Reduced-order state reconstruction for nonlinear dynamical systems in the presence of model uncertainty Nikolaos Kazantzis a,⇑ , Vasiliki Kazantzi b a Department of Chemical Engineering, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA b School of Business and Economics, Department of Project Management, Technological Educational Institute, Larissa 41110, Greece article info Keywords: Nonlinear dynamical systems Nonlinear state estimation Reduced-order state estimation Singular PDEs Invariance Model uncertainty abstract A new approach to the reduced-order state reconstruction problem for nonlinear dynam- ical systems in the presence of model uncertainty is proposed. The problem of interest is conveniently formulated and addressed within the context of singular first-order non- homogeneous partial differential equations (PDE) theory, leading to a reduced-order non- linear state estimator that is constructed through the solution of a system of singular PDEs. A set of necessary and sufficient conditions is derived that ensure the existence and uniqueness of a locally analytic solution to the above system of PDEs, and a series solution method is developed that is easily programmable with the aid of a symbolic software pack- age such as MAPLE. Furthermore, the convergence of the estimation error or the mismatch between the actual unmeasurable states and their estimates is analyzed and characterized in the presence of model uncertainty. Finally, the performance of the proposed reduced- order state reconstruction method is evaluated in a case study involving a biological reac- tor that exhibits nonlinear dynamic behavior. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction Broad classes of systems and processes exhibit nonlinear dynamic behavior and are typically modeled by systems of nonlinear differential equations [1,5,7,10,11,13,24]. These dynamic models aim at capturing the actual behavior of the sys- tem of interest as faithfully as possible, and are now extensively used (simulated) in order to describe and characterize dynamic system responses to various external stimuli (random/unexpected and/or intentional), as well as reliably monitor its performance [1,5,7,10,11,13,14,24]. Furthermore, in order to meet the above objectives and characterize the system’s dynamic behavior, the explicit use of such a dynamic model (in various degrees of complexity and descriptive accuracy) is often complemented by sensor measurements related to measurable physical and chemical quantities [3,5,11,14]. How- ever, it is a rare occasion in practice for all variables to be available for direct on-line measurement due to physical and/or technical limitations pertaining to the current state of sensor technology [3,5,11,14]. In most cases there is a substantial need for an accurate estimation and dynamic reconstruction of key unmeasurable physical and chemical variables, espe- cially when they are used for system performance monitoring purposes and in the design of advanced control systems [3,5,11,14]. For this particular task, a state estimator (state observer) or ‘‘software sensor’’ is usually employed and appro- priately designed in order to accurately reconstruct the aforementioned unmeasurable variables. In the world of linear sys- tems, both the well-known Kalman filter and its deterministic analogue realized by Luenberger’s observer [3,5,14,20], offer a full comprehensive solution to the problem. In the case of nonlinear systems, the traditional practical approach in 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.05.063 ⇑ Corresponding author. E-mail addresses: nikolas@wpi.edu (N. Kazantzis), kazantzi@teilar.gr (V. Kazantzi). Applied Mathematics and Computation 218 (2012) 11708–11718 Contents lists available at SciVerse ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc