相关论文
2003
Stability and dissipativity theory for discrete-time non-negative and compartmental dynamical systems

Non-negative and compartmental dynamical systems are derived from mass and energy balance considerations that involve dynamic states whose values are non-negative. These models are widespread in engineering, biomedicine and ecology. In this paper we develop several results on stability, dissipativity and stability of feedback interconnections of discrete-time linear and non-linear non-negative dynamical systems. Specifically, using linear Lyapunov functions we first develop necessary and sufficient conditions for Lyapunov stability and asymptotic stability for non-negative systems. In addition, using linear and non-linear storage functions with linear supply rates we develop new notions of dissipativity theory for non-negative dynamical systems. Finally, these results are used to develop general stability criteria for feedback interconnections of non-negative dynamical systems.

2001 - Journal of Number Theory
On the Number of Cycles of p-adic Dynamical Systems

We found the asymptotics, p→∞, for the number of cycles for iteration of monomial functions in the fields of p-adic numbers. This asymptotics is closely connected with classical results on the distribution of prime numbers.

1998
Relative Entropy and Identification of Gibbs Measures in Dynamical Systems

In this work we explore the idea of using the relative entropy of ergodic measures for the identification of Gibbs measures in dynamical systems. The question we face is how to estimate the thermodynamic potential (together with a grammar) from a sample produced by the corresponding Gibbs state.

1999 - Oper. Res. Lett.
Projected dynamical systems in a complementarity formalism

Projected dynamical systems have been introduced by Dupuis and Nagurney as dynamic extensions of variational inequalities. In the systems and control literature, complementarity systems have been studied as input/output dynamical systems whose inputs and outputs are connected through complementarity conditions. We show here that, under mild conditions, projected dynamical systems can be written as complementarity systems.

2005 - Journal of Differential Equations
Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations

Abstract The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state. To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold.

1995
The fundamental theorem of dynamical systems

We propose the title of The Fundamental Theorem of Dynamical Systems for a theorem of Charles Conley concerning the decomposition of spaces on which dynamical systems are defined. First, we briefly set the context and state the theorem. After some definitions and preliminary results, based both on Conley's work and modifications to it, we present a sketch of a proof of the result in the setting of the iteration of continuous functions on compact metric spaces. Finally, we claim that this theorem should be called The Fundamental Theorem of Dynamical Systems.

2019 - ArXiv
Learning Linear Dynamical Systems with Semi-Parametric Least Squares

We analyze a simple prefiltered variation of the least squares estimator for the problem of estimation with biased, semi-parametric noise, an error model studied more broadly in causal statistics and active learning. We prove an oracle inequality which demonstrates that this procedure provably mitigates the variance introduced by long-term dependencies. We then demonstrate that prefiltered least squares yields, to our knowledge, the first algorithm that provably estimates the parameters of partially-observed linear systems that attains rates which do not not incur a worst-case dependence on the rate at which these dependencies decay. The algorithm is provably consistent even for systems which satisfy the weaker marginal stability condition obeyed by many classical models based on Newtonian mechanics. In this context, our semi-parametric framework yields guarantees for both stochastic and worst-case noise.

2017
DCMIP2016: A Review of Non-hydrostatic Dynamical Core Design and Intercomparison of Participating Models

Abstract. Atmospheric dynamical cores are a fundamental component of global atmospheric modeling systems and are responsible for capturing the dynamical behavior of the Earth's atmosphere via numerical integration of the Navier–Stokes equations. These systems have existed in one form or another for over half of a century, with the earliest discretizations having now evolved into a complex ecosystem of algorithms and computational strategies. In essence, no two dynamical cores are alike, and their individual successes suggest that no perfect model exists. To better understand modern dynamical cores, this paper aims to provide a comprehensive review of 11 non-hydrostatic dynamical cores, drawn from modeling centers and groups that participated in the 2016 Dynamical Core Model Intercomparison Project (DCMIP) workshop and summer school. This review includes a choice of model grid, variable placement, vertical coordinate, prognostic equations, temporal discretization, and the diffusion, stabilization, filters, and fixers employed by each system.

