CN113625647A - Nonlinear system event driver and DOFSS controller joint design method - Google Patents

Nonlinear system event driver and DOFSS controller joint design method Download PDF

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CN113625647A
CN113625647A CN202110962406.8A CN202110962406A CN113625647A CN 113625647 A CN113625647 A CN 113625647A CN 202110962406 A CN202110962406 A CN 202110962406A CN 113625647 A CN113625647 A CN 113625647A
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matrix
dofss
event driver
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李富强
郜丽赛
祁诗阳
谷小青
张益维
郑宝周
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Henan Agricultural University
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Abstract

The invention discloses a nonlinear system event driver and DOFSS controller joint design method, which comprises the following steps: a, establishing a nonlinear object model, an actuator saturation model and an event driver based on nonlinear object measurement output; b, establishing a random deception attack model, a DOFSS controller model and a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters; and C, designing joint design conditions of the nonlinear system event driver and the DOFSS controller under the influence of random deception attack, actuator saturation and network induced delay, solving parameters of the event driver and a gain matrix of the equivalent DOFSS controller, and finally obtaining the event driver and the DOFSS controller. The invention can solve the problem that the existing nonlinear system cannot be stable under the influence of random deception attack, actuator saturation and network induced delay.

Description

Nonlinear system event driver and DOFSS controller joint design method
Technical Field
The invention relates to the field of networked control systems, in particular to a combined design method of a nonlinear system event driver and a dynamic output feedback fuzzy saturated security (DOFSS) controller under random spoofing attack.
Background
The networked control system introduces a shared communication network into a control closed loop, has the advantages of high flexibility, low cost, convenience in installation and maintenance and the like, and is widely applied to the fields of smart power grids, smart traffic and the like. Networked control systems typically employ well-developed periodic sampling control strategies, and the sampling rate is typically set higher in order to guarantee system performance in the worst case. However, in practice, the worst case situation is rare, and the large amount of redundant data generated by the high sampling rate causes the waste of system-limited resources such as network bandwidth, and the like, and greatly influences the system performance.
Unlike the periodic control strategy which neglects the system dynamic to perform on-time control, the event-driven control strategy performs on-demand control only when the system dynamic meets the event-driven conditions, thereby saving the system limited resources such as network bandwidth and the like. However, existing event drivers typically employ a continuous-time event-driven mechanism that requires the addition of dedicated monitoring hardware and complex pre-computation to avoid the sesno phenomenon (i.e., driving samples an infinite number of times in a finite time).
Although the shared communication network brings great convenience to the networked control system, network-induced delay which degrades system performance is also introduced, and the system faces network attack threats. Network attacks are roughly divided into denial of service attacks and spoofing attacks, and the denial of service attacks prevent data packets from being delivered by blocking a communication network; the deception attack generates false data packets through data tampering, has strong concealment and great harm, and is an attack type researched by the invention. However, existing efforts focus on how to design event-driven mechanisms to conserve more system resources, with less simultaneous consideration of spoofing attacks and network-induced latency effects. In addition, existing event-driven control system analysis and synthesis methods generally assume that there is no attack threat, and such methods generally cannot be directly applied to event-triggered control system analysis under the influence of a spoofing attack.
In reality, actuator saturation is a non-linear phenomenon commonly existing in a control system. And if the input quantity of the actuator exceeds the saturation threshold value, the actuator enters a saturation state. In the saturated state, further increasing the actuator input amount does not have any effect on the actuator output. However, existing efforts are less likely to consider both actuator saturation and spoofing attack effects. Furthermore, existing efforts typically assume that the system state is fully measurable and state feedback controllers are designed, however in practice the system state is typically not directly accessible.
In order to solve the problems and simultaneously consider the influences of random spoofing attack, actuator saturation, network induced delay and incapability of directly measuring the object state, the invention provides a nonlinear system event driver and DOFSS controller joint design method.
Disclosure of Invention
The invention aims to provide a nonlinear system event driver and DOFSS controller joint design method, which can solve the problem that the conventional nonlinear system cannot be stable under the influences of random deception attack, actuator saturation and network induced delay, can effectively save system limited resources such as network bandwidth and the like, and can remove the hypothesis limitation on the complete testability of the system state.
The invention adopts the following technical scheme:
a nonlinear system event driver and dofss controller joint design method, comprising the steps of:
a, establishing a nonlinear object model and an actuator saturation model, and designing an event driver based on nonlinear object measurement output;
b, establishing a random deception attack model and a DOFSS controller model, and establishing a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters;
designing the joint design conditions of the nonlinear system event driver and the DOFS controller under the influence of random deception attack, actuator saturation and network induced delay to solve the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure BDA0003222564250000021
And gain matrix of equivalent DOFSS controller
Figure BDA0003222564250000022
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
In the step A, the nonlinear object model is as follows:
Figure BDA0003222564250000031
wherein the content of the first and second substances,
Figure BDA0003222564250000032
is the derivative of x (t), x (t) represents the object state, x (t) is an n-dimensional real number,
Figure BDA0003222564250000033
representing a control input affected by actuator saturation,
Figure BDA00032225642500000316
is nuDimensional real number, y (t) representing measurement output, y (t) being nyDimensional real number, t represents time, Ai,BiAnd CiRepresenting a gain matrix; the number of the object fuzzy rules is r, and i represents the serial number of the object fuzzy rules; substitution type
Figure BDA0003222564250000034
And is
Figure BDA0003222564250000035
Figure BDA0003222564250000036
And thetag(t) represents the 1 st and the 1 st, respectively
Figure BDA0003222564250000037
The individual and the g-th antecedent variables, g representing the number of antecedent variables,
Figure BDA0003222564250000038
the sequence number of the front-piece variable is shown,
Figure BDA0003222564250000039
representing a front-part variable
Figure BDA00032225642500000310
Membership in fuzzy sets
Figure BDA00032225642500000311
Represents the accumulation and multiplication operations, respectively.
