Modal LMI-based observer design with performance guarantees for semilinear parabolic PDEs: Application to bioreactor system

This paper presents a systematic modal LMI-based framework for infinite-dimensional state observer design of semilinear parabolic partial differential equation (PDE) systems. The proposed methodology exploits the Riesz-spectral decomposition property of parabolic operators to partition the infinite-dimensional dynamics into finite-dimensional slow and infinite-dimensional fast subsystems, enabling late-lumping observer synthesis with guaranteed exponential stability. A modal output injection operator is designed through systematic eigenvalue assignment, while Lipschitz continuity assumptions on the nonlinear terms facilitate rigorous stability analysis of the estimation error dynamics. The design conditions are formulated as computationally tractable linear matrix inequalities (LMIs) derived from Lyapunov stability theory, directly incorporating prescribed decay rate specifications and observer gain magnitude constraints. To address practical implementation considerations, we develop a modal truncation strategy supported by center manifold theory and establish ellipsoidal stability regions through polytopic constraint analysis. The effectiveness of the proposed approach is demonstrated through application to a tubular bioreactor system, where numerical simulations confirm exponential convergence with prescribed performance and validate the practical viability of the infinite-dimensional observer design methodology for distributed parameter systems.