2010 - Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Stability analysis and controller synthesis for hybrid dynamical systems

Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially the case in many technological systems in which logic decision-making and embedded control actions are combined with continuous physical processes. Also for many mechanical, biological, electrical and economical systems the use of hybrid models is essential to adequately describe their behaviour. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. This paper presents an overview from the perspective of the control community on modelling, analysis and control design for hybrid dynamical systems and surveys the major research lines in this appealing and lively research area.

2003 - SIAM Rev.
Approved for public release; further dissemination unlimited Error Estimation for Reduced Order Models of Dynamical Systems ∗

The use of reduced-order models to describe a dynamical system is pervasive in science and engineering. Often these models are used without an estimate of their error or range of validity. In this paper we consider dynamical systems and reduced models built using proper orthogonal decomposition. We show how to compute estimates and bounds for these errors by a combination of small sample statistical condition estimation and error estimation using the adjoint method. Most important, the proposed approach allows the assessment of regions of validity for reduced models, i.e., ranges of perturbations in the original system over which the reduced model is still appropriate. Numerical examples validate our approach: the error norm estimates approximate well the forward error, while the derived bounds are within an order of magnitude.

1991
Time series prediction by adaptive networks: a dynamical systems perspective

The links between adaptive layered networks, functional interpolation and dynamical systems are considered and applied to the nonlinear predictive analysis of time series. The ability of networks to produce interpolation surfaces to generators of data (i.e. differential equations, iterative maps) is used to analyse a variety of time series. If network may be trained to approximate a static) generator of data, the network may be iterated on its own output to produce a time series with the same characteristics as the training waveform. However, since iterated networks are one example of nonlinear dynamical systems, this raises problems of sensitive dependence upon initial conditions leading ultimately to deterministic chaos. An introduction to the relevant concepts is presented and illustrations are provided from simple chaotic maps, nonlinear differential equations, and stock-market prediction. The latter example is included to illustrate the problems which often occur in real-world data due to noise, undersampling, high dimensionality and insufficient data.

2014 - SIAM J. Sci. Comput.
A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems

In this work an efficient approach for a posteriori error estimation for POD-DEIM reduced nonlinear dynamical systems is introduced. The considered nonlinear systems may also include time- and parameter-affine linear terms as well as parametrically dependent inputs and outputs. The reduction process involves a Galerkin projection of the full system and approximation of the system's nonlinearity by the DEIM method [S. Chaturantabut and D. C. Sorensen, SIAM J. Sci. Comput., 32 (2010), pp. 2737--2764]. The proposed a posteriori error estimator can be efficiently decomposed in an offline/online fashion and is obtained by a one-dimensional auxiliary ODE during reduced simulations. Key elements for efficient online computation are partial similarity transformations and matrix-DEIM approximations of the nonlinearity Jacobians. The theoretical results are illustrated by application to an unsteady Burgers equation and a cell apoptosis model.

2007 - NIPS
A Constraint Generation Approach to Learning Stable Linear Dynamical Systems

Stability is a desirable characteristic for linear dynamical systems, but it is often ignored by algorithms that learn these systems from data. We propose a novel method for learning stable linear dynamical systems: we formulate an approximation of the problem as a convex program, start with a solution to a relaxed version of the program, and incrementally add constraints to improve stability. Rather than continuing to generate constraints until we reach a feasible solution, we test stability at each step; because the convex program is only an approximation of the desired problem, this early stopping rule can yield a higher-quality solution. We apply our algorithm to the task of learning dynamic textures from image sequences as well as to modeling biosurveillance drug-sales data. The constraint generation approach leads to noticeable improvement in the quality of simulated sequences. We compare our method to those of Lacy and Bernstein [1, 2], with positive results in terms of accuracy, quality of simulated sequences, and efficiency.

