CN106257873A - A kind of uncatalyzed coking H ∞ fault tolerant control method of nonlinear network networked control systems - Google Patents

Abstract

The invention discloses the uncatalyzed coking H of a kind of nonlinear network networked control systems_∞Fault tolerant control method, consider that nonlinear network networked control systems is in the case of Parameter Perturbation, time delay, packet loss and executor occur random fault, initially set up the nonlinear networked control system model of closed loop, reconstruct the Lyapunov function including packet loss information, utilize Lyapunov Theory of Stability and LMI to analyze method, obtain nonlinear network networked control systems Stochastic stable and H_∞The sufficient condition that fault-tolerant controller exists, Matlab LMI workbox is utilized to solve, providing uncatalyzed coking fault-tolerant controller gain matrix is to provide the condition that minimal disturbances suppression ratio γ can optimize, it is thus achieved that the controller gain matrix K optimized under minimal disturbances suppression ratio^*.The present invention considers executor and the situation of random fault occurs, and the probability of happening of random fault meets BerRoulli distribution, has more practical significance.

Description

A kind of uncatalyzed coking H of nonlinear network networked control systems∞Fault tolerant control method

Technical field

The present invention relates to network control system and faults-tolerant control, particularly relate to a kind of non-thread with time delay and packet loss The uncatalyzed coking H of property network control system_∞Fault tolerant control method.

Background technology

The closed-loop control system formed by communication network is referred to as network control system (networked control Systems, be abbreviated NCSs), NCSs have convenient for installation and maintenance, motility is high and is prone to the advantages such as reconstruct.But, communication network Introducing result in system and there is problems in that 1) network delay: data when communication network transmission because network blockage or The reasons such as external interference so that there is network delay problem in network control system；2) packet loss: in data transmission procedure because The problem that the reason such as network blockage and resource contention can cause data-bag lost.The most extraneous uncertain factor may result in Systematic function reduces even unstability.Therefore, make NCSs have fault-tolerant ability and to keep preferable interference free performance to have the heaviest The theory significance wanted and more practical value.

For time delay and packet loss problem present in NCSs, a lot of scholars and expert have done numerous studies.Zheng Ying etc. exist In paper " robust Fault-Tolerant Control of uncertain network networked control systems ", obtained by Time Delay Estimation Method and the online time delay method that obtains Obtain time-vary delay system, and drive controller, the method for event-driven executor by clock, use STATE FEEDBACK CONTROL strategy to grind Study carefully Robust Integrity problem；Li Wei etc. are at the paper " H of nonlinear network networked control systems based on T-S fuzzy model_∞Robust Faults-tolerant control " in, for there is the non-linear NCSs of time delay and packet loss, modeling based on T-S fuzzy model, have studied at sensor Or when there is failure of removal in executor, H_∞The method for designing of Robust Fault-tolerant Controller.In research in terms of above-mentioned faults-tolerant control only Consider fixing controller, have certain limitation.Also some changes can be produced in view of controller parameter is interfered by outside Changing, take non-fragiie control to seem particularly significant, non-fragile controller can make system keep rapidly stable.Horse is defended the country etc. at paper " the uncatalyzed coking H of networking nonlinear system_∞Control " in, for there is the time delay being less than a sampling period and meeting Ma Erke The non-linear NCSs of husband's drop probabilities, have studied uncatalyzed coking H_∞The method for designing of controller, but this research does not accounts for performing The problem of device generation failure of removal.Cao Hui is superfine in paper " the saturated uncertain NCS uncatalyzed coking robust Fault-Tolerant Control of executor ", For the saturated situation of executor, give the method for designing of uncatalyzed coking Robust Fault-tolerant Controller, but this is complete only for executor Lost efficacy or the most normal situation, can not meet the needs of actual state.

Summary of the invention

For above-mentioned problems of the prior art, the invention provides the non-of a kind of nonlinear network networked control systems Fragile H_∞Fault tolerant control method.Consider nonlinear network networked control systems Parameter Perturbation, time delay, packet loss and executor occur with Under machine failure condition, devise uncatalyzed coking feedback of status H_∞Fault-tolerant controller so that nonlinear network networked control systems is above-mentioned In the case of remain to keep stable, and can be by Disturbance Rejection at given level.

