351 research outputs found
Some Applications of Polynomial Optimization in Operations Research and Real-Time Decision Making
We demonstrate applications of algebraic techniques that optimize and certify
polynomial inequalities to problems of interest in the operations research and
transportation engineering communities. Three problems are considered: (i)
wireless coverage of targeted geographical regions with guaranteed signal
quality and minimum transmission power, (ii) computing real-time certificates
of collision avoidance for a simple model of an unmanned vehicle (UV)
navigating through a cluttered environment, and (iii) designing a nonlinear
hovering controller for a quadrotor UV, which has recently been used for load
transportation. On our smaller-scale applications, we apply the sum of squares
(SOS) relaxation and solve the underlying problems with semidefinite
programming. On the larger-scale or real-time applications, we use our recently
introduced "SDSOS Optimization" techniques which result in second order cone
programs. To the best of our knowledge, this is the first study of real-time
applications of sum of squares techniques in optimization and control. No
knowledge in dynamics and control is assumed from the reader
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
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