CN112180725A - Fuzzy proportional-integral state estimation method for nonlinear system with redundant time-delay channel - Google Patents

Fuzzy proportional-integral state estimation method for nonlinear system with redundant time-delay channel Download PDF

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CN112180725A
CN112180725A CN202011054145.1A CN202011054145A CN112180725A CN 112180725 A CN112180725 A CN 112180725A CN 202011054145 A CN202011054145 A CN 202011054145A CN 112180725 A CN112180725 A CN 112180725A
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王业政
王子栋
邹磊
宋保业
赵忠义
郭继跃
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Shandong University of Science and Technology
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Abstract

The invention discloses a fuzzy proportional-integral state estimation method of a nonlinear system with redundant time-lag channels, which is used for solving the state estimation problem of the nonlinear system under the influence of one type of redundant time-lag channels and specifically comprises the following steps: establishing a system state space model based on a Takagi-Sugeno fuzzy modeling technology; establishing a redundant channel transmission model considering a time lag phenomenon; constructing a fuzzy proportional-integral filter based on the available measurement output; then, constructing a performance index auxiliary function, and solving a linear matrix inequality according to the auxiliary function to obtain a value of the gain of the filter; and finally, calculating to obtain an estimated value of the fuzzy proportional-integral filter based on the obtained filter gain. The method combines two technologies of fuzzy state estimation and proportional-integral state estimation, has good estimation effect, and ensures easy execution of the algorithm because the key variables adopt a method for solving the matrix inequality.

Description

Fuzzy proportional-integral state estimation method for nonlinear system with redundant time-delay channel
Technical Field
The invention relates to a fuzzy proportional-integral state estimation method of a nonlinear system with redundant time-lag channels.
Background
The state estimation is one of basic research problems of control theory and has important significance in practical engineering. The basic idea of state estimation is to obtain an estimated value of the real state of the system under a given performance index by using the obtained output information and adopting a proper filtering algorithm. In order to meet different industrial requirements since the last decades, many filtering algorithms have been proposed to achieve state estimation and to achieve satisfactory results in practice. The nonlinear phenomenon widely exists in life, almost all practical systems present nonlinear characteristics, and the method has important practical significance for state estimation of the nonlinear systems.
Among many filtering algorithms for nonlinear systems, fuzzy filtering proves to be a very effective state estimation method, which mainly benefits from the advantages of fuzzy sets and fuzzy systems, such as greater introduction of human knowledge, concise models, definite physical meanings, easy implementation and the like. On the other hand, the integral term is introduced into the filtering algorithm to eliminate the steady-state error and improve the robustness of the filter, so that the application of the proportional-integral filtering algorithm generally brings better estimation effect.
In modern industrial production, information interaction between elements in a system is typically accomplished by a communication network. The use of a communication network brings many advantages such as reduced cost, ease of maintenance, flexibility of transmission, etc. However, in practice, the bandwidth of the communication network is limited, and when a large amount of data is transmitted in the network at the same time, data congestion is easily caused, and then a series of adverse effects, such as transmission skew, data packet loss, etc., are brought about, and the generation of these phenomena greatly affects the filtering effect.
In order to minimize the adverse effects caused by data congestion, a number of strategies have been proposed by those skilled in the art to plan data transmission, wherein the use of redundant channels is a more efficient method. Under a redundant channel mechanism, a plurality of channels are commonly used for data transmission, so that the reliability of data transmission can be improved, and the effect of state estimation is further improved. However, most of the existing filtering algorithms for the nonlinear system aim at the state estimation problem of the nonlinear system under a single channel, so that the state estimation problem of the nonlinear system under the influence of redundant time-lag channels cannot be processed.
Disclosure of Invention
The invention aims to provide a fuzzy proportional-integral state estimation method of a nonlinear system with redundant time-lag channels, which aims to solve the problem that the existing filtering method cannot process the state estimation of the nonlinear system under the influence of the redundant time-lag channels.
