## 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

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

Throughout the chapter, the following notations are used:

## 2. System description

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

where

The variables

and [8]

where

## 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] *If* *has no imaginary eigenvalues, the* *norm of the system transfer function matrix*

is

while the

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

*If* *and* *are regular matrices, then the system matrix factorization can be realized such that*

and the transform matrix

where

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* *and a given positive scalar* *the square transfer function matrix* *of the fault detection filter satisfies the conditions*

where

*Proof.* Considering the fault transfer function matrix of dimension

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

respectively, where

Specifying the following matrix product

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

Presetting

where

Then,

and it is evident 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

and

implies Eq. (16).

Regarding the transfer function matrix

then with Eq. (29), it is

Since

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

Thus, defining the fault detection filter transfer function matrix as

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

*Evidently, writing the fault residual vector as*

where

and

the autonomous observer error equation is

where

*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*

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

*It is important to note the fact that the eigenvalues of* *and of* *are the same whenever* *is related to* *as* *for any invertible* *. But this does not mean that if eigenvalues of the matrix* *are stable then eigenvalues of the matrix* *are also stable. Thus, as well as for a stable system, it can lead to an unstable matrix* *, 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

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

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

where the matrix

*Then,* *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.

## 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

where

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* *and* *, and a given positive scalar* *every square transfer function matrix* *of the fault detection filter satisfies for all* *the conditions*

while

*is the residual filter gain matrix,* *is the maximal singular value of* *, the polynomial* *of order* *is stable, and* *and* *are left orthogonal complements to the fault input matrix*

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

*Since, independently on* *, the condition (52) is satisfied (**), all sub-filter transfer function matrices have the same H**norm, i.e.,*

*Moreover, considering that* *, then*

*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,* *implies 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* *to be such, when transformed using* *that* *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*

*Applying the duality principle and inserting the additive observer gain component* *obtained as a solution of the Riccati equation (45) for* *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

*The price for such an additional stabilization is that if* *signs are changing in eigenvalues of* *to obtain the stable* *, also* *eigenvalues* *of* *change their signs and the resulting matrix* *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.*

## 5. Illustrative example

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

where the measured output variables

The model parameters of the system are considered as:

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

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

The set of real scalars,

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 (

with the derived parameters

Note that in this case all

Applying Eq. (56), the following structure of

and

Choosing

where the eigenvalue spectrum of

respectively. It is evident that all diagonal elements of

Analogously, designing for the matrix

Since

Solving also for

It is evident that matrix

Thus, defining the weighting matrices of appropriate dimensions as

and solving the dual LQ control problem to change the sign of unstable eigenvalue of

It can be easily verified that

while, evidently,

Note that the same structure of

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

Since the matrices

where

while

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

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

It can verify that TS fuzzy fault detection filters created for the couple pairs (

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

## 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.