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

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 param- eters 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.


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 ∞ =H À optimization 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: x T and X T denote the transpose of the vector x and the matrix X, respectively; for a square matrix X ≥ 0 means that X is a symmetric positive semi-definite matrix; the symbol I n indicates the nth-order unit matrix; I R denotes the set of real numbers; and I R n and I R nÂr refer to the set of all n-dimensional real vectors and n Â r real matrices.

System description
The considered class of the Takagi-Sugeno dynamic systems with additive faults is described as the following: where q t ð Þ ∈ I R n , u t ð Þ ∈ I R r , and y t ð Þ ∈ I R m stand for state, control input, and measurable output, respectively; f t ð Þ ∈ I R p is an additive fault vector; A i ∈ I R nÂn , B i ∈ I R nÂr , F i ∈ I R nÂp , C ∈ I R mÂn , and m ¼ p and the matrix products V i ¼ CF i and V i ∈ I R mÂm are regular matrices for all i.
The variables θ j t ð Þ and j ¼ 1, 2, …, o, bound with the sector TS model, span the o-dimensional vector of premise variables: and [8]

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] If A has no imaginary eigenvalues, the H ∞ norm of the system transfer function matrix is while the kth singular value σ h of the complex matrix G jω ð Þ is the nonnegative square root of the kth largest eigenvalue ε k of G * jω ð ÞG jω ð Þ, G * jω ð Þ is the adjoint of G jω ð Þ, and σ 1 is the largest singular value. The singular values of the transfer function matrix G s ð Þ are evaluated on the imaginary axis, and it is assumed that the singular values are ordered such that σ k ≥ σ kþ1 , k ¼ 1, 2, …, n À 1.
To apply in design methodology, the following result from [4] is quoted. Lemma 1 If m ¼ p and V ¼ CF are regular matrices, then the system matrix factorization can be realized such that and the transform matrix T ∈ I R nÂn takes the form where V À1 C ∈ I R mÂn , F ⊥ ∈ I R nÀm ð Þ Â n , and F ⊥ are 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.
Theorem 1 A linear fault detection filter to the system [(5), (6)] is stable and unitary if for regular V ¼ CF and a given positive scalar s o ∈ I R the square transfer function matrix G r s ð Þ of the fault detection filter satisfies the conditions where J ∈ I R nÂr is the residual filter gain matrix, σ 1 is the maximal singular value of G r s ð Þ, the polynomial P o s ð Þ of order n À m ð Þis stable, and G r 0 ð Þ ∈ I R mÂm .
Proof. Considering the fault transfer function matrix of dimension m Â m as and then regrouping terms using Eqs. (9) and (10), it yields immediately the expressions respectively, where A o is given in Eq. (15).

Specifying the following matrix product
and, with the block matrix structure of Eqs. (15) and (21), it can be defined as Presetting where s o ∈ I R is 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, and it is evident that ΔA o is stable if A o22 is Hurwitz, denoting here that Rewriting the set of Eq. (22) to admit a stable solution where then Eqs. (20) and (21) must satisfy the following conditions: Therefore, the observer system matrix A e takes the form and implies Eq. (16).
Regarding the transfer function matrix G e s ð Þ of the state error estimate as follows Since Substituting Eq. (34) into Eq. (32), it can obtain Thus, defining the fault detection filter transfer function matrix as G r s and Eq. (36) implies Eq. (14). This concludes the proof.
Corollary 1 Evidently, writing the fault residual vector as and r t ð Þ ∈ I R m is the vector of residual signals, then based on the following observer structure the autonomous observer error equation is where q e t ð Þ ∈ I R n is the observer state, y e t ð Þ ∈ I R m is the estimated system output, and J ∈ I R nÂm is the observer gain matrix; the fault detection filter (37), (39) is stable and unitary if for given positive scalar s o ∈ I R and the Hurwitz matrix A o22 the 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: and, consequently, it yields Another option is to design the observer sensor subsystem so that V ¼ I m .
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 f t ð Þ and the second one, whose dynamics is not affected explicitly by the fault f t ð Þ, exists.
Remark 1 It is important to note the fact that the eigenvalues of A and of A o are the same whenever 11]. But this does not mean that if eigenvalues of the matrix A o are stable then eigenvalues of the matrix A o22 are also stable. Thus, as well as for a stable system, it can lead to an unstable matrix A o22 , 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 A o22 results such that A e be stable without loss of unitarity.
Lemma 2 [7,12] To change signs of unstable eigenvalues of the system matrix A, the gain matrix K ∈ I R nÂr of the state feedback additive stabilization is a solution of the continuous-time algebraic Riccati equation (CARE) where the matrix Q ∈ I R nÂn is null matrix and R ∈ I R rÂr and R ¼ R T > 0 are positive definite symmetric matrices.
Then, K is 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.

