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

Abstract

Abstract. 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. Mathematics Subject Classification.

Non‐quadratic exponential stabilisation of non‐linear hyperbolic partial differential equation systems

“…By pre-and post-multiplying (13) and (27) (31), (32) and (33) lead to (21), (22) and (23), respectively. Pre-and post-multiplying (30) by diag{M −1 , M −1 , M −1 } and diag{M −T , M −T , M −T } and using the same procedure as applied on (29) to derive (23), LMI (24) will be obtained.…”

“…Among other control methodologies, model predictive control has also been proposed for hyperbolic distributed parameter systems in a number of publications [3,[19][20][21]. Recently, LQ optimal control has been proposed for different classes of hyperbolic PDE model using matrix Riccati differential equation [22,23] and using spectral factorisation and operator Diophantine equation [24,25]. These methods [22][23][24][25] are only suitable for a neighbourhood of operating condition [1].…”

“…Recently, LQ optimal control has been proposed for different classes of hyperbolic PDE model using matrix Riccati differential equation [22,23] and using spectral factorisation and operator Diophantine equation [24,25]. These methods [22][23][24][25] are only suitable for a neighbourhood of operating condition [1].…”

Linear quadratic optimal boundary control of a diffusion-convection-reaction system

“…Subsequently, the infinite dimensional controllers are formulated for the infinite dimensional space realization of the system [Curtain and Zwart (1995), Callier and Winkin (1990), Callier and Winkin (1992)]. Within the class of distributed parameter systems, Aksikas et al(2008Aksikas et al( , 2009) studied the solution of LQ control problem for hyperbolic systems by solving an operator Riccati equation. Under some assumptions, the operator Riccati equation can be converted to an equivalent Matrix Riccati equation, which can be solved numerically.…”

Distributed optimal control of a Dimethyl Ether (DME) catalytic distillation column

“…The second methodology involves solving an operator Riccati equation (ORE) for a given state-space model [16]. This method was used in [17] for a particular class of hyperbolic PDEs. The methodology was then extended to a more general class of hyperbolic system by using an infinite-dimensional Hilbert state-space setting with distributed input and output [18].…”