This paper proposes a new approach of observer design for nonlinear systems described by a Takagi-Sugeno model. Its main contribution concerns models with premise variables depending on the system states which are completely or partially unknown. This case is more difficult than when the premise variables are known or measured. Indeed, in this case, weighting functions of the observer depend on state estimates and the state estimation error is then governed by a Lipschitz nonlinear system. Here, two main results are established. Firstly, relaxed stability conditions are provided, using a nonquadratic Lyapunov function, to guarantee asymptotic stability of the observer. This aims to reduce the conservativeness compared to the existing works and enhance the maximal admissible Lipschitz constant for which the linear matrix inequality (LMI) conditions are feasible. Secondly, the Input-to-State Stability concept combined to a nonquadratic Lyapunov function are used to guarantee a bounded state estimation error which relaxes the conservativeness related to the Lipschitz constant. The robustness aspect is dealt with respect to some bounded modeling uncertainties and additive bounded perturbations. The stability conditions are expressed in terms of LMI.