58 research outputs found
Active Target Defense Differential Game with a Fast Defender
This paper addresses the active target defense differential game where an
Attacker missile pursues a Target aircraft. A Defender missile is fired by the
Target's wingman in order to intercept the Attacker before it reaches the
aircraft. Thus, a team is formed by the Target and the Defender which cooperate
to maximize the distance between the Target aircraft and the point where the
Attacker missile is intercepted by the Defender missile, while the Attacker
tries to minimize said distance. The results shown here extend previous work.
We consider here the case where the Defender is faster than the Attacker. The
solution to this differential game provides optimal heading angles for the
Target and the Defender team to maximize the terminal separation between Target
and Attacker and it also provides the optimal heading angle for the Attacker to
minimize the said distance.Comment: 9 pages, 8 figures. A shorter version of this paper will be presented
at the 2015 American Control Conferenc
Escape Regions of the Active Target Defense Differential Game
The active target defense differential game is addressed in this paper. In
this differential game an Attacker missile pursues a Target aircraft. The
aircraft is however aided by a Defender missile launched by, say, the wingman,
to intercept the Attacker before it reaches the Target aircraft. Thus, a team
is formed by the Target and the Defender which cooperate to maximize the
separation between the Target aircraft and the point where the Attacker missile
is intercepted by the Defender missile, while the Attacker simultaneously tries
to minimize said distance. This paper focuses on characterizing the set of
coordinates such that if the Target's initial position belong to this set then
its survival is guaranteed if both the Target and the Defender follow their
optimal strategies. Such optimal strategies are presented in this paper as
well.Comment: 19 pages, 9 figures. arXiv admin note: text overlap with
arXiv:1502.0274
On the Synthesis of Optimal Control Laws
In this paper we advocate for Isaacs\u27 method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin\u27s canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we correctly require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; achieving tight tolerances is expensive, and from a theoretical point of view, constraining the end state to a terminal manifold of co-dimension n-1 renders the optimal control problem well-posed. Thus, the correct solution of the optimal control problem is obtained. In this respect, two target sets are considered: a smooth circular target and a square target with corners; obviously, the size of the target sets can be shrunk to become very small. Closed-loop state-feedback control laws are developed which drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs\u27 method for the solution of differential games, which entails Dynamic Programming (DP), working backward from the Usable Part (UP) of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions provided by Pontryagin\u27s Maximum Principle (PMP). Special attention is given to the critical UP of the target set in the process of obtaining the global solution of the optimal control problem at hand. In this paper, Isaacs\u27 method for the solution of differential games is applied to the solution of optimal control problems and the juxtaposition of the PMP and DP is undertaken
Nonadiabatic extension of the Heisenberg model
The localized states within the Heisenberg model of magnetism should be
represented by best localized Wannier functions forming a unitary
transformation of the Bloch functions of the narrowest partly filled energy
bands in the metals. However, as a consequence of degeneracies between the
energy bands near the Fermi level, in any metal these Wannier functions cannot
be chosen symmetry-adapted to the complete paramagnetic group M^P. Therefore,
it is proposed to use Wannier functions with the reduced symmetry of a magnetic
subgroup M of M^P [case (a)] or spin dependent Wannier functions [case (b)].
