In this paper, we propose a finite-time structure-preserving model reduction method based on the shifted Legendre polynomials for K-power bilinear systems. First, it aims to transform the K-power bilinear system into a general bilinear system and use the shifted Legendre polynomials expansions to obtain the approximate low-rank decomposition factors of the controllability and observability Gramians, which are calculated by the recurrence formulas. Second, the approximate balanced system of the K-power bilinear system is constructed by the corresponding projection transformation of each subsystem. Then, the reduced-order model is constructed by truncating the states corresponding to the small approximate singular values. Combined with the dominant subspace projection method, the reduction process is modified to alleviate the shortcomings that above method may unexpectedly result in unstable systems although the original one is stable. Finally, two numerical experiments are given to demonstrate the effectiveness of the proposed algorithms.

在本文中，我们提出了一种基于 K 次双线性系统的移位勒让德多项式的有限时间结构保持模型简化方法。首先，其目的是将K次双线性系统转化为一般双线性系统，并利用平移勒让德多项式展开式得到可控性和可观性Gramians的近似低秩分解因子，并通过递推公式计算得到。其次，通过各子系统相应的投影变换构建K次双线性系统的近似平衡系统。然后，通过截断小近似奇异值对应的状态来构造降阶模型。结合主导的子空间投影方法，对约简过程进行了修改，以缓解上述方法虽然原始系统是稳定的但可能意外地导致系统不稳定的缺点。最后，给出两个数值实验来证明所提出算法的有效性。