Consider two fractional chaotic systems
$D^\alpha x=f(x)$ as master system
and
$D^\beta y=g(y)$ as slave system.($0 \lt \beta \le \alpha \le 1$)
that $ x, y ∈ \mathbb R^n$ are the n-dimensional state vectors for drive and response systems, respectively; $f, g : \mathbb R^n \to \mathbb R^n$ are continuous vector functions and U(x, y) is the controller that added to slave system.
We want to design our control so that $e=||x-y||\to 0$.We decompose the controller function $U(x, y)$ into two sub-controllers $U_I(x, y)$ and $U_{II}(x, y)$, i.e., $U(x, y) = U_I(x, y) + U_{II}(x, y)$ and propose the following form for the subcontroller $U_I(x, y)$:
$U_I(x, y) = (D^{−(\alpha-\beta)} − 1)[ g(y)]$. So we can rewrite the response system as follows: $D^αy = D^{−(\alpha-\beta)}g(y) + U_{II}(x, y)$.
Applying a Laplace transform to the above system and letting $Y (s) = L(y(t)) $, we obtain $s^\beta Y (s) − s^{\beta - 1}y(0) = s^{−(\alpha-\beta)}L(g(y)) + L(U_{II}(x, y))$. (1)
Multiplying both the left-hand and right-hand sides of Eq. (1) by $s^{(\alpha-\beta)}$
$s^\alpha Y (s) − s^{\alpha - 1}y(0) = L(g(y)) +s^{(\alpha-\beta)} L(U_{II}(x, y))$.
And again applying the inverse Laplace transform to the result.
$D^\alpha y=g(y)+?$ My question is whether the following equality is correct?
$L^{-1}(s^{(\alpha-\beta)} L(U_{II}(x, y)))=D^{(\alpha-\beta)}U_{II}(x, y)$.