Fault separation and fault-tolerant control method of non-Gaussian random distribution control system for sensor and actuator faults

Abstract

The invention provides a fault separation and fault-tolerant control method of a non-Gaussian random distribution control system for sensor and actuator faults. The method comprises the steps that a state-space equation of a discrete SDC system with the sensor and actuator faults is established; constructing an augmentation system according to the state-space equation, and constructing a fault separation observer by using the augmentation system; determining an error system according to a residual signal in the fault separation observer, and designing a fault estimation law; calculating parameters of an observer matrix and a fault estimation law, and verifying system stability; and designing a fault-tolerant controller according to an incremental system of the state-space equation of the discrete SDC system. According to the method, the sensor fault is virtually represented in the form of the actuator fault, and the time-varying fault is separated by adopting a plurality of fault isolation observers with the same number of sensors and actuators, so that the information of each fault can be obtained.

Description

Technical Field

[0001] The present invention relates to the technical field of non-Gaussian random distribution control systems, and particularly to a fault separation and fault-tolerant control method for a non-Gaussian random distribution control system with sensor and actuator faults. Background Art

[0002] As modern control systems become increasingly complex, they are affected by various external environments. Therefore, the reliability of control systems is easily reduced and even affected by certain types of faults, which may cause certain damage to the systems. Once a fault occurs in the system, the fault must be detected as early as possible, located, and the severity of the fault must be judged as accurately as possible. Fault separation and fault estimation are usually carried out separately. The residuals of each observer are obtained through a fault separation observer, and then these residuals are used to design a fault estimation law for estimation. Accurately estimating the fault magnitude is crucial for effective fault detection, separation, and subsequent fault-tolerant control. The SDC (Stochastic Distribution Control) system mainly focuses on controlling the shape of the output probability density function (PDF) of the process variation. In such a system, the relationship between the system input and the system output is considered during the modeling process, rather than the relationship between the system input and output considered in ordinary systems. When fault information is obtained through fault separation and estimation algorithms, the fault-tolerant control scheme should be able to mitigate the impact of the fault and make the output PDF track the desired PDF. The MPC (Model Predictive Control) algorithm has achieved a combination with the stochastic distribution system and obtained good fault-tolerant control effects.

[0003] The invention patent with the application number 202010025435.7 discloses an asynchronous fault-tolerant control method for a switched system with actuator and sensor faults. First, by transformation, the actuator and sensor faults are regarded as part of the state to construct an augmented system; second, a fault estimation observer is proposed for the augmented system, and at the same time, it is assumed that there is an inevitable lag between the switching signal of the observer and the original system, resulting in asynchronous switching between the original system and the observer; furthermore, to solve the asynchronous switching problem, sufficient conditions for the asymptotic stability of the error system and satisfaction of the H ∞ performance index are given; finally, based on the fault estimation information, a state feedback controller based on the observer is designed to ensure the stability of the closed-loop system. The above invention can not only accurately estimate the state and faults of the system, but also ensure that the closed-loop system is stable under actuator and sensor faults and external disturbances. However, the above patent does not consider the simultaneous presence of time-varying faults in sensors and actuators. Summary of the Invention

[0004] In the existing research, there are few studies on the fault separation and fault-tolerant control of randomly distributed control systems. Considering the technical problem that there are also few studies on the time-varying faults of sensors and actuators simultaneously, the present invention proposes a fault separation and fault-tolerant control method for a non-Gaussian randomly distributed control system with sensor and actuator faults. After obtaining the fault information, a model predictive fault-tolerant control scheme is proposed to reduce the impact of faults on the system.

[0005] To achieve the above object, the technical solution of the present invention is realized as follows: A fault separation and fault-tolerant control method for a non-Gaussian randomly distributed control system with sensor and actuator faults, the steps are as follows:

[0006] Step 1: Establish the state-space equation of a discrete SDC system with sensor and actuator faults;

[0007] Step 2: Construct an augmented system according to the state-space equation, and construct a fault separation observer by using the augmented system;

[0008] Step 3: Determine the error system according to the residual signal in the fault separation observer, and design a fault estimation law;

[0009] Step 4: Calculate the parameters of the observer matrix and the fault estimation law, and verify the system stability;

[0010] Step 5: Design a fault-tolerant controller according to the incremental system of the state-space equation of the discrete SDC system.

