# Fuzzy Filter-Based FDD Design for Non-Gaussian Stochastic Distribution Processes Using T-S Fuzzy Modeling

Authors: Yang Yi; Yue-Yue Zhao; Song-Yin Cao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper studies the fuzzy modeling problem and the fault detection and diagnosis (FDD) algorithm for non-Gaussian stochastic distribution systems based on the nonlinear fuzzy filter design. Following spline function approximation for output probability density functions (PDFs), the T-S fuzzy model is built as a nonlinear identifier to describe the dynamic relationship between the control input and the weight vector. By combining the designed filter and the threshold value, the fault in T-S weight model can be detected and the stability of error system can also be guaranteed. Moreover, the novel adaptive fuzzy filter based on stochastic distribution function is designed to estimate the size of system fault. Finally, the simulation results can well verify the effectiveness of the proposed algorithm for
the constant fault and the time-varying fault, respectively.


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## Body

## 1. Introduction

In order to improve the stability and the security of system, the fault detection and diagnosis (FDD) algorithm for the complex systems has been an important part in the field of control engineering. Many significant approaches have also been presented and applied to practical processes successfully (see [1–4] and references therein). On the other hand, a series of modeling and control strategies which control the shape of output probability density functions (PDFs) for non-Gaussian stochastic processes have received considerable attention (see [5–8]). However, most of the existing FDD results for stochastic systems were only concerned with Gaussian variables, where mean or variance was the objective for optimization and control [9]. Since 2000, the FDD algorithm for non-Gaussian stochastic distribution systems has begun to be discussed based on filter theory [10]. In these results, the modeling problem is often ignored and only linear model or common nonlinear model is considered in FDD (see [10–12]).In recent years, the well-known T-S fuzzy model was viewed as a popular and powerful modeling tool since it is a powerful solution that bridges the gap between linear and nonlinear control systems (see [13–15]). By introducing a family of fuzzy IF-THEN rules that represent local linear input-output relations of the system, many complex nonlinear models, such as descriptor systems [16], stochastic systems [17], and time-delay model [14], can be described or approximated by T-S fuzzy modeling. Meanwhile, some typical control problems, including dynamic tracking control [14], sliding-mode control [18], and filter design [19, 20], have also been considered through T-S fuzzy modeling.In this paper, we provide a novel FDD approach for non-Gaussian stochastic distribution systems. Based on B-spline approximation and T-S fuzzy modeling, the concerned FDD problem of the dynamic non-Gaussian systems can be transferred into a special nonlinear FDD problem for deterministic T-S weight dynamics. Instead of common nonlinear observer or filter in [10–12], the nonlinear fuzzy filter and the adaptive filter are conducted by involving the measured output PDFs with the T-S fuzzy weight dynamics. Moreover, by optimizing a series of linear matrix inequalities (LMIs), the fault existing in the stochastic processes can be detected with a defined threshold and the satisfactory estimation value for the size of fault can also be guaranteed. It is noted that the T-S fuzzy model is first applied into the FDD for stochastic distribution systems which solves the nonlinear modeling difficulty in previous results [10–12]. This represents a significant extension of the previous results that only common linear/nonlinear weight dynamic models are considered and has also an independent significance in the field of fuzzy FDD.In this paper, for a square matrixM, we denote that sym(M)=M+MT. The identity and zero matrices are expressed by I and 0, respectively. For a symmetric matrix M, the notationM>(≥)0is used to denote that M is positive definite (positive semidefinite). The case forM<(≤)0follows similarly. Moreover, for a vector v(t), define Euclidean norm by ∥v(t)∥=vT(t)v(t).


