The performance index

## 1. Introduction

The problem of filter design for descriptor systems system has been intensively studied by a number of researchers for the past three decades; see Ref.[1]-[6]. This is due not only to theoretical interest but also to the relevance of this topic in control engineering applications. Descriptor systems or so called singularly perturbed systems are dynamical systems with multiple time-scales. Descriptor systems often occur naturally due to the presence of small “parasitic” parameter, typically small time constants, masses, etc.

The main purpose of the singular perturbation approach to analysis and design is the alleviation of high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamics modes. The separation of states into slow and fast ones is a nontrivial modelling task demanding insight and ingenuity on the part of the analyst. In state space, such systems are commonly modelled using the mathematical framework of singular perturbations, with a small parameter, say

In the last few years, many researchers have studied the

Fuzzy system theory enables us to utilize qualitative, linguistic information about a highly complex nonlinear system to construct a mathematical model for it. Recent studies show that a fuzzy linear model can be used to approximate global behaviors of a highly complex nonlinear system; see for example, Ref.[7]-[19]. In this fuzzy linear model, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by “blending" these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling where a single model is used to describe the global behaviour of a system, the fuzzy modelling is essentially a multi-model approach in which simple sub-models (linear models) are combined to describe the global behaviour of the system.

What we intend to do in this paper is to design a robust

This paper is organized as follows. In Section 2, system descriptions and definitions are presented. In Section 3, based on an LMI approach, we develop a technique for designing a robust

## 2. System descriptions

In this section, we generalize the TS fuzzy system to represent a TS fuzzy descriptor system with parametric uncertainties. As in Ref.[19], we examine a TS fuzzy descriptor system with parametric uncertainties as follows:

where

**Assumption 1**

where

for any known positive constant

Next, let us recall the following definition.

**Definition 1*** Suppose γis a given positive number. A system (1) is said to have an L2-gain less than or equal to γif*

## 3. Robust H ∞ fuzzy filter design

Without loss of generality, in this section, we assume that

We are now aiming to design a full order dynamic

where

### 3.1. Case I–ν ( t ) is available for feedback

The premise variable of the fuzzy model

Before presenting our next results, the following lemma is recalled.

**Lemma 1** Consider the system (4). Given a prescribed

where

with

then the prescribed

where

Proof. It can be shown by employing the same technique used in Ref.[18]-[19].

**Remark 1** The LMIs given in Lemma 1 may become ill-conditioned when

**Theorem 1** Consider the system (4). Given a prescribed

where

with

then there exists a sufficiently small

where

with

Proof. Suppose the inequalities (17)-(19) hold, then the matrices

Clearly,

where

Now, we need to show that

By the Schur complement, it is equivalent to showing that

Substituting (28) and (29) into the left hand side of (32), we get

The Schur complement of (17) is

According to (34), we learn that

Using (35) and the Schur complement, it can be shown that there exists a sufficiently small

Next, employing (28), (29) and (30), the controller’s matrices given in (16) can be re-expressed as follows:

Substituting (28), (29), (30) and (36) into (14) and (15), and pre-post multiplying by

where the

Note that the

Employing (20)-(22) and knowing the fact that for any given negative definite matrix

### 3.2. Case II–ν ( t ) is unavailable for feedback

The fuzzy filter is assumed to be the same as the premise variables of the fuzzy system model. This actually means that the premise variables of fuzzy system model are assumed to be measurable. However, in general, it is extremely difficult to derive an accurate fuzzy system model by imposing that all premise variables are measurable. In this subsection, we do not impose that condition, we choose the premise variables of the filter to be different from the premise variables of fuzzy system model of the plant. In here, the premise variables of the filter are selected to be the estimated premise variables of the plant. In the other words, the premise variable of the fuzzy model

where

with

Note that the above technique is basically employed in order to obtain the plant’s premise variable to be the same as the filter’s premise variable; e.g. [17]. Now, the premise variable of the system is the same as the premise variable of the filter, thus we can apply the result given in Case I. By applying the same technique used in Case I, we have the following theorem.

**Theorem 2** Consider the system (4). Given a prescribed

where

with

then there exists a sufficiently small

where

with

*Proof.* It can be shown by employing the same technique used in the proof for Theorem 1.

## 4. Example

Consider the tunnel diode circuit shown in Figure 1 where the tunnel diode is characterized by

Assuming that the inductance,

where

For the sake of simplicity, we will use as few rules as possible. Assuming that

**Plant Rule 1:** IF

**Plant Rule 2:** IF

where

Now, by assuming that

Note that the plot of the membership function Rules 1 and 2 is the same as in Figure 2. By employing the results given in Lemma 1 and the Matlab LMI solver, it is easy to realize that

Case I-

In this case,

Hence, the resulting fuzzy filter is

where

Case II:

In this case,

The resulting fuzzy filter is

where

The performance index | ||

Fuzzy Filter in Case I | Fuzzy Filter in Case II | |

.0001 | 0.141 | 0.283 |

.1 | 0.316 | 0.509 |

.25 | 0.479 | 0.596 |

.26 | 0.500 | |

.30 | 0.591 | |

.31 |

**Remark 2** The ratios of the filter error energy to the disturbance input noise energy are depicted in Figure 3 when

## 5. Conclusion

The problem of designing a robust