## 2 Adaptive Time-Delayed Control Schemes
### The Proposed Adaptive Time-Delayed Control Scheme  
In this section, we propose an ATDC with a new adaptive law that is based on the sliding-mode [26]. The sliding variable \(s_{t}\) used in the sliding-mode yields  
\[s_{t}=\dot{e}_{t}+K_{s}e_{t} \tag{8}\]  
where \(K_{s}\) is the sliding gain. As \(K_{s}\) is a positive design parameter, \(s_{t}\) serves to improve the speed of reaching equilibrium point when \(K_{s}\) increases. By using the sliding variable \(s_{t}\), we construct the proposed ATDC based on the sliding-mode (ATDC-SM) as below:  
\[\tau_{t}^{B}=-\bar{M}_{t}\dot{q}_{t-L}+\tau_{t-L}^{B}+\bar{M}_{t}\big{(}\dot{q }_{d,t}+K_{s}\dot{e}_{t}+\beta s_{t}+K\text{sgn}(s_{t})\big{)} \tag{9}\]where \(\tau_{t}^{B}\) is a control input of the proposed ATDC-SM. The time-varying control gain \(\hat{M}_{t}\) of the proposed ATDC-SM is defined as  
\[\hat{M}_{t}=\hat{M}(1+a^{B}\omega_{t}) \tag{10}\]  
where \(a^{B}\) is a positive value for adjusting the adaptation speed of the control gain. The derivative of the update parameter \(\omega_{t}\) is defined as  
\[\dot{\omega}_{t}=\left\{ \begin{array}{lcl}-\bar{a}^{B}|s_{t}|&\quad\text{if} &\omega_{t}\geq\omega^{*}\\ \bar{a}^{B}|s_{t}|^{\text{sgn}(\theta_{t})}\text{sgn}(\theta_{t})&\quad\text{ if}&0<\omega_{t}<\omega^{*}\\ \bar{a}^{B}|s_{t}|&\quad\text{if}&\omega_{t}=0\end{array} \right. \tag{11}\]  
where \(\bar{a}^{B}\) is a positive design value for adjusting the speed of the update parameter \(\omega_{t}\) and \(\theta_{t}=|s_{t}|-\tilde{\theta}_{t}\). The TVB \(\tilde{\theta}_{t}\) is defined as \(\frac{1}{a|s_{t}|^{5}+\epsilon}\) where \(\alpha\), \(\epsilon\), and \(\zeta\) are tunable positive gains for adjusting \(\theta_{t}\). From these values, it is noted that the update parameter \(\omega_{t}\) is tuned for achieving a suitable control gain \(\hat{M}_{t}\). As shown in Equation (11), the proposed adaptive law \(\dot{\omega}_{t}\) is based on the \(\tilde{\theta}_{t}\), which can adjust the increasing or decreasing points of the update parameter \(\omega_{t}\). These points vary in position around the sliding manifold. In other words, \(\tilde{\theta}_{t}\) can be used to adjust the magnitude of the update parameter \(\omega_{t}\) that increases or decreases appropriately. For \(0<\omega_{t}<\omega^{*}\), the update parameter \(\omega_{t}\) has two adaptation rules according to the \(\text{sgn}(|s_{t}|-\tilde{\theta}_{t})\): \(|s_{t}|\geq\tilde{\theta}_{t}\) and \(|s_{t}|<\tilde{\theta}_{t}\). * When \(|s_{t}|\geq\tilde{\theta}_{t}\), the update parameter \(\omega_{t}\) increases until \(|s_{t}|<\tilde{\theta}_{t}\). Subsequently, \(\tilde{\theta}_{t}\) is a time-varying parameter, which is inversely proportional to \(|s_{t}|\). This implies that \(\tilde{\theta}_{t}\) becomes smaller when the sliding variable \(s_{t}\) moves away from the sliding manifold. They exhibit an inverse relationship: when the time-varying parameter \(\tilde{\theta}_{t}\) is small, the update parameter \(\omega_{t}\) provides a fast response and will be large consequently. On the other hand, when the sliding variable \(s_{t}\) is in the vicinity of the sliding manifold, a large update parameter may cause undesired side effects, e.g., chattering or fluctuation, owing to excessive control torque. To avoid this problem, \(\tilde{\theta}_{t}\) in Equation (11) is increased to a point where it will quickly create a region in which the update parameter \(\omega_{t}\) decreases. * When \(|s_{t}|<\tilde{\theta}_{t}\), the update parameter \(\omega_{t}\) is decreased to overcome the undesired side effects. However, it is difficult to avoid chattering or fluctuation if the adaptation speed of the update parameter \(\omega_{t}\) is slow. In order to solve this problem, this paper proposes \(|s_{t}|^{\text{sgn}(\theta_{t})}\) which is inversely proportional to the magnitude of the sliding variable \(s_{t}\). It implies that the control gain \(\hat{M}_{t}\) provides fast adaptation speed and hence can reduce chattering and fluctuation.