IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 12, NO. 1, JANUARY 2004 101 A Nonlinear Observer Design for Fuel Cell Hydrogen Estimation Murat Arcak, Member, IEEE, Haluk Görgün, Lars Malcolm Pedersen, and Subbarao Varigonda Abstract—We present an observer design to estimate the partial pressure of hydrogen in the anode channel of a fuel cell. A pre- cise knowledge of this pressure is of importance to ensure reliable and efficient operation of the fuel cell power system. Our design makes use of a monotonic nonlinear growth property of the voltage output on hydrogen partial pressures at the inlet and at the exit of the channel. By treating the slowly varying inlet partial pressure as an unknown parameter, an adaptive observer is developed that employs a nonlinear voltage injection term. We then study the ro- bustness of this observer against variations in the inlet partial pres- sure, and analyze its sensitivity to other modeling errors. We also prove a robustness property of the observer against the parameter estimation error, which means that it can be implemented with al- ternative parameter estimator designs. Index Terms—Adaptive observers, fuel cells, robustness. I. INTRODUCTION F UEL cell technology offers high efficiency and low emis- sions, and holds great promise for future power generation systems. Recent developments in polymer electrolyte membrane (PEM) and catalyst technology have dramatically increased the power density of fuel cells, and made them viable for vehicular and portable power applications, as well as for stationary power plants. A typical fuel cell power system consists of numerous in- terconnected components, as presented comprehensively in the books [1], [6], [10], and more concisely in the survey paper [3]. The anode side of the cell stack is fed by the fuel processing system (FPS) which reforms natural gas, gasoline, methanol, or other hydrocarbons into hydrogen. In the cell stack hydrogen from the anode reacts with oxygen from the cathode to generate electricity. Feedback control of fuel cell power systems have recently started to attract attention. Several control problems for fuel cell-powered electric vehicles are outlined in Powers and Nicastri [13], Mays et al. [11], and Boettner et al. [2]. More recently, Pukrushpan et al. [14] have derived a lumped dynamic model of the cell stack, and regulated the net power output and the cathode oxygen excess ratio by controlling the air supply to the cathode. Among the most challenging feedback problems in fuel cell power systems is the regulation of hydrogen supplied from the Manuscript received January 10, 2003; revised July 29, 2003. Manuscript received in final form October 7, 2003. Recommended by Associate Editor A. Stefanopoulou. The work of M. Arcak and H. Görgün was supported in part by the National Science Foundation by Grants ECS-0226094 and ECS-0238268. M. Arcak and H.Görgün are with the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180 USA (e-mail: arcakm@rpi.edu). L. M. Pedersen and S. Varigonda are with the United Technologies Research Center, East Hartford, CT 06108 USA. Digital Object Identifier 10.1109/TCST.2003.821958 FPS to the anode channel of the cell stack. Insufficient supply causes “starvation” of the cell—a phenomenon encountered in various types of fuel cells [12], [16], [17] in which the plat- inum catalyst starts consuming the graphite used in the flow fields. While starvation reduces the life of the cell, excessive hydrogen output from FPS reduces its efficiency, and is also un- desirable. Achieving satisfactory load-following performance while, at the same time, meeting these dynamic constraints on hydrogen supply, requires advanced feedback designs that reg- ulate the cooperation of the cell stack and the FPS (see, e.g., Pukrushpan et al. [15] for significant steps in this direction). A major obstacle to implementability of such designs, however, is the absence of reliable measurements of hydrogen partial pres- sure in the cell stack. Existing hydrogen sensors are not suitable for use with these controllers because, as discussed in [8], most of them suffer from slow response times, low accuracy, high cost, and sensitivity to variations in gas composition. In this paper, we design a nonlinear observer that estimates the hydrogen partial pressure at the inlet and at the exit of the anode channel, from available voltage, current, and total pres- sure measurements. Because the inlet partial pressure is gov- erned by the FPS dynamics which are slower than the dynamics in the cell stack, we treat it as a slowly varying parameter. We then design an adaptive observer which estimates this param- eter from voltage measurements, and employs it in the observer for exit partial pressure. We prove that this adaptive observer is robust to disturbances and to variations in the inlet partial pressure, within the input-to-state stability (ISS) framework of Sontag [18]. A study of the ISS gains shows that increasing the adaptation gain decreases the sensitivity to parameter variations. It also points to an increased sensitivity to output disturbances when the voltage is high. As we shall further discuss, this sensi- tivity is due to weaker observability properties at higher voltage levels. Simulation studies show that the speed of response and ac- curacy of the observer compare favorably to the available hy- drogen sensors. Unlike the simplified model used in our design, the simulations are carried out with a high-order, detailed, simu- lation model, which was developed at United Technologies Re- search Center. Although our simulation studies are for PEM fuel cells, we wish to emphasize that the observer design is appli- cable to other types, such as phosporic acid and alkaline fuel cells, that use hydrogen as fuel. In Section II we present our adaptive observer design, and prove its stability and convergence in the absence of modeling errors. In Section III we discuss in further detail the simplifying assumptions in the model, and study their effect on the conver- gence properties of the observer. In Section IV, we prove that the 1063-6536/04$20.00 © 2004 IEEE