Quasi-continuous high-order sliding-mode controllers for reduced-order chaos synchronization

Adaptive Fuzzy Synchronization for Uncertain Chaotic Systems with Different Dimensions and Disturbances

“…Till now, some investigations on the synchronization between chaotic systems with different dimensions have been published [2,14,[17][18][19][20][21][22][23][24][25]. In References [2,18,[20][21][22][23][24][25], the reduced-order synchronization was investigated, where the slave system with the lower dimension could synchronize with the canonical projection of the master system possessing a higher dimension. References [14,17] and [19] proposed control schemes for both reduced-order synchronization and increased-order synchronization (which indicated that the master system possessed the lower dimension), which broke the restrictions on the dimension relationship of the master and slave systems.…”

“…References [14,17] and [19] proposed control schemes for both reduced-order synchronization and increased-order synchronization (which indicated that the master system possessed the lower dimension), which broke the restrictions on the dimension relationship of the master and slave systems. But, it should be pointed out that in [14], the reduced-order synchronization was achieved by the similar method to that in [2,[20][21][22][23][24][25], and increasedorder synchronization was obtained via constructing auxiliary state vector. Thus, in essence, the conception of synchronization in [2,14,18,[20][21][22][23][24][25] is similar to those with the same dimension.…”

“…But, it should be pointed out that in [14], the reduced-order synchronization was achieved by the similar method to that in [2,[20][21][22][23][24][25], and increasedorder synchronization was obtained via constructing auxiliary state vector. Thus, in essence, the conception of synchronization in [2,14,18,[20][21][22][23][24][25] is similar to those with the same dimension. Different from [2,14,18,[20][21][22][23][24][25], by introducing a continuously differentiable function, References [17,19] defined and discussed the finite-time generalized synchronization and generalized synchronization for the chaotic systems with different orders, respectively.…”

“…Thus, in essence, the conception of synchronization in [2,14,18,[20][21][22][23][24][25] is similar to those with the same dimension. Different from [2,14,18,[20][21][22][23][24][25], by introducing a continuously differentiable function, References [17,19] defined and discussed the finite-time generalized synchronization and generalized synchronization for the chaotic systems with different orders, respectively. However, both of them neglected the possible and inevitable uncertainties in the master and slave systems.…”

“…Investigation for this kind of master-slave systems is extremely important and necessary in practice, especially in the industrial processes, because it is usually too complex or expensive to model the structure of the physical systems [27]. References [17][18][19][20][21][22][23] did not consider the uncertainties at all. Although References [2,14,24,25] discussed the synchronization problem for the chaotic systems with fully unknown parameters, all of their synchronization controllers employed the nonlinear functions in the master and/or slave systems, that is, for the kind of master-slave systems discussed in this paper, they are out of use.…”