CN106529479A - Non-fragile dissipative filtering method of nonlinear networked control system - Google Patents

Abstract

The present invention discloses a non-fragile dissipative filtering method of a nonlinear networked control system. The method comprises the steps of firstly establishing a nonlinear networked filtering error system model on the conditions of considering the time delay and the packet loss of the nonlinear networked control system and the perturbation of the filter parameters, then constructing a Lyapunov function, and then utilizing a Lyapunov stability theory and a linear matrix inequality analysis method to obtain the sufficient conditions of the mean square exponential stability of a nonlinear networked filtering error system and the existence of a non-fragile dissipative filter, utilizing a Matlab LMI tool kit to solve, and definding a non-fragile dissipative filter parameter matrix. The method of the present invention considers the random time delay and the pocket loss situations between the sensors and the filters, is suitable for the general dissipative filtering including the H-infinite filtering, and enables the conservatism of the non-fragile dissipative filter design to be reduced. Moreover, a non-modeling state of the system is considered when a full-order filter is designed, thereby being able to reduce the calculation burdens and the design cost.

Description

A kind of uncatalyzed coking dissipation filtering method of nonlinear network networked control systems

Technical field

The present invention relates to nonlinear network networked control systems and dissipation filtering, more particularly to a kind of to have time delay and packet loss Nonlinear network networked control systems uncatalyzed coking dissipation filtering method.

Background technology

Network control system (networked control are referred to as by the closed-loop control system that communication network is formed Systems, is abbreviated NCSs), NCSs has the advantages that convenient for installation and maintenance, flexibility is high and is easy to reconstruct.However, communication network Introducing result in system and there is problems with：1) network delay：Data in communication network transmission because network blockage or The reasons such as external interference so that there is network delay in network control system；2) packet loss：In data transmission procedure because The reason such as network blockage and resource contention can cause the problem of data-bag lost.Extraneous uncertain factor may result in simultaneously Systematic function reduces even unstability.Therefore, make NCSs that there is fault-tolerant ability and preferable interference free performance is kept with very heavy The theory significance wanted and more practical value.

For time delay present in NCSs and packet loss problem, many scholars and expert have done numerous studies.Horse leap etc. In paper " robust dissipation filterings of unknown time-delay Discrete-time Nonlinear Systems ", time delay is have studied to the stability of a system and dissipation The impact of performance.Lin Qiongbin etc. have studied packet loss pair in " having the dissipation fuzzy filter of many data packetloss nonlinear systems " The impact of the stability of a system and dissipative performance, Zhang Peng etc. are at paper " the robust dissipation filtering devices of Linear uncertain time-delay systems " In, the impact of Study system parameter uncertainty and time delay to the stability and dissipative performance of system.Above-mentioned dissipation filtering The research of aspect has considered only time delay or packet loss, and is not subject to external interference in view of wave filter inherent parameters Some changes can be also produced, but time delay and packet loss are simultaneous and wave filter inherent parameters in an actual situation Can be interfered change, so seeming particularly significant using uncatalyzed coking filtering, it can make system immediate stability, Disturbance Rejection More preferably, filtering estimation effect is more preferable for level.Li Xiuying etc. " has network system H of a step stochastic Time-Delay and many packet losses in paper_∞ Wave filter is designed ", packet loss and time lag are studied to the stability of a system and H_∞The impact of performance, and system model is augmented, Higher-dimension wave filter is devised, so substantially increases the computation burden that the wave filter of design higher-dimension can bring, so design wave filter When consider system and do not model state, can make system that there is stronger robustness, and to design full rank wave filter be to keep away The higher-dimension wave filter estimation of augmented state is exempted from, computation burden and design cost can have been significantly reduced.

The content of the invention

For the problem that above-mentioned technology is present, the invention provides a kind of uncatalyzed coking dissipation filtering of network control system Method.Exist in the case of time delay, packet loss and wave filter have Parameter Perturbation in view of network control system, devising non- Fragile dissipation filtering device so that network control system remains to keep Stochastic stable, and strict dissipativity in these cases.

The technical solution adopted in the present invention is：A kind of uncatalyzed coking dissipation filtering side of nonlinear network networked control systems Method, comprises the following steps：

1) set up nonlinear networked filtering error system model：

Wherein：x(k)∈RⁿIt is the state vector of system, x_f(k) ∈RⁿIt is state estimation, y (k) ∈ R^rIt is that the measurement that wave filter is received is exported, w (k) ∈ R^pIt is external interference signals, f (k, x (k)) meet Lipschitz condition Nonlinear Vector items | | f (k, x (k)) | |≤| | Wx (k) | |, w (k) ∈ l₂[0, ∞) to disturb Dynamic input, e (k)=z (k)-z_fK () is filtering error, z (k) ∈ R^mIt is to be estimated signal, z_f(k)∈R^mIt is the defeated of wave filter Go out；

