Positive Linear Systems

Farina, Lorenzo; Rinaldi, Sergio

doi:10.1002/9781118033029

Cited by 1,375 publications

(430 citation statements)

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“…Note also that A(X) is a Metzler matrix (all components, except the ones in the diagonal, are non-negative [25]), which implies that the solution stays in the positive orthant provided that the initial condition is given in the positive orthant. Moreover, straighforward computations show that the solution is bounded.…”

Section: Temporal Modeling Of the Wild Mosquito Populationmentioning

confidence: 99%

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

Dufourd

Dumont

2013

Computers & Mathematics with Applications

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The aim of this paper is to develop a mathematical model to simulate mosquito dispersal and its control taking into account environmental parameters, like wind, temperature, or landscape elements. We particularly focus on Aedes albopictus mosquito which is now recognized as a major vector of human arboviruses, like chikungunya, dengue, or yellow fever. One way to prevent those epidemics is to control the vector population. Biological control tools, like the Sterile Insect Technique (SIT), are of great interest as an alternative to chemical control tools which are very detrimental to environment. The success of SIT is based on a good knowledge of the biology of the insect, but also on an accurate modeling of the insect's distribution. We consider a compartmental approach and derive temporal and spatio-temporal models, using Advection-Diffusion-Reaction equations to model mosquito dispersal. Periodic releases of sterilized males are modeled with an impulse differential equation. Finally, using the splitting operator approach, and well-suited numerical methods for each operator, we provide numerical simulations for mosquito spreading, and test different vector control scenarios. We show that environmental parameters, like vegetation, can have a strong influence on mosquito distribution and in the efficiency of vector control tools, like SIT.keyword Advection diffusion reaction equation, impulse differential equation, mosquito dispersal, sterile insect technique, operator splitting, nonstan-

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Section: Temporal Modeling Of the Wild Mosquito Populationmentioning

confidence: 99%

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

Dufourd

Dumont

2013

Computers & Mathematics with Applications

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“…Continuous-time and discrete-time positive systems have been studied in detail in recent years [1][2][3][4][5][6][7][8][9][10]. If the state and output possess the positivity property under nonnegative initial conditions and controls, the positivity is said to be internal or, simply, the system is positive.…”

Section: Introductionmentioning

confidence: 99%

“…The property of asymptotic hyperstability generalizes that of absolute stability [13][14][15] which generalizes the most basic concept of stability of dynamic systems. See, for instance [2,3,11,[13][14][15][16][17][18][19][20][21][22][23][24][25][26], and references therein. It is well known that closed-loop hyperstability is, by nature, a powerful version of closed-loop stability since it refers to the stability of a hyperstable linear feed-forward plant (in the sense of the positive realness of the associated transfer matrix) under a wide class of feedback controllers applied.…”

Section: Introductionmentioning

confidence: 99%

“…If the output possesses such a property, the system is said to be externally positive. Thus, positive systems are intrinsically interesting to describe some problems like Markov chains, queuing problems, certain distillation columns and biological, and other physical compartmental problems where populations or concentrations cannot be negative [2,3]. It is well known that time-invariant dynamic linear systems which are externally positive, while having positive real or strictly positive real transfer matrices, are, in addition, hyperstable or asymptotically hyperstable, i.e., globally Lyapunov stable for any nonlinear and/or time-varying feedback device satisfying Popov's type inequality for all time, [11,12].…”

Section: Introductionmentioning

confidence: 99%

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On the Asymptotic Hyperstability of Linear Time-Invariant Continuous-Time Systems under a Class of Controllers Satisfying Discrete-Time Popov’s Inequality

Sen

2018

Mathematical Problems in Engineering

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This paper is concerned with the property of asymptotic hyperstability of a continuous-time linear system under a class of continuous-time nonlinear and perhaps time-varying feedback controllers belonging to a certain class with two main characteristics; namely, (a) it satisfies discrete-type Popov's inequality at sampling instants and (b) the control law within the intersample period is generated based on its value at sampling instants being modulated by two design weighting auxiliary functions. The closed-loop continuous-time system is proved to be asymptotically hyperstable, under some explicit conditions on such weighting functions, provided that the discrete feed-forward transfer function is strictly positive real.

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“…Positive dynamical systems arise as models in a diverse range of fields from biology, chemistry, ecology, and economics to genetics, medicine, and engineering [36, p. xv], and are understandably well studied, with textbooks [37,38,39]. Control of positive dynamical systems leads to positive input control systems [40], where the input variables are assumed to be positive as well. In a pest management context, the input variable is naturally allowed to take negative values, provided that a nonnegative number or distribution of pests remains, leading to the concept of so-called positive state systems [41].…”

mentioning

confidence: 99%

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

Guiver¹,

Edholm²,

Jin³

et al. 2016

SIAM J. Appl. Math.

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Abstract. Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows for the possibility of rotating between different courses of management action. In developing our control strategies we appeal to comparison and monotonicity arguments. Interestingly, componentwise nonnegativity of the model, combined with an irreducibility assumption, implies that several issues typically associated with high-gain adaptive controllers do not arise and usual high-gain structural assumptions are not required.

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Positive Linear Systems

Cited by 1,375 publications

References 0 publications

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control

On the Asymptotic Hyperstability of Linear Time-Invariant Continuous-Time Systems under a Class of Controllers Satisfying Discrete-Time Popov’s Inequality

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

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