Estimate of the region of attraction for a class of nonlinear time delay systems: a leukemia post-transplantation dynamics example

Villafuerte, R.; Mondié, Sabine

doi:10.1109/cdc.2007.4434724

Cited by 3 publications

(5 citation statements)

References 10 publications

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“…Assuming that this condition is met, a complete-type L-K functional can be constructed for this system and used for robustness analysis, when a drift term f ( 16 The complete type L-K functional has since been applied to a biological problem. 18 However, the estimate on the region of attraction obtained was found to be somewhat conservative. 18 Further, the analysis has been extended to construct a complete-type L-K functional which had a cross term in the time derivative.…”

Section: Introductionmentioning

confidence: 93%

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Attitude Stabilization with Network Delay in Feedback Control Implementation

Chunodkar

Akella²

2009

AIAA Guidance, Navigation, and Control Conference

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This paper considers the problem of stabilizing attitude dynamics with an unknown constant delay in feedback. It is assumed that a strict upper bound of the delay is known. A novel modification to the concept of the complete type Lyapunov-Krasovskii (L-K) functional ensures robustness to time-delay in the control design. The governing dynamics is partitioned to form a nominal system with a perturbation term. Frequency domain analysis is employed in order to construct necessary and sufficient stability conditions for the nominal system. Consequently, a complete type L-K functional is constructed. As an intermediate step, an analytical solution for the underlying Lyapunov matrix is obtained. The perturbation term is defined such that it is a function of the state at current time alone, which allows simplifications to be made in the L-K functional construction. Departing from previous approaches, where parameter values are arbitrarily chosen to satisfy the sufficient conditions obtained from robustness analysis, a systematic numerical optimization process is used to choose control parameters so that the region of attraction is maximized. The estimate of the region of attraction is directly related to the initial angular velocity norm and the closed-loop system is shown to be stable for a large set of initial attitude orientations. This is to our best knowledge the first result that provides stable closed-loop control design for the attitude dynamics problem with an unknown delay in feedback. Moreover, the results obtained are found to be tighter compared to some other sufficient condition based results documented existing in literature that assume perfect knowledge of the delay.

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Section: Introductionmentioning

confidence: 93%

“…18 However, the estimate on the region of attraction obtained was found to be somewhat conservative. 18 Further, the analysis has been extended to construct a complete-type L-K functional which had a cross term in the time derivative. 17 This generalization reduced the conservatism of the estimate.…”

Section: Introductionmentioning

confidence: 93%

Attitude Stabilization with Network Delay in Feedback Control Implementation

Chunodkar

Akella²

2009

AIAA Guidance, Navigation, and Control Conference

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Section: Introductionmentioning

confidence: 99%

“…Therefore, it is necessary to consider these in mathematical modeling. [21][22][23][24][25][26] Some members of the scientific community have argued that time delays should be omitted or disregarded from mathematical models since they may cause undesirable/poor behavior in the system response or even compromise its stability. However, this interpretation is wrong.…”

Section: Introductionmentioning

confidence: 99%

Mathematical model with time‐delay and delayed controller for a bioreactor

Villafuerte-Segura

Itzá-Ortiz

López‐Pérez

et al. 2022

Math Methods in App Sciences

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In this paper, a fractional Lotka-Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time-delay 𝜏 due to the dead-time (non-trivial) that the microbe needed to metabolize the substrate. A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay 𝜏 as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators, or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller u(t) = k p x(t), which is shown to fail to stabilize the obtained model under the proposed analysis.Another suitable choice is the use of a delayed controller u(t) = k r x(t − h) which successfully stabilizes the model even when it is unstable. The delay h in the feedback control is due to the dead-time necessary to obtain the measurement of the biomass in the bioreactor by dry weight. Finally, the proposed theoretical results are corroborated through numerical simulations.

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“…Naturally, delays are always present in bioprocesses, since there are dead times between biological reactions/interactions, metabolization of substances, considerations of past population rates, among others. Therefore, it is necessary to consider these in mathematical modeling [10,25,31,39,41,43]. Some members of the scientific community have argued that time delays should be omitted or disregarded from mathematical models since they may cause undesirable/poor behavior in the system response or even compromise its stability.…”

Section: Introductionmentioning

confidence: 99%

Fractional Lotka-Volterra model with time-delay and delayed controller for a bioreactor

Villafuerte-Segura,

Itzá-Ortiz,

López-Pérez

et al. 2020

Preprint

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In this paper, a fractional Lotka-Volterra mathematical model for a bioreactor is proposed and used to fit the data provided by a bioprocess known as continuous fermentation of Zymomonas mobilis. The model contemplates a time-delay τ due to the dead-time in obtaining the measurement of biomass x(t). A Hopf bifurcation analysis is performed to characterize the inherent self oscillatory experimental bioprocess response. As consequence, stability conditions for the equilibrium point together with conditions for limit cycles using the delay τ as bifurcation parameter are obtained. Under the assumptions that the use of observers, estimators or extra laboratory measurements are avoided to prevent the rise of computational or monetary costs, for the purpose of control, we will only consider the measurement of the biomass. A simple controller that can be employed is the proportional action controller u(t) = kpx(t), which is shown to fail to stabilize the obtained model under the proposed analysis. Another suitable choice is the use of a delayed controller u(t) = krx(t − h) which successfully stabilizes the model even when it is unstable. Finally, the proposed theoretical results are corroborated through numerical simulations.

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Estimate of the region of attraction for a class of nonlinear time delay systems: a leukemia post-transplantation dynamics example

Cited by 3 publications

References 10 publications

Attitude Stabilization with Network Delay in Feedback Control Implementation

Attitude Stabilization with Network Delay in Feedback Control Implementation

Mathematical model with time‐delay and delayed controller for a bioreactor

Fractional Lotka-Volterra model with time-delay and delayed controller for a bioreactor

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