Improved Bilinear Balanced Truncation for Order Reduction of the ‎High-Order Bilinear System Based on Linear Matrix Inequalities ‎

This paper proposes a new Model Order ‎Reduction (MOR) method based on the Bilinear Balanced Truncation ‎‎(BBT) approach. In the BBT method, solving the generalized Lyapunov ‎equations is necessary to determine the bilinear system's controllability ‎and observability Gramians. Since the bilinear systems are generally of ‎high order, the computation of the Gramians of controllability and ‎observability have huge computational volumes. In addition, the accuracy ‎of reduced-order model obtained by BT is relatively low. In fact, the ‎balanced truncation method is only available for local energy bands due ‎to the use of type I Gramians. In this paper, BBT based on type II ‎controllability and observability Gramians would be considered to fix ‎these drawbacks.‎ At first, a new iterative method is proposed for determining ‎the proper order for the reduced-order bilinear model, which is related to ‎the number of Hankel singular values of the bilinear system whose real ‎parts are closest to origin and have the most significant amount of ‎energy. Then, the problem of determining of type II controllability and ‎observability Gramians of the high-order bilinear system have been ‎formulated as a constrained optimization problem with some Linear ‎Matrix Inequality (LMI) constraints for an intermediate middle-order ‎system. Then, the achieved Gramians are applied to the BBT method to ‎determine the reduced-order model of the bilinear system. Next, the ‎steady state accuracy of the reduced model would be improved via ‎employing a tuning factor.‎ Using the concept of type II Gramians and via the proposed ‎method, the accuracy of the proposed bilinear BT method is increased. ‎For validation of the proposed method, three high-order bilinear models ‎are approximated. The achieved results are compared with some well-‎known MOR approaches such as bilinear BT, bilinear Proper Orthogonal ‎Decomposition (POD) and Bilinear Iterative Rational Krylov subspace ‎Algorithm (BIRKA) methods.‎ According to the obtained results, the proposed MOR ‎method is superior to classical bilinear MOR methods, but is almost ‎equivalent to BIRKA. It is out-performance respecting to BIRKA is its ‎guaranteed stability and convergence. ‎