System Of Vector Fields Research Articles

Simultaneous local normal forms of dynamical systems with singular underlying geometric structures

Abstract The aim of this paper is to develop, for the first time, a general theory of simultaneous local normalisation of couples ( X , G ) , where X is a dynamical system (vector field) and G is an underlying geometric structure preserved by X, even if both have singularities. Such couples appear naturally in many problems, e.g. Hamiltonian dynamics, where G is a symplectic structure and one has the theory of Birkhoff normal forms, or constrained dynamics, where G is a smooth, in general singular, distribution of tangent subspaces, etc. In this paper, the geometric structure G is of the following types: volume form, symplectic form, contact form, Poisson tensor, as well as their singular versions. The paper addresses mainly the more difficult situations when both X and G are singular at a point and its results prove the existence of natural simultaneous normal forms in these cases. In general, the normalisation is only formal, but when G and X are (real or complex) analytic and X is analytically or Darboux integrable, then the simultaneous normalisation is also analytic. Our theory is based on a new approach, called the Toric Conservation Principle, as well as the classical step-by-step normalisation technique, and the equivariant path method.

Inference on the Macroscopic Dynamics of Spiking Neurons.

The process of inference on networks of spiking neurons is essential to decipher the underlying mechanisms of brain computation and function. In this study, we conduct inference on parameters and dynamics of a mean-field approximation, simplifying the interactions of neurons. Estimating parameters of this class of generative model allows one to predict the system's dynamics and responses under changing inputs and, indeed, changing parameters. We first assume a set of known state-space equationsand address the problem of inferring the lumped parameters from observed time series. Crucially, we consider this problem in the setting of bistability, random fluctuations in system dynamics, and partial observations, in which some states are hidden. To identify the most efficient estimation or inversion scheme in this particular system identification, we benchmark against state-of-the-art optimization and Bayesian estimation algorithms, highlighting their strengths and weaknesses. Additionally, we explore how well the statistical relationships between parameters are maintained across different scales. We found that deep neural density estimators outperform other algorithms in the inversion scheme, despite potentially resulting in overestimated uncertainty and correlation between parameters. Nevertheless, this issue can be improved by incorporating time-delay embedding. We then eschew the mean-field approximation and employ deep neural ODEs on spiking neurons, illustrating prediction of system dynamics and vector fields from microscopic states. Overall, this study affords an opportunity to predict brain dynamics and responses to various perturbations or pharmacological interventions using deep neural networks.

Extreme events and extreme multistability in a nearly conservative system.

This study investigates the emergence of extreme events in a complex variable dynamical system. In the absence of an external forcing, the model exhibits nearly Hamiltonian dynamics. When we set the system to a nearly conservative state and perturb it with external forcing, the formation of the onset of the extreme events was detected. By applying nullcline analysis and the system's vector field, we explored the underlying mechanism that leads to extreme events. Furthermore, we have conducted a thorough investigation to show the dynamic origins of extreme amplitude events and their transitions. The hardware electronic experiment is used to validate the numerical results of the onset of extreme events, and the results obtained are in good agreement with one another.

Necessary conditions for local controllability of a particular class of systems with two scalar controls

We consider control-afflne systems with two scalar controls, such that one control vector field vanishes at an equilibrium state. We state two necessary conditions for local controllability around this equilibrium, involving the iterated Lie brackets of the system vector fields, with controls that are either bounded, small in L∞ or small in W1,∞. These results are illustrated with several examples.

On the discontinuous dynamics of a class of 2-DOF frictional vibration systems with asymmetric elastic constraints

<abstract><p>In this paper, the discontinuous dynamic behavior of a two-degree-of-freedom frictional collision system including intermediate elastic collision and unilateral elastic constraints subjected to periodic excitation is studied by using flow switching theory. In this system, given that the motion of each object might have a velocity that is either greater than or less than zero and each object experiences a periodic excitation force that has negative feedback, because the kinetic and static friction coefficients differ, the flow barrier manifests when the object's speed is zero. Based on the discontinuity or nonsmoothness of the oscillator's motion generated by elastic collision and friction, the motion states of the oscillator in the system are divided into 16 cases and the absolute and relative coordinates are used to define various boundaries and domains in the oscillator motion's phase space. On the basis of this, the G-function and system vector fields are used to propose the oscillator motion's switching rules at the displacement and velocity boundaries. Finally, some dynamic behaviors for the 2-DOF oscillator are demonstrated via numerical simulation of the oscillator's stick, grazing, sliding and periodic motions and the scene of sliding bifurcation. The mechanical system's optimization designs with friction and elastic collision will benefit from this investigation's findings.</p></abstract>

