Trajectories Of Linear System Research Articles

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

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.

Parametric Iteration Method for Solving Linear Optimal Control Problems

This article presents the Parametric Iteration Method (PIM) for finding optimal control and its corresponding trajectory of linear systems. Without any discretization or transformation, PIM provides a sequence of functions which converges to the exact solution of problem. Our emphasis will be on an auxiliary parameter which directly affects on the rate of convergence. Comparison of PIM and the Variational Iteration Method (VIM) is given to show the preference of PIM over VIM. Numerical results are given for several test examples to demonstrate the applicability and efficiency of the method.

Minimum‐time control of systems with Coulomb friction: near global optima via mixed integer linear programming

AbstractThis work presents a method of finding near global optima to minimum‐time trajectory generation problems for systems that would be linear if it were not for the presence of Coulomb friction. The required final state of the system is assumed to be maintainable by the system, and the input bounds are assumed to be large enough so that the role of maintaining zero acceleration during finite time intervals of zero velocity (the role of static friction) can always be assumed by the input. Other than the previous work for generating minimum‐time trajectories for robotic manipulators for which the path in joint space is already specified, this work represents, to the best of our knowledge, the first approach for generating near global optima for minimum‐time problems involving a non‐linear class of dynamic systems. The reason the optima generated are near global optima instead of exactly global optima is due to a discrete‐time approximation of the system (which is usually used anyway to simulate such a system numerically). The method closely resembles previous methods for generating minimum‐time trajectories for linear systems, where the core operation is the solution of a Phase I linear programming problem. For the non‐linear systems considered herein, the core operation is instead the solution of a mixed integer linear programming problem. Copyright © 2001 John Wiley & Sons, Ltd.

Extremal property of the initial value of the adjoint variable in the minimization problem of the energy functional on trajectories of linear systems with constrained control

Extremal property of the initial value of the adjoint variable in the minimization problem of the energy functional on trajectories of linear systems with constrained control

Parametric insensitivity and controlled invariance

Parametric insensitivity of state and output trajectories of linear systems is approached by employing an analysis method recently developed for imbedding a large class of structural properties usually expressed in terms of matrices into a unified geometric framework. Necessary and sufficient conditions useful for determining maximal state and output subspaces, where parameter insensitive trajectories range, are given. Moreover, a procedure for the synthesis of controllers which discriminate the trajectories according to their insensitivity to various parameters is proposed. It is also shown that such controllers are consistent with completely arbitrary pole assignment.