In the step a, the event driver based on the nonlinear object measurement output is:
Figure BDA00032225642500000312
wherein, delta epsilon (0,1) is a driver threshold parameter,
Figure BDA00032225642500000317
is a positive definite matrix, h denotes the sampling period, tkh denotes the kth drive time, tkh is t of the sampling periodkMultiple, tk+1h denotes the (k + 1) th driving time, tk+1h is t of the sampling periodk+1The lower subscript k denotes the drive time number,
Figure BDA00032225642500000313
which is indicative of the current sampling instant,
Figure BDA00032225642500000314
is tkAfter h is first
Figure BDA00032225642500000318
One sampling period, y (t)kh) And
Figure BDA00032225642500000319
respectively represent tkh and
Figure BDA00032225642500000320
and (3) outputting corresponding nonlinear object measurement, wherein min { } represents a minimum function, and a right upper corner mark T of the matrix represents the transposition of the matrix.
In the step A, the actuator saturation model is as follows:
Figure BDA00032225642500000315
wherein the content of the first and second substances,
Figure BDA00032225642500000321
denotes the number u (t)
Figure BDA00032225642500000322
Dimensional components, u (t) representing object control inputs without regard to actuator saturation effects, u (t) being nuThe number of the dimensional real number is,
Figure BDA00032225642500000323
the number of dimensions of u (t) is shown,
Figure BDA00032225642500000324
and
Figure BDA00032225642500000325
respectively representing the maximum and minimum allowable output values of the actuator,
Figure BDA00032225642500000326
to represent
Figure BDA00032225642500000327
The corresponding actuator saturation function value, sat (), represents the actuator saturation function. In the step B, the random deception attack model is as follows:
Figure BDA0003222564250000041
wherein, the first and second guide rollers are arranged in a row,
Figure BDA0003222564250000042
representing a spoofing attack function, wherein a (t) epsilon {0,1} represents a Bernoulli distribution random variable, and when a (t) is 1, the spoofing attack is activated and the controller input is tampered; when a (t) is 0, the spoofing attack sleeps, and the controller input is not tampered;
Figure BDA0003222564250000043
g is an attack energy limiting matrix;
Figure BDA00032225642500000416
y (t- η (t)) represents a nonlinear object measurement output corresponding to time t- η (t), where t- η (t) is tkh+nkh，e(t)＝y(tkh)-y(tkh+nkh)，y(tkh+nkh) Representing the sampling instant tkh+nkh corresponding to the non-linear object measurement output.
In step B, the dofss controller model is:
Figure BDA0003222564250000044
wherein the content of the first and second substances,
Figure BDA00032225642500000417
is xcDerivative of (t), xc(t) denotes controller status, xc(t) is an n-dimensional real number, xc(t- η (t)) represents the controller state for t- η (t),
Figure BDA0003222564250000045
and
Figure BDA0003222564250000046
is a gain matrix; the number of fuzzy rules of the controller is r, j represents the serial number of the fuzzy rules of the controller, and the alternative formula
Figure BDA0003222564250000047
And is
Figure BDA0003222564250000048
Figure BDA0003222564250000049
Representing a front-part variable
Figure BDA00032225642500000410
Membership in fuzzy sets
Figure BDA00032225642500000411
Membership function of (c).
In the step B, the closed-loop system model organically integrating random deception attack, event drivers, actuator saturation and network induced delay parameters is as follows:
Figure BDA00032225642500000412
in the formula (I), the compound is shown in the specification,
Figure BDA00032225642500000413
is the derivative of x (t),
Figure BDA00032225642500000414
representing the closed-loop system state, x (t-eta (t)) representing the closed-loop system state corresponding to t-eta (t),
Figure BDA00032225642500000415
and
Figure BDA0003222564250000051
representing closed-loop system gain matrices, alternatively
Figure BDA0003222564250000052
Figure BDA0003222564250000053
And
Figure BDA0003222564250000054
respectively represents u1,
Figure BDA00032225642500000514
And
Figure BDA0003222564250000055
corresponding function value of actuator dead zone u1,
Figure BDA00032225642500000515
And
Figure BDA00032225642500000513
respectively represent the 1 st and the 1 st dimensions of u (t)
Figure BDA00032225642500000517
And nuThe component of the dimension(s) is,
Figure BDA00032225642500000516
representing the actuator dead band function.
The step C comprises the following specific steps:
c1: determining a nonlinear system asymptotic stability condition under the influence of random deception attack, an event driver, actuator saturation and network induced delay based on the Lyapunov stability theory;
c2 obtaining the combined design condition of the event driver and the DOFSS controller by utilizing the linear matrix inequality technology based on the nonlinear system asymptotic stable condition obtained in the step C1, namely obtaining the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure BDA00032225642500000518
And gain matrix of equivalent DOFSS controller
Figure BDA0003222564250000056
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
The nonlinear system asymptotic stability condition under the influence of random deception attack, event driver, actuator saturation and network induced delay is as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure BDA00032225642500000519
An actuator saturation parameter ε, an attack energy definition matrix G, and an event driver threshold parameter δ ∈ (0,1), if there is a positive definite matrix
Figure BDA00032225642500000520
R＞0,S＞0,Q1＞0,Q2＞0,Q3> 0, and matrix U2,U3Satisfy the following requirements
Figure BDA0003222564250000057
And
Figure BDA0003222564250000058
the closed loop system is asymptotically stable under the influence of random deception attack, event drivers, actuator saturation and network induced delay; in the above stability conditions, the following alternative formula is used:
Figure BDA0003222564250000059
Figure BDA00032225642500000510
Figure BDA00032225642500000511
Figure BDA00032225642500000512
Figure BDA0003222564250000061
Figure BDA0003222564250000062
Figure BDA0003222564250000063
Figure BDA0003222564250000064
wherein the content of the first and second substances,
Figure BDA00032225642500000613
the mathematical expectation for a (t) is shown,
Figure BDA00032225642500000614
is a mathematical expectation function. He { } denotes the sum of the matrix and its transpose, denotes the symmetric terms in the symmetric matrix,
Figure BDA00032225642500000615
the matrix represents a zero matrix, the upper right corner mark-1 of the matrix represents an inverse matrix, I is an identity matrix, col represents a column matrix, and diag represents a diagonal matrix.
