An Adaptive Observer Design Methodology for Bounded Nonlinear Processes Naira Hovakimyan 1 , Anthony J. Calise 2 , and Venkatesh K. Madyastha 3 School of Aerospace Engineering, Georgia Institute of Technology Atlanta, GA 30332 Abstract In this paper we address the problem of augmenting a linear observer with an adaptive element. The de- sign of the adaptive element employs two nonlinearly parameterized neural networks, the input and output layer weights of which are adapted on line. The goal is to improve the performance of the linear observer when applied to a nonlinear system. The networks’ teaching signal is generated using a second linear observer of the nominal system’s error dynamics. Boundedness of sig- nals is shown through Lyapunov’s direct method. The approach is robust to unmodeled dynamics and distur- bances. Simulations illustrate the theoretical results. 1 Introduction Design of adaptive observers for nonlinear systems is an intensively addressed topic in the recent litera- ture. This is an important aspect in the problems of state estimation, system identification or output feed- back control, e.g. [1, 2, 3, 4, 5]. However, most of the available results impose assumptions that severely limit their domain of applicability, such as to systems that are linear with respect to unknown parameters or systems that can be transformed to output feedback form. The universal approximation property of neu- ral networks (NNs) has motivated NN based identifica- tion and estimation schemes, like the ones reported in [6, 7, 8, 9, 10, 11], that relax these assumptions. The main challenge lies in defining an error signal for up- dating the NN weights. The observer developed in [9] introduces a strictly positive real (SPR) filter that en- ables writing the NN weights adaptive laws in terms of only the available measurement error signal. How- ever, the filter needed to satisfy the SPR condition may not always exist, particularly for systems with multiple outputs. In [10] this restriction has been relaxed, and an approach is laid out for general nonlinear processes. However, a major difference is that the approach in [9] 1 Research Scientist II, Senior Member IEEE, e-mail: naira.hovakimyan@ae.gatech.edu 2 Professor, Member IEEE 3 Graduate Student augments an existing linear observer, whereas the ap- proach in [10] does not. The adaptive laws developed in both approaches are limited to adapting only the NN output layer weights. Our approach is an augmenting approach, similar to the approach in [9], but does not impose an SPR condition. Also, both input and out- put layer weights of the NNs are adapted. We propose a simple linear filter to generate a teaching signal for the adaptive laws. Ultimate boundedness of error sig- nals is shown through Lyapunov’s direct method. The approach is robust to unmodeled dynamics and distur- bances. The paper is organized as follows. In Section 2 we state the assumptions about the dynamic system, recall uni- versal approximation properties of neural networks and define the observer for the case in which the measure- ment equation is linear and exactly known. In Section 3 the teaching signal for the NN is derived, in Section 4 we give sufficient conditions for ultimate bounded- ness of error signals. In Section 5 an extension to the case where the measurement equation is either nonlin- ear or imprecisely known is considered. This includes the case in which the measurement equation might in- volve states that are not modeled in the nominal ob- server design step. In Section 6 we present an example to illustrate the performance of the adaptive observer when applied to a nonlinear process with unmodeled dynamics. 2 System Description and Observer Structure 2.1 Problem Formulation Let the dynamics of an observable and bounded nonlin- ear process be given by the following equations: ˙ x 0 = f 0 (x 0 , u, v) y = g 0 (x 0 , u, v) , (1) where x 0 ∈ Ω 0 ⊂ R n0 is the state of the sys- tem, u ∈ R m , y ∈ R l are the system input (con- trol) and output (measurement) signals, respectively, v ∈ R k is a bounded and unknown disturbance input, f 0 (·, ·, ·), g 0 (·, ·, ·) are partially known continuous func- tions, and f 0 satisfies Lipschitz conditions with respect