CN111538244B - Net cage lifting control method based on distributed event trigger strategy - Google Patents

Net cage lifting control method based on distributed event trigger strategy Download PDF

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CN111538244B
CN111538244B CN202010410371.2A CN202010410371A CN111538244B CN 111538244 B CN111538244 B CN 111538244B CN 202010410371 A CN202010410371 A CN 202010410371A CN 111538244 B CN111538244 B CN 111538244B
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CN111538244A (en
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汪星一
钟智雄
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Minjiang University
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Abstract

A net cage lifting control method based on a distributed event trigger strategy is disclosed. The invention relates to a cage underwater communication of multi-buoy linkage control, and particularly provides a lifting control method based on a distributed event trigger strategy, which comprises the steps of firstly analyzing the motion principle of a cage, regarding each buoy of the cage as a rigid mass point, modeling, and expressing the rigid mass point as a type II T-S fuzzy model; secondly, an output feedback fuzzy controller based on distributed events for reducing data communication is provided; on the basis, the proposed controller is further designed by combining an accessible set analysis method based on a Lyapunov method. The control method provided by the invention can ensure the lifting reliability and anti-interference performance of the multi-buoy linkage type net cage.

Description

Net cage lifting control method based on distributed event trigger strategy
Technical Field
The invention relates to multi-buoy linkage control underwater communication of a net cage, in particular to a net cage lifting control method based on a distributed event trigger strategy.
Background
Deep water gravity type active net cages are usually controlled by multi-buoy linkage to carry out underwater communication, and sea storms bring certain influence and interference on settlement of the net cages and signal transmission among the multi-buoys. In order to avoid serious accidents such as easy toppling and communication obstacle of the net cage in the sedimentation process under the action of sea waves, an output feedback fuzzy controller based on distributed events for reducing data communication is needed to ensure the reliability and anti-interference performance of the lifting of the multi-buoy linkage net cage.
Disclosure of Invention
In view of this, the present invention provides a method for controlling the ascending and descending of a net cage based on a distributed event triggering strategy, so as to ensure the reliability and anti-interference of the ascending and descending of a multi-buoy linked net cage.
In order to achieve the purpose, the invention adopts the following technical scheme: a net cage lifting control method based on a distributed event trigger strategy is characterized by comprising the following steps:
step S1: analyzing the movement principle of the net cage, regarding each buoy of the net cage as a rigid mass point, modeling, and expressing the rigid mass point as a type II T-S fuzzy model;
step S2: a distributed event based output feedback fuzzy controller for reducing data communication is proposed;
and step S3: on the basis, the output feedback fuzzy controller is further designed based on a Lyapunov method and combined with an accessible set analysis method, so that the reliability and the anti-interference performance of the multi-buoy linkage type net cage lifting are ensured.
In an embodiment of the present invention, the step S1 specifically includes:
step S11: analyzing the movement principle of the net cage, regarding each buoy of the net cage as a rigid particle and modeling:
Figure BDA0002492901740000011
Figure BDA0002492901740000012
D(V)＝diag{X u | u | |u|+X u ，Y v | v | |v|+Y v ，Z w | w | |w|+Z w } (3)
Figure BDA0002492901740000021
in the formula, M represents a net cage quality matrix; m is the total mass of the net cage; x is the longitudinal resultant force of the net cage; y is the transverse resultant force of the net cage; z is the vertical resultant force of the net cage;
Figure BDA0002492901740000022
representing the hydrodynamic coefficient of the net cage; v represents a net cage motion matrix; k is z (V) is the cage motion variable; the C (V) matrix represents the resultant force of Coriolis force, centripetal force and moment thereof generated by the inherent mass of the mobile net cage;pthe transverse inclination angle speed of the net cage;qthe speed of the longitudinal inclination angle of the net cage;rthe net cage yaw rate; d (V) is a hydrodynamic damping matrix suffered by the movement of the mobile net cage;uis the longitudinal direction of the net cageThe moving speed;vthe transverse moving speed of the net cage;wthe vertical moving speed of the net cage is obtained; non-viable cellsu|、|v|、|wRespectively representing the absolute values of the two; x u 、Y v 、Z w Is the linear damping coefficient; x u | u | 、Y v | v | 、Z w | w | Is the fourth order damping coefficient; g (E) is a vector formed by the inherent gravity, the inherent buoyancy and the interaction resultant moment of the net cage; g and B respectively represent the gravity and buoyancy of the net cage;
Figure BDA0002492901740000023
and theta respectively represents a roll angle and a pitch angle of the rotation of the net cage;
further simplifying a net cage motion equation to obtain a system control design model:
Figure BDA0002492901740000024
wherein [ x y ψ] T ＝x(t)，
Figure BDA0002492901740000025
First derivatives of x, y, ψ, respectively; [ tau ] to u τ v τ r ] T X and y are net cage positions (x and y) in a terrestrial coordinate system, psi is net cage bow angle, and tau u Thrust generated by a propulsion system when the net cage moves longitudinally; tau. v Thrust generated by the propulsion system when the net cage moves transversely; tau is r The moment is generated when the dragging net cage rotates; the non-linear equation for the net cage system is then as follows:
Figure BDA0002492901740000026
step S12: then, according to the formula (6), establishing a singular fuzzy system equation of the single buoy, as shown in the formula (7):
Figure BDA0002492901740000027
wherein E (h) is non-singular and satisfies
Figure BDA0002492901740000028
Wherein
Figure BDA0002492901740000029
Figure BDA00024929017400000210
E s Representing a non-singular system matrix, A l Representing the system state variable, r e And r f The inference rule numbers respectively represent the left side and the right side; h is s [ζ(t)]And mu l [ζ(t)]Is a function of normalized membership to a normalized degree,
Figure BDA0002492901740000031
represents the derivative of the system state variable, ω (t) = -C (V) -g (E), which is an external disturbance; they satisfy the following conditions:
Figure BDA0002492901740000032
Figure BDA0002492901740000033
wherein h is sφ [ζ φ (t)]And mu lφ [ζ φ (t)]Is the degree of membership, definition h s ：＝h s [ζ(t)]And mu l ：＝μ l [ζ(t)]To simplify the description; g represents the number of fuzzy members;
step S13: regarding each buoy of the net cage as a rigid mass point, establishing a large nonlinear singular system, wherein the system consists of N interconnected subsystems, and establishing a singular nonlinear equation of the net cage system controlled by the linkage of a plurality of buoys:
Figure BDA0002492901740000034
wherein
Figure BDA0002492901740000035
Is the number of subsystems, { E i (ζ i (t))，A ii (ζ i (t))，B i (ζ i (t))，C i ，D i (ζ i (t)) } is a medium having a measurable quantity nonlinear dynamics ζ i (t) system matrix, A ij (ζ i (t)，ζ j (t)) represents an interconnection matrix between the ith subsystem and the jth subsystem;
Figure BDA0002492901740000036
and
Figure BDA0002492901740000037
indicating system state, control inputs, disturbances and outputs, n xi 、n ui、 n ωi、 n yi Is the order of the matrix;
step S14: describing the singular nonlinear equation of the multi-buoy linked deep water net cage lifting system considered in (8) by a T-S model of type II, as follows:
Figure BDA0002492901740000038
wherein E is i (h i ) Is non-singular and satisfies
Figure BDA0002492901740000039
Figure BDA00024929017400000310
Figure BDA00024929017400000311
And
Figure BDA00024929017400000312
a set of fuzzy inference rule sets representing left and right sides, respectively; n is i 、r i 、r j Is the number of fuzzy rule sets;
Figure BDA00024929017400000313
and
Figure BDA00024929017400000314
are normalized membership functions that satisfy the following condition:
Figure BDA00024929017400000315
Figure BDA0002492901740000041
wherein the content of the first and second substances,
Figure BDA0002492901740000042
and the number of the first and second electrodes,
Figure BDA0002492901740000043
wherein the content of the first and second substances,
Figure BDA0002492901740000044
Figure BDA0002492901740000045
respectively representing the lower limit of a membership function, the upper limit of the membership function, the lower limit of a fuzzy member membership function and the upper limit of the fuzzy member membership function;
Figure BDA0002492901740000046
Figure BDA0002492901740000047
in an embodiment of the present invention, the step S2 specifically includes:
step S21: first, define
Figure BDA0002492901740000048
For system output, an output feedback fuzzy controller with event-based distributed broadcasting is built as follows:
Figure BDA0002492901740000049
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00024929017400000410
wherein
Figure BDA00024929017400000411
Is the gain of the controller to be designed,
Figure BDA00024929017400000412
representing a normalized membership function in the fuzzy controller;
front part variable ζ i (T) is measurable, in type II T-S model set, degree of membership
Figure BDA0002492901740000051
Upper limit and degree of membership of
Figure BDA0002492901740000052
Is a priori, but is a non-linear function
Figure BDA0002492901740000053
And
Figure BDA0002492901740000054
is unknown;thus, the normalized membership function of the fuzzy controller depends on
Figure BDA0002492901740000055
And
Figure BDA0002492901740000056
namely, it is
Figure BDA0002492901740000057
To obtain a minimum boundary
Figure BDA0002492901740000058
Step S22: an event trigger mechanism is proposed to identify whether or not to transmit the system output signal y i (t) causing data communication to be reduced; to implement the desired event-based control problem, the event triggering conditions are as follows:
event trigger conditions:
Figure BDA0002492901740000059
wherein sigma i ≧ 0 is a selected scalar; according to the execution event trigger condition (15), the system outputs the strategy based on the event trigger as follows:
Figure BDA00024929017400000510
step S23: in the proposed event trigger condition (15), the system output y is only sent when a specific event is triggered i (t); however, a measurable precursor variable ζ i (t) transmission needs to be triggered in time, which results in a reduction of partial data transmission; to further reduce the communication data, another event trigger condition is proposed for the front-end variables, as follows:
event trigger conditions:
Figure BDA00024929017400000511
wherein e i ≧ 0 is a selected scalar; the strategy of the front-part variable based on event trigger is as follows:
Figure BDA00024929017400000512
two event trigger conditions are used in the event-based policies proposed by equations (15) and (17) to verify when the precursor variables and system outputs can be transmitted through the network, reducing data transmission in the communication network;
step S24: variable η of equation (13) i Is dependent on a antecedent variable ζ i (t) and the variables of equation (17)
Figure BDA00024929017400000517
Is dependent on a front-part variable based on an event trigger condition
Figure BDA00024929017400000514
The event-based distributed IT-2 fuzzy controller in (13) is thus updated as follows:
Figure BDA00024929017400000515
wherein the content of the first and second substances,
Figure BDA00024929017400000516
in an embodiment of the present invention, the step S3 specifically includes:
step S31: the fuzzy singular model of the multi-buoy linkage type net cage system (9) is rewritten as follows:
Figure BDA0002492901740000061
wherein the content of the first and second substances,
Figure BDA0002492901740000062
definition of
Figure BDA0002492901740000063
Then there are:
Figure BDA0002492901740000064
step S32: the closed-loop fuzzy control system consisting of (13) and (23) can be rewritten as:
Figure BDA0002492901740000065
wherein
Figure BDA0002492901740000066
Step S33: the Lyapunov matrix is selected as follows:
Figure BDA0002492901740000067
wherein
Figure BDA0002492901740000068
Is easy to obtain, and has the advantages of easy acquisition,
Figure BDA0002492901740000069
definition of
Figure BDA00024929017400000610
And assuming that the unknown measurement noise is bounded, this can be satisfied:
Figure BDA00024929017400000611
wherein
Figure BDA0002492901740000071
Is a positive scalar quantity;
step S34: designing (19) the event-based distributed fuzzy controller such that the state trajectory of the resulting closed-loop control system in (24) is limited by the following reachable set:
Figure BDA0002492901740000072
the ellipsoid defined by the reachable set of the closed-loop control system in (24) is given by:
Figure BDA0002492901740000073
wherein
Figure BDA0002492901740000074
Consider the following Lyapunov function:
Figure BDA0002492901740000075
wherein
Figure BDA0002492901740000076
Easy to obtain V (t) = V T (t)≥0.
Defining:
Figure BDA0002492901740000077
the following can be obtained:
Figure BDA0002492901740000078
from (24) can be obtained:
Figure BDA0002492901740000079
wherein
Figure BDA00024929017400000710
And matrix multiplication
Figure BDA00024929017400000711
Due to the fact that
Figure BDA00024929017400000712
Wherein
Figure BDA00024929017400000713
And scalar k > 0.n is the matrix order;
given a positive definite symmetric matrix
Figure BDA0002492901740000081
And
Figure BDA0002492901740000082
from (31) and (32) follows:
Figure BDA0002492901740000083
and
Figure BDA0002492901740000084
and
Figure BDA0002492901740000085
it can be easily seen from (33) - (35):
Figure BDA0002492901740000086
then, a positive definite symmetric matrix is given
Figure BDA0002492901740000087
From the policy (16) based on the event trigger mechanism it follows:
Figure BDA0002492901740000088
from (30) - (37), the following are known:
Figure BDA0002492901740000091
definition of
Figure BDA0002492901740000092
Further, the following functions are defined:
Figure BDA0002492901740000093
wherein α ∈ [0,1].
