## 2 Problem Formulation
### Control Problem Formulation
#### DIT-SMC Design Based NDOB
According to the Hypothesis 1, (27), the second term and the third term can be deduced by  
\[ \begin{array}{l}s_{k}^{T}\phi_{ik}-\delta_{ik}k_{eik}\|\ \phi_{ik}\ \|s_{ik}^{T}I_{ik}H\hat{d}_{ik}^{-1}A_{\pi ik}^{-1}\frac{s_{ik}}{\|s_{ik}\|_{ \infty}}\\ \leq\ \|\ \phi_{ik}\ \|\ \|s_{ik}^{T}\|-k_{eik}\|\ \phi_{ik}\ \|\ \|s_{ik}^{T}\|\\ +k_{e}\|\ \phi_{ik}\ \|\ \|(A_{\pi ik}-J_{ik})A_{\pi ik}^{-1}\ \|\ \|s_{ik}\ \|\\ \leq\ \|\ \phi_{ik}\ \|\ \|s_{ik}\ \|-k_{eik}(1-\chi_{ik})\|\ \phi_{ik}\ \|\ \|s_{ik}\ \|\\ \leq 0\end{array} . \tag{33}\]  
\(\square\)  
In (33), the condition \(k_{eik}>1/(1-\chi_{ik})\) can be picked up so that the \(\|\ \phi_{ik}\ \|\ \|s_{ik}\ \|-k_{eik}(1-\chi_{ik})\|\ \phi_{ik}\ \|\ \|s_{ik}\ \|\leq 0\) holds true. The first term of \(\dot{V}_{ik}\) has the following form of  
\[ \begin{array}{l}\frac{s_{k}^{T}}{\|s_{ik}\|_{2}}diag\Big{[}\frac{p_{11}}{q_{1 1}},\frac{p_{12}}{q_{12}}\Big{]}I_{ik}\Big{\{}Z(x_{ik},d_{ik})-\hat{d}_{ik}-k_ {ik}\text{sgn}(s_{ik})-\eta_{ik}s_{ik}\Big{\}}\\ \leq\min\Big{\{}diag\Big{[}\frac{p_{11}}{q_{11}},\frac{p_{12}}{q_{12}}\Big{]} I_{ik}\Big{\}}\left(-k_{k}\frac{\|s_{ik}\|_{1}}{\|s_{ik}\|_{2}}-\eta_{ik} \frac{\|s_{ik}\|_{2}^{2}}{\|s_{ik}\|_{2}}+\frac{s^{T}}{\|s_{ik}\|_{2}}\Big{[} \begin{array}{l}\|\ e_{dik}\ \|_{\infty}\\ \|\ e_{dik}\ \|_{\infty}\end{array}\right]\Big{)}\\ \leq\min\Big{\{}diag\Big{[}\frac{p_{11}}{q_{11}},\frac{p_{12}}{q_{12}}\Big{]} I_{ik}\Big{\}}\left(-k_{k}\frac{\|s_{ik}\|_{1}}{\|s_{ik}\|_{2}}-\eta_{ik} \frac{\|s_{ik}\|_{2}^{2}}{\|s_{ik}\|_{2}}+\frac{\|s_{ik}\|_{1}}{\|s_{ik}\|_{2}} \|\ e_{dik}\ \|_{\infty}\right)\\ =\min\Big{\{}diag\Big{[}\frac{p_{11}}{q_{11}},\frac{p_{12}}{q_{12}}\Big{]}I_{ ik}\Big{\}}\left(-(k_{ik}-\|\ e_{dik}\ \|_{\infty})\frac{\|s_{ik}\|_{1}}{\|s_{ik}\|_{2}}-\eta_{ik} \frac{\|s_{ik}\|_{2}^{2}}{\|s_{ik}\|_{2}}\right)\end{array} \right.\end{array}. \tag{34}\]  
\(\eta_{ik}>0\), \(k_{ik}>\|\ e_{dik}\ \|_{\infty}\) can be selected in the control design in order to ensure \(\frac{s_{k}^{T}}{\|s_{ik}\|_{2}}diag\Big{[}\frac{p_{11}}{q_{11}},\frac{p_{12} }{q_{12}}\Big{]}I_{ik}\left\{ \begin{array}{l}Z(x_{ik},d_{ik})-\hat{d}_{ik}\\ -k_{ik}\text{sgn}(s_{ik})\\ -\eta_{ik}s_{ik}\end{array} \right\}\leq 0\) is held true. \(\dot{V}_{ik}<0\) can be picked up by deducing from (32)-(34). That illustrates the control law can asymptotically stabilize the leader-follower formation system by the derivative and integral terminal sliding mode. Therefore, the follower robots can trace the leader robot with the desired distance and angle steadily. The characteristics of DIT-SMC method are as follows: (1) the convergence time \(T_{k}\) can be adjusted by the parameters of the control law; (2) the formation system starts on the derivative and integral terminal sliding surface; (3) the singular problem of T-SMC is avoided; (4) the derivative term can weaken the nonlinear effect. In (31), the parameter must be assigned as a conservative value to guarantee the formation system stability. From (30), \(\textbf{e}_{dik}\) can be exponentially convergent to \(\textbf{0}_{2\times 1}\) by selecting \(\textbf{L}_{ik}\), meaning that \(\kappa_{ik}\) can be very small. Even if \(\kappa_{ik}\) is assigned from a conservative perspective, its value may not be very large. That illustrates the DIT-SMC based NDOB control law protects the formation from the high switching frequency problem, and can substantially alleviate the chattering problem.