CN106257873B - A kind of uncatalyzed coking H ∞ fault tolerant control method of nonlinear network networked control systems - Google Patents

Abstract

The invention discloses a kind of uncatalyzed coking H of nonlinear network networked control systems_∞Fault tolerant control method, consider nonlinear network networked control systems when random fault occurs for Parameter Perturbation, time delay, packet loss and actuator, initially set up the nonlinear networked control system model of closed loop, reconstruct include packet loss information Lyapunov function, using Lyapunov Theory of Stability and linear matrix inequality analysis method, nonlinear network networked control systems Stochastic stable and H are obtained_∞Adequate condition existing for fault-tolerant controller is solved using the tool box Matlab LMI, is provided uncatalyzed coking fault-tolerant controller gain matrix and isThe condition that minimal disturbances inhibiting rate γ can optimize is provided, minimal disturbances inhibiting rate is obtainedThe controller gain matrix K of lower optimization^*.The present invention considers actuator there is a situation where random fault, and the probability of happening of random fault meets BerRoulli distribution, is more of practical significance.

Description

Non-fragile H of nonlinear networked control system∞Fault tolerant control method

Technical Field

The invention relates to a networked control system and fault-tolerant control, in particular to a non-fragile H of a nonlinear networked control system with time delay and packet loss_∞Provided is a fault-tolerant control method.

Background

A closed-loop control system formed through a communication network is called a Networked Control Systems (NCSs), and the NCSs have the advantages of convenience in installation and maintenance, high flexibility, easiness in reconfiguration and the like. However, the introduction of the communication network causes the following problems to the system: 1) network delay: when data is transmitted in a communication network, the network delay problem exists in a networked control system due to network blockage or external interference and other reasons; 2) packet loss: the data packet loss problem is caused by network congestion, resource competition and other reasons in the data transmission process. Meanwhile, external uncertain factors can cause the performance of the system to be reduced and even unstable. Therefore, the NCSs have fault-tolerant capability and keep better anti-interference performance, and the method has very important theoretical significance and practical value.

Aiming at the problems of time delay and packet loss in NCSs, a plurality of scholars and a plurality of scholarsExperts have conducted a great deal of research. Zhengying and so on in the thesis of 'robust fault-tolerant control of uncertain networked control system', obtain time-varying delay through a delay estimation method and an online acquisition delay method, and research the robust integrity problem through a clock driving controller and an event driving actuator method by adopting a state feedback control strategy; li Wei et al thesis "T-S fuzzy model based nonlinear networked control system H_∞In robust fault-tolerant control, a T-S fuzzy model is used for modeling aiming at nonlinear NCSs with time delay and packet loss, and H is researched when a sensor or an actuator fails_∞A design method of a robust fault-tolerant controller. In the research on the aspect of fault-tolerant control, only a fixed controller is considered, and the fault-tolerant control method has certain limitations. Considering that the controller parameters are also subject to some variation due to external interference, it is important to adopt non-fragile control, and the non-fragile controller can quickly keep the system stable. Mawei et al in the thesis "non-fragile H of networked nonlinear systems_∞In the control, aiming at the nonlinear NCSs with the time delay less than one sampling period and the packet loss probability according with Markov, the non-fragile H is researched_∞The design method of the controller, but the research does not consider the problem that the actuator fails. In the article "uncertain NCS non-fragile robust fault-tolerant control of actuator saturation", caochien et al, a design method of a non-fragile robust fault-tolerant controller is provided for the actuator saturation condition, but this only addresses the situation that the actuator is completely failed or completely normal, and the requirement of the actual condition cannot be met.

Disclosure of Invention

In view of the above problems in the prior art, the present invention provides a non-fragile H of a non-linear networked control system_∞Provided is a fault-tolerant control method. Considering the condition that a nonlinear networked control system has parameter perturbation, time delay, packet loss and random failure of an actuator, designing non-fragile state feedback H_∞And the fault-tolerant controller enables the nonlinear networked control system to still keep stable under the conditions and can restrain the disturbance at a given level.