2002
CHAOTIC ATTRACTORS OF AN INFINITE-DIMENSIONAL DYNAMICAL SYSTEM

We study the chaotic attractors of a delay differential equation. The dimension of several attractors computed directly from the definition agrees to experimental resolution ~vith the dimension computed from the spectrum of Lyapunov exponents according to a conjecture of Kaplan and Yorke. Assuming this conjecture to be valid, as the delay parameter is varied, from computations of the spectrum of Lyapunov exponents, we observe a roughly linear increase from two to twenty in the dimension, while lhe metric entropy remains roughly constant. These results are compared to a linear analysis, and the asymptotic behavior of the Lyapunov exponents is derived.

2003 - DMCS
Predecessor and Permutation Existence Problems for Sequential Dynamical Systems

A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was introduced in [BR99] as a formal model for analyzing simulation systems. Here, we address the complexity of two basic problems and their generalizations for SDSs.Given an SDS $\mathcal{S}$ and a configuration $\mathcal{C}$, the PREDECESSOR EXISTENCE (or PRE) problem is to determine whether there is a configuration $\mathcal{C}'$ such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$. Our results provide separations between efficiently solvable and computationally intractable instances of the PRE problem. For example, we show that the PRE problem can be solved efficiently for SDSs with Boolean state values when the node functions are symmetric and the underlying graph is of bounded tree width. In contrast, we show that allowing just one non-symmetric node function renders the problem $\mathbf{NP}$-complete even when the underlying graph is a star (which has a tree width of 1). Our results extend some of the earlier results by Sutner [Su95] and Green [Gr87] on the complexity of the PREDECESSOR EXISTENCE problem for 1-dimensional cellular automata.Given two configurations $\mathcal{C}$ and $\mathcal{C}'$ of a partial SDS $\mathcal{S}$, the PERMUTATION EXISTENCE (or PME) problem is to determine whether there is a permutation of nodes such that $\mathcal{S}$ has a transition from $\mathcal{C}'$ to $\mathcal{C}$ in one step. We show that the PME problem is $\mathbf{NP}$-complete even when the function associated with each node is a simple-threshold function. We also show that the problem can be solved efficiently for SDSs whose underlying graphs are of bounded degree and bounded tree width. We consider a generalized version (GEN-PME) of the PME problem and show that the problem is $\mathbf{NP}$-complete for SDSs where each node function is NOR and the underlying graph has a maximum node degree of 3. When each node computes the OR function or when each node computes the AND function, we show that the GEN-PME problem is solvable in polynomial time.

1996
A Dynamical System Approach to Stochastic Approximations

It is known that some problems of almost sure convergence for stochastic approximation processes can be analyzed via an ordinary differential equation (ODE) obtained by suitable averaging. The goal of this paper is to show that the asymptotic behavior of such a process can be related to the asymptotic behavior of the ODE without any particular assumption concerning the dynamics of this ODE. The main results are as follows: a) The limit sets of trajectory solutions to the stochastic approximation recursion are, under classical assumptions, almost surely nonempty compact connected sets invariant under the flow of the ODE and contained in its set of chain-recurrence. b) If the gain parameter goes to zero at a suitable rate depending on the expansion rate of the ODE, any trajectory solution to the recursion is almost surely asymptotic to a forward trajectory solution to the ODE.

2011 - IEEE Transactions on Automatic Control
Reaching an Optimal Consensus: Dynamical Systems That Compute Intersections of Convex Sets

In this paper, multi-agent systems minimizing a sum of objective functions, where each component is only known to a particular node, is considered for continuous-time dynamics with time-varying interconnection topologies. Assuming that each node can observe a convex solution set of its optimization component, and the intersection of all such sets is nonempty, the considered optimization problem is converted to an intersection computation problem. By a simple distributed control rule, the considered multi-agent system with continuous-time dynamics achieves not only a consensus, but also an optimal agreement within the optimal solution set of the overall optimization objective. Directed and bidirectional communications are studied, respectively, and connectivity conditions are given to ensure a global optimal consensus. In this way, the corresponding intersection computation problem is solved by the proposed decentralized continuous-time algorithm. We establish several important properties of the distance functions with respect to the global optimal solution set and a class of invariant sets with the help of convex and non-smooth analysis.

