A contractive approach to separable Lyapunov functions for monotone systems

Coogan, Samuel

doi:10.1016/j.automatica.2019.05.001

Cited by 36 publications

(48 citation statements)

References 44 publications

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“…We note that analogous results to Theorem 1 for 1contractivity for monotone nonlinear compartmental continuous-time systems were proven in Como etal [15],[16]. (e.g., see[15, Lemma 1]), and that similar ideas underlie the differential Finsler-Lyapunov framework of Forni and Sepulchre[17],[18] as well as work on monotone and hierarchical systems[19],[20],[21]. While our approach in this paper is to derive conditions on the differential map δ → Q(p) T δ so as to guarantee 1-contractivity of the map…”

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confidence: 54%

Stability Theory of Stochastic Models in Opinion Dynamics

Askarzadeh

Halder

et al. 2020

IEEE Trans. Automat. Contr.

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We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, that are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the 1 -metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear, respectively-both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

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confidence: 54%

Stability Theory of Stochastic Models in Opinion Dynamics

Askarzadeh

Halder

et al. 2020

IEEE Trans. Automat. Contr.

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“…satisfies 22,35 (J(x, t)) ≤ −2 uniformly in t, where (⋅) is a matrix measure § as shown by Russo et al, 36 Forni and Sepulchre, 22 and Coogan. 35…”

Section: 11mentioning

confidence: 98%

“…Assume that the Jacobian matrices̃q (q v ) and m q m (q mv ) are symmetric and assume that the products Π (q v , t)̃q (q v ) and Π m (q mv , t) m q m (q mv ) commute. Then the closed-loop variational system preserves the structure of the variational pH-like system (35) t). Under above hypotheses the products Π (q v , t)̃q (q v ) and…”

Section: Structural Propertiesmentioning

confidence: 99%

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A family of virtual contraction based controllers for tracking of flexible‐joints port‐Hamiltonian robots: Theory and experiments

Reyes‐Báez

Schaft

Jayawardhana

et al. 2020

Intl J Robust & Nonlinear

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In this work, we present a constructive method to design a family of virtual contraction based controllers that solve the standard trajectory tracking problem of flexible-joint robots in the port-Hamiltonian framework. The proposed design method, called virtual contraction based control, combines the concepts of virtual control systems and contraction analysis. It is shown that under potential energy matching conditions, the closed-loop virtual system is contractive and exponential convergence to a predefined trajectory is guaranteed. Moreover, the closed-loop virtual system exhibits properties such as structure preservation, differential passivity, and the existence of (incrementally) passive maps. The method is later applied to a planar RR robot, and two nonlinear tracking control schemes in the developed controllers family are designed using different contraction analysis approaches. Experiments confirm the theoretical results for each controller. K E Y W O R D Scontraction, flexible-joints robots, port-Hamiltonian systems, tracking control, virtual control systems INTRODUCTIONControl problems in rigid robots have been widely studied in the literature due to they are instrumental in modern manufacturing systems. However, as pointed out in Nicosia and Tomei 1 the elasticity in the joints often cannot be neglected for accurate position tracking. For every joint, that is, actuated by a motor, we basically need two degrees of freedom (dof) instead of one. Such flexible-joint robots (FJRs) are therefore underactuated mechanical systems. In the work of Spong 2 two state feedback control laws based, respectively, on feedback linearization and singular perturbation theory are presented for a simplified FJRs model. Similarly, in de Wit et al 3 a dynamic feedback controller for a more detailed model is presented. In Loría and Ortega 4 a computed-torque controller for FJRs is designed, which does not need jerk measurements. In Ailon and Ortega 5 and Brogliato et al 6 passivity-based control (PBC) schemes are proposed. The first one is an observer-based controller, which requires only motor position measurements. In the latter one, a PBC controller is designed and compared with backstepping and decoupling techniques. For further details on PBC of FJRs, we refer Partial results were presented in the IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control 2018. Int J Robust Nonlinear Control. 2020;30:3269-3295. wileyonlinelibrary.com/journal/rnc

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“…These works are mostly motivated by the fundamental observation that an asymptotically stable positive LTI system always admits linear (also called separable) Lyapunov functions [9], [10], which is essentially related to the Perron-Frobenius theorem on the dominant eigenvalue of a positive matrix [11] and its variant on Metzler matrices [2]. Since almost all practical applications of monotone systems mentioned above such as biological systems contain nonlinearity, stability analysis of monotone nonlinear systems using separable functions has also been studied [12]- [16].…”

Section: Introductionmentioning

confidence: 99%

Contraction Analysis of Monotone Systems via Separable Functions

Kawano

Besselink

Cao

2020

IEEE Trans. Automat. Contr.

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In this paper, we study incremental stability of monotone nonlinear systems through contraction analysis. We provide sufficient conditions for incremental asymptotic stability in terms of the Lie derivatives of differential one-forms or Lie brackets of vector fields. These conditions can be viewed as sum-or max-separable conditions, respectively. For incremental exponential stability, we show that the existence of such separable functions is both necessary and sufficient under standard assumptions for the converse Lyapunov theorem of exponential stability. As a by-product, we also provide necessary and sufficient conditions for exponential stability of positive linear time-varying systems. The results are illustrated through examples.

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A contractive approach to separable Lyapunov functions for monotone systems

Cited by 36 publications

References 44 publications

Stability Theory of Stochastic Models in Opinion Dynamics

Stability Theory of Stochastic Models in Opinion Dynamics

A family of virtual contraction based controllers for tracking of flexible‐joints port‐Hamiltonian robots: Theory and experiments

Contraction Analysis of Monotone Systems via Separable Functions

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