The joint design conditions of the event driver and the DOFSS controller are as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure BDA00032225642500000616
The saturation parameter epsilon of the actuator and the attack energy limit matrix G, and the real number epsilon1＞0,∈2＞0,∈3> 0, if real numbers are present
Figure BDA00032225642500000617
Positive definite matrix
Figure BDA00032225642500000618
Figure BDA0003222564250000065
And a matrix
Figure BDA0003222564250000066
Satisfy the requirement of
Figure BDA0003222564250000067
Figure BDA0003222564250000068
And
Figure BDA0003222564250000069
the closed loop system is asymptotically stable while obtaining event driver parameters
Figure BDA00032225642500000610
And gain matrix of equivalent DOFSS controller
Figure BDA00032225642500000611
As follows
Figure BDA00032225642500000612
Under the above conditions, the following alternative formulae were used
Figure BDA0003222564250000071
Figure BDA0003222564250000072
Figure BDA0003222564250000073
Figure BDA0003222564250000074
Figure BDA0003222564250000075
Figure BDA0003222564250000076
Figure BDA0003222564250000077
Figure BDA0003222564250000078
Figure BDA0003222564250000079
Figure BDA00032225642500000710
Is a positive definite matrix, and N is an N multiplied by N dimensional real matrix.
The invention can solve the problem that the existing nonlinear system cannot be stable under the influence of random deception attack, actuator saturation and network induced delay, can effectively save system limited resources such as network bandwidth and the like, and can remove the hypothesis limitation of completely measurable system states.
Drawings
FIG. 1 is a schematic diagram of feedback control of event-driven output of a nonlinear system under spoofing attack in the present invention;
FIG. 2 is a schematic flow chart of the present invention.
Detailed Description
The invention is described in detail below with reference to the following figures and examples:
as shown in fig. 1, a feedback control system for event-driven output of a nonlinear system under random spoofing attack includes a sensor for periodically sampling measurement output of a nonlinear object, a sensor sampling data is sent to an event driver, and the event driver determines whether an event-driven condition is satisfied: if yes, sending the sampling data; otherwise, the sampled data is discarded. The DOFSS controller receives the zero-order keeper data and calculates a control signal, and the actuator adjusts the state of an object according to the control signal and considers the saturation influence of the actuator.
As shown in fig. 2, the method for designing a nonlinear system event driver and dofss controller in combination according to the present invention includes the following steps:
a, establishing a nonlinear object model and an actuator saturation model, and designing an event driver based on nonlinear object measurement output;
wherein, when describing the nonlinear object model as a T-S (Takagi-Sugeno) fuzzy system:
let the ith fuzzy rule of the nonlinear object be expressed as follows: if theta1(t) is Mi1,...,
Figure BDA0003222564250000085
Is composed of
Figure BDA0003222564250000086
..., θg(t) is MigThen, then
Figure BDA0003222564250000081
In the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000087
is the derivative of x (t), x (t) represents the object state, x (t) is an n-dimensional real number,
Figure BDA0003222564250000088
representing a control input affected by actuator saturation,
Figure BDA0003222564250000089
is nuDimensional real number, y (t) representing measurement output, y (t) being nyDimensional real number, t represents time, Ai,BiAnd CiRepresenting a gain matrix; the number of the object fuzzy rules is r, and i represents the serial number of the object fuzzy rules; theta1(t),
Figure BDA00032225642500000810
And thetag(t) denotes 1 st, k-th and g-th antecedent variables, respectively, g denotes the number of antecedent variables,
Figure BDA00032225642500000811
indicating the sequence number of the antecedent variable, Mi1,
Figure BDA00032225642500000812
And MigRespectively represent the 1 st and the 1 st fuzzy rules of the object
Figure BDA00032225642500000813
And the g fuzzy set.
The nonlinear object model represented by the T-S fuzzy system is obtained by using product fuzzy inference, a center average deblurring device and a single-point fuzzifier as follows
Figure BDA0003222564250000082
In the formula, an alternative
Figure BDA0003222564250000083
And is
Figure BDA0003222564250000084
Figure BDA00032225642500000814
Representing a front-part variable
Figure BDA00032225642500000816
Membership in fuzzy sets
Figure BDA00032225642500000815
Represents the accumulation and multiplication operations, respectively.
In designing an event driver based on nonlinear object measurement output:
based on the measured output of the nonlinear object, the event driver is designed as follows:
Figure BDA0003222564250000091
wherein, delta epsilon (0,1) is a driver threshold parameter,
Figure BDA00032225642500000911
is a positive definite matrix, h denotes the sampling period, tkh denotes the kth drive time, tkh is t of the sampling periodkMultiple, tk+1h denotes the (k + 1) th driving time, tk+1h is t of the sampling periodk+1The lower subscript k denotes the drive time number.
Figure BDA0003222564250000092
Which is indicative of the current sampling instant,
Figure BDA0003222564250000093
is tkAfter h is first
Figure BDA00032225642500000912
One sampling period, y (t)kh) And
Figure BDA00032225642500000913
respectively represent tkh and
Figure BDA00032225642500000914
corresponding non-linear object measurement output, min { } denotes the minimum function, and the upper right corner of the matrix, T, denotes the transpose of the matrix.
In the invention, an event driver judges and judges the event driving condition in the formula (3) at each period sampling point, and if the condition is met, the event driver sends sampling data; if the condition is not met, the sampled data is discarded. Therefore, the event driver transmits only the sample data satisfying the event-driven condition, and system-limited resources such as network bandwidth can be saved. In addition, the event driver only uses the periodic sampling value output by the nonlinear object measurement, the software implementation is easy, the Chino phenomenon is avoided in principle, and the assumed limit that most achievements can completely measure the object state is released.
The actuator saturation model is established as follows:
Figure BDA0003222564250000094
in the formula (I), the compound is shown in the specification,
Figure BDA00032225642500000915
denotes the number u (t)
Figure BDA00032225642500000916
Dimensional components, u (t) representing object control inputs without regard to actuator saturation effects, u (t) being nuThe number of the dimensional real number is,
Figure BDA00032225642500000917
the number of dimensions of u (t) is shown,
Figure BDA00032225642500000918
and
Figure BDA00032225642500000919
respectively representing the maximum and minimum allowable output values of the actuator,
Figure BDA00032225642500000920
to represent
Figure BDA00032225642500000921
The corresponding actuator saturation function value, sat (), represents the actuator saturation function.
