In this work, we establish different control design approaches for discrete-time systems, which build upon the notion of finite-step control Lyapunov functions (fs-CLFs). The design approaches are formulated as optimization problems and solved in a model predictive control (MPC) fashion. In particular, we establish contractive multi-step MPC with and without reoptimization and compare it to classic MPC. The idea behind these approaches is to use the fs-CLF as running cost. These new design approaches are particularly relevant in situations where information exchange between plant and controller cannot be ensured at all time instants. An example shows the different behavior of the proposed controller design approaches.
This paper is concerned with relationships of Lyapunov exponents with sensitivity and stability for non-autonomous discrete systems. Some new concepts are introduced for non-autonomous discrete systems, including Lyapunov exponents, strong sensitivity at a point and in a set, Lyapunov stability, and exponential asymptotical stability. It is shown that the positive Lyapunov exponent at a point implies strong sensitivity for a class of non-autonomous discrete systems. Furthermore, the uniformly positive Lyapunov exponents in a totally invariant set imply strong sensitivity in this set under cert…
We revisit the canonical continuous-time and discrete-time matrix algebraic and matrix differential equations that play a central role in Lyapunov based stability arguments. The goal is to generalize and extend these types of equations and subsequent analysis to dynamical systems on domains other than $\R$ or $\Z$, e.g. nonuniform discrete domains or domains consisting of a mixture of discrete and continuous components. We compare and contrast the standard theory with the theory in this general case.
We address the classic problem of stability and asymptotic stability in the sense of Lyapunov of the equilibrium point of autonomic differential equations using discrete approach. This new approach includes a consideration of a family of hypersurfaces instead of the Lyapunov functions, and conditions on the right part of the differential equation instead of conditions on a Lyapunov function along trajectories of the equation.
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.
Lyapunov functions are used to prove stability of equilibria, or to indicate a gradient-like structure of a dynamical system. Zelenyak (1968) and Matano (1988) constructed a Lyapunov function for quasilinear parabolic equations. We modify Matano's method to construct a Lyapunov function for fully nonlinear parabolic equations under Dirichlet and mixed nonlinear boundary conditions of Robin type.
We prove that impulsive systems, which possess an ISS Lyapunov function, are ISS for time sequences satisfying the fixed dwell-time condition. If an ISS Lyapunov function is the exponential one, we provide a stronger result, which guarantees uniform ISS of the whole system over sequences satisfying the generalized average dwell-time condition. Then we prove two small-gain theorems that provide a construction of an ISS Lyapunov function for an interconnection of impulsive systems, if the ISS-Lyapunov functions for subsystems are known. The construction of local ISS Lyapunov functions via linear…
This paper focuses on the fractional difference of Lyapunov functions related to Riemann-Liouville, Caputo and Grunwald-Letnikov definitions. A new way of building Lyapunov functions is introduced and then five inequalities are derived for each definition. With the help of the developed inequalities, the sufficient conditions can be obtained to guarantee the asymptotic stability of the nabla discrete fractional order nonlinear systems. Finally, three illustrative examples are presented to demonstrate the validity and feasibility of the proposed theoretical results.
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.
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…
Title: Global stabilization of control nonlinear system "inverted pendulum on a cart" using method of two Lyapunov functions Authors: B. L. Mazov (Nizhny Novgorod Technical University) Comments: 10 pages Subj-class: Dynamical Systems MSC-class: 34D23 (Primary); 93H10 (Secondary) In this work a mathematical apparatus for method of several global Lyapunov functions is applied to study the stability properties of nonlinear model of concrete singularly perturbed mechanical system "inverted pendulum on a cart" with discontinuous relay-type control. In this analysis, the concept of solution in sense…
This paper is concerned with stability analysis of nonlinear time-varying systems by using Lyapunov function based approach. The classical Lyapunov stability theorems are generalized in the sense that the time-derivative of the Lyapunov functions are allowed to be indefinite. The stability analysis is accomplished with the help of the scalar stable functions introduced in our previous study. Both asymptotic stability and input-to-state stability are considered. Particularly, for asymptotic stability, several concepts such as uniform and non-uniform asymptotic stability, and uniform and non-uni…
We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent results of X. Dai on the asymptotic growth rate of such systems, we introduce the concept of multinorm as an algebraic tool for stability analysis. We conjugate this tool with two families of multiple quadratic Lyapunov functions, parameterized by an integer T >= 1, and obtain converse Lyapunov Theorems for each. Lyapunov functions of the first family associ…
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.…
This paper investigates the problem of synchronization for nonlinear systems. Following a Lyapunov approach, we firstly study global synchronization of nonlinear systems in canonical control form with both distributed proportional-derivative and proportional-integral-derivative control actions of any order. To do so, we develop a constructive methodology and generate in an iterative way inequality constraints on the coupling matrices which guarantee the solvability of the problem or, in a dual form, provide the nonlinear weights on the coupling links between the agents such that the network sy…
Finite-time Lyapunov exponents and vectors are used to define and diagnose boundary-layer type, two-timescale behavior in the tangent linear dynamics and to determine the associated manifold structure in the flow of a finite-dimensional nonlinear autonomous dynamical system. Two-timescale behavior is characterized by a slow-fast splitting of the tangent bundle for a state space region. The slow-fast splitting is defined using finite-time Lyapunov exponents and vectors, guided by the asymptotic theory of partially hyperbolic sets, with important modifications for the finite-time case; for examp…
An approach for computing Lyapunov functions for nonlinear continuous-time differential equations is developed via a new, Massera-type construction. This construction is enabled by imposing a finite-time criterion on the integrated function. By means of this approach, we relax the assumptions of exponential stability on the system dynamics, while still allowing integration over a finite time interval. The resulting Lyapunov function can be computed based on any K infinity function of the norm of the solution of the system. In addition, we show how the developed converse theorem can be used to …
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.