The technical solution adopted in the present invention is: the uncatalyzed coking H of a kind of nonlinear network networked control systems_∞Faults-tolerant control side Method, comprises the following steps:

1) the nonlinear networked control system model of closed loop is set up:

$\left\{\begin{array}{c}\tilde{x}(k+1)={\widetilde{\mathrm{\Φ}}}_{\mathrm{\σ}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{R}}_{\mathrm{\σ}\left(k\right)}w\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)={\tilde{C}}_{\mathrm{\σ}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{D}}_{\mathrm{\σ}\left(k\right)}w\left(k\right)\end{array}\right.,\mathrm{\σ}\left(k\right)=0,1$

Wherein, during σ (k)=0, represent that current time data are transmitted normally in a network；During σ (k)=1, represent current time number Lose according to when being transmitted by network；

Wherein,x(k)∈RⁿFor state vector, u (k) ∈ R^PFor controlling Input quantity, w (k) ∈ R^lExternal disturbance and w (k) ∈ L for finite energy₂[0, ∞), z (k) ∈ R^qFor controlling output, f (k, x (k)) meet Lipschitz condition Nonlinear Vector item, | | f (k, x (k)) | |≤| | F₁x(k)||；A∈R^n×n、B₀∈R^n×p、B₁ ∈R^n×p、C∈R^q×n、D∈R^q×l、R∈R^q×lAnd F₁∈R^n×nFor constant matrices；ΔA∈R^n×n、ΔB₀∈R^n×pWith Δ B₁∈R^n×p It is time delay and the uncertain part of systematic parameter perturbation, there is following form:

[ΔA ΔB₀ ΔB₁]=D₁F(k)[E₁ E₂ E₃]

Wherein, D₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pFor constant matrices, F (k) ∈ R^n×nFor meeting following bar The unknown uncertain matrix of part, its element Lebesgue can survey and bounded F (k)^TF(k)≤I；M=diag{m₁,m₂,…,m_n, m₁,m₂,…,m_nFor n orthogonal stochastic variable, m_i=1 is that executor is normal, m_i=0 thoroughly lost efficacy for executor, when 0 ＜ m_iDuring ＜ 1, representing executor's partial failure, executor occurs the probability of random fault to meet Bernoulli distribution, thus may be used ? Variance be The state feedback controller is taked to beK∈R^p×nFor controlling gain battle array, Δ K is for controlling gain perturbation battle array.Δ K=D₁F(k)E₄, E₄∈R^n×pFor constant matrices.

1) structure includes the Lyapunov function of packet loss information:

$V\[\tilde{x}\left(k\right),\mathrm{\σ}\left(k\right)\]={\tilde{x}}^T\left(k\right)P_i\tilde{x}\left(k\right)$

Wherein, P_i=diag{P_i11,P_i22, i=0,1, P_i11∈R^n×nAnd P_i22∈R^p×pFor unknown positive definite symmetric matrices, number Can describe by the Markov chain with two states according to transmitting procedure, its state-transition matrix is P=[p_ij], p_ij= Pr{ σ (k+1)=j | σ (k)=i},

2) utilize Lyapunov Theory of Stability and LMI to analyze method, obtain nonlinear networked control System Stochastic stable and H_∞The sufficient condition that fault-tolerant controller exists:

For following linear MATRIX INEQUALITIES:

$\left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0$

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0$

Wherein,Λ₂₂=diag{-ε₁I,-ε₃I,-ε₂I+H₁,

Λ₄₄=diag{-ε₂I,-ε₁I,-ε₁I}, Λ₅₅=-γ²I,

Λ₆₅=[R^T 0 R^T 0]^T

${\mathrm{\&Lambda;}}_{66}=diag\{-p_{00}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{00}^{-1}{P}_{022}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T,-p_{01}^{-1}{P}_{111}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{01}^{-1}{P}_{122}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T\}$

Λ₇₅=D, Λ₇₇=-I

$H_1={\mathrm{\&epsiv;}}_3\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right){\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right)}^T,H_2={\mathrm{\&epsiv;}}_3\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right){\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right)}^T$

Θ₂₂=diag{-ε₁I,-ε₄I},

Θ₄₄=diag{-ε₄I,-ε₁I,-ε₁I}, Θ₅₅=-γ²I,

Θ₆₅=[R^T 0 R^T 0]^T

${\mathrm{\&Theta;}}_{66}=diag\{-p_{10}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{10}^{-1}{P}_{022}^{-1},-p_{01}^{-1}{P}_{111}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{11}^{-1}{P}_{122}^{-1}\}$

Θ₇₅=D, Θ₇₇=-I, γ are Disturbance Rejection rate；

P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, ε_iAnd matrix Y ∈ R (i=1,2,3,4)^p×nFor known variables, its dependent variable is all known , can draw according to systematic parameter or directly give, utilize Matlab LMI workbox to solve, if there is symmetry just Set matrix P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, matrix Y and scalar ε_i＞ 0 (i=1,2,3,4), then nonlinear network networked control systems be with Machine stable and there is H_∞Performance γ, uncatalyzed coking fault-tolerant controller gain matrix isAnd step 4 can be proceeded)； If above-mentioned known variables does not solve, then nonlinear network networked control systems is not Stochastic stable and does not have H_∞Performance γ, Uncatalyzed coking fault-tolerant controller gain matrix can not be obtained, also cannot carry out step 4)；

