6,974 research outputs found
A Fixed Time Convergent Dynamical System to Solve Linear Programming
The aim of this paper is to present a new dynamical system which solves linear programming. Its design is considered as a sliding mode control problem, where its structure is based on the Karush-Kuhn-Tucker optimality conditions, and its multipliers are the control inputs to be implemented by using fixed time stabilizing terms with vectorial structure, based on the unit control, instead of common terms used in other approaches. Thus, the main features of the proposed system are the fixed convergence time to the programming solution and the fixed parameters number despite of the optimization problem dimension. That is, there is a time independent to the initial conditions in which the system converges to the solution and, the proposed structure can be easily scaled from a small to a higher dimension problem. The applicability of the proposed scheme is tested on real-time optimization of an electrical Microgrid prototype.Consejo Nacional de Ciencia y Tecnologí
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
Observer design for systems with an energy-preserving non-linearity
Observer design is considered for a class of non-linear systems whose
non-linear part is energy preserving. A strategy to construct convergent
observers for this class of non-linear system is presented. The approach has
the advantage that it is possible, via convex programming, to prove whether the
constructed observer converges, in contrast to several existing approaches to
observer design for non-linear systems. Finally, the developed methods are
applied to the Lorenz attractor and to a low order model for shear fluid flow
Robust-to-Dynamics Optimization
A robust-to-dynamics optimization (RDO) problem is an optimization problem
specified by two pieces of input: (i) a mathematical program (an objective
function and a feasible set
), and (ii) a dynamical system (a map
). Its goal is to minimize over the
set of initial conditions that forever remain in
under . The focus of this paper is on the case where the
mathematical program is a linear program and the dynamical system is either a
known linear map, or an uncertain linear map that can change over time. In both
cases, we study a converging sequence of polyhedral outer approximations and
(lifted) spectrahedral inner approximations to . Our inner
approximations are optimized with respect to the objective function and
their semidefinite characterization---which has a semidefinite constraint of
fixed size---is obtained by applying polar duality to convex sets that are
invariant under (multiple) linear maps. We characterize three barriers that can
stop convergence of the outer approximations from being finite. We prove that
once these barriers are removed, our inner and outer approximating procedures
find an optimal solution and a certificate of optimality for the RDO problem in
a finite number of steps. Moreover, in the case where the dynamics are linear,
we show that this phenomenon occurs in a number of steps that can be computed
in time polynomial in the bit size of the input data. Our analysis also leads
to a polynomial-time algorithm for RDO instances where the spectral radius of
the linear map is bounded above by any constant less than one. Finally, in our
concluding section, we propose a broader research agenda for studying
optimization problems with dynamical systems constraints, of which RDO is a
special case
Evolution, dynamics, and fixed points
Sign-compatible dynamics describe changes in the composition of a population driven by differences in fitness. A saturated equilibrium is a fixed point for sign-compatible dynamics where each subgroup with positive population share has highest fitness. An evolutionary stable equilibrium is a saturated equilibrium attracting all trajectories nearby, such that the Euclidean distance to it decreases monotonically. We address existence, multiplicity, and dynamical stability of fixed points of sign-compatible dynamics. A saturated equilibrium may be approximated by using a variable dimension restart algorithm for solving the nonlinear complementarity problem. Journal of Economic Literature Classification Numbers: C62, C68, C72, C73. Keywords: Sign-compatible population dynamics, saturated equilibrium, evolutionary stable equilibrium, dynamic stability, nonlinear complementarity problem.mathematical economics and econometrics
Help on SOS
In this issue of IEEE Control Systems Magazine, Andy Packard and friends respond to a query on determining the region of attraction using sum-of-squares methods
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