A_fd=A_f+ΔA_f, B_fd=B_f+ΔB_f, C_fd=C_f+ΔC_f

Wherein：A_f,B_f,C_fIt is filter parameter matrix, A ∈ R^n×n, B ∈ R^n×p, C ∈ R^r×n, D ∈ R^r×p, L₁∈R^m×n, L₂ ∈R^m×p, 0 and I is for null matrix and unit matrix, A_fd=A_f+ΔA_f, B_fd=B_f+ΔB_f, C_fd=C_f+ΔC_f；ΔA_f=H₁F₁(k) E₁, Δ B_f=H₂F₂(k)E₂, Δ C_f=H₃F₃(k)E₃For filter parameter perturbation matrices, A_f∈R^n×n, B_f∈R^n×r, C_f∈R^m×nFor Filter parameter matrix, H₁∈R^n×d, H₂∈R^n×h, H₃∈R^m×a, E₁∈R^d×n, E₂∈R^h×r, E₃∈R^a×n, F₁(k)∈R^d×d, F₂ (k)∈R^h×h, F₃(k)∈R^a×a；

Definition：

Introduce：ξ (k)=(1- θ (k)) δ (k+1),

Then there is following statistical property (E represents mathematic expectaion)：

Wherein：δ_kAnd θ_kIt is uncorrelated random variables,

By model：

Wherein：y(k)∈R^rIt is measurement output；

Time delay probability of happening is

Drop probabilities are

The probability of acceptance is data on timey(k)∈R^rIt is measurement output；

2) Lyapunov functions are constructed；

Wherein：P is positive definite symmetric matrices；

3) calculate uncatalyzed coking dissipation filtering device parameter matrix A_f, B_f, C_fWith system performance index γ, system Stochastic stable and Uncatalyzed coking dissipative control device exist adequate condition be：

For following linear MATRIX INEQUALITIES：

Wherein：

Ψ₃=diag {-ε₁I,-ε₁I,-ε₂I,-ε₂I,-ε₃I,-ε₃I,-ε₄I,-ε₄I}

Π₃=diag {-Π ,-Π ,-Π ,-Π }

Π₄=diag {-P₂,-P₂,-P₂,-P₂}

Wherein, W, V are nonsingular constant matrices, are met, WV^T=I-XZ^-1；

X∈R^n×n, Z ∈ R^n×n,P2∈R^2r×2r, ε_i＞ 0 (i=1,2,3,4) is For known variables, other variables are known, can be drawn according to systematic parameter or directly be given, using Matlab LMI works Tool case is solved, if there is symmetric positive definite matrix X, Z, P₂And matrixWith scalar ε i ＞ 0 (i=1,2,3, 4), then networking filtering error system is Stochastic stable and has strict dissipativity, and uncatalyzed coking filter parameter matrix is γ=Σ (| | e (k) | |)/Σ (| | w (k) | |), and can be after It is continuous to carry out step 4)；If above-mentioned known variables are without solution, networking filtering error system is not Stochastic stable and is unsatisfactory for tight Lattice dissipativeness, it is impossible to obtain the parameter matrix of uncatalyzed coking wave filter, cannot also carry out step 4)；

4) calculate uncatalyzed coking H_∞Filter parameter matrix A_f, B_f, C_f, each matrix parameter is taken as：Q=-I, R=γ²I, S= 0, H_∞The lower Optimal Disturbance Rejection of filtering compares γ_optThe condition of optimization is：

Make e=γ²If following optimization problem is set up：

X=X^T＞ 0, Z=Z^T＞ 0, X-Z ＞ 0,ε_i＞ 0 (i=1,2,3,4)

The minimal disturbances inhibiting rate of systemThe parameter matrix of uncatalyzed coking dissipation filtering device can also be optimised for simultaneously

Compared with prior art, the present invention has following Advantageous Effects：

1) present invention is directed to the nonlinear network networked control systems with time delay and packet loss, while considering filter parameter Perturbation and the impact of external disturbance, establish networking filtering error system model, the corresponding stability of a system and dissipation filtering Solution；

2) present invention considers the probability of happening of random loss and time delay, random loss and time delay and meets Bernoulli point Cloth, more practical significance；

3) present invention take into account the perturbation of filter parameter, optimize system performance index so that control based on network system System is with more preferable interference free performance；

4) present invention is filtered suitable for Dissipative, including H_∞Filtering reduces the guarantor of the uncatalyzed coking filter design method Keeping property.

5) system is considered during present invention design wave filter and do not model state, make system that there is stronger robustness, and The wave filter of design is full rank, it is to avoid the higher-dimension wave filter of augmented state is estimated, can significantly reduce computation burden and set Meter cost.

Description of the drawings

Accompanying drawing 1 is the flow chart of nonlinear network networked control systems uncatalyzed coking dissipation filtering method.

When accompanying drawing 2 has general uncatalyzed coking (Q, S, R) dissipation filtering device, variable z (k) to be estimated is estimated with whichResponse Figure.

Accompanying drawing 3 has uncatalyzed coking H_∞During wave filter, variable z (k) to be estimated is estimated with whichResponse diagram.

Specific embodiment

Below in conjunction with the accompanying drawings the specific embodiment of the present invention is described further.