3) condition that minimal disturbances suppression ratio γ can optimize is given:

Make e=γ²If following optimization problem is set up:

$\begin{array}{cc}\min e& s.t.\left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0\end{array}$

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0$

P₀₁₁＞ 0, P₀₂₂＞ 0, P₁₁₁＞ 0, P₁₂₂＞ 0, ε_i＞ 0 (i=1,2,3,4)

Then can obtain closed loop nonlinear network networked control systems and meet uncatalyzed coking H_∞Under the conditions of faults-tolerant control, system is Microvariations suppression ratioUncatalyzed coking fault-tolerant controller gain matrix K also can be optimised for simultaneously

Compared with prior art, the invention have the advantages that

1) the present invention is directed to the nonlinear network networked control systems with time delay and packet loss, simultaneously take account of model parameter Uncertainty, controller gain perturbations and the impact of external disturbance, establish the nonlinear networked control system model of closed loop, give Stability and the H of system when executor occurs random fault are gone out_∞The solution of faults-tolerant control.

2) present invention considers executor the situation of random fault occurs, and the probability of happening of random fault meets Bernoulli is distributed, and has more practical significance.

3) present invention optimizes minimal disturbances suppression ratio γ so that nonlinear network networked control systems has the most anti-dry Immunity energy.

Accompanying drawing explanation

Accompanying drawing 1 is the flow chart that nonlinear network networked control systems fault-tolerant controller solves；

Accompanying drawing 2 is carried out γ after device does not occur random fault and optimizes^*The nonlinear networked control of closed loop when=0.8228 System mode response diagram processed；

Accompanying drawing 3 is γ after an executor occurs random fault and optimizes^*Closed loop when=0.8210 is nonlinear networked Control system condition responsive figure；

Accompanying drawing 4 is γ after executor's complete failure and optimization^*The nonlinear networked control of closed loop when=0.8677 System mode response diagram.

Detailed description of the invention

Below in conjunction with the accompanying drawings the detailed description of the invention of the present invention is described further.

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

Step 1: set up the nonlinear networked control system model of closed loop

Network control system data transmission procedure can describe by the Markov chain with two states, its state Transfer matrix is P=[p_ij], p_ij=Pr{ σ (k+1)=j | σ (k)=i},σ (k)=0 Transmit for data normal in a network, lose when σ (k)=1 is transmitted by network for data.

When σ (k)=0, and when network inducement delay is less than a sampling period, it is contemplated that the uncertainty of systematic parameter, Then the discrete time model of controlled system is

$\left\{\begin{array}{c}x(k+1)=\hat{A}x\left(k\right)+{\hat{B}}_{0}u\left(k\right)+{\hat{B}}_{1}u(k-1)Rw\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)=Cx\left(k\right)+Dw\left(k\right)\end{array}\right.---\left(1\right)$

Wherein,x(k)∈RⁿFor state vector, u (k) ∈ R^PFor controlling Input quantity, w (k) ∈ R^lExternal disturbance and w (k) ∈ L for finite energy₂[0, ∞), z (k) ∈ R^qFor controlling output, f (k, x (k)) meet Lipschitz condition Nonlinear Vector item, | | f (k, x (k)) | |≤| | F₁x(k)||；A∈R^n×n、B₀∈R^n×p、B₁ ∈R^n×p、C∈R^q×n、D∈R^q×l、R∈R^q×lAnd F₁∈RⁿFor constant matrices；ΔA∈R^n×n、ΔB₀∈R^n×pWith Δ B₁∈R^n×pIt is Time delay and the uncertain part of systematic parameter perturbation, have a following form:

[ΔA ΔB₀ ΔB₁]=D₁F(k)[E₁ E₂ E₃]

Wherein, D₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pFor constant matrices, F (k) ∈ R^n×nFor meeting following bar The unknown uncertain matrix of part, in order to describe the random fault of the generation of executor, introduces ffault matrix M, and its form is

M=diag{m₁,m₂,…,m_n} (2)

Wherein, m₁,m₂,…,m_nFor n orthogonal stochastic variable, m_i=1 is that executor is normal, m_i=0 is executor Thoroughly lost efficacy, as 0 ＜ m_iDuring ＜ 1, represent executor's partial failure.Executor occurs the probability of random fault to meet Bernoulli is distributed, and thus can obtain Variance beIn addition Can also obtain:

Wherein,

Design point feedback controller is

$u\left(k\right)=(K+\mathrm{\&Delta;}K)x\left(k\right)=\hat{K}x\left(k\right)---\left(3\right)$

Wherein, K ∈ R^p×nFor controlling gain battle array, Δ K is for controlling gain perturbation battle array.Δ K=D₁F(k)E₄, E₄∈R^n×pFor often Matrix number.

