Joseph J. Winkin

Dynamical analysis of distributed parameter tubular reactors

This paper is dedicated to the dynamical analysis of tubular reactor models, namely plug-flow and axial dispersion reactors involving sequential reactions for which the kinetics only depends on the concentrations of the reactants involved in the reaction. This analysis is performed by describing these models as infinite-dimensional state-space systems, with bounded observation and control operators. It is shown that these systems are exponentially stable, and that (a) the plug-flow reactor model is observable when the component concentrations are measured at the reactor output and observable with respect to its physical domain when only the product concentration is measured, and it is reachable with respect to its physical domain when it is controlled at the reactor input; and (b) any finite set of dominant modes of the axial dispersion reactor model is observable, when either all the component concentrations or only the product concentration are measured, and reachable, by using appropriate sensors and actuators. The dynamical properties of related finite-dimensional models are also discussed.

Robust boundary control of systems of conservation laws

The stability problem of a system of conservation laws perturbed by non-homogeneous terms is investigated. These non-homogeneous terms are assumed to have a small C1-norm. By a Riemann coordinates approach a sufficient stability criterion is established in terms of the boundary conditions. This criterion can be interpreted as a robust stabilization condition by means of a boundary control, for systems of conservation laws subject to external disturbances. This stability result is then applied to the problem of the regulation of the water level and the flow rate in an open channel. The flow in the channel is described by the Saint-Venant equations perturbed by small non-homogeneous terms that account for the friction effects as well as external water supplies or withdrawals.

Brief paper: LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor

A general linear controller design method for a class of hyperbolic linear partial differential equation (PDEs) systems is presented. This is achieved by using an infinite-dimensional Hilbert state-space description with infinite-dimensional (distributed) input and output. 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 linearized fixed-bed reactor model, it is applied to the original nonlinear model and the resulting closed-loop stability is analyzed. Numerical simulations are performed to show the performance of the designed controller.

LQ-optimal control of infinite-dimensional systems by spectral factorization

Abstract We consider the standard LQ-optimal control of an exponentially stabilizable and detectable infinite-dimensional semigroup state space system with bounded sensing and control. It is reported that the reachable restriction of the LQ-optimal state feedback operator can be identified (1) by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper-stable transfer function algebra Â, and (2) by obtaining any constant solution of an operator diophantine equation over Â. The properties of the restricted solution feedback operator are discussed. It is shown in particular that the LQ-optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable. These theoretical results are applied to a simple model of heat diffusion with “collocated” sensor and actuator, leading to an approximation procedure converging exponentially fast to the LQ-optimal state feedback operator. This procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus.

Spectral factorization and LQ-optimal regulation for multivariable distributed systems

A necessary and sufficient condition is proved for the existence of a bistable spectral factor (with entries in the distributed proper-stable transfer function algebra 𝒜-) in the context of distributed multivariable convolution systems with no delays; a by-product is the existence of a normalized coprime fraction of the transfer function of such a possibly unstable system (with entries in the algebra ℬ of fractions over 𝒜-). We next study semigroup state-space systems SGB with bounded sensing and control (having a transfer function with entries in ℬ) and consider its standard LQ-optimal regulation problem having an optimal state feedback operator K0. For a system SGB, a formula is given relating any spectral factor of a (transfer function) coprime fraction power spectral density to K0; a by-product is the description of any normalized coprime fraction of the transfer function in terms of K0. Finally, we describe an alternative way of finding the solution operator K0 of the LQ-problem using spectral factor...

Trajectory analysis of nonisothermal tubular reactor nonlinear models

The existence and uniqueness of the state trajectories (temperature and reactant concentration) are analyzed for nonisothermal plug flow and axial dispersion tubular reactor models. It is mainly shown that these trajectories exist on the whole (nonnegative real) time axis and the set of all physically feasible state values is invariant under the dynamics equations. The main nonlinearity in the model originates from the Arrhenius-type kinetics term in the model equations. The analysis essentially uses Lipschitz and dissipativity properties of the nonlinear operator involved in the dynamics and the concept of state trajectory positivity.

Optimal LQ-Feedback Regulation of a Nonisothermal Plug Flow Reactor Model by Spectral Factorization

The linear-quadratic (LQ) optimal temperature and reactant concentration regulation problem is studied for a partial differential equation model of a nonisothermal plug flow tubular reactor by using a nonlinear infinite dimensional Hilbert state space description. First the dynamical properties of the linearized model around a constant temperature equilibrium profile along the reactor are studied: it is shown that it is exponentially stable and (approximately) reachable. Next the general concept of LQ-feedback is introduced. It turns out that any LQ-feedback is optimal from the input-output viewpoint and stabilizing. For the plug flow reactor linearized model, a state LQ-feedback operator is computed via the solution of a matrix Riccati differential equation (MRDE) in the space variable. Thanks to the reachability property, the computed LQ-feedback is actually the optimal one. Then the latter is applied to the nonlinear model, and the resulting closed-loop system dynamical performances are analyzed. A criterion is given which guarantees that the constant temperature equilibrium profile is an asymptotically stable equilibrium of the closed-loop system. Moreover, under the same criterion, it is shown that the control law designed previously is optimal along the nonlinear closed-loop model with respect to some cost criterion. The results are illustrated by some numerical simulations.

Distributed system transfer functions of exponential order

σo is a real number. We construct a transfer function algebra of fractions, viz. F˚(σo), for modelling possibly unstable distributed systems such that (i) [fcirc] in F˚(σo) is holomorphic in Re s ≧ σo, (i.e. is σo-stable), iff [fcirc] is σo-exponentially stable, and (ii) we allow delay in the direct input-output transmission of the system. This algebra is (a) a restriction of the algebra B˚(σo) developed by Callier and Desoer (1978, 1980 a), (b) an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained. The algebra F˚ (σo) can be used for modelling and feedback system design. It is shown that standard semigroup systems are better modelled by a transfer function in F˚(σo) rather than B˚(σo).

Power-shaping control: Writing the system dynamics into the Brayton–Moser form

Power-shaping control is an extension of energy-balancing passivity-based control that is based on a particular form of the dynamics, the Brayton–Moser form. One of the main difficulties in this control approach is to write the dynamics in the suitable form since this requires the solution of a partial differential equation (PDE) system with an additional sign constraint. Here a general methodology is described for solving this partial differential equation system. The set of all solutions to the PDE system is given as the solution of a linear equation system. Furthermore a necessary condition is given so that a solution of the linear system which meets the sign condition exists. This methodology is illustrated on the case of a chemical reactor, where the physical knowledge of the system is used to find a suitable solution.

Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor

The concept of asymptotic stability is studied for a class of infinite-dimensional semilinear Banach state space (distributed parameter) systems. Asymptotic stability criteria are established, which are based on the concept of strictly m-dissipative operator. These theoretical results are applied to a nonisothermal plug flow tubular reactor model, which is described by semilinear partial differential equations, derived from mass and energy balances. In particular it is shown that, under suitable conditions on the model parameters, some equilibrium profiles are asymptotically stable equilibriums of such model.