Cone-Copositive Lyapunov Functions for Complementarity Systems: Converse Result and Polynomial Approximation

This article establishes the existence of Lyapunov functions for analyzing the stability of a class of state-constrained systems, and it describes algorithms for their numerical computation. The system model consists of a differential equation coupled with a set-valued relation which introduces discontinuities in the vector field at the boundaries of the constraint set. In particular, the set-valued relation is described by the subdifferential of the indicator function of a closed convex cone, which results in a cone-complementarity system. The question of analyzing stability of such systems is addressed by constructing cone-copositive Lyapunov functions. As a first analytical result, we show that exponentially stable complementarity systems always admit a continuously differentiable cone-copositive Lyapunov function. Putting some more structure on the system vector field, such as homogeneity, we can show that the aforementioned functions can be approximated by a rational function of cone-copositive homogeneous polynomials. This later class of functions is seen to be particularly amenable for numerical computation as we provide two types of algorithms for precisely that purpose. These algorithms consist of a hierarchy of either linear or semidefinite optimization problems for computing the desired cone-copositive Lyapunov function. Some examples are given to illustrate our approach.

Cooperative Dynamic Fuzzy Perimeter Surveillance: Modeling and Fluid-Based Framework

In this article, considering the performance limitations of the sensors used in real missions, the cooperative dynamic fuzzy perimeter surveillance (CDFPS) problem and its fluid-based solution framework are proposed for the first time. First, the two-dimensional concentration diffusion distribution function is used to imitate an irregular dynamic perimeter. Then, according to the onboard sensor resolution ratio, a fuzzification process is applied for the perimeter area. On this basis, the detected fuzzy perimeter area is modeled with virtual obstacles (VOs). Considering an obstacle set composed of VOs, fixed obstacles and agents, we transform the CDFPS problem into a unified obstacle avoidance problem for the first time and a fluid-based path planning method is designed based on restrained interfered fluid dynamical system and Lyapunov guidance vector field. To quantify the surveillance effect, two indexes for the CDFPS problem are proposed: the perimeter neutrality index and the phase angle uniformity index. Around these proposed indexes, the attractive/repulsive effects and the multiagent cooperative mechanism are constructed. Finally, the receding horizon control is introduced to further improve the surveillance effect. Simulations show that the proposed framework can drive agents to persistently track the perimeter and establish relatively stable phase angle offsets between adjacent agents.

Convergence of Nonlinear Observers on <inline-formula> <tex-math notation="LaTeX">${\mathbb{R}}^{n}$</tex-math> </inline-formula> With a Riemannian Metric (Part II)

In [1], it is established that a convergent observer with an infinite gain margin can be designed for a given nonlinear system when a Riemannian metric showing that the system is differentially detectable (i.e., the Lie derivative of the Riemannian metric along the system vector field is negative in the space tangent to the output function level sets) and the level sets of the output function are geodesically convex is available. In this paper, we propose techniques for designing a Riemannian metric satisfying the first property in the case where the system is strongly infinitesimally observable (i.e., each time-varying linear system resulting from the linearization along a solution to the system satisfies a uniform observability property) or where it is strongly differentially observable (i.e. the mapping state to output derivatives is an injective immersion) or where it is Lagrangian. Also, we give results that are complementary to those in [1]. In particular, we provide a locally convergent observer and make a link to the existence of a reduced order observer. Examples illustrating the results are presented.