If augmentation is vectorialWhen executor occurs random fault, uncertain closed loop is non-linear Network control system is

$\left\{\begin{array}{c}\tilde{x}(k+1)={\widetilde{\mathrm{\&Phi;}}}_0\tilde{x}\left(k\right)+{\tilde{R}}_{0}w\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)={\tilde{C}}_0\tilde{x}\left(k\right)+{\tilde{D}}_{0}w\left(k\right)\end{array}\right.---\left(4\right)$

Wherein,

When σ (k)=1, packet is lost in network transmits, and now uses the value of previous moment, i.e. u (k)=u (k- 1), then network control system model is

$\left\{\begin{array}{c}\tilde{x}(k+1)={\widetilde{\mathrm{\&Phi;}}}_1\tilde{x}\left(k\right)+{\tilde{R}}_{1}w\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)={\tilde{C}}_1\tilde{x}\left(k\right)+{\tilde{D}}_{1}w\left(k\right)\end{array}\right.---\left(5\right)$

Wherein,

Composite type (4) and formula (5), in the case of executor occurs random fault, closed loop nonlinear network networked control systems can To be described as following Markov jump system:

$\left\{\begin{array}{c}\tilde{x}(k+1)={\widetilde{\mathrm{\&Phi;}}}_{\mathrm{\&sigma;}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{R}}_{\mathrm{\&sigma;}\left(k\right)}w\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)={\tilde{C}}_{\mathrm{\&sigma;}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{D}}_{\mathrm{\&sigma;}\left(k\right)}w\left(k\right)\end{array}\right.,\mathrm{\&sigma;}\left(k\right)=0,1---\left(6\right)$

Step 2: structure includes the Lyapunov function of packet loss information

$V\&lsqb;\tilde{x}\left(k\right),\mathrm{\&sigma;}\left(k\right)\&rsqb;={\tilde{x}}^T\left(k\right)P_i\tilde{x}\left(k\right)---\left(7\right)$

Wherein, P_i=diag{P_i11,P_i22, i=0,1, P_i11∈R^n×nAnd P_i22∈R^p×pFor unknown positive definite symmetric matrices.

Due to f^T(k)f(k)≤x^T(k)F₁ ^TF₁X (k), accordingly, there exist ε₁＞ 0, makes ε₁x^T(k)F₁ ^TF₁x(k)-ε₁f^T(k)f (k)≥0。

When external disturbance w (k)=0:

$\mathrm{\&Delta;}V\&lsqb;\tilde{x}\left(k\right),\mathrm{\&sigma;}\left(k\right)\&rsqb;=E\{V\&lsqb;\tilde{x}(k+1),\mathrm{\&sigma;}(k+1)\&rsqb;|\tilde{x}\left(k\right),\mathrm{\&sigma;}\left(k\right)=i\}-V\&lsqb;\tilde{x}\left(k\right),\mathrm{\&sigma;}\left(k\right)=i\&rsqb;\&le;{\mathrm{\&xi;}}^T\left(k\right){\stackrel{\&OverBar;}{\mathrm{\&Psi;}}}_i\mathrm{\&xi;}\left(k\right)$

Wherein,

${\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_0=\left[\begin{array}{cc}\hat{A}+{\hat{B}}_0\stackrel{\&OverBar;}{M}\hat{K}& {\hat{B}}_1\stackrel{\&OverBar;}{M}\\ \hat{K}& 0\end{array}\right],{\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_1=\left[\begin{array}{cc}\hat{A}& ({\hat{B}}_0+{\hat{B}}_1)\stackrel{\&OverBar;}{M}\\ 0& I\end{array}\right]$

Step 3: utilize Lyapunov Theory of Stability and LMI to analyze method, obtain nonlinear networked Control system Stochastic stable and H_∞The sufficient condition that fault-tolerant controller exists, step is as follows:

Step 3.1: based on step 2) the Lyapunov function that constructs, utilize Lyapunov Theory of Stability and linear matrix Inequality analyzes method, first determines whether the stochastic stability of nonlinear network networked control systems, obtains nonlinear networked control The sufficient condition of system Stochastic stable.

AssumeAs i=0, according to Schur lemma:

$\left[\begin{array}{cccc}-P_0+H& *& *& *\\ 0& -{\mathrm{\&epsiv;}}_{1}I& *& *\\ {\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_0& I& -p_{00}^{-1}{P}_0^{-1}& *\\ {\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_0& I& 0& -p_{01}{P}_1^{-1}\end{array}\right]<0---\left(8\right)$

Δ A, Δ B₀, Δ B₁, the expression formula of Δ K substitutes into formula (8), and after converting according to Schur lemma, inequality is same Shi Zuocheng and the right side are taken advantage ofObtain

