Compensating the distributed effect of a wave PDE in the actuation or sensing path of MIMO LTI systems

The problem of compensation of infinite-dimensional actuator or sensor dynamics of more complex type than pure delay was solved recently using the backstepping method for PDEs. In this paper we construct an explicit feedback law for a multi-input LTI system which compensates the wave PDE dynamics in its input and stabilizes the overall system. Our design is based on a novel infinite-dimensional backstepping–forwarding transformation. We illustrate the effectiveness of our design with a simulation example of a single-input second order system, in which the wave input enters the system through two different channels, each one located at a different point in the domain of the wave PDE. Finally, we consider a dual problem where we design an exponentially convergent observer that compensates the distributed effect of the wave sensor dynamics.

Research highlights
► Stabilization of multi-input LTI systems with distributed inputs that satisfy wave PDEs, is achieved.
► An observer for multi-output LTI systems is designed, when the sensor dynamics are of wave type and their affect to the outputs is distributed.
► Novel backstepping–forwarding transformations are introduced for the control design of coupled systems of ODEs and PDEs with distributed coupling.