An Impulsive Input Approach to Short Time Convergent Control for Linear Systems
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. A rendez-vous problem for satellites is used as an example for the practical applicability of the techniques presented here.