The original Heisenberg model is reinterpreted in order to understand the
pronounced symmetry of these Wannier functions. While the original model
assumes that there is exactly one electron at each atom, the extended model
postulates that in narrow bands there are as many as possible atoms occupied by
exactly one electron. However, this state with the highest possible atomiclike
character cannot be described within the adiabatic (or Born-Oppenheimer)
approximation because in the (true) nonadiabatic system the electrons move on
localized orbitals that are still symmetric on the average of time, but not at
any moment. The nonadiabatic states have the same symmetry as the adiabatic
states and determine the commutation properties of the nonadiabatic Hamiltonian
H^n. The nonadiabatic Heisenberg model is a purely group- theoretical model
which interprets the commutation properties of H^n that are explicitly given in
this paper for the two important cases (a) and (b). There is evidence that the
occurrence of these two types of Wannier functions in the band structure of a
metal is connected with the occurrence of magnetism and superconductivity,
respectively
Optimal Policy for Sequential Stochastic Resource Allocation
A gambler in possession of R chips/coins is allowed N(\u3eR) pulls/trials at a slot machine. Upon pulling the arm, the slot machine realizes a random state i ɛ{1, ..., M} with probability p(i) and the corresponding positive monetary reward g(i) is presented to the gambler. The gambler can accept the reward by inserting a coin in the machine. However, the dilemma facing the gambler is whether to spend the coin or keep it in reserve hoping to pick up a greater reward in the future. We assume that the gambler has full knowledge of the reward distribution function. We are interested in the optimal gambling strategy that results in the maximal cumulative reward. The problem is naturally posed as a Stochastic Dynamic Program whose solution yields the optimal policy and expected cumulative reward. We show that the optimal strategy is a threshold policy, wherein a coin is spent if and only if the number of coins r exceeds a state and stage/trial dependent threshold value. We illustrate the utility of the result on a military operational scenario
The Barrier Surface in the Cooperative Football Differential Game
This paper considers the blocking or football pursuit-evasion differential
game. Two pursuers cooperate and try to capture the ball carrying evader as far
as possible from the goal line. The evader wishes to be as close as possible to
the goal line at the time of capture and, if possible, reach the line. In this
paper the solution of the game of kind is provided: The Barrier surface that
partitions the state space into two winning sets, one for the pursuer team and
one for the evader, is constructed. Under optimal play, the winning team is
determined by evaluating the associated Barrier function.Comment: 5 pages, 1 figur
Optimal Guidance of a Relay Aircraft to Extend Small Unmanned Aircraft Range
This paper developed guidance laws to optimally and autonomously position a relay Micro Aerial Vehicle (MAV) to provide an operator with real-time Intelligence, Surveillance, and Reconnaissance (ISR) by relaying communication and video signals from a rover MAV to the base, thus extending the rover\u27s reach. The ISR system is comprised of two MAVs, the Relay and the Rover, and a Base. The Relay strives to position itself so as to minimize the radio frequency (RF) power required for maintaining communications between the Rover and the Base, while the Rover performs the ISR mission, which may maximize the required RF power. The optimal control of the Relay MAV then entails the solution of a differential game. Applying Pontryagin\u27s Maximum Principle yields a standard, albeit nonlinear, Two-Point Boundary Value Problem (TPBVP). Suboptimal solutions are first obtained as an aid in solving the TPBVP which yields the solution of the differential game. One suboptimal approach is based upon the geometry of the ISR system: The midpoint between the Rover and the Base is the ideal location which minimizes the RF power required, so the Relay heads toward that point—assuming that the Rover is stationary. At the same time, to maximize the rate of required RF power, the Rover moves in the opposite direction of the Relay—assuming the Relay is stationary. These are optimal strategies in the end-game, but it is suboptimal to use them throughout the game. Another suboptimal approach investigated envisions the Rover to remain stationary and solves for the optimal path for the Relay to minimize the RF power requirement. This one-sided optimization problem is analyzed using a Matlab-based optimization program, GPOCS, which uses the Gauss pseudospectral method of discretization. The results from GPOCS corroborated with the geometry-based suboptimal Relay strategy of heading straight toward the midpoint between the Rover and the Base. The suboptimal solutions are readily implementable for real-time operation and are used to facilitate the solutions of the TPBVP
Surveillance of a Faster Fixed-Course Target
The maximum surveillance of a target which is holding course is considered,
wherein an observer vehicle aims to maximize the time that a faster target
remains within a fixed-range of the observer. This entails two coupled phases:
an approach phase and observation phase. In the approach phase, the observer
strives to make contact with the faster target, such that in the observation
phase, the observer is able to maximize the time where the target remains
within range. Using Pontryagin's Minimum Principle, the optimal control laws
for the observer are found in closed-form. Example scenarios highlight various
aspects of the engagement.Comment: 12 pages, 8 figure
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