Synchronization Of Three Dimensional Cancer Model Case Study
Synchronization of Three Dimensional Cancer Model with Chen System using a Robust Adaptive Sliding Mode Controller with application to secure communications
Abstract: In this computational study, we synchronize the Three Dimensional Cancer Model (TDCM) with Chen System (CS) using a Robust Adaptive Sliding Mode Controller (RASMC) together with uncertainties, external disturbances and fully unknown parameters. A simple suitable sliding surface, which includes synchronization errors, is constructed and appropriate update laws are used to tackle the uncertainties, external disturbances and unknown parameters. All simulations to achieve the synchronization for the proposed technique for the two non-identical systems under consideration are being …show more content…
The master and slave systems can be rewritten in the form of (2.1) and (2.2) as follows: (3.4) (3.5)
Therefore, using (2.7), the error dynamics can be expressed …show more content…
4. Numerical Simulation
In the proposed study, we have chosen , , , , , , , and the initial values of the update vector parameters , , and are , , and respectively. Furthermore, The vector of switching gains for , the coefficient and the sliding surfaces are , , . The TDCM and Chen system are started with the initial conditions as follows: and , , .
In this regard, we consider a Lyapunov function (that is a positive definite function also) as follows: (4.1)
Figure 10 shows that the derivative of Lyapunov function (4.1) is less than or equal to zero for t is bigger than zero which strongly support that the synchronization is stable on the selected sliding surface i.e figure 10 verify theorem 1.
Figure 1 illustrates the synchronization errors of the TDCM and Chen system, as one can see the all synchronization errors converge to the zero, which implies that the chaos synchronization between the TDCM and Chen system is realized. The time responses of the update vector parameters , , and are depicted in Figures 2 - 5, respectively. It is very well clear that all of the update parameters approach to some
Abstract: In this computational study, we synchronize the Three Dimensional Cancer Model (TDCM) with Chen System (CS) using a Robust Adaptive Sliding Mode Controller (RASMC) together with uncertainties, external disturbances and fully unknown parameters. A simple suitable sliding surface, which includes synchronization errors, is constructed and appropriate update laws are used to tackle the uncertainties, external disturbances and unknown parameters. All simulations to achieve the synchronization for the proposed technique for the two non-identical systems under consideration are being …show more content…
The master and slave systems can be rewritten in the form of (2.1) and (2.2) as follows: (3.4) (3.5)
Therefore, using (2.7), the error dynamics can be expressed …show more content…
4. Numerical Simulation
In the proposed study, we have chosen , , , , , , , and the initial values of the update vector parameters , , and are , , and respectively. Furthermore, The vector of switching gains for , the coefficient and the sliding surfaces are , , . The TDCM and Chen system are started with the initial conditions as follows: and , , .
In this regard, we consider a Lyapunov function (that is a positive definite function also) as follows: (4.1)
Figure 10 shows that the derivative of Lyapunov function (4.1) is less than or equal to zero for t is bigger than zero which strongly support that the synchronization is stable on the selected sliding surface i.e figure 10 verify theorem 1.
Figure 1 illustrates the synchronization errors of the TDCM and Chen system, as one can see the all synchronization errors converge to the zero, which implies that the chaos synchronization between the TDCM and Chen system is realized. The time responses of the update vector parameters , , and are depicted in Figures 2 - 5, respectively. It is very well clear that all of the update parameters approach to some