# Research on Tracking Control of Urban Rail Trains Based on Improved Disturbance Observer
## 3 Main Results
### The Improved Disturbance Observer Design  
The nonlinear disturbance observer proposed in [20; 27] is simple in structure, has few computations, and can obtain highly accurate estimates. Therefore, for the URTs system (12), the following improved state-space disturbance observer is proposed to estimate the complex unknown lumped disturbances. \[\left\{ \begin{array}{rl}\hat{d}_{L}(t)&=P(t)+LX(t)\\ \hat{P}(t)&=-LBP(t)-LBLX(t)\\ &-L\big{(}AX(t)+B_{\rho}U(t)\big{)}-c\text{sgn}(\text{e}_{\text{d}}(t))\end{array} \right. \tag{13}\]  
Figure 3: Block diagram of the URTs under IDOBC. where \(\hat{d}_{L}(t)\) is the estimate of the lumped disturbance \(d_{L}(t)\). \(L=[L_{A},L_{B}]\) is the gain matrix of the observer to be designed. \(c\text{sgn}(\text{e}_{\text{d}}(\text{t}))\) is the observer error compensation term and \(c\) is a constant. \(P(t)\) is the state variable inside the observer. Here, some mild assumptions are given to prove the stability of the disturbance observer system (13). **Assumption 1**. : \(\left(A,B_{\rho}\right)\) _is controllable._  
**Assumption 2**. : _The lumped disturbance \(d_{L}(t)\) to the URTs is not a slow time-varying disturbance, i.e., it satisfies \(\left\|\hat{d}_{L}(t)\right\|\leq\varepsilon\), where \(\varepsilon>0\) is an unknown constant._  
The convergence of the improved disturbance observer is given by Theorem 1 below. **Theorem 1**. : _The URTs system (12) is assumed to satisfy Assumptions 1 and 2. When the observer gain matrix \(L\) is chosen such that \(-L_{B}\) is Hurwitz, the disturbance estimate \(\hat{d}_{L}(t)\) obtained from the observer (13) could track the unknown lumped disturbance \(d_{L}(t)\) asymptotically._  
Proof of Theorem 1.: Define interference observer estimation error  
\[e_{d}(t)=\hat{d}_{L}(t)-d_{L}(t) \tag{14}\]  
Combining the URTs system (12), the disturbance observer (13), and the estimation error (14), one has  
\[\left\{ \begin{array}{rl}\dot{e}_{d}(t)&=\ddot{\hat{d}}(t)-\dot{d}_{L}(t)\\ &=\dot{P}(t)+L\ddot{X}(t)-\dot{d}_{L}(t)\\ &=-LB\Big{(}\hat{d}_{L}(t)-d_{L}(t)\Big{)}-\dot{d}_{L}(t)-c\text{sgn}(\text{e} _{\text{d}}(\text{t}))\\ &=-LB\dot{e}_{d}(t)-\dot{d}_{L}(t)-c\text{sgn}(\text{e}_{\text{d}}(\text{t})) \end{array} \right. \tag{15}\]  
Since \(-L_{B}\) is Hurwitz, according to the form of the matrix \(B\), then \(-LB\) is also Hurwitz. Then it means that there exist positive symmetric matrices \(\Phi\) and \(Q\) satisfying the following form  
\[(-LB)^{T}\Phi+\Phi(-LB)^{T}=-Q \tag{16}\]  
For (16), define the Lyapunov function \(V_{a}(t)\)  
\[V_{a}(t)=e_{d}^{T}(t)\Phi e_{d}(t) \tag{17}\]  
According to the state trajectory of (15), the derivative of \(V_{a}\) is obtained  
\[\left\{ \begin{array}{rl}\dot{V}_{a}(t)&=\dot{e}_{d}^{T}(t)\Phi e_{d}(t)+e_{d }^{T}(t)\Phi\dot{e}_{d}(t)\\ &=\left(-LB\dot{e}_{d}(t)-\dot{d}_{L}(t)-c\text{sgn}(\text{e}_{\text{d}}(\text {t}))\right)^{T}\Phi e_{d}(t)\\ &+e_{d}^{T}(t)\Phi\big{(}-LB\dot{e}_{d}(t)-d_{L}(t)-c\text{sgn}(\text{e}_{\text {d}}(\text{t}))\big{)}\\ &=-e_{d}^{T}(t)Qe_{d}(t)-2e_{d}^{T}(t)\Phi\big{(}\dot{d}_{L}(t)+c\text{sgn}( \text{e}_{\text{d}}(\text{t}))\big{)}\end{array} \right. \tag{18}\]  
From Assumption 2, the derivative of the \(\dot{V}_{a}(t)\) satisfies  
\[\left\{ \begin{array}{rl}\dot{V}_{a}(t)&\leq-\sigma_{\min}(Q)\|e_{d}(t)\|^{2 }+2(\varepsilon+|c|)\sigma_{\max}(\Phi)\|e_{d}(t)\|\\ &\leq-(1-\theta)\sigma_{\min}(Q)\|e_{d}(t)\|^{2},\forall\|e_{d}(t)\|\geq\frac{2( \varepsilon+|c|)\sigma_{\max}(\Phi)}{\delta\sigma_{\min}(Q)}\end{array} \right. \tag{19}\]  
where \(0<\vartheta<1\). It shows that when \(\varepsilon+|c|\) is used as the system input and \(e_{d}(t)\) is used as the state, the estimation error of the observer is input-to-state stable [28]. **Remark 3**. : _The disturbance observer operates with only the internal state variable \(P\) and does not need to calculate the differentiation of the system state variable \(X(t)\), which has the advantage of not causing noise amplification and is easy to implement in the practical UKTs system._  
**Remark 4**.