[0011] The state-space equation in Step 1 is:

[0012]

[0013] where, x(k) ∈ R n×1 , v(k) ∈ R q1×1 are the state vector and the control output vector respectively; u(k) ∈ R p×1 represents the control input vector; n represents the dimension of the state vector, q1 is the dimension of the control output vector, and p is the dimension of the control input vector; f a (k) = [f1(k) f2(k)... f r (k)] T is the actuator fault that occurs in the system, and f s (k) = [f r+1 (k) f r+2 (k)... f r+m (k)] T is the sensor fault that occurs in the system; r is the number of actuator faults, and m - r is the number of sensor faults; A, B, D, G a , G sis a constant system matrix of appropriate dimension; R(y) represents a set of basis functions,

[0014] The output probability density function γ(y, u(k)) of the random distribution system is approximated using linear B-splines, and the basis functions R1(y), R2(y),..., R n (y) are selected as n pre-determined basis functions of the linear B-spline in the interval y ∈ [a, b], and ω1, ω2,..., ω n are the corresponding n weights. Then the output probability density function of the random distribution system is:

[0015] where the set of basis functions the i1-th basis coefficient

[0016] The augmented system is:

[0017]

[0018] where, and represent the state vectors of the augmented system at times k + 1 and k respectively, F(k) is the augmented fault vector, is the augmented control output vector, is the augmented probability density function, represents the parameter matrix of the augmented system; a new state variable x s ∈R p×1 , and

[0019]

[0020] The state variable x at time k + 1 s (k + 1) = -A s x s (k) + A s v(k); where A s is a selected Hurwitz matrix, and I represents the identity matrix;

[0021] When performing fault separation, the faults other than the fault to be separated are regarded as disturbances, and the augmented system is expressed as:

[0022]

[0023] In the formula, g i is the i-th column of the fault parameter matrix , f i (k) is the fault vector of the i-th fault to be separated, and the fault parameter matrix Remove the $i$-th column to obtain the parameter matrix $M$ of the $i$-th fault i , and remove the fault vector $\mathbf{f}_i(k)$ of the $i$-th fault from the augmented fault vector $\mathbf{F}(k)$ i to obtain the disturbance $\mathbf{d}(k)$ i (k).

[0024] The method for constructing the fault separation observer is as follows: According to the augmented system, introduce state variables to construct a set of fault separation observers as follows:

[0025]

[0026] where and represent the state, control output vector, and estimated value of the output probability density function at time $k$ respectively; $\boldsymbol{\varepsilon}_i(k)$ i represents the residual signal of the system at the $i$-th fault, and $\mathbf{L}_i$ i represents the $i$-th gain matrix to be determined, represents the estimated value of the fault vector $\mathbf{f}_i(k)$ i (k).

[0027] The method for determining the error system based on the residual signal in the fault separation observer is as follows: Define the error The $i$-th residual signal is expressed as:

[0028]

[0029] where $\boldsymbol{\sigma}(y)$ is a weight vector predefined on the interval $[a, b]$,

[0030] The $i$-th estimated dynamic error system at time $k + 1$ is:

[0031]

[0032] where represents the fault error of the $i$-th fault vector,

[0033] The method for designing the fault estimation law is as follows: The $i$-th residual signal $\boldsymbol{\varepsilon}_i(k)$ i is affected by the $i$-th fault vector $\mathbf{f}_i(k)$. When the $i$-th sensor fails, it satisfies the fault separation logic: i (k). When the $i$-th sensor fails, it satisfies the fault separation logic:

[0034] where $\tau_i$ i is the separation threshold of the $i$-th fault separation observer;

[0035] Obtain the $i$-th fault amplitude information based on the $i$-th residual signal. The adaptive fault estimation law for the $i$-th fault is:

[0036]

[0037] Among them, represents the parameter matrix in the fault estimation law.

[0038] If there exists a positive definite symmetric matrix P i , select the Lyapunov function. When the difference between two sampling points is less than 0, the estimated dynamic error system is stable. The parameter matrices L i and λ make the following linear matrix inequality

[0039] hold, then the fault separation observer and the dynamic error system are stable; where, η3 represents an arbitrary real number, and I represents the identity matrix.

[0040] The proof method for the stability of the fault separation observer and the dynamic error system is as follows:

[0041] Select the Lyapunov function as: where, e i (k) represents the error signal of the i-th fault; represents the fault error of the i-th fault vector;

[0042] Then the Lyapunov functions V1(k + 1) and are expressed as:

[0043]

[0044] where, the intermediate variable

[0045] Therefore, the difference between two adjacent sampling points of the discrete system is

[0046]

[0047] where, the intermediate variable Intermediate variable

[0048] where,

[0049] The intermediate parameter matrix is obtained through the Schur complement lemma:

[0050]

[0051] Obviously, Φ2 = Φ1 + λI < 0, λ is a pre-specified positive constant, and ΔV1 ≤ -λ||q|| 2 +2||θ||||N i (k)||||q|| + ||Ni (k)|| 2 ; where || || is the norm of the matrix;

[0052] When , ΔV2 < 0; when a fault occurs, the observation error system is stable.

