## 2 Problem Formulation
### Control Problem Formulation
#### DIT-SMC Design Based NDOB  
In order to resolve the above problem, the nonlinear disturbance is proposed to estimate the uncertainties \(d_{ik}\) in the leader-follower formation system (13). At first, the following assumption is taken into account. **Hypothesis 3**. : _The unmatched uncertainties possess a slow change rate, meaning that \(\dot{d}_{ik}\approx 0_{2\times 1}\), where \(0_{2\times 1}=\left[ \begin{array}{cc}0&0\end{array} \right]^{T}\)._  
Considering the formation dynamic model (6), the nonlinear disturbance observer is formulated by  
\[\left\{ \begin{array}{c}\dot{z}_{ik}=-L_{ik}B_{ik,2}p_{ik}-L_{ik}(B_{ik,2}L_{ ik}x_{ik}+A_{ik}x_{ik}+B_{ik,1}(1+\Delta_{k})u_{k})\\ \dot{d}_{ik}=z_{ik}+L_{ik}x_{ik}\end{array} \right.. \tag{29}\]  
Here \(z_{ik}\in R^{2\times 1}\), \(L_{ik}\in R^{2\times 1}\), \(\hat{d}_{ik}\in R^{2\times 1}\) are the state vector of nonlinear disturbance observer, the observer gain matrix set by designer, and the estimated value of unmatched uncertainties respectively. Define the estimate error vector as  
\[e_{dik}=d_{ik}-\hat{d}_{ik}.\]  
Differentiate \(e_{dik}\) with respect to time \(t\) and take the Hypothesis 3 into account. Furthermore, the dynamics of \(e_{dik}\) is presented as  
\[ \begin{array}{l}\dot{e}_{dik}=\dot{d}_{ik}-\dot{d}_{ik}\\ =-\dot{p}_{ik}-L_{ik}\dot{x}_{ik}\\ =L_{ik}B_{ik,2}p_{ik}+L_{ik}(B_{ik,2}L_{ik}x_{ik}+A_{ik}x_{ik}+B_{ik,1}(I+ \Delta_{k})u_{k})\\ -L_{ik}(A_{ik}x_{ik}+B_{ik,1}(I+\Delta_{k})u_{k}+B_{ik,2}d_{ik})\\ =L_{ik}B_{ik,2}\Big{(}\hat{d}_{ik}-L_{ik}x_{ik}\Big{)}+L_{ik}(B_{ik,2}L_{ik}x_ {ik}+A_{ik}x_{ik}+B_{ik,1}(I+\Delta_{k})u_{k})\\ -L_{ik}(B_{ik,2}d_{ik}+A_{ik}x_{ik}+B_{ik,1}(I+\Delta_{k})u_{k})\\ =L_{ik}B_{ik,2}\Big{(}\hat{d}_{ik}-d_{ik}\Big{)}=-L_{ik}B_{ik,2}e_{dik}\end{array} . \tag{30}\]  
The solution of (30) is \(e_{dik}=\exp(-L_{ik}B_{ik,2}t)e_{dik}(0)\), which indicates the estimate error will exponentially converge to zero as \(t\rightarrow\infty\) if \(L_{ik}B_{ik,2}\) is set as a positive constant. Here, \(e_{dik}(0)\) is the initial state of \(e_{dik}\). Considering input-output dynamics (13) and observer (29), the control law based on NDOB is determined by  
\[ \begin{array}{l}u=\hat{H}_{dik}^{-1}\Big{[}\hat{y}_{ik}^{d}-k_{ik}\|\ \phi_{ik}\ \|diag\Big{[}\frac{q_{11}}{p_{11}},\frac{q_{12}}{p_{12}}\Big{]}A_{\pi k}^{-1} \frac{s_{k}}{\|s_{ik}\|_{1}}\Big{]}\\ -H_{ik}^{-1}\Big{(}\hat{d}_{ik}(t)-k_{ik}sign(s_{ik})-\eta_{ik}s_{ik}\Big{)} \end{array} . \tag{31}\]  
**Theorem 2**. : _Consider the dynamic model of leader-follower formation system (6), take the assumption 1, 2, 3, 4 into account, adopt the input-output model (13), design the derivative and integral terminal sliding mode surface (17) and nonlinear disturbance observer (29). If the derivative and integral terminal control law is set as (31), the leader-follower formation system with unmatched uncertainties is asymptotically stable when \(k_{ik}>e_{d}^{*}\)._  
**Proof. ** Selecting the Lyapunov function as \(V_{ik}=\frac{1}{2}\)s\(\dot{s}\), differentiating \(V_{ik}\) with the respect to time \(t\) and substituting the \(\dot{s}_{ik}\) into the derivative of \(V_{ik}\) yields  
\[\left. \begin{array}{l}\dot{V}_{ik}=\frac{s_{k}^{T}}{\|s_{ik}\|_{2}}diag \Big{[}\frac{p_{11}}{q_{11}},\frac{p_{12}}{q_{12}}\Big{]}I_{ik}\left\{\begin{array} []{l}d_{ik}-\hat{d}_{ik}-\eta_{ik}s_{ik}\\ -k_{ik}\text{sgn}(s_{ik})\end{array}\right\}\\ +\frac{s_{k}^{T}}{\|s_{ik}\|_{2}}\phi_{ik}-\delta_{ik}k_{eik}\|\ \phi_{ik}\ \|s_{ ik}^{T}H\hat{d}_{ik}^{-1}I_{ik}A_{\pi ik}^{-1}\frac{s_{ik}}{\|s_{ik}\|_{1}\|s_{ik}\|_{2}} \end{array} \right. \tag{32}\]  
Here, \(J_{ik}=diag\Big{[}\dot{e}_{D01}^{(P_{11}/q_{11}-1)},\dot{e}_{D02}^{(P_{12}/q_{ 12}-1)}\Big{]}\). Since \(p_{1K}>0\), \(q_{1K}>0\), \(p_{2K}>0\), \(q_{2K}>0\), \(p_{1K}>q_{1K}\), \(p_{2K}>q_{2K}\) exist in the controller, \(\dot{e}_{D01}^{(P_{11}/q_{11}-1)}\) and \(\dot{e}_{D02}^{(P_{12}/q_{12}-1)}\) hold true for all \(\dot{e}_{D01}^{(P_{11}/q_{11}-1)}\
eq 0\), \(\dot{e}_{D02}^{(P_{12}/q_{12}-1)}\
eq 0\).