Discrete-time output observers for boundary control systems

The paper studies the output observer design problem for a linear infinite-dimensional control plant modelled as an abstract boundary input/output control system. It is known that such models lead to an equivalent state space description with unbounded control (input) and observation (output) operators. For this class of infinite-dimensional systems we use the Cayley transform to approximate the sophisticated infinite-dimensional continuous-time model by a discrete-time infinite-dimensional one with all involved operators bounded. This significantly simplifies mathematical aspects of the observer design procedure. As is well known, the essential feature of the Cayley transform is that it preserves various system theoretic properties of the control system model, which may be useful in analysis. As an illustration, we consider an example of designing an output observer for the one-dimensional heat equation with measured controls (inputs) in the Neumann boundary conditions, measured outputs in the Dirichlet boundary conditions and an unmeasured output at a fixed point within the domain. Numerical simulations of this example show that the interpolated continuous-time signal, obtained from the discrete-time observer, can be successfully used for tracking the continuous-time plant output.