Local Decompositions of Second Order Infinite‐Dimensional Systems

AbstractThis article deals with the local system decomposition of infinite‐dimensional systems, which are described by second‐order nonlinear partial differential equations. It will be shown that the existence of a certain codistribution allows a local triangular decomposition of the generalized system vector field and the boundary conditions. Based on this triangular decomposition, non‐accessibility follows by a simple structural analysis. Throughout the article differential geometric methods are applied, highlighting the geometric picture behind the system description. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Local Decomposition and Accessibility of Nonlinear Infinite-Dimensional Systems

This article deals with the local system decomposition of infinite-dimensional systems, which are described by second-order nonlinear partial differential equations. We show that if there exists a certain codistribution which is invariant under the generalized system vector field, a local triangular decomposition can be obtained. Furthermore, we draw connections to a different approach which is based on transformation groups. Throughout the article we apply differential geometric methods, highlighting the geometric picture behind the system description. The article is closed with a nonlinear example.

Approximation of dynamic system vector field using neural networks in approximate aircraft attainability domain construction problem

Approximation of dynamic system vector field using neural networks in approximate aircraft attainability domain construction problem

Control Design for a Class of Nonholonomic Systems Via Reference Vector Fields and Output Regulation

This paper presents procedural guidelines for the construction of discontinuous state feedback controllers for driftless, kinematic nonholonomic systems, with extensions to a class of dynamic nonholonomic systems with drift. Given an n-dimensional kinematic nonholonomic system subject to κ Pfaffian constraints, system states are partitioned into “leafwise” and “transverse,” based on the structure of the Pfaffian constraint matrix. A reference vector field F is defined as a function of the leafwise states only in a way that it is nonsingular everywhere except for a submanifold containing the origin. The induced decomposition of the configuration space, together with requiring the system vector field to be aligned with F, suggests choices for Lyapunov-like functions. The proposed approach recasts the original nonholonomic control problem as an output regulation problem, which although nontrivial, may admit solutions based on standard tools.

LMI stability conditions for uncertain rational nonlinear systems

SUMMARYThis paper presents LMI conditions for local, regional, and global robust asymptotic stability of rational uncertain nonlinear systems. The uncertainties are modeled as real time varying parameters with magnitude and rate of variation bounded by given polytopes and the system vector field is a rational function of the states and uncertain parameters. Sufficient LMI conditions for asymptotic stability of the origin are given through a rational Lyapunov function of the states and uncertain parameters. The case where the time derivative of the Lyapunov function is negative semidefinite is also considered and connections with the well known LaSalle's invariance conditions are established. In regional stability problems, an algorithm to maximize the estimate of the region of attraction is proposed. The algorithm consists of maximizing the estimate for a given target region of initial states. The size and shape of the target region are recursively modified in the directions where the estimate can be enlarged. The target region can be taken as a polytope (convex set) or union of polytopes (non‐convex set). The estimates of the region of attraction are robust with respect to the uncertain parameters and their rate of change. The case of global and orthant stability problems are also considered. Connections with some results found in sum of squares based methods and other related methods found in the literature are established. The LMIs in this paper are obtained by using the Finsler's Lemma and the notion of annihilators. The LMIs are characterized by affine functions of the state and uncertain parameters, and they are tested at the vertices of a polytopic region. It is also shown that, with some additional conservatism, the use of the vertices can be avoided by modifying the LMIs with the S‐Procedure. Several numerical examples found in the literature are used to compare the results and illustrate the advantages of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.

On Stability of Measure Driven Differential Equations

We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with the inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions present a trade-off between the Lebesgue-integrable and the measure-driven components of the system. In case the system is not asymptotically stable, we derive weaker conditions such that the norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems.

Realization of Parameters Identification in Only Locally Lipschitzian Dynamical Systems with Multiple Types of Time Delays

Most of the existing results for identifying parameters in dynamical systems through adaptive coupling techniques always require a uniform Lipschitz condition on system vector fields. However, it is not this requirement but the only locally Lipschitz condition that is more suitable for many nonlinear systems in real applications. Also time delays sometimes are essential for modeling real systems, so that in addition to the discrete form, time delays in the distributed form could be considered. This paper thus considers the problem of parameters identification in only locally Lipschitzian dynamical systems with multiple types of time delays through adaptive coupling techniques. In particular, the paper proposes some linear independence conditions for realization of parameters identification in both nonautonomous and autonomous systems and draws a comparison between these conditions and the persistent of excitation condition which is used frequently in the literature. Moreover, the paper proposes an index to determine the nonlinear degree of the considered systems, which enables us to distinguish the systems in which the parameters identification can be globally or only locally achieved. The paper also uses some representative and elaborate examples to illustrate the practical usefulness of the obtained theoretical results. During the discussion, the paper takes into full consideration the property of infinite dimensions for nonautonomous or autonomous dynamical systems with time delays.

