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[1] 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 [2, 3, 1]. Importantly, QLMOR, unlike NORM, totally avoids explicit moment calculations (A
ⁱ
B 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[2, 3, 1]. 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 [2, 3, 1] 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 [2, 3, 1]. Indeed, QLMOR is able to reduce systems that Taylor-expansion based methods fail to reduce due to passivity loss and impracti-cally 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 [4]) and nonlinear-projection based methods (e.g., maniMOR [5]).