Controller and observer design for first order LTI systems with unknown dynamics

The design of controllers and observers often relies on first order models of the system in question. These models are often obtained either through step-response tests, through on-line or off-line identification, or through developing a mathematical model. When the system in question has unknown or uncertain parameters, the developed model also contains uncertainties and the controller/observer design may result in bad performance or even instability. In this paper, we present a combined design of a controller and an observer for scalar linear time-invariant systems with unknown parameters. We combine a model reference adaptive controller, which does not require a model of the system, with a Luenberger observer which uses the desired closed-loop dynamics as its model. The method is given the name MRACO. Our proposed method is similar to what is known as closed-loop reference model adaptive control, but the key difference is that our method does not use a closed-loop reference model. We show through Lyapunov theory and by application of Barbâlat's lemma that all error states in the closed-loop system converge to zero and that all signals are bounded. Several simulations are performed to support the proofs.