[0053] The method for designing the fault-tolerant controller is as follows:

[0054] Make the error system output track the desired output through the fault-tolerant controller, and the desired distribution is: γ g (y) = R(y)v g + L(y); where v g is the desired weight;

[0055] The state-space representation of the discrete SDC system at the previous moment is:

[0056]

[0057] The incremental system of the state space is:

[0058]

[0059] where the state vector increment Δx(k + 1) = x(k + 1) - x(k), the control input vector increment Δu(k + 1) = u(k + 1) - u(k), and the actuator fault increment Δf a (k + 1) = f a (k + 1) - f a (k);

[0060] The prediction horizon is p, the control horizon is m and p > m, and the p-step prediction control output vector and the m-step control input vector increment are respectively expressed as

[0061]

[0062] The prediction equation of the control output vector for the next p steps of the system is:

[0063] V(k + 1) = S x Δx(k) + S v v(k) + S u ΔU(k) + S a Δf a (k) + S s Δf s (k);

[0064] where Δf s (k) represents the sensor fault increment, and the coefficient matrix

[0065]

[0066] To ensure the stable operation of the discrete SDC system, the objective function is selected as:

[0067]

[0068] where Γ v,j is the weight factor of the jth predicted output error, and Γ u,j is the weight factor of the jth control input. The predicted output error weight matrix Γ v = diag(Γ v,1 , Γ v,2 ,..., Γ v,p ), and the control input weight matrix Γ u = diag(Γ u,1 , Γ u,2 ,..., Γ u,m );

[0069] Define the error matrix E(k) = V g - ((S x + S v D)Δx(k) + S as ΔF(k)), where the desired weight matrix V g = [v g v g ... v g , the coefficient matrix S as = [S a S s , and the fault matrix ΔF(k) = [f a (k) f s (k)] T . Rewrite the objective function as a matrix vector:

[0070]

[0071] The open-loop optimal solution of the system is obtained by solving

[0072]

[0073] The increment of the optimal control sequence at time k can be obtained as

[0074] Take the first element of the obtained open-loop optimal control sequence increment as the increment of the system input

[0075] Δu(k) = [I 0...0]ΔU * (k) = K mpc E(k);

[0076] Replace the system state with the estimated value and design the final predicted fault-tolerant controller:

[0077]

[0078] Among them, the state vector x(k) and the fault vector f(k) in the error matrix E(k) are respectively replaced with their estimated values and to obtain the estimated value of the error matrix Intermediate matrix Error matrix E(k) = V g -((S x + S v D)Δx(k)+ S as ΔF(k)).

[0079] The model of the closed-loop incremental system is obtained as

[0080]

[0081] If the stability of the closed-loop incremental system is based on all the eigenvalues of A - BK mpc (S x + S v D) being located in the left half-plane, then the closed-loop system is regarded as stable, where S x and S v respectively represent the coefficient matrices of the state vector and the control output vector.

[0082] The beneficial effects of the present invention: For a non-Gaussian random distribution control system with actuator and sensor faults, a fault isolation, estimation and fault-tolerant control algorithm is proposed. The sensor fault is virtually represented in the form of an actuator fault. Multiple fault isolation observers with the same number of sensors and actuators are used to separately isolate the time-varying faults, and information about each fault can be obtained, such as the location and time of each fault occurrence; then a fault estimation scheme is proposed, and the size information of the fault is obtained by using the generated residuals, and the fault can be estimated to obtain the fault amplitude information. After obtaining the fault information, a model predictive fault-tolerant control scheme is proposed to reduce the impact of the fault on the system. Finally, the feasibility of the algorithm is verified through simulation examples. The advantages of the present invention compared with the prior art are:

[0083] (1) The present invention proposes an FDI (Fault Detection and Isolation) algorithm for discrete SDC systems with actuator and sensor faults, which can identify the time, magnitude, and shape of faults, obtain fault information, and use it in subsequent fault-tolerant controller design. A set of separation observers is designed to detect and isolate faults, and an adaptive fault estimation algorithm is designed to obtain the magnitude information of actuator and sensor faults, and can accurately estimate faults based on the separation residuals of the system.

[0084] (2) The present invention adopts the MPC (Model Predictive Control) strategy to reduce the impact of faults, compensates for actuator and sensor faults according to the fault estimation information. When faults occur in sensors and actuators simultaneously, the PDF can still track the given distribution, enabling the system output PDF to track the desired PDF and achieving fault-tolerant control. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0086] Figure 1 is the flowchart of the present invention.

[0087] Figure 2 is the schematic diagram of the closed-loop control system of the reactor.

[0088] Figure 3 is the curve graph of the separation residuals when faults occur in the actuator and sensor of the system of the present invention at different times.

[0089] Figure 4 is the schematic diagram of the actuator and sensor fault information obtained by the present invention through the fault estimation rate, where (a) is the actuator and (b) is the sensor.

[0090] Figure 5 is the curve graph of the separation residuals when faults occur in the actuator and sensor of the system of the present invention at the same time.

[0091] Figure 6 is the schematic diagram of the actuator and sensor fault information obtained by the present invention through the fault estimation rate, where (a) is the actuator and (b) is the sensor.

