Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (5): 1907-1922.doi: 10.1007/s42967-024-00415-9

• ORIGINAL PAPERS • Previous Articles    

The Eigenvalue Assignment for the Fractional Order Linear Time-Invariant Control Systems

Bin-Xin He1, Hao Liu1,2   

  1. 1. School of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211106, Jiangsu, China;
    2. Shenzhen Research Institute, Nanjing University of Aeronautics and Astronautics, Shenzhen, 518063, Guangdong, China
  • Received:2023-11-17 Revised:2024-04-13 Accepted:2024-04-16 Online:2024-09-04 Published:2024-09-04
  • Contact: Hao Liu,E-mail:hliu@nuaa.edu.cn E-mail:hliu@nuaa.edu.cn
  • Supported by:
    This research was supported in part by the National Natural Science Foundation of China (Grant Nos. 11401305 and 11571171) and Shenzhen Science and Technology Program (Grant No. JCYJ20230807142002006).

Abstract: The eigenvalue assignment for the fractional order linear time-invariant control systems is addressed in this paper and the existence of the solution to this problem is also analyzed based on the controllability theory of the fractional order systems. According to the relationship between the solution to this problem and the solution to the nonlinear matrix equation, we propose a numerical algorithm via the matrix sign function method based on the rational iteration for solving this nonlinear matrix equation, which can circumvent the limitation of the assumption of linearly independent eigenvectors. Moreover, the proposed algorithm only needs to solve the linear system with multiple right-hand sides and it converges quadratically. Finally, the efficiency of the proposed approach is shown through numerical examples.

Key words: Fractional order systems, Eigenvalue assignment problem, Nonlinear matrix equation, Matrix sign function