Robust Boundary Controller Design with Proportional Integral Observer for a Linear 1D Heat Equation

The problem of state estimation with boundary disturbance reconstruction and output regulation for linear 1D parabolic systems with a boundary measurement and unknown constant disturbance input on the opposite boundary is addressed. As the disturbance is unmatched it cannot be directly compensated by the measurement. The problem is addressed with an approach that consists of a backstepping observer for the nominal part with an asymptotic compensation term for the disturbance reconstruction and control offset compensation. The main difference to similar studies relies in the fact that the state space does not need to be extended, leading to a reduced-order observer setup. The effectiveness of the approach is illustrated using numerical simulations.