In order to achieve the purpose, the invention adopts the following technical scheme:
a fuzzy proportional-integral state estimation method for a nonlinear system with redundant dead-time channels, comprising the steps of:
s1, establishing a system state space model based on a T-S fuzzy modeling technology, as shown in a formula (1):
Figure BDA0002710414610000021
wherein x (k) represents the state variable of the nonlinear system at the time k, and x (k) is nxA dimension column vector;
y (k) represents the measured output signal of the nonlinear system at time k, y (k) being nyA dimension column vector;
z (k) represents the signal to be estimated at time k, z (k) is nzA dimension column vector;
ω (k) represents energy-bounded noise, ω (k) being nwA dimension column vector; n isx、ny、nz、nwRepresents a known positive integer;
r represents the total number of fuzzy rules;
θ (k) represents a vector consisting of detectable states or measurement outputs; mu.si(theta (k)) is a membership function of the fuzzy system and represents the weight of the ith fuzzy mode in the whole nonlinear system at the moment k;
Airepresenting the system matrix, C the output matrix, EiRepresenting the process noise matrix, FiRepresenting a signal matrix to be estimated; wherein, the matrix Ai、C、Ei、FiAre all known constant matrices;
s2, establishing a signal transmission model under a redundant channel mechanism to realize data transmission between the nonlinear system and the filter;
setting q transmission channels between the nonlinear system and the filter for data transmission;
definition of
Figure BDA0002710414610000022
Representing the signal transmitted by the i-th channel received by the filter at the time k, while considering the influence of random transmission time lag, based on the output equation in the system state space model established in step S1, then
Figure BDA0002710414610000023
Expressed as:
Figure BDA0002710414610000024
wherein the content of the first and second substances,
Figure BDA0002710414610000025
Figure BDA0002710414610000026
representing variable definition symbols;
Figure BDA0002710414610000027
for random transmission skew, d represents the upper bound of the skew, dl(k) The value of (a) represents the length of time lag of the ith channel at the moment k; dlA known channel noise matrix;
event dl(k) The probability of occurrence of t is Pr { d }l(k)＝t}＝plt，0≤plt1 ≦ is a known scalar;
vl(k) channel noise, v, representing the energy-bounded of the ith channell(k) Is nvlVector of dimension, nvlRepresents a known positive integer;
s3, constructing an auxiliary function and calculating the proportional gain K of the filterilAnd integral gain Lil：
Wherein, KilThe proportional gain of the measurement output signal corresponding to the ith fuzzy mode of the filter through the ith channel is represented; l isilThe integral gain of the measurement output signal corresponding to the ith fuzzy mode of the filter through the ith channel is represented;
specifically, the following auxiliary functions are constructed according to the system state space model in step S1 and the signal transmission model in step S2:
Figure BDA0002710414610000031
Figure BDA0002710414610000032
wherein V (k) represents a selected Lyapunov functional, and J (iota) represents a difference between an estimation error and an energy of external noise;
P、Rcin order to determine the positive matrix to be solved,
Figure BDA00027104146100000317
represents a mathematical expectation;
Figure BDA0002710414610000033
which is indicative of an estimation error that is,
Figure BDA0002710414610000034
Figure BDA0002710414610000035
represents an estimated value of z (k);
gamma > 0 is a given scalar quantity,
Figure BDA0002710414610000036
the noise vector after dimension increase is obtained;
tau represents an intermediate variable, and eta (k) and eta (tau) respectively represent a state estimation error and a vector after the dimension of an integral vector is increased;
the formula (3) is used for analyzing the stability of eta (k), and the formula (4) is used for testing the disturbance resistance of eta (k);
