# Robust Adaptive Synchronization of a Class of Uncertain Chaotic Systems with Unknown Time-Delay
## 2 Preliminaries and Problem Formulation
: _The non-linear functions \(f(x(t-\tau_{1}),t),\ g(y(t-\tau_{2}),t)\in R\) containing the unknown time delays in the master and slave chaotic systems defined in general forms (1) and (2), for each \(x(t),\ y(t)\in R\), provide the following Lipschitz conditions:_  
\[\begin{array}{l}\big{|}f(x(t-\tau_{1}))-f(x(t-\hat{\tau}_{1}))\big{|}\leq l _{1}\|x(t-\tau_{1})-x(t-\hat{\tau}_{1})\|\leq m_{1}\big{|}(t-\tau_{1})-(t-\hat{ \tau}_{1})\big{|}\\ \qquad\qquad\qquad\  
**Proof. ** According to the proposed structure based on (7), \(n-1\) controlled synchronization error dynamics are presented in the following linear form:  
\[\left\{ \begin{array}{c}\dot{e}_{1}=e_{2},\\ \dot{e}_{2}=e_{3},\\ \vdots\\ \dot{e}_{n-2}=e_{n-1},\\ \dot{e}_{n-1}=e_{n}+u_{0}(t).\end{array} \right. \tag{9}\]  
If \(u_{0}(t)\) is designed in the form (8) then the linear time invariant (LTI) dynamic system (9) is converted to the following form:  
\[\left\{ \begin{array}{c}\dot{e}_{1}=e_{2},\\ \dot{e}_{2}=e_{3},\\ \vdots\\ \dot{e}_{n-2}=e_{n-1},\\ \dot{e}_{n-1}=K_{1}e_{1}+K_{2}e_{2}+\ldots+K_{n-1}e_{n-1}.\end{array} \right. \tag{10}\]  
It is clear that according to [PERSON]'s stability theory in LTI systems, if \(K_{i}\) values for \(i~{}=~{}1,~{}2,\ldots,~{}n~{}-~{}1\), are chosen that all eigenvalues \(\lambda_{i}\) of the LTI system (10), satisfy conditions \(Real(\lambda_{i})\leq 0\) then, by designing \(u_{0}(t)\) based on Theorem 1 conditions, all \(n-1\) synchronization error dynamics in the system (7) are stable, and this stability is a necessary condition for robust synchronization of chaotic systems with synchronization error in form (7). \(\Box\)  
**Theorem 2. **_Robust synchronization of the master and slave chaotic systems with uncertainty and unknown time delays described in forms (1) and (2) is fulfilled if the controller \(u_{0}(t)\) characterized in form (8) satisfies the condition of Theorem 1 and the adaptive controller \(u(t)\) is designed as follows:_  
\[u(t)=f(x(t-\hat{\tau}_{1}))-g(y(t-\hat{\tau}_{2}))-sign(e_{n})(\hat{\gamma}_{ 1}+\hat{\gamma}_{2}+\hat{\alpha}_{1}h_{1}(x)+\hat{\alpha}_{2}h_{2}(y))-k_{0}e _{n}, \tag{11}\]  
_so that \(k_{0}\) is a desired positive real value and the adaptation laws in the controller (11) are determined as follows:_  
\[ \begin{array}{c}\dot{\widetilde{\alpha}}_{1}=-|e_{n}|h_{1}(x)-\eta_{1} \widetilde{\alpha}_{1},~{}~{}\dot{\widetilde{\alpha}}_{2}=-|e_{n}|h_{2}(y)- \eta_{2}\widetilde{\alpha}_{2},\\ \dot{\widetilde{\gamma}}_{1}=-|e_{n}|-\eta_{3}\widetilde{\gamma}_{1},~{}~{} \dot{\widetilde{\gamma}}_{2}=-|e_{n}|-\eta_{4}\widetilde{\gamma}_{2},\\ \dot{\widetilde{\tau}}_{1}=-|e_{n}|sign(\overline{\tau}_{1})-\eta_{5} \overline{\tau}_{1},~{}~{}\dot{\widetilde{\tau}}_{2}=-|e_{n}|sign(\overline{ \tau}_{2})-\eta_{6}\overline{\tau}_{2},\end{array}  \tag{12}\]  
_where \(\eta_{i}\) for \(i=1,~{}2,~{}\ldots,~{}6\) are constant and positive values and \(\widetilde{\alpha}_{1},\widetilde{\alpha}_{2},~{}~{}\widetilde{\gamma}_{1},~ {}~{}\widetilde{\gamma}_{3},~{}\overline{\tau}_{1},\overline{\tau}_{2}\) denote the estimation errors of uncertain and unknown parameters in the master and slave systems of (1) and (2)._  
**Proof.