State-of-the-art advancements in the realm of industrial process control and monitoring often require accurate descriptions of complex processes and their dynamical behaviours. Usually, many industrial processes are described by partial differential equations (PDE) or ordinary differential equations (ODE) depending on whether their dynamics evolve spatio-temporally or temporally, and thus are classified as distributed parameter systems (DPS) and lumped parameter systems (LPS), respectively. In this thesis, discrete-time estimator, soft sensor, and regulator designs are proposed for linear distributed and lumped parameter systems.

Considering the unavailability of full state information or prohibitive cost for installing spatially-distributed sensors, state estimation is often necessary for the regulation (or control) problems and/or for the monitoring purpose. To estimate the spatio-temporal state, the discrete-time Luenberger observer and Kalman filter are proposed for linear infinite-dimensional systems. Specifically, the discrete-time observer gain can be solved from continuous- or discrete-time Riccati equations numerically. To account for the output and disturbance constraints in the estimation, moving horizon estimation (MHE), as an optimization-based approach, is developed for a rather general class of DPSs, namely regular linear infinite-dimensional systems, by extending the MHE theory of LPSs. Stability and optimality are proved for the proposed MHE design.

Towards the unavailability of accurate model parameters and/or model structures, three estimators are further proposed, ranging from the mode-based MHE for output/state and mode estimation of switching regular linear infinite-dimensional systems, to the hybrid estimator development for pipeline leak detection, localization, and estimation of distributed pipeline systems based on discrete-time Luenberger observer and support vector machine (SVM), to the entirely model-free (i.e., data-driven) transfer learning (TL) based soft sensor design of linear finite-dimensional systems using variational Bayesian inference. The stability of the proposed advanced MHE design is given.

For the sake of disturbance rejection and reference tracking, the discrete-time output regulator design is developed for linear distributed parameter systems. In particular, the Cayley-Tustin (CT) bilinear transformation is applied to approximate the continuous system by a discrete-time infinite-dimensional system with essential model properties being preserved. Specifically, two types of output regulator designs are presented, namely, state-feedback regulator design and error-feedback regulator design, by exploring the internal model principle. Discrete regulator equations are formulated, and their solvability is proved and linked to the continuous counterparts. To ensure the stability of the discrete-time closed-loop system, the design of stabilizing feedback gain and its dual problem of stabilizing output injection gain design are provided using Riccati equations.

The effectiveness of the developed discrete-time Luenberger observer, Kalman filter, MHE, and advanced MHE methods are demonstrated on the transport-reaction processes, wave, Schrodinger and beam equations, and a heat exchanger system. The proposed soft sensor algorithm (i.e., transfer slow feature analysis) is validated through a numerical example, the Tennessee Eastman (TE) benchmark dataset, and a steam-assisted gravity drainage (SAGD) industrial process. The applicability of the proposed discrete-time output regulator designs is verified through heat equation, transport equation, pipeline networks, and two fluid flow systems (i.e., Kuramoto-Sivashinsky and Ginzburg-Landau equations).