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.

There are many industrial and biological reactiondiffusion 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 timevarying 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 time invariant 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 infinitedimensional 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.

The asymptotic behaviour is studied for a class of non-linear distributed parameter timevarying dissipative systems. This is achieved by using time-varying infinite-dimensional Banach state space description. Stability criteria are established, which are based on the dissipativity of the system in addition to another technical condition. The general development is applied to semi-linear systems with time varying nonlinearity. Stability criteria are extracted from the previous conditions. These theoretical results are applied to a class of transport-reaction processes. Different types of nonlinearity are studied by adapting the criteria given in the early portions of the paper.

This paper deals with boundary optimal control problem for coupled parabolic PDEODE systems. The problem is studied using innite-dimensional state space representation of the coupled PDE-ODE system. Linearization of the non-linear system is established around a steady state prole. Using some state transformations, the linearized system is formulated as a well-posed innite-dimensional system with bounded input and output operators. It has been shown that the resulting system is a Riesz Spectral system. The LQ-control problem is studied on the basis of the solution of the corresponding eigenvalues problem. The results have been applied to the case study of catalytic cracking reactor with catalyst deactivation. Numerical simulations are performed to illustrate the performances of the developed controller.

In this work characteristics-based model predictive control (CBMPC) of a fixed bed reactor with catalyst deactivation is studied. Performance of CBMPC has been analyzed for two cases: one that incorporates the catalyst deactivation within the reactor model and another that ignores the deactivation. Simulation results show that the performance of first controller that incorporates the catalyst deactivation is better than the controller that ignores the deactivation.

A linear infinite dimensional state space representation of a catalytic flow reversal reactor is used to formulate a state LQ-feedback operator via the solution of a Riccati differential equation. Flow velocity is used to keep the temperature and mole fraction of reactant in the reactor moving along a desired stationary state. Numerical simulations are used to show the performance of the formulated controller.

The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a time-varying plug flow tubular reactor. First some properties of the linearized model around a specific equilibrium profile are studied. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. The controller is applied to the nonlinear reactor system and tested numerically.

This contribution addresses the development of a Linear Quadratic regulator (LQ) for controlling concentration profiles along a catalytic distillation column which is modelled by a set of coupled hyperbolic Partial Differential and Algebraic Equations (PDAEs). The methodology is based on an infinite-dimensional representation of the system and solving the related Operator Riccati Equation (ORE). The performance of the designed control policy is assessed through a numerical simulation.

In this work, the boundary control of a distributed parameter system (DPS) modeled by parabolic partial differential equations with spatially varying coefficients is studied. An infinite dimensional state space setting is formulated and an exact transformation of the boundary actuation is realized to obtain an evolutionary model. The evolutionary model is used for subsequent linear quadratic regulator synthesis which incorporates the spatially varying coefficients of the underlying set of the PDEs. The formulated LQR controller is applied to the nonlinear model of the system and its performance is studied.

In this paper an infinite-dimensional LQR control-based design for a system containing linear hyperbolic partial differential equations coupled with linear ordinary differential equations is presented. The design is based on an infinite-dimensional Hilbert state-space representation of the coupled system. The feedback control gain is obtained by solving algebraic and differential matrix Riccati equations that result from an operator Riccati equation solution. The designed LQR control is applied to a system containing a continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) in series with the recycle-rate from PFR to CSTR as controlled variable. The LQR controller’s performance is evaluated by numerical simulation of the original nonlinear system.

This paper deals with linear-quadratic control problem for a catalytic flow reversal reactor using an infinite dimensional Hilbert space representation of the system. A LQ-controller is developed on the basis of the catalytic reactor with unidirectional flow. The controller is formulated to keep the distribution of the temperature along the axis of the reactor at stationary state by using the fluid flow velocity. We study the application of the controller on the catalytic reactor with reverse flow operation. We take advantage of the two-time scale characteristic of catalytic tubular reactors to develop a controller that requires only the measurement of the temperature along the axis of the reactor. Using the infinite dimensional state space, a state LQ-feedback operator is computed via the solution of a Riccati differential equation. The developed controller is tested numerically for the catalytic combustion of lean methane emissions in CFRR unit and implemented for a reactor configuration at the CANMET Energy Technology Centre Varennes, Quebec, Canada, and currently experimental tests are underway.

The linear-quadratic (LQ) optimal control problem is studied for a class of first-order hyperbolic partial differential equation models by using a nonlinear infinite-dimensional Hilbert state-space description. First the dynamical properties of the linearized model around some equilibrium profile are studied. Next the LQ-feedback operator is computed by using the corresponding operator Riccati algebraic equation whose solution can be obtained via a related matrix Riccati differential equation in the space variable. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed.

