Finite-time fuzzy reliable controller design for fractional-order tumor system under chemotherapy
Introduction
An abnormal cell growth in a particular organ or part of the body is called cancer. Cancer is the world's worst killer disease, especially in western countries. The tumor can occur in any part of the body. Depending on the location of tumor, it has over 100 types; like lung cancer, colon cancer, skin cancer, lymphoma, breast cancer, prostate cancer, etc. The mathematical model of cancer contains three trajectories such as (i) population of tumor cells (TC), (ii) population of effector-immune-cells (EIC) and (iii) population of lymphocyte (see [1]). Although the TC are killed by the EIC via mass-action dynamics, the growth of TC will be in progress due to the reverse action (i.e.) EIC are inactivated by TC. Therefore, it is important to treat the TC with some treatment. There are several types of treatment, including chemotherapy, surgery and radiation. Among them, chemotherapy is an effective treatment that treats tumors through medication. The mathematical model and control problems of chemotherapy for tumors are discussed in [1], [2], [3], [4], [6], [5]. In the meantime, it is obvious that the tumor system is a nonlinear system, T-S fuzzy provides a great algorithm to linearize the nonlinear systems [7]. In T-S fuzzy systems, a nonlinear system with the help of fuzzy rules is divided into simple local subsystems [8], [9], [10], [11]. They are then linked via normalized membership functions. The authors Mani et al. [12] discussed the designing of permanent magnet synchronous motor through T-S fuzzy rules. And most recently, Zhang et al. [11] investigated the process of normalization rectangular Fractional-Order(FO) fuzzy systems.
On the other hand, FO derivatives (refer [13], [14], [15], [16]) help us to design a real-time practical system more accurately and flexibly. Due to its pliability when modeling a system, it attracts many researchers (see [4], [19], [20], [21]). So, the rate of change in FO over time turns the tumor system more flexible [23], [25]. Over the two past decades, researchers have paid more attention to the control problem of the FO system [19], [23], [24], [25], [26], [28]. The stability analysis is the most important part in control problems [4], [29], [30]. Lyapunov's indirect method gives an effectual algorithm to stabilize the required system without computing the solution. Rooka et al. [17] investigated the stability problem of FO tumor system and Mahdy et al. [18] designed memristor-based control for FO tumor system. In [30], the authors discussed the finite-time stabilization of T-S fuzzy systems via intermediate observer. The existence and stabilization in a finite-time boundary of discrete FO system is reported in [31]. An investigation on finite-time stability of impulsive nonlinear system through settling time estimation is made in [32]. A novel set of Lyapunov constraints of time-varying delayed system for finite-time stability is discussed by the authors in [33]. Meanwhile, in the modern lifestyle, due to the addiction of smoking or tobacco products, cell phone tower radiation, usage of plastic carriers like water bottles, the pollutions are increased. These external factors are considered for increasing the TC (i.e.) in mathematical term, these noises may also induce instability of tumor system. Hence, the extended dissipative which is a generalized performance of [34], passivity [35], dissipativity [36] and [37] performance is considered to reclaim the optimum level of stability [38], [39].
Further, in chemotherapy, drug is assumed to be the controller, so it can only be given to victim via intravenous or pharmaceutical drugs. Therefore, there may be some losses in drug when it passes through veins or oesophagus to the affected part. Due to the above discussion, it is necessary to take into account of the actual drug dosage that reaches the affected part (i.e.) it is important to introduce a quantized input for our proposed system. In [40], the authors discussed an intermediate fault-estimator controller for a FO system using quantized output. Further, an adaptive and global-feedback quantized input are designed in [41] and [42], respectively. Nevertheless, till now, the issue of finite-time based reliable control under extended dissipativity for FO T-S fuzzy tumor system subject to external disturbances using quantized input has not been examined yet. Inspired by the above analysis, this article investigates a novel set of finite-time based extended dissipative stability constraints for tumor systems under chemotherapy with external disturbance is computed by reliable state-feedback quantized input. The main developments of this article are summarized below:
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A novel model of fuzzy fractional-order tumor system along with the drug toxicity and drug concentration of chemotherapy is constructed. Then, a boundedness criterion is derived to ensure the constructed system is finite-time bounded.
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A reliable controller is designed to reduce the effect of fault in controller.
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Then the impact of quantized input and outside disturbances are taken into account and a prescribed extended dissipative performance is included to reduce the conservatism.
The rest of this paper is given as follows. The model description is shown in Section 2. Section 3 is devoted to the finite-time boundedness and quantized reliable controller synthesis. Simulation studies are given to show the effectiveness of the proposed methods in Section 4. In Section 5, conclusions are presented.
Section snippets
Problem formulation
This section explains the formulation of the addressed problem with the help of reliable controller. For this, we need the following definition: Definition 1 [22] For a non autonomous function , the Caputo fractional derivative with the order is defined as where is Gamma function and is the initial condition.
Main results
In this division, a fuzzy reliable controller scheme (8) for the considered fractional-order fuzzy tumor system subject to quantized-input is investigated. Firstly, a new set of sufficient constraints for the finite-time boundedness analysis of fractional-order fuzzy tumor system is establish with known gain matrices which is presented in the following theorem. Theorem 1 Let the Assumption 1 and Assumption 3 hold. For the known controller gains 's and known parameters , , Ω, there exist positive
Numerical example
A numerical example is given in this section to reveal the efficacy of proposed result. The generalized model (3) takes the value of parameters from the table displayed below (refer [5]):
Parameter Definition Numerical value a2 Fractional TC kill by chemotherapy 0.3 day−1 b1 1/b1 is TC carrying capacity 1 cell−1 b2 1/b2 is NC carrying capacity 1 cell−1 c1 Fractional EIC cell kill by TC cells 1 cell−1day−1 c2 Fractional TC kill by EIC 0.5 cell−1day − 1 c3 Fractional TC kill by NC 1 cell−1day−1 c4 Fractional NC kill by
Conclusion
A stability analysis of fractional-order tie-up with tumor-immune system under chemotherapy and Lyapunov's theory is done. In addition, the finite-time stabilization of constructed system with the effect of quantified input is studied. Moreover, the nonlinear tumor model is reconstructed as T-S fuzzy model. Furthermore, on the basis of Lyapunov's stability theory, the mandatory conditions are constructed with Lyapunov's candidate. In particular, to get a better performance of controller even in
Declaration of Competing Interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
This research work of S. Senpagam was financially supported by Department of Science and Technology, Government of India, New Delhi, through Promotion of University Research and Scientific Excellence Phase-II, Government of India, New Delhi.
Acknowledgements
This research work of S. Senpagam is financially supported by DST-PURSE Phase-II(BU/DSTPURSE(II)/APPOINTMENT/07), Government of India, New Delhi.
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