The technical scheme adopted by the inventionThe method comprises the following steps: non-fragile H of nonlinear networked control system_∞The fault-tolerant control method comprises the following steps:

1) establishing a closed-loop nonlinear networked control system model:

when σ (k) is 0, the data at the current moment are normally transmitted in the network; when sigma (k) is 1, the data loss is shown when the data at the current moment are transmitted through the network;

wherein,x(k)∈Rⁿis a state vector, u (k) e R^PTo control the input, w (k) e R^lIs an external perturbation of finite energy and w (k) e L₂[0,∞)，z(k)∈R^qIn order to control output quantity, F (k, x (k)) meets the nonlinear vector term of the Lipschitz condition, and | | | F (k, x (k)) | | | is less than or equal to | | 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×nIs a constant matrix; Δ A ∈ R^n×n、ΔB₀∈R^n×pAnd Δ B₁∈R^n×pIs an uncertain part of time delay and system parameter perturbation and has the following form:

[ΔA ΔB₀ ΔB₁]＝D₁F(k)[E₁ E₂ E₃]

wherein D is₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pIs a constant matrix, F (k) ε R^n×nTo satisfy the followingAn unknown uncertainty matrix of elements Lebesgue measurable and bounded F (k)^TF(k)≤I；M＝diag{m₁,m₂,…,m_n}，m₁,m₂,…,m_nFor n mutually uncorrelated random variables, m_i1 is actuator normal, m_iWhen the actuator is completely failed, 0 is more than m_iWhen the frequency is less than 1, the actuator is partially failed, and the probability of the actuator having random failure meets Bernoulli distribution, so that the method can obtain Has a variance of Adopts a state feedback controller ofK∈R^p×nTo control the gain array, Δ K is a control gain perturbation array. Δ K ═ D₁F(k)E₄，E₄∈R^n×pIs a constant matrix.

1) Constructing a Lyapunov function containing packet loss information:

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, the data transfer process can be described by a Markov chain with two states, the state transition matrix of which is P ═ P_ij]，p_ij＝Pr{σ(k+1)＝j|σ(k)＝i}，

2) Obtaining the random stability sum H of the nonlinear networked control system by using the Lyapunov stability theory and a linear matrix inequality analysis method_∞Sufficient conditions for the fault tolerant controller to exist:

for the following linear matrix inequality:

wherein,Λ₂₂＝diag{-ε₁I,-ε₃I,-ε₂I+H₁}，

Λ₄₄＝diag{-ε₂I,-ε₁I,-ε₁I}，Λ₅₅＝-γ²I，

Λ₆₅＝[R^T 0 R^T 0]^T

Λ₇₅＝D，Λ₇₇＝-I

Θ₂₂＝diag{-ε₁I,-ε₄I}，

Θ₄₄＝diag{-ε₄I,-ε₁I,-ε₁I}，Θ₅₅＝-γ²I，

Θ₆₅＝[R^T 0 R^T 0]^T

Θ₇₅＝D，Θ₇₇-I, γ is the perturbation suppression ratio;

P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂，ε_i(i ═ 1,2,3,4) and the matrix Y ∈ R^p×nIs unknown variable, other variables are known, can be obtained according to system parameters or directly given, and is solved by using a Matlab LMI tool box if a symmetric positive definite matrix P exists₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂Y matrix and e scalar_i> 0(i ═ 1,2,3,4), the nonlinear networked control system is randomly stable and has H_∞Performance gamma, non-vulnerable fault-tolerant controller gain matrix ofAnd step 4) may be continued; if the unknown variable has no solution, the nonlinear networked control system is not randomly stable and has no H_∞Performance γ, no non-fragile fault-tolerant controller gain matrix can be obtained, step 4) may not be performed either;

3) given the conditions that the minimum disturbance rejection γ can be optimized:

let e be γ²If the following optimization problem holds:

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

the closed-loop nonlinear networked control system can be obtained to conform to the non-fragile H_∞Minimum disturbance rejection rate of system under fault-tolerant control conditionWhile the gain matrix K of the non-vulnerable fault-tolerant controller is also optimized

Compared with the prior art, the invention has the following advantages:

1) aiming at the nonlinear networked control system with time delay and packet loss, the invention simultaneously considers the uncertainty of model parameters, the influence of controller gain perturbation and external disturbance, establishes a closed-loop nonlinear networked control system model, and provides the stability and H of the system when the actuator has random fault_∞A solution for fault tolerant control.

2) The invention considers the condition that the actuator has random faults, and the probability of the random faults meets Bernoulli distribution, thereby having practical significance.

3) The invention optimizes the minimum disturbance rejection rate gamma, so that the nonlinear networked control system has better anti-interference performance.