[0092] Figure 7 is the shape graph of the desired output PDF of the present invention.

[0093] Figure 8 It is a comparison diagram of the actual trajectory and the desired trajectory of the faulty system at different times for fault tolerance control of the present invention.

[0094] Figure 9 It is a comparison diagram of the actual trajectory and the desired trajectory of the faulty system at different times without fault tolerance control of the present invention.

[0095] Figure 10 It is a comparison diagram of the actual trajectory and the desired trajectory of the faulty system at the same time for fault tolerance control of the present invention.

[0096] Figure 11 It is a comparison diagram of the actual trajectory and the desired trajectory of the faulty system at the same time without fault tolerance control of the present invention.

[0097] Figure 12 It is a comparison diagram of the initial PDF, the final PDF and the desired PDF of the present invention for fault tolerance control at different times.

[0098] Figure 13 It is a comparison diagram of the initial PDF, the final PDF and the desired PDF of the present invention for fault tolerance control at the same time. Specific embodiments

[0099] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0100] As Figure 1 shown, for a fault separation and fault tolerance control method of a non-Gaussian random distribution control system with sensor and actuator faults, the technical problem to be solved is to achieve fault separation and fault tolerance control of actuator and sensor faults. The implementation method is as follows: for a non-Gaussian random distribution control system with actuator and sensor faults, a fault separation observer, a fault estimation law and a fault tolerance controller are designed to quickly and accurately separate faults and ensure that the system output can still track the given output after faults occur. The technical solution of the present invention: the sensor fault is virtually represented in the form of an actuator fault; multiple fault isolation observers with the same number of sensors and actuators are used to estimate the faults respectively, a fault separation observer is designed, the observer gain matrix is obtained by solving LMI, and a fault estimation law is designed; then a fault estimation scheme is proposed to obtain the size information of the fault by using the residuals generated by the fault separation observer; after obtaining the fault information, a model predictive fault tolerance control scheme is proposed to reduce the impact of faults on the system. The specific implementation steps of the present invention are as follows:

[0101] Step 1: Establish the state-space equation of the discrete SDC system with sensor and actuator faults. The expression of the state-space equation is as follows:

[0102]

[0103] Where: x(k)∈R n×1 ,v(k)∈R q1×1 are the state vector and the control output vector respectively; u(k)∈R p×1 represents the control input vector; n represents the dimension of the state vector, q1 is the dimension of the control output vector, and p is the dimension of the control input vector. a (k)=[f1(k)f2(k)...f r (k)] T is the actuator failure that occurs in the system, f i (k) represents the fault of the i-th actuator. f s (k)=[f r+1 (k)f r+2 (k)...f r+m (k)] T is the sensor failure that occurs in the system. r is the number of actuator failures, and mr is the number of sensor failures. A,B,D,G a ,G s is a constant system matrix of appropriate dimension. R(y) represents the basis function set, Basis functions R1(y), R2(y), ..., R n (y) are n predetermined basis functions on the interval y∈[a,b].

[0104] Use linear B-spline to approximate the output probability density function γ(y,u(k)) of the random distribution system and select R1(y), R2(y), ..., R n (y) as the n basis functions predetermined on the interval [a,b] by linear B-spline, ω1,ω2,...,ω n are the corresponding n weights, then the output probability density function of the random distribution system can be expressed as:

[0105]

[0106] Among them, the basis function set The i-th basis function coefficient

[0107] Step 2: Construct an augmented system based on the state space equation, and use the augmented system to construct a fault separation observer.

[0108] The constructed augmented system is shown below:

[0109]

[0110] Among them, and represent the state vectors of the augmented system at times \(k + 1\) and \(k\) respectively, \(F(k)\) is the augmented fault vector, is the augmented weight vector, i.e., the control output vector, is the augmented probability density function, represents the parameter matrix of the augmented system. To facilitate the subsequent design of the system observer, a new state variable \(x\) s ∈R p×1 is introduced, and

[0111]

[0112] . This method solves the problem that there are strong constraints on parameter selection when directly designing a sensor fault diagnosis observer for an ordinary system, and some methods are not applicable to randomly distributed systems. The state variable at time \(k + 1\)

[0113] x s (k + 1)= -A s x s (k)+A s v(k)

[0114] where \(A\) s is a Hurwitz matrix to be selected. \(I\) represents the identity matrix.

[0115] When performing fault separation, faults other than the fault to be separated are regarded as disturbances. The augmented system can be expressed as:

[0116]

[0117] In the formula, \(g\) i is the \(i\)-th column of the fault parameter matrix , \(f\) i (k) is the fault vector to be separated, and the fault vector \(f\) i (k) is the \(i\)-th fault in the augmented fault vector \(F(k)\), including actuator and sensor faults. Removing the \(i\)-th column \(g\) from the fault parameter matrix i obtains the parameter matrix \(M\) i of the \(i\)-th fault. Removing the fault vector \(f\) i (k) of the \(i\)-th fault from the augmented fault vector \(F(k)\) obtains the disturbance \(d\) i (k).

