Open access peer-reviewed chapter

Fuzzy Fault Detection Filter Design for One Class of Takagi-Sugeno Systems

By Dušan Krokavec, Anna Filasová, Jakub Kajan and Tibor Kočík

Submitted: November 10th 2017Reviewed: January 23rd 2018Published: February 28th 2018

DOI: 10.5772/intechopen.74328

Downloaded: 293

Abstract

The constrained unitary formalism to fuzzy fault detection filter synthesis for one class of nonlinear systems, representable by continuous-time Takagi-Sugeno fuzzy models, is presented in the chapter. In particular, a way to produce the special set of matrix parameters of the fuzzy filter is proposed to obtain the desired H ∞ norm properties of the filter transfer function matrix. The significance of the treatment in relation to the systems under influence of actuator faults is analyzed in this context, and relations to corresponding setting of singular values of filters are discussed.

Keywords

  • multiple models
  • continuous-time Takagi-Sugeno fuzzy models
  • fuzzy fault detection filters
  • fuzzy state observers

1. Introduction

Since the work of Hou and Patton [1], there has been much interest in the design of fault residuals for linear systems that use H/Hoptimization principle in transfer function matrix of fault detection filter designed to scale up fault detection punctuality and high sensitivity to faults [2]. While retaining these features, a novel class of fault detection filters are proposed in [3, 4], preserving the unitary implementation of the fault detection filter transfer function matrix and receipting residual signal directional properties. However, the use of this methodology for Takagi-Sugeno (TS) fuzzy systems hits the boundaries of the working sectors and requires special adaptations.

Considering the properties of TS fuzzy models [5, 6], and some specifics in frequency characteristic evaluation of multiple model structures, the approach proposed in the chapter reformulates the H norm technique suitable in TS fuzzy fault detection filter design. The problem is solved via unitary modal technique when every linear TS fuzzy filter part is designed to have the same singular values of the transfer function matrix. Since working sector constraints may cause that the stable linear filter component cannot be obtained for a linear part in TS fuzzy model, to maintain H norm of the filter, the LQ modal control principle [7] is used for additional stabilization. Because additional stabilization aggravates directional properties of the applied linear part, in general, if additional stabilization is necessary, the residuals are only quasi-directional. It is immediately apparent that the formulated problem is related to forcing the singular values conditioned as state observer dynamics. The chosen model of the system is selected for this chapter to be sufficiently complex in illustration of all these specifics of synthesis.

Throughout the chapter, the following notations are used: xTand XTdenote the transpose of the vector xand the matrix X, respectively; for a square matrix X0means that Xis a symmetric positive semi-definite matrix; the symbol Inindicates the nth-order unit matrix; IRdenotes the set of real numbers; and IRnand IRn×rrefer to the set of all n-dimensional real vectors and n×rreal matrices.

2. System description

The considered class of the Takagi-Sugeno dynamic systems with additive faults is described as the following:

q̇t=i=1shiθtAiqt+Biut+FiftE1
yt=CqtE2

where qtIRn, utIRr, and ytIRmstand for state, control input, and measurable output, respectively; ftIRpis an additive fault vector; AiIRn×n, BiIRn×r, FiIRn×p, CIRm×n, and m=pand the matrix products Vi=CFiand ViIRm×mare regular matrices for all i.

The variables θjtand j=1,2,,o, bound with the sector TS model, span the o-dimensional vector of premise variables:

θt=θ1tθ2tθotE3

and [8]

i=1shiθt=1E4

where hiθt,i=1,2,,sis the set of normalized membership function. It is supposed that the measurable premise variables, the nonlinear sectors, and the normalized membership functions are chosen in such a way that the pairs AiBiare controllable and the pairs AiCare observable for all i.

3. Basic preliminaries from linear systems

Let the state-space description of a linear continuous-time dynamic systems take the form with equivalent meanings and dimensions as they are described in Section 2. The nature of the characterization of expected solutions to the system [(5), (6)] is given by the following results.

q̇t=Aqt+But+FftE5
yt=CqtE6

Definition 1 [9, 10] If Ahas no imaginary eigenvalues, the Hnorm of the system transfer function matrix

Gs=CsInA1BE7

is

Gs=supωIRσ1G=supωIRε1GGE8

while the kth singular value σhof the complex matrix Gis the nonnegative square root of the kth largest eigenvalue εkof GG, Gis the adjoint of G, and σ1is the largest singular value. The singular values of the transfer function matrix Gsare evaluated on the imaginary axis, and it is assumed that the singular values are ordered such that σkσk+1,k=1,2,,n1.