Drawings

FIG. 1 is a flow chart of a solution for a fault-tolerant controller of a nonlinear networked control system;

FIG. 2 shows that the actuator does not have random failure and is optimized to be gamma^*0.8228, a closed-loop nonlinear networked control system state response diagram;

FIG. 3 shows an actuator with random failure and optimized gamma^*0.8210, a closed-loop nonlinear networked control system state response diagram;

FIG. 4 is a graph of one actuator fully failed and optimized γ^*0.8677, the state response diagram of the closed-loop nonlinear networked control system.

Detailed Description

The following further describes the embodiments of the present invention with reference to the drawings.

Referring to FIG. 1, a non-vulnerable H of a non-linear networked control system_∞The fault-tolerant control method comprises the following steps:

step 1: establishing a closed-loop nonlinear networked control system model

The data transmission process of the networked control system can be described by a Markov chain with two states, and the state transition matrix is P ═ P_ij]，p_ij＝Pr{σ(k+1)＝j|σ(k)＝i}，And sigma (k) is 0, which means that data is normally transmitted in the network, and sigma (k) is 1, which means that data is lost when the data is transmitted through the network.

When σ (k) is 0 and the network-induced delay is less than one sampling period, considering the uncertainty of the system parameters, the discrete time model of the controlled system is

Wherein,x(k)∈Rⁿis a state vector, u (k) e R^PTo control the input, w (k) e R^lIs an external perturbation of finite energy and w (k) e L₂[0,∞)，z(k)∈R^qIn order to control output quantity, F (k, x (k)) meets the nonlinear vector term of the Lipschitz condition, and | | | F (k, x (k)) | | | is less than or equal to | | 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∈RⁿIs a constant matrix; Δ A ∈ R^n×n、ΔB₀∈R^n×pAnd Δ B₁∈R^n×pIs an uncertain part of time delay and system parameter perturbation and has the following form:

[ΔA ΔB₀ ΔB₁]＝D₁F(k)[E₁ E₂ E₃]

wherein D is₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pIs a constant matrix, F (k) ε R^n×nIn order to satisfy the unknown uncertainty matrix of the following condition, in order to describe the random fault of the actuator, a fault matrix M is introduced, the form of which is

M＝diag{m₁,m₂,…,m_n} (2)

Wherein m is₁,m₂,…,m_nFor n mutually uncorrelated random variables, m_i1 is actuator normal, m_iWhen the actuator is completely failed, 0 is more than m_iIf < 1, the actuator is partially disabled. The probability of the random fault of the actuator meets the Bernoulli distribution, so that the probability of the random fault of the actuator can be obtained Has a variance ofFurthermore, it is possible to obtain:

wherein,

the design state feedback controller is

Wherein K ∈ R^p×nTo control the gain array, Δ K is a control gain perturbation array. Δ K ═ D₁F(k)E₄，E₄∈R^n×pIs a constant matrix.

Setting an augmented vectorWhen the actuator has random fault, the uncertain closed-loop nonlinear networked control system is

Wherein,

when σ (k) is 1, the data packet is lost in network transmission, and the value of the previous time, i.e. u (k) -u (k-1), is adopted, the networked control system model is that

Wherein,

by combining equation (4) and equation (5), in the event of a random failure of the actuator, the closed-loop nonlinear networked control system can be described as a markov jump system as follows:

step 2: constructing Lyapunov function containing packet loss information

Wherein, P_i＝diag{P_i11,P_i22}，i＝0,1，P_i11∈R^n×nAnd P_i22∈R^p×pA symmetric matrix is determined for the unknown positive.

Due to f^T(k)f(k)≤x^T(k)F₁ ^TF₁x (k) and thus, ε is present₁> 0, let ε₁x^T(k)F₁ ^TF₁x(k)-ε₁f^T(k)f(k)≥0。

When external disturbance w (k) is 0:

wherein,

and step 3: obtaining the random stability sum H of the nonlinear networked control system by using the Lyapunov stability theory and a linear matrix inequality analysis method_∞The method comprises the following steps of:

step 3.1: based on the Lyapunov function constructed in the step 2), firstly, judging the random stability of the nonlinear networked control system by utilizing a Lyapunov stability theory and a linear matrix inequality analysis method to obtain a sufficient condition for the random stability of the nonlinear networked control system.

