Absolute Stability Analysis Using the Liénard Equation: A Study Derived From Control of Fuel Cell Ultracapacitor Hybrids

Load-following in solid oxide fuel cells (SOFCs), hybridized with an ultracapacitor for energy storage, refers to an operating mode where the fuel cell's generated power follows the variable power demand, delivering the total demanded power at steady-state. Implementing this operating mode presents a rich set of problems in dynamical systems and control. This paper focuses on state-of-charge (SOC) control of the ultracapacitor during load-following, under transient constraints, and in the presence of an unknown nonlinearity. The problem is generalized to stabilization of a plant containing a cascaded connection of a driver and a driven dynamics, where the former is nonlinear and largely unknown. Closed-loop stability of the system is studied as a Lur'e problem and via energy-based Lyapunov equations, but both impose conservative conditions on the nonlinearity. An alternate approach is developed, where the closed-loop dynamics are formulated as a class of Liénard equations. The corresponding analysis, which is based on the nonlinear characteristics of the Liénard equation, yields more definitive and less conservative stability criteria. Additional conditions that lead to limit cycles are also derived, and a bifurcation pattern is revealed. The generality of the proposed approach indicates applicability to a variety of nonlinear systems.