2,161 research outputs found
Converse Lyapunov Theorems for Switched Systems in Banach and Hilbert Spaces
We consider switched systems on Banach and Hilbert spaces governed by
strongly continuous one-parameter semigroups of linear evolution operators. We
provide necessary and sufficient conditions for their global exponential
stability, uniform with respect to the switching signal, in terms of the
existence of a Lyapunov function common to all modes
Symbolic Models for Stochastic Switched Systems: A Discretization and a Discretization-Free Approach
Stochastic switched systems are a relevant class of stochastic hybrid systems
with probabilistic evolution over a continuous domain and control-dependent
discrete dynamics over a finite set of modes. In the past few years several
different techniques have been developed to assist in the stability analysis of
stochastic switched systems. However, more complex and challenging objectives
related to the verification of and the controller synthesis for logic
specifications have not been formally investigated for this class of systems as
of yet. With logic specifications we mean properties expressed as formulae in
linear temporal logic or as automata on infinite strings. This paper addresses
these complex objectives by constructively deriving approximately equivalent
(bisimilar) symbolic models of stochastic switched systems. More precisely,
this paper provides two different symbolic abstraction techniques: one requires
state space discretization, but the other one does not require any space
discretization which can be potentially more efficient than the first one when
dealing with higher dimensional stochastic switched systems. Both techniques
provide finite symbolic models that are approximately bisimilar to stochastic
switched systems under some stability assumptions on the concrete model. This
allows formally synthesizing controllers (switching signals) that are valid for
the concrete system over the finite symbolic model, by means of mature
automata-theoretic techniques in the literature. The effectiveness of the
results are illustrated by synthesizing switching signals enforcing logic
specifications for two case studies including temperature control of a six-room
building.Comment: 25 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1302.386
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
Geometric and asymptotic properties associated with linear switched systems
Consider continuous-time linear switched systems on R^n associated with
compact convex sets of matrices. When the system is irreducible and the largest
Lyapunov exponent is equal to zero, there always exists a Barabanov norm (i.e.
a norm which is non increasing along trajectories of the linear switched system
together with extremal trajectories starting at every point, that is
trajectories of the linear switched system with constant norm). This paper
deals with two sets of issues: (a) properties of Barabanov norms such as
uniqueness up to homogeneity and strict convexity; (b) asymptotic behaviour of
the extremal solutions of the linear switched system. Regarding Issue (a), we
provide partial answers and propose four open problems motivated by appropriate
examples. As for Issue (b), we establish, when n = 3, a Poincar\'e-Bendixson
theorem under a regularity assumption on the set of matrices defining the
system. Moreover, we revisit the noteworthy result of N.E. Barabanov [5]
dealing with the linear switched system on R^3 associated with a pair of
Hurwitz matrices {A, A + bcT }. We first point out a fatal gap in Barabanov's
argument in connection with geometric features associated with a Barabanov
norm. We then provide partial answers relative to the asymptotic behavior of
this linear switched system.Comment: 37 page
Tools for Stability of Switching Linear Systems: Gain Automata and Delay Compensation.
The topic of this paper is the analysis of stability for a class of switched linear systems, modeled by hybrid automata. In each location of the hybrid automaton the dynamics is assumed to be linear and asymptotically stable; the guards on the transitions are hyperplanes in the state space. For each location an estimate is made of the gain via a Lyapunov function for the dynamics in that location, given a pair of ingoing and outgoing transitions. It is shown how to obtain the best possible estimate by optimizing the Lyapunov function. The estimated gains are used in defining a so-called gain automaton that forms the basis of an algorithmic criterion for the stability of the hybrid automaton. The associated gain automaton provides a systematic tool to detect potential sources of instability as well as an indication on to how to stabilize the hybrid systems by requiring appropriate delays for specific transitions
- …