QLMOR: A new projection-based approach for nonlinear model order reduction

We present a new projection-based nonlinear model order reduction method, named QLMOR (MOR via quadratic-linear systems). QL-MOR employs two novel ideas: (1) we show that DAEs (differential-algebraic equations) with many commonly-encountered nonlinear kernels can be re-written equivalently into a special format, QL-DAEs (quadratic-linear differential algebraic equations, i.e., DAEs that are quadratic in their state variables and linear in their inputs); (2) we adapt the moment-matching reduction technique of NORM to reduce these QLDAEs into QLDAEs of much smaller size. Because of the generality of the QLDAE form, QLMOR has significantly broader applicability than Taylor-expansion based methods. Importantly, QLMOR, unlike NORM, totally avoids explicit moment calculations (AⁱB terms), hence it has improved numerical stability properties as well. Because the reduced model has only quadratic nonlinearities (i.e., no cubic and higher-order terms), its computational complexity is less than that of similar prior methods. We also prove that QLMOR-reduced models preserve local passivity, and provide an upper bound on the size of the QLDAEs derived from a polynomial system. We compare QLMOR against prior methods on a circuit and a biochemical reaction-like system, and demonstrate that QLMOR-reduced models retain accuracy over a significantly wider range of excitation than Taylor-expansion based methods. Indeed, QLMOR is able to reduce systems that Taylor-expansion based methods fail to reduce due to passivity loss and impractically high computational costs. QLMOR therefore demonstrates that Volterra-kernel based nonlinear MOR techniques can in fact have far broader applicability than previously suspected, possibly being competitive with trajectory-based methods (e.g., TPWL) and nonlinear-projection based methods (e.g., maniMOR).