## 2. Problem Formulation and T-S Fuzzy Modeling

For a complex non-Gaussian stochastic process, we denote thatu(t)∈Rm is the control input, y(t)∈[a,b] is the stochastic output of the concerned system, and F represents the fault to be detected and is supposed as a constant vector. The conditional probability of output y(t) lying inside [a,β] can be defined as follows:
(1)P{a≤y(t)<β}=∫aβ‍γ(z,u(t),F)dz,
where γ(z,u(t),F) stands for the output PDF with fault term under the effect of the control input u(t).Similarly, with [7, 10, 11], it is supposed that the output PDF γ(z,u(t),F), as the control objective, can be measured or estimated. For the PDF γ(z,u(t),F), the square root B-spline expansion is given as
(2)γ(z,u(t),F)=∑i=1n‍vi(u(t),F)bi(z)+ω(z,u(t),F),
where bi(z) are the prespecified basis functions and vi(u(t),F) are the corresponding weight values with fault. ω(z,u(t),F) represents the model uncertainty or the error term of the approximation of PDFs, which is assumed to satisfy the inequality ∥ω(z,u(t),F)∥≤ξ, where ξ is a known positive constant.Due to the PDF constraint condition∫ab‍γ(z,u(t),F)dz=1, only n-1 weights are independent. Thus, the output PDF can be further expressed as
(3)γ(z,u(t),F)=B(z)V(u(t),F)+h(V(u(t),F))bn(z)+ω(z,u(t),F),
where V(u(t),F)=[v1(u(t),F),…,vn-1(u(t),F)]T, B(z)=[b1(z),…,bn-1(z)]. Similarly, with [7], the nonlinear term h(V(t)) satisfies the following equality:
(4)h(V(t))=Λ3-VT(t)Λ0V(t)-Λ2V(t)Λ3,
where Λ0=Λ1-Λ3-1Λ2TΛ2 and
(5)Λ1=∫ab‍BT(z)B(z)dz,Λ2=∫ab‍B(z)bn(z)dz,Λ3=∫ab‍bn2(z)dz≠0.It is obvious that the nonlinear termh(V(t)) satisfies the Lipschitz condition within its operation region; that is, for any V1(t) and V2(t), there exists a known matrix U1 such that
(6)∥h(V1(t))-h(V2(t))∥≤∥U1(V1(t)-V2(t))∥.In the following, we will find the dynamic relationship between the control inputu(t) and the weight vectors V(u(t),F), which corresponds to a further modeling procedure. It is well known that the T-S fuzzy model is a powerful solution for identifying complex nonlinear dynamics by a blending of some local linear system models. Compared with those results that only consider common linear/nonlinear weight dynamic model (see [10–12]), we will use the T-S model to describe the nonlinear weight dynamics and the ith rule of the T-S fuzzy model is of the following form.Plant Rule
i
. If θ1 is μi1,θ2 is μi2,θp is μip, then
(7)x˙(t)=Aix(t)+Gix(t-τ(t))+Hiu(t)+JiF,V(t)=Eix(t),
where V(t)≔V(u(t),F)∈Rn-1 is the independent weight vector. x(t)∈Rm is the unmeasured state and x(t-τ(t)) represents the time-delay state. The time-varying delay τ(t) is assumed to satisfy 0<τ˙(t)<β<1, where β is a known constant. Ai, Gi, Hi, Ji, and Ei are known constant real matrices of appropriate dimension. θj and uij(i=1,…,r,j=1,…,p) are, respectively, the premise variables and the fuzzy sets, r is the number of the If-Then rules, and p is the number of the premise variables. By fuzzy blending, the overall fuzzy model can be defined as follows:
(8)x˙(t)=∑i=1r‍hi(θ)(Aix(t)+Gix(t-τ(t))+Hiu(t)+JiF),V(t)=∑i=1r‍hi(θ)Eix(t),
where
(9)θ=[θ1,⋯,θp],hi(θ)=σi(θ)∑i=1rσi(θ),σi(θ)=∏j=1p‍μij(θj).
Moreover, we have
(10)σi(θ)≥0,i=1,…,r,∑i=1r‍σi(θ)>0
for any θ. Hence, hi(θ) satisfies
(11)hi(θ)≥0,i=1,…,r,∑i=1r‍hi(θ)=1.


