Interpolation-Based Model Order Reduction for Quadratic-Bilinear Systems and $${\mathcal {H}_{2}}$$ H 2 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 $${\mathcal {H}_{2}}$$ H 2 optimal model approximation problem for this class of systems. To solve the model order reduction problem, a notion of generalized transfer functions and the $${\mathcal {H}_{2}}$$ H 2 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 $${\mathcal {H}_{2}}$$ H 2 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.