14 research outputs found
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
Error estimates for a tree structure algorithm solving finite horizon control problems
In the Dynamic Programming approach to optimal control problems a crucial
role is played by the value function that is characterized as the unique
viscosity solution of a Hamilton-Jacobi-Bellman (HJB) equation. It is well
known that this approach suffers of the "curse of dimensionality" and this
limitation has reduced its practical in real world applications. Here we
analyze a dynamic programming algorithm based on a tree structure. The tree is
built by the time discrete dynamics avoiding in this way the use of a fixed
space grid which is the bottleneck for high-dimensional problems, this also
drops the projection on the grid in the approximation of the value function. We
present some error estimates for a first order approximation based on the
tree-structure algorithm. Moreover, we analyze a pruning technique for the tree
to reduce the complexity and minimize the computational effort. Finally, we
present some numerical tests
A HJB-POD approach for the control of nonlinear PDEs on a tree structure
The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced. The method allows to work without a structured grid and avoids any interpolation. Here, we aim at testing the algorithm for nonlinear two dimensional PDEs. We apply model order reduction to decrease the computational complexity since the tree structure algorithm requires to solve many PDEs. Furthermore, we prove an error estimate which guarantees the convergence of the proposed method. Finally, we show efficiency of the method through numerical tests
Optimal Trajectories of a UAV Base Station Using Hamilton-Jacobi Equations
We consider the problem of optimizing the trajectory of an Unmanned Aerial
Vehicle (UAV). Assuming a traffic intensity map of users to be served, the UAV
must travel from a given initial location to a final position within a given
duration and serves the traffic on its way. The problem consists in finding the
optimal trajectory that minimizes a certain cost depending on the velocity and
on the amount of served traffic. We formulate the problem using the framework
of Lagrangian mechanics. We derive closed-form formulas for the optimal
trajectory when the traffic intensity is quadratic (single-phase) using
Hamilton-Jacobi equations. When the traffic intensity is bi-phase, i.e. made of
two quadratics, we provide necessary conditions of optimality that allow us to
propose a gradient-based algorithm and a new algorithm based on the linear
control properties of the quadratic model. These two solutions are of very low
complexity because they rely on fast convergence numerical schemes and closed
form formulas. These two approaches return a trajectory satisfying the
necessary conditions of optimality. At last, we propose a data processing
procedure based on a modified K-means algorithm to derive a bi-phase model and
an optimal trajectory simulation from real traffic data.Comment: 30 pages, 10 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1812.0875