According to the actuator saturation model (4), when
Figure BDA0003222564250000095
When the saturation function of the actuator is output as the maximum allowable output value of the actuator, i.e. the maximum allowable output value of the actuator
Figure BDA0003222564250000096
When in use
Figure BDA0003222564250000097
When the actuator saturation function is output as the minimum allowable output value of the actuator, i.e.
Figure BDA0003222564250000098
When in use
Figure BDA0003222564250000099
When, the actuator saturation function is output as
Figure BDA00032225642500000923
Using an actuator saturation model (4), control inputs in the non-linear object model (2) affected by actuator saturation are entered
Figure BDA00032225642500000922
Expressed as:
Figure BDA00032225642500000910
in the formula, an alternative
Figure BDA0003222564250000101
u1And
Figure BDA0003222564250000102
1 st and n th of u (t), respectivelyuDimensional component, sat: (u1) And
Figure BDA0003222564250000103
respectively represents u1And
Figure BDA0003222564250000104
the corresponding actuator saturation function value. Substitution type
Figure BDA0003222564250000105
And
Figure BDA0003222564250000106
respectively represents u1,
Figure BDA00032225642500001022
And
Figure BDA0003222564250000107
the corresponding function value of the dead zone of the actuator,
Figure BDA00032225642500001018
representing the actuator dead band function.
From the actuator saturation model (4), the following equation holds
Figure BDA0003222564250000108
In the formula, real number
Figure BDA0003222564250000109
To represent
Figure BDA00032225642500001019
Max { } denotes the maximum function.
B, establishing a random deception attack model and a DOFSS controller model, and establishing a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters;
first, the random spoofing attack effect is not consideredDOFSS controller input with zero-order keeper
Figure BDA00032225642500001023
Is shown as
Figure BDA00032225642500001010
In the formula, [ t ]kh+τk,tk+1h+τk+1) Indicating the zero-order keeper hold time,
Figure BDA00032225642500001011
and
Figure BDA00032225642500001012
respectively represent tkh and tk+1h corresponding to the network induced delay, τ and
Figure BDA00032225642500001020
respectively representing the lower and upper bounds of the network induced delay.
Definition of
Figure BDA00032225642500001021
Dividing the zero order keeper hold time as follows
Figure BDA00032225642500001013
In the formula, nkIndicates the number of the division subintervals,
Figure BDA00032225642500001014
is nkMaximum value, U, being a union symbol, subinterval
Figure BDA00032225642500001015
Is shown below
Figure BDA00032225642500001016
In dividing sub-intervals
Figure BDA00032225642500001017
Above, the function is defined as follows
e(t)＝y(tkh)-y(tkh+nkh),η(t)＝t-(tkh+nkh) (10)；
In the formula, y (t)kh+nkh) Representing the sampling instant tkh+nkh corresponding to the non-linear object measurement output.
Using equation (10), the dofss controller input (7) is represented as:
Figure BDA0003222564250000111
in the formula, y (t- η (t)) represents a nonlinear object measurement output corresponding to time t- η (t), and t- η (t) ═ t is obtained from (10)kh+nkh。
Secondly, a random spoofing attack model is established as follows
Figure BDA0003222564250000112
In the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000113
representing a spoofing attack function, wherein a (t) epsilon {0,1} represents a Bernoulli distribution random variable, and when a (t) is 1, the spoofing attack is activated and the controller input is tampered; when a (t) is 0, the spoofing attack sleeps, and the controller input is not tampered; since attack energy is usually limited in practice, i.e.
Figure BDA00032225642500001111
Satisfies the following formula
Figure BDA0003222564250000114
Where G is an attack energy definition matrix.
Using equations (11) and (12), controller inputs under random spoofing attacks are derived
Figure BDA00032225642500001112
As follows
Figure BDA0003222564250000115
Subsequently, a dofss controller model is built as follows:
let the jth fuzzy rule of the controller be expressed as: if theta1(t) is Mj1,...,
Figure BDA00032225642500001113
Is composed of
Figure BDA00032225642500001114
...,θg(t) is MjgThen, then
Figure BDA0003222564250000116
In the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000117
is xcDerivative of (t), xc(t) denotes controller status, xc(t) is an n-dimensional real number, xc(t- η (t)) represents the controller state for t- η (t),
Figure BDA0003222564250000118
and
Figure BDA0003222564250000119
is a gain matrix; the number of fuzzy rules of the controller is r, j represents the serial number of the fuzzy rules of the controller, Mj1,
Figure BDA00032225642500001115
And MjgRespectively representing the 1 st fuzzy set, the kth fuzzy set and the g fuzzy set under the jth fuzzy rule of the controller.
Using product fuzzy inference, center-mean deblurring, and single-point fuzzifier, a dofss controller model was obtained as follows
Figure BDA00032225642500001110
In the formula, an alternative
Figure BDA0003222564250000121
And is
Figure BDA0003222564250000122
Figure BDA0003222564250000123
Representing a front-part variable
Figure BDA0003222564250000124
Membership in fuzzy sets
Figure BDA0003222564250000125
Membership function of (c).
In summary, the nonlinear object model (2) and the dofss controller model (16) are combined to establish a closed-loop system model as follows:
Figure BDA0003222564250000126
in the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000127
the state of the closed-loop system is represented,
Figure BDA0003222564250000128
is the derivative of chi (t), and chi (t-eta (t)) represents the closed-loop system state corresponding to t-eta (t),
Figure BDA0003222564250000129
and
Figure BDA00032225642500001210
representing a closed loop system gain matrix.
Designing the joint design conditions of the nonlinear system event driver and the DOFS controller under the influence of random deception attack, actuator saturation and network induced delay to solve the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure BDA00032225642500001211
And gain matrix of equivalent DOFSS controller
Figure BDA00032225642500001212
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
The step C comprises the following two specific steps:
c1: based on the Lyapunov stability theory, the asymptotic stability condition of the nonlinear system under the influence of random deception attack, event drivers, actuator saturation and network induced delay is determined.
