13,628 research outputs found

    Data-driven approximations of dynamical systems operators for control

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    The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.Comment: 37 pages, 4 figure

    Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions

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    We consider polynomial differential equations and make a number of contributions to the questions of (i) complexity of deciding stability, (ii) existence of polynomial Lyapunov functions, and (iii) existence of sum of squares (sos) Lyapunov functions. (i) We show that deciding local or global asymptotic stability of cubic vector fields is strongly NP-hard. Simple variations of our proof are shown to imply strong NP-hardness of several other decision problems: testing local attractivity of an equilibrium point, stability of an equilibrium point in the sense of Lyapunov, invariance of the unit ball, boundedness of trajectories, convergence of all trajectories in a ball to a given equilibrium point, existence of a quadratic Lyapunov function, local collision avoidance, and existence of a stabilizing control law. (ii) We present a simple, explicit example of a globally asymptotically stable quadratic vector field on the plane which does not admit a polynomial Lyapunov function (joint work with M. Krstic). For the subclass of homogeneous vector fields, we conjecture that asymptotic stability implies existence of a polynomial Lyapunov function, but show that the minimum degree of such a Lyapunov function can be arbitrarily large even for vector fields in fixed dimension and degree. For the same class of vector fields, we further establish that there is no monotonicity in the degree of polynomial Lyapunov functions. (iii) We show via an explicit counterexample that if the degree of the polynomial Lyapunov function is fixed, then sos programming may fail to find a valid Lyapunov function even though one exists. On the other hand, if the degree is allowed to increase, we prove that existence of a polynomial Lyapunov function for a planar or a homogeneous vector field implies existence of a polynomial Lyapunov function that is sos and that the negative of its derivative is also sos.Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1112.0741, arXiv:1210.742

    Decomposition of Nonlinear Dynamical Networks via Comparison Systems

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    In analysis and control of large-scale nonlinear dynamical systems, a distributed approach is often an attractive option due to its computational tractability and usually low communication requirements. Success of the distributed control design relies on the separability of the network into weakly interacting subsystems such that minimal information exchange between subsystems is sufficient to achieve satisfactory control performance. While distributed analysis and control design for dynamical network have been well studied, decomposition of nonlinear networks into weakly interacting subsystems has not received as much attention. In this article we propose a vector Lyapunov functions based approach to quantify the energy-flow in a dynamical network via a model of a comparison system. Introducing a notion of power and energy flow in a dynamical network, we use sum-of-squares programming tools to partition polynomial networks into weakly interacting subsystems. Examples are provided to illustrate the proposed method of decomposition.Comment: to be presented at ECC 201

    Vector Contraction Analysis for Nonlinear Dynamical Systems

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    This paper derives new results for the analysis of nonlinear systems by extending contraction theory in the framework of vector distances. A new tool, vector contraction analysis utilizing a notion of the vector-valued norm which evidently induces a vector distance between any pair of trajectories of the system, offers an amenable framework as each component of vector-valued norm function satisfies fewer strict conditions as that of standard contraction analysis. Particularly, every element of vector-valued norm derivative need not be strictly negative definite for convergence of any pair of trajectories of the system. Moreover, vector-valued norm derivative satisfies a component-wise inequality employing some comparison system. In fact, the convergence analysis is performed by comparing the relative distances between any pair of the trajectories of the original nonlinear system and the comparison system. Comparison results are derived by utilizing the concepts of quasi-monotonicity property of the function and vector differential inequalities. Moreover, the results are also derived in the framework of the cone ordering instead of utilizing component-wise inequalities between vectors. In addition, the proposed framework is illustrated by examples

    A Local Characterization of Lyapunov Functions and Robust Stability of Perturbed Systems on Riemannian Manifolds

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    This paper proposes several Converse Lyapunov Theorems for nonlinear dynamical systems defined on smooth connected Riemannian manifolds and characterizes properties of corresponding Lyapunov functions in a normal neighborhood of an equilibrium. We extend the methods of constructing of Lyapunov functions for ordinary differential equations on \mathds{R}^{n} to dynamical systems defined on Riemannian manifolds by employing the differential geometry. By employing the derived properties of Lyapunov functions, we obtained the stability of perturbed dynamical systems on Riemannian manifolds. The results are obtained by employing the notions of normal neighborhoods, the injectivity radius on Riemannian manifolds and existence of bump functions on manifolds