Combining (30) to (40), it is possible to obtain:
Figure BDA0002492901740000094
easily seen, inequality pi i (h i ，μ i ，η i ) Meaning J (t) <0, so the inequality in (28) can be directly found;
since J (t) <0, it is possible to obtain:
Figure BDA0002492901740000095
this means that:
V(k+1)-1＜α(V(k)-1). (43)
from (43), it is easy to derive:
V(k)＜α k (V(0)-1)+1. (44)
from (28):
Figure BDA0002492901740000101
wherein
Figure BDA0002492901740000102
Thus, considering a large-scale IT-2 fuzzy system using the event-based distributed fuzzy controller of (19) in (9), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (26), if there is a symmetric positive definite matrix
Figure BDA0002492901740000103
Figure BDA0002492901740000104
Sum matrix
Figure BDA0002492901740000105
Figure BDA0002492901740000106
Sum matrix multiplier
Figure BDA0002492901740000107
Sum positive scalar quantity
Figure BDA0002492901740000108
So that for all subsystems
Figure BDA0002492901740000109
The following matrix inequality holds:
Π i (h i， μ i， η i )＜0， (46)
and the number of the first and second electrodes,
Figure BDA00024929017400001010
where Sym means the sum of the matrix and its transpose, e.g. Sym (A) = A + A T (ii) a I is an identity matrix;
Figure BDA00024929017400001011
in addition, the reachable set estimates satisfy the following boundary,
Figure BDA00024929017400001012
thereby deriving sufficient conditions to ensure that there is an event-based distributed fuzzy controller that can drive the state traces within reachable set boundaries;
step S35: definition of
Figure BDA0002492901740000111
By applying the cone-complement theorem, one can obtain:
Figure BDA0002492901740000112
wherein the content of the first and second substances,
Figure BDA0002492901740000113
since given an interconnect matrix
Figure BDA0002492901740000114
And have pairs of compatible dimensionsWeighting and determining matrix
Figure BDA0002492901740000115
The following equation holds true:
Figure BDA0002492901740000116
the following can be obtained:
Figure BDA0002492901740000117
then, extracting fuzzy antecedent variables yields:
Figure BDA0002492901740000118
wherein
Figure BDA0002492901740000119
Is defined in formula (53);
it should be noted that existing relaxation techniques
Figure BDA00024929017400001110
Figure BDA00024929017400001111
Is no longer suitable for fuzzy controller synthesis because
Figure BDA00024929017400001112
Figure BDA0002492901740000121
In practice, a positive scalar is always found from (14) and (18)
Figure BDA0002492901740000122
Compliance
Figure BDA0002492901740000123
Wherein
Figure BDA0002492901740000124
Similar to the asynchronous relaxation technique, assume that
Figure BDA0002492901740000125
Wherein
Figure BDA0002492901740000126
Is a symmetric matrix, one can derive:
Figure BDA0002492901740000127
by passing through
Figure BDA0002492901740000128
Thereafter, the conditions in (49) and (50) can be directly obtained using the existing relaxation method applied to (54);
therefore, consider the large IT-2 fuzzy system in (9), if a symmetric positive definite matrix exists
Figure BDA0002492901740000129
Figure BDA00024929017400001210
The event-based distributed fuzzy controller in (19) may ensure that the reachable set of the closed-loop system (24) is limited by the ellipse boundaries in (26),
Figure BDA00024929017400001211
sum matrix
Figure BDA00024929017400001212
Figure BDA00024929017400001213
And matrix product
Figure BDA00024929017400001214
And symmetric matrices with compatible dimensions
Figure BDA00024929017400001215
Sum positive scalar quantity
Figure BDA00024929017400001216
So as to all
Figure BDA00024929017400001217
The following matrix inequality holds:
Figure BDA00024929017400001218
Figure BDA00024929017400001219
Figure BDA00024929017400001220
wherein
Figure BDA00024929017400001221
Figure BDA0002492901740000131
Step S36: the inequalities in equations (56) - (58) are not non-linear matrix inequalities; to facilitate fuzzy controller design, a surrogate descriptor representation will be introduced in (23) for the proposed controller, as follows:
Figure BDA0002492901740000132
thus, the augmented system can be rewritten as:
Figure BDA0002492901740000133
wherein
Figure BDA0002492901740000134
Considering the large IT-2 fuzzy system in (9) and the event-based distributed fuzzy controller in (19) based on the fuzzy singular system in (61), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (27), if there is a positive definite symmetric matrix
Figure BDA0002492901740000135
Matrix array
Figure BDA0002492901740000136
Figure BDA0002492901740000137
Figure BDA0002492901740000141
Symmetric matrix of compatible dimensions
Figure BDA0002492901740000142
And all of
Figure BDA0002492901740000143
Positive scalar quantity of
Figure BDA0002492901740000144
Then the following non-linear matrix inequality holds:
Figure BDA0002492901740000145
Figure BDA0002492901740000146
Figure BDA0002492901740000147
wherein
Figure BDA0002492901740000148
Figure BDA0002492901740000151
Further, the controller gain matrix in (15) may be calculated by:
Figure BDA0002492901740000152
compared with the prior art, the invention has the following beneficial effects:
the invention adopts a net cage lifting control method based on a distributed event triggering strategy, and can ensure the reliability and anti-interference of the multi-buoy linkage type net cage lifting.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
The invention is further explained by the following embodiments in conjunction with the drawings.
Referring to fig. 1, the present invention provides a method for controlling the ascending and descending of a net cage based on a distributed event trigger strategy, which is characterized by comprising the following steps:
step S1: analyzing the motion principle of the net cage, regarding each buoy of the net cage as a rigid mass point, modeling, and expressing the rigid mass point as a type II T-S fuzzy model;
step S2: a distributed event based output feedback fuzzy controller for reducing data communication is proposed;
and step S3: on the basis, the output feedback fuzzy controller is further designed based on a Lyapunov method and combined with an accessible set analysis method, so that the reliability and the anti-interference performance of the multi-buoy linkage type net cage lifting are ensured.