To apply in design methodology, the following result from [4] is quoted.

If m=pand V=CFare regular matrices, then the system matrix factorization can be realized such that

C=V0T,TF=Im0E9

and the transform matrix TIRn×ntakes the form

T=V1CF,E10

where V1CIRm×n, FIRnm×n, and Fare the left orthogonal complements to F.

The idea of the following condition was derived originally as an approximation in the frequency domain for the fault transfer function matrix reflecting Eqs. (5) and (6) from [12]. Here, it is demonstrated that it can be simply adapted for fault residual filter design.

A linear fault detection filter to the system [(5), (6)] is stable and unitary if for regular V=CFand a given positive scalar soIRthe square transfer function matrix Grsof the fault detection filter satisfies the conditions

Ps=detsInAJC=s+somPos,E11
σ1=σ2==σm,limω0σh=so,E12
Gr0=diagso1so1so1,E13
Grs=V1CsInAJC1F=s+so1Im,E14
Ao=TAT1=Ao11Ao12Ao21Ao22,E15
J=T1LoV1,Lo=soIm+Ao11Ao21,E16

where JIRn×ris the residual filter gain matrix, σ1is the maximal singular value of Grs, the polynomial Posof order nmis stable, and Gr0IRm×m.

Proof. Considering the fault transfer function matrix of dimension m×mas

Gfs=CsInA1FE17

and then regrouping terms using Eqs. (9) and (10), it yields immediately the expressions

Gfs=CT1TsInA1T1TF=CT1(sInTAT11TF,E18
Gfs=V0sInAo1Ip0,E19

respectively, where Aois given in Eq. (15).

Specifying the following matrix product Ao=TMV1CT1, where MIRn×mis a real matrix, it yields

Ao=TMV1CT1=V1CFMV1V0=V1CM0FM0E20

and, with the block matrix structure of Eqs. (15) and (21), it can be defined as

ΔAo=AoAo=Ao11V1CMAo12Ao21FMAo22.E21

Presetting

Ao11V1CM=soIm,Ao21FM=0,E22

where soIRis a prescribed positive real value. The plus sign is introduced for the purposes that come to light in the stability ensuing development of the observer system matrix.

Then,

ΔAo=soImAo120Ao22E23

and it is evident that ΔAois stable if Ao22is Hurwitz, denoting here that

Pos=detsInmAo22.E24

Rewriting the set of Eq. (22) to admit a stable solution

soIm+Ao11Ao21=V1CFM=TM=TT1Lo=Lo,E25

where

M=T1Lo,E26

then Eqs. (20) and (21) must satisfy the following conditions:

Ao=TMV1CT1=TJCT1,E27
ΔAo=AoAo=TAJCT1=TAeT1.E28

Therefore, the observer system matrix Aetakes the form

Ae=AJC=AMV1CE29

and

J=MV1=T1LoV1E30

implies Eq. (16).

Regarding the transfer function matrix Gesof the state error estimate as follows

Ges=CsInAe1F,E31

then with Eq. (29), it is

Ges=CT1sInTAeT11TF=V0sInΔAo1Ip0.E32

Since

sInΔAo=s+soImAo120sInmAo22,E33
sInΔAo1=s+so1Ims+so1Ao12sInmAo2210sInmAo221,E34

Substituting Eq. (34) into Eq. (32), it can obtain

Ges=Vs+so1Im=Vs+so.E35

Thus, defining the fault detection filter transfer function matrix as Grs=V1Ges, then

Grs=V1Ges=s+so1ImE36

and Eq. (36) implies Eq. (14). This concludes the proof.

Evidently, writing the fault residual vector as

rt=V1Cet=V1Cqtqet,E37

where

et=qtqetE38

and rtIRmis the vector of residual signals, then based on the following observer structure

q̇et=Aqet+But+JCqtqet,E39
yet=Cqet,E40

the autonomous observer error equation is

ėt=AJCet,E41

where qetIRnis the observer state, yetIRmis the estimated system output, and JIRn×mis the observer gain matrix; the fault detection filter (37), (39) is stable and unitary if for given positive scalar soIRand the Hurwitz matrix Ao22the conditions (15) and (16) are satisfied.

Practically, with understanding Eq. (30), the observer sensor subsystem for the fault detection filter can be designed as follows:

ezt=ztzet=V1CqtqetE42

and, consequently, it yields

q̇et=Aqet+But+MCqtqet.E43

Another option is to design the observer sensor subsystem so that V=Im.

