# Research on Tracking Control of Urban Rail Trains Based on Improved Disturbance Observer
## 3 Main Results
### Controller Design and Stability Proof Based on Improved Disturbance Observer  
In order to compensate for the effects of unknown lumped disturbances and partial actuator faults, the control law can have the form as  
\[U(t)=KX(t)-\rho^{-1}\dot{d}_{L}(t) \tag{20}\]  
where \(K\) is the controller gain matrix to be designed. Bringing the control law (20) into the UKTs system (12), combined with the disturbance observer estimation error as (15), yields the augmented system state space expression (21). \[\dot{\xi}(t)=\bar{M}\xi(t)+\bar{N}\varphi(t) \tag{21}\]  
with \(\xi^{T}(t)=[X^{T}(t),e_{d}^{T}(t)]\), \(\varphi(t)=d_{L}(t)+c\text{sgn}(\text{e}_{\text{d}}(t))\), \(\bar{M}=\left[ \begin{array}{cc}A+B_{\rho}K&-B\\ 0_{n\times 2n}&-LB\end{array} \right]\) and \(\bar{N}=\left[ \begin{array}{c}0_{2n\times n}\\ -I_{n}\end{array} \right]\). Define the reference output of the augmentation system (21) as follows:  
\[Z(t)=C_{1}X(t)+C_{2}e_{d}(t)=\bar{C}\xi(t) \tag{22}\]  
where \(\bar{C}=[C_{1},C_{2}]\). The stability of the closed-loop system consisting of the augmented system (21) and the reference output (22) is given by Theorem 2. **Theorem 2**. : _Consider the augmentation system (21) and the reference output (22). For a given positive constant \(\lambda\), if there exist matrices \(H_{1}>0\), \(H_{2}>0\), and \(H_{3}\) satisfying the matrix inequality shown as (23), then there exists a control law (20) based on an improved disturbance observer that ensures the closed-loop system is asymptotically stable and satisfies the \(H_{\infty}\) performance index \(J=\int_{0}^{+}\big{(}Z^{T}(t)Z(t)-\lambda\varphi^{T}(t)\varphi(t)\big{)}dt\)._  
\[\left[ \begin{array}{cccc}\Gamma_{1}+\Gamma_{2}&-B&0&0&H_{1}C_{1}^{T}\\ *&\Gamma_{3}&-H_{2}&0&C_{2}^{T}\\ *&*&-\lambda I&0&0\\ *&*&*&-I&0\\ *&*&*&*&-I\end{array} \right]<0 \tag{23}\]  
where \(\Gamma_{1}=Sym(AH_{1})\), \(\Gamma_{2}=Sym(B_{\rho}H_{3})\), and \(\Gamma_{3}=Sym(-H_{2}LB)\). Proof of Theorem 2.: Construct the candidate Lyapunov function \(V_{b}\). \[V_{b}(t)=X^{T}(t)P_{1}X(t)+e_{d}^{T}(t)P_{2}e_{d}(t) \tag{24}\]  
with \(P_{1}>0\), \(P_{2}>0\). The derivative of the above Lyapunov function can be obtained by collation as (25). \[\left[ \begin{array}{cc}\dot{V}_{b}(t)&=X^{T}(t)\Big{(}P_{1}(A+B_{\rho}K)+(A+B_{ \rho}K)^{T}P_{1}\Big{)}X(t)\\ &\quad-Sym\big{(}X^{T}(t)P_{1}B_{\sigma}(t)+e_{d}^{T}(t)P_{2}\varphi(t)\big{)} \\ &\quad-e_{d}^{T}(t)\Big{(}P_{2}LB+(LB)^{T}P_{2}\Big{)}e_{d}(t)\end{array} \right. \tag{25}\]  
According to (25) and the \(H_{\infty}\) performance index, one has  
\[J=\int_{0}^{t}\Big{(}Z^{T}(t)Z(t)-\lambda\varphi^{T}(t)\varphi(t)+\dot{V}_{b}( t)\Big{)}dt-V_{b}(t) \tag{26}\]  
(26) could be further written as  
\[J=\int_{0}^{\tau}\psi^{T}(t)\Xi\psi(t)dt-V_{b}(t) \tag{27}\]  
where \(\psi(t)=[ \begin{array}{cc}X^{T}(t)&e_{d}^{T}(t)&\varphi^{T}(t)\end{array} ]^ {T}\),  
\[\Xi=\left[ \begin{array}{cc}Sym\big{(}P_{1}(A+B_{\rho}K)\big{)}+C_{1}^{T}C_{ 1}&-P_{1}B+C_{1}^{T}C_{2}&0\\ *&-Sym(P_{2}LB)+C_{2}^{T}C_{2}&-P_{2}\\