33 research outputs found
Optimal Stabilization using Lyapunov Measures
Numerical solutions for the optimal feedback stabilization of discrete time
dynamical systems is the focus of this paper. Set-theoretic notion of almost
everywhere stability introduced by the Lyapunov measure, weaker than
conventional Lyapunov function-based stabilization methods, is used for optimal
stabilization. The linear Perron-Frobenius transfer operator is used to pose
the optimal stabilization problem as an infinite dimensional linear program.
Set-oriented numerical methods are used to obtain the finite dimensional
approximation of the linear program. We provide conditions for the existence of
stabilizing feedback controls and show the optimal stabilizing feedback control
can be obtained as a solution of a finite dimensional linear program. The
approach is demonstrated on stabilization of period two orbit in a controlled
standard map
Submodular Function Maximization for Group Elevator Scheduling
We propose a novel approach for group elevator scheduling by formulating it
as the maximization of submodular function under a matroid constraint. In
particular, we propose to model the total waiting time of passengers using a
quadratic Boolean function. The unary and pairwise terms in the function denote
the waiting time for single and pairwise allocation of passengers to elevators,
respectively. We show that this objective function is submodular. The matroid
constraints ensure that every passenger is allocated to exactly one elevator.
We use a greedy algorithm to maximize the submodular objective function, and
derive provable guarantees on the optimality of the solution. We tested our
algorithm using Elevate 8, a commercial-grade elevator simulator that allows
simulation with a wide range of elevator settings. We achieve significant
improvement over the existing algorithms.Comment: 10 pages; 2017 International Conference on Automated Planning and
Scheduling (ICAPS
Covering orthogonal polygons with star polygons: The perfect graph approach
AbstractThis paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n3) algorithm