Model reduction design for continuous systems with ﬁnite frequency speciﬁcations

This paper is concerned with the problem of model reduction design for continuous systems in Takagi-Sugeno fuzzy model. Through the deﬁned FF H ∞ gain performance, sufﬁcient conditions are derived to design model reduction and to assure the fuzzy error system to be asymptotically stable with a FF H ∞ gain performance index. The explicit conditions of fuzzy model reduction are developed by solving linear matrix inequalities. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. This is an open access article under the CC BY-SA license.


INTRODUCTION
In the last few decades, many researchers have investigated model reduction including continuous and discrete settings as these systems have great applications in engineering fields. The problem is to design a low-order model to approach a higher order model given according to some specified criteria. Indeed, many results-based model reduction approach were presented [1]- [10].
In the practical, systems are always more or less disturbed, therefore, model reduction issues for nonlinear systems have been extensively discussed through the T-S fuzzy model approach, see [11]- [16]. Among the most of the existed literature on model reduction problems, the disturbances are considered in the entire frequency (EF) domain, which will bring overdesign in the filtering design. While many practical engineering problems are more suitable to be considered in finite frequency (FF) ranges [17]- [26].
The main objective of this paper is to design a model reduction for continuous T-S fuzzy systems with disturbance in FF domain. Through the defined FF H ∞ gain performance, sufficient conditions are derived to design model reduction and to assure the fuzzy error system to be asymptotically stable with a FF H ∞ gain performance index. The explicit conditions of fuzzy FF are developed by solving linear matrix inequalities. A systematic model reduction design scheme is proposed, which could reduce the conservatism of the results compared to the one considered in EF domain. Finally, a simulation example demonstrates the usefulness of the proposed method.
Notations: The notation A > 0 (A ≤ 0) means that A is positive definite (positive semi-definite). A −1 , A T , A * denote the inverse, the transpose and the complex conjugate transpose of matrix A, respectively. Symbol * represents the term originated by conjugate symmetry in a matrix.

PROBLEM FORMULATION
The plant under consideration is a continuous T-S fuzzy system described by its i-th rule as follows: where x(t) ∈ R n is the state vector; y(t) ∈ R p is the measured output vector; u(t) ∈ R m is the external noise signal of the following frequency sets, where LF, MF and HF stand for low-, middle-, and high-frequency ranges, respectively. Via using inference product, singleton fuzzifer and center-average defuzzifer, nonlinear system (1) can be described by:ẋ where and In this paper, we are interested in approximating the T-S fuzy system (4) by a stablenth-order (n < n) reduced-order T-S model.
wherex(t) ∈ Rn(n < n) is the state of the reduced-order model,ŷ(t) ∈ R p is the output of the reduced-order model. Then, the fuzzy reduced-order model as (8), Then, we have the error model: whereĀ Next, given a scalar γ and a rectangular FF domain, the error system (Σ e ) is said to have a FF H ∞ performance if it satisfies the following inequality holds, where The following inequalities hold [27]: Let ∈ R n , J ∈ R n×n , X ∈ R m×n , rank (X ) = r < n and X ⊥ ∈ X n×(n−r) such that X X ⊥ = 0 [28], so that the following conditions are equivalent: Error system (10) is stable and the FF H ∞ performance (12) is satisfied if there exist P = P T , Q = Q T > 0 such that [29].
First. Define: which is nothing but (17). On the other hand, Let: then, we obtain (16). Error system in (8) is stable with an H ∞ performance bound γ, if there exist Ì ISSN: 2088-8708 Built on Th. 3., we pick of parameters F, G, L , H: Finally, by applying Lemma 2, we have Th. 3. Moreover, under the above conditions, we can obtain a state-space realization of model reduction (8) with the following parameters as (27).

NUMERICAL EXAMPLE
Consider tunnel diode circuit shown in Figure 1 with two rules [30]: where  The membership functions: when x 1 (t) is almost ±3 and 0.
We propose in Table 1 shows the values of γ obtained in different frequency ranges. We can see from Table 1 shows the values of γ obtained with the approaches existing in [5], [22], [23] and Theorem 3. We can see that the proposed method provides better results than the existing ones.

CONCLUSION
This paper has concerned with the problem of the model reduction design for continuous T-S fuzzy systems with FF disturbances. Assuming the disturbances is dominated in a known FF domain. Through applying a more general linearization procedure, systematic methods have been proposed for model reduction design, which guarantees the asymptotic stability and the FF H ∞ gain performance of the error system. A simulation example has been given to illustrate the effectiveness of the proposed method.