Given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure BDA00032225642500001215
An actuator saturation parameter ε, an attack energy definition matrix G, and an event driver threshold parameter δ ∈ (0,1), if there is a positive definite matrix
Figure BDA00032225642500001216
R＞0,S＞0,Q1＞0,Q2＞0,Q3> 0, and matrix U2,U3Satisfy the following requirements
Figure BDA00032225642500001213
And
Figure BDA00032225642500001214
the closed loop system (17) is asymptotically stable under the influence of random spoofing attacks, event drivers, actuator saturation and network induced delays.
In the above stability conditions, the following alternative formula is used:
Figure BDA0003222564250000131
Figure BDA0003222564250000132
Figure BDA0003222564250000133
Figure BDA0003222564250000134
Figure BDA0003222564250000135
Figure BDA0003222564250000136
Figure BDA0003222564250000137
Figure BDA0003222564250000138
wherein the content of the first and second substances,
Figure BDA0003222564250000139
the mathematical expectation for a (t) is shown,
Figure BDA00032225642500001310
is a mathematical expectation function. He { } denotes the sum of the matrix and its transpose, denotes the symmetric terms in the symmetric matrix,
Figure BDA00032225642500001311
the matrix represents a zero matrix, the upper right corner mark-1 of the matrix represents an inverse matrix, I is an identity matrix, col represents a column matrix, and diag represents a diagonal matrix.
And (3) proving that: the structure of Lyapunov functional is as follows
Figure BDA00032225642500001312
In the formula, an alternative
Figure BDA00032225642500001313
ρ＝0.5(η3-η1)，
Figure BDA00032225642500001314
Representing integral variables, χ(s), χ (s- η)1) And χ (s- η)2) Respectively represent s, s-eta1And s-eta2The corresponding closed-loop system state is,
Figure BDA00032225642500001315
the derivative of χ(s) is represented.
Derived from Lyapunov functional
Figure BDA0003222564250000141
In the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000142
is the derivative of v (t),
Figure BDA0003222564250000143
and
Figure BDA0003222564250000144
respectively representing alternative expressions corresponding to t and t-rho
Figure BDA0003222564250000145
χ(t-η1) Represents t-eta1Corresponding closed loop system state, alternative ζ1,ζ2,ζ3Is represented as follows:
Figure BDA0003222564250000146
in the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000147
representing integral variables
Figure BDA00032225642500001418
The corresponding closed loop system state derivative.
Consider the following two cases:
(1) if eta (t) belongs to [ eta ∈ [ ]1,η2) ζ in pair (21)1,ζ2,ζ3Using the Qisheng inequality, and using
Figure BDA0003222564250000148
Further on ζ2Obtained by using a mutual convex method
Figure BDA0003222564250000149
In the formula, an alternative
Figure BDA00032225642500001410
And
Figure BDA00032225642500001411
and χ (t- η)3) Respectively represent t-eta2And t-eta3Corresponding closed loop system state.
(2) If η (t) is equal to [ η [ ]2,η3]ζ in pair (21)1,ζ2,ζ3Using the Qisheng inequality, and using
Figure BDA00032225642500001412
Further on ζ3Obtained by using a mutual convex method
Figure BDA00032225642500001413
In the formula, an alternative formula is used
Figure BDA00032225642500001414
And
Figure BDA00032225642500001415
using (22) and (23), the mathematical expectation is given to the derivative of the Lyapunov functional (20)
Figure BDA00032225642500001416
In the formula (I), the compound is shown in the specification,
Figure BDA00032225642500001417
as an alternative.
Using equation (10), derived from the event driver (3) and the attack energy limited condition (13)
Figure BDA0003222564250000151
Obtained from formula (24) using formula (25)
Figure BDA0003222564250000152
Wherein the following alternative formula is used
Figure BDA0003222564250000153
Obtained by applying the Schur supplement theory to the formula (18)
Figure BDA0003222564250000154
By substituting formula (27) for formula (26) to obtain
Figure BDA0003222564250000155
According to Lyapunov's stability theory, the closed loop system (17) is asymptotically stable. After the syndrome is confirmed.
In the above system stable condition, there is an event driver positive definite matrix
Figure BDA0003222564250000156
Inverse matrix of
Figure BDA0003222564250000157
And DOFSS controller gain matrix
Figure BDA0003222564250000158
Coupled with the positive definite matrix P, this condition cannot therefore be used directly for event driver and controller designs. To solve this problem, the present invention will further present an event driver and dofss controller joint design method.
C2 obtaining the combined design condition of the event driver and the DOFSS controller by utilizing the linear matrix inequality technology based on the nonlinear system asymptotic stable condition obtained in the step C1, namely obtaining the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure BDA0003222564250000159
And gain matrix of equivalent DOFSS controller
Figure BDA00032225642500001510
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
First, the prior art (theorem 1) used in step C2 is given as follows:
theorem 1. if a positive definite matrix Q > 0 is given, the matrix L and the real number e > 0, then the inequality (L-e Q) Q-1(L-epsilon Q) is equal to or greater than 0, namely inequality-LQ-1L≤∈2Q-2 ∈ L holds.
Then, the event driver and dofss controller joint design conditions are given as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure BDA00032225642500001511
The saturation parameter epsilon of the actuator and the attack energy limit matrix G, and the real number epsilon1＞0,∈2＞0,∈3> 0, if real numbers are present
Figure BDA0003222564250000161
Positive definite matrix
Figure BDA0003222564250000162
Figure BDA0003222564250000163
And a matrix
Figure BDA0003222564250000164
Satisfy the requirement of
Figure BDA0003222564250000165
Figure BDA0003222564250000166
And
Figure BDA0003222564250000167
the closed loop system (17) is asymptotically stable while obtaining event driver parameters
Figure BDA0003222564250000168
And a gain matrix of the equivalent DOFSS controller (34)
Figure BDA0003222564250000169
As follows
Figure BDA00032225642500001610
Under the above conditions, the following alternative formulae were used
Figure BDA00032225642500001611
Figure BDA00032225642500001612
Figure BDA00032225642500001613
Figure BDA00032225642500001614
Figure BDA00032225642500001615
Figure BDA00032225642500001616
Figure BDA00032225642500001617
Figure BDA00032225642500001618
Figure BDA00032225642500001619
Figure BDA00032225642500001620
X is greater than 0, Y is greater than 0 and is positive definite matrix, N is N X N dimension real number matrix.
