Revisiting totally positive differential systems: A tutorial and new results

Margaliot, Michael; Sontag, Eduardo D.

doi:10.1016/j.automatica.2018.11.016

Cited by 55 publications

(48 citation statements)

References 39 publications

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“…The final section concludes. In passing we note that our results are part of a growing body of research on the applications of sign-regularity (and, in particular, total positivity) to dynamical systems [1], [4], [17], [20], [28], [30].…”

Section: Introductionmentioning

confidence: 99%

Discrete-Time $k$-Positive Linear Systems

Alseidi

Margaliot

Garloff

2021

IEEE Trans. Automat. Contr.

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Positive systems play an important role in systems and control theory and have found many applications in multi-agent systems, neural networks, systems biology, and more. Positive systems map the nonnegative orthant to itself (and also the nonpositive orthant to itself). In other words, they map the set of vectors with zero sign variations to itself. In this note, discrete-time linear systems that map the set of vectors with up to k − 1 sign variations to itself are introduced. For the special case k = 1 these reduce to discrete-time positive linear systems. Properties of these systems are analyzed using tools from the theory of sign-regular matrices. In particular, it is shown that almost every solution of such systems converges to the set of vectors with up to k − 1 sign variations. Also, the operation of such systems on k-dimensional parallelotopes are studied.

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Section: Introductionmentioning

confidence: 99%

Discrete-Time $k$-Positive Linear Systems

Alseidi

Margaliot

Garloff

2021

IEEE Trans. Automat. Contr.

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“…where B [2] denotes the 2nd additive compound of the matrix B (see, e.g. [24], [16]). Recall (see e.g.…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

“…This was further developed by Smith [26] who considered time-varying and periodic nonlinear systems of a similar form. The recent paper [16] pointed out the connection between this work and the notion of a totally positive differential system (TPDS) introduced by Schwarz.…”

Section: Introductionmentioning

confidence: 95%

Compact attractors of an antithetic integral feedback system have a simple structure

Margaliot

Sontag²

2019

Preprint

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Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a twodimensional linear system (making the closed-loop system a nonlinear four-dimensional system) has been analyzed in much detail. This system has a unique equilibrium but, depending on parameters, it may exhibit periodic orbits. An interesting open question is whether other dynamical behaviors, such as chaotic attractors, might be possible for some parameter choices. This note shows that, for any parameter choices, every bounded trajectory satisfies a Poincaré-Bendixson property. The analysis is based on the recently introduced notion of k-cooperative dynamical systems. It is shown that the model is a strongly 2-cooperative system, implying that the dynamics in the omegalimit set of any precompact solution is conjugate to the dynamics in a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors. Index TermsPoincaré-Bendixson property, k-cooperative dynamical systems, sign-regular matrices, synthetic biology, antithetic feedback, synthetic biology, nonlinear control theory.MM is with the School of EE-Systems, Tel Aviv University, Israel 69978.

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“…Schwarz used the VDP to show that the number of sign variations in x(t) is a (integer-valued) Lyapunov function for the TPDS (2). It was recently shown [16] that TPDSs have important applications in the stability analysis of continuous-time nonlinear cooperative dynamical systems with a tridiagonal Jacobian.…”

Section: Introductionmentioning

confidence: 99%

Entrainment to subharmonic trajectories in oscillatory discrete-time systems

2020

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A matrix A is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix A is called oscillatory if it is TN and some power of A is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time k is oscillatory. We analyze the properties of n-dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-varying and T -periodic then any trajectory either leaves any compact set or converges to an (n − 1)T -periodic trajectory, that is, a subharmonic trajectory. These results hold for any dimension n. The analysis of such systems requires establishing that a line integral of the Jacobian of the nonlinear system is an oscillatory matrix. This is non-trivial, as the sum of two oscillatory matrices is not necessarily oscillatory, and this carries over to integrals. We derive several new sufficient conditions guaranteeing that the line integral of a matrix is oscillatory, and demonstrate how this yields interesting classes of discrete-time nonlinear systems that admit a well-ordered behavior.

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Revisiting totally positive differential systems: A tutorial and new results

Cited by 55 publications

References 39 publications

Discrete-Time $k$-Positive Linear Systems

Discrete-Time $k$-Positive Linear Systems

Compact attractors of an antithetic integral feedback system have a simple structure

Entrainment to subharmonic trajectories in oscillatory discrete-time systems

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