according to the helper function, Lyapunov's stability theory and H∞Theoretically, the following steps are performed to find the gain of the filter:
s31. solving the following set of linear matrix inequalities (5) to obtain a set of initial solutions { P, X, R }c，Lil，Kil}：
Figure BDA0002710414610000037
Where X is the intermediate variable matrix, P, X, Rc、Kil、LilAll are matrix variables to be solved;
Figure BDA0002710414610000038
is a diagonal matrix;
wherein the content of the first and second substances,
Figure BDA0002710414610000039
defining an augmented matrix of a system matrix
Figure BDA00027104146100000310
Wherein the dimension increasing matrix of the system matrix
Figure BDA00027104146100000311
Wherein 0 represents a zero matrix and I represents an identity matrix;
defining an intermediate variable matrix
Figure BDA00027104146100000312
Wherein the content of the first and second substances,
Figure BDA00027104146100000313
defining an intermediate variable matrix
Figure BDA00027104146100000314
Then
Figure BDA00027104146100000315
Wherein the content of the first and second substances,
Figure BDA00027104146100000316
representing an intermediate matrix after the system noise matrix and the channel noise matrix are subjected to dimension increase;
Figure BDA0002710414610000041
Figure BDA0002710414610000042
definition of
Figure BDA0002710414610000043
Represents the diagonal matrix after the dimension of the intermediate variable matrix X is increased, wherein,
Figure BDA0002710414610000044
order to
Figure BDA0002710414610000045
Figure BDA0002710414610000046
Wherein:
Figure BDA0002710414610000047
Figure BDA0002710414610000048
Figure BDA0002710414610000049
wherein, s is 1, 2, …, d;
Figure BDA00027104146100000410
wherein p isl0Represents the probability that the ith channel does not have time lag, i is 1, 2, …, q; p is a radical oflsRepresents the probability of the occurrence of the time lag with the length s of the l-th channel, wherein s is 1, 2, … and d;
order to
Figure BDA00027104146100000411
t is 0, given a sufficiently small number ρ > 0;
wherein the content of the first and second substances,
Figure BDA00027104146100000412
represents an initial value;
s32, solving an optimization problem under the constraint of a formula (5): min tr (PX)(t)+P(t)X)；
Wherein min represents the minimum function, and tr represents the trace of the matrix;
s33, replacing X in the first matrix inequality on the left side in the formula (5) with P, and recording a newly obtained inequality as H < 0; if the variables { P, X, R ] solved in step S32 arec，Kil，LilH < 0 can be satisfied and:
Figure BDA0002710414610000057
the obtained gain matrix K is outputil，LilAnd exiting; otherwise, it is heldStep S34 is performed;
where tr (PX) represents a trace of matrix PX,
Figure BDA0002710414610000051
s34, if t is larger than a given maximum step length, exiting; otherwise, let t be t +1, P(t)＝P，X(t)＝X，
Figure BDA0002710414610000052
Return to perform step S32;
the proportional gain K of the filter is obtained through the steps S31 to S34ilAnd integral gain Lil；
S4, obtaining the proportional gain L of the filter according to calculationilAnd integral gain LilCalculating a state estimation value of the system;
according to the system state space model in the step S1, the signal transmission model in the step S2 and the filter gain in the step S3, the following fuzzy-proportional integral filter is constructed by fully considering the influence of the redundant time lag channel, as shown in the formula (6);
Figure BDA0002710414610000053
wherein the content of the first and second substances,
Figure BDA0002710414610000054
is an estimate of state x (k);
(a) representing a kronecker function, when a is 0, (a) is 1, otherwise, (a) is 0;
hl(k) for the integral vector corresponding to the l channel, hl(0) Is hl(k) An initial value of (1);
Figure BDA0002710414610000055
for the estimation of the signal z (k) to be estimated,
Figure BDA0002710414610000056
is the initial value of the filter;
and calculating a state estimation value of the nonlinear system with the redundant time-lag channel according to the fuzzy-proportional-integral filter (6).