Referring to the drawings 1, a kind of uncatalyzed coking H of nonlinear network networked control systems_∞Fault tolerant control method, including following step Suddenly：

Step 1：Set up networking filtering error system model

Consider following nonlinear network networked control systems

Wherein:x(k)∈RⁿIt is the state vector of system, z (k) ∈ R^mIt is to be estimated signal,It is that measurement is exported, w (k)∈R^pIt is external interference signals, f (k, x (k)) meets Lipschitz condition Nonlinear Vector items | | f (k, x (k)) | |≤| | Wx(k)||；w(k)∈l₂[0, ∞), matrix A, B, C, D, L₁,L₂Respectively there is the known constant matrix of corresponding dimension.

Assume that sensor is that clock drives,Wave filter is transmitted through the network to after packing, due to communicate bandwidth and The reasons such as network congestion, packet can occur inevitable time lag in the transmission and even lose, this phenomenon be it is random, one As using the random sequence of Bernoulli distributions describing these situations.

When having random delay：

When having random loss：

Wherein：y(k)∈R^PIt is that the measurement that wave filter is received is exported, θ_kIt is that the Bernoulli that value is 0 and 1 is distributed Random sequence, meets

By the stochastic variable for introducing 2 Bernoulli distribution, by random delay and packet loss phenomenon with a model retouching State：

Wherein：δ_kAnd θ_kIt is uncorrelated random variables,By model (2) understand, time delay probability of happening is

Drop probabilities are

The probability of acceptance is data on time

Designing wave filter is：

WhereinIt is state estimation, z_f(k)∈R^qIt is the estimation of z (k), A_fd,B_fd,C_fdWith filter parameter It is uncertain and meet following form：

Wherein：x_f(k)∈RⁿIt is state estimation, z_f(k)∈R^mIt is the output of wave filter, A_fd, B_fd, C_fdIt is not true with parameter It is qualitative and meet following form：

Wherein：A_f,B_f,C_fIt is filter parameter matrix, H_iAnd E_i(i=1,2,3) it is known normal matrix；F_iK () meets：

F_i(k)^TF_i(k)≤I, i=1,2,3,4 (6)

Definition filtering error is e (k)=z (k)-z_f(k).In order to further obtain filtering error system, new change is introduced Amount, order

ξ (k)=(1- θ (k)) δ (k+1) (7)

Then there is following statistical property (E represents mathematic expectaion)：

Formula (7) is substituted into formula (2) to obtain

Definition：

θ (k) ξ (k)=ξ (k) (1- ξ (k)) δ (k+1)=0 is noticed, then by formula (1) and formula (3), filtering as follows can be obtained and missed Difference system：

Wherein：

It is as follows that supply function E (w, z, T) can be measured in definition：

Wherein：Q∈R^m×m, R ∈ R^p×pFor known symmetrical matrix and Q ＜ 0, S ∈ R^m×pFor known constant matrix.

Step 2：Construction Lyapunov functions

Wherein：P is symmetric positive definite matrix.

When w (k)=0,

Utilize：

Due to f^T(k)f(k)≤x^T(k)W^TWx (k), accordingly, there exist τ ＞ 0, τ x^T(k)W^TWx(k)-τf^T(k)f(k)≥0

Wherein：

Γ₂=[G^T 0 0 0]^T

M_1,2=M₁-M₂,

N_1,2=N₁-N₂

Step 3：Using Lyapunov Theory of Stability and LMI analysis method, control based on network system is obtained The solution of adequate condition and filter parameter that system Stochastic stable and uncatalyzed coking dissipation filtering device are present, step are as follows：

Step 3.1：Based on the Lyapunov functions that step 2 is constructed, using Lyapunov Theory of Stability and linear matrix Inequality analysis method, first determines whether the Stochastic stable and strict dissipativity of networking filtering error system, obtains networking filter The adequate condition that wave error system Stochastic stable and dissipation filtering device are present.

Lemma is mended by Schur to obtain：Φ ＜ 0 are equivalent to:

I.e.Wherein, μ is the minimal eigenvalue of-Φ.Thus can obtain, So filtering error system (10) is Stochastic stable.

When w (k) ≠ 0, constructing Lyapunov functions in the same manner can obtain：

Due to：f^T(k)f(k)≤x^T(k)W^TWx (k), therefore, deposit 0 τ x of τ ＞^T(k)W^TWx(k)-τf^T(k)f(k)≥0

Wherein：

G₂=[G^T 0 0 0 0]^T

As Λ ＜ 0, sufficiently small a ＞ 0 are certainly existed so that (Λ+adiag (0, I)) ＜ 0, so：

Formula (15) k is sued for peace from 0 to T available:

Because filtering error system (10) is Stochastic stable, it is initial zero under conditions ofAndInstitute With：To arbitrary T ＞ 0 and all non-zeros w (k),Meet strict dissipativity.Mended using Schur and drawn Reason, Λ ＜ 0 are equivalent to

The adequate condition that networking filtering error system Stochastic stable shown in system (10) and dissipation filtering device are present is：When During external disturbance w (k) ≠ 0, it is known that Q, S, R and Q ＜ 0 and A_fd,B_fd,C_fd, there is positive definite symmetric matrices P and formula (16) set up, When the adequate condition of step 3.1 is set up, then execution step 3.2；If the adequate condition of step 3.1 is false, system is not Stochastic stable and uncatalyzed coking dissipation filtering device just do not exist, it is impossible to execution step 3.2.