## 3. Fault Detection for T-S Fuzzy Weight Model

To detect the fault existing in the output stochastic distribution, we construct the following fuzzy filter:(12)x^˙(t)=∑i=1r‍hi(θ)(Aix^(t)+Gix^τ(t)+Hiu(t)+Lε(t)),
where x^(t) is the estimated state, L is the gain to be determined later, and ε(t) represents the residual signal and is defined as follows:
(13)ε(t)=∫ab‍σ(z)(γ(z,u(t),F)-γ^(z,u(t)))dz,γ^(z,u(t))=∑i=1r‍hi(θ)B(z)Eix^(t)+h(∑i=1r‍hi(θ)Eix^(t))bn(z),
where σ(z) can be regarded as the prespecified vector defined on [a,b]. By defining e(t)=x(t)-x^(t), E-=∑i=1r‍hi(θ)Ei, the estimated error system can be expressed as
(14)e˙(t)=∑i=1r‍hi(θ)[(Ai-LΓ1)e(t)+Gieτ(t)-LΓ2(h(E-x(t))-h(E-x^(t)))+JiF-LΔ(t)],
where
(15)Γ1=∫ab‍σ(z)B(z)E-dz,Γ2=∫ab‍σ(z)bndz,Δ(t)=∫ab‍σ(z)ω(z,u(t),F)dz.It can be seen that∥Δ(t)∥≤ξ~, where ξ~=ξ∥∫ab‍σ(z)dz∥ is a known constant. Meanwhile, the residual signal can be expressed as
(16)ε(t)=∫ab‍σ(z)B(z)E-e(t)dz+∫ab‍σ(z)[h(E-x(t))-h(E-x^(t))]bn(z)dz+∫ab‍σ(z)ω(z,u(t),F)dz=Γ1e(t)+Γ2[h(E-x(t))-h(E-x^(t))]+Δ(t).In the fault detection phase, our objective is to find the gainL such that the estimated error system (14) is stable if F=0, which can be formulated in the following theorem.Theorem 1.
For the known parametersλ>0, η>0, suppose that there exist matrices P>0, Q>0, R, such that the following LMIs
(17)[Πi+ηIPGiRΓ2GiTP-(1-β)Q+ηI0Γ2TRT0-λ-2I]<0,i=1,…,r
are solvable, where
(18)Πi=sym(PAi-RΓ1)+Q+λ-2E-TUTUE-;
then the error system (14) is stable when F=0, and, for any t∈[-τ(t),+∞), the error variable e(t) satisfies
(19)∥e(t)∥≤α0=max{∥em∥,2η-1ξ~∥R∥},
where the gain L can be computed by L=P-1R.Proof.
Denote the Lyapunov function candidate as(20)Φ(e,x,x^,t)=eT(t)Pe(t)+∫t-τ(t)t‍eT(τ)(t)Qe(τ)dτ+1λ2∫0t‍[∥UE-e(τ)∥2-∥h(E-x(τ))-h(E-x(τ))∥2]dτ,
when F=0, and, based on (14), we can get
(21)Φ˙≤∑i=1n‍hi(θ){eT(t)(sym(PAi-RΓ1)+Q)e(t)}+2∑i=1r‍hi(θ)eTPGieτ(t)-(1-β)eτT(t)Qeτ(t)+λ-2eT(t)E-TUTUE-e(t)-2eT(t)PLΔ(t)+λ2eT(t)PLΓ2Γ2TLTPe(t)=∑i=1r‍hi(θ)ξT(t)Ωiξ(t)-2eT(t)PLΔ(t),
where ξ(t)=[eT(t),eτT(t)]T,
(22)Ωi=[ΞiPGiGiTP-(1-β)Q],Ξi=sym(PAi-RΓ1)+Q+λ-2E-TUTUE-+λ2PLΓ2Γ2TLTP.
By definingL=P-1R and using Schur complement formula with respect to (17), we have
(23)Φ˙≤-η∥e(t)∥2+2δ~∥e(t)∥∥R∥=-η∥e(t)∥(∥e(t)∥-2η-1∥R∥δ~).
Thus, when ∥e(t)∥>2η-1∥R∥δ~, Φ˙<0 can be guaranteed. So we can get that the system (14) is stable and the estimated error satisfies
(24)∥e(t)∥≤max{∥em∥,2η-1ξ~∥R∥},
where ∥em∥=max-τ(t)<t<0∥e(t)∥.
It is noted that Theorem1 gives a necessary condition for fault detection. Based on the conclusion of Theorem 1, when the fault F=0, the residual error signal satisfies the following inequality:
(25)∥ε(t)∥≤∥Γ1∥∥e(t)∥+∥Γ2∥∥U∥∥e(t)∥+∥Δ(t)∥≤α0(∥Γ1∥+∥Γ2∥∥U∥)+δ~.
By defining the thresholdα=α0(∥Γ1∥+∥Γ2∥∥U∥)+δ~, we can conclude that the system has no fault, when the residual error signal is less than or equal to the threshold. If the residual signal is greater than the threshold, the system causes the existence of fault.


