A Systematic Design of Backstepping-Based State Feedback Controllers for ODE-PDE-ODE Systems

Abstract

The backstepping design of stabilizing state feedback controllers is addressed for bidirectionally coupled ODE-PDE-ODE systems, that naturally arise for infinite-dimensional systems where actuator or sensor dynamics are taken into account. A common framework for both linear parabolic and hyperbolic partial differential equations with spatially varying coefficients is presented, that solely relies on the strict feedback form of these systems. Without imposing any limitation on the dimension of the ODEs and PDEs involved, a multi-step design algorithm is suggested, that makes use of classical concepts (such as integrator backstepping and output zeroing) known for the control of ODE systems. By that, backstepping is brought back to its ODE origins and a systematic design of backstepping controllers for ODE-PDE-ODE systems is offered, with a unified treatise for parabolic and hyperbolic PDEs.