Finite-time output regulation for linear 2×2 hyperbolic systems using backstepping

This contribution presents the backstepping design of output feedback regulators for boundary controlled linear 2×2 hyperbolic systems, that achieve regulation in finite time. It is assumed that the disturbances can act in-domain, at both boundaries and at the output to be controlled. The latter need not be available for measurement and consists of in-domain pointwise, distributed or boundary outputs. Firstly, a solution of the finite-time state feedback regulator problem is given on the basis of the regulator equations. They are formulated in backstepping coordinates so that a solution is attainable in closed-form. This leads to a very straightforward regulator design for 2×2 hyperbolic systems with a general class of outputs. Then, a finite-dimensional reference observer that converges in finite-time is introduced, which consists of two observers and a delay. This result is extended to the backstepping design of finite-time disturbance observers for 2×2 hyperbolic systems with a collocated measurement. In particular, two backstepping disturbance observers are determined so that after introducing a delay the disturbance model and plant states can be estimated in finite-time. Hence, by combining the state feedback regulator with these observers a finite-time output feedback regulator is obtained. For the state feedback regulator and the disturbance observer existence conditions are derived in terms of the plant transfer behaviour. A simple example with an in-domain pointwise and distributed output illustrates the theoretical results.