002988410470000034
wherein:
denotes the transpose of the symmetric position matrix,
Figure FDA0002988410470000035
Figure FDA0002988410470000036
Ψ3＝diag{-I,-I,-I,P-G-GT,P-G-GT,P-G-GT,P-G-GT,P-G-GT}，
Figure FDA0002988410470000037
Figure FDA0002988410470000038
Figure FDA0002988410470000039
Figure FDA00029884104700000310
Figure FDA0002988410470000041
Figure FDA0002988410470000042
Figure FDA0002988410470000043
Figure FDA0002988410470000044
Figure FDA0002988410470000045
Figure FDA0002988410470000046
Figure FDA0002988410470000047
Figure FDA0002988410470000048
M＝[I 0]，
Figure FDA0002988410470000049
Qd＝diag{Q1,Q2,…,Qq}，
Figure FDA00029884104700000410
Figure FDA00029884104700000411
wherein:
Figure FDA00029884104700000412
is an unknown matrix, λ1> 0 is a variable that is not known,
Figure FDA00029884104700000413
is a given constant, gamma > 0 is a given index;
given a positive scalar quantity
Figure FDA00029884104700000414
Solving inequality (7) by using an index with gamma larger than 0 and using a Matlab LMI tool box; when the inequality (7) has a solution, there is a positive definite matrix P, QjJ-1, …, q, matrix G,
Figure FDA00029884104700000415
and a positive scalar λ1The mean square index of the filtering error system (3) is stable and meets the requirement of H∞Performance indexes, namely, parameters of the fault detection filter can be obtained, and the step 5) is carried out; when inequality (7) is not solvedWhen the mean square index of the filtering error system (3) is stable, the parameters of the fault detection filter cannot be obtained, and the process is finished;
5) calculating optimal fault detection filter parameters
According to
Figure FDA00029884104700000416
And (3) solving a performance index gamma, and solving an optimization problem by using a Matlab LMI tool box:
Figure FDA00029884104700000417
when formula (8) has a solution, the optimum H∞The performance index is gammaminThe parameters of the optimal fault detection filter are obtained as follows:
Figure FDA0002988410470000051
wherein:
Figure FDA0002988410470000052
is a non-singular matrix; turning to step 6);
when the formula (8) is not solved, the optimal fault detection filter cannot be obtained, and the process is finished;
6) networked control system random fault detection
According to the input of the fault detection filter obtained when the networked control system actually operates
Figure FDA0002988410470000053
Obtaining residual signals r (k) output by the fault detection filter according to a fault detection filter formula (2), and then obtaining a residual evaluation function J (k) and a threshold value J through calculation according to formulas (4) and (5)thAnd finally, judging whether the random fault occurs or not according to the formula (6).
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