Suppose thatWhen i ═ 0, according to Schur lemma:

a, B₀，ΔB₁The expression of delta K is substituted into the formula (8), and after the expression is transformed according to Schur theorem, the inequality is simultaneously multiplied by left and rightTo obtain

Wherein,

∏₄₄＝diag{-ε₂I,-ε₁I,-ε₁I}，

similarly, when i ═ 1, the inequality can be obtained:

wherein,

Ω₄₄＝diag{-ε₄I,-ε₁I,-ε₁I}，

according to the Lyapunov stability theory, the sufficient condition for the random stability of the nonlinear networked control system shown in the formula (6) is as follows: when the external disturbance w (k) is 0, a symmetric positive definite matrix P exists₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂The matrix Y is formed by R^p×nAnd scalar quantitySo that the linear matrix inequalities (9) and (10) hold. When the sufficient condition of the step 3.1 is met, the step 3.2 is executed again; if the sufficiency of step 3.1 is not met, then the system is not randomly stable and H_∞The fault tolerant controller is not present and step 3.2 cannot be performed.

Step 3.2: at zero initial conditions, define:

wherein,

if it is notWhen i is 0, according to the Schur theorem, the following inequality can be obtained:

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

Similarly, when i ═ 1, the inequality can be obtained:

wherein, theta₁₁＝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 the Lyapunov stability theory, the nonlinear networked control system shown in the formula (6) has H_∞Sufficient conditions for the fault tolerant controller to exist are: when N → ∞ is reached,this is true.

Solving by using a Matlab LMI tool box if a symmetric positive definite matrix P exists₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂The matrix Y is formed by R^p×nAnd a scalar ε_iIf the linear matrix inequalities (12) and (13) are satisfied > 0(i → 1,2,3,4), then N → ∞ can be obtainedThe nonlinear networked control system (6) is randomly stable and has H_∞Performance gamma, non-fragile fault-tolerant controller gain matrixAnd step 4 can be continued; if the unknown variable has no solution, the nonlinear networked control system is not randomly stable and has no H_∞Performance γ, no non-fragile fault-tolerant controller gain matrix can be obtained, nor can step 4 be performed.

And 4, step 4: giving the condition that the minimum disturbance rejection rate γ can be optimized.

Let e be γ²If the following optimization problem holds:

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

a closed loop system (6) is obtained that is compliant with the non-vulnerable H_∞Minimum disturbance rejection rate of system under fault-tolerant control conditionWhile the gain matrix K of the non-vulnerable fault-tolerant controller is also optimized

Example (b):

the invention provides a non-fragile H adopting a nonlinear networked control system_∞In the fault-tolerant control method, when w (k) is 0 without external disturbance, the nonlinear closed-loop nonlinear networked control system is randomly stable. When external disturbance exists, the system is also random and stable and has certain anti-interference capability. The specific implementation method comprises the following steps:

step 1: the controlled object is a closed-loop nonlinear networked control system, the state space model of the controlled object is a formula (6), and the system parameters are given

C＝[0.1 0.1]，D＝0.6

Assuming that the disturbance signal is

Markov chain state transition probability matrix of

Assuming that the system has 2 actuators, the expectation and variance of random failures for 3 cases were chosen. Case 1: namely, the actuator is completely normal and does not have random fault; case 2: that is, one actuator has random fault, and the other actuator is completely normal; case 3:i.e. one actuator is completely disabled and the other actuator is completely normal.

Step 2: solving by using a Matlab LMI tool box to obtain a symmetric positive definite matrix P under different random fault conditions₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂Y matrix and e scalar_i> 0(i ═ 1,2,3,4), see table 1; non-vulnerable controllers K and H can be found from Table 1_∞Performance index γ, see table 2; and optimizing the obtained result to obtain K^*And gamma^*The results are shown in Table 3.

TABLE 1 feasible solution of unknown parameters of system under different random fault parameters

TABLE 2 controller parameters before System optimization under different random Fault parameters and H_∞Performance index

TABLE 3 controller parameters and H after System optimization under different random Fault parameters_∞Performance index

And step 3: giving an initial state of x (0) to [1, -0.5%]^TAnd (3) simulating the state response of the closed-loop nonlinear networked control system under different random fault conditions by using the result solved by the Matlab LMI toolbox in the step (2), as shown in the figures 2 to 4.

