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