Step 3.2：The solution of filter parameter.

By P=diag { P₁,P₂Inequality (16) is substituted into, then to inequality (16) both sides premultiplication diag { Σ₂,I,I,I, Σ₁,I,Σ₁,I,Σ₁,I,Σ₁,I,I}^T, diag { Σ are taken advantage of on the right side₂,I,I,I,Σ₁,I,Σ₁,I,Σ₁,I,Σ₁, I, I } pass through again Equivalence transformation can obtain formula (17)：

Further can be calculated：

Wherein：

Ξ_c=[L₁ L₁Z^-1 -C_fV^T],Ξ_d=L₂

Uncatalyzed coking filter parameter matrix and Parameter Perturbation matrix can be tried to achieve by following formula：

Wherein：W, V are nonsingular constant matrices, are met：WV^T=I-XZ^-1。

Diag { I, Z, I, I, I, I, Z, I, Z, I, Z, I, Z, I, I, I, I, I } is taken advantage of to formula (18) premultiplication, the right side in conjunction with formula (19) obtain Ψ₁, then and filter joint parameter (4) and (5), by the indeterminate of formula with determine item and separate, utilization Schur mends lemma and obtains formula (20).

Wherein：

Ψ₃=diag {-ε₁I,-ε₁I,-ε₂I,-ε₂I,-ε₃I,-ε₃I,-ε₄I,-ε₄I}

Π₃=diag {-Π ,-Π ,-Π ,-Π }

Π₄=diag {-P₂,-P₂,-P₂,-P₂}

X∈R^n×n, Z ∈ R^n×n, P₂∈P^2r×2r,ε_i＞ 0 (i=1,2,3,4) is known variables, its Its variable is known, can be drawn according to systematic parameter or directly be given, and is solved using Matlab LMI tool boxes, If there is symmetric positive definite matrix X, Z, P₂MatrixWith scalar ε_i(i=1,2,3,4), then networking is filtered ＞ 0 Error system is Stochastic stable and has strict dissipativity, and uncatalyzed coking filter parameter matrix is And step 4 can be proceeded)；If above-mentioned known variables are without solution, networking filtering Error system is not Stochastic stable and is unsatisfactory for strict dissipativity, it is impossible to obtains the parameter matrix of uncatalyzed coking wave filter, can not yet With carry out step 4)；

Step 4：Q, S are worked as in consideration, the dissipation filtering problem of system, wherein H when R chooses different value_∞Filtering can be considered as one As dissipation filtering a kind of special case.If Dissipative filtering, then asked using γ=Σ (| | e (k) | |)/Σ (| | w (k) | |) Go out corresponding system performance index γ；If the H of standard_∞Filtering, i.e., each matrix parameter are taken as：Q=-I, R=γ²I, S=0, provides H_∞The lower Optimal Disturbance Rejection of filtering compares γ_optThe condition of optimization is：

Make e=γ²If following optimization problem is set up：

X=X^T＞ 0, Z=Z^T＞ 0,ε_i＞ 0 (i=1,2,3,4)

The Optimal Disturbance Rejection ratio of systemThe parameter matrix of uncatalyzed coking dissipation filtering device can also be optimised for simultaneously

Embodiment：

Using a kind of uncatalyzed coking dissipation filtering method of network control system proposed by the present invention, in no external disturbance In the case of i.e. w (k)=0 when, networking filtering error system is Stochastic stable.When there is external disturbance, system is also Stochastic stable and with certain antijamming capability.Concrete methods of realizing is as follows：

Step 1：Consider following network control system：

Y (k)=[2-3 5] x (k)+2w (k)

Z (k)=[- 0.1 0.3-0.2] x (k)

Consideration is given below parameter：

H₃=[1 0 0], E₂=0.02

Here takeWhen there is Parameter Perturbation F₁(k)=F₂(k)=F₃(k)=F₄During (k)=I, it is assumed that System disturbance input w (k)=1/k², it is considered to work as Q, S, when R chooses different value, the dissipation filtering problem of system, wherein H_∞Filtering It is considered as a kind of special case of Dissipative filtering.

Step 2：Each parameter of Dissipative filtering is taken as Q=-0.9, S=0.5, R=12.According to formula (20), utilize Matlab LMI tool boxes uncatalyzed coking filtering parameter, and obtained with γ=∑ (| | e (k) | |)/∑ (| | w (k) | |) respectively and be Unite corresponding performance indications γ,

C_f=[- 0.0854 0.2958-0.2085]

γ=0.7331

However, making Δ A_f, Δ B_f, Δ C_fPerformance indications γ for drawing traditional wave filter when being zero are 0.7375, contrast It was found that, performance indications γ of uncatalyzed coking wave filter are substantially little than traditional performance of filter index γ, this explanation institute of the present invention Uncatalyzed coking wave filter has more preferable Disturbance Rejection performance compared with conventional filter.

Due to the time delay that there is uncertainty probability in network control system and packet loss, can obtain under different performance index Dissipation filtering device such as table 1.