[0118] According to the augmented system, a set of fault separation observers are constructed by introducing state variables as follows:

[0119] ​​​​​​​​

[0120] Among them, and respectively represent the state, output weight, and estimated value of the output PDF at time k. ε i (k) represents the residual signal of the system during the i-th fault, and L i represents the i-th gain matrix to be determined. By performing subsequent Lyapunov stability proofs, linear matrix inequalities are obtained and solved using inequalities. represents the estimated value of the fault vector f i (k), and the estimated value needs to be obtained by designing a fault estimation law subsequently.

[0121] Step 3: Determine the error system based on the residual signal in the fault isolation observer and design a fault estimation law.

[0122] Define the error The i-th residual signal can be expressed as:

[0123]

[0124] Among them, σ(y) is a weight vector predefined on the interval [a, b], and the summation symbol simplifies the solution process.

[0125] The i-th estimated dynamic error system at time k + 1 is shown as follows:

[0126]

[0127] Among them, represents the fault error of the i-th fault vector,

[0128] The i-th residual signal ε i (k) is only affected by one fault, namely the i-th fault vector f i (k). Therefore, when the i-th sensor fails, the fault isolation logic is satisfied:

[0129] ||ε i (k)|| > τ i , the fault f i (k) has occurred

[0130] ||ε i (k)|| ≤ τ i , the fault f i (k) has not occurred

[0131] Among them, τ i is the isolation threshold of the i-th fault isolation observer.

[0132] The i-th fault amplitude information is obtained from the i-th residual, and the adaptive fault estimation law for the i-th fault is as follows:

[0133]

[0134] where, denotes the parameter matrix in the fault estimation law, which is obtained by solving the linear matrix inequality through the following formula (8). The selected Lyapunov function contains formula (6).

[0135] Step 4: Calculate the observer matrix and the parameters of the fault estimation law, and verify the system stability.

[0136] If there exists a positive definite symmetric matrix P i , for Lyapunov stability analysis, select the Lyapunov function. When the difference between two sampling points is less than 0, the estimated dynamic error system is stable, and the parameter matrices L i and λ make the following linear matrix inequality hold, then the fault separation observer and the dynamic error system are stable:

[0137]

[0138] where, η3 is an arbitrary real number. Lemma: For any matrix of appropriate dimensions, for any γ > 0, the following inequality holds X T Y + Y T X ≤ γX T X + γ -1 Y T Y, and I represents the identity matrix.

[0139] Proof: Select the Lyapunov function as:

[0140]

[0141] where, e i (k) represents the error signal of the i-th fault. represents the fault error of the i-th fault vector.

[0142] Then the difference between two adjacent sampling points of the discrete system can be obtained can be expressed as:

[0143]

[0144] where, the intermediate variable Define the intermediate variable to make the calculation more intuitive.

[0145] Therefore, the difference between two adjacent sampling points of the discrete system can be obtained

[0146]

[0147] Among them, the intermediate variable Intermediate variable Among them Combined with the famous Schur complement lemma, the following equation can be obtained through the Schur complement lemma:

[0148]

[0149] Among them, Φ2, S 11 , S 12 , S 22 respectively represent intermediate parameter matrices.

[0150] Obviously, Φ2 = Φ1 + λI < 0, where λ is a pre-specified normal number. It can be obtained that

[0151] ΔV1 ≤ -λ||q|| 2 +2||θ||||N i (k)||||q|| + ||N i (k)|| 2 (14)

[0152] Among them, || || is the norm of the matrix.

[0153] Obviously, when ΔV2 < 0. Therefore, when a fault occurs, the observation error system is stable, and when the formula (8) is satisfied, the system is stable.

[0154] Step Five: Design a fault-tolerant controller according to the incremental system of the state-space equation of the discrete SDC system.

[0155] After obtaining the fault information, that is, the formula (7), using the fault separation and estimation scheme (fault estimation law), it is necessary to construct a model predictive (MPC) fault-tolerant controller to ensure that when faults occur in the actuators and sensors, the output probability density function of the discrete SDC system can track the given PDF. The system output is made to track the desired output through the fault-tolerant controller, and the desired distribution is as follows:

[0156] γ g (y) = R(y)v g +L(y) (15)

[0157] Among them, v g is the desired weight.

[0158] The state-space equation of the discrete SDC system at the previous moment can be expressed as:

[0159]

[0160] Therefore, by taking the difference between Equation (1) and Equation (16), the incremental system can be obtained as follows:

[0161]

[0162] Among them, the state vector increment Δx(k + 1) = x(k + 1) - x(k), the control input vector increment Δu(k + 1) = u(k + 1) - u(k), and the actuator fault increment Δf a (k + 1) = f a (k + 1) - f a (k).

