Interpolation-Based Model Order Reduction for Quadratic-Bilinear Systems and H2 Optimal Approximation

The work of this paper focuses on model order reduction for a special class of nonlinear dynamical systems, that is, the class of quadratic-bilinear dynamical systems. This kind of systems can be used to represent other nonlinear dynamical systems with strong nonlinearities such as exponent and high-order polynomials. This paper addresses the H2 optimal model approximation problem for this class of systems. To solve the model order reduction problem, a notion of generalized transfer functions and the H2 norm are first discussed. A Volterra series interpolation scheme is proposed to interpolate the system from both the input-to-output and the output-to-input directions. In contrast to existing methods, we propose to interpolate all Volterra kernels, which can be achieved by solving Sylvester equations. The necessary H2 optimality conditions are fulfilled by the proposed interpolation scheme. A fixed point method is applied to solve the nonlinear Sylvester equations. A numerical example demonstrates the effectiveness of the proposed methods.