Performance indications under 1 different delay of table and packet loss

Note 2：Time delay probability is equal toThe probability of packet loss is equal to

As it can be seen from table 1 the probability of packet loss and time delay can be changed by changing the expectation of θ (k) and δ (k), with losing The probability of bag and time delay increases, and performance indications can also become big, and the estimation effect of uncatalyzed coking dissipation filtering device can be deteriorated, and illustrate packet loss The performance of system is had an impact with time delay.

Step 3：H_∞Each parameter of filtering is taken as：Q=-I, R=γ²I, S=0.In the same manner, according to formula (21), utilize Matlab LMI tool boxes obtain uncatalyzed coking filtering parameter, and obtain corresponding Optimal Disturbance Rejection than γ o_pt,

C_f=[0.1911-0.3255 0.0739]

γ_opt=3.2713

However, making Δ A_f, Δ B_f, Δ C_fPerformance indications γ for drawing traditional wave filter when being zero are 4.0021, contrast It was found that, the Optimal Disturbance Rejection of uncatalyzed coking wave filter compares γ_optThan traditional performance of filter index γ_optLittle, this explanation is originally Invention uncatalyzed coking wave filter used has more preferable antijamming capability compared with conventional filter.

Due to the time delay that there is uncertainty probability in network control system and packet loss, different Optimal Disturbance Rejections can be obtained Uncatalyzed coking H than under_∞Wave filter such as table 2.

Performance indications under 2 different delay of table and packet loss

From table 2 it can be seen that the probability of packet loss and time delay can be changed by changing the expectation of θ (k) and δ (k), with losing The probability of bag and time delay increases, and Optimal Disturbance Rejection can become big than also, and the antijamming capability of system can be deteriorated, uncatalyzed coking H_∞Filtering The estimation effect of device can be deteriorated, and illustrate that packet loss and time delay are had an impact to the performance of system.

Step 4：Using in step 2 and 3When Matlab LMI tool boxes solve result, use Matlab simulates corresponding variable z (k) to be estimated of network control system and estimates z with which_fThe response of (k), such as accompanying drawing 2 and attached Shown in Fig. 3.

By accompanying drawing 2 and accompanying drawing 3 as can be seen that variable z (k) to be estimated of system estimates z with which_fK () is bounded stability, Illustrate the design of uncatalyzed coking dissipation filtering device and uncatalyzed coking H when there is certainty time delay and packet loss in a network environment_∞Wave filter Design is effective.

It is more than presently preferred embodiments of the present invention, not makees any pro forma restriction, every foundation to the present invention The technical spirit of the present invention belongs to inventive technique to any simple modification made for any of the above embodiments, equivalent variations and modification In the range of scheme.

Claims (1)

1. a kind of uncatalyzed coking dissipation filtering method of nonlinear network networked control systems, it is characterised in that specifically include following step Suddenly：

1) set up nonlinear networked filtering error system model：

$\left\{\begin{array}{c}x(k+1)=Ax\left(k\right)+Bw\left(k\right)+Gf(k,x(k\left)\right)\\ e\left(k\right)=Cx\left(k\right)+Dw\left(k\right)\end{array}\right.$

Wherein：X (k+1)=[x (k+1)^T x_f(k+1)^T Y(k)^T y(k)^T]^T, x (k) ∈ RⁿIt is the state vector of system, x_f(k) ∈RⁿIt is state estimation, y (k) ∈ R^rIt is that the measurement that wave filter is received is exported, w (k) ∈ R^pIt is external interference signals, f (k, x (k)) meet Lipschitz condition Nonlinear Vector items | | f (k, x (k)) | |≤| | Wx (k) | |, w (k) ∈ l₂[0, ∞) to disturb Dynamic input, e (k)=z (k)-z_fK () is filtering error, z (k) ∈ R^mIt is to be estimated signal, z_f(k)∈R^mIt is the defeated of wave filter Go out；

$A=\stackrel{\‾}{A}+\widetilde{A_1}+\widetilde{A_2},B=\stackrel{\‾}{B}+\widetilde{B_1}+\widetilde{B_2}$

$\widetilde{A_2}=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ \left(\mathrm{\ξ}\right(k)-\stackrel{\‾}{\mathrm{\ξ}})C& 0& -\left(\mathrm{\ξ}\right(k)-\stackrel{\‾}{\mathrm{\ξ}})I& 0\\ 0& 0& 0& 0\end{array}\right]$

A_fd=A_f+ΔA_f, B_fd=B_f+ΔB_f, C_fd=C_f+ΔC_f

A∈R^n×n, B ∈ R^n×p, C ∈ R^r×n, D ∈ R^r×p, L₁∈R^m×n, L₂∈R^m×pFor systematic parameter matrix；

A_f∈R^n×n, B_f∈R^n×r, C_f∈R^m×nFor filter parameter matrix, Δ A_f=H₁F₁(k)E₁, Δ B_f=H₂F₂(k)E₂, Δ C_f =H₃F₃(k)E₃For filter parameter perturbation matrices；H₁∈R^n×d, H₂∈R^n×h, H₃∈R^m×a, E₁∈R^d×n, E₂∈R^h×r, E₃∈R^{a ×n}, F₁(k)∈R^d×d, F₂(k)∈R^h×h, F₃(k)∈R^a×a；

