Distributed control of linear partial integro-differential equations based on the input-output linearisation approach

Abstract

In this paper, the input-output linearisation control approach is extended to distributed parameter systems whose dynamical behaviour is described by a partial integro-differential equation. The design of the infinite dimensional state feedback controller is achieved using the late lumping approach, i.e., using the partial integro-differential equation model without any prior reduction or approximation. Thus, based on the notion of the characteristic index as a generalisation of the relative degree, a distributed state feedback controller is designed by evaluating the successive time derivatives of the controlled output. The designed controller yields in closed loop a first order lumped parameter system where the time constant is a design parameter that fixes the desired dynamical behaviour. The stability of the closed loop system is investigated, based on semi-group theory, by employing the perturbation theorem of the bounded linear operators, and the sufficient condition for exponential stability in L2-norm is derived. This condition yields the upper bound for the design parameter, i.e., the time constant. Both output tracking and stabilisation capabilities of the developed state feedback are demonstrated through numerical simulation by considering three application examples: Volterra, Fredholm and Fredholm-Volterra PIDEs. The effectiveness of the developed controller is shown by simulation.