State Estimation of Fuzzy Sugeno Systems with Local Nonlinear Rules and Unmeasurable Premise Variables H. Moodi, M. Farrokhi Department of Electrical Engineering, Iran University of Science and Technology, Tehran 1684613114, Iran Email {moodi, farrokhi}@iust.ac.ir Abstract—This paper considers the design of observers for a class of continuous and discrete-time nonlinear systems presented by Takagi-Sugeno (T-S) model with nonlinear subsystems and unmeasurable premise variables. As a result, the proposed T-S structure reduces the number of rules in the Sugeno model by using local nonlinear rules. Moreover, it can represent larger class of nonlinear systems as compared to the measurable premise variable case. The proposed observer guarantees exponential convergence of state estimation error by Lyapunov stability analysis and linear matrix inequality (LMI) formulation. Numerical examples illustrate effectiveness of the proposed method. I. INTRODUCTION TAKAGI-SUGENO (T-S) fuzzy model is a well-known tool for nonlinear system modeling with increasing interest in recent years. Because T-S model is a universal approximator, it can model any smooth nonlinear system with any degree of accuracy [1]. Furthermore, local linear subsystems of this model allow one to use powerful linear systems tools, such as LMIs, to analyze and synthesize T-S fuzzy systems. However, as complexity of the system increases, the number of rules in the model and hence, the number and dimension of LMIs increases and becomes harder to solve. One possible solution is to reduce the accuracy in the model, which decreases the model complexity; however, the convergence of fuzzy controller or observer is not guaranteed in this case. To solve this problem, one may use nonlinear local subsystems for the T- S model. This will decrease the number of rules while increasing the model accuracy. A very simple form of these nonlinear Sugeno model is used in [2] that has used a linear form for the consequence part plus a sinusoidal term. A more advanced work is done by Dong in [3] and [4], where he used sector-bounded functions in the subsystems. In [5] and [6], Tanaka et al. have proposed the T-S model with polynomial subsystems; for stability analysis he used sum of squares (SOS) approach. This was the first use of SOS instead of LMI in fuzzy systems analysis. In [7] and [8], Sala represents a similar form of Sugeno model and used the Taylor series expansion of the system for construction of the polynomial subsystems. He states that the nonlinear consequent in the T- S model not only reduces the number of rules but also reduces the conservativeness in the controller design. Design of fuzzy observers for such systems is briefly discussed here. Fuzzy observers were first introduced in [9] in 1994 and ever since has been an active research issue. Different types of fuzzy observers have been discussed in literature. The separation property of fuzzy observer and controller was first discussed in [10] and more completely in [11]. When uncertainty exists in the model, the separation property cannot be proved in general. Later, in this regard, robust fuzzy observers were discussed in different papers such as [12] and [13]. Adaptive fuzzy observers also have attracted researchers' attention in recent years [14] and [15]. Sliding mode fuzzy observers were first presented in [16]. Robust sliding mode observers for systems with unknown input are discussed in several papers such as [17]. For estimating the states of T–S systems, two cases for the premise variable can be distinguished. First, the premise variable vector does not depend on the estimated states; and second when the premise variable depends on some of the states to be estimated. The latter structure can represent a larger class of non-linear systems. Unfortunately, the developed methods for fuzzy observer design with measured premise variables are not directly applicable for the systems with premise variables as a function of states [18]. Most works in literatures are based on the first case. The second case was first discussed in [19] in 2001. Many recent works are also working on obtaining less conservative conditions for this case [18], [20] and [21]. However, for T-S systems with nonlinear consequent parts, no such work has been reported, which is the main contribution of this paper. In other words, in this paper, a state observer for T-S fuzzy systems with nonlinear consequent part is designed when the premise variables depend on the estimated states. The paper is organized as follows: In Section 2 the nonlinear Sugeno model is described. The proposed observer and its convergence proof for discrete systems are presented in Section 3 and for continuous case in Section 4. In Section 5, two numerical examples are given to show effectiveness of the proposed method. II. SUGENO MODEL WITH NONLINEAR SUBSYSTEMS Consider a class of discrete-time nonlinear system described by ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) a b ya yb xk f xk f xk xk g xk uk yk f xk f xk xk + = + + = + ϕ ϕ (1) where () xk is the state, () uk is the control input, () y k is the measurable output, { } ( ( )) ,, , ∈ n f xk n a b ya yb are