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 theor...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.展开更多
A new easy method has been presented to calculate the variable intervals corresponding to the stable univariant curves and to discriminate the stabilities of invariant points. This method and the one reported previous...A new easy method has been presented to calculate the variable intervals corresponding to the stable univariant curves and to discriminate the stabilities of invariant points. This method and the one reported previously constitute a simple and universal theory for the computer-plotting of the equilibrium phase diagrams of a multisystem—sign function matrix (SFM) discrimination method. Its main steps are: determining the stable univariant scheme according to the derivative (or difference) of AIG m ; grouping the univariant curves by comparisons of the mutual relations among them; determining the existing intervals of the variables for the stable curves by comparisons of coordinate values of the curves about the invariant point; determining the stabilities of invariant points by comparisons of relations between the common curves and the invariant points. This method is suitable for any kind of phase diagram of closed or open systems in a phase diagram “space” with either 2 or more than 2 dimensions.展开更多
基金National Natural Science Foundation of China(Grant Nos.11401305,11571171)Shenzhen Science and Technology Program(Grant No.JCYJ20230807142002006).
文摘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.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.4907017849773200).
文摘A new easy method has been presented to calculate the variable intervals corresponding to the stable univariant curves and to discriminate the stabilities of invariant points. This method and the one reported previously constitute a simple and universal theory for the computer-plotting of the equilibrium phase diagrams of a multisystem—sign function matrix (SFM) discrimination method. Its main steps are: determining the stable univariant scheme according to the derivative (or difference) of AIG m ; grouping the univariant curves by comparisons of the mutual relations among them; determining the existing intervals of the variables for the stable curves by comparisons of coordinate values of the curves about the invariant point; determining the stabilities of invariant points by comparisons of relations between the common curves and the invariant points. This method is suitable for any kind of phase diagram of closed or open systems in a phase diagram “space” with either 2 or more than 2 dimensions.