$\left[\begin{array}{ccccc}{\&Pi;}_{11}& *& *& *& *\\ {\&Pi;}_{21}& {\&Pi;}_{22}& *& *& *\\ 0& 0& {\&Pi;}_{33}& *& *\\ 0& 0& {\&Pi;}_{43}& {\&Pi;}_{44}& *\\ {\&Pi;}_{51}& 0& {\&Pi;}_{53}& {\&Pi;}_{45}& {\&Pi;}_{55}\end{array}\right]<0---\left(9\right)$

Wherein,

∏₄₄=diag{-ε₂I,-ε₁I,-ε₁I},

${\&Pi;}_{55}=diag\{-p_{00}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{00}^{-1}{P}_{022}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T,-p_{01}^{-1}{P}_{111}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{01}^{-1}{P}_{122}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T\},$

$H_1={\mathrm{\&epsiv;}}_3\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right){\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right)}^T,H_2={\mathrm{\&epsiv;}}_3\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right){\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right)}^T$

In like manner, as i=1, inequality can be obtained:

$\left[\begin{array}{ccccc}{\mathrm{\&Omega;}}_{11}& *& *& *& *\\ {\mathrm{\&Omega;}}_{21}& {\mathrm{\&Omega;}}_{22}& *& *& *\\ 0& 0& {\mathrm{\&Omega;}}_{33}& *& *\\ 0& 0& {\mathrm{\&Omega;}}_{43}& {\mathrm{\&Omega;}}_{44}& *\\ {\mathrm{\&Omega;}}_{51}& 0& {\mathrm{\&Omega;}}_{53}& {\mathrm{\&Omega;}}_{45}& {\mathrm{\&Omega;}}_{55}\end{array}\right]<0---\left(10\right)$

Wherein,

Ω₄₄=diag{-ε₄I,-ε₁I,-ε₁I},

${\mathrm{\&Omega;}}_{55}=diag\{-p_{10}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{10}^{-1}{P}_{022}^{-1},-p_{01}^{-1}{P}_{111}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{11}^{-1}{P}_{122}^{-1}\}$

According to Lyapunov Theory of Stability, filling of the nonlinear network networked control systems Stochastic stable shown in formula (6) Point condition is: when external disturbance w (k)=0, there is symmetric positive definite matrix P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, matrix Y ∈ R^p×nAnd scalarLMI (9) and (10) are set up.When the sufficient condition of step 3.1 is set up, then perform Step 3.2；If the sufficient condition of step 3.1 is false, then system is not Stochastic stable and H_∞Fault-tolerant controller is not deposited , it is impossible to perform step 3.2.

Step 3.2: under zero initial condition, definition:

$J_N=E\{z^T\left(k\right)z\left(k\right)-{\mathrm{\&gamma;}}^2{w}^T\left(k\right)w\left(k\right)\}<E\left\{{\&Sigma;}_{k=0}^N{\mathrm{\&eta;}}^T\left(k\right){\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}_i\mathrm{\&eta;}\left(k\right)\right\}---\left(11\right)$

Wherein,

${\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}_{i21}=\underset{j=0}{\overset{1}{\&Sigma;}}{p}_{ij}{P}_j{\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_i,={\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}_{i22}=\underset{j=0}{\overset{1}{\&Sigma;}}{p}_{ij}{P}_j-{\mathrm{\&epsiv;}}_{1}I,{\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}_{i31}={\tilde{R}}_i^T\underset{j=0}{\overset{1}{\&Sigma;}}{p}_{ij}{P}_j{\stackrel{\&OverBar;}{\mathrm{\&Phi;}}}_i+{\tilde{D}}_i^T{\tilde{C}}_i,{\widetilde{\mathrm{\&Lambda;}}}_{i32}={\tilde{R}}_i^T\underset{j=0}{\overset{1}{\&Sigma;}}{p}_{ij}{P}_j,$

${\stackrel{\&OverBar;}{\mathrm{\&Lambda;}}}_{i33}={\tilde{R}}_i^T\underset{j=0}{\overset{1}{\&Sigma;}}{p}_{ij}{P}_j{\tilde{R}}_i+{\tilde{D}}_i^T{\tilde{D}}_i-{\mathrm{\&gamma;}}^{2}I$

IfAs i=0, according to Schur lemma, can obtain with lower inequality:

$\left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0---\left(12\right)$

Wherein, Λ₁₁=∏₁₁, Λ₂₁=∏₂₁, Λ₂₂=∏₂₂, Λ₃₃=∏₃₃, Λ₄₃=∏₄₃, Λ₄₄=∏₄₄, Λ₆₆= ∏₅₅, Λ₆₁=∏₅₁, Λ₆₃=∏₅₃, Λ₆₄=∏₅₄, Λ₆₆=∏₅₅, Λ₅₅=-γ²I, Λ₆₅=[R^T 0 R^T 0]^T,Λ₇₅=D, Λ₇₇=-I

In like manner, as i=1, inequality can be obtained:

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0---\left(13\right)$

Wherein, Θ₁₁=Ω₁₁, Θ₂₁=Ω₂₁, Θ₂₂=Ω₂₂, Θ₃₃=Ω₃₃, Θ₄₃=Ω₄₃, Θ₄₄=Ω₄₄, Θ₆₁= Ω₅₁, Θ₆₃=Ω₅₃, Θ₆₄=Ω₅₄, Θ₆₆=Ω₅₅, Θ₆₅=[R^T 0 R^T 0]^T, Θ₅₅=-γ²I,Θ₇₅=D, Θ₇₇=-I.