The invention has the following advantages:
as described above, the invention provides a fuzzy proportional-integral state estimation method of a nonlinear system with a redundant time-lag channel, which considers the redundant channel transmission strategy and the random transmission time-lag phenomenon in network communication at the same time and combines two technologies of fuzzy state estimation and proportional-integral state estimation to construct a fuzzy proportional-integral filter, and the filter can effectively solve the state estimation problem of the nonlinear system under the influence of the redundant time-lag channel and has better estimation effect. In addition, the method for solving the matrix inequality is adopted in the solving process of the key variable (filter gain), so that the method is easy to execute.
Drawings
FIG. 1 is a flow chart of a fuzzy proportional-integral state estimation method with redundant dead-time channel nonlinear system according to the present invention;
FIG. 2 shows the actual signal trace x in the present invention1(k) And an estimated trajectory comparison map thereof;
FIG. 3 shows an actual signal trace x in the present invention2(k) And an estimated trajectory comparison map thereof;
FIG. 4 shows an actual signal trace x in the present invention3(k) And an estimated trajectory comparison map thereof;
FIG. 5 shows the state estimation error in the present invention
Figure BDA0002710414610000061
A trace plot of the norm.
Detailed Description
The invention is described in further detail below with reference to the following figures and detailed description:
examples
The embodiment describes a fuzzy proportional-integral state estimation method for a nonlinear system with redundant channels, which is based on a fuzzy proportional-integral technology to solve the state estimation problem of the nonlinear system under the influence of redundant time-lag channels.
As shown in fig. 1, a fuzzy proportional-integral state estimation method with redundant channel nonlinear system includes the following steps:
s1, establishing a system state space model based on a T-S fuzzy modeling technology, as shown in a formula (1):
Figure BDA0002710414610000062
wherein x (k) represents the state variable of the nonlinear system at the time k, and x (k) is nxA dimension column vector; y (k) represents the measured output signal of the nonlinear system at time k, y (k) being nyA dimension column vector; z (k) represents the signal to be estimated at time k, z (k) is nzA dimension column vector; ω (k) represents energy-bounded noise, ω (k) being nwA dimension column vector; n isx、ny、nz、nwRepresents a known positive integer;
r represents the total number of fuzzy rules; θ (k) represents a vector of detectable states or measurement outputs, typically related to the nonlinear term of a nonlinear system; mu.si(theta (k)) is a membership function of the fuzzy system and represents the weight of the ith fuzzy mode in the whole nonlinear system at the moment k; a. theiRepresenting the system matrix, C the output matrix, EiRepresenting the process noise matrix, FiRepresenting a signal matrix to be estimated; wherein, the matrix Ai、C、Ei、FiAre all known constant matrices.
And S2, establishing a signal transmission model under a redundant channel mechanism to realize data transmission between the nonlinear system and the filter.
Setting q transmission channels between the nonlinear system and the filter for data transmission;
definition of
Figure BDA0002710414610000063
Indicating that the filter received at time kThe signals transmitted by the I channels, considering the influence of random transmission time lag, based on the output equation in the system state space model established in the step S1, then
Figure BDA0002710414610000071
Expressed as:
Figure BDA0002710414610000072
wherein the content of the first and second substances,
Figure BDA0002710414610000073
Figure BDA0002710414610000074
representing variable definition symbols.
Figure BDA0002710414610000075
For random transmission skew, d represents the upper bound of the skew, dl(k) The value of (a) represents the length of time lag of the ith channel at the moment k; dlA known channel noise matrix;
event dl(k) The probability of occurrence of t is Pr { d }l(k)＝t}＝plt，0≤plt1 ≦ is a known scalar;
vl(k) channel noise, v, representing the energy-bounded of the ith channell(k) Is nvlVector of dimension, nvlAre known positive integers.
S3, constructing an auxiliary function and calculating the proportional gain K of the filterilAnd integral gain Lil：
Wherein, KilThe proportional gain of the measurement output signal corresponding to the ith fuzzy mode of the filter through the ith channel is represented; l isilIndicating that the ith blurring mode of the filter corresponds to the integral gain of the measured output signal through the ith channel.
