Output Feedback Boundary Control of a Heat PDE Sandwiched Between Two ODEs

Abstract

We present designs for exponential stabilization of an ODE-heat PDE-ODE coupled system where the control actuation only acts in one ODE. The combination of PDE backstepping and ODE backstepping is employed in a state-feedback control law and in an observer that estimates PDE and two ODE states only using one PDE boundary measurement. Based on the state-feedback control law and the observer, the outputfeedback control law is then proposed. The exponential stability of the closed-loop system and the boundedness and exponential convergence of the control law are proved via Lyapunov analysis. Finally, numerical simulations validate the effectiveness of this method for the "sandwiched" system.

Output feedback control of general linear heterodirectional hyperbolic ODE–PDE–ODE systems

“…Remark 7: The fact that the state feedback controller (33) depends on boundary values and a time derivative thereof is not unexpected. A similar result is obtained in [4], [6], [24], [23] for the control of ODE-PDE-ODE systems. It should be remarked, however, that the implementation of this state feedback controller poses no serious problems in applications.…”

“…So far, only few results exist for the backstepping control of parabolic ODE-PDE-ODE systems in the literature. A first approach is presented in [23] for a single parabolic PDE. Therein, the backstepping state feedback control of a heat equation cascaded with an actuating ODE at one boundary and a bidirectionally coupled ODE at the other boundary is presented.…”

Ein systematischer Backstepping-Zugang zur Regelung gekoppelter ODE-PDE-ODE-Systeme

“…für n i(k) (z) = N(z)φ i(k) , wobeī ergeben. Die inversen Transformationen von (57) mit den Integralkernen K 1,I (z, ζ ) ∈ ℝ n×n und K 2,I (z, ζ ) ∈ ℝ n×n sind analog zu (20) definiert. Wählt man den Kern K 1 (z, ζ ) ∈ ℝ n×n als Lösung der klassischen Kerngleichungen Λ(z) z K 1 (z, ζ ) + ζ K 1 (z, ζ )Λ(ζ ) = K 1 (z, ζ )A(ζ ) (59a)…”

“…B. [20] oder der PDE-ODE-Fall in [17]). Im Gegensatz zu diesen Einschrittentwürfen wird in [7] und [6] die Verwendung mehrerer Transformationsschritte vorgeschlagen, also ein Mehrschrittverfahren.…”

Modeling and adaptive control for a spatial flexible spacecraft with unknown actuator failures

“…In recent decades, the control problems of flexible structures described by infinite dimensional systems have received considerable attention. Several control schemes have been developed for heat equations [10,11], beam equations [12][13][14], string equations [15][16][17][18], and other common equations employed in flexible systems. In particular, flexible spacecraft are generally described using Euler-Bernoulli beam models.…”