According to Lyapunov Theory of Stability, the nonlinear network networked control systems shown in formula (6) has H_∞Fault-tolerant control The sufficient condition that device processed exists is: as N → ∞,Set up.

Matlab LMI workbox is utilized to solve, if there is symmetric positive definite matrix P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, matrix Y∈R^p×nWith scalar ε_i＞ 0 (i=1,2,3,4) so that LMI (12) and (13) are set up, then as N → ∞, can ?Then nonlinear network networked control systems (6) is Stochastic stable and has H_∞Performance γ, uncatalyzed coking fault-tolerant controller gain matrixAnd step 4 can be proceeded；If above-mentioned known variables Do not solve, then nonlinear network networked control systems is not Stochastic stable and does not have H_∞Performance γ, it is impossible to obtain uncatalyzed coking and hold Wrong controller gain matrix, also cannot carry out step 4.

Step 4: provide the condition that minimal disturbances suppression ratio γ can optimize.

Make e=γ²If following optimization problem is set up:

$\begin{array}{cc}\min e& s.t.\left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0\end{array}$

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0$

P₀₁₁＞ 0, P₀₂₂＞ 0, P₁₁₁＞ 0, P₁₂₂＞ 0, ε_i＞ 0 (i=1,2,3,4) (14)

Then can obtain closed loop system (6) and meet uncatalyzed coking H_∞Under the conditions of faults-tolerant control, the minimal disturbances suppression ratio of systemUncatalyzed coking fault-tolerant controller gain matrix K also can be optimised for simultaneously

Embodiment:

Use the uncatalyzed coking H of a kind of nonlinear network networked control systems of present invention proposition_∞Fault tolerant control method, is not having When being w (k)=0 in the case of external disturbance, nonlinear closed loop's nonlinear network networked control systems is Stochastic stable.Work as existence During external disturbance, system is also Stochastic stable and has certain capacity of resisting disturbance.Concrete methods of realizing is as follows:

Step 1: controlled device is closed loop nonlinear network networked control systems, its state-space model is formula (6), given Its systematic parameter is

$A=\left[\begin{array}{cc}0.371& 0.232\\ 0.512& 0.438\end{array}\right],B_0=\left[\begin{array}{cc}0.415& 1.125\\ 0.216& 0.106\end{array}\right],B_1=\left[\begin{array}{cc}0.212& 0.120\\ 0.175& 0.117\end{array}\right],D_1=\left[\begin{array}{cc}0.1& 0.2\\ 0.15& 0.2\end{array}\right],E_1=\left[\begin{array}{cc}0.1& 0.2\\ 0.3& 0.1\end{array}\right]$

$E_2=\left[\begin{array}{cc}0.2& 0.1\\ 0.3& 0.2\end{array}\right],E_3\left[\begin{array}{cc}0.2& 0.2\\ 0.1& 0.2\end{array}\right],E_4=\left[\begin{array}{cc}0.01& 0.02\\ 0.01& 0.02\end{array}\right],F_1=\left[\begin{array}{c}0.3\sin \left(0.01k\right)\\ 0.3\cos \left(0.01k\right)\end{array}\right]/k,R=\left[\begin{array}{c}0.5\\ 0\end{array}\right]$

C=[0.1 0.1], D=0.6

Assume that disturbing signal is

$w\left(k\right)=\left\{\begin{array}{cc}0.3& 20\&le;k\&le;25\\ 0& otherwise\end{array}\right.$

Markov chain state transition probability matrix is

$P=\left[\begin{array}{cc}0.8& 0.2\\ 0.7& 0.3\end{array}\right]$

Assume that system has 2 executors, choose expectation and the variance of 3 kinds of situation random faults.Situation 1: I.e. executor is completely normal, and random fault does not occur；Situation 2: There is random fault in i.e. one executor, another one executor is complete Normally；Situation 3:I.e. one executor's complete failure, another one executor The most normal.

Step 2: utilize Matlab LMI workbox to solve, under different random failure situations, try to achieve symmetric positive definite matrix P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, matrix Y and scalar ε_i＞ 0 (i=1,2,3,4), is shown in Table 1；Can be in the hope of non-fragiie control according to table 1 Device K and H_∞Performance indications γ, are shown in Table 2；And the result obtained is optimized obtains K^*And γ^*, concrete outcome, it is shown in Table 3.