With existence of the system parameter transformation, the above structures really mean that the subset of transformed state variables whose dynamics is explicitly affected by the additive fault ftand the second one, whose dynamics is not affected explicitly by the fault ft, exists.

It is important to note the fact that the eigenvalues of Aand of Aoare the same whenever Aois related to Aas Ao=TAT1for any invertible T[11]. But this does not mean that if eigenvalues of the matrix Aoare stable then eigenvalues of the matrix Ao22are also stable. Thus, as well as for a stable system, it can lead to an unstable matrix Ao22, and any additional stabilization is required.

To apply the above results, it is necessary to be able to design fault residual filter if an unstable Ao22results such that Aebe stable without loss of unitarity.

Lemma 2 [7, 12] To change signs of unstable eigenvalues of the system matrix A, the gain matrix KIRn×rof the state feedback additive stabilization

ut=KqtE44

is a solution of the continuous-time algebraic Riccati equation (CARE)

PA+ATPPBR1BTP+Q=0,E45

where the matrix QIRn×nis null matrix and RIRr×rand R=RT>0are positive definite symmetric matrices.

Then, Kis given as

K=R1BTP.E46

It is in that form that is able to be exploit for specific properties of the problem in TS fuzzy fault detection filter design.

In view of the above, these results hold for continuous-time linear systems, and, in principle, Theorem 1 gives a practical method to design unitary fault residual filters for the given linear system. Similar results are obtained for unitary TS fuzzy fault detection filter design in the following section.

4. TS fuzzy fault detection filters

Using the same set of membership functions, the fuzzy fault detection filter is built on the TS fuzzy observer

q̇et=i=1shiθtAiqt+Biuit+JiCqtqetE47
yet=CqetE48

where qetIRnis the observer state vector, yetIRmis the estimated system output vector, and JiIRn×mand i=1,2,,sare the sets of the observer gain matrices. Additionally, the output vector of the residual TS fuzzy filter is defined as

rt=i=1shiθtrit=i=1shiθtVi1CetE49
rit=Vi1CetE50
et=qtqetE51

where rt,ritIRm, ViIRm×m. Evidently, Vi=CFihas to be a regular matrix for all i.

Formally, the following result can be simply derived.

A TS fuzzy fault detection filter to the system [(1), (2)] is stable and unitary if for the set of regular matrices Vi=CFiand i=1,2,,s, and a given positive scalar soIRevery square transfer function matrix Grisof the fault detection filter satisfies for all ithe conditions

σ1=σ2==σm,limω0σh=so,E52
Gri0=diagso1so1so1,E53
Gris=Vi1CsInAiJiC1Fi=s+so1Im,E54

while

Ti=Vi1CFi,E55
Aoi=TiAiTi1=Ao11iAo12iAo21iAo22i,E56
Ji=Ti1LioVi1,Lio=soIm+Ao11iAo21i,E57
Pis=detsInAiJiC=s+somPois,E58
Pois=detsInmAo22i.E59

JiIRn×ris the residual filter gain matrix, σ1is the maximal singular value of Gris, the polynomial Poisof order nmis stable, and Gri0IRm×mand FiIRnm×nare left orthogonal complements to the fault input matrix Fi.

Proof. Because every sub-model in Eq. (47) is described by linear equations, Eqs. (15) and (16) imply directly the conditions (56) and (57), and Eq. (58) is given by Eq. (11). This concludes the proof.

In practice, an additive fault typically enters through a matrix Fthat does not depend on the sectoral boundaries defining the TS model. In this case, the synthesis is substantially simplified because Vis a constant matrix, and so it yields

T=V1CF,E60
Aoi=TAiT1=Ao11iAo12iAo21iAo22i,E61
Ji=T1LioV1,Lio=soIm+Ao11iAo21i.E62

Since, independently on i, the condition (52) is satisfied (σ1=σ2==σm), all sub-filter transfer function matrices have the same Hnorm, i.e.,

Gris=Grosforalli.E63

Moreover, considering that i=1shiθt=1, then

Grs=i=1shiθtGris=GrosE64

That is, the Hnorm of the transfer function matrix of such defined TS fuzzy fault detection filter is independent on the system working point. Of course, this cannot be said about the dynamics of the time response of the sub-filter components.

Moreover, Gri0implies that all residual components of TS fuzzy fault detection filter have the same directional properties, which ensure unitary properties of the filter.

