2,281 research outputs found
Monotone numerical schemes and feedback construction for hybrid control systems
International audienceHybrid systems are a general framework which can model a large class of control systems arising whenever a collection of continuous and discrete dynamics are put together in a single model. In this paper, we study the convergence of monotone numerical approximations of value functions associated to control problems governed by hybrid systems. We discuss also the feedback reconstruction and derive a convergence result for the approximate feedback control law. Some numerical examples are given to show the robustness of the monotone approximation schemes
Finite volume schemes for diffusion equations: introduction to and review of modern methods
We present Finite Volume methods for diffusion equations on generic meshes,
that received important coverage in the last decade or so. After introducing
the main ideas and construction principles of the methods, we review some
literature results, focusing on two important properties of schemes (discrete
versions of well-known properties of the continuous equation): coercivity and
minimum-maximum principles. Coercivity ensures the stability of the method as
well as its convergence under assumptions compatible with real-world
applications, whereas minimum-maximum principles are crucial in case of strong
anisotropy to obtain physically meaningful approximate solutions
An Efficient Policy Iteration Algorithm for Dynamic Programming Equations
We present an accelerated algorithm for the solution of static
Hamilton-Jacobi-Bellman equations related to optimal control problems. Our
scheme is based on a classic policy iteration procedure, which is known to have
superlinear convergence in many relevant cases provided the initial guess is
sufficiently close to the solution. In many cases, this limitation degenerates
into a behavior similar to a value iteration method, with an increased
computation time. The new scheme circumvents this problem by combining the
advantages of both algorithms with an efficient coupling. The method starts
with a value iteration phase and then switches to a policy iteration procedure
when a certain error threshold is reached. A delicate point is to determine
this threshold in order to avoid cumbersome computation with the value
iteration and, at the same time, to be reasonably sure that the policy
iteration method will finally converge to the optimal solution. We analyze the
methods and efficient coupling in a number of examples in dimension two, three
and four illustrating its properties
On a small-gain approach to distributed event-triggered control
In this paper the problem of stabilizing large-scale systems by distributed
controllers, where the controllers exchange information via a shared limited
communication medium is addressed. Event-triggered sampling schemes are
proposed, where each system decides when to transmit new information across the
network based on the crossing of some error thresholds. Stability of the
interconnected large-scale system is inferred by applying a generalized
small-gain theorem. Two variations of the event-triggered controllers which
prevent the occurrence of the Zeno phenomenon are also discussed.Comment: 30 pages, 9 figure
Isochronous Partitions for Region-Based Self-Triggered Control
In this work, we propose a region-based self-triggered control (STC) scheme
for nonlinear systems. The state space is partitioned into a finite number of
regions, each of which is associated to a uniform inter-event time. The
controller, at each sampling time instant, checks to which region does the
current state belong, and correspondingly decides the next sampling time
instant. To derive the regions along with their corresponding inter-event
times, we use approximations of isochronous manifolds, a notion firstly
introduced in [1]. This work addresses some theoretical issues of [1] and
proposes an effective computational approach that generates approximations of
isochronous manifolds, thus enabling the region-based STC scheme. The
efficiency of both our theoretical results and the proposed algorithm are
demonstrated through simulation examples
Multigrid methods for two-player zero-sum stochastic games
We present a fast numerical algorithm for large scale zero-sum stochastic
games with perfect information, which combines policy iteration and algebraic
multigrid methods. This algorithm can be applied either to a true finite state
space zero-sum two player game or to the discretization of an Isaacs equation.
We present numerical tests on discretizations of Isaacs equations or
variational inequalities. We also present a full multi-level policy iteration,
similar to FMG, which allows to improve substantially the computation time for
solving some variational inequalities.Comment: 31 page
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
Guaranteed Control of Sampled Switched Systems using Semi-Lagrangian Schemes and One-Sided Lipschitz Constants
In this paper, we propose a new method for ensuring formally that a
controlled trajectory stay inside a given safety set S for a given duration T.
Using a finite gridding X of S, we first synthesize, for a subset of initial
nodes x of X , an admissible control for which the Euler-based approximate
trajectories lie in S at t [0,T]. We then give sufficient conditions
which ensure that the exact trajectories, under the same control, also lie in S
for t [0,T], when starting at initial points 'close' to nodes x. The
statement of such conditions relies on results giving estimates of the
deviation of Euler-based approximate trajectories, using one-sided Lipschitz
constants. We illustrate the interest of the method on several examples,
including a stochastic one
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