And (3) proving that: the positive definite matrix P is decomposed as follows
Figure BDA0003222564250000171
In the formula, X is more than 0, Y is more than 0 and is a positive definite matrix, and N is an N multiplied by N dimensional real number matrix. From the Sull complement theorem, a positive definite matrix P > 0 is equivalent to
Figure BDA0003222564250000172
The system stability condition in step C1 is changed as follows
Figure BDA0003222564250000173
Figure BDA0003222564250000174
Wherein the following alternative formulae are used
Ψ1＝diag{Φ1,Φ1},Ψ2＝diag{Φ1,Φ1,Φ1,Φ1,Φ1,I,I,I,Ψ3,I,I,I},
Figure BDA0003222564250000175
Figure BDA0003222564250000176
In pair (32)
Figure BDA0003222564250000177
And
Figure BDA0003222564250000178
obtained by Using theorem 1 in (28)
Figure BDA0003222564250000179
And
Figure BDA00032225642500001710
therefore, the closed loop system (17) is asymptotically stable if given conditions are satisfied. Simultaneous acquisition of event driver parameters
Figure BDA00032225642500001711
And the gain matrix of the DOFSS controller (16) are as follows
Figure BDA00032225642500001712
To process (33) the unknown matrix N, a linear transformation is used
Figure BDA00032225642500001713
The equivalent DOFSS controller is obtained as follows
Figure BDA00032225642500001714
In the formula (I), the compound is shown in the specification,
Figure BDA00032225642500001715
representing the equivalent dofss controller state,
Figure BDA00032225642500001716
is a real number in the n-dimension,
Figure BDA00032225642500001717
is composed of
Figure BDA00032225642500001718
The derivative of (a) of (b),
Figure BDA0003222564250000181
representing equivalent DOFSS controller states, gain matrices, corresponding to t- η (t)
Figure BDA0003222564250000182
Obtained from (29). After the syndrome is confirmed.
Through the combined design method of the event driver and the DOFSS controller, a user can determine each parameter one by one according to specific design requirements, and the event driver and the DOFSS controller are obtained according to the steps, so that the event driver can reduce the data transmission rate, and system limited resources such as network bandwidth and the like are saved; the DOFSS controller enables the nonlinear system to be asymptotically stable under the influence of random spoofing attack, event drivers, actuator saturation and network-induced delay.
Application scenarios of the present invention are exemplified as follows: in recent years, network attacks against practical industrial control systems have been frequent, such as: in 2015, the ukraine power system was attacked by malware, and about 140 million people were affected by the power outage. Aiming at the scene, the related method of the invention is applied, the centrifugal machine system and the power system of the nuclear power station are modeled into a nonlinear object, an event driver and an actuator saturation model are designed, a random deception attack model, a DOFSS controller model and a closed-loop system model organically integrating random deception attack, event driver, actuator saturation and network induced delay parameters are established, the asymptotic stable condition of the closed-loop system is deduced, the combined design condition of the event driver and the DOFSS controller is given, and the event driver and the DOFSS controller meeting the requirements are obtained at the same time.
The present invention is described in detail below with reference to examples:
step A: establishing a nonlinear object model, and designing an event driver based on nonlinear object measurement output:
wherein, the nonlinear object takes a mass spring damping system as an example, and the kinetic equation is described as
Figure BDA0003222564250000183
In the formula (I), the compound is shown in the specification,
Figure BDA0003222564250000184
a displacement from a reference point is indicated,
Figure BDA0003222564250000185
and
Figure BDA0003222564250000186
respectively represent
Figure BDA0003222564250000187
The first and second derivatives of (a), m is mass,
Figure BDA0003222564250000188
it is indicated that the friction force is,
Figure BDA0003222564250000189
the elastic force of the spring is shown,
Figure BDA00032225642500001810
representing a control input affected by actuator saturation,
Figure BDA00032225642500001811
are real numbers. Parameter setting is as follows
Figure BDA00032225642500001812
c is 2 n.m/s,
Figure BDA00032225642500001813
Figure BDA00032225642500001814
defining object states
Figure BDA00032225642500001815
The mass-spring damping system (35) can be described as a non-linear object (2), where μ1(θ(t))＝-(θ1(t)+8)/2.88,μ2(θ(t))＝1-μ1(θ(t)),
Figure BDA0003222564250000191
r 2, g 1, the gain matrix is as follows:
Figure BDA0003222564250000192
designing an event driver (3) based on the measurement output of a nonlinear object, wherein the sampling period h is 100 milliseconds, the threshold parameter delta of the driver and a positive definite matrix
Figure BDA00032225642500001913
The result of the event driver design condition in conjunction with the dofss controller in step C2.
Establishing an actuator saturation model (4): wherein the maximum allowable output value of the actuator
Figure BDA0003222564250000193
uiMaximum value of
Figure BDA0003222564250000194
Real number
Figure BDA0003222564250000195
And B: establishing a random deception attack and DOFSS controller model, and establishing a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters;
first, a random spoofing attack model is established as
Figure BDA0003222564250000196
Wherein the content of the first and second substances,
Figure BDA0003222564250000197
for spoofing attack functions, tanh represents a non-linear hyperbolic tangent function, a (t) e {0,1} is a Bernoulli distributed random variable, and the mathematical expectation is
Figure BDA0003222564250000198
The attack energy limit matrix is G ═ 0.1 (the one-dimensional matrix is equally real).
Next, a dofss controller model is built (16).
Then, a closed-loop system model (17) organically integrating random spoofing attacks, event drivers, actuator saturation and network-induced delay parameters is established, wherein the network-induced delay lower boundτ10 ms and upper bound
Figure BDA0003222564250000199
Milliseconds.
And C: designing the joint design conditions of the event driver and the DOFSS controller under the influence of random deception attack, actuator saturation and network induced delay to obtain the parameters of the event driver
Figure BDA00032225642500001910
And gain matrix of equivalent DOFSS controller
Figure BDA00032225642500001911
And finally obtaining the event driver and the DOFSS controller which meet the system requirements.
