Minimization of Peak Effect in the Free Motion of Linear Systems with Restricted Control

Abstract

A peak effect minimization problem in the free motion of linear systems is considered in the paper. The paper proposes an iterative procedure for the peak effect minimization using a combination of the recently proposed gramian-based approach and the theory of using the condition number of an eigenvectors matrix for the upper bound estimations of the system state processes. Minimization of control costs is based on the analysis of the singular value decomposition of a gramian of control costs, followed by the formation of major and minor estimations of the gramian. Minimization of peak effect in the trajectories of free movement of systems is carried out by minimizing the condition number of the eigenvectors matrix of the matrix of a stable closed-loop system, while the state matrix with the desired eigenvalues and eigenvectors is designed with the generalized modal control. The development of an iterative algorithm for the peak effect minimization in the trajectories of linear systems under any non-zero initial conditions with restricted control is based on an aggregated index. The index takes into account both the estimate of the gramian of control costs and the condition number of the eigenvectors matrix of the stable closed-loop system. Minimization of the aggregated index makes it possible to ensure minimal deviations in the trajectories of free movement of systems of the considered class. The procedure is applied to a system of two satellites with restricted control, where peak effects in satellites relative trajectories are minimized. Two cases of peak affect minimization are considered. In the first case, the peak effect minimization in the trajectories of free movement of satellites is carried out only by minimizing the gramian of control costs. In the second case, the peak effect minimization is realized using the developed algorithm. The results illustrate the efficiency of the procedure and indicate the decrease of the peak effect for the satellites relative trajectories.