Specifically, the following auxiliary functions are constructed according to the system state space model in step S1 and the signal transmission model in step S2:
Figure BDA0002710414610000076
Figure BDA0002710414610000077
wherein V (k) represents a selected Lyapunov functional, and J (iota) represents a difference between an estimation error and an energy of external noise;
P、Rcpositive definite matrix is to be solved; e represents a mathematical expectation;
Figure BDA0002710414610000078
which is indicative of an estimation error that is,
Figure BDA0002710414610000079
Figure BDA00027104146100000710
represents an estimated value of z (k);
gamma > 0 is a given scalar quantity,
Figure BDA00027104146100000711
the noise vector after dimension increase is obtained;
tau represents an intermediate variable, and eta (k) and eta (tau) respectively represent a state estimation error and a vector after the dimension of an integral vector is increased;
equation (3) is used to analyze the stability of η (k), and equation (4) is used to test the anti-disturbance capability of η (k).
According to the helper function, Lyapunov's stability theory and H∞Theoretically, the following steps are performed to find the gain of the filter:
s31. solving the following set of linear matrix inequalities (5) to obtain a set of initial solutions { P, X, R }c，Lil，Kil}：
Figure BDA00027104146100000712
Where X is the intermediate variable matrix, P, X, Rc、Kil、LilAre all matrix variables to be solved.
Figure BDA0002710414610000081
Is a diagonal matrix;
wherein the content of the first and second substances,
Figure BDA0002710414610000082
defining an augmented matrix of a system matrix
Figure BDA0002710414610000083
Wherein the dimension increasing matrix of the system matrix
Figure BDA0002710414610000084
Middle 0 represents a zero matrix; i denotes an identity matrix.
Defining an intermediate variable matrix
Figure BDA0002710414610000085
Wherein the content of the first and second substances,
Figure BDA0002710414610000086
defining an intermediate variable matrix
Figure BDA0002710414610000087
Then
Figure BDA0002710414610000088
Wherein the content of the first and second substances,
Figure BDA0002710414610000089
and representing the intermediate matrix after the system noise matrix and the channel noise matrix are subjected to dimension increase.
Figure BDA00027104146100000810
Figure BDA00027104146100000811
Definition of
Figure BDA00027104146100000812
Represents the increased diagonal matrix of X, wherein,
Figure BDA00027104146100000813
order to
Figure BDA00027104146100000814
Figure BDA00027104146100000815
Wherein:
Figure BDA00027104146100000816
Figure BDA00027104146100000817
Figure BDA00027104146100000818
wherein, s is 1, 2, …, d;
Figure BDA0002710414610000091
wherein p isl0Represents the probability that the ith channel does not have time lag, i is 1, 2, …, q; p is a radical oflsThe probability of the occurrence of the time lag with the length s of the l-th channel is shown, wherein s is 1, 2, … and d.
Order to
Figure BDA0002710414610000092
t is 0, given a sufficiently small number ρ > 0; wherein the content of the first and second substances,
Figure BDA0002710414610000093
indicating an initial value.
S32, solving an optimization problem under the constraint of a formula (5): min tr (PX)(t)+P(t)X); wherein min represents the minimum function, and tr represents the trace of the matrix.
S33, replacing X in the first matrix inequality on the left side in the formula (5) with P, and recording a newly obtained inequality as H < 0;
if the variables { P, X, R ] solved in step S32 arec，Kil，LilH < 0 can be satisfied and:
Figure BDA0002710414610000098
the obtained gain matrix K is outputil，LilAnd exiting; otherwise, step S34 is executed.
Where tr (PX) represents a trace of matrix PX,
Figure BDA0002710414610000094
s34, if t is larger than a given maximum step length, exiting; otherwise, let t be t +1, P(t)＝P，X(t)＝X，
Figure BDA0002710414610000095
The execution returns to step S32.
The proportional gain K of the filter is obtained through the steps S31 to S34ilAnd integral gain Lil。
S4, obtaining the proportional gain L of the filter according to calculationilAnd integral gain LilAnd calculating the state estimation value of the system.
