Iterative method for estimating the robust domains of attraction of non-linear systems: Application to cancer chemotherapy model with parametric uncertainties

Riah, Rachid; Fiacchini, Mirko; Alamir, Mazen

doi:10.1016/j.ejcon.2018.12.002

Cited by 8 publications

(10 citation statements)

References 19 publications

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“…Considering the multiplication terms of β and s 1 in (27) in this case, this is also a two-line relation. Therefore, the optimization program (39) is not adopted in the solution process of this case.…”

Section: Examples and Analysismentioning

confidence: 96%

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Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

et al. 2020

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We present a sum of squares (SOS) method for the synthesis of nonlinear polynomial control systems. As an emerging numerical solution method in recent years, SOS targets polynomials as the research object. It guarantees that the polynomial we solve for is always nonnegative. In this paper, we give a generalized S-procedure to solve the SOS problem. As an illustration of how the SOS method can be used, the region of attraction (ROA) in a nonlinear polynomial system is analyzed in detail. The method of determining decision variables is given in the SOS problem. We discuss the determination and solution of set-containment constraints and the conservatism problem in solving the SOS problem. SOS provides a convenient numerical method to solve nonlinear problems that are not easy to solve analytically. INDEX TERMS Sum of squares, nonlinear polynomial system, region of attraction, sum of squares program, set-containment constraint.

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Section: Examples and Analysismentioning

confidence: 96%

“…In the reference [26], estimation of the regions of attraction for autonomous nonlinear systems is studied. In the reference [27], attraction domain estimation of the model with parametric uncertainties is studied. In the reference [28], the Maximal Lyapunov Function Method is used to estimate the domain.…”

Section: Introductionmentioning

confidence: 99%

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

et al. 2020

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“…Seminal prior works in this convexity-related line are; [19][20][21][22] they show that set-based LDI ideas can be generalized to the so-called convex differential inclusions (CDI), which are more general models capable of representing nonlinear and uncertain systems with lower conservatism (overbounding) than LDI, being these a particular case. Furthermore, they present iterative scaling/shooting algorithms to compute contractive sets, both in analysis and robust control design settings.…”

Section: Introductionmentioning

confidence: 99%

Set‐based gain‐scheduled control via quasi‐convex difference inclusions

Sala

Ariño

Robles

2020

Intl J Robust & Nonlinear

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A nonlinear system with sector-bounded nonlinearities may be expressed as a quasi-linear parameter-varying (LPV) system (convex combination of linear models), being this a well-known fact. The convex difference inclusion (CDI) modeling framework proposed by Fiacchini and coworkers in several of their works generalizes the quasi-LPV modeling procedure and proposes robust controllers enlarging polytopic domain of attraction estimates. This works further generalizes the CDI approach to a gain-scheduled case including, also, some quasi-convex cases. Controller design is based on convexity properties of two set valued maps describing (with some uncertainty) the state evolution and the state-dependent set where scheduling variables take values. As most set-based approaches, the proposal is tractable in low-dimensional cases. The presented results encompass prior quasi-LPV and CDI models as particular cases.

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“…In details, one can see previous studies. [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53] Here, we must point out that all the mentioned publications about Hopf bifurcation of fractional-order systems only involve the impact of one or the sum of multiple delays on Hopf bifurcation. However, up till now, no scholar has studied the impact of two different delays on Hopf bifurcation of fractional-order differential models.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

Liao

et al. 2019

Math Methods in App Sciences

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Cancer is one of the most serious diseases in the world. The investigation on cancer treatment has attracted great attention from medical workers, mathematical researchers, and scholars in various fields. To understand the intrinsic characteristics of cancer, numerous scholars have established cancer treatment models and discussed the dynamical properties. However, the main work of many scholars only focuses on the integer-order cancer treatment models, while the study on fractional-order ones is quite a few. In the present article, on the basis of previous publications, we will put up a new fractional-order chemotherapy model with two different delays. By applying the stability theory and the Hopf bifurcation of fractional-order differential equations, we obtain a series of new stability criteria of the involved model and some sufficient conditions to ensure the existence of Hopf bifurcation of the involved model. Furthermore, the impact of two different delays and the fractional order on the stability and the emergence of Hopf bifurcation of involved model is presented.To illustrate the validity of analytical predictions, we perform computer simulations with Matlab software. The theoretical results in the manuscript are innovative and play an important role in cancer treatment.

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Iterative method for estimating the robust domains of attraction of non-linear systems: Application to cancer chemotherapy model with parametric uncertainties

Cited by 8 publications

References 19 publications

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

Application of Sum-of-Squares Method in Estimation of Region of Attraction for Nonlinear Polynomial Systems

Set‐based gain‐scheduled control via quasi‐convex difference inclusions

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

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