3,055 research outputs found
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
A graph theoretic approach to input-to-state stability of switched systems
This article deals with input-to-state stability (ISS) of discrete-time
switched systems. Given a family of nonlinear systems with exogenous inputs, we
present a class of switching signals under which the resulting switched system
is ISS. We allow non-ISS systems in the family and our analysis involves
graph-theoretic arguments. A weighted digraph is associated to the switched
system, and a switching signal is expressed as an infinite walk on this
digraph, both in a natural way. Our class of stabilizing switching signals
(infinite walks) is periodic in nature and affords simple algorithmic
construction.Comment: 14 pages, 1 figur
Verification of Uncertain POMDPs Using Barrier Certificates
We consider a class of partially observable Markov decision processes
(POMDPs) with uncertain transition and/or observation probabilities. The
uncertainty takes the form of probability intervals. Such uncertain POMDPs can
be used, for example, to model autonomous agents with sensors with limited
accuracy, or agents undergoing a sudden component failure, or structural damage
[1]. Given an uncertain POMDP representation of the autonomous agent, our goal
is to propose a method for checking whether the system will satisfy an optimal
performance, while not violating a safety requirement (e.g. fuel level,
velocity, and etc.). To this end, we cast the POMDP problem into a switched
system scenario. We then take advantage of this switched system
characterization and propose a method based on barrier certificates for
optimality and/or safety verification. We then show that the verification task
can be carried out computationally by sum-of-squares programming. We illustrate
the efficacy of our method by applying it to a Mars rover exploration example.Comment: 8 pages, 4 figure
Approximately bisimilar symbolic models for incrementally stable switched systems
Switched systems constitute an important modeling paradigm faithfully
describing many engineering systems in which software interacts with the
physical world. Despite considerable progress on stability and stabilization of
switched systems, the constant evolution of technology demands that we make
similar progress with respect to different, and perhaps more complex,
objectives. This paper describes one particular approach to address these
different objectives based on the construction of approximately equivalent
(bisimilar) symbolic models for switched systems. The main contribution of this
paper consists in showing that under standard assumptions ensuring incremental
stability of a switched system (i.e. existence of a common Lyapunov function,
or multiple Lyapunov functions with dwell time), it is possible to construct a
finite symbolic model that is approximately bisimilar to the original switched
system with a precision that can be chosen a priori. To support the
computational merits of the proposed approach, we use symbolic models to
synthesize controllers for two examples of switched systems, including the
boost DC-DC converter.Comment: 17 page
Koopman operator-based model reduction for switched-system control of PDEs
We present a new framework for optimal and feedback control of PDEs using
Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a
linear but infinite-dimensional operator which describes the dynamics of
observables. A numerical approximation of the Koopman operator therefore yields
a linear system for the observation of an autonomous dynamical system. In our
approach, by introducing a finite number of constant controls, the dynamic
control system is transformed into a set of autonomous systems and the
corresponding optimal control problem into a switching time optimization
problem. This allows us to replace each of these systems by a K-ROM which can
be solved orders of magnitude faster. By this approach, a nonlinear
infinite-dimensional control problem is transformed into a low-dimensional
linear problem. In situations where the Koopman operator can be computed
exactly using Extended Dynamic Mode Decomposition (EDMD), the proposed approach
yields optimal control inputs. Furthermore, a recent convergence result for
EDMD suggests that the approach can be applied to more complex dynamics as
well. To illustrate the results, we consider the 1D Burgers equation and the 2D
Navier--Stokes equations. The numerical experiments show remarkable performance
concerning both solution times and accuracy.Comment: arXiv admin note: text overlap with arXiv:1801.0641
Synthesis of Switching Protocols from Temporal Logic Specifications
We propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high-level specifications in linear temporal logic. The synthesized protocols are robust against exogenous disturbances on the continuous dynamics. Two types of finite transition systems, namely under- and over-approximations, that abstract the behavior of the underlying continuous dynamics are defined. In particular, we show that the discrete synthesis problem for an under-approximation can be formulated as a model checking problem, whereas that for an over-approximation can be transformed into a two-player game. Both of these formulations are amenable to efficient, off-the-shelf software tools. By construction, existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a continuous switching protocol for the continuous synthesis problem, which can be implemented at the continuous level to ensure the correctness of the nonlinear switched system. Moreover, the proposed framework can be straightforwardly extended to accommodate specifications that require reacting to possibly adversarial external events. Finally, these results are illustrated using three examples from different application domains
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