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The present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation. [ABSTRACT FROM AUTHOR]

This manuscript addresses the output regulation problem for a class of scalar boundary controlled first-order hyperbolic partial integro-differential equation (PIDE) systems with Fredholm integrals. In particular, with the advantage of the backstepping approach, simple structure systems can be obtained such that regulator equations for the state feedback regulator design are analyzed and solved in backstepping coordinates. Moreover, the finite time output regulation is achieved. In the observer-based output feedback regulator design, it is not necessary that the outputs to be controlled belong to the available output measurements and these outputs can be distributed, point-wise and/or boundary in nature, while the boundary placed measurements are used for regulator design. For the observer gains design, a transformation of the ODE–PDE system into an ODE–PDE cascade is considered. It is also shown that the separation principle holds for the output feedback regulator design and the exponential output regulation is realized for the resulting stable closed-loop system. Finally, the output regulation results are illustrated with two numerical simulations: a Korteweg–de Vries-like equation and a PDE–ODE interconnected system. [ABSTRACT FROM AUTHOR]

This manuscript addresses the output regulation problem for linear distributed parameter systems (DPSs) with bounded input and unbounded output operators. We introduce novel methods for the design of the output feedback and error feedback regulators. In the output feedback regulator design, the measurements available for the regulator do not belong to the set of controlled outputs. The proposed output feedback regulator with the injection of the measurement y m ( t ) and reference y r ( t ) can realize both the plant and the exosystem states estimation, disturbance rejection and reference signal tracking, simultaneously. Moreover, new design approach provides an alternative choice for seeking the output injection gain in a traditional error feedback regulator design. The regulator parameters are easily configured to solve the output regulation problems, and to ensure the stability of the closed-loop systems. The results are demonstrated via computer simulation in two types of representative systems: the parabolic partial differential equation (PDE) system and the first order hyperbolic PDE system. [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]

This paper extends the backstepping-based observer design to the state estimation of parabolic PDEs with time-dependent spatial domain. The design is developed for the stabilization of a collocated boundary measurement and actuation of an unstable 1D heat equation with the application to the temperature distribution regulation in Czochralski crystal growth process. The PDE system that describes the estimation error dynamics is transformed to an exponentially stable target system through invertible transformations to obtain the time-varying kernel PDE defined on the time-varying triangular-shape domain. The exponential stability of the closed-loop system with an observer-based output-feedback controller is established by the use of a Lyapunov function. Finally, numerical solutions to the kernel PDEs and simulations are given to demonstrate successful stabilization of the unstable system. [ABSTRACT FROM AUTHOR]

In this work, an observer based controller synthesis is proposed based on the linear matrix inequality (LMI) framework to achieve stabilization of the parabolic PDE in the presence of input constraints. A novel feature of the proposed synthesis is to construct the LMI formulation within the modal space by converting the original parabolic PDE into an infinite-dimensional abstract state space setting. The state feedback controller and the Luenberger observer are developed by accounting for the entire infinite number of modal states, thereby stabilizing the system and reconstructing the state rigorously. The input constraints naturally existing in realistic applications are considered in the design framework. Finally, since initial modal states are hardly known in advance, the LMI formulation, based on the augmented modal state space, including unstable modal states, its estimation error and fast modal output, is developed to maximize the region of attraction (ROA) for the real process state, such that the controller and the observer are robust enough to the error in the initial state guess. By considering a numerical example of an unstable parabolic PDE, we demonstrate that if feasible, the state feedback controller and the observer, generated by the LMI formulation, have capability to drive the process state to the equilibrium steady state. [ABSTRACT FROM AUTHOR]

The thermal regulation problem for a lithium ion (Li-ion) battery with boundary control actuation is considered. The model of the transient temperature dynamics of the battery is given by a nonhomogeneous parabolic partial differential equation (PDE) on a two-dimensional spatial domain which accounts for the time-varying heat generation during the battery discharge cycle. The spatial domain is given as a disk with radial and angular coordinates which captures the nonradially symmetric heat-transfer phenomena due to the application of the control input along a portion of the spatial domain boundary. The Li-ion battery model is formulated within an appropriately defined infinite-dimensional function space setting which is suitable for spectral controller synthesis. The key challenges in the output feedback model-based controller design addressed in this work are: the dependence of the state on time-varying system parameters, the restriction of the input along a portion of the battery domain boundary, the observer-based optimal boundary control design where the separation principle is utilized to demonstrate the stability of the closed loop system, and the realization of the outback feedback control problem based on state measurement and interpolation of the temperature field. Numerical results for simulation case studies are presented. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3782-3796, 2013 [ABSTRACT FROM AUTHOR]

Abstract: Alternation of normal action-potential morphology in the myocardium is a condition with a beat-to-beat oscillation in the length of the electric wave which is linked through electromechanical coupling to the cardiac muscle contraction, and is believed to be the first manifestation of the onset of life threatening ventricular arrhythmias and sudden cardiac death. In this work, the effects of electrical and mechanical stimuli are utilized in alternans annihilation problem. Electrical stimuli that alter the action-potential morphology are represented by a pacer located at the domain''s boundary, while mechanical stimuli are distributed within the spatial domain and affect the action potential by altering intracellular calcium kinetics. Alternation of action potential is described by the small amplitude of alternans parabolic partial differential equation (PDE). Spatially uniform unstable steady state of the alternans amplitude PDE is stabilized by optimal control methods through boundary and spatially distributed actuation. Mixed boundary and spatially distributed actuation is manipulated by a linear quadratic regulator (LQR) in the full-state-feedback control structure and in a compensator design with a finite-dimensional Luenberger-type observer, and it achieves exponential stabilization in a finite size tissue cable length. The proposed control problem formulation and the performance and robustness of the closed-loop system under the proposed linear controller are evaluated through simulations. [Copyright &y& Elsevier]

This paper develops a novel observer-based output-feedback control law for a class of interconnected first-order hyperbolic partial integral differential equations (PIDEs) with in-domain, non-local coupling terms and measurement delay compensation. A transport PDE is introduced to account for in-domain recycle delay which results in an extended spatial domain with the PIDE and transport PDE being cascaded. Since standard backstepping approach cannot be applied, an affine Volterra integral transform is introduced which allows for the construction of a boundary controller for the PIDE and delay compensation. An observer backstepping transformation containing four kernels is realized, reconstructing the states throughout the delayed measurement. By the use of the backstepping method, a set of coupled kernel equations where one of the kernels has two boundary conditions is established. The existence and invertibility for the control and observer transformations are proved and the finite-time stability for the output system is established while results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]