In this embodiment, the step S1 specifically includes:
step S11: analyzing the movement principle of the net cage, regarding each buoy of the net cage as a rigid particle and modeling:
Figure BDA0002492901740000161
Figure BDA0002492901740000162
D(V)＝diag{X u | u | |u|+X u ，Y v | v | |v|+Y v ，Z w | w | |w|+Zw} (3)
Figure BDA0002492901740000163
in the formula, M represents a net cage quality matrix; m is the total mass of the net cage; x is the longitudinal resultant force of the net cage; y is the transverse resultant force of the net cage; z is the vertical resultant force of the net cage;
Figure BDA0002492901740000164
representing the hydrodynamic coefficient of the net cage; v represents a net cage motion matrix; k is z (V) is the cage motion variable; c (V) matrix represents the resultant force of Coriolis force, centripetal force and moment thereof generated by the inherent mass of the mobile net cage;pthe transverse inclination angle speed of the net cage;qthe net cage longitudinal inclination angle speed;rthe net cage yaw rate; d (V) is a hydrodynamic damping matrix applied to the movement of the movable net cage;uthe longitudinal moving speed of the net cage;vthe transverse moving speed of the net cage;wthe vertical moving speed of the net cage is obtained; non-viable cellsu|、|v|、lwRespectively express taking itThe absolute values of these; x u 、Y v 、Z w Is the linear damping coefficient; x u | u | 、Y v | v | 、Z w | w | Is the fourth order damping coefficient; g (E) is a vector formed by the inherent gravity, the inherent buoyancy and the interaction resultant moment of the net cage; g and B respectively represent the gravity and the buoyancy of the net cage;
Figure BDA0002492901740000165
and theta respectively represents a roll angle and a pitch angle of the rotation of the net cage;
further simplifying a net cage motion equation to obtain a system control design model:
Figure BDA0002492901740000171
wherein [ x y ψ] T ＝x(t)，
Figure BDA0002492901740000172
First derivatives of x, y, psi, respectively; [ tau ] to u τ v τ r ] T The position (x, y) of the net cage under the terrestrial coordinate system is defined as = u (t). X, y, the yawing angle of the net cage is defined as psi, and tau is defined as u Thrust generated by the propulsion system when the net cage moves longitudinally; tau. v Thrust generated by the propulsion system when the net cage moves transversely; tau. r The moment is generated when the towing net cage rotates; the non-linear equation for the net cage system is then as follows:
Figure BDA0002492901740000173
step S12: then, according to the formula (6), establishing a singular fuzzy system equation of the single buoy, as shown in the formula (7):
Figure BDA0002492901740000174
wherein E (h) is non-singular and satisfies
Figure BDA0002492901740000175
Wherein
Figure BDA0002492901740000176
Figure BDA0002492901740000177
E s Representing a non-singular system matrix, A l Representing the system state variable, r e And r f Respectively representing the inference rule numbers on the left side and the right side; h is a total of s [ζ(t)]And mu l [ζ(t)]Is a function of normalized membership to a normalized degree,
Figure BDA0002492901740000178
represents the derivative of the system state variable, ω (t) = -C (V) -g (E), as an external disturbance; they satisfy the following conditions:
Figure BDA0002492901740000179
Figure BDA00024929017400001710
wherein h is sφ [ζ φ (t)]And mu lφ [ζ φ (t)]Is degree of membership, definition h s ：＝h s [ζ(t)]And mu l ：＝μ l [ ζ (death)]To simplify the description; g represents the number of fuzzy members;
step S13: regarding each buoy of the net cage as a rigid particle, establishing a large nonlinear singular system, wherein the system consists of N interconnected subsystems, and establishing a singular nonlinear equation of the net cage system under the linkage control of the plurality of buoys:
Figure BDA00024929017400001711
wherein
Figure BDA00024929017400001712
Is the number of subsystems, { E i (ζ i (t))，A ii (ζ i (t))，Bi(ζi(t))，C i ，D i (ζ i (t)) } is a medium having a measurable quantity nonlinear dynamics ζ i System matrix of (deaths), A ij (ζ i (t)，ζ j (death)) represents an interconnection matrix between the ith subsystem and the jth subsystem;
Figure BDA0002492901740000181
and
Figure BDA0002492901740000182
indicating system state, control inputs, interference and outputs, n xi 、n ui、 n ωi、 n yi Is the order of the matrix;
step S14: describing the singular nonlinear equation of the multi-buoy linked deep water net cage lifting system considered in (8) by a T-S model of type II, as follows:
Figure BDA0002492901740000183
wherein, E i (h i ) Is non-singular and satisfies
Figure BDA0002492901740000184
Figure BDA0002492901740000185
Figure BDA0002492901740000186
And
Figure BDA0002492901740000187
respectively represent the left side anda set of fuzzy inference rule sets on the right side; n is i 、r i 、r j Is the number of fuzzy rule sets;
Figure BDA0002492901740000188
and
Figure BDA0002492901740000189
are normalized membership functions that satisfy the following condition:
Figure BDA00024929017400001810
wherein the content of the first and second substances,
Figure BDA00024929017400001811
and also,
Figure BDA00024929017400001812
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA00024929017400001813
Figure BDA00024929017400001814
respectively representing the lower limit of a membership function, the upper limit of the membership function, the lower limit of a fuzzy member membership function and the upper limit of the fuzzy member membership function;
Figure BDA0002492901740000191
Figure BDA0002492901740000192
in this embodiment, the step S2 specifically includes:
step S21: first, define
Figure BDA0002492901740000193
For system output, an output feedback fuzzy controller with event-based distributed broadcasting is built as follows:
Figure BDA0002492901740000194
wherein the content of the first and second substances,
Figure BDA0002492901740000195
wherein
Figure BDA0002492901740000196
Is the gain of the controller to be designed,
Figure BDA0002492901740000197
representing a normalized membership function in the fuzzy controller;
front variable ζ i (T) is measurable, in type II T-S model set, degree of membership
Figure BDA0002492901740000198
Upper limit and degree of membership of
Figure BDA0002492901740000199
Is a priori, but is a non-linear function
Figure BDA00024929017400001910
And
Figure BDA00024929017400001911
is unknown; thus, the normalized membership function of the fuzzy controller depends on
Figure BDA00024929017400001912
Medicine for curing cancer
Figure BDA00024929017400001913
Namely that
Figure BDA00024929017400001914
To obtain a minimum boundary
Figure BDA00024929017400001915
Step S22: an event trigger mechanism is proposed to identify whether or not to transmit the system output signal y i (deaths) resulting in reduced data communication; to implement the desired event-based control problem, the event triggering conditions are as follows:
event trigger conditions are as follows:
Figure BDA00024929017400001916
wherein sigma i ≧ 0 is a selected scalar; according to the execution event trigger condition (15), the system outputs the strategy based on the event trigger as follows:
Figure BDA00024929017400001917
step S23: in the proposed event trigger condition (15), the system output y is only sent when a specific event is triggered i (t); however, a measurable precursor variable ζ i (t) the need to transmit with a time-triggered strategy, which results in a reduction of partial data transmission; to further reduce the communication data, another event trigger condition is proposed for the front-piece variable, as follows:
event trigger conditions:
Figure BDA00024929017400001918
wherein e i ≧ 0 is a selected scalar; the strategy of the front-part variable based on event trigger is as follows:
Figure BDA0002492901740000201
two event trigger conditions are used in the event-based policies proposed by equations (15) and (17) to verify when the precursor variables and system outputs can be transmitted through the network, reducing data transmission in the communication network;
step S24: variable η of equation (13) i Is dependent on a antecedent variable ζ i (t) and the variables of equation (17)
Figure BDA0002492901740000202
Is dependent on a front-part variable based on an event trigger condition
Figure BDA0002492901740000203
The event-based distributed IT-2 fuzzy controller in (13) is thus updated as follows:
Figure BDA0002492901740000204
wherein the content of the first and second substances,
Figure BDA0002492901740000205
in this embodiment, the step S3 specifically includes:
step S31: the fuzzy singular model of the multi-buoy linkage type net cage system (9) is rewritten as follows:
Figure BDA0002492901740000206
wherein the content of the first and second substances,
Figure BDA0002492901740000207
definition of
Figure BDA0002492901740000208
Then there are:
Figure BDA0002492901740000209
step S32: the closed-loop fuzzy control system consisting of (13) and (23) can be rewritten as:
Figure BDA0002492901740000211
wherein
Figure BDA0002492901740000212
Step S33: the Lyapunov matrix is selected as follows:
Figure BDA0002492901740000213
wherein
Figure BDA0002492901740000214
Is easy to obtain, and can be used for preventing,
Figure BDA0002492901740000215
definition of
Figure BDA0002492901740000216
And assuming that the unknown measurement noise is bounded, this can be satisfied:
Figure BDA0002492901740000217
wherein
Figure BDA0002492901740000218
Is a positive scalar quantity;
step S34: designing (19) the event-based distributed fuzzy controller such that the state trajectory of the resulting closed-loop control system in (24) is limited by the following reachable set:
Figure BDA0002492901740000219
the ellipsoid defined by the reachable set of the closed-loop control system in (24) is given by:
Figure BDA00024929017400002110
wherein
Figure BDA00024929017400002111
Consider the following Lyapunov function:
Figure BDA00024929017400002112
wherein
Figure BDA00024929017400002113
Easy to obtain V (t) = V T (t)≥0.
Defining:
Figure BDA0002492901740000221
the following can be obtained:
Figure BDA0002492901740000222
from (24) can be obtained:
Figure BDA0002492901740000223
wherein
Figure BDA0002492901740000224
And matrix multiplication
Figure BDA0002492901740000225
Due to the fact that
Figure BDA0002492901740000226
Wherein
Figure BDA0002492901740000227
And scalar k > 0.n is the matrix order;
given a positive definite symmetric matrix
Figure BDA0002492901740000228
And
Figure BDA0002492901740000229
from (31) and (32):
Figure BDA00024929017400002210
and
Figure BDA00024929017400002211
and
Figure BDA00024929017400002212
it is readily seen from (33) to (35):
Figure BDA0002492901740000231
then, a positive definite symmetric matrix is given
Figure BDA0002492901740000232
From the policy (16) based on the event trigger mechanism it follows:
Figure BDA0002492901740000233
from (30) to (37), it is known that:
Figure BDA0002492901740000234
definition of
Figure BDA0002492901740000235
Further, the following functions are defined:
Figure BDA0002492901740000236
wherein alpha is ∈ [0,1].
In combination with (30) to (40), there can be obtained:
Figure BDA0002492901740000237
inequality II i (h i ，μ i ，η i ) Meaning J (t) <0, so the inequality in (28) can be directly found;
since J (t) <0, it is possible to obtain:
Figure BDA0002492901740000241
this means that:
V(k+1)-1＜α(V(k)-1). (43)
from (43), it is easy to derive:
V(k)＜α k (V(0)-1)+1. (44)
from (28):
Figure BDA0002492901740000242
wherein
Figure BDA0002492901740000243
Thus, considering a large-scale IT-2 fuzzy system using the event-based distributed fuzzy controller of (19) in (9), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (26), if there is a symmetric positive definite matrix
Figure BDA0002492901740000244
Figure BDA0002492901740000245
Sum matrix
Figure BDA0002492901740000246
Figure BDA0002492901740000247
Sum matrix multiplier
Figure BDA0002492901740000248
Sum positive scalar quantity
Figure BDA0002492901740000249
So that for all subsystems
Figure BDA00024929017400002410
The following matrix inequality holds:
Π i (h i ，μ i ，η i )＜0， (46)
and the number of the first and second electrodes,
Figure BDA00024929017400002411
where Sym means the sum of the matrix and its transpose, e.g. Sym (A) = A + A T (ii) a I is an identity matrix;
Figure BDA00024929017400002412
Figure BDA0002492901740000251
in addition, the reachable set estimates satisfy the following boundary,
Figure BDA0002492901740000252
thereby deriving sufficient conditions to ensure that there is an event-based distributed fuzzy controller that can drive the state traces within reachable set boundaries;
step S35: definition of
Figure BDA0002492901740000253
By applying the cone-complement theorem, we can obtain:
Figure BDA0002492901740000254
wherein the content of the first and second substances,
Figure BDA0002492901740000255
since given an interconnect matrix
Figure BDA0002492901740000256
And having a symmetric positive definite matrix of compatible dimensions
Figure BDA0002492901740000257
The following equation holds true:
Figure BDA0002492901740000258
the following can be obtained:
Figure BDA0002492901740000259
then, extracting fuzzy antecedent variables yields:
Figure BDA0002492901740000261
wherein
Figure BDA0002492901740000262
Is defined in formula (53);
it should be noted that prior art relaxation techniques
Figure BDA0002492901740000263
Figure BDA0002492901740000264
Is no longer suitable for fuzzy controller synthesis because
Figure BDA0002492901740000265
Figure BDA0002492901740000266
In practice, a positive scalar is always found from (14) and (18)
Figure BDA0002492901740000267
Compliance
Figure BDA0002492901740000268
Wherein
Figure BDA0002492901740000269
Similar to the asynchronous relaxation technique, assume
Figure BDA00024929017400002610
Wherein
Figure BDA00024929017400002611
Is a symmetric matrix, one can derive:
Figure BDA00024929017400002612
by re-passing
Figure BDA00024929017400002613
Thereafter, the conditions in (49) and (50) can be directly obtained using the existing relaxation method applied to (54);
therefore, consider the large IT-2 fuzzy system in (9), if a symmetric positive definite matrix exists
Figure BDA00024929017400002614
Figure BDA00024929017400002615
The event-based distributed fuzzy controller in (19) may ensure that the reachable set of the closed-loop system (24) is limited by the ellipse boundaries in (26),
Figure BDA00024929017400002616
sum matrix
Figure BDA00024929017400002617
Figure BDA00024929017400002618
And matrix product
Figure BDA00024929017400002619
And symmetric matrices with compatible dimensions
Figure BDA00024929017400002620
Sum positive scalar quantity
Figure BDA00024929017400002621
So as to all
Figure BDA00024929017400002622
The following matrix inequality holds:
Figure BDA00024929017400002623
Figure BDA00024929017400002624
Figure BDA00024929017400002625
wherein
Figure BDA0002492901740000271
Step S36: the inequalities in equations (56) - (58) are not non-linear matrix inequalities; to facilitate fuzzy controller design, a surrogate descriptor representation will be introduced in (23) for the proposed controller, as follows:
Figure BDA0002492901740000272
thus, the augmented system can be rewritten as:
Figure BDA0002492901740000273
wherein
Figure BDA0002492901740000274
Considering the large IT-2 fuzzy system in (9) and the event-based distributed fuzzy controller in (19) based on the fuzzy singular system in (61), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (27), if there is a positive definite symmetric matrix
Figure BDA0002492901740000281
Matrix of
Figure BDA0002492901740000282
Figure BDA0002492901740000283
Figure BDA0002492901740000284
Symmetric matrix of compatible dimensions
Figure BDA0002492901740000285
And all of
Figure BDA0002492901740000286
Positive scalar quantity of
Figure BDA0002492901740000287
The following non-linear matrix inequality holds:
Figure BDA0002492901740000288
Figure BDA0002492901740000289
Figure BDA00024929017400002810
wherein
Figure BDA00024929017400002811
Figure BDA0002492901740000291
Further, the controller gain matrix in (15) may be calculated by:
Figure BDA0002492901740000292
the above description is only a preferred embodiment of the present invention, and all the equivalent changes and modifications made according to the claims of the present invention should be covered by the present invention.

Claims (2)

1. A net cage lifting control method based on a distributed event trigger strategy is characterized by comprising the following steps:
step S1: analyzing the movement principle of the net cage, regarding each buoy of the net cage as a rigid mass point, modeling, and expressing the rigid mass point as a type II T-S fuzzy model;
step S2: a distributed event based output feedback fuzzy controller for reducing data communication is proposed;
and step S3: on the basis, the output feedback fuzzy controller is further designed based on a Lyapunov method and combined with an accessible set analysis method so as to ensure the reliability and anti-interference performance of the multi-buoy linkage type net cage lifting;
the step S1 specifically comprises the following steps:
step S11: analyzing the movement principle of the net cage, regarding each buoy of the net cage as a rigid particle and modeling:
Figure FDA0003827049870000011
Figure FDA0003827049870000012
D(V)＝diag{X u，|u| |u|+X u ，Y v，|v| |v|+Y v ，Z w，|w| |w|+Z w } (3)
Figure FDA0003827049870000013
in the formula, M represents a net cage quality matrix; m is the total mass of the net cage; x is the longitudinal resultant force of the net cage; y is the transverse resultant force of the net cage; z is the vertical resultant force of the net cage;
Figure FDA0003827049870000014
representing the hydrodynamic coefficient of the net cage; v represents a net cage motion matrix; k is z (V) is the cage motion variables; c (V) matrix represents the resultant force of Coriolis force, centripetal force and moment thereof generated by the inherent mass of the mobile net cage;pthe transverse inclination angle speed of the net cage;qthe net cage longitudinal inclination angle speed;rthe net cage yaw rate; d (V) is a hydrodynamic damping matrix suffered by the movement of the mobile net cage;uthe longitudinal moving speed of the net cage;vthe transverse moving speed of the net cage;wthe vertical moving speed of the net cage is obtained; non-viable cellsu|、|v|、|wRespectively representing the absolute values of the two; x u 、Y v 、Z w Is the linear damping coefficient; x u|u| 、Y v|v| 、Z w|w| Is the fourth order damping coefficient; g (E) is a vector formed by the inherent gravity, the inherent buoyancy and the interaction resultant moment of the net cage; g and B respectively represent the gravity and the buoyancy of the net cage;
Figure FDA0003827049870000015
and theta respectively represents a roll angle and a pitch angle of the rotation of the net cage;
further simplifying a net cage motion equation to obtain a system control design model:
Figure FDA0003827049870000021
Figure FDA0003827049870000022
Figure FDA0003827049870000023
wherein [ x y ψ] T ＝x(t)，
Figure FDA0003827049870000024
First derivatives of x, y, ψ, respectively; [ tau ] of u τ v τ r ] T = u (t); x and y are net cage positions (x and y) in a terrestrial coordinate system, psi is net cage bow roll angle, and tau u Thrust generated by the propulsion system when the net cage moves longitudinally; tau is v Thrust generated by the propulsion system when the net cage moves transversely; tau. r The moment is generated when the dragging net cage rotates; the non-linear equation for the net cage system is then as follows:
Figure FDA0003827049870000025
step S12: then, according to the formula (6), establishing a singular fuzzy system equation of the single buoy, as shown in the formula (7):
Figure FDA0003827049870000026
wherein E (h) is non-singular and satisfies
Figure FDA0003827049870000027
Wherein
Figure FDA0003827049870000028
Figure FDA0003827049870000029
E s Representing a non-singular system matrix, A l Representing the system matrix, r e And r f The inference rule numbers respectively represent the left side and the right side; h is s [ζ(t)]And mu l [ζ(t)]Is a function of normalized membership to a normalized degree,
Figure FDA00038270498700000210
represents the derivative of the system state variable, ω (t) = -C (V) -g (E), as an external disturbance; they satisfy the following conditions:
Figure FDA00038270498700000211
Figure FDA00038270498700000212
wherein h is sφ [ζ φ (t)]And mu lφ [ζ φ (t)]Is degree of membership, definition h s :＝h s [ζ(t)]And mu l :＝μ l [ζ(t)]To simplify the description; g represents the number of fuzzy members;
step S13: regarding each buoy of the net cage as a rigid particle, establishing a large nonlinear singular system, wherein the system consists of N interconnected subsystems, and establishing a singular nonlinear equation of the net cage system under the linkage control of the plurality of buoys:
Figure FDA00038270498700000213
wherein
Figure FDA00038270498700000214
Is aNumber of systems, { E i (ζ i (t)),A ii (ζ i (t)),B i (ζ i (t)),C i ,D i (ζ i (t)) } is a linear dynamic having measurable zeta potentials i (t) system matrix, A ij (ζ i (t),ζ j (t)) represents an interconnection matrix between the ith subsystem and the jth subsystem;
Figure FDA0003827049870000031
and
Figure FDA0003827049870000032
representing system state, control inputs, disturbances and outputs, n xi 、n ui 、n ωi 、n yi Is the order of the matrix;
step S14: the singular non-linear equation of the multi-buoy linked deep water cage hoisting system considered in (8) is described by a type II T-S model, as follows:
Figure FDA0003827049870000033
wherein, E i (h i ) Is non-singular and satisfies
Figure FDA0003827049870000034
Figure FDA0003827049870000035
Figure FDA0003827049870000036
And
Figure FDA0003827049870000037
a set of fuzzy inference rule sets representing left and right sides, respectively; n is a radical of an alkyl radical i 、r i 、r j Is the number of fuzzy rule sets;
Figure FDA0003827049870000038
and
Figure FDA0003827049870000039
are normalized membership functions that satisfy the following condition:
Figure FDA00038270498700000310
Figure FDA00038270498700000311
wherein the content of the first and second substances,
Figure FDA00038270498700000312
Figure FDA00038270498700000313
Figure FDA00038270498700000314
Figure FDA00038270498700000315
and the number of the first and second electrodes,
Figure FDA00038270498700000316
Figure FDA00038270498700000317
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA00038270498700000318
Figure FDA00038270498700000319
respectively representing the lower limit of a membership function, the upper limit of the membership function, the lower limit of a fuzzy member membership function and the upper limit of the fuzzy member membership function;
Figure FDA0003827049870000041
the step S2 specifically includes:
step S21: first, define
Figure FDA0003827049870000042
For system output, an output feedback fuzzy controller with event-based distributed broadcasting is built as follows:
Figure FDA0003827049870000043
wherein the content of the first and second substances,
Figure FDA0003827049870000044
Figure FDA0003827049870000045
wherein
Figure FDA0003827049870000046
Is the gain of the controller to be designed,
Figure FDA0003827049870000047
representing a normalized membership function in the fuzzy controller;
front part variable ζ i (T) is measurable, in type II T-S model set, degree of membership
Figure FDA0003827049870000048
Upper limit and degree of membership of
Figure FDA0003827049870000049
Is a priori, but is a non-linear function
Figure FDA00038270498700000410
And
Figure FDA00038270498700000411
is unknown; thus, the normalized membership function of the fuzzy controller depends on
Figure FDA00038270498700000412
And
Figure FDA00038270498700000413
namely, it is
Figure FDA00038270498700000414
To obtain a minimum boundary
Figure FDA00038270498700000415
Step S22: an event trigger mechanism is proposed to identify whether or not to transmit the system output signal y i (t) causing data communication to be reduced; to implement the desired event-based control problem, the event triggering conditions are as follows:
event trigger conditions are as follows:
Figure FDA00038270498700000416
wherein sigma i ≧ 0 is a selected scalar; according to the execution event trigger condition (15), the system outputs the strategy based on the event trigger as follows:
Figure FDA00038270498700000417
step S23: in the event trigger condition (15), the system output y is transmitted only when a specific event is triggered i (t); however, the measurable precursor variable ζ i (t) transmission needs to be triggered in time, which results in a reduction of partial data transmission; to further reduce the communication data, another event trigger condition is proposed for the front-end variables, as follows:
event trigger conditions:
Figure FDA00038270498700000418
wherein e i ≧ 0 is a selected scalar; the strategy of the front-piece variable based on event trigger is as follows:
Figure FDA0003827049870000051
two event trigger conditions are used in the event-based policies proposed by equations (15) and (17) to verify when to transmit the precursor variables and system outputs over the network, reducing data transmission in the communication network;
step S24: variable η of equation (13) i Is dependent on a antecedent variable ζ i (t) and the variables of equation (17)
Figure FDA0003827049870000052
Is dependent on a front-piece variable based on an event trigger condition
Figure FDA0003827049870000053
Thus is paired with(13) The distributed IT-2 fuzzy controller based on the event carries out the following updating:
Figure FDA0003827049870000054
wherein the content of the first and second substances,
Figure FDA0003827049870000055
Figure FDA0003827049870000056
2. the method for controlling ascending and descending of a net cage based on the distributed event triggering strategy according to claim 1, wherein the step S3 is specifically:
step S31: the fuzzy singular model of the multi-buoy coordinated type net cage system (9) is rewritten as follows:
Figure FDA0003827049870000057
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003827049870000058
Figure FDA0003827049870000059
Figure FDA00038270498700000510
Figure FDA00038270498700000511
Figure FDA00038270498700000512
Figure FDA00038270498700000513
definition of
Figure FDA00038270498700000514
Then there are:
Figure FDA0003827049870000061
step S32: the closed-loop fuzzy control system consisting of (13) and (23) is rewritten as:
Figure FDA0003827049870000062
wherein
Figure FDA0003827049870000063
Step S33: the Lyapunov matrix is selected as follows:
Figure FDA0003827049870000064
wherein
Figure FDA0003827049870000065
Is easy to obtain, and has the advantages of easy acquisition,
Figure FDA0003827049870000066
definition of
Figure FDA0003827049870000067
And assuming that the unknown measurement noise is bounded, this satisfies:
Figure FDA0003827049870000068
wherein
Figure FDA0003827049870000069
Is a positive scalar;
step S34: designing (19) an event-based distributed fuzzy controller such that the state trajectory of the resulting closed-loop control system in (24) is limited by the following reachable set:
Figure FDA00038270498700000610
the ellipsoid defined by the reachable set of the closed-loop control system in (24) is given by:
Figure FDA00038270498700000611
wherein
Figure FDA00038270498700000612
Consider the following Lyapunov function:
Figure FDA00038270498700000613
wherein
Figure FDA00038270498700000614
V (is easily obtained)t)＝V T (t)≥0；
Defining:
Figure FDA0003827049870000071
obtaining:
Figure FDA0003827049870000072
obtained from (24):
Figure FDA0003827049870000073
wherein
Figure FDA0003827049870000074
And matrix multiplication
Figure FDA0003827049870000075
