Reduced-Order Approximation of Bilinear Systems Using a New Hybrid Method based on Balanced Truncation and Iterative Rational Krylov Algorithms

Abstract

In this work, a hybrid approach is proposed for the reduced order approximation of the bilinear system by combining the Balanced Truncation (BT) and Bilinear Iterative Rational Krylov Algorithm (BIRKA). Bilinear BT (BBT) has low accuracy but guarantees stability, while BIRKA convergence suffers from sensitivity to initial choice of reduced-order system. To start, the proposed approach minimizes the Integral Square Error (ISE) index to specify the order of the reduced bilinear approximation. To assurance BIRKA convergence, two approaches, BBT and Linear BT (LBT), are applied to prepare the initial guess of the reduced-order approximation.Although BBT prepare a good stable initial guess for BIRKA, solving the generalized Lyapunov equations to find the solution is very computationally expensive. The initial guess is provided by LBT through solving the Lyapunov equations, which decreases computational complexity. Furthermore, the eigenvalues are replaced by the condition number in BIRKA to decrease complexity. To verify the efficiency of the proposed approach, three bilinear test systems are being examined. Finally, the performance of the proposed approach is compared with several classical approaches. The finding 2 indicate that the convergence probability of BIRKA increases. Also, the time for the determining the Model Order Reduction (MOR) decreases.