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An impulsive input approach to short time convergent control for linear systems

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Author: Weiss, M. · Shtessel, Y.
Type:bookPart
Date:2013
Source:Advances in Aerospace Guidance, Navigation and Control, Springer, Berlin/Heidelberg, 2013, ISBN 978-3-642-38252-9, p.99-119, 99-119
Identifier: 488343
doi: doi:10.1007/978-3-642-38253-6_8
Keywords: Aviation · Linear systems · Defence Research · Defence, Safety and Security · Mechatronics, Mechanics & Materials · WS - Weapon Systems · TS - Technical Sciences

Abstract

The paper considers the problem of bringing the state of a controllable linear system to the origin in a very short time. It takes the approach of considering an “ideal” control input consisting of a linear combination of the Dirac delta function and its derivatives that realizes this goal instantaneously. Three schemes are introduced to approximate the impulsive input with physically realizable functions: a smooth approximation with compact support, a Gaussian function approximation and a step approximation. It is shown using a numerical example that all approximations work reasonably well, with the Gaussian approximation providing slightly worse results. It is also shown that a direct approach to obtain a state nulling input by solving an integral equation runs quicker into numerical problems than the impulsive input approach as the convergence time decreases. Finally, an application to an orbital rendez-vous problem is presented.