54,612 research outputs found
The moment exponential stability of infinite-dimensional linear stochastic switched systems
This paper studies the 2nd-moment exponential stability of a class of infinite-dimensional linear stochastic switched systems comprising two unstable subsystems. We first construct an algebraic sufficient condition on the existence of multiple Lyapunov functions. Then, two switching strategies are designed to stabilize infinite-dimensional linear stochastic switched systems in terms of the multiple Lyapunov function method. Moreover, the system possesses good robust stability of the switching time with our switching strategies
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
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
Switching in heteroclinic networks
We study the dynamics near heteroclinic networks for which all eigenvalues of
the linearization at the equilibria are real. A common connection and an
assumption on the geometry of its incoming and outgoing directions exclude even
the weakest forms of switching (i.e. along this connection). The form of the
global transition maps, and thus the type of the heteroclinic cycle, plays a
crucial role in this. We look at two examples in , the House and
Bowtie networks, to illustrate complex dynamics that may occur when either of
these conditions is broken. For the House network, there is switching along the
common connection, while for the Bowtie network we find switching along a
cycle
Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions
This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS
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