Ilyasse Aksikas

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 linear–quadratic control of coupled parabolic–hyperbolic PDEs

ABSTRACT This paper focuses on the optimal control design for a system of coupled parabolic–hypebolic partial differential equations by using the infinite-dimensional state-space description and the corresponding operator Riccati equation. Some dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the linear–quadratic (LQ)-optimal control problem. A state LQ-feedback operator is computed by solving the operator Riccati equation, which is converted into a set of algebraic and differential Riccati equations, thanks to the eigenvalues and the eigenvectors of the parabolic operator. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ-optimal controller designed in the early portion of the paper is implemented for the original nonlinear model. Numerical simulations are performed to show the controller performances.

LQ (optimal) control of hyperbolic PDAEs

The linear quadratic control synthesis for a set of coupled first-order hyperbolic partial differential and algebraic equations is presented by using the infinite-dimensional Hilbert state-space representation of the system and the well-known operator Riccati equation (ORE) method. Solving the algebraic equations and substituting them into the partial differential equations (PDEs) results in a model consisting of a set of pure hyperbolic PDEs. The resulting PDE system involves a hyperbolic operator in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. The C0-semigroup generation property of such an operator is proven and it is shown that the generated C0-semigroup is exponentially stable and, consequently, the ORE has a unique and non-negative solution. Conversion of the ORE into a matrix Riccati differential equation allows the use of a numerical scheme to solve the control problem.

Asymptotic stability of time-varying distributed parameter semi-linear systems

Abstract The asymptotic behaviour is studied for a class of non-linear distributed parameter time-varying 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.

Asymptotic behaviour of contraction non-autonomous semi-flows in a Banach space

The asymptotic behaviour is studied for a class of non-autonomous infinite-dimensional non-linear dissipative systems. This is achieved by using the concept of contraction semi-flow, which is a generalization of contraction non-linear semigroup. Conditions are presented under which the solution of the abstract differential equation converges to the omega limit set (the equilibrium profile, respectively). The general development is applied to semi-linear systems with time-varying non-linearity. Asymptotic behaviour and stability criteria are established on the basis of the conditions given in the early portion of the paper. The theoretical results are applied to a general class of first-order hyperbolic time-varying semi-linear PDEs.

Model predictive control of selective catalytic reduction in diesel-powered vehicles

This paper proposes a method to synthesize an optimal controller for the SCR section of the diesel exhaust after-treatment 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.

Optimal control of coupled parabolic–hyperbolic non-autonomous PDEs: infinite-dimensional state-space approach

ABSTRACT This paper deals with the design of an optimal state-feedback linear-quadratic (LQ) controller for a system of coupled parabolic–hypebolic non-autonomous partial differential equations (PDEs). The infinite-dimensional state space representation and the corresponding operator Riccati differential equation are used to solve the control problem. Dynamical properties of the coupled system of interest are analysed to guarantee the existence and uniqueness of the solution of the LQ-optimal control problem and also to guarantee the exponential stability of the closed-loop system. Thanks to the eigenvalues and eigenfunctions of the parabolic operator and also the fact that the hyperbolic-associated operator Riccati differential equation can be converted to a scalar Riccati PDE, an algorithm to solve the LQ control problem has been presented. The results are applied to a non-isothermal packed-bed catalytic reactor. The LQ optimal controller designed in the early portion of the paper is implemented for the original non-linear model. Numerical simulations are performed to show the controller performances.

Boundary optimal control design for a system of parabolic–hyperbolic PDEs coupled with an ODE

ABSTRACT This paper deals with the design of a boundary optimal controller for a general model of parabolic–hyperbolic PDEs coupled with an ODE. The augmented infinite-dimensional state-space representation has been used in order to solve the optimal state-feedback control problem. By using the perturbation theorem, it has been shown that the system generates a -semigroup on the augmented state space. Also, the dynamical properties of both the original and the augmented systems have been studied. Under some technical conditions, it has been shown that the augmented system generates an exponentially stabilisable and detectable -semigroups. The linear-quadratic control problem has been solved for the augmented system. A decoupling technique has been implemented to decouple and solve the corresponding Riccati equation. Monolithic catalyst reactor model has been used to test the performances of the developed controller through numerical simulations.

Single-step full-state feedback control design for nonlinear hyperbolic PDEs

ABSTRACT 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 Lyapunovs 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.

Optimal control of a time-varying system of coupled parabolic-hyperbolic PDEs

This paper is devoted to design an optimal linear quadratic controller for a time-varying system of coupled parabolic and hyperbolic partial differential equations (PDEs). Infinite-dimensional state space approach is adopted to solve the control problem via the well-known operator Riccati equation. The latter is converted into a scalar partial differential equation coupled with a set of ordinary differential equations. The main algorithm to solve the Riccati equation is presented. The designed controller is tested on a model of catalytic packed-bed chemical reactor to show the performances of the developed controller.