Wherein the nonlinear system asymptotic steady condition is obtained from step C1, and further, the event driver and DOFSS controller joint design condition is obtained from step C2, wherein ∈1＝∈2＝∈31. Solving the joint design condition to obtain event driver parameters
Figure BDA00032225642500001912
And equivalent DOFSS controller gain matrix as follows
Figure BDA0003222564250000201
In the embodiment, under the action of a designed DOFSS controller, the mass spring damping system can be gradually stabilized under the influence of random deception attack, event drivers, actuator saturation and network induced delay. In addition, the sensor samples 100 data cycles in simulation time [0,10 seconds ], with the event driver transmitting 71 data, at a data transmission rate of 71%. Compared with the data sending rate of the periodic sampler of 100 percent, the event driver saves 29 percent of system resources on the premise of ensuring the system performance. In addition, 15 data out of 71 data sent by the event driver were tampered with by random spoofing attack, and the attack rate was 21%.
The embodiment shows that, on one hand, the event driver can reduce the data transmission rate to 71%, and system limited resources such as 29% of network bandwidth are saved. On the other hand, although up to 21% of event driver transmission data is tampered with by random spoofing attacks, the nonlinear system can be asymptotically stabilized by the dofss controller. In addition, the method is designed based on the nonlinear object measurement output, and the hypothesis limit that most achievements can completely measure the object state is removed.

Claims (10)

1. A method for joint design of a nonlinear system event driver and a dofss controller, comprising the steps of:
a, establishing a nonlinear object model and an actuator saturation model, and designing an event driver based on nonlinear object measurement output;
b, establishing a random deception attack model and a DOFSS controller model, and establishing a closed-loop system model organically integrating random deception attack, an event driver, actuator saturation and network induced delay parameters;
designing the joint design conditions of the nonlinear system event driver and the DOFS controller under the influence of random deception attack, actuator saturation and network induced delay to solve the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure FDA0003222564240000011
And gain matrix of equivalent DOFSS controller
Figure FDA0003222564240000012
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
2. The nonlinear system event driver and dofss controller joint design method according to claim 1, wherein in the step a, the nonlinear object model is:
Figure FDA0003222564240000013
wherein the content of the first and second substances,
Figure FDA0003222564240000014
is the derivative of x (t), x (t) represents the object state, x (t) is an n-dimensional real number,
Figure FDA0003222564240000015
representing a control input affected by actuator saturation,
Figure FDA0003222564240000016
is nuDimensional real number, y (t) representing measurement output, y (t) being nyDimensional real number, t represents time, Ai,BiAnd CiRepresenting a gain matrix; the number of the object fuzzy rules is r, and i represents the serial number of the object fuzzy rules; substitution type
Figure FDA0003222564240000017
And is
Figure FDA0003222564240000018
Figure FDA0003222564240000019
And thetag(t) represents the 1 st and the 1 st, respectively
Figure FDA00032225642400000110
The individual and the g-th antecedent variables, g representing the number of antecedent variables,
Figure FDA00032225642400000111
the sequence number of the front-piece variable is shown,
Figure FDA00032225642400000112
representing a front-part variable
Figure FDA00032225642400000113
Membership in fuzzy sets
Figure FDA00032225642400000114
Represents the accumulation and multiplication operations, respectively.
3. The nonlinear system event driver and dofss controller joint design method according to claim 2, wherein in step a, the event driver based on the nonlinear object measurement output is:
Figure FDA0003222564240000021
wherein, delta epsilon (0,1) is a driver threshold parameter,
Figure FDA0003222564240000022
is a positive definite matrix, h denotes the sampling period, tkh denotes the kth drive time, tkh is t of the sampling periodkMultiple, tk+1h denotes the (k + 1) th driving time, tk+1h is t of the sampling periodk+1The lower subscript k denotes the drive time number,
Figure FDA0003222564240000023
which is indicative of the current sampling instant,
Figure FDA0003222564240000024
is tkAfter h is first
Figure FDA0003222564240000025
One sampling period, y (t)kh) And
Figure FDA0003222564240000026
respectively represent tkh and
Figure FDA0003222564240000027
and (3) outputting corresponding nonlinear object measurement, wherein min { } represents a minimum function, and a right upper corner mark T of the matrix represents the transposition of the matrix.
4. The nonlinear system event driver and dofss controller joint design method according to claim 3, wherein in step a, the actuator saturation model is:
Figure FDA0003222564240000028
wherein the content of the first and second substances,
Figure FDA0003222564240000029
denotes the number u (t)
Figure FDA00032225642400000210
Dimensional components, u (t) representing object control inputs without regard to actuator saturation effects, u (t) being nuThe number of the dimensional real number is,
Figure FDA00032225642400000211
the number of dimensions of u (t) is shown,
Figure FDA00032225642400000212
and
Figure FDA00032225642400000213
respectively representing the maximum and minimum allowable output values of the actuator,
Figure FDA00032225642400000214
to represent
Figure FDA00032225642400000215
The corresponding actuator saturation function value, sat (), represents the actuator saturation function.
5. The nonlinear system event driver and dofss controller joint design method according to claim 4, wherein in the step B, the stochastic spoofing attack model is:
Figure FDA00032225642400000216
wherein the content of the first and second substances,
Figure FDA00032225642400000217
representing a spoofing attack function, wherein a (t) epsilon {0,1} represents a Bernoulli distribution random variable, and when a (t) is 1, the spoofing attack is activated and the controller input is tampered; when a (t) is 0, the spoofing attack sleeps, and the controller input is not tampered;
Figure FDA00032225642400000218
g is an attack energy limiting matrix;
Figure FDA00032225642400000219
y (t- η (t)) represents a nonlinear object measurement output corresponding to time t- η (t), where t- η (t) is tkh+nkh，e(t)＝y(tkh)-y(tkh+nkh)，y(tkh+nkh) Watch (A)Indicating the sampling time tkh+nkh corresponding to the non-linear object measurement output.
