PROXIMITY DEGREE FOR SIMPLE AND MULTIPLE STRUCTURES OF THE EIGENVALUES: OVERSHOOT MINIMIZATION FOR FREE MOTION TRAJECTORIES OF APERIODIC SYSTEM

T. A. Akunov, N. A. Dudarenko, N. A. Polinova, A. V. Ushakov


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Abstract

The paper deals with steady aperiodic continuous system, state matrix of which has a real spectrum of the eigenvalues which absolute value is less than unity. The latest authors’ works show that for such absolute values and multiple structure of eigenvalues on the free motion trajectories of the system by norm of the state vector the significant overshoot is detected, alternated by monotonous motion toward a state of rest. In order to minimize the overshoot value, it is proposed to modify the structure of the eigenvalues, transforming it into a simple one. The result of structure modification is the following: initial eigenvalue and shifted along the real axis of the complex plane to the left by a fixed value relative to the adjacent eigenvalues; each of them has unit multiplicity. Such modification gives the possibility to form the estimation of the proximity degree of eigenvalues simple structure to the multiple one. Moreover, it can be defined in a relative form, which guarantees the reduction of the above overshoot for the free motion trajectory. Results of computer experiments illustrate the issues of the paper.


Keywords: aperiodic system, proximity degree of eigenvalues to multiplicity, norm, trajectory

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