According to the system state space model in the step S1, the signal transmission model in the step S2 and the filter gain in the step S3, the following fuzzy-proportional integral filter is constructed by fully considering the influence of the redundant time lag channel, as shown in the formula (6);
Figure BDA0002710414610000096
wherein the content of the first and second substances,
Figure BDA0002710414610000097
is an estimate of state x (k);
(a) representing a kronecker function, when a is 0, (a) is 1, otherwise, (a) is 0;
hl(k) for the integral vector corresponding to the l channel, hl(0) Is hl(k) An initial value of (1);
Figure BDA0002710414610000101
for the estimation of the signal z (k) to be estimated,
Figure BDA0002710414610000102
is the initial value of the filter;
and calculating a state estimation value of the nonlinear system with the redundant time-lag channel according to the fuzzy-proportional-integral filter (6).
As can be seen from the above formula (6), the fuzzy-pi filter designed by the present invention fully considers the influence of the redundant channels, can simultaneously use the information transmitted by each redundant channel, and correspondingly designs different proportional gains and integral gains, thereby greatly improving the design freedom of the filter and improving the robustness of the filter.
In addition, compared with the existing method in the prior art, the method has the following advantages:
compared with the traditional fuzzy state estimation method based on fuzzy-proportion, the method of the invention introduces an integral link, thereby eliminating steady-state errors and improving the robustness of the filter, thereby ensuring better estimation effect.
Compared with the traditional non-fuzzy proportional-integral state estimation method, the method of the invention is more suitable for a non-linear system because a fuzzy algorithm is added.
The method is used for detecting the running state of the industrial system in real time and can better meet the application requirements of the actual industry.
The fuzzy proportional-integral state estimation method with the redundant channel nonlinear system provided by the invention is explained in combination with experiments to verify the effectiveness of the method provided by the invention.
In the experiment process, the experiment step length is 200, a redundant channel and a random transmission time lag are added to a semi-physical simulation platform for obtaining the real-time state of the system, and the system output given by the platform is transmitted to a computer to be used as the input of a fuzzy proportional-integral estimator.
The state estimation value is generated by the method provided by the invention and is compared with the real value of the system state provided by the platform, and in the experiment, the system state is a 3-dimensional column vector.
Firstly, the first component x of the system real state vector is obtained by a computer1(k) And the corresponding estimated values, using a computer Matlab software drawing tool to obtain fig. 2. As can be seen from fig. 2:
the method of the invention is used for the first component x of the system state1(k) The estimation effect is good, and the estimation value can quickly approach to the true value.
In the same way, the second component x of the system real state vector is obtained by computer2(k) And the corresponding estimated values, using a computer Matlab software drawing tool to obtain fig. 3. As can be seen from fig. 3:
the method of the invention is applied to the second component x of the system state2(k) The estimation effect is good, and the estimation value can quickly approach to the true value.
In the same way, the third component x of the system real state vector is obtained by the computer3(k) And corresponding estimated value, using the Matlab software of computer to drawThe drawing tool gets fig. 4. As can be seen from fig. 4:
the method of the invention is used for the third component x of the system state3(k) The estimation effect is good, and the estimation value can quickly approach to the true value.
As can be seen from fig. 2 to fig. 4, in this experiment, the original system state is divergent, that is, the system is unstable, and in this case, the method proposed by the present invention can still obtain a better estimation effect, which proves the effectiveness of the present invention.
The obtained state estimation error is input into a computer by respectively providing a single transmission channel, a double transmission channel and a three transmission channel in a simulation platform, and the state estimation error is calculated by utilizing Matlab software and a corresponding image is drawn to obtain a graph 5.
As can be seen from FIG. 5, the error is estimated for three transmission channels
Figure BDA0002710414610000111
The amplitude is minimum, the convergence speed is fastest, and the estimation method has good estimation effect on the fuzzy proportional-integral state estimation of the nonlinear system with a plurality of redundant channels.