[0163] The prediction horizon is p, the control horizon is m and p > m. The p-step prediction output weights (control output vector) and the m-step control input vector increment are respectively expressed as

[0164]

[0165] The matrix subscripts m and p do not represent the dimensions of the matrix, but the number of vectors in the matrix. To predict the control output of the system in the next p steps, the prediction equation of the output weights is obtained by combining Equation (17) and (18) as:

[0166] V(k + 1) = S x Δx(k) + S v v(k) + S u ΔU(k) + S a Δf a (k) + S s Δf s (k) (19)

[0167] Among them, Δf s (k) represents the sensor fault increment, and the coefficient matrix

[0168]

[0169] To ensure the stable operation of the discrete SDC system, the objective function is selected as:

[0170]

[0171] Among them, Γ v,j is the weight factor of the jth prediction output error, and Γ u,j is the weight factor of the jth control input. Therefore, the prediction output error weight matrix Γ v = diag(Γ v,1 , Γ v,2 ,..., Γ v,p ), and the control input weight matrix Γ u = diag(Γ u,1 , Γu,2 ,...,Γ u,m )。

[0172] Define the error matrix \(E(k)=V\) g -((S x +S v D)\(\Delta x(k)+S\) as \(\Delta F(k)\)), where the expected weight matrix \(V\) g \(=[v\) g \(v\) g ...v\) g , the coefficient matrix \(S\) as \(=[S\) a \(S\) s , the fault matrix \(\Delta F(k)=[f\) a (k)f\) s (k)] T , the objective function can be rewritten as a matrix vector:

[0173]

[0174] The open-loop optimal solution of the system can be obtained by solving

[0175]

[0176] So the increment of the optimal control sequence at time \(k\) is

[0177]

[0178] Take the first element of the obtained open-loop optimal control sequence increment as the increment of the system input

[0179] \(\Delta u(k)=[I\ 0...0]\Delta U\) * (k)=K\) mpc E(k)\ (24)

[0180] Replace the system state with the estimated value and design the final predicted fault-tolerant controller:

[0181]

[0182] Among them, replace the state vector \(x(k)\) and the fault vector \(f(k)\) in the error matrix \(E(k)\) with their estimated values and to obtain the estimated value of the error matrix

[0183] Intermediate matrix Error matrix

[0184] E(k)=V\) g -((Sx +S v D)Δx(k)+S as ΔF(k))

[0185] The model of the closed-loop incremental system can be obtained

[0186]

[0187] If the stability of the closed-loop incremental system is based on

[0188] A - BK mpc (S x +S v D)(27)

[0189] all the eigenvalues of which are located in the left half-plane, then the closed-loop system, i.e., formula (26), can be regarded as stable. Therefore, after the predictive controller design is completed, it is necessary to verify whether all the eigenvalues of the matrix A - BK mpc (S x +S v D) are in the left half-plane to ensure the stability of the designed control system.

[0190] Example:

[0191] A fault isolation and fault-tolerant control method for a non-Gaussian random distribution control system with sensor and actuator faults, the steps are as follows: Step 1: Use the shape of the molecular weight distribution PDF in the continuous stirred tank reactor (CSTR) dynamic process to verify the feasibility of fault isolation and fault-tolerant control for a non-Gaussian random distribution control system with multiple sensors and actuators. Figure 2 It is a schematic diagram of the closed-loop control system of the reactor, where F m 、F i respectively represent the inputs of the monomer and the initiator. The system parameter matrix is shown as follows:

[0192]

[0193] The output PDF of the discrete SDC system is approximated using linear B-spline basis functions as follows:

[0194] φ1 = 0.5(y - 2) 2 M1 + (-y 2 + 7y - 11.5)M2 + 0.5(y - 5) 2 M3

[0195] φ2 = 0.5(y - 3) 2 M2 + (-y 2 + 9y - 19.5)M3 + 0.5(y - 6) 2 M4

[0196] φ3 = 0.5(y - 4)2 M3 + (-y 2 + 11y - 29.5)M4 + 0.5(y - 7) 2 M5

[0197] where M1, M2, M3, M4, and M5 respectively represent

[0198] Step 2: Use MATLAB to simulate the fault separation and fault-tolerant control results of the equivalent system, such as Figures 3 - 11 .

[0199] From Figure 3 it can be seen that according to the separation residual, i.e., formula (6), the fault occurs at k = 300, Figure 3 which shows the FDI results of actuator and sensor faults at different times. According to the separation residual, the actuator faults f1(k) and f2(k) occur at k = 300 and k = 500, and the sensor faults f3(k) and f4(k) occur at k = 300 and k = 500. The estimated magnitudes of the actuator and sensor faults are as Figure 4 shown, and through Figure 4 it can be seen the time-varying fault information of the actuator and sensor that do not occur simultaneously, such as the occurrence time and fault amplitude. Figure 5 which shows the FDI results of simultaneous actuator and sensor faults. According to the separation residual, the actuator faults f1(k) and f2(k) and the sensor faults f3(k) and f4(k) occur at k = 300. The estimated values of the actuator faults and sensor faults in formula (6) are as Figure 6 shown, and through Figure 6 the time-varying fault information of the simultaneous actuator and sensor.

[0200] Figure 7 is the desired system output PDF shape, Figure 8 which is that when faults occur at different times, under the action of the designed fault-tolerant controller, it can ensure that the actual output can track the desired output, and the system output PDF has the same shape as the desired PDF under the action of the designed fault-tolerant controller. Figure 9 is the output of the system without fault-tolerant control. It can be seen that the actual output of the system is unstable. Figure 10 is that when faults occur at the same time, under the action of the designed fault-tolerant controller, it can ensure that the actual output can track the desired output. Figure 11 is the output of the system without fault-tolerant control. It can be seen that the actual output of the system is unstable. Through the above comparison, it can be seen that the fault-tolerant controller can perform fault-tolerant control on the faulty system, indicating that after fault separation and fault-tolerant control when a fault occurs, the system output fluctuates less, can effectively contain the impact of the fault, and ensure the stable operation of the system.

[0201] Figure 12 and Figure 13 are the comparisons of the initial PDF, the final PDF and the expected PDF of the system in two cases. It can be seen from the two figures that after the fault tolerance control is carried out, the final PDF of the system can track the expected PDF.

[0202] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principle of the present invention shall be included within the protection scope of the present invention.

Claims

1. A fault separation and fault-tolerant control method for a non-Gaussian random distribution control system with sensor and actuator faults, characterized in that, The steps are as follows: Step 1: Establish the state - space equation of a discrete SDC system with sensor and actuator faults; Step 2: Construct an augmented system according to the state - space equation, and construct a fault - separation observer using the augmented system; Step 3: Determine the error system according to the residual signal in the fault - separation observer, and design a fault - estimation law; Step 4: Calculate the parameters of the observer matrix and the fault - estimation law, and verify the system stability; Step 5: Design a fault - tolerant controller according to the incremental system of the state - space equation of the discrete SDC system.

2. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 1, characterized in that, The state - space equation of Step 1 is: where \(x(k)\in\mathbb{R}\) n×1 , \(v(k)\in\mathbb{R}\) q1×1 are the state vector and the control output vector respectively; \(u(k)\in\mathbb{R}\) p×1 represents the control input vector; \(n\) denotes the dimension of the state vector, \(q_1\) is the dimension of the control output vector, and \(p\) is the dimension of the control input vector; \(f\) a (k)=[f1(k) f2(k) \(\cdots\) f r (k)] T is the actuator fault that occurs in the system, and \(f\) s (k)=[f r+1 (k) f r+2 (k) \(\cdots\) f r+m (k)] T is the sensor fault that occurs in the system; \(r\) is the number of actuator faults, and \(m - r\) is the number of sensor faults; \(A\), \(B\), \(D\), \(G\) a , \(G\) s are constant system matrices of appropriate dimensions; \(R(y)\) represents the set of basis functions, The output probability density function γ(y, u(k)) of a random distribution system is approximated using linear B-splines. The basis functions R1(y), R2(y),..., R n (y) are taken as n pre-determined basis functions of the linear B-spline over the interval y ∈ [a, b], and ω1, ω2,..., ω n are the corresponding n weights. Then the output probability density function of the random distribution system is: Among them, the set of basis functions The i1-th basis coefficient 3. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensors and actuators according to claim 2, characterized in that, The augmented system is: Among them, and represent the state vectors of the augmented system at the (k + 1)-th and k-th moments respectively, F(k) is the augmented fault vector, is the augmented control output vector, is the augmented probability density function, represents the parameter matrix of the augmented system; a new state variable x s ∈R p×1 is introduced, and The state variable x at time k+1 s (k+1) = -A s x s (k) + A s v(k); where A s is a selected Hurwitz matrix, and I represents the identity matrix; When performing fault separation, faults other than the fault to be separated are regarded as disturbances, and the augmented system is expressed as: where g i is the i-th column of the fault parameter matrix , f i (k) is the fault vector of the i-th fault to be separated, and the fault parameter matrix obtains the parameter matrix M of the i-th fault by removing the i-th column i . By removing the fault vector f i (k) of the i-th fault from the augmented fault vector F(k), the disturbance d i (k) is obtained.

4. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 3, characterized in that, The method for constructing the fault - separation observer is: According to the augmented system, introduce state variables to construct a set of fault - separation observers as: Among them, and represent the state, control output vector, and estimated value of the output probability density function at time k, respectively; ε i (k) represents the residual signal of the system at the i-th fault, and L i represents the i-th gain matrix to be determined, represents the fault vector f i (k).

5. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 4, characterized in that, The method for determining the error system based on the residual signal in the fault isolation observer is as follows: Define the error The i-th residual signal is expressed as: wherein, σ(y) is a weight vector predefined on the interval [a, b] The estimated dynamic error system of the i - th at time k + 1 is: Among them, represents the fault error of the i-th fault vector, 6. The fault separation and fault tolerance control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 5, characterized in that, The method for designing the fault estimation law is as follows: The i-th residual signal ε i (k) is affected by the i-th fault vector f i (k). When the i-th sensor fails, the fault isolation logic is satisfied: where τ i is the separation threshold of the i-th fault separation observer; Obtain the i - th fault amplitude information according to the i - th residual signal, and the adaptive fault - estimation law of the i - th fault is: Among them, represents the parameter matrix in the fault estimation law.

7. The fault separation and fault tolerance control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 5, characterized in that, If there exists a positive definite symmetric matrix P i , select the Lyapunov function. When the difference between two sampling points is less than 0, it is estimated that the dynamic error system is stable. The parameter matrices L i and λ make the following linear matrix inequality hold, then the fault separation observer and the dynamic error system are stable; where, η3 represents an arbitrary real number, and I represents the identity matrix.

8. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 7, characterized in that, The proof method for the stability of the fault - separation observer and the dynamic error system is: Choose the Lyapunov function as follows: where e i (k) represents the error signal of the i-th fault; represents the fault error of the i-th fault vector; Then the Lyapunov function V1(k + 1) and is expressed as: wherein, intermediate variable Therefore, the difference between two adjacent sampling points of the discrete system is Among them, intermediate variable Intermediate variable Where, Obtain the intermediate parameter matrix through the Schur complement lemma: Obviously, Φ2 = Φ1 + λI < 0, where λ is a pre-specified positive number, and we get ΔV1 ≤ -λ||q|| 2 + 2||θ||||N i (k)||||q|| + ||N i (k)|| 2 ; where || || is the norm of a matrix; When ΔV2 < 0; when a fault occurs, the observation error system is stable.

9. The fault separation and fault-tolerant control method for a non-Gaussian random distribution control system with sensor and actuator faults according to any one of claims 2-8, characterized in that, The method for designing the fault - tolerant controller is: The error system output is made to track the desired output through a fault-tolerant controller, and the desired distribution is: γ g (y) = R(y)v g + L(y); where, v g is the desired weight; The state - space representation of the discrete SDC system at the previous moment is: The incremental system of the state - space is: where the state vector increment Δx(k + 1) = x(k + 1) - x(k), the control input vector increment Δu(k + 1) = u(k + 1) - u(k), and the actuator fault increment Δf a (k + 1) = f a (k + 1) - f a (k); The prediction horizon is p, the control horizon is m and p>m. The p - step prediction control output vector and the m - step control input vector increment are respectively expressed as The prediction equation of the control output vector for the next p steps of the system is: V(k + 1) = S x Δx(k) + S v v(k) + S u ΔU(k) + S a Δf a (k) + S s Δf s (k); where, Δf s (k) represents the sensor fault increment, and the coefficient matrix To ensure the stable operation of the discrete SDC system, select the objective function as: where, Γ v,j is the weight factor of the j-th predicted output error, and Γ u,j is the weight factor of the j-th control input. The predicted output error weight matrix Γ v = diag(Γ v,1 , Γ v,2 ,..., Γ v,p ), and the control input weight matrix Γ u = diag(Γ u,1 , Γ u,2 ,..., Γ u,m ); Define the error matrix \(E(k)=V\) g -((S x +S v D)\(\Delta x(k)+S as \Delta F(k))\), where the expected weight matrix \(V g = [v g v g ...v g , the coefficient matrix \(S as = [S a S s , the fault matrix \(\Delta F(k)=[f a (k)f s (k)] T , rewrite the objective function as a matrix vector: The open-loop optimal solution of the system is obtained by solving The increment of the optimal control sequence at time k can be obtained as Take the first element of the obtained open - loop optimal control sequence increment as the increment of the system input Δu(k) = [I 0...0]ΔU * (k) = K mpc E(k); Replace the system state with the estimated value and design the final predicted fault - tolerant controller: Among them, the state vector x(k) and the fault vector f(k) in the error matrix E(k) are respectively replaced with their estimated values and to obtain the estimated value of the error matrix Intermediate matrix Error matrix E(k) = V g -((S x + S v D)Δx(k)+ S as ΔF(k)).

10. The fault separation and fault-tolerant control method for the non-Gaussian random distribution control system of sensor and actuator faults according to claim 9, characterized in that, Obtain the model of the closed - loop incremental system as If the stability of the closed-loop incremental system is based on A - BK mpc (S x + S v D), and all eigenvalues of (S x and S v represent the coefficient matrices of the state vector and the control output vector respectively, then the closed-loop system is considered stable.

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