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

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.