## 4. Fault Diagnosis for T-S Fuzzy Weight Model

Based on the results regarding fault detection, this part will estimate the size of the fault if the system has fault. We construct the following adaptive fuzzy filter:(26)x^˙(t)=∑i=1r‍hi(θ)(Aix^(t)+Gix^τ(t)+Hiu(t)+Lε(t)+JiF^(t)),F^˙=-C1F^(t)+C2ε(t),ε(t)=∫ab‍(γ(z,u(t),F)-γ^(z,u(t)))dz,γ^(z,u(t))=B(z)E-x^(t)+h(E-x^(t))bn(z),
where F^ is the estimate value of the fault F,Ci>0,(i=1,2) are the designed learning rate. Denoting e(t)=x(t)-x^(t), F~(t)=F-F^(t), the estimated error system can be expressed as follows:
(27)e˙(t)=∑i=1r‍hi(θ)[(h(E-x(t))-h(E-x^(t)))(Ai-LΓ1)e(t)+Gieτ(t)-LΓ2(h(E-x(t))-h(E-x^(t)))+JiF~-LΔ(t)],F~˙=-C1F~+C1F-C2ε(t).Theorem 2.
Suppose that∥F∥≤M/2, ∥F^∥≤M/2; then one has ∥F~∥≤M. For the known parameter λ>0 and matrices Ci, (i=1,2), there exist matrices P>0, Q>0, R and constants k1>0, θj>0, (j=1,2,3) satisfying
(28)[Π1iΠ2iPGiE-TUTΠ2iT-2C1T+Θ1TΘ100GiTP0-(1-β)Q0UE-00-θ3I]<0,
where
(29)Π1i=Πi+kI+Θ2TΘ2Π2i=PJi-Γ1TC2TΘ1=[θ2C2θ3C2Γ2],Θ2=[λRΓ2θ1R];
then the error system (27) is stable in the presence of F. The estimation error satisfies
(30)∥e(t)∥2≤max{∥em∥2,k-1((θ1-2+θ2-2)δ~2)+∥C1∥M2}
for all t∈[-τ,∞). The gain L of diagnosis filter (26) can be computed by L=P-1R.Proof.
Define the Lyapunov function as follows:(31)Υ(e,x,x^,F~,t)=Φ(e,x,x^,t)+F~T(t)F~(t),
where Φ(e,x,x^,t) is defined in (20), and it can be concluded that
(32)Φ˙≤∑i=1r‍hi(θ)ξiT(t)Ω-iξi(t)+θ1-2ΔT(t)Δ(t)+2∑i=1r‍hi(θ)eT(t)PJiF~(t),
where
(33)Ω-i=[Ξi+θ12RRTPGiGiTP-(1-β)Q].
Furthermore, we can get that(34)Υ˙≤∑i=1r‍hi(θ)ξiT(t)Ωiξi(t)+θ1-2ΔT(t)Δ(t)+2∑i=1r‍hi(θ)eT(t)PJiF~(t)+2F~T(t)F~˙(t)≤∑i=1r‍hi(θ)ξ-T(t)Ψiξ-(t)+(θ1-2+θ2-2)δ~2+2FT(t)C1TF~(t),
where
(35)ξ-(t)=[eT(t),edT(t),F~T(t)]T,ϖ=-2C1T+θ22C2C2T+θ32C2Γ2Γ2TC2T,Ψi=[Ξi+θ12RRT+θ3-2E-TUTUE-PGiΠ13GiTP(β-1)Q0Π13T0ϖ].
By using Schur complement, the inequality (28) is equivalent to Ψi<diag{-kI,0,0}. Then we can get that
(36)Υ˙≤-k∥e(t)∥2+(θ1-2+θ2-2)δ~2+2F~T(t)C1F^(t)≤-k∥e(t)∥2+(θ1-2+θ2-2)δ~2+∥C1∥M2.
SoΥ˙<0, if k∥e(t)∥2≥(θ1-2+θ2-2)δ~2+∥C1∥M2 holds. Similarly, with Theorem 1, the estimated error ∥e(t)∥ satisfies the inequality (30). It can be validated that system (27) is stable in the presence of F, which also implies the good fault diagnosis performance.