This contribution addresses the development of a Linear Quadratic Regulator (LQR) for a set of time-varying hyperbolic PDEs coupled with a set of time-varying ODEs through the boundary. The approach is based on an infinite-dimensional Hilbert state-space description of the system and the well-known operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of equivalent differential and algebraic matrix Riccati equations. The feedback operator can then be found by solving the resulting matrix Riccati equations. The performance of the designed control policy is assessed by applying it to a system of interconnecting continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) through a numerical simulation.

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.

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.

This paper proposes a method to synthesize an optimal controller for the SCR section of the diesel exhaust aftertreatment system, which is based on a system model consisting of coupled hyperbolic and parabolic partial differential equations (PDEs). This results in a boundary control problem, where the control objectives are to reduce the amount of NOx emissions and ammonia slip to the fullest extent possible using the inlet concentration of ammonia as the manipulated variable. The proposed method combines the method of characteristics and spectral decomposition to produce a non-linear model predictive control (NMPC) approach. The results show that the proposed NMPC is able to achieve a very high level of control performance in terms of NOx and ammonia slip reduction.

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 domain’s 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.

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.

This work deals with the linear-quadratic control problem for a non-isothermal packed-bed catalytic reactor, which is described by coupled hyperbolic and parabolic partial differential equations model. The classical Riccati equation approach, in the infinite-dimensional setting, is adopted. An optimal LQ-feedback is computed via the eigenvalues and the eigenfunctions of the parabolic subsystem, while a differential Riccati equation is derived for the hyperbolic subsystem. Numerical simulations are performed to show the performance of the designed controller on non-linear reactor system.

The paper deals with the linear-quadratic control problem for a time-varying partial differential equation model of a catalytic fixed-bed reactor. The classical Riccati equation approach, for time-varying infinite-dimensional systems, is extended to cover the two-time scale property of the fixed-bed reactor. Dynamical properties of the linearized model are analyzed by using the concept of evolution systems. An optimal LQ-feedback is computed via the solution of a matrix Riccati partial differential equation. Numerical simulations are performed to show the performance of the designed controller on the fixed-bed reactor.

In this contribution a linear quadratic (LQ) control design for a partial differential and algebraic equation (PDAE) system which represents a catalytic distillation process, is presented. The model involves a set of coupled partial differential equations (PDEs), ordinary differential equations (ODEs), and algebraic equations (AEs). The design is based on an infinitedimensional state-space representation of the system in a Hilbert space and the well-known operator Riccati equation (ORE) method. The underlying PDE-ODE-AE system is converted to one containing coupled PDEs-ODEs, in which the PDE part involves a hyperbolic operator with a space-varying and non-symmetric velocity matrix whose eigenvalues are not necessarily negative through of the domain. Moreover, in the resulting PDE-ODE system, the control variable acts through the ODE part on the boundaries of the PDE section. Using the boundary control transformation method, the PDE-ODE system is represented in an infinite-dimensional state-space with a homogeneous boundary condition. The stabilizability and the detectability properties of the resulting system are explored, which provide guarantees of the existence and uniqueness of the solution to the resulting ORE. The ORE is solved by converting it to a set of equivalent matrix Riccati equations. The designed LQ controller is implemented on the process and its performance is evaluated.

The present work proposes an extension of singlestep feedback linearization with pole-placement formulation to the class of nonlinear hyperbolic systems. In particular, the mathematical formulation in the context of singular PDE theory is utilized via system of first order quasi-linear singular PDEs within the nonlinear hyperbolic PDE setting to obtain single step state nonlinear transformation and feedback control law with prescribed closed loop dynamics. The solution of quasi linear singular PDE is guaranteed by the Lyapunov’s auxiliary theorem and locally invertible analytic transformation is applied by the full state feedback law to yield desired stable hyperbolic PDE system with assignable dynamics. The simultaneous state transformation and feedback linearization are realized in one step, avoiding the restrictions existing in other approaches.

A stability analysis is performed for a linearized plug flow reactor model. The dynamics of the linearized process are described by linear partial differential equations (PDE’s) with spatial-dependent coefficients. The analysis is performed on an equivalent triangularized model, that is described by a triangular system of PDE’s. Then the stability of the linearized model is established, by using the invariance of stability under system equivalence.

The concept of asymptotic stability is studied for a class of dissipative semi-linear infinite-dimensional Banach state space systems with nonlinearity defined on a convex subset of the state space. Stability criteria are established, which are based on a weaker technical condition of the m-dissipativity. These theoretical results are applied to a class of transport-reaction processes. Different types of nonlinearity are studied by adapting the criteria given in the early portions of the paper.

A general linear controller design method for a class of hyperbolic partial differential equations (PDE’s) systems is presented, which uses infinite-dimensional Hilbert state-space description. A state LQ-feedback operator is computed via the solution of a matrix Riccati differential equation in the space variable. The proposed method is applied to a fixed-bed chemical reactor control problem where one elementary reaction takes place. An optimal controller is designed for the linearized fixed-bed reactor model, it is applied to the nonlinear original model and the resulting closed-loop stability is analyzed.