Convergence of Nonlinear Observers on $\BBR^{n}$ With a Riemannian Metric (Part I)

We study how convergence of an observer whose state lives in a copy of the given system's space can be established using a Riemannian metric. We show that the existence of an observer guaranteeing the property that a Riemannian distance between system and observer solutions is nonincreasing implies that the Lie derivative of the Riemannian metric along the system vector field is conditionally negative. Moreover, we establish that the existence of this metric is related to the observability of the system's linearization along its solutions. Moreover, if the observer has an infinite gain margin then the level sets of the output function are geodesically convex. Conversely, we establish that, if a complete Riemannian metric has a Lie derivative along the system vector field that is conditionally negative and is such that the output function has a monotonicity property, then there exists an observer with an infinite gain margin.

Single Integration Optimization of Linear Time-Varying Switched Systems

This technical note considers the switching time optimization of time-varying linear switched systems subject to quadratic cost-also potentially time-varying. The problem is formulated so that only a single set of differential equations need to be solved prior to optimization. Once these differential equations have been solved, the cost may be minimized over an arbitrary number of modes and mode sequences without requiring additional simulation. The differential equations that need to be simulated are as smooth as the system's vector fields, despite the fact that the optimization itself is nonsmooth. The technique is illustrated in examples from the literature.

Synchronization Between Adaptively Coupled Systems With Discrete and Distributed Time-Delays

This paper investigates complete synchronization of unidirectionally and adaptively coupled systems with discrete and distributed time delays. Instead of the conventional hypothesis of a uniform Lipschitz condition on the system's vector fields, only a local Lipschitz condition is adopted. It is proved that the local complete synchronization can be achieved through a unidirectional and adaptive coupling, and that the global complete synchronization can be realized when the nonlinear degree of the vector fields is smaller than some derived critical value. The results are illustrated in some representative models with time delays. Also considered is complete synchronization with an exponential convergence rate on the adaptively coupled time-delayed systems with vector fields that are one-sided uniformly Lipschitz (systems of this type can admit time-varying discrete and distributed delays). All the results can be further generalized to various types of synchronization between bidirectionally coupled time-delayed systems or among delayed kinetic systems of a complex network.

Finite-time state observation for non-linear uncertain systems via higher-order sliding modes

This article deals with the problem of finite-time state estimation for a class of non-linear systems possibly affected by modelling uncertainties and/or unknown inputs. The proposed method, based on the high-order sliding mode control approach, does not require the system to be transformed to any normal form, which can be difficult to achieve in the presence of model uncertainties. The sufficient conditions for observability are derived in terms of certain geometric restrictions imposed on the system's vector fields. Methods for the approximate and exact reconstruction of the unknown inputs are given and simulation results are provided and commented.

Stabilization of trajectories for systems on Lie groups. Application to the rolling sphere.

This paper addresses the stabilization of admissible reference trajectories generated with constant inputs for driftless systems on Lie groups. The general expression of the linear approximation of the tracking error system is derived from the system's constants of structure and a necessary condition for the controllability of this approximation is specified in terms of the growth of the filtration of the Lie Algebra generated by the system's vector fields. This condition is illustrated with examples of mechanical systems whose control inputs correspond to velocity variables. By contrast with nonholonomic mobile robots whose kinematic equations can be transformed into the chained form, the linearized system associated with the rolling sphere is never controllable. Consequences of this lack of controllability as for stabilization problems are discussed from a general viewpoint and addressed more specifically for the rolling sphere. Finally, a practical stabilizer for this system based on the transverse function approach is proposed.