James Ng

Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

Optimal control of a class of linear nonautonomous parabolic PDE via two-parameter semigroup representation

This paper considers the two-parameter semigroup representation of a class of parabolic partial differential equation (PDE) with time and spatially dependent coefficients. The properties of the PDE which are necessary for the initial and boundary value problem to be posed as a linear nonautonomous evolution equation on an appropriately defined infinite-dimensional function space are presented. Using these properties, the associated nonautonomous operator generates a two-parameter semigroup which yields the generalized solution of the initial and boundary value problem. The explicit expression of the two-parameter semigroup is provided and enables the application of optimal control theory for infinite dimensional systems.

Application of optimal boundary control to reaction-diffusion system with time-varying spatial domain

This paper considers the optimal boundary control of a parabolic partial differential equation (PDE) with time-varying spatial domain which is coupled to a second order ordinary differential equation (ODE) describing the time evolution of the domain boundary. The infinite-dimensional state space representation of the PDE yields a linear non autonomous evolution system with an operator which generates a two-parameter semigroup with analytic expression provided in this work. The nonautonomous evolution system is trans formed into an extended system which enables the optimal boundary control problem to be considered. The optimal control law of the extended system is determined and numerical results of the closed-loop feedback system are provided.

Model predictive control formulation for a class of time-varying linear parabolic PDEs

This paper considers the model predictive control (MPC) formulation for a class of discrete time-varying linear state-space model representations of parabolic partial differential equations (PDEs) with time-dependent parameters. The time-dependence of the parameters are due to the changes in physical properties or operating conditions of the system such as phase transformation, reactor catalyst fouling, and/or domain deformations which arise in many industrial processes. The MPC formulation is constructed for the low dimensional discrete finite-dimensional state space representation of the PDE system and constraints on input and infinite-dimensional state evolution are incorporated in the convex optimization algorithm. The underlying MPC synthesis is utilizing the appropriately defined model representation of the PDE and yields convex quadratic optimization problem which includes input and PDE state constraints. Using the illustrative example of a crystal growth process in which the time-varying property is associated with the evolution of grown crystal, the proposed time-varying MPC formulation is implemented for the optimal crystal temperature regulation problem under the presence of input and state constraints.

Optimal control of transport-reaction system with time varying spatial domain

This paper deals with the optimal control of multi-dynamics process described by the rigid body dynamics equation and time-varying parabolic PDE. The optimal control is realized for the crystal growth process described by the underlying dynamics of the transport-reaction process given by the parabolic partial differential equations (PDEs) with the time varying spatial domain that is coupled with the rigid body dynamics representing the pulling of the pure crystal out of melt. The underlying transport-reaction system is developed from the first principles and the associated dynamics is analyzed in the appropriate functional state space setting. The complete description of the evolutionary parabolic domain time varying PDE is provided in the operator form and exploited within the optimal control setting, together with the optimal control of crystal pulling out of melt. Numerical simulations demonstrate a realization of optimal control law and its effects on both the temperature profile in the crystal with the time varying domain and crystal domain time evolution.

Aspects of controllability and observability for time-varying PDE systems

There are many industrial and biological reaction-diffusion systems which exhibit time-varying features where certain parameters of the system change during the process. The underlying transport-phenomena are often modelled using parabolic partial differential equations (PDEs) with time-varying coefficients which describe the dynamics of the process. Often it is of interest to control this dynamical behaviour such as the regulation of temperature or concentration, and one approach is the use of infinite-dimensional systems theory to represent the PDE models, with time-varying process parameters, as abstract nonautonomous evolution equations on appropriately defined function spaces. In contrast to timeinvariant control problems, the theory for controllability and observability for time-varying systems is less well established. In this work, we consider some pertinent aspects regarding the controllability and observability of nonautonomous infinite-dimensional systems. An example is considered for which the conditions for exact, null, and approximate controllability and observability are verified, and some observations regarding the influence of time-varying input and measurement operators are provided.

Multiscale optimal control of transport-reaction system with time varying spatial domain

This paper deals with the multi-scale optimal control of transport-reaction systems with the underlying dynamics governed by the second order rigid body dynamics, coupled with the parabolic partial differential equations (PDEs) with time-varying spatial domains, developed by considering the first principles dynamical equations for continuum mechanics. A functional theory is employed to explore the process model time-varying features, which lead to the characterization of the time varying spatial operator as a Riesz-spectral operator. This characterization facilitates the formulation of the optimal control problem where the infinite-dimensional system associated with the time-varying spatial operator is coupled with a finite-dimensional system describing the motion of the domain. The temperature control of the underlying transportreaction dynamics is realized through the optimal control law regulating the trajectory of the domain boundary coupled with the optimal heating input applied along the domain. The optimal control law associated with the domains boundary is obtained as a solution to the algebraic Riccati equation, while the optimal control law associated with the temperature regulation is obtained as a solution of a time-dependent Ricatti equation.

Discrete mechanics optimal control (DMOC) and model predictive control (MPC) synthesis for reaction-diffusion process system with moving actuator

This paper is concerned with the computation of optimal motion control as well as the optimal input injection policy of an actuator arm regulating the temperature in a reaction-diffusion system. The system has two dynamical components consisting of the arm mechanics with inertial, elastic and damping properties, which is driven by bounded mechanical actuation controls and an underlying reaction-diffusion system described by the parabolic PDE. The state of the actuator arm parametrizes the input injection operator of the parabolic PDE systems model and causes coupling between the two dynamical systems generally operating at different time scales. The method proposed in this paper is aimed at solving this coupled problem. The actuator mechanics and its control are achieved in the discrete mechanics and optimal control (DMOC) framework, while the input injection for the reaction diffusion system is calculated by the modal model predictive control (MMPC) algorithm suitable for the dissipative systems. The actuation arm policy and input to the parabolic PDE system include in its realization low-order discrete representation of the parabolic PDE evolution and incorporate optimality with respect to both the state of the PDE and the actuator displacement cost from current to some more optimal control position as well as naturally present input and PDE state constraints. The proposed actuation arm policy and optimal stabilization of the unstable reaction-diffusion system in the presence of constraints in the full state-feedback controller realization have been evaluated through simulations.

Model predictive control of Czochralski crystal growth process

This paper presents a model predictive control (MPC) formulation for the Czochralski (CZ) crystal growth which is given by the discrete finite-dimensional system representation of a class of parabolic partial differential equations (PDEs) with time-varying features. The motivation behind this work is to address the problem of stabilizing, infinite horizon, linear quadratic regulator synthesis for PDEs with time-dependent parameters which affect the dynamics of the underlying system and appear in the state-space representation. The modal decomposition of the PDE facilitates its approximation by a finite-dimensional linear state-space system. A receding horizon regulator is proposed which incorporates the time-dependence of the PDE parameters and numerical simulations are carried out which demonstrate the optimal stabilization of the process under state and input constraints.