Adaptive observers with exponential rate of convergence

The problem of observing the state of an unknown, time invariant linear system from measurements of its input and output is considered. Instead of adapting the parameters in a Luenberger observer to solve the problem, as was done by earlier authors, the approach taken here proceeds from a so-called parametrized observer, which is only an alternative, equivalent representation of the Luenberger observer. However, the parametrized observer has a different structure where the state estimate is a linear function (and not a functional) of its parameters. Therefore, adapting the parameters in the parametrized observer results in a complete separation of the observer dynamics from the adaptive loop which substantially simplifies the design of suitable parameter adaptation schemes. Three such schemes are presented and proven to be globally exponentially rather than asymptotically convergent. In particular, the second and the third adaptation schemes allow the construction of adaptive observers with arbitrarily high (exponential) rates of convergence.