Reliable solution of special event location problems for ODEs

Shampine, L. F.; Gladwell, I.; Brankin, R. W.

doi:10.1145/103147.103149

Cited by 72 publications

(50 citation statements)

References 8 publications

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“…The boundary point is accurately detected, and then, the integrated vector field is switched according to the governing laws of the system. In practice, this can be implemented by means of the standard MAT-LAB ODE solvers together with their built-in event location routines [32,33], as suggested in [34].…”

Section: Capsule Modeling With Position Feedback Controlmentioning

confidence: 99%

Controlling multistability in a vibro-impact capsule system

Liu

Chávez

2017

Nonlinear Dyn

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This work concerns the control of multistability in a vibro-impact capsule system driven by a harmonic excitation. The capsule is able to move forward and backward in a rectilinear direction, and the main objective of this work is to control such motion in the presence of multiple coexisting periodic solutions. A position feedback controller is employed in this study, and our numerical investigation demonstrates that the proposed control method gives rise to a dynamical scenario with two coexisting solutions, corresponding to forward and backward progression. Therefore, the motion direction of the system can be controlled by suitably perturbing its initial conditions, without altering the system parameters. To study the robustness of this control method, we apply numerical continuation methods in order to identify a region in the parameter space in which the proposed controller can be applied. For this purpose, we employ the MATLAB-based numerical platform COCO, which

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Section: Capsule Modeling With Position Feedback Controlmentioning

confidence: 99%

Controlling multistability in a vibro-impact capsule system

Liu

Chávez

2017

Nonlinear Dyn

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“…In our case, these interruptions are defined by the impulse times at which the pest control actions are carried out. In order to get reliable numerical simulations of the model behavior, the impulse times need to be accurately detected, which can be achieved by means of the standard MATLAB ODE solvers together with their built-in event location routines [60,61], as suggested in [62]. In this way, direct numerical integration will be implemented in the present work.…”

Section: Numerical Analysis Of a Pest Control Scheme With Impulsive Ementioning

confidence: 99%

Modeling and Analysis of Integrated Pest Control Strategies via Impulsive Differential Equations

Chávez

Jungmann

Siegmund

2017

International Journal of Differential Equations

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The paper is concerned with the development and numerical analysis of mathematical models used to describe complex biological systems in the framework of Integrated Pest Management (IPM). Established in the late 1950s, IPM is a pest management paradigm that involves the combination of different pest control methods in ways that complement one another, so as to reduce excessive use of pesticides and minimize environmental impact. Since the introduction of the IPM concept, a rich set of mathematical models has emerged, and the present work discusses the development in this area in recent years. Furthermore, a comprehensive parametric study of an IPM-based impulsive control scheme is carried out via path-following techniques. The analysis addresses practical questions, such as how to determine the parameter values of the system yielding an optimal pest control, in terms of operation costs and environmental damage. The numerical study concludes with an exploration of the dynamical features of the impulsive model, which reveals the presence of codimension-1 bifurcations of limit cycles, hysteretic effects, and period-doubling cascades, which is a precursor to the onset of chaos.

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“…More formally: problem Given f : R n → R n , x 0 = x(t 0 ) ∈ R n and g : R n → R such that g(x 0 ) < 0, simulatė x = f (x), for the time interval [t 0 , t * ] where t * must be computed as the first time instant such that g(x(t)) ≥ 0. problem We assume the guard set has a non-empty interior and is described as Guard = {x : g(x) ≥ 0} where g(x) is a continuously differentiable. See [12] for an interesting discussion of the unique difficulties associated with solving such problems. It is well known that systems of differential equations with nonlinear guards can be transformed to a equivalent systems with linear guards by appending a new state variable z = g(x) then the new system isẋ…”

Section: Motivation and Previous Workmentioning

confidence: 99%

“…As a result they fail to detect an event when there are multiple transitions in a single step. Building on this work, Shampine and his colleagues [12] exploit the fact that interpolation polynomials can be generated for the guard dynamics and are able to correctly identify event occurrences using Strum sequences when the guards are of polynomial expressions but do not use this information to select step sizes. Several similar algorithms for event detection in differential algebraic equations were evaluated in [15].…”

Section: Motivation and Previous Workmentioning

confidence: 99%

Accurate Event Detection for Simulating Hybrid Systems

Esposito

Kumar

Pappas

2001

Lecture Notes in Computer Science

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Abstract. It has been observed that there are a variety of situations in which the most popular hybrid simulation methods can fail to properly detect the occurrence of discrete events. In this paper, we present a method for detecting discrete which, using techniques borrowed from control theory, selects integration step sizes in such a way that the simulation slows down as the state approaches a set which triggers an event (a guard set). Our method guarantees that the state will approach the boundary of this set exponentially; and in the case of linear or polynomial guard descriptions, terminating on it, without entering it. Given that any system with a nonlinear guard description can be transformed to an equivalent system with a linear guard description, this technique is applicable to a broad class of systems. Even in situations where nonlinear guards have not been transformed to the canonical form, the method is still increases the chances of detecting and event in practice. We show how to extend the method to guard sets which are constructed from many simple sets using boolean operators (e.g. polyhedral or semi-algebraic sets) . The technique is easily used in combination with existing numerical integration methods and does not adversely affect the underlying accuracy or stability of the algorithms.

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Reliable solution of special event location problems for ODEs

Cited by 72 publications

References 8 publications

Controlling multistability in a vibro-impact capsule system

Controlling multistability in a vibro-impact capsule system

Modeling and Analysis of Integrated Pest Control Strategies via Impulsive Differential Equations

Accurate Event Detection for Simulating Hybrid Systems

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