Sectoral boundaries may cause a matrix Aito be such, when transformed using Tithat Ao22iwill not be Hurwitz matrix. Because the transfer function matrix of the corresponding filter linear component in this case is unstable, maintaining the unitary property requires changes in the signs of the unstable eigenvalues of the associated Aei°=AiJiC.

Applying the duality principle and inserting the additive observer gain component KsiTobtained as a solution of the Riccati equation (45) for Aei°T, according to the scheme given in Lemma 2, the observer gain matrix is changed as

Ji°=Ji+KsiT,Aei°=AiJi°C.E65

This additive stabilization results that the consequential characteristic polynomial, taking also the form

Pis=detsInAei=s+somPois,E66

is stable since Poisis now stable.

The price for such an additional stabilization is that if jsigns are changing in eigenvalues of Ao22ito obtain the stable Ao22i, also jeigenvalues soof Gri0change their signs and the resulting matrix Gri0will not be diagonal. According to Eq. (8), this does not result in a change in Hnorm, but such filter component will arrive at the unitary directional residual properties.

5. Illustrative example

The three-tank system is described by the set of Eqs. [13, 14] as

dq1tdt=u1tF1α1signq1tq2t2gq1tq2tF1i=13λiqiti=13λiqit,
dq2tdt=u2tF2α22gq2tF2q2tq2t+α1signq1tq2t2gq1tq2tF1i=13λiqiti=13λiqit+α3signq3tq2t2gq3tq2tF3i=13ηiqiti=13ηiqit,
dq3tdt=u3tF3α3signq3tq2t2gq3tq2tF3i=13ηiqiti=13ηiqit,
ykt=Fkqkt,k=1,2,3,

where the measured output variables yktare water levels in tanks qktm,k=1,2,3and the incoming flows are considered as the inputs variables uktm3/s,k=1,2,3; the bounds of the state and input variables are

q1max=q3max=1.00m,q2max=0.90m,u1,2,3min=0m3/s,q1min=q3min=0.02m,q2min=0.01m,u1,2,3max=0.005m3/s.

λk,ηkIRare positive scalars and signis the sign function.

The model parameters of the system are considered as:

gFkα1α3α2the gravitational acceleration9.80665m/s2,thesamesection of tanks0.25m2,the equivalent section of the pipe between the first and second tank6.5×104m2,the equivalent section of the pipe between the third and second tank6.5×104m2,the equivalent section of the outlet pipe from the second tank6.5×103m2,

Minimizing the number of premise variables and excluding switching modes in controller work, the premise variables are chosen as follows

θ1t=α1signq1tq2t2gq1tq2tF1i=13λiqit,
θ2t=α22gq2tF2q2t=α2F22gq2t,
θ3t=α3signq3tq2t2gq3tq2tF3i=13ηiqit.

Computed from the input variable bounds, the sector bounds of the premise variables imply the numbering:

i=1θ1maxθ2maxθ3max,i=2θ1maxθ2maxθ3min,i=3θ1maxθ2minθ3max,i=4θ1maxθ2minθ3min,i=5θ1minθ2maxθ3max,i=6θ1minθ2maxθ3min,i=7θ1minθ2minθ3max,i=8θ1minθ2minθ3min,

which is used in the system state matrix construction

Ai=λ1θ1iλ2θ1iλ3θ1iλ1θ1i+η1θ3iλ2θ1i+η2θ3iθ2iλ3θ1i+η3θ3iη1θ3iη2θ3iη3θ3i,B=F11000F21000F31,C=F1000F2000F3

and prescribed, moreover, that the matrix Cis given in such a way that the product CBis the identity matrix. This regularizes the residual design conditions if Band Care diagonal matrices.

The sector functions are trapezoidal, and the membership functions are constructed as product of three sector functions with the same ordering as Ai.

The set of real scalars, λk, ηk, and k=1,2,3, is interactively optimized under limitations that all couples AiBand AiCare controllable and observable for the given set of indices i, where

λ1=0.1992,λ2=0.6894,λ3=0.1618,η1=0.6891,η2=0.3646,η3=0.0569.

