Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

This article deals with the backstepping design of observer-based compensators for parabolic ODE–PDE–ODE systems. The latter consist of
$\boldsymbol {n}$
coupled parabolic PDEs with distinct diffusion coefficients and spatially-varying coefficients, that are bidirectionally coupled to ODEs at both boundaries. The actuation and sensing appears through these ODEs resulting in a challenging control problem. For this setup a systematic backstepping approach is proposed, in order to determine a state feedback controller and an observer. In particular, the state feedback loop and the observer error dynamics are mapped into stable ODE–PDE–ODE cascades by making use of a sequence of transformations. With this, the design can be traced back to the solution of kernel equations already found in the literature as well as initial and boundary value problems, that can be solved numerically. Exponential stability of the closed-loop system is verified, wherein the decay rate can be directly specified in the design. The results of the article are illustrated by the output feedback control of an unstable ODE–PDE–ODE system with two coupled parabolic PDEs.