LQ control design of a class of hyperbolic PDE systems: Application to fixed-bed reactor (original) (raw)

Optimal linear–quadratic control of coupled parabolic–hyperbolic PDEs

International Journal of Control, 2016

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 packedbed 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 for a class of time-varying coupled PDEs-ODEs system

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 infinitedimensional Hilbert state-space realization of the system and operator Riccati equation (ORE). In order to solve the optimal control problem, the ORE is converted to a set of differential and algebraic matrix Riccati equations. The feedback gain can then be found by solving the resulting matrix Riccati equations. The control policy is applied to a system of continuous stirred tank reactor (CSTR) and a plug flow reactor (PFR) in series and the controller performance is evaluated by numerical simulation.

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

International Journal of Systems Science, 2018

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 hyperbolicassociated 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.

LQ control of coupled hyperbolic PDEs and ODEs: Application to a CSTR-PFR system

IFAC Proceedings Volumes, 2010

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.

Optimal control design for time-varying catalytic reactors: a Riccati equation-based approach

International Journal of Control, 2009

The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a timevarying catalytic reactor. First, the dynamical properties of the linearised model 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. Finally, the designed controller is applied to the non-linear reactor system and tested numerically.

Optimal LQ-Control of a PDAE Model of a Catalytic Distillation Process

IFAC Proceedings Volumes, 2012

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.

Feedback control of hyperbolic PDE systems

AIChE Journal, 1996

This article deals with distributed parameter systems described by first-order hyperbolic partial differential equations (PDEs), for which the manipulated input, the controlled output, and the measured output are distributed in space. For these systems, a general output-feedback control methodology is developed employing a combination of theory of PDEs and concepts ffom geometric control. A concept of characteristic index is introduced and used for the synthesis of distributed state-feedback laws that guarantee output tracking in the closed-loop system. Analytical formulas of distributed outputfeedback controllers are derived through combination of appropriate distributed state observers with the developed state-feedback controllers. Theoretical analogies between our approach and available results on stabilization of linear hyperbolic PDEs are also identified. The developed control methodology is implemented on a nonisothermal plug-flow reactor and its performance is evaluated through simulations. P. D. Christofides is presently at the approach limits the controller performance, and may Iead to unacceptable control quality.

Optimal control of a time-varying catalytic fixed bed reactor with catalyst deactivation

2009 American Control Conference, 2009

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 fixedbed 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.

Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

Esaim-control Optimisation and Calculus of Variations, 2008

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 (distributed parameter) 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 is 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.

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

International Journal of Control, 2018

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.