Systems & Control Letters 60 (2011) 771–777 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Estimation of states and parameters for linear systems with nonlinearly parameterized perturbations Håvard Fjær Grip a,b,∗ , Ali Saberi a , Tor A. Johansen b a School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, United States b Department of Engineering Cybernetics, Norwegian University of Science and Technology, O.S. Bragstads plass 2D, NO-7491 Trondheim, Norway article info Article history: Received 20 July 2009 Received in revised form 18 January 2011 Accepted 29 March 2011 Available online 5 July 2011 Keywords: Estimation Nonlinear parameterization Observers abstract We consider systems that can be described by a linear part with a nonlinear perturbation, where the perturbation is parameterized by a vector of unknown, constant parameters. Under a set of technical assumptions about the perturbation and its relationship to the outputs, we present a modular design technique for estimating the system states and the unknown parameters. The design consists of a high- gain observer that estimates the states of the system together with the full perturbation, and a parameter estimator constructed by the designer to invert a nonlinear equation. We illustrate the technique on a simulated dc motor with friction. © 2011 Elsevier B.V. All rights reserved. 1. Introduction A common problem in model-based control and estimation is the presence of uncertain perturbations. These perturbations may be the result of external disturbances or internal plant changes, such as a configuration change, system fault, or changes in plant characteristics. The uncertainty associated with the perturbations can in many cases be characterized by a vector of unknown, constant parameters. Unknown parameters are often dealt with by introducing pa- rameter estimates that are updated online in a suitable manner. Adaptive techniques for linearly parameterized systems have been treated in a large body of work (see, e.g., [1,2]), but only a few specialized techniques have been developed for nonlinearly param- eterized systems. Some of these methods are based on convex- ity or concavity of the parameterization [3–5]; these can also be extended to parameterizations that can be convexified through reparameterization (see [6]). Other methods apply to first-order systems with fractional parameterizations (e.g., [7,8]). An ap- proach that applies to higher-order systems with matrix fractional parameterizations – where an auxiliary estimate of the full ∗ Corresponding author at: School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752, United States. Tel.: +1 509 715 9195. E-mail address: grip@ieee.org (H.F. Grip). perturbation is used in the estimation of the unknown parame- ters – was presented by Qu [9]. Tyukin et al. [10] used the idea of virtual algorithms, which are designed as though the time deriva- tive of the measurements were available, to construct a family of adaptation laws for monotonically parameterized perturbations. Other available methods include a hierarchical approach based on gridding the parameter space with a set of discrete candidate parameters [11] and a feedback-domination design utilizing a linearly parameterized bound on the nonlinearly parameterized terms [12]. In a recent paper, Liu et al. [13] presented a method based on Immersion & Invariance [14], where monotonic parame- terizations are generated through the solution of a partial differen- tial equation (see also [15]). Grip et al. [16] have recently presented a design methodology for estimating unknown parameters in systems of the form ˙ x = f (t , x) + B(t , x)v(t , x) + φ, where φ = B(t , x)g (t , x,θ) is a pertur- bation parameterized by the parameter vector θ . This methodology is based on the observation that, if the perturbation φ were directly available, then identifying θ would be a matter of inverting the nonlinear equation φ = B(t , x)g (t , x,θ) with respect to θ . This line of thought leads to a modular design consisting of a parameter estimator and a perturbation estimator. The parameter estimator is designed as though φ were known, to dynamically invert the ex- pression φ = B(t , x)g (t , x,θ) with respect to θ . The perturbation estimator is designed to produce an estimate of φ to be used by the parameter estimator in lieu of the actual perturbation. The param- eter estimate is in turn fed back to the perturbation estimator. 0167-6911/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2011.03.012