## 5. An Illustrative Example

Suppose that the output PDFs can be approximated using the following B-spline model:(37)bi={|sin2πz|,z∈[0.5(i-1),0.5i]0,z∈[0.5(j-1),0.5j]i≠j,
where i=1,2,3, z∈[0,1.5].In the simulation, it is supposed thatδ~=∥Λ1∥-1=5. The fault F is defined as
(38)F(t)={0,t≤51,t>5.
For the T-S fuzzy weight dynamics, the rule i=2 and the model parameters are given as follows:
(39)A1=[-511-5],G1=[0110],H1=H2=[11],(40)A2=[-222-2],G2=[0110],E1=E2=[4114].Furthermore, we choose the following Gaussian type functions as our member functions:(41)Mi=exp(-(v2±1)2/σ2)exp(-(v2+1)2/σ2)+exp(-(v2-1)2/σ2),i=1,2,
where σ=0.8.By definingλ=η=1, θ1=0.1, and θ2=θ3=0.05 and solving LMIs (17), we can get that
(42)L=[6.06721.52131.64946.6723].The responses of the residual signal are shown in Figure1. It can be clearly seen that when t>5, the fault will occur. Figure 2 is the fault and its estimated value that shows the estimation error can be converged in a small field. Figure 3 shows the 3D mesh plot of the output PDFs and we can find that the fault occurs in front of two peaks. Moreover, when the fault F is redefined as a time-varying function
(43)F(t)={0,t≤51+0.2sint,t>5
the responses of time-varying fault and its estimated value are shown in Figure 4.Figure 1
The responses of residual signal.Figure 2
Fault and its estimation value.Figure 3
3D mesh plot of the PDFs.Figure 4
Time-varying fault and its estimation value.


## 6. Conclusions

This paper presents a novel FDD algorithm for non-Gaussian stochastic distribution systems based on T-S fuzzy modeling and nonlinear fuzzy filter design. A series of LMIs based solution is presented such that the estimation error system is stable and the fault can be detected through a known threshold. Moreover, the adaptive filter based on the T-S fuzzy model is designed to estimate the size of system fault by optimizing the solutions for the concerned LMIs. Simulations are given to demonstrate the efficiency of the proposed approach.



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