Annual Conference of the Prognostics and Health Management Society, 2010 Efficient Tracking of Behavior in Complex Hybrid Systems via Hybrid Bond Graphs Benjamin Podgursky 1 , Gautam Biswas 1 , and Xenofon Koutsoukos 1 1 Dept of EECE/ISIS, Vanderbilt University, Nashville, TN, 37235, United States benjamin.t.podgursky@vanderbilt.edu gautam.biswas@vanderbilt.edu xenofon.koutsoukos@vanderbilt.edu ABSTRACT For many real-world systems, which exhibit complex, nonlinear, and hybrid behavior, it is important to accu- rately track and monitor the state and health of these sys- tems. The continuous state estimation problem has been well studied, and a number of extensions of the Kalman filter to nonlinear systems have been proposed. Hy- brid state estimation poses an additional challenge, be- cause the model must be quickly updated during a mode change to facilitate accurate, real time tracking. This paper discusses an approach to minimize the amount of equation regeneration necessary when the system under- goes hybrid mode changes. These equations are used as the state update equation for an Unscented Kalman Fil- ter that tracks the system’s state. We demonstrate the effectiveness of our approach by tracking the hybrid be- haviors of NASA’s ADAPT test bench. Results show that our algorithm scales well for tracking large nonlinear and hybrid systems. 1. INTRODUCTION Many mission critical systems, such as aircraft and power generation systems, exhibit complex nonlinear and hybrid behaviors. Hybrid systems are characterized by intervals of continuous behavior interspersed with discrete changes. With increased needs for safety and reliability, it is becoming important to monitor system behavior online, and couple the monitoring system with accurate fault detection and isolation mechanisms. In an- other example, accurate and robust tracking is the pri- mary functionality of aircraft avionics systems. Accurate tracking of complex, nonlinear hybrid be- haviors is in itself a major challenge. In realistic sit- uations, this challenge is further compounded by noisy sensors and inaccuracies in the system models. Kalman filters and their extensions (Lefebvre, Bruyninckx & This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Schutter, 2004) have been used extensively for track- ing online behaviors. We briefly discuss two of the for- malisms: (1) the Extended Kalman Filter (EKF) (Welsh & Bishop, 2006), and (2) the Unscented Kalman Filter (UKF) (Julier & Uhlmann, 1997) in the next section. Tracking the hybrid system mode and state adds addi- tional challenges to the nonlinear continuous state esti- mation problem since the modes, i.e., the configurations of the system, and therefore, the dynamic system model change during system behavior evolution. For example, aircraft operate in different modes that include taxi, take- off, cruise, descend, and land. Traditionally hybrid sys- tems have been modeled as Hybrid Automata (Cuijpers & Reniers, 2005), (Henzinger, 1996), where each mode of operation is described as a state of the automaton, and the automata specifies the transitions between different modes of operation. Within each mode, the continuous state of the system is modeled by a set of differential equations. This representation works well for modeling small systems; however, modeling of large, complex sys- tems requires complete enumeration of all of the system modes. Pre-enumerating all of these modes is wasteful in both space and time, because most modes may not oc- cur at runtime. On some large systems, complete mode enumeration may be completely infeasible. A more efficient method for tracking hybrid behav- ior is to generate models for each mode only when that mode is visited. However, model generation after a mode change must occur quickly, before the state estimate di- verges from the true state. In this paper, we adopt the Hybrid Bond Graph (HBG) approach to model nonlinear hybrid systems (Mosterman & Biswas, 1998). Real world systems are large, nonlinear, and may have lots of switching com- ponents. HBGs represent a paradigm for representing these large complex systems in a compact form. This is because in the HBG representation, system modes do not have to be pre-enumerated, instead they are generated at runtime when a transition occurs to that mode. We use the equations generated from the HBG model in a mode to formulate the equations for the Unscented Kalman Fil- ter, which tracks the behavior of the system. Section 2 defines the filtering problem and discusses the strengths and weaknesses of the Extended Kalman 1