678 research outputs found
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
Semidefinite Relaxations for Stochastic Optimal Control Policies
Recent results in the study of the Hamilton Jacobi Bellman (HJB) equation
have led to the discovery of a formulation of the value function as a linear
Partial Differential Equation (PDE) for stochastic nonlinear systems with a
mild constraint on their disturbances. This has yielded promising directions
for research in the planning and control of nonlinear systems. This work
proposes a new method obtaining approximate solutions to these linear
stochastic optimal control (SOC) problems. A candidate polynomial with variable
coefficients is proposed as the solution to the SOC problem. A Sum of Squares
(SOS) relaxation is then taken to the partial differential constraints, leading
to a hierarchy of semidefinite relaxations with improving sub-optimality gap.
The resulting approximate solutions are shown to be guaranteed over- and
under-approximations for the optimal value function.Comment: Preprint. Accepted to American Controls Conference (ACC) 2014 in
Portland, Oregon. 7 pages, colo
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
On feedback stabilization of linear switched systems via switching signal control
Motivated by recent applications in control theory, we study the feedback
stabilizability of switched systems, where one is allowed to chose the
switching signal as a function of in order to stabilize the system. We
propose new algorithms and analyze several mathematical features of the problem
which were unnoticed up to now, to our knowledge. We prove complexity results,
(in-)equivalence between various notions of stabilizability, existence of
Lyapunov functions, and provide a case study for a paradigmatic example
introduced by Stanford and Urbano.Comment: 19 pages, 3 figure
Recommended from our members
Mini-Workshop: Applied Koopmanism
Koopman and Perron–Frobenius operators are linear operators that encapsulate dynamics of nonlinear dynamical systems without loss of information. This is accomplished by embedding the dynamics into a larger infinite-dimensional space where the focus of study is shifted from trajectory curves to measurement functions evaluated along trajectories and densities of trajectories evolving in time. Operator-theoretic approach to dynamics shares many features with an optimization technique: the Lasserre moment–sums-of-squares (SOS) hierarchies, which was developed for numerically solving non-convex optimization problems with semialgebraic data. This technique embeds the optimization problem into a larger primal semidefinite programming (SDP) problem consisting of measure optimization over the set of globally optimal solutions, where measures are manipulated through their truncated moment sequences. The dual SDP problem uses SOS representations to certify bounds on the global optimum. This workshop highlighted the common threads between the operator-theoretic dynamical systems and moment–SOS hierarchies in optimization and explored the future directions where the synergy of the two techniques could yield results in fluid dynamics, control theory, optimization, and spectral theory
Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis
This thesis settles a number of questions related to computational complexity
and algebraic, semidefinite programming based relaxations in optimization and
control.Comment: PhD Thesis, MIT, September, 201
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