Consequently, the TS model matrix parameters are

A1=0.01630.05630.01320.12251.03920.02200.10620.05620.0088,A2=0.01630.05630.01320.00541.10690.01140.02170.01150.0018,A3=0.01630.05630.01320.12250.00890.02200.10620.05620.0088,A4=0.01630.05630.01320.00540.07660.01140.02170.01150.0018,A5=0.00340.01190.00280.10281.10730.00600.10620.05620.0088,A6=0.00340.01190.00280.02511.17500.00460.02170.01150.0018,A7=0.00340.01190.00280.10280.07710.00600.10620.05620.0088,A8=0.00340.01190.00280.02510.14470.00460.02170.01150.0018.
B=400040004,C=0.250000.250000.25.

Since the orthogonal complement to a square matrix does not exist, three fault detection filters can be considered for single actuator fault detection. To illustrate the design procedure, the TS fuzzy fault detection filter for the pair (C23, B23) is considered, i.e.,

CC23=00.250000.25,FB23=004004,

with the derived parameters

V=CF=1001,V1C=00.250000.25,F=100,T=00.250000.25100.

Note that in this case all Aiwith index higher than 4 lead to an unstable structure of Ao22i°and the resulting observer matrices Aeineed to be additionally stabilized, applying the principle given in Lemma 2.

Applying Eq. (56), the following structure of Ao1for the initial matrix A1is computed:

Ao1=1.03920.02200.03060.05620.00880.02660.22500.05280.0163,Ao111=1.03920.02200.05620.0088,Ao121=0.03060.0266,Ao211=0.22500.0528,Ao221=0.0163,

and Ao221=0.0163implies that the associated TS fuzzy fault detection filter linear component can be designed directly.

Choosing so=5, it is resulting from Eqs. (57) and (58) that

L1°=3.96080.02200.05624.99120.22500.0528,J1=0.22500.052815.84320.08790.224819.9649,Ae1=0.0163000.12255.000.106205.0,

where the eigenvalue spectrum of Ae1and the steady-state value of the TS fuzzy fault detection filter transfer function matrix Gr10are

ρAe1=0.01635.05.0,Gr10=V1CAe11F=0.20.2,

respectively. It is evident that all diagonal elements of Gr10take the value so1=0.2. The same structure of Gr0is obtained solving with Alfor l=1,2,3,4.

Analogously, designing for the matrix A5, it can be seen that

Ao5=1.10730.00600.02570.05620.00880.02660.04750.01110.0034,Ao511=1.10730.00600.05620.0088,Ao512=0.02570.0266,Ao521=0.04750.0111,Ao522=0.0034.

Since Ao222=0.0034, evidently, the associated TS fuzzy fault detection filter linear component with the unitary transfer function matrix has to be stabilized additively.

Solving also for so=5, then

L5°=3.89270.00600.05624.99120.04750.0111,J5=0.04750.011115.57070.02390.224819.9649,Ae5=0.0034000.10285.000.106205.0.

It is evident that matrix Fe5is not Hurwitz and has to be additively stabilized.

Thus, defining the weighting matrices of appropriate dimensions as

Q=0,S=0,R=VVT=I2

and solving the dual LQ control problem to change the sign of unstable eigenvalue of Fe5using the MATLAB function Ks5=careFe2TQRSI3, then

Ks5T=0.64560.66710.01330.01370.01370.0142,Aes5=Ae5Ks5TC=0.00340.16140.16680.10285.00330.00340.10620.00345.0035.

It can be easily verified that

ρAes5=0.00345.05.0,
Gr50=V1CAes51F=0.00660.19990.19990.0066,ρGr50=0.20000.2000.

while, evidently, Gr50is not diagonal and the eigenvalues of Gr50are ±0.2=±s01.

Note that the same structure of Grl0is obtained solving with the system matrices Aland l=5,6,7,8when additional stabilization is required. Evidently, elements of this set of TS fuzzy residual filter linear components are stable, non-unitary, and without directional residual properties. Nevertheless, these properties guarantee the same singular values of the linear transfer function matrix components; as follows the result of Definition 1, the TS fuzzy residual filter will have all the singular values the same. To document this, the singular value plot of the TS fuzzy fault detection filter, as well as of all its linear parts, is equal to that presented in Figure 1. With respect to the structure of the matrices Band C, the comparable results are obtainable for the matrix pairs (C12, B12)and (C13, B13).

Figure 1.

TS fuzzy fault detection filter singular value plot.

The rest of gain matrices of the stable TS fault detection filter is as follows:

J2=0.22500.052815.57250.04560.045920.0072,J3=0.22500.052819.96430.08790.224819.9649,J4=0.22500.052819.69350.04560.045920.0072,

J5=0.69300.656015.58400.01020.238519.9791,J6=3.08092.712615.31570.03190.032420.0189,
J7=0.69300.656019.70510.01020.238519.9791,J8=3.08092.712619.43670.03190.032420.0189.