As can be seen from fig. 2 to 4, the system has the anti-interference performance γ^*Minimum or gamma^*0.8210, the system state response curve of fig. 3 converges faster than fig. 2 and 4; as can be seen from fig. 3 and 4, when the actuator has a random fault, even if a disturbance occurs outside the system, the system can still maintain gradual stability under the action of the controller, and the system has good anti-interference performance.

The present invention is not intended to be limited to the particular embodiments shown above, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (1)

1. Non-fragile H of nonlinear networked control system_∞The fault-tolerant control method is characterized by specifically comprising the following steps of:

1) establishing a closed-loop nonlinear networked control system model:

when σ (k) is 0, the data at the current moment are normally transmitted in the network; when sigma (k) is 1, the data loss is shown when the data at the current moment are transmitted through the network;

wherein x (k) e RⁿIn the form of a state vector, the state vector,u(k)∈R^Pto control the input, w (k) e R^lIs an external perturbation of finite energy and w (k) e L₂[0,∞)，z(k)∈R^qIn order to control output quantity, F (k, x (k)) meets the nonlinear vector term of the Lipschitz condition, and | | | F (k, x (k)) | | | is less than or equal to | | 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×nIs a constant matrix; Δ A ∈ R^n×n、ΔB₀∈R^n×pAnd Δ B₁∈R^n×pIs an uncertain part of time delay and system parameter perturbation and has the following form:

[ΔA ΔB₀ ΔB₁]＝D₁F(k)[E₁ E₂ E₃]

wherein D is₁∈R^n×n、E₁∈R^n×n、E₂∈R^n×pAnd E₃∈R^n×pIs a constant matrix, F (k) ε R^n×nFor an unknown uncertainty matrix whose elements Lebesgue are measurable and bounded F (k)^TF(k)≤I；M＝diag{m₁,m₂,…,m_n}，m₁,m₂,…,m_nFor n mutually uncorrelated random variables, m_i1 is actuator normal, m_iWhen the actuator is completely failed, 0 is more than m_iWhen the frequency is less than 1, the actuator is partially failed, the probability of the actuator having random faults meets Bernoulli distribution, has a variance of

The state feedback controller isK∈R^p×nFor controlling the gain matrix, Δ K is a control gain perturbation matrix, Δ K ═ D₁F(k)E₄，E₄∈R^n×pIs a constant matrix;

2) constructing a Lyapunov function containing packet loss information:

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, the data transfer process can be described by a Markov chain with two states, the state transition matrix of which is P ═ P_ij]，p_ij＝Pr{σ(k+1)＝j|σ(k)＝i}，

3) Obtaining the random stability sum H of the nonlinear networked control system by using the Lyapunov stability theory and a linear matrix inequality analysis method_∞Sufficient conditions for the fault tolerant controller to exist:

for the following linear matrix inequality:

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

Λ₆₅＝[R^T 0 R^T 0]^T

Λ₇₅＝D，Λ₇₇＝-I

Θ₂₂＝diag{-ε₁I,-ε₄I}，

Θ₄₄＝diag{-ε₄I,-ε₁I,-ε₁I}，Θ₅₅＝-γ²I，

Θ₆₅＝[R^T 0 R^T 0]^T

Θ₇₅＝D，Θ₇₇-I, γ is the perturbation suppression ratio;

P₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂，ε_i(i ═ 1,2,3,4) and the matrix Y ∈ R^p×nIs unknown variable, other variables are known, can be obtained according to system parameters or directly given, and is solved by using a Matlab LMI tool box if a symmetric positive definite matrix P exists₀₁₁,P₀₂₂,P₁₁₁,P₁₂₂Y matrix and e scalar_i> 0(i ═ 1,2,3,4), the nonlinear networked control system is randomly stable and has H_∞Performance gamma, non-vulnerable fault-tolerant controller gain matrix ofAnd step 4) may be continued; if the unknown variable has no solution, the nonlinear networked control system is not randomly stable and has no H_∞Performance gamma, if the gain matrix of the non-fragile fault-tolerant controller cannot be obtained, step 4) cannot be carried out;

4) given the conditions that the minimum disturbance rejection γ can be optimized:

let e be γ²If the following optimization problem holds:

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

the closed-loop nonlinear networked control system can be obtained to conform to the non-fragile H_∞Minimum disturbance rejection rate of system under fault-tolerant control conditionWhile the gain matrix K of the non-vulnerable fault-tolerant controller is also optimized