On numerical observers. Application to a simple academic adaptive control example

Numerical differentiation techniques and the so-called regulation model method are used in the design of a compensation law which makes the system y = u + y² not only stable but follow exogenous inputs. The stabilization of this simple academic system, where is assumed to be a constant with unknown value, is known as challenging. The compensator is shown to comply with the following conditions: 1. the measurements are available only at sampling instants, 2. and the measurements are known through an additive random noise. The regulation model method consists in the specification of the way the plant output should follow the reference input while the numerical differentiation ingredients avail the observer design task. The main design parameters consist of those of a second order constant linear dynamics. There choices govern the stability of the. compensated system, the noise filtering, and the speed at which the output is steered to the reference input. The stability analysis reveals the noteworthy facts that, for fixed design parameters, the compensator is able to handle a quite significant amount of measurements noise, as well as slowly varying.