The feasible solution of system unknown parameter under table 1 different random fault parameter

Controller parameter before system optimization and H under table 2 different random fault parameter_∞Performance indications

Controller parameter after system optimization and H under table 3 different random fault parameter_∞Performance indications

Step 3: given original state is x (0)=[1 ,-0.5]^T, utilize Matlab LMI workbox in step 2 to solve As a result, the closed loop nonlinear network networked control systems condition responsive of different random failure condition is simulated with Matlab, such as Fig. 2 extremely Shown in Fig. 4.

By Fig. 2 to Fig. 4 it can be seen that work as system rejection to disturbance performance γ^*Minimum i.e. γ^*When=0.8210, the system shape of Fig. 3 State response curve convergence rate is faster than Fig. 2 and Fig. 4；By Fig. 3 and Fig. 4 it can be seen that when executor occurs random fault, even if There is disturbance in its exterior, and under the effect of controller, system is maintained to asymptotically stability, and system has good Interference free performance.

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

Claims (1)

1. the uncatalyzed coking H of a nonlinear network networked control systems_∞Fault tolerant control method, it is characterised in that specifically include following Step:

1) the nonlinear networked control system model of closed loop is set up:

$\left\{\begin{array}{c}\tilde{x}(k+1)={\widetilde{\mathrm{\&Phi;}}}_{\mathrm{\&sigma;}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{R}}_{\mathrm{\&sigma;}\left(k\right)}w\left(k\right)+f(k,x(k\left)\right)\\ z\left(k\right)={\tilde{C}}_{\mathrm{\&sigma;}\left(k\right)}\tilde{x}\left(k\right)+{\tilde{D}}_{\mathrm{\&sigma;}\left(k\right)}w\left(k\right)\end{array}\right.,\mathrm{\&sigma;}\left(k\right)=0,1$

Wherein, during σ (k)=0, represent that current time data are transmitted normally in a network；During σ (k)=1, represent that current time data are led to Lose when crossing network transmission；

Wherein, x (k) ∈ RⁿFor state vector, u (k) ∈ R^PFor controlling input quantity, w (k) ∈ R^lExternal disturbance for finite energy And w (k) ∈ L₂[0, ∞), z (k) ∈ R^qFor controlling output, f (k, x (k) meet Lipschitz condition Nonlinear Vector item, | | f(k,x(k))||≤||F₁x(k)||；A∈Rⁿ、B₀∈R^n×p、B₁∈R^{n ×p}、C∈R^q×n、D∈R^q×l、R∈R^q×lAnd F₁∈R^n×nFor constant matrices；ΔA∈R^n×n、ΔB₀∈R^n×pWith Δ B₁∈R^n×pWhen being Prolong and the uncertain part of systematic parameter perturbation, there is following form:

[ΔA ΔB₀ ΔB₁]=D₁F(k)[E₁ E₂ E₃]

Wherein, D₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pFor constant matrices, F (k) ∈ R^n×nFor meeting following condition not Knowing uncertain matrix, its element Lebesgue can survey and bounded F (k)^TF(k)≤I；M=diag{m₁,m₂,…,m_n, m₁,m₂,…,m_n For n orthogonal stochastic variable, m_i=1 is that executor is normal, m_i=0 thoroughly lost efficacy for executor, as 0 ＜ m_iDuring ＜ 1, table Showing executor's partial failure, executor occurs the probability of random fault to meet Bernoulli distribution, Variance be

State feedback controller isK∈R^p×nFor controlling gain battle array, Δ K is for controlling gain perturbation Battle array, Δ K=D₁F(k)E₄, E₄∈R^n×pFor constant matrices；

2) structure includes the Lyapunov function of packet loss information:

$V\&lsqb;\tilde{x}\left(k\right),\mathrm{\&sigma;}\left(k\right)\&rsqb;={\tilde{x}}^T\left(k\right)P_i\tilde{x}\left(k\right)$

Wherein, P_i=diag{P_i11,P_i22, i=0,1, P_i11∈R^n×nAnd P_i22∈R^p×pFor unknown positive definite symmetric matrices, data pass Defeated process can describe by the Markov chain with two states, and its state-transition matrix is P=[p_ij], p_ij=Pr{ σ (k+1)=j | σ (k)=i},

3) utilize Lyapunov Theory of Stability and LMI to analyze method, obtain nonlinear network networked control systems Stochastic stable and H_∞The sufficient condition that fault-tolerant controller exists:

For following linear MATRIX INEQUALITIES:

$\left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0$

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0$

Wherein,Λ₂₂=diag{-ε₁I,-ε₃I,-ε₂I+H₁,Λ₄₄=diag{-ε₂I,-ε₁I,-ε₁I}, Λ₅₅=-γ²I,

Λ₆₅=[R^T 0 R^T 0]^T

${\mathrm{\&Lambda;}}_{66}=diag\{-p_{\mathrm{00}}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{\mathrm{00}}^{-1}{P}_{022}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T,-p_{\mathrm{01}}^{-1}{P}_{\mathrm{111}}^{-1}+{\mathrm{\&epsiv;}}_2{D}_1{D}_1^T+H_2,-p_{\mathrm{01}}^{-1}{P}_{\mathrm{122}}^{-1}+{\mathrm{\&epsiv;}}_3{D}_1{D}_1^T\}$

Λ₇₅=D, Λ₇₇=-I

$H_1={\mathrm{\&epsiv;}}_3\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right){\left(E_2\stackrel{\&OverBar;}{M}{D}_1\right)}^T,H_2={\mathrm{\&epsiv;}}_3\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right){\left(B_0\stackrel{\&OverBar;}{M}{D}_1\right)}^T$

Θ₂₂=diag{-ε₁I,-ε₄I},

Θ₄₄=diag{-ε₄I,-ε₁I,-ε₁I}, Θ₅₅=-γ²I,

Θ₆₅=[R^T 0 R^T 0]^T

${\mathrm{\&Theta;}}_{66}=diag\{-p_{10}^{-1}{P}_{011}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{10}^{-1}{P}_{022}^{-1},-p_{01}^{-1}{P}_{111}^{-1}+{\mathrm{\&epsiv;}}_4{D}_1{D}_1^T,-p_{11}^{-1}{P}_{1\mathrm{22}}^{-1}\}$

Θ₇₅=D, Θ₇₇=-I, γ are Disturbance Rejection rate；

P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, ε_iAnd matrix Y ∈ R (i=1,2,3,4)^p×nFor known variables, its dependent variable is all known, Can draw according to systematic parameter or directly give, utilize Matlab LMI workbox to solve, if there is symmetric positive definite Matrix P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂, matrix Y and scalar ε_i＞ 0 (i=1,2,3,4), then nonlinear network networked control systems is random Stable and there is H_∞Performance γ, uncatalyzed coking fault-tolerant controller gain matrix isAnd step 4 can be proceeded)；As The most above-mentioned known variables does not solve, then nonlinear network networked control systems is not Stochastic stable and does not have H_∞Performance γ, no Uncatalyzed coking fault-tolerant controller gain matrix can be obtained, also cannot carry out step 4)；

4) condition that minimal disturbances suppression ratio γ can optimize is given:

Make e=γ²If following optimization problem is set up:

$\begin{array}{ccc}\min e& s.t.& \left[\begin{array}{ccccccc}{\mathrm{\&Lambda;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Lambda;}}_{21}& {\mathrm{\&Lambda;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Lambda;}}_{43}& {\mathrm{\&Lambda;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Lambda;}}_{55}& *& *\\ {\mathrm{\&Lambda;}}_{61}& 0& {\mathrm{\&Lambda;}}_{63}& {\mathrm{\&Lambda;}}_{64}& {\mathrm{\&Lambda;}}_{65}& {\mathrm{\&Lambda;}}_{66}& *\\ {\mathrm{\&Lambda;}}_{71}& 0& 0& 0& {\mathrm{\&Lambda;}}_{75}& 0& {\mathrm{\&Lambda;}}_{77}\end{array}\right]<0\end{array}$

$\left[\begin{array}{ccccccc}{\mathrm{\&Theta;}}_{11}& *& *& *& *& *& *\\ {\mathrm{\&Theta;}}_{21}& {\mathrm{\&Theta;}}_{22}& *& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{33}& *& *& *& *\\ 0& 0& {\mathrm{\&Theta;}}_{43}& {\mathrm{\&Theta;}}_{44}& *& *& *\\ 0& 0& 0& 0& {\mathrm{\&Theta;}}_{55}& *& *\\ {\mathrm{\&Theta;}}_{61}& 0& {\mathrm{\&Theta;}}_{63}& {\mathrm{\&Theta;}}_{64}& {\mathrm{\&Theta;}}_{65}& {\mathrm{\&Theta;}}_{66}& *\\ {\mathrm{\&Theta;}}_{71}& 0& 0& 0& {\mathrm{\&Theta;}}_{75}& 0& {\mathrm{\&Theta;}}_{77}\end{array}\right]<0$

P₀₁₁＞ 0, P₀₂₂＞ 0, P₁₁₁＞ 0, P₁₂₂＞ 0, ε_i＞ 0 (i=1,2,3,4)

Then can obtain closed loop nonlinear network networked control systems and meet uncatalyzed coking H_∞Under the conditions of faults-tolerant control, the minimum of system is disturbed Dynamic suppression ratioUncatalyzed coking fault-tolerant controller gain matrix K also can be optimised for simultaneously