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

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.     

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

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.     

Infinite-dimensional LQ optimal control of a dimethyl ether (DME) catalytic distillation column

This contribution addresses the development of a linear quadratic (LQ) regulator in order to control the concentration profiles along a catalytic distillation column, which is modelled by a set of coupled hyperbolic partial differential and algebraic equations (PDAEs). The proposed method is based on an infinite-dimensional state-space representation of the PDAE system which is generated by a transport operator. The presence of the algebraic equations, makes the velocity matrix in the transport operator, spatially varying, non-diagonal, and not necessarily negative through of the domain. The optimal control problem is treated using operator Riccati equation (ORE) approach. The existence and uniqueness of the non-negative solution to the ORE are shown and the ORE is converted into a matrix Riccati differential equation which allows the use of a numerical scheme to solve the control problem. The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. The performance of the designed control policy is assessed through a numerical study.     

Optimal control of an advection-dominated catalytic fixed-bed reactor with catalyst deactivation

The paper focuses on 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 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 evaluate the closed loop performance of the designed controller on the fixed-bed reactor. The performance of the proposed controller is compared to performance of an infinite dimensional controller formulated by ignoring the catalyst deactivation. Simulation results show that the performance of the proposed controller is better compared to the controller ignoring the catalyst deactivation when the deactivation time is close to the resident time of the reactor.     

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.     

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.