We study convergence of nonlinear systems in the presence of an `almost Lyapunov' function which, unlike the classical Lyapunov function, is allowed to be nondecreasing---and even increasing---on a nontrivial subset of the phase space. Under the assumption that the vector field is free of singular points (away from the origin) and that the subset where the Lyapunov function does not decrease is sufficiently small, we prove that solutions approach a small neighborhood of the origin. A nontrivial example where this theorem applies is constructed.
In this short report, a new Lyapunov function for the Moog voltage-controlled filter is demonstrated, under zero-input conditions, and under nonlinear autonomous conditions (i.e. when parameters are not time-varying). The new definition allows for a proof of stability over the entire allowable range of parameters (cutoff frequency and resonance), and can be used as a starting point for Hamiltonian-based numerical simulation methods.
Motivated by models that arise in controlled ship maneuvering, we analyze Hopf bifurcations in systems with piecewise smooth nonlinear part. In particular, we derive explicit formulas for the generalization of the first Lyapunov coefficient to this setting. This generically determines the direction of branching (super- versus sub-criticality), but in general this differs from any fixed smoothing of the vector field. We focus on nonsmooth nonlinearities of the form $u_i|u_j|$, but our results are formulated in broader generality for systems in any dimension with piecewise smooth nonlinear part.…
In this contribution, we study an homogeneous class of nonlinear time delay systems with time-varying perturbations. Using the Lyapunov-Krasovskii approach, we introduce a functional that leads to perturbation conditions matching those obtained previously in the Razumikhin framework. The functionals are applied to the estimation of the domain of attraction and of the system solutions. An illustrative example is given.
For a pendulum suspended below a vibrating block with white noise forcing, the solution in which the pendulum remains vertical is called the single mode solution. When this solution becomes unstable there is energy transfer from the block to the pendulum, helping to absorb the vibrations of the block. We study the Lyapunov exponent governing the almost-sure stability of the process linearized along the single mode solution. The linearized equation is excited by a combination of white and colored noise processes, which makes the evaluation of the Lyapunov exponent non trivial. We obtain an expl…
Learning for control of dynamical systems with formal guarantees remains a challenging task. This paper proposes a learning framework to simultaneously stabilize an unknown nonlinear system with a neural controller and learn a neural Lyapunov function to certify a region of attraction (ROA) for the closed-loop system. The algorithmic structure consists of two neural networks and a satisfiability modulo theories (SMT) solver. The first neural network is responsible for learning the unknown dynamics. The second neural network aims to identify a valid Lyapunov function and a provably stabilizing …
Linear Lyapunov Functions for Nonlinear Compartmental Systems
Chalmers University of Technology（查尔姆斯理工大学）；Mines Paris – PSL University（巴黎矿业高等学院-巴黎文理研究大学）；University of Gothenburg（哥德堡大学）；Pázmány Péter Catholic University（帕兹马尼·彼得天主教大学）；Université Paris-Saclay（巴黎萨克雷大学）
This paper considers the problem of approximating the "maximal" region of attraction (the set that contains all asymptotically stable sets) of any given set of locally exponentially stable nonlinear Ordinary Differential Equations (ODEs) with a sufficiently smooth vector field. Given a locally exponential stable ODE with a differentiable vector field, we show that there exists a globally Lipschitz continuous converse Lyapunov function whose 1-sublevel set is equal to the maximal region of attraction of the ODE. We then propose a sequence of d-degree Sum-of-Squares (SOS) programming problems th…
Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions
ACCESS Linnaeus Centre（ACCESS林奈乌斯中心）；Department of Automatic Control, School of Electrical Engineering, KTH Royal Institute of Technology（瑞典皇家理工学院电气工程学院自动控制系）
Kernel Methods for the Construction of Certified Lyapunov Functions via Approximate Koopman Eigenfunctions
University of Sussex（萨塞克斯大学）；University of Iceland（冰岛大学）；Caltech（加州理工学院）；Alan Turing Institute（艾伦·图灵研究所）；Ca’ Foscari University of Venice（威尼斯卡福斯卡里大学）；Clemson University（克莱姆森大学）
A method for constructing homogeneous Lyapunov functions of degree 1 from polynomial invariant sets is presented for linear time varying systems, homogeneous dynamic systems and the class of nonlinear systems that can be represented as such. The method allows the development of Lyapunov functions that are not necessarily symmetric about the origin, unlike the polynomial, homogeneous Lyapunov functions discussed in the prior literature. The work is illustrated by very simple examples.
In this contribution, the estimates for the response of time delay systems with nonlinear homogeneous right-hand side of degree strictly greater than one are constructed. The existing results obtained via the Lyapunov--Razumikhin approach are reminded. Their proofs, revisited in the appendix, lead to explicit expressions of the involved constants. Based on a recently introduced Lyapunov--Krasovskii functional and known estimates of the domain of attraction, we present new estimates of the system response. We compare both approaches and discuss the illustrative examples.