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 q e t ð Þ ∈ I R n is the observer state vector, y e t ð Þ ∈ I R m is the estimated system output vector, and J i ∈ I R nÂm and i ¼ 1, 2, …, s are the sets of the observer gain matrices. Additionally, the output vector of the residual TS fuzzy filter is defined as where r t ð Þ, r i t ð Þ ∈ I R m , V i ∈ I R mÂm . Evidently, V i ¼ CF i has to be a regular matrix for all i.
Formally, the following result can be simply derived. while J i ∈ I R nÂr is the residual filter gain matrix, σ 1 is the maximal singular value of G ri s ð Þ, the polynomial P oi s ð Þ of order n À m ð Þis stable, and G ri 0 ð Þ ∈ I R mÂm and F ⊥ i ∈ I R nÀm ð Þ Â n are left orthogonal complements to the fault input matrix F i .
Corollary 2 In practice, an additive fault typically enters through a matrix F that does not depend on the sectoral boundaries defining the TS model. In this case, the synthesis is substantially simplified because V is a constant matrix, and so it yields Corollary 3 Since, independently on i, the condition (52) is satisfied ( That is, the H ∞ norm 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, G ri 0 ð Þ implies that all residual components of TS fuzzy fault detection filter have the same directional properties, which ensure unitary properties of the filter.
Remark 2 Sectoral boundaries may cause a matrix A i to be such, when transformed using T i that A o22i will 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 A Applying the duality principle and inserting the additive observer gain component K T si obtained as a solution of the Riccati equation (45) for A T ei , 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 P oi s ð Þ is now stable.
The price for such an additional stabilization is that if j signs are changing in eigenvalues of A o22i to obtain the stable A o22i , also j eigenvalues s o of G ri 0 ð Þ change their signs and the resulting matrix G ri 0 ð Þ will not be diagonal. According to Eq. (8), this does not result in a change in H ∞ norm, but such filter component will arrive at the unitary directional residual properties.

Illustrative example
The three-tank system is described by the set of Eqs. [13,14] as where the measured output variables y k t ð Þ are water levels in tanks q k t ð Þ m ½ , k ¼ 1; 2; 3 and the incoming flows are considered as the inputs variables u k t ð Þ m 3 =s Â Ã , k ¼ 1; 2; 3; the bounds of the state and input variables are λ k , η k ∈ I R are positive scalars and sign Á ð Þ is the sign function.
The model parameters of the system are considered as: -the gravitational acceleration 9:80665 m=s 2 Â Ã , -the same ð Þsection of tanks 0:25 m 2 Â Ã , -the equivalent section of the pipe between the first and second tank 6:5 Â 10 À4 m 2 Â Ã , -the equivalent section of the pipe between the third and second tank 6:5 Â 10 À4 m 2 Â Ã , -the equivalent section of the outlet pipe from the second tank 6:5 Â 10 À3 m 2 Â Ã , 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 C is given in such a way that the product CB is the identity matrix. This regularizes the residual design conditions if B and C are diagonal matrices.
The sector functions are trapezoidal, and the membership functions are constructed as product of three sector functions with the same ordering as A i .
The set of real scalars, λ k , η k , and k ¼ 1; 2; 3, is interactively optimized under limitations that all Þare controllable and observable for the given set of indices i, where with the derived parameters Note that in this case all A i with index higher than 4 lead to an unstable structure of A o22i and the resulting observer matrices A ei need to be additionally stabilized, applying the principle given in Lemma 2.
Applying Eq. (56), the following structure of A o1 for the initial matrix A 1 is computed:  where the eigenvalue spectrum of A e1 and the steady-state value of the TS fuzzy fault detection filter transfer function matrix G r1 0 ð Þ are r A e1 ð Þ ¼ À0:0163 À5:0 À5: respectively. It is evident that all diagonal elements of G r1 0 ð Þ take the value s À1 o ¼ 0:2. The same structure of G r * 0 ð Þ is obtained solving with A l for l ¼ 1; 2; 3; 4.
Analogously, designing for the matrix A 5 , it can be seen that Solving also for s o ¼ 5, then It is evident that matrix F e5 is not Hurwitz and has to be additively stabilized.
Thus, defining the weighting matrices of appropriate dimensions as 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 (C 13 , B 13 ) naturally do not have directional properties, since the second column of K is 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.

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 quasidirectional 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.