Since the matrices Aiof the TS fuzzy system are not Hurwitz, the system in simulations is stabilized using the local-state feedback control laws, acting in the forced modes. Adapting the method presented in [14] to design the control law parameters, the local controller parameters are computed as

K1=0.17800.00830.01500.00830.07010.00410.01500.00430.1798,K2=0.17800.00790.00080.00750.08690.00280.00080.00270.1824,
K3=0.17800.00840.01500.00820.18420.00410.01500.00420.1798,K4=0.17800.00780.00080.00760.16750.00270.00080.00280.1824,
K5=0.18290.01420.01310.01410.08700.00610.01300.00630.1798,K6=0.18290.00200.00270.00170.10370.00080.00270.00070.1824,
K7=0.18290.01430.01310.01400.16740.00610.01300.00630.1798,K8=0.18290.00190.00270.00180.15070.00070.00270.00070.1824,
W1=0.18210.02240.01170.02230.18970.00960.01150.00980.1820,W2=0.18210.00620.00410.00610.18990.00010.00470.00020.1819,
W3=0.18210.02250.01170.02240.18650.00960.01150.00980.1820,W4=0.18210.00630.00410.00620.18670.00010.00470.00010.1819,
W5=0.18200.01120.01380.01160.18990.00760.01350.00770.1820,W6=0.18200.00490.00210.00460.19010.00190.00270.00220.1819,
W7=0.18200.01140.01380.01170.18670.00760.01350.00780.1820,W8=0.18200.00480.00210.00450.18690.00190.00270.00210.1819,

where

Wi=CAiBKi1B1,
uit=Kiqt+Wiwo,

while woIRnis the vector of the desired steady-state system outputs.

If necessary for any more complex system, PDS controller principle can be applied to stabilize the plant (see, e.g., authors’ publications [15, 16] or other references [17, 18]).

To display simulations in the MATLAB and Simulink environment, the forced mode control is established with local controller parameter given as above for the system initial conditions qT0=0.20.30.2and woT=0.60.50.4. Fault detection filter is constructed on the couple (C23, B23) and the set of matrices Aiand i=1,2,8.

As the results, Figure 2 shows the TS fuzzy system output responses, illustrating their asymptotic convergence to the steady states, and Figure 3 presents the TS fuzzy fault detection filter response, reflecting a steplike 90% gain loss of the second actuator at the time instant t=60s. These examples illustrate the power that can be invoked through the prescribed H norm properties.

Figure 2.

System output responses.

Figure 3.

Residual signal responses.

It can verify that TS fuzzy fault detection filters created for the couple pairs (C12, B12) and (C13, B13) have similar properties as that defined for the couple (C23, B23). The difference is, for example, that in the occurrence of a single fault of the second actuator the responses of TS fuzzy fault detection filter defined for the couple (C13, B13) naturally do not have directional properties, since the second column of Kis not included in its construction.

As can be seen from the solution, the sector functions defined in this way cannot create a unitary TS fuzzy fault detection filter, but the obtained orthogonal properties of the residual signals are sufficient to detect and isolate actuator faults.

6. Concluding remarks

The problem of designing the TS fuzzy fault detection filters for highly nonlinear mechanical systems representable by the TS fuzzy model is considered, to achieve the desired filter H norm property in all working point belonging to the assigned work sectors. The proposed method exploits features offered in TS fuzzy system models to design TS fuzzy fault detection filters. The rules and formulation are developed to generate residual signals with quasi-directional properties and to make the TS filter transfer function matrix with prescribed H norm properties. By a convenient choose of the sector functions, this purpose is reached using a relative small number of membership functions. If unitary definition for TS fuzzy fault detection filters is satisfied, the design methodology provides new opportunities for fault detection and isolation rules in fault tolerant nonlinear control systems, their analysis, and optimization.

Acknowledgments

The work presented in this chapter was supported by VEGA, the Grant Agency of the Ministry of Education, and the Academy of Sciences of Slovak Republic, under Grant No. 1/0608/17. This support is very gratefully acknowledged.

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Dušan Krokavec, Anna Filasová, Jakub Kajan and Tibor Kočík (February 28th 2018). Fuzzy Fault Detection Filter Design for One Class of Takagi-Sugeno Systems, Nonlinear Systems - Modeling, Estimation, and Stability, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/intechopen.74328. Available from:

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