Application of Luenberger's observer in RFPT-based adaptive control — A case study

The traditional way of thinking in controller design prefers the use of the “state space representation” introduced by R. Kalman in the early sixties of the past century. This system description is in close relationship with linear or at least partly linear system in which the linear part can be used in forming a quadratic Lyapunov function in the stability proof. In the standard model of such systems it is assumed that the state of the system is not directly observable, only certain linear functions of the state variable are directly measurable. Since such approaches introduce certain feedback gains for the state variable, observers are needed that calculate the estimation of the state variable on the basis of directly measurable quantities. The Luenberger observers solve this task via introducing a differential equation for the estimated state. In order to avoid the mathematical difficulties of Lyapunov's “direct method” the “Robust Fixed Point Transformations (RFPT)” were introduced in a novel adaptive technique that instead of the state space representation directly utilized the available approximate model of the system to estimate its “response function”. In this approach it was assumed that the system's response is directly observable and an iterative sequence was generated by the use of “Banach's Fixed Point Theorem” that converged to an appropriate deformation of the rough initial model to obtain precise trajectory tracking. In the present paper it is shown that the Luenberger observers and the RFPT-based mathod can be combined in a more conventional approach of the adaptive controllers that are designed on the basis of finding appropriate feedback gains. Illustrative simulation examples are presented to substantiate this statement.