Due to the fact that
Figure FDA0003827049870000076
Wherein
Figure FDA0003827049870000077
And a scalar k>0, n is the matrix order;
given a positive definite symmetric matrix
Figure FDA0003827049870000078
And
Figure FDA0003827049870000079
from (31) and (32):
Figure FDA00038270498700000710
and
Figure FDA00038270498700000711
and
Figure FDA00038270498700000712
it can be easily seen from (33) - (35):
Figure FDA0003827049870000081
Figure FDA0003827049870000082
Figure FDA0003827049870000083
then, a positive definite symmetric matrix is given
Figure FDA0003827049870000084
From the policy (16) based on the event trigger mechanism it follows:
Figure FDA0003827049870000085
from (30) - (37) we obtained:
Figure FDA0003827049870000086
definition of
Figure FDA0003827049870000087
Figure FDA0003827049870000088
Figure FDA0003827049870000089
Further, the following functions are defined:
Figure FDA00038270498700000810
wherein α ∈ [0,1];
combining (30) - (40) to obtain:
Figure FDA00038270498700000811
easily seen, inequality pi i (h i ,μ i ,η i ) Meaning J (t)<0, so the inequality in (28) is directly found;
since J (t) <0, we get:
Figure FDA0003827049870000091
this means that:
V(k+1)-1<α(V(k)-1) (43)
from (43), it is easy to derive:
V(k)<α k (V(0)-1)+1 (44)
from (28) are obtained:
Figure FDA0003827049870000092
wherein
Figure FDA0003827049870000093
Thus, considering a large-scale IT-2 fuzzy system using the event-based distributed fuzzy controller of (19) in (9), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (26), if there is a symmetric positive definite matrix
Figure FDA0003827049870000094
Figure FDA0003827049870000095
Sum matrix
Figure FDA0003827049870000096
And matrix multiplier
Figure FDA0003827049870000097
Sum positive scalar quantity
Figure FDA0003827049870000098
So that for all subsystems
Figure FDA0003827049870000099
The following matrix inequality holds:
Π i (h i ,μ i ,η i )<0, (46)
and also,
Figure FDA00038270498700000910
where Sym means the sum of the matrix and its transpose, e.g. Sym (A) = A + A T (ii) a I is an identity matrix;
Figure FDA00038270498700000911
Figure FDA00038270498700000912
Figure FDA0003827049870000101
Figure FDA0003827049870000102
Figure FDA0003827049870000103
in addition, the reachable set estimates satisfy the following boundary,
Figure FDA0003827049870000104
sufficient conditions are thus derived to ensure that there is an event-based distributed fuzzy controller that can drive the state traces within reachable set boundaries;
step S35: definition of
Figure FDA0003827049870000105
Figure FDA0003827049870000106
By applying the cone complement theorem, the following results are obtained:
Figure FDA0003827049870000107
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003827049870000108
Figure FDA0003827049870000109
Figure FDA00038270498700001010
Figure FDA00038270498700001011
Figure FDA00038270498700001012
since given an interconnect matrix
Figure FDA00038270498700001013
And having a symmetric positive definite matrix of compatible dimensions
Figure FDA00038270498700001014
The following equation holds true:
Figure FDA00038270498700001015
obtaining:
Figure FDA0003827049870000111
then, extracting fuzzy antecedent variables yields:
Figure FDA0003827049870000112
wherein
Figure FDA0003827049870000113
Is defined in formula (53);
it should be noted that existing relaxation techniques
Figure FDA0003827049870000114
Figure FDA0003827049870000115
Is no longer suitable for fuzzy controller synthesis because
Figure FDA0003827049870000116
Figure FDA0003827049870000117
In practice, a positive scalar is always found from (14) and (18)
Figure FDA0003827049870000118
Compliance
Figure FDA0003827049870000119
Wherein
Figure FDA00038270498700001110
Similar to the asynchronous relaxation technique, assume that
Figure FDA00038270498700001111
Wherein
Figure FDA00038270498700001112
Is a symmetric matrix, giving:
Figure FDA00038270498700001113
by re-fixing
Figure FDA00038270498700001114
Thereafter, the conditions in (49) and (50) are directly obtained using the existing relaxation method applied to (54);
therefore, consider the large IT-2 blur system in (9), if a symmetric positive definite matrix exists
Figure FDA00038270498700001115
Figure FDA00038270498700001116
The event-based distributed fuzzy controller in (19) ensures that the reachable set of the closed-loop system (24) is limited by the ellipse boundaries in (26),
Figure FDA00038270498700001117
sum matrix
Figure FDA00038270498700001118
Figure FDA00038270498700001119
And matrix product
Figure FDA00038270498700001120
And symmetric matrices with compatible dimensions
Figure FDA00038270498700001121
Sum positive scalar quantity
Figure FDA00038270498700001122
So as to all
Figure FDA00038270498700001123
The following matrix inequality holds:
Figure FDA00038270498700001124
Figure FDA00038270498700001125
Figure FDA0003827049870000121
wherein
Figure FDA0003827049870000122
Figure FDA0003827049870000123
Figure FDA0003827049870000124
Figure FDA0003827049870000125
Figure FDA0003827049870000126
Figure FDA0003827049870000127
Figure FDA0003827049870000128
Figure FDA0003827049870000129
Step S36: the inequalities in equations (56) - (58) are not non-linear matrix inequalities; to facilitate fuzzy controller design, a surrogate descriptor representation will be introduced in (23) for the proposed controller, as follows:
Figure FDA00038270498700001210
thus, the augmented system is rewritten as:
Figure FDA00038270498700001211
wherein
Figure FDA00038270498700001212
Figure FDA00038270498700001213
Figure FDA00038270498700001214
Figure FDA0003827049870000131
Considering the large IT-2 fuzzy system in (9) and the event-based distributed fuzzy controller in (19) based on the fuzzy singular system in (61), the reachable set of the closed-loop system (24) is limited by the elliptical boundaries in (27), if there is a positive definite symmetric matrix
Figure FDA0003827049870000132
Matrix array
Figure FDA0003827049870000133
Figure FDA0003827049870000134
Symmetric matrix of compatible dimensions
Figure FDA0003827049870000135
And all of
Figure FDA0003827049870000136
Positive scalar quantity of
Figure FDA0003827049870000137
Then the following non-linear matrix inequality holds:
Figure FDA0003827049870000138
Figure FDA0003827049870000139
Figure FDA00038270498700001310
wherein
Figure FDA00038270498700001311
Figure FDA00038270498700001312
Figure FDA00038270498700001313
Figure FDA00038270498700001314
Figure FDA00038270498700001315
Figure FDA00038270498700001316
Figure FDA00038270498700001317
Figure FDA0003827049870000141
Figure FDA0003827049870000142
Figure FDA0003827049870000143
Figure FDA0003827049870000144
Figure FDA0003827049870000145
Figure FDA0003827049870000146
Figure FDA0003827049870000147
Figure FDA0003827049870000148
Figure FDA0003827049870000149
Figure FDA00038270498700001410
Figure FDA00038270498700001411
Further, the controller gain matrix in (15) is calculated by:
Figure FDA00038270498700001412
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