We study asymptotic stability of continuous-time systems with mode-dependent guaranteed dwell time. These systems are reformulated as special cases of a general class of mixed (discrete-continuous) linear switching systems on graphs, in which some modes correspond to discrete actions and some others correspond to continuous-time evolutions. Each discrete action has its own positive weight which accounts for its time-duration. We develop a theory of stability for the mixed systems; in particular, we prove the existence of an invariant Lyapunov norm for mixed systems on graphs and study its stru…
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 …
A Path-Complete Lyapunov Function is an algebraic criterion composed of a finite number of functions, called its pieces, and a directed, labeled graph defining Lyapunov inequalities between these pieces. It provides a stability certificate for discrete-time switching systems under arbitrary switching. In this paper, we prove that the satisfiability of such a criterion implies the existence of a Common Lyapunov Function, expressed as the composition of minima and maxima of the pieces of the Path-Complete Lyapunov function. The converse, however, is not true even for discrete-time linear systems…
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）
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）
The paper studies numerical methods that preserve a Lyapunov function of a dynamical system, i.e. numerical approximations whose energy decreases, just like in the original differential equation. With this aim, a discrete gradient method is implemented for numerical integration of a system of ordinary differential equations. In principle, this procedure yields first order methods, but the analysis paves the way to the design of higher-order methods. As a case in point, the proposed method is applied to the Duffing equation without external forcing, considering that in this case, preserving the…
Lyapunov functions provide a tool to analyze the stability of nonlinear systems without extensively solving the dynamics. Recent advances in sum-of-squares methods have enabled the algorithmic computation of Lyapunov functions for polynomial systems. However, for general large-scale nonlinear networks it is yet very difficult, and often impossible, both computationally and analytically, to find Lyapunov functions. In such cases, a system decomposition coupled to a vector Lyapunov functions approach provides a feasible alternative by analyzing the stability of the nonlinear network through a re…
We prove that input-to-state stability (ISS) of nonlinear systems over Banach spaces is equivalent to existence of a coercive Lipschitz continuous ISS Lyapunov function for this system. For linear infinite-dimensional systems, we show that ISS is equivalent to existence of a non-coercive ISS Lyapunov function and provide two simpler constructions of coercive and non-coercive ISS Lyapunov functions for input-to-state stable linear systems.
In this paper we introduce the fractional-order variant of a Gompertz-like discrete system. The chaotic behavior is suppressed with an impulsive control algorithm. The numerical integration and the Lyapunov exponent are obtained by means of the discrete fractional calculus. To verify numerically the obtained results, beside the Lyapunov exponent, the tools offered by the 0-1 test are used.
Taming discrete rough paths via strong Lyapunov functions
University of Klagenfurt（克拉根福大学）；Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany（德国莱比锡马克斯·普朗克科学计量学研究所）；Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam（越南科学技术研究院数学研究所）
We characterize synchronization phenomenon in discrete-time, discrete-state random dynamical systems, with random and probabilistic Boolean networks as particular examples. In terms of multiplicative ergodic properties of the induced linear cocycle, we show such a random dynamical system with finite state synchronizes if and only if the Lyapunov exponent $0$ has simple multiplicity. For the case of countable state space, characterization of synchronization is provided in terms of the spectral subspace corresponding to the Lyapunov exponent $-\infty$. In addition, for both cases of finite and c…
We first study the discrete Schrödinger equations with analytic potentials given by a class of transformations. It is shown that if the coupling number is large, then its logarithm equals approximately to the Lyapunov exponents. When the transformation becomes the skew-shift, we prove that the Lyapunov exponent is week Hölder continuous, and the spectrum satisfies Anderson Localization and contains large intervals. Moreover, all of these conclusions are non-perturbative.
For a dynamical system, it is known that the existence of a Lyapunov-type density function, called Lyapunov density or Rantzer's density function, implies convergence of Lebesgue almost all solutions to an equilibrium. Using the duality between Frobenius-Perron and Koopmann operators, we generalize this result from equilibrium to invariant sets, both for continuous- and discrete-time. Furthermore, some redundant assumptions that exist in the literature of almost everywhere stability for discrete-time, such as the local stability of the attractor and the compactness of the state space, has been…
We discuss the existence of solutions with oblique asymptotes to a class of second order nonlinear ordinary differential equations by means of Lyapunov functions. The approach is new in this field and allows for simpler proofs of general results regarding Emden-Fowler like equations.
Empirically defining some constant probabilistic orbits of f(x) and g(x) iterated high-order functions, the stability of these functions in possible entangled interaction dynamics of the environment through its orbit's connectivity (open sets) provides the formation of an exponential dynamic fixed point b = S(n+1) as a metric space (topological property) between both iterated functions for short time lengths. However, the presence of a dynamic fixed point f(g(x)) (x+1) can identify a convergence at iterations for larger time lengths of b (asymptotic stability in Lyapunov sense). Qualitative (Q…
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.
