867 research outputs found
Region of Attraction Estimation Using Invariant Sets and Rational Lyapunov Functions
This work addresses the problem of estimating the region of attraction (RA)
of equilibrium points of nonlinear dynamical systems. The estimates we provide
are given by positively invariant sets which are not necessarily defined by
level sets of a Lyapunov function. Moreover, we present conditions for the
existence of Lyapunov functions linked to the positively invariant set
formulation we propose. Connections to fundamental results on estimates of the
RA are presented and support the search of Lyapunov functions of a rational
nature. We then restrict our attention to systems governed by polynomial vector
fields and provide an algorithm that is guaranteed to enlarge the estimate of
the RA at each iteration
Stability and Boundedness of Solutions to Some Non-autonomous Multidimensional Nonlinear Systems
Assessment of degree of boundedness and stability of multidimensional
nonlinear systems with time-dependent and especially nonperiodic coefficients
is an important applied problem which has no adequate resolution yet. Most of
the known techniques mostly provide computationally intensive and conservative
stability criteria in this area which frequently fail to gage the degrees of
stability and especially boundedness of solutions to the corresponding systems.
Recently, we outline a new approach to this task resting on analysis of
solutions to a scalar auxiliary equation bounding from above time-histories of
the norms of solutions to the original systems. This paper develops a new
technique casting the auxiliary equation in a simplified form which, in turn,
amplifies its application domain and reduces the computational hamper of our
prior approach. Consequently, we develop novel boundedness and stability
criteria and estimated the trapping and stability regions for some
multidimensional nonlinear systems with time - dependent coefficients. This let
us to assess in target simulations the degree of boundedness and stability of
multidimensional nonlinear and non-autonomous systems which were intractable to
our prior methodolog
LMI techniques for optimization over polynomials in control: A survey
Numerous tasks in control systems involve optimization problems over polynomials, and unfortunately these problems are in general nonconvex. In order to cope with this difficulty, linear matrix inequality (LMI) techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems. This survey aims to provide the reader with a significant overview of the LMI techniques that are used in control systems for tackling optimization problems over polynomials, describing approaches such as decomposition in sum of squares, Positivstellensatz, theory of moments, Plya's theorem, and matrix dilation. Moreover, it aims to provide a collection of the essential problems in control systems where these LMI techniques are used, such as stability and performance investigations in nonlinear systems, uncertain systems, time-delay systems, and genetic regulatory networks. It is expected that this survey may be a concise useful reference for all readers. © 2006 IEEE.published_or_final_versio
Can chaotic quantum energy levels statistics be characterized using information geometry and inference methods?
In this paper, we review our novel information geometrodynamical approach to
chaos (IGAC) on curved statistical manifolds and we emphasize the usefulness of
our information-geometrodynamical entropy (IGE) as an indicator of chaoticity
in a simple application. Furthermore, knowing that integrable and chaotic
quantum antiferromagnetic Ising chains are characterized by asymptotic
logarithmic and linear growths of their operator space entanglement entropies,
respectively, we apply our IGAC to present an alternative characterization of
such systems. Remarkably, we show that in the former case the IGE exhibits
asymptotic logarithmic growth while in the latter case the IGE exhibits
asymptotic linear growth. At this stage of its development, IGAC remains an
ambitious unifying information-geometric theoretical construct for the study of
chaotic dynamics with several unsolved problems. However, based on our recent
findings, we believe it could provide an interesting, innovative and
potentially powerful way to study and understand the very important and
challenging problems of classical and quantum chaos.Comment: 21 page
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
On the performance of nonlinear dynamical systems under parameter perturbation
AbstractWe present a method for analysing the deviation in transient behaviour between two parameterised families of nonlinear ODEs, as initial conditions and parameters are varied within compact sets over which stability is guaranteed. This deviation is taken to be the integral over time of a user-specified, positive definite function of the difference between the trajectories, for instance the L2 norm. We use sum-of-squares programming to obtain two polynomials, which take as inputs the (possibly differing) initial conditions and parameters of the two families of ODEs, and output upper and lower bounds to this transient deviation. Equality can be achieved using symbolic methods in a special case involving Linear Time Invariant Parameter Dependent systems. We demonstrate the utility of the proposed methods in the problems of model discrimination, and location of worst case parameter perturbation for a single parameterised family of ODE models
Derivative-free Kalman Filter-based Control of Nonlinear Systems with Application to Transfemoral Prostheses
Derivative-free Kalman filtering (DKF) for estimation-based control of a special class of nonlinear systems is presented. The method includes a standard Kalman filter for the estimation of both states and unknown inputs, and a nonlinear system that is transformed to controllable canonical state space form through feedback linearization (FL). A direct current (DC) motor with an input torque that is a nonlinear function of the state is considered as a case study for a nonlinear single-input-single-output (SISO) system. A three degree-of-freedom (DOF) robot / prosthesis system, which includes a robot that emulates human hip and thigh motion and a powered (active) transfemoral prosthesis disturbed by ground reaction force (GRF), is considered as a case study for a nonlinear multi-input-multi-output (MIMO) system. A PD/PI control term is used to compensate for the unknown GRF. Simulation results show that FL can compensate for the system\u27s nonlinearities through a virtual control term, in contrast to Taylor series linearization, which is only a first-order linearization method. FL improves estimation performance relative to the extended Kalman filter, and in some cases improves the initial condition region of attraction as well. A stability analysis of the DKF-based control method, considering both estimation and unknown input compensation, is also presented. The error dynamics are studied in both frequency and time domains. The derivative of the unknown input plays a key role in the error dynamics and is the primary limiting factor of the closed-loop estimation-based control system stability. It is shown that in realistic systems the derivative of the unknown input is the primary determinant of the region of convergence. It is shown that the tracking error asymptotically converges to the derivative of the unknown input
Stability of Ecological Systems: A Theoretical Review
The stability of ecological systems is a fundamental concept in ecology,
which offers profound insights into species coexistence, biodiversity, and
community persistence. In this article, we provide a systematic and
comprehensive review on the theoretical frameworks for analyzing the stability
of ecological systems. Notably, we survey various stability notions, including
linear stability, sign stability, diagonal stability, D-stability, total
stability, sector stability, structural stability, and higher-order stability.
For each of these stability notions, we examine necessary or sufficient
conditions for achieving such stability and demonstrate the intricate interplay
of these conditions on the network structures of ecological systems. Finally,
we explore the future prospects of these stability notions
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