6. The nonlinear system event driver and dofss controller joint design method according to claim 5, wherein in the step B, the dofss controller model is:
Figure FDA0003222564240000031
wherein the content of the first and second substances,
Figure FDA0003222564240000032
is xcDerivative of (t), xc(t) denotes controller status, xc(t) is an n-dimensional real number, xc(t- η (t)) represents the controller state for t- η (t),
Figure FDA0003222564240000033
and
Figure FDA0003222564240000034
is a gain matrix; the number of fuzzy rules of the controller is r, j represents the serial number of the fuzzy rules of the controller, and the alternative formula
Figure FDA0003222564240000035
And is
Figure FDA0003222564240000036
Figure FDA0003222564240000037
Figure FDA0003222564240000038
Representing a front-part variable
Figure FDA0003222564240000039
Membership in fuzzy sets
Figure FDA00032225642400000310
Membership function of (c).
7. The nonlinear system event driver and dofss controller joint design method according to claim 6, wherein in the step B, the closed-loop system model organically integrating stochastic spoofing attack, event driver, actuator saturation and network-induced delay parameters is:
Figure FDA00032225642400000311
in the formula (I), the compound is shown in the specification,
Figure FDA00032225642400000312
is the derivative of x (t),
Figure FDA00032225642400000313
representing the closed-loop system state, x (t-eta (t)) representing the closed-loop system state corresponding to t-eta (t),
Figure FDA00032225642400000314
and
Figure FDA00032225642400000315
representing closed-loop system gain matrices, alternatively
Figure FDA00032225642400000316
Figure FDA00032225642400000317
And
Figure FDA00032225642400000318
respectively represent
Figure FDA00032225642400000319
And
Figure FDA00032225642400000320
the corresponding function value of the dead zone of the actuator,
Figure FDA00032225642400000321
and
Figure FDA00032225642400000322
respectively represent the 1 st and the 1 st dimensions of u (t)
Figure FDA00032225642400000323
And nuThe component of the dimension(s) is,
Figure FDA00032225642400000324
representing the actuator dead band function.
8. The nonlinear system event driver and dofss controller joint design method according to claim 1, wherein the step C comprises the following specific steps:
c1: determining a nonlinear system asymptotic stability condition under the influence of random deception attack, an event driver, actuator saturation and network induced delay based on the Lyapunov stability theory;
c2 obtaining the combined design condition of the event driver and the DOFSS controller by utilizing the linear matrix inequality technology based on the nonlinear system asymptotic stable condition obtained in the step C1, namely obtaining the event driver parameters meeting the communication and control requirements of the nonlinear system
Figure FDA0003222564240000041
And gain matrix of equivalent DOFSS controller
Figure FDA0003222564240000042
And finally obtaining the event driver and the DOFSS controller which meet the communication and control requirements of the nonlinear system.
9. The nonlinear system event driver and dofss controller joint design method in accordance with claim 8, wherein: the nonlinear system asymptotic stability condition under the influence of random deception attack, event driver, actuator saturation and network induced delay is as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure FDA0003222564240000043
An actuator saturation parameter ε, an attack energy definition matrix G, and an event driver threshold parameter δ ∈ (0,1), if there is a positive definite matrix
Figure FDA0003222564240000044
P＞0,R＞0,S＞0,Q1＞0,Q2＞0,Q3> 0, and matrix U2,U3Satisfy the following requirements
Figure FDA0003222564240000045
And
Figure FDA0003222564240000046
the closed loop system is asymptotically stable under the influence of random deception attack, event drivers, actuator saturation and network induced delay; in the above stability conditions, the following alternative formula is used:
Figure FDA0003222564240000047
Π21＝col{η10Λ1,η21Λ1,η32Λ1,η10Λ2,η21Λ2,η32Λ2},Π31＝CiE1e3+e6,
Figure FDA0003222564240000048
Π51＝G(CiE1e3+e6),
Figure FDA0003222564240000049
Figure FDA00032225642400000410
Π44＝-ε-1,Π55＝-I,
Figure FDA00032225642400000411
Figure FDA00032225642400000412
Figure FDA00032225642400000413
η10＝η1-η0,η21＝η2-η1,η32＝η3-η2,η0＝0,η1＝τ,η2＝0.5(η3+η1),
Figure FDA0003222564240000051
Figure FDA0003222564240000052
Figure FDA0003222564240000053
wherein the content of the first and second substances,
Figure FDA0003222564240000054
the mathematical expectation for a (t) is shown,
Figure FDA0003222564240000055
is a mathematical expectation function; he { } denotes the sum of the matrix and its transpose, denotes the symmetric terms in the symmetric matrix,
Figure FDA0003222564240000056
the matrix represents a zero matrix, the upper right corner mark-1 of the matrix represents an inverse matrix, I is an identity matrix, col represents a column matrix, and diag represents a diagonal matrix.
10. The nonlinear system event driver and dofss controller joint design method in accordance with claim 9, wherein: the joint design conditions of the event driver and the DOFSS controller are as follows:
given a sampling period h, the lower bound of network induced delayτAnd upper bound
Figure FDA0003222564240000057
The saturation parameter epsilon of the actuator and the attack energy limit matrix G, and the real number epsilon1＞0,∈2＞0,∈3> 0, if real numbers are present
Figure FDA0003222564240000058
Positive definite matrix
Figure FDA0003222564240000059
Figure FDA00032225642400000510
And a matrix
Figure FDA00032225642400000511
Satisfy the requirement of
Figure FDA00032225642400000512
Figure FDA00032225642400000513
And
Figure FDA00032225642400000514
the closed loop system is asymptotically stable while obtaining event driver parameters
Figure FDA00032225642400000515
And gain matrix of equivalent DOFSS controller
Figure FDA00032225642400000516
As follows
Figure FDA00032225642400000517
Under the above conditions, the following alternative formulae were used
Figure FDA00032225642400000518
Figure FDA00032225642400000519
Figure FDA00032225642400000520
Figure FDA00032225642400000521
Figure FDA00032225642400000522
Figure FDA0003222564240000061
Figure FDA0003222564240000062
Figure FDA0003222564240000063
Figure FDA0003222564240000064
Ψ1＝diag{Φ1,Φ1},
Figure FDA0003222564240000065
X is greater than 0, Y is greater than 0 and is positive definite matrix, N is N X N dimension real number matrix.
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