UC San Diego

Motivated by tactical missile guidance, in which there exists a short, finite amount of time to intercept a moving target, we introduce a new approach to stabilization and estimation in finite time which allows the user to prescribe the time by which the estimation and control objectives will be achieved. We accomplish this by employing time-varying feedback gains which tend to infinity as time approaches the prescribed time. Remarkably, despite the gains going to infinity, the overall control inputs and output estimation error injections remain finite, thereby allowing for potential use in real-world finite-time control applications such as tactical missile guidance. The first step in the design process involves extending existing stability concepts for nonlinear systems by introducing new definitions for finite-time stability using class K and class KL functions. This is done by compressing the usual infinite time interval to a finite one through the use of a time-varying, monotonic scaling function which tends to infinity as time approaches the prescribed time. Then, time-varying state coordinate transformations are used to design two feedback controllers for nonlinear systems in the normal form which stabilize the state in a finite time that is independent of initial conditions and freely prescribed by the user. The first controller is appropriate for systems having no uncertainties, while the second is intended for systems having matched uncertainties and possibly unknown control input coefficient. Prescribed-time observers for linear systems in the observer canonical form are then introduced using a similar time-varying state coordinate transformation design, resulting in observers akin to the Luenberger observer but with time-varying gains that tend to infinity as time approaches the prescribed time. Then, the prescribed-time uncertainty-free controller and prescribed-time observer are combined, and it is shown that a finite-time separation principle exists between the controller and the observer. The presentation is concluded with a series of examples in which the new prescribed-time controllers and observers are applied to provide new solutions to canonical problems in tactical missile guidance.