Sign and stability of equilibria in quasi-monotone positive nonlinear systems

Silva-Navarro, G.; Álvarez-Gallegos, Jaime

doi:10.1109/9.557585

Cited by 27 publications

(18 citation statements)

References 4 publications

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“…Assumption 1: The matrix is asymptotically stable. Define the error signal as , then the augmented observing system is given by (5) For general linear systems, it is only required that system (5) is asymptotically stable. For positive linear systems, however, this specification is not enough, since not only the stability of (5) but also the positivity of should be guaranteed.…”

Section: Lemma 2 ([45])mentioning

confidence: 99%

“…P OSITIVE systems are dynamic systems whose state variables are constrained to be positive (at least nonnegative) at all times. Such systems abound in various fields, e.g., biomedicine [1], [2], pharmacokinetics [3], ecology [4], chemical engineering [5], industrial engineering [6], economics, and so on. Recently, an interesting application of positive system model to TCP-like Internet congestion control has also appeared [7], [8].…”

Section: Introductionmentioning

confidence: 99%

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Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems

Shu

Lam

Gao

et al. 2008

IEEE Trans. Circuits Syst. I

228

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This paper is concerned with the design of observers and dynamic output-feedback controllers for positive linear systems with interval uncertainties. The continuous-time case and the discrete-time case are both treated in a unified linear matrix inequality (LMI) framework. Necessary and sufficient conditions for the existence of positive observers with general structure are established, and the desired observer matrices can be constructed easily through the solutions of LMIs. An optimization algorithm to the error dynamics is also given. Furthermore, the problem of positive stabilization by dynamic output-feedback controllers is investigated. It is revealed that an unstable positive system cannot be positively stabilized by a certain dynamic output-feedback controller without taking the positivity of the error signals into account. When the positivity of the error signals is considered, an LMI-based synthesis approach is provided to design the stabilizing controllers. Unlike other conditions which may require structural decomposition of positive matrices, all proposed conditions in this paper are expressed in terms of the system matrices, and can be verified easily by effective algorithms. Two illustrative examples are provided to show the effectiveness and applicability of the theoretical results.

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Section: Lemma 2 ([45])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems

Shu

Lam

Gao

et al. 2008

IEEE Trans. Circuits Syst. I

228

View full text Add to dashboard Cite

show abstract

“…Such systems are frequently encountered in various field, for instance, pharmacokinetics [7], chemical reaction [8] and internet congestion control [9]. Since the states of positive systems are confined within a "cone" located in the positive orthant rather than in whole state spaces, many elegant results for general linear systems connot be simply applied in positive systems.…”

Section: Introductionmentioning

confidence: 99%

Exponential stability of discrete-time linear singular positive time-delay systems

Liu

Tong

2015

The 27th Chinese Control and Decision Conference (2015 CCDC)

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This paper is concerned with the problem of α-exponential stability for discrete-time linear singular positive time-delay systems. Firstly, sufficient and necessary conditions for the positivity of the system are investigated based on the singular value decomposition and monomial coordinate transformation method. Then in virtue of the obtained results, a new exponential stability condition is established. Moreover, all the obtained results are formulated in terms of algebraic matrix inequalities and therefore are numerically tractable. Finally, a numerical simulation is given to show the effectiveness of the proposed techniques.

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“…The absolute stability problem is that, although there is a large measure of arbitrariness in the functional form of the non-linearity, the admissible class } should have su ciently well de® ned features for the zero solution of the system to be globally asymptotically stable for all f 2 } (Aizerman and Gantmacher 1964, Popov 1973, Somolinos 1977, Krasnosel' skii and Pokrovskii 1978, GruijicÂ 1981, Singh 1982, Molchanov and Pyatniskii 1986, Miyagi and Yamashita 1992, Silva-Navarro and Alvarez-Gallegos 1997.…”

Section: Introductionmentioning

confidence: 99%

“…The main results are stated in } 3, which is devoted to systems which are positive or monotone in some sense (for example, Narendra and Neumann 1966, Haddad and Kapila 1995, Silva-Navarro and Alvarez-Gallegos 1997. Section 4 contains an example, and proofs of the theorems are given in the last section.…”

Section: Introductionmentioning

confidence: 99%

Absolute stability of positive systems with differential constraints

Diamond¹,

Opoitsev²

2000

International Journal of Control

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We consider discrete-time systems x… k ‡ 1 †ˆAx… k † ‡ f … x… k † † where the matrix A of the linear part is known and positive, the non-linearity f is unknown but belongs to a class for which A ‡ f 0 … x † is positive with spectral radius < 1 for all x 2 R n . This, with the additional property that x ¡ Ax ¡ f … x † is proper, is su cient for global stability of the system. The results are applied to the continuous system _ xˆAx ‡ B'… C T x † by considering the translation operator along trajectories and studying the resulting discrete system.

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Sign and stability of equilibria in quasi-monotone positive nonlinear systems

Cited by 27 publications

References 4 publications

Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems

Positive Observers and Dynamic Output-Feedback Controllers for Interval Positive Linear Systems

Exponential stability of discrete-time linear singular positive time-delay systems

Absolute stability of positive systems with differential constraints

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