H₂-Quasi-Optimal Model Order Reduction for Quadratic-Bilinear Control Systems

We investigate the optimal model reduction problem for large-scale quadratic-bilinear (QB) control systems. Our contributions are threefold. First, we discuss the variational analysis and the Volterra series formulation for QB systems. We then define the H₂-norm for a QB system based on the kernels of the underlying Volterra series and propose a truncated H₂-norm as well. Next, we derive first-order necessary conditions for an optimal approximation, where optimality is measured in terms of the truncated H₂-norm of the error system. We then propose an iterative model reduction algorithm, which upon convergence yields a reduced-order system that approximately satisfies the newly derived optimality conditions. We also discuss an efficient computation of the reduced Hessian, using the special Kronecker structure of the Hessian of the system. We illustrate the efficiency of the proposed method by means of several numerical examples resulting from semidiscretized nonlinear partial differential equations and show its competitiveness with existing model reduction schemes such as moment-matching and balanced truncation for QB systems by comparing accuracy in the time-domain simulations and in the truncated H₂-norm.