It should be understood, however, that the description herein of specific embodiments is not intended to limit the invention to the particular forms disclosed, but on the contrary, the intention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the appended claims.

Claims (1)

1. A fuzzy proportional-integral state estimation method of a nonlinear system with redundant time-lapse channels is characterized in that,
the method comprises the following steps:
s1, establishing a system state space model based on a T-S fuzzy modeling technology, as shown in a formula (1):
Figure FDA0002710414600000011
wherein x (k) represents the state variable of the nonlinear system at the time k, and ω (k) is nxA dimension column vector;
y (k) represents the measured output signal of the nonlinear system at time k, y (k) being nyA dimension column vector;
z (k) represents the signal to be estimated at time k, z (k) is nzA dimension column vector;
ω (k) represents energy-bounded noise, ω (k) being nwA dimension column vector; n isx、ny、nz、nwRepresents a known positive integer;
r represents the total number of fuzzy rules;
θ (k) represents a vector consisting of detectable states or measurement outputs; mu.si(theta (k)) is a membership function of the fuzzy system and represents the weight of the ith fuzzy mode in the whole nonlinear system at the moment k;
Airepresenting the system matrix, C the output matrix, EiRepresenting the process noise matrix, FiRepresenting a signal matrix to be estimated; wherein, the matrix Ai、C、Ei、FiAre all known constant matrices;
s2, establishing a signal transmission model under a redundant channel mechanism to realize data transmission between the nonlinear system and the filter;
setting q transmission channels between the nonlinear system and the filter for data transmission;
definition of
Figure FDA0002710414600000012
Representing the signal transmitted by the i-th channel received by the filter at the time k, while considering the influence of random transmission time lag, based on the output equation in the system state space model established in step S1, then
Figure FDA0002710414600000013
Expressed as:
Figure FDA0002710414600000014
wherein the content of the first and second substances,
Figure FDA0002710414600000015
Figure FDA0002710414600000016
representing variable definition symbols;
Figure FDA0002710414600000017
for random transmission skew, d represents the upper bound of the skew, dl(k) The value of (a) represents the length of time lag of the ith channel at the moment k; dlA known channel noise matrix;
event dl(k) The probability of occurrence of t is Pr { d }l(k)＝t}＝plt，0≤plt1 ≦ is a known scalar;
vl(k) channel noise, v, representing the energy-bounded of the ith channell(k) Is nvlVector of dimension, nvlRepresents a known positive integer;
s3, constructing an auxiliary function and calculating the proportional gain K of the filterilAnd integral gain Lil：
Wherein, KilThe proportional gain of the measurement output signal corresponding to the ith fuzzy mode of the filter through the ith channel is represented; l isilThe integral gain of the measurement output signal corresponding to the ith fuzzy mode of the filter through the ith channel is represented;
specifically, the following auxiliary functions are constructed according to the system state space model in step S1 and the signal transmission model in step S2:
Figure FDA0002710414600000021
Figure FDA0002710414600000022
wherein V (k) represents a selected Lyapunov functional, and J (iota) represents a difference between an estimation error and an energy of external noise;
P、Rcin order to determine the positive matrix to be solved,
Figure FDA0002710414600000023
represents a mathematical expectation;
Figure FDA0002710414600000024
which is indicative of an estimation error that is,
Figure FDA0002710414600000025
Figure FDA0002710414600000026
represents an estimated value of z (k);
gamma > 0 is a given scalar quantity,
Figure FDA0002710414600000027
the noise vector after dimension increase is obtained;
τ represents an intermediate variable, η (k),
Figure FDA00027104146000000215
Respectively representing the state estimation error and the vector after the dimension of the integral vector is increased;