0 and I is for null matrix and unit matrix；

Y (k)=ξ (k) y (k)+(1- ξ (k)) y (k), ξ (k)=(1- θ (k)) δ (k+1), there is following statistical property, E represents mathematics Expect：

$E\[\mathrm{\θ}\left(k\right)\]=\stackrel{\‾}{\mathrm{\θ}},$

$E\[\mathrm{\δ}\left(k\right)\]=\stackrel{\‾}{\mathrm{\δ}},$

$E\[\mathrm{\ξ}\left(k\right)\]=(1-\stackrel{\‾}{\mathrm{\θ}})\stackrel{\‾}{\mathrm{\δ}}=\stackrel{\‾}{\mathrm{\ξ}},$

$E\[{\left(\mathrm{\θ}\right(k)-\stackrel{\‾}{\mathrm{\θ}})}^2\]=\stackrel{\‾}{\mathrm{\θ}}(1-\stackrel{\‾}{\mathrm{\θ}})={\mathrm{\σ}}_1^2,$

$E\[{\left(\mathrm{\ξ}\right(k)-\stackrel{\‾}{\mathrm{\ξ}})}^2\]=\stackrel{\‾}{\mathrm{\ξ}}(1-\stackrel{\‾}{\mathrm{\ξ}})={\mathrm{\σ}}_2^2,$

$E\[\left(\mathrm{\ξ}\right(k)-\stackrel{\‾}{\mathrm{\ξ}})\left(\mathrm{\θ}\right(k)-\stackrel{\‾}{\mathrm{\θ}})\]=-\stackrel{\‾}{\mathrm{\θ}}\stackrel{\‾}{\mathrm{\ξ}}=-{\mathrm{\σ}}_3^2,$

Wherein：δ_kAnd θ_kIt is uncorrelated random variables,By model

Y (k)=θ_ky(k)+(1-θ_k)(1-θ_k-1)δ_ky(k-1)+(1-θ_k)[1-(1-θ_k-1)δ_k)]y(k-1)

Wherein：y(k)∈R^rIt is measurement output；

Time delay probability of happening is

Drop probabilities are

The probability of acceptance is data on time

2) Lyapunov functions are constructed；

$V\left(\tilde{x}\right(k\left)\right)=\tilde{x}{\left(k\right)}^{T}P\tilde{x}\left(k\right)$

Wherein P is positive definite symmetric matrices；

3) calculate uncatalyzed coking dissipation filtering device parameter matrix A_f, B_f, C_fWith system performance index γ, system meansquare exponential stability and Uncatalyzed coking dissipative control device exist adequate condition be：

For following linear MATRIX INEQUALITIES：

$\left[\begin{array}{cc}{\mathrm{\&Psi;}}_1& {\mathrm{\&Psi;}}_2^T\\ {\mathrm{\&Psi;}}_2& {\mathrm{\&Psi;}}_3\end{array}\right]<0$

Wherein：

${\mathrm{\&Psi;}}_1=\left[\begin{array}{ccccccc}-\mathrm{\&Pi;}+{\mathrm{\&Pi;}}_{11}& *& *& *& *& *& *\\ 0& -P_2& *& *& *& *& *\\ 0& 0& -\mathrm{\&tau;}I& *& *& *& *\\ -{\mathrm{\&Pi;}}_1& 0& 0& {\mathrm{\&Pi;}}_2& *& *& *\\ {\mathrm{\&Theta;}}_{11}& {\mathrm{\&Theta;}}_{12}& {\mathrm{\&Theta;}}_{13}& {\mathrm{\&Theta;}}_{14}& {\mathrm{\&Pi;}}_3& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& {\mathrm{\&Theta;}}_{23}& {\mathrm{\&Theta;}}_{24}& 0& {\mathrm{\&Pi;}}_4& *\\ {\mathrm{\&Theta;}}_c& 0& 0& {\mathrm{\&Theta;}}_d& 0& 0& Q^{-1}\end{array}\right]$

${\mathrm{\&Psi;}}_2=\left[\begin{array}{ccccccccccccccccccccccc}0& 0& 0& 0& 0& 0& {\hat{H}}_1^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& {\mathrm{\&epsiv;}}_1{\hat{E}}_1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& {\hat{H}}_3^{T}S& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -{\hat{H}}_3^T\\ 0& {\mathrm{\&epsiv;}}_2{\hat{E}}_3& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \stackrel{\&OverBar;}{\mathrm{\&theta;}}{\hat{H}}_2^T& 0& {\mathrm{\&beta;}}_1{\hat{H}}_2^T& 0& 0& 0& {\mathrm{\&sigma;}}_3{\hat{H}}_2^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ {\mathrm{\&epsiv;}}_3{\hat{E}}_{2}C& {\mathrm{\&epsiv;}}_3{\hat{E}}_{2}C& 0& 0& 0& {\mathrm{\&epsiv;}}_3{\hat{E}}_{2}D& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& \left(1-\stackrel{\&OverBar;}{\mathrm{\&theta;}}\right){\hat{H}}_2^T& 0& -{\mathrm{\&beta;}}_1{\hat{H}}_2^T& 0& 0& 0& -{\mathrm{\&sigma;}}_3{\hat{H}}_2^T& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& {\mathrm{\&epsiv;}}_4{\hat{E}}_2& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]$

