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

Abstract

This paper proposes a hybrid method for order reduction of the bilinear system model using 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. The proposed method first determines the order of the reduced bilinear model by minimizing the index of Integral Square Error (ISE). Then, the initial guess of reduced-order system is provided via two approaches, BBT and Linear BT (LBT), to guarantee the convergence of BIRKA. The result of BBT is a good stable initial guess for BIRKA, but it is very computationally expensive to solve the generalized Lyapunov equations to find the solution. LBT decreases the computational complexity by providing the initial guess via solving the Lyapunov equations. To further decrease the complexity, the condition number is substituted in place of the eigenvalues in BIRKA. Three bilinear test systems are considered to show the efficiency of proposed method. Finally, the performance of the proposed method is compared with some classical methods. The results show that the convergence probability of BIRKA increases. Also, the time for the determining the model order reduction decreases.