2011 - IEEE Transactions on Robotics
Learning Stable Nonlinear Dynamical Systems With Gaussian Mixture Models

This paper presents a method to learn discrete robot motions from a set of demonstrations. We model a motion as a nonlinear autonomous (i.e., time-invariant) dynamical system (DS) and define sufficient conditions to ensure global asymptotic stability at the target. We propose a learning method, which is called Stable Estimator of Dynamical Systems (SEDS), to learn the parameters of the DS to ensure that all motions closely follow the demonstrations while ultimately reaching and stopping at the target. Time-invariance and global asymptotic stability at the target ensures that the system can respond immediately and appropriately to perturbations that are encountered during the motion. The method is evaluated through a set of robot experiments and on a library of human handwriting motions.

2005
Discrete Dynamical Systems

This manuscript analyzes the fundamental factors that govern the qualitative behavior of discrete dynamical systems. It introduces methods of analysis for stability analysis of discrete dynamical systems. The analysis focuses initially on the derivation of basic propositions about the factors that determine the local and global stability of discrete dynamical systems in the elementary context of a one dimensional, first-order, autonomous, systems. These propositions are subsequently generalized to account for stability analysis in a multi-dimensional, higher-order, non-autonomous, nonlinear, dynamical systems.

2012
Hybrid Dynamical Systems: Modeling, Stability, and Robustness

Hybrid dynamical systems exhibit continuous and instantaneous changes, having features of continuous-time and discrete-time dynamical systems. Filled with a wealth of examples to illustrate concepts, this book presents a complete theory of robust asymptotic stability for hybrid dynamical systems that is applicable to the design of hybrid control algorithms--algorithms that feature logic, timers, or combinations of digital and analog components. With the tools of modern mathematical analysis, Hybrid Dynamical Systems unifies and generalizes earlier developments in continuous-time and discrete-time nonlinear systems. It presents hybrid system versions of the necessary and sufficient Lyapunov conditions for asymptotic stability, invariance principles, and approximation techniques, and examines the robustness of asymptotic stability, motivated by the goal of designing robust hybrid control algorithms. This self-contained and classroom-tested book requires standard background in mathematical analysis and differential equations or nonlinear systems. It will interest graduate students in engineering as well as students and researchers in control, computer science, and mathematics.

The Complete Proof Theory of Hybrid Systems

André Platzer
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André Platzer

Abstract:Hybrid systems are a fusion of continuous dynamical systems and discrete dynamical systems. They freely combine dynamical features from both worlds. For that reason, it has often been claimed that hybrid systems are more challenging than continuous dynamical systems and than discrete systems. We now show that, proof-theoretically, this is not the case. We present a complete proof-theoretical alignment that interreduces the discrete dynamics and the continuous dynamics of hybrid systems. We give a sound and complete axiomatization of hybrid systems relative to continuous dynamical systems and a sound and complete axiomatization of hybrid systems relative to discrete dynamical systems. Thanks to our axiomatization, proving properties of hybrid systems is exactly the same as proving properties of continuous dynamical systems and again, exactly the same as proving properties of discrete dynamical systems. This fundamental cornerstone sheds light on the nature of hybridness and enables flexible and provably perfect combinations of discrete reasoning with continuous reasoning that lift to all aspects of hybrid systems and their fragments.

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引用
Formal Methods for Industrial Critical Systems: 25th International Conference, FMICS 2020, Vienna, Austria, September 2–3, 2020, Proceedings
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2020
Automated Reasoning
Lecture Notes in Computer Science
2016
Logical Foundations of Cyber-Physical Systems
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2018
Dynamic Logics of Dynamical Systems
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2012
Logics of Dynamical Systems
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2012
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Differential Equation Axiomatization: The Impressive Power of Differential Ghosts
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Tactical contract composition for hybrid system component verification
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A Survey on Analog Models of Computation
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2012
A Formally Verified Plasma Vertical Position Control Algorithm
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A Method for Invariant Generation for Polynomial Continuous Systems
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