Ψ₃=diag {-ε₁I,-ε₁I,-ε₂I,-ε₂I,-ε₃I,-ε₃I,-ε₄I,-ε₄I}

$\mathrm{\&Pi;}=\left[\begin{array}{cc}X& Z\\ Z& Z\end{array}\right],P_2=\left[\begin{array}{cc}{Y}_1& Y_3\\ Y_3& Y_2\end{array}\right],{\mathrm{\&Pi;}}_{11}=\left[\begin{array}{cc}{\mathrm{\&tau;W}}^{T}W& {\mathrm{\&tau;W}}^{T}W\\ {\mathrm{\&tau;W}}^{T}W& {\mathrm{\&tau;W}}^{T}W\end{array}\right]$

${\mathrm{\&Pi;}}_1=\left[\begin{array}{cc}-S^T{L}_1& -S^T{L}_1+S^T{\hat{C}}_f\end{array}\right],{\mathrm{\&Pi;}}_2=-D^{T}S-S^{T}D-R$

Π₃=diag {-Π ,-Π ,-Π ,-Π }, Π₄=diag {-P₂,-P₂,-P₂,-P₂}

${\mathrm{\&Theta;}}_{11}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{A_1}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_1}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_3}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_{1,3}}^T\end{array}\right]}^T,{\mathrm{\&Theta;}}_{12}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{A_2}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_2}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_4}^T& {\mathrm{\&Theta;}}_{{\hat{M}}_{2,4}}^T\end{array}\right]}^T,$

${\mathrm{\&Theta;}}_{14}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{B_1}^T& {\mathrm{\&Theta;}}_{{\hat{N}}_1}^T& {\mathrm{\&Theta;}}_{{\hat{N}}_2}^T& {\mathrm{\&Theta;}}_{{\hat{N}}_{1,2}}^T\end{array}\right]}^T,{\mathrm{\&Theta;}}_{21}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{A_3}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_1}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_3}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_{1,3}}^T\end{array}\right]}^T$

${\mathrm{\&Theta;}}_{22}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{A_4}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_2}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_4}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_{2,4}}^T\end{array}\right]}^T,{\mathrm{\&Theta;}}_{24}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{B_2}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_1}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_2}^T& {\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_{1,2}}^T\end{array}\right]}^T$

${\mathrm{\&Theta;}}_{13}={\left[\begin{array}{cccc}{\mathrm{\&Theta;}}_{c_1}^T& 0& 0& 0\end{array}\right]}^T,{\mathrm{\&Theta;}}_{23}={\left[\begin{array}{cccc}0& 0& 0& 0\end{array}\right]}^T$

${\mathrm{\&Theta;}}_c=\left[\begin{array}{cc}{L}_1& L_1-{\hat{C}}_f\end{array}\right],{\mathrm{\&Theta;}}_d=L_2,{\mathrm{\&Theta;}}_{c_1}={\left[\begin{array}{cc}{X}^T& Z^T\end{array}\right]}^T$

${\mathrm{\&Theta;}}_{A_1}=\left[\begin{array}{cc}XA+\stackrel{\&OverBar;}{\mathrm{\&theta;}}{\hat{B}}_{f}C& XA+\stackrel{\&OverBar;}{\mathrm{\&theta;}}{\hat{B}}_{f}C+{\hat{A}}_f\\ ZA& ZA\end{array}\right],{\mathrm{\&Theta;}}_{A_2}=\left[\begin{array}{cc}\left(1-\stackrel{\&OverBar;}{\mathrm{\&theta;}}\right){\hat{B}}_f& 0\\ 0& 0\end{array}\right]$

${\mathrm{\&Theta;}}_{A_3}=P_2\left[\begin{array}{cc}(\stackrel{\&OverBar;}{\mathrm{\&theta;}}+\stackrel{\&OverBar;}{\mathrm{\&xi;}})C& (\stackrel{\&OverBar;}{\mathrm{\&theta;}}+\stackrel{\&OverBar;}{\mathrm{\&xi;}})C\\ \stackrel{\&OverBar;}{\mathrm{\&theta;}}C& \stackrel{\&OverBar;}{\mathrm{\&theta;}}C\end{array}\right],{\mathrm{\&Theta;}}_{A_4}=P_2\left[\begin{array}{cc}(1-\stackrel{\&OverBar;}{\mathrm{\&theta;}}-\stackrel{\&OverBar;}{\mathrm{\&xi;}})I& 0\\ (1-\stackrel{\&OverBar;}{\mathrm{\&theta;}})I& 0\end{array}\right]$

${\mathrm{\&Theta;}}_{{\hat{M}}_1}={\mathrm{\&beta;}}_1\left[\begin{array}{cc}{\hat{B}}_{f}C& {\hat{B}}_{f}C\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{M}}_3}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{M}}_{1,3}}={\mathrm{\&sigma;}}_3\left[\begin{array}{cc}{\hat{B}}_{f}C& {\hat{B}}_{f}C\\ 0& 0\end{array}\right]$

