Final Glide-back Envelope Computation for Reusable Launch Vehicle Using Reachability

Scalable calculation of reach sets and tubes for nonlinear systems with terminal integrators

“…In the fully implicit formulation with a single Hamiltonian such as (6), D z ψ(t, z) (the costate of the corresponding optimal control problem) provides a definitive choice of u. In the MIE formulation, only D y φ i (t, y) and D y φ i (t, y) are available for i = 1, 2, .…”

“…For systems in which the dynamics decouple, the high dimensional PDE can be broken naturally into multiple, lower-dimensional PDEs. Complete decoupling of the dynamics occurs rarely, so a projection-based scheme whereby coupling terms could be treated as disturbances was proposed in [12] (see section 6 for a discussion of how that approach can be applied to the class of dynamics studied in this paper), and a time-based decoupling where the system dynamics could be separated into fast and slow components was proposed in [6]. The exponential cost of HJ PDE based schemes arises from the requirement in traditional PDE solvers to grid the state space, so [4] proposes to reduce that cost by using a neural network representation of the solution instead.…”

A viability approach for the stabilization of an underactuated underwater vehicle in the presence of current disturbances

“…Besides, it is preferable to underapproximate the viability kernel rather than overapproximate it. Therefore we give the advantage to the disturbance v r (·), which tries to make the viable set larger, by allowing the control input u u u 1 1 1 (·) to use only non-anticipative strategies, as presented in [24]. Consequently, computing the viability kernel for (4) is equivalent with computing the set of initial states for which the control input u u u 1 1 1 (·) wins the game.…”

“…Given the estimation for the bounds of u r , v r , r as We follow the formulation of a differential game with two players [24]. The control input u u u 1 1 1 (·) is the first player who tries to keep the vehicle into the safe set S S S, whereas the disturbance v r (·) is the second player who tries to drive the vehicle out of S S S. Furthermore, it is important to define what information the players know about each others decisions.…”

Safe Set Protection and Restoration for Unimpaired and Impaired Aircraft

“…Recent flight control research contains many examples of safe set envelope protection schemes. [1][2][3][4][5][6][7] The idea of a safe 3 or viable 2 invariant set derives from a decades old control problem in which the plant controls are restricted to a bounded set U and it is desired to keep the system state within a convex, not necessarily bounded, subset C of the state space. Feuer 8 studied the question: under what conditions does there exist for each initial state in C an admissible control producing a trajectory that remains in C for all t > 0?…”