6. Finite-Time LQR and SDRE for Satellite Formation Flying

Even though the results of the infinite-time LQR, SDRE, DI approaches are seemingly easier to understand and implement and can lead to acceptable results with appropriate tuning, it is very important to understand that such approaches are not strongly recommended for satellite formation flying problems in general. This is because in infinite-time formulations, the error is usually driven to zero asymptotically. However, as one can see from Chapter 2, the satellite formation flying problem is fundamentally a problem where one should ensure formation flying in two neighboring ‘orbits’ (unlike the formation flying of aerial vehicles). Hence, the right problem formulation should ensure that the relative desired position and velocity vectors are achieved at a ‘particular time’ (not earlier, not later). Once that is achieved, from that time onward, the deputy satellite remains in the desired orbit with respect to the chief satellite. Hence, such a problem formulation should ideally be done under the ‘finite-time’ optimal control paradigm instead. To address such finite-time terminal constraints problems, fortunately a few advanced techniques are also available in the literature. The present and next chapters deal with a few of these techniques, and demonstrate the suitability and applicability of these methods for satellite formation flying problems. In this chapter, an overview of the finite-time LQR and SDRE techniques is presented, followed by their usage for the satellite formation flying application.