125,192 research outputs found
Stability of switched linear differential systems
We study the stability of switched systems where the dynamic modes are
described by systems of higher-order linear differential equations not
necessarily sharing the same state space. Concatenability of trajectories at
the switching instants is specified by gluing conditions, i.e. algebraic
conditions on the trajectories and their derivatives at the switching instant.
We provide sufficient conditions for stability based on LMIs for systems with
general gluing conditions. We also analyse the role of positive-realness in
providing sufficient polynomial-algebraic conditions for stability of two-modes
switched systems with special gluing conditions
On the D-Stability of Linear and Nonlinear Positive Switched Systems
We present a number of results on D-stability
of positive switched systems. Different classes of linear and
nonlinear positive switched systems are considered and simple
conditions for D-stability of each class are presented
Common Lyapunov Function Based on Kullback–Leibler Divergence for a Switched Nonlinear System
Many problems with control theory have led to investigations into
switched systems. One of the most urgent problems related to the analysis of the
dynamics of switched systems is the stability problem. The stability of a switched
system can be ensured by a common Lyapunov function for all switching modes under
an arbitrary switching law. Finding a common Lyapunov function is still an interesting
and challenging problem. The purpose of the present paper is to prove the stability of
equilibrium in a certain class of nonlinear switched systems by introducing a common
Lyapunov function; the Lyapunov function is based on generalized Kullback–Leibler
divergence or Csiszár's I-divergence between the state and equilibrium. The switched
system is useful for finding positive solutions to linear algebraic equations, which
minimize the I-divergence measure under arbitrary switching. One application of the
stability of a given switched system is in developing a new approach to reconstructing
tomographic images, but nonetheless, the presented results can be used in numerous
other areas
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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Fault detection for switched positive systems under successive packet dropouts with application to the Leslie matrix model
In this paper, the problem of fault detection filter design is dealt with for a class of switched positive systems
with packet dropouts on the channel between the sensors and the filters. The phenomena of packet dropouts
are governed by a Bernoulli process, and a stochastic switched positive system is established based on the
augmented states of the plants and filters. Two criteria are developed to evaluate the performance of the
fault detection for the system under investigation. Sufficient conditions are established on the existence
of the desired filters for the mean-square stability with an L1 disturbance attenuation level, and an index
for the L fault sensitivity is also derived through constructing a switched Lyapunov function in term of
linear programming. Two illustrative examples, one of which is concerned with the Leslie matrix model, are
provided to show the effectiveness and applicability of the proposed results. Copyright © 2015 John Wiley
& Sons, Ltd.National Natural Science Foundation of China under Grants 61104114, 61201035, 61374070, 61473055. Fundamental Research Funds for the Central Universities in China under Grants DUT14QY14, DUT14QY31, and the Natural Science Foundation of Liaoning under Grants L2014026, 2015020075
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