${\mathrm{\&Theta;}}_{{\hat{M}}_2}={\mathrm{\&beta;}}_1\left[\begin{array}{cc}-{\hat{B}}_f& 0\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{M}}_4}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{M}}_{2,4}}={\mathrm{\&sigma;}}_3\left[\begin{array}{cc}-{\hat{B}}_f& 0\\ 0& 0\end{array}\right]$

${\mathrm{\&Theta;}}_{{\hat{N}}_1}={\mathrm{\&beta;}}_1\left[\begin{array}{c}{\hat{B}}_{f}D\\ 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{N}}_2}={\mathrm{\&beta;}}_1\left[\begin{array}{c}0\\ 0\end{array}\right],{\mathrm{\&Theta;}}_{{\hat{N}}_{1,2}}={\mathrm{\&sigma;}}_3\left[\begin{array}{c}{\hat{B}}_{f}D\\ 0\end{array}\right]$

${\mathrm{\&Theta;}}_{B_1}={\mathrm{\&beta;}}_1\left[\begin{array}{c}XB+\stackrel{\&OverBar;}{\mathrm{\&theta;}}{\hat{B}}_{f}D\\ ZB\end{array}\right],{\mathrm{\&Theta;}}_{B_2}=P_2\left[\begin{array}{c}\left(\stackrel{\&OverBar;}{\mathrm{\&theta;}}+\stackrel{\&OverBar;}{\mathrm{\&xi;}}\right)D\\ \stackrel{\&OverBar;}{\mathrm{\&theta;}}D\end{array}\right]$

${\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_1}={\mathrm{\&beta;}}_1{P}_2\left[\begin{array}{cc}C& C\\ C& C\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_3}={\mathrm{\&beta;}}_1{P}_2\left[\begin{array}{cc}C& C\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_{1,3}}={\mathrm{\&sigma;}}_3{P}_2\left[\begin{array}{cc}0& 0\\ C& C\end{array}\right]$

${\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_2}={\mathrm{\&beta;}}_1{P}_2\left[\begin{array}{cc}-I& 0\\ -I& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_4}={\mathrm{\&beta;}}_2{P}_2\left[\begin{array}{cc}-I& 0\\ 0& 0\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{M}}_{2,4}}={\mathrm{\&sigma;}}_3{P}_2\left[\begin{array}{cc}0& 0\\ -I& 0\end{array}\right]$

${\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_1}={\mathrm{\&beta;}}_1{P}_2\left[\begin{array}{c}D\\ D\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_2}={\mathrm{\&beta;}}_2{P}_2\left[\begin{array}{c}D\\ 0\end{array}\right],{\mathrm{\&Theta;}}_{{\stackrel{\&OverBar;}{N}}_{1,2}}={\mathrm{\&sigma;}}_3{P}_2\left[\begin{array}{c}0\\ D\end{array}\right]$

$\begin{array}{cccccc}{A}_f=W^{-1}{\hat{A}}_f{Z}^{-1}{V}^{-T},& B_f=W^{-1}{\hat{B}}_f,& C_f={\hat{C}}_f{Z}^{-1}{V}^{-T},& H_1=W^{-1}{\hat{H}}_1,& E_1={\hat{E}}_1{Z}^{-1}{V}^{-T}& H_2=W^{-1}{\hat{H}}_2\end{array},$

$E_3={\hat{E}}_3{Z}^{-1}{V}^{-T},H_3={\hat{H}}_3,E_2={\hat{E}}_2;$

Wherein：W, V are nonsingular constant matrices, are met, WV^T=I-XZ^-1；

X∈R^n×n, Z ∈ R^n×n,P₂∈R^2r×2r, ε_i＞ 0 (i=1,2,3,4) is not Know variable, other variables are known, can draw according to systematic parameter or directly give, using Matlab LMI tool boxes Solved, if there is symmetric positive definite matrix X, Z, P₂And matrixWith scalar ε_i＞ 0 (i=1,2,3,4), then Networking filtering error system is meansquare exponential stability and has strict dissipativity, and uncatalyzed coking filter parameter matrix isγ=Σ (| | e (k) | |)/Σ (| | w (k) | |), and can be after It is continuous to carry out step 4)；If above-mentioned known variables are without solution, networking filtering error system is not meansquare exponential stability and is discontented with Sufficient strict dissipativity, it is impossible to obtain the parameter matrix of uncatalyzed coking wave filter, cannot also carry out step 4)；

4) calculate uncatalyzed coking H_∞Filter parameter matrix A_f, B_f, C_f, each matrix parameter is taken as：Q=-I, R=γ²I, S=0, H_∞Filter Under ripple, Optimal Disturbance Rejection compares γ_optThe condition of optimization is：

Make e=γ²If following optimization problem is set up：

$\begin{array}{ccc}\min e& s.t.& \left[\begin{array}{cc}{\mathrm{\&Psi;}}_1& {\mathrm{\&Psi;}}_2^T\\ {\mathrm{\&Psi;}}_2& {\mathrm{\&Psi;}}_3\end{array}\right]<0\end{array}$

X=X^T＞ 0, Z=Z^T＞ 0, X-Z ＞ 0,ε_i＞ 0 (i=1,2,3,4)

The minimal disturbances inhibiting rate of systemThe parameter matrix of uncatalyzed coking dissipation filtering device is optimised for simultaneously