This paper is concerned with the observer design of multivariable PDE systems in 1D. The proposed method uses a proper orthogonal decomposition (POD) algorithm to extract the dominant modes of the states distribution whereby a reduced-order model (ROM) is obtained. A state estimator design method is then proposed by leveraging the Lipschitz properties of the ROM with a robust Luenberger-type observer of the system states to reduce measurement sensors requirements. The Lyapunov method used in this work provides sufficient conditions in terms of standard linear matrix inequalities (LMIs) to ensure the exponential convergence of the estimation error with a prescribed decay rate. The online estimation performance is further improved by means of a stability region analysis algorithm. The performance of the proposed method is analyzed to estimate a physico-chemical system exhibiting a significant nonlinear behavior.