866 research outputs found
Robust Region-of-Attraction Estimation
We propose a method to compute invariant subsets of the region-of-attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameter-independent Lyapunov functions are used to characterize invariant subsets of the robust region-of-attraction. A branch-and-bound type refinement procedure reduces the conservatism. We demonstrate the method on an example from the literature and uncertain controlled short-period aircraft dynamics
A Sums-of-Squares Extension of Policy Iterations
In order to address the imprecision often introduced by widening operators in
static analysis, policy iteration based on min-computations amounts to
considering the characterization of reachable value set of a program as an
iterative computation of policies, starting from a post-fixpoint. Computing
each policy and the associated invariant relies on a sequence of numerical
optimizations. While the early research efforts relied on linear programming
(LP) to address linear properties of linear programs, the current state of the
art is still limited to the analysis of linear programs with at most quadratic
invariants, relying on semidefinite programming (SDP) solvers to compute
policies, and LP solvers to refine invariants.
We propose here to extend the class of programs considered through the use of
Sums-of-Squares (SOS) based optimization. Our approach enables the precise
analysis of switched systems with polynomial updates and guards. The analysis
presented has been implemented in Matlab and applied on existing programs
coming from the system control literature, improving both the range of
analyzable systems and the precision of previously handled ones.Comment: 29 pages, 4 figure
Discrete time piecewise affine models of genetic regulatory networks
We introduce simple models of genetic regulatory networks and we proceed to
the mathematical analysis of their dynamics. The models are discrete time
dynamical systems generated by piecewise affine contracting mappings whose
variables represent gene expression levels. When compared to other models of
regulatory networks, these models have an additional parameter which is
identified as quantifying interaction delays. In spite of their simplicity,
their dynamics presents a rich variety of behaviours. This phenomenology is not
limited to piecewise affine model but extends to smooth nonlinear discrete time
models of regulatory networks. In a first step, our analysis concerns general
properties of networks on arbitrary graphs (characterisation of the attractor,
symbolic dynamics, Lyapunov stability, structural stability, symmetries, etc).
In a second step, focus is made on simple circuits for which the attractor and
its changes with parameters are described. In the negative circuit of 2 genes,
a thorough study is presented which concern stable (quasi-)periodic
oscillations governed by rotations on the unit circle -- with a rotation number
depending continuously and monotonically on threshold parameters. These regular
oscillations exist in negative circuits with arbitrary number of genes where
they are most likely to be observed in genetic systems with non-negligible
delay effects.Comment: 34 page
Model predictive control techniques for hybrid systems
This paper describes the main issues encountered when applying model predictive control to hybrid processes. Hybrid model predictive control (HMPC) is a research field non-fully developed with many open challenges. The paper describes some of the techniques proposed by the research community to overcome the main problems encountered. Issues related to the stability and the solution of the optimization problem are also discussed. The paper ends by describing the results of a benchmark exercise in which several HMPC schemes were applied to a solar air conditioning plant.Ministerio de Eduación y Ciencia DPI2007-66718-C04-01Ministerio de Eduación y Ciencia DPI2008-0581
Backstepping controller synthesis and characterizations of incremental stability
Incremental stability is a property of dynamical and control systems,
requiring the uniform asymptotic stability of every trajectory, rather than
that of an equilibrium point or a particular time-varying trajectory. Similarly
to stability, Lyapunov functions and contraction metrics play important roles
in the study of incremental stability. In this paper, we provide
characterizations and descriptions of incremental stability in terms of
existence of coordinate-invariant notions of incremental Lyapunov functions and
contraction metrics, respectively. Most design techniques providing controllers
rendering control systems incrementally stable have two main drawbacks: they
can only be applied to control systems in either parametric-strict-feedback or
strict-feedback form, and they require these control systems to be smooth. In
this paper, we propose a design technique that is applicable to larger classes
of (not necessarily smooth) control systems. Moreover, we propose a recursive
way of constructing contraction metrics (for smooth control systems) and
incremental Lyapunov functions which have been identified as a key tool
enabling the construction of finite abstractions of nonlinear control systems,
the approximation of stochastic hybrid systems, source-code model checking for
nonlinear dynamical systems and so on. The effectiveness of the proposed
results in this paper is illustrated by synthesizing a controller rendering a
non-smooth control system incrementally stable as well as constructing its
finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure
Review on computational methods for Lyapunov functions
Lyapunov functions are an essential tool in the stability analysis of dynamical systems, both in theory and applications. They provide sufficient conditions for the stability of equilibria or more general invariant sets, as well as for their basin of attraction. The necessity, i.e. the existence of Lyapunov functions, has been studied in converse theorems, however, they do not provide a general method to compute them. Because of their importance in stability analysis, numerous computational construction methods have been developed within the Engineering, Informatics, and Mathematics community. They cover different types of systems such as ordinary differential equations, switched systems, non-smooth systems, discrete-time systems etc., and employ di_erent methods such as series expansion, linear programming, linear matrix inequalities, collocation methods, algebraic methods, set-theoretic methods, and many others. This review brings these different methods together. First, the different types of systems, where Lyapunov functions are used, are briefly discussed. In the main part, the computational methods are presented, ordered by the type of method used to construct a Lyapunov function
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