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: 819


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.


  • 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:


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:


and [8]


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.


Definition 1[9, 10] IfAhas no imaginary eigenvalues, theHnorm of the system transfer function matrix




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.

Ifm=pandV=CFare regular matrices, then the system matrix factorization can be realized such that


and the transform matrix TIRn×ntakes the form


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 regularV=CFand a given positive scalarsoIRthe square transfer function matrixGrsof the fault detection filter satisfies the conditions


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


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


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

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


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




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.



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


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




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


Therefore, the observer system matrix Aetakes the form




implies Eq. (16).

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


then with Eq. (29), it is




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


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


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

Evidently, writing the fault residual vector as




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


the autonomous observer error equation is


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 understandingEq. (30), the observer sensor subsystem for the fault detection filter can be designed as follows:


and, consequently, it yields


Another option is to design the observer sensor subsystem so thatV=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 ofAand ofAoare the same wheneverAois related toAasAo=TAT1for any invertibleT[11]. But this does not mean that if eigenvalues of the matrixAoare stable then eigenvalues of the matrixAo22are also stable. Thus, as well as for a stable system, it can lead to an unstable matrixAo22, 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 matrixA, the gain matrixKIRn×rof the state feedback additive stabilization


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


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

Then,Kis given as


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


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


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 matricesVi=CFiandi=1,2,,s, and a given positive scalarsoIRevery square transfer function matrixGrisof the fault detection filter satisfies for allithe conditions




JiIRn×ris the residual filter gain matrix,σ1is the maximal singular value ofGris, the polynomialPoisof ordernmis stable, andGri0IRm×mandFiIRnm×nare left orthogonal complements to the fault input matrixFi.

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 matrixFthat does not depend on the sectoral boundaries defining the TS model. In this case, the synthesis is substantially simplified becauseVis a constant matrix, and so it yields


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


Moreover, considering thati=1shiθt=1, then


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 matrixAito be such, when transformed usingTithatAo22iwill 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 associatedAei°=AiJiC.

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


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


is stable since Poisis now stable.

The price for such an additional stabilization is that ifjsigns are changing in eigenvalues ofAo22ito obtain the stableAo22i, alsojeigenvaluessoofGri0change their signs and the resulting matrixGri0will not be diagonal. According toEq. (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


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


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


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


which is used in the system state matrix construction


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


Consequently, the TS model matrix parameters are


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.,


with the derived parameters


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:


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


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


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


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


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

Thus, defining the weighting matrices of appropriate dimensions as


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


It can be easily verified that


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:



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




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= woT= 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.



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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