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The present manuscript considers the observer design problem for a class of scalar semi-linear hyperbolic partial differential equation (PDE) systems with a recycle loop through the boundary point. The design method of Kazantzis and Kravaris [(1998). Nonlinear observer design using Lyapunov's auxiliary theorem. Systems & Control Letters, 34(5), 241–247] developed for the nonlinear finite dimensional systems observer design is extended to semi-linear hyperbolic PDE systems. The observer design problem is tackled through a first-order associated PDE. Due to the existence of spatial partial derivative operator, Lyapunov's Auxiliary Theorem originated from finite-dimensional systems can be no longer applied to seek conditions ensuring solvability of the associated PDE. In this manuscript, a new theorem is formulated and proved to ensure the solvability. The solution for the associated PDE is locally analytic nonlinear coordinate transformation which provides foundation for observer realization and a series solution approach is developed. Simulation examples are presented to study the performance of the proposed observer. [ABSTRACT FROM AUTHOR]

The work addresses the output regulation problem for coupled linear multiple input multiple output (MIMO) hyperbolic partial integro-differential equation systems with disturbances affecting the systems through the space and boundary input. The exosystems are extended to generate ramp signals and general family of polynomial signals. The system decomposition is applied through the state transformation and yields a decoupled equivalent system. Based on the decoupled form, the backstepping transformation is applied and then in the new coordinate, the full state and output-feedback regulators are designed, respectively. For the state feedback regulator, the corresponding regulator equation is obtained and its solvability conditions are provided to facilitate the regulator design and feasibility. The design of observer-based regulator is based on the decoupling of the observer error system into a PDE subsystem and an ODE subsystem so that the backstepping approach achieves stabilisation by eigenvalue assignment leading to design of observer stabilizing gains. [ABSTRACT FROM AUTHOR]

In this paper, the output regulation problem is addressed for a class of linear hyperbolic infinite-dimensional systems with spatially varying coefficients modelling a large class of convection-dominated transport reaction systems. In particular, distributed parameter systems with bounded input and unbounded output operators are considered. First, we demonstrate a general conclusion about the exponential stability of the considered system by relating the stability to the solution of an associated differential equation. Based on the assumption that the hyperbolic system satisfies the exponential stability conditions, the main manuscript contribution is the development of two novel finite-dimensional regulators, output and error feedback regulators, such that the controlled output of the plant tracks a reference signal generated by a known signal process (exosystem). In order to guarantee the feasibility of the proposed regulators, the solvability of the corresponding Sylvester equations is discussed and the solvability conditions are provided. Finally, simulations of output regulation of an axial dispersion reactor and a relevant numerical example illustrate the main results and performance of the proposed regulators synthesis. [ABSTRACT FROM AUTHOR]

This paper investigates the output regulation problem for a class of regular first-order hyperbolic partial differential equation (PDE) systems. A state feedback and an error feedback regulator are considered to force the output of the hyperbolic PDE plant to track a periodic reference trajectory generated by a neutrally stable exosystem. A new explanation is given to extend the results in the literature to solve the regulation problem associated with the first-order hyperbolic PDE systems. Moreover, in order to provide the closed-loop stability condition for the solvability of the regulator problems, the design of stabilising feedback gain and its dual problem design of stabilising output injection gain are considered in this paper. This paper develops an easy method to obtain an adjustable stabilising feedback gain and stabilising output injection gain with the aid of the operator Riccati equation. [ABSTRACT FROM AUTHOR]