    On the Difficulty of Deciding Asymptotic Stability of Cubic Homogeneous Vector Fields

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    It is well-known that asymptotic stability (AS) of homogeneous polynomial vector fields of degree one (i.e., linear systems) can be decided in polynomial time e.g. by searching for a quadratic Lyapunov function. Since homogeneous vector fields of even degree can never be AS, the next interesting degree to consider is equal to three. In this paper, we prove that deciding AS of homogeneous cubic vector fields is strongly NP-hard and pose the question of determining whether it is even decidable. As a byproduct of the reduction that establishes our NP-hardness result, we obtain a Lyapunov-inspired technique for proving positivity of forms. We also show that for asymptotically stable homogeneous cubic vector fields in as few as two variables, the minimum degree of a polynomial Lyapunov function can be arbitrarily large. Finally, we show that there is no monotonicity in the degree of polynomial Lyapunov functions that prove AS; i.e., a homogeneous cubic vector field with no homogeneous polynomial Lyapunov function of some degree dd can very well have a homogeneous polynomial Lyapunov function of degree less than dd.Comment: 7 page

    Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems

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    The notion of Lyapunov function plays a key role in design and verification of dynamical systems, as well as hybrid and cyber-physical systems. In this paper, to analyze the asymptotic stability of a dynamical system, we generalize standard Lyapunov functions to relaxed Lyapunov functions (RLFs), by considering higher order Lie derivatives of certain functions along the system's vector field. Furthermore, we present a complete method to automatically discovering polynomial RLFs for polynomial dynamical systems (PDSs). Our method is complete in the sense that it is able to discover all polynomial RLFs by enumerating all polynomial templates for any PDS.Comment: 6 pages, one algorith

    Stability Analysis of Monotone Systems via Max-separable Lyapunov Functions

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    We analyze stability properties of monotone nonlinear systems via max-separable Lyapunov functions, motivated by the following observations: first, recent results have shown that asymptotic stability of a monotone nonlinear system implies the existence of a max-separable Lyapunov function on a compact set; second, for monotone linear systems, asymptotic stability implies the stronger properties of D-stability and insensitivity to time-delays. This paper establishes that for monotone nonlinear systems, equivalence holds between asymptotic stability, the existence of a max-separable Lyapunov function, D-stability, and insensitivity to bounded and unbounded time-varying delays. In particular, a new and general notion of D-stability for monotone nonlinear systems is discussed and a set of necessary and sufficient conditions for delay-independent stability are derived. Examples show how the results extend the state-of-the-art

    Global Stability Results for Systems Under Sampled-Data Control

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    In this work sufficient conditions expressed by means of single and vector Lyapunov functions of Uniform Input-to-Output Stability (UIOS) and Uniform Input-to-State Stability (UISS) are given for finite-dimensional systems under feedback control with zero order hold.Comment: Submitted for possible publication to the International Journal of Robust and Nonlinear Contro

    Data-driven feedback stabilization of nonlinear systems: Koopman-based model predictive control

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    In this work, a predictive control framework is presented for feedback stabilization of nonlinear systems. To achieve this, we integrate Koopman operator theory with Lyapunov-based model predictive control (LMPC). The main idea is to transform nonlinear dynamics from state-space to function space using Koopman eigenfunctions - for control affine systems this results in a bilinear model in the (lifted) function space. Then, a predictive controller is formulated in Koopman eigenfunction coordinates which uses an auxiliary Control Lyapunov Function (CLF) based bounded controller as a constraint to ensure stability of the Koopman system in the function space. Provided there exists a continuously differentiable inverse mapping between the original state-space and (lifted) function space, we show that the designed controller is capable of translating the feedback stabilizability of the Koopman bilinear system to the original nonlinear system. Remarkably, the feedback control design proposed in this work remains completely data-driven and does not require any explicit knowledge of the original system. Furthermore, due to the bilinear structure of the Koopman model, seeking a CLF is no longer a bottleneck for LMPC. Benchmark numerical examples demonstrate the utility of the proposed feedback control design.Comment: 11 pages, corrected typos, removed commented out line
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