A Self–Tuning Velocity Observer Formulation for a Class of Nonlinear Systems Baris Bidikli, Enver Tatlicioglu*, and Erkan Zergeroglu Abstract— This work presents the design and the corre- sponding stability analysis of a model free velocity observer formulation for nonlinear systems modeled by Euler-Lagrange formulation. The observation gains of the proposed formulation are tuned online according to an update algorithm removing the burden of observation gain tuning. Lyapunov based arguments are applied to prove the overall system stability. Performance of the observer proposed is illustrated via extensive simulation studies. Experimental studies are also utilized to demonstrate the viability of the proposed formulation. I. I NTRODUCTION In this paper, we present a novel velocity observer un- der the restriction that the system under consideration has dynamical uncertainties. Thus the exact knowledge of dy- namical parameters of the system are unavailable for the observer design. Under the standard assumptions that the system dynamics are bounded and first–order differentiable, we proposed a model free velocity observer formulation that achieves asymptotic velocity observation error regulation. In the literature, there are several observer formulations [1], [2], [3], [4], [5], [6], [7] where some of them are model– free [5], [6], [7]. Different from the available model–free observers, the proposed formulation is self–tuning. That is the observation gains used our formulation are updated online according to an update rule dictated by the stability analysis . To our best knowledge, the self tuning approach used in our formulation is the first presented in the literature. In assistance of the time varying nature of the observer gains, we also were able to remove the requirement of the prior knowledge of the upper bounds of uncertain system dynamics. The stability and convergence properties of the proposed velocity observer are supported by Lyapunov– based arguments. The performance of the designed velocity observer was demonstrated via simulation studies. Experi- mental studies are also presented to illustrate the effective- ness of the proposed formulation. B. Bidikli is with the Department of Mechatronics Engineering, The Graduate School of Natural and Applied Sciences, Dokuz Ey- lul University, Tinaztepe Campus, Buca, Izmir, 35390 Turkey (Email: baris.bidikli@deu.edu.tr). E. Tatlicioglu is with the Department of Electrical & Electronics Engineer- ing, Izmir Institute of Technology, Izmir, 35430 Turkey (Phone: +90 (232) 7506536; Fax: +90 (232) 7506599; E-mail: envertatlicioglu@iyte.edu.tr). E. Zergeroglu is with the Department of Computer Engineering, Gebze Institute of Technology, 41400, Gebze, Kocaeli, Turkey (Email: e.zerger@gtu.edu.tr). E. Tatlicioglu is funded by The Scientific and Technological Research Council of Turkey via grant number 115E726. *To whom all the correspondence should be addressed. II. SYSTEM MODEL AND I TS PROPERTIES The general model for Euler–Lagrange systems is given as ¨ x = H + Gτ (1) where x (t), ˙ x (t) and ¨ x (t) ∈ R m denote the state vector and its first and second time derivatives, respectively. H (x, ˙ x) ∈ R m and G (x, ˙ x) ∈ R m×m are nonlinear functions, τ (t) ∈ R m is the control input vector. It is also assumed that the following assumption is satisfied for the model in (1). Assumption 1: H (x, ˙ x) and G (x, ˙ x) are C 1 functions. The control input is a C 1 function and τ (t), ˙ τ (t) ∈L ∞ . The system states are bounded for all time (i.e., x (t), ˙ x (t) ∈ L ∞ ). Remark 1: The mathematical model for an m degree–of– freedom, revolute joint, direct drive, robot manipulator is given as [8] M (q)¨ q + V m (q, ˙ q)˙ q + g (q)+ F d ˙ q = τ (2) where q (t), ˙ q (t) and ¨ q (t) ∈ R m denote the link position, velocity and acceleration, respectively, M (q) ∈ R m×m represents the positive–definite, symmetric inertia matrix, V m (q, ˙ q) ∈ R m×m represents the centripetal Coriolis matrix, g (q) ∈ R m is the gravitational vector, F d ∈ R m×m de- notes the constant, diagonal positive–definite, viscous friction matrix, and τ (t) being the control input torque, can be rearranged as a model that is given in (1) by selecting x  q, H  −M -1 (V m ˙ q + g + F d ˙ q) and G  M -1 . III. OBSERVER DESIGN Estimating the unavailable velocity signal ˙ x (t) by design- ing an observer having online gain tuning methodology is the main objective of this study. We realize our design under the restriction that H (x, ˙ x), G (x, ˙ x) and τ (t) are unavailable. To quantify this objective, the velocity observation error denoted by ˙ ˜ x (t) ∈ R m is defined as ˙ ˜ x  ˙ x − ˙ ˆ x (3) where ˙ ˆ x (t) ∈ R m is the observed velocity. In view of (3), the position observation error, ˜ x (t) ∈ R m , is defined in the following manner ˜ x  x − ˆ x. (4) The velocity observer is designed in the following form ˙ ˆ x = p +[k (t)+ I m ]˜ x (5) 2016 IEEE 55th Conference on Decision and Control (CDC) ARIA Resort & Casino December 12-14, 2016, Las Vegas, USA 978-1-5090-1837-6/16/$31.00 ©2016 IEEE 3751