Design of Proportional Integral Adaptive Observers

Adaptive observers were first introduced by Carroll and Lindorff in [1], and later on modified by Kudva and Narendra in [2]. For a more detailed analysis and design of adaptive systems see [3], [4], [5]. In many applications where the parameters are unknown and states are not accessible, adaptive observer appears to be a valuable method in estimation of both parameters and states of the system. Literature reports several useful applications in which adaptive observers have successfully been employed [6] [11]. In almost all of these applications the conventional structure of adaptive observer (PAO) is used. A new type of observer, known as proportional Integral Observer (PIO) has been extensively used over the past several years for various purposes. The PIO has been originally introduced in connection to LTR problem, in which time and frequency recoveries have been realized independently [12]. In [13], PIO is used for fault detection and it was further analyzed in [9]. In a recent paper [14] the authors provide a new formulation for PIO design by using the integral gain to stabilize the noise free error dynamics while the proportional gain is used to decouple the disturbance. In spite of various designs available for PIO, its generalization to adaptive case has not sufficiently been explored. Although an ad hoc design of PI Adaptive Observer (PIAO) has been reported in [15] and [16], there is no systematic design approach available for it. The difficulty encountered in previous publications was due to the fact that a direct generalization of PAO design to PIAO was not possible and, consequently, the observer gains were adjusted until satisfactory convergence results were achieved. This paper attempts to make use of conventional Proportional Adaptive Observer (PAO) and extend it to Proportional Integral Adaptive Observer (PIAO). This step is not trivial and extra care is necessary to accomplish the design procedure. Therefore the main result of this paper is to provide a systematic design procedure for PIAO to make this