the formula (3) is used for analyzing the stability of eta (k), and the formula (4) is used for testing the disturbance resistance of eta (k);
according to the helper function, Lyapunov's stability theory and H∞Theoretically, the following steps are performed to find the gain of the filter:
s31. solving the following set of linear matrix inequalities (5) to obtain a set of initial solutions { P, X, R }c，Lil，Kil}：
Figure FDA0002710414600000028
Where X is the intermediate variable matrix, P, X, Rc、Kil、LilAll are matrix variables to be solved;
Figure FDA0002710414600000029
is a diagonal matrix;
wherein the content of the first and second substances,
Figure FDA00027104146000000210
defining an augmented matrix of a system matrix
Figure FDA00027104146000000211
Figure FDA00027104146000000212
Wherein the dimension increasing matrix of the system matrix
Figure FDA00027104146000000213
Wherein 0 represents a zero matrix and I represents an identity matrix;
defining an intermediate variable matrix
Figure FDA00027104146000000214
Wherein the content of the first and second substances,
Figure FDA0002710414600000031
defining an intermediate variable matrix
Figure FDA0002710414600000032
Then
Figure FDA0002710414600000033
Wherein the content of the first and second substances,
Figure FDA0002710414600000034
representing an intermediate matrix after the system noise matrix and the channel noise matrix are subjected to dimension increase;
Figure FDA0002710414600000035
Figure FDA0002710414600000036
definition of
Figure FDA0002710414600000037
Represents the diagonal matrix after the dimension of the intermediate variable matrix X is increased, wherein,
Figure FDA0002710414600000038
order to
Figure FDA0002710414600000039
Figure FDA00027104146000000310
Wherein:
Figure FDA00027104146000000311
Figure FDA00027104146000000312
Figure FDA00027104146000000313
wherein, s is 1, 2, …, d;
Figure FDA00027104146000000314
wherein p isl0Represents the probability that the ith channel does not have time lag, i is 1, 2, …, q; p is a radical oflsRepresents the probability of the occurrence of the time lag with the length s of the l-th channel, wherein s is 1, 2, … and d;
order to
Figure FDA00027104146000000315
t is 0, given a sufficiently small number ρ > 0;
wherein the content of the first and second substances,
Figure FDA00027104146000000316
represents an initial value;
s32, solving an optimization problem under the constraint of a formula (5): mintr (PX)(t)+P(t)X)；
Wherein min represents the minimum function, and tr represents the trace of the matrix;
s33, replacing X in the first matrix inequality on the left side in the formula (5) with P, and recording a newly obtained inequality as H < 0; if the variables { P, X, R ] solved in step S32 arec，Kil，LilH < 0 can be satisfied and:
Figure FDA0002710414600000041
the obtained gain matrix K is outputil，LilAnd exiting; otherwise, go to step S34;
where tr (PX) represents a trace of matrix PX,
Figure FDA0002710414600000042
s34, if t is larger than a given maximum step length, exiting; otherwise, let t be t +1, P(t)＝P，X(t)＝X，
Figure FDA0002710414600000043
Return to perform step S32;
the proportional gain K of the filter is obtained through the steps S31 to S34ilAnd integral gain Lil；
S4, obtaining the proportional gain L of the filter according to calculationilAnd integral gain LilCalculating a state estimation value of the system;
according to the system state space model in the step S1, the signal transmission model in the step S2 and the filter gain in the step S3, the following fuzzy-proportional integral filter is constructed by fully considering the influence of the redundant time lag channel, as shown in the formula (6);
Figure FDA0002710414600000044
wherein the content of the first and second substances,
Figure FDA0002710414600000045
is an estimate of state x (k);
(a) representing a kronecker function, when a is 0, (a) is 1, otherwise, (a) is 0;
hl(k) for the integral vector corresponding to the l channel, hl(0) Is hl(k) An initial value of (1);
Figure FDA0002710414600000046
for the estimation of the signal z (k) to be estimated,
Figure FDA0002710414600000047
is the initial value of the filter;
and calculating a state estimation value of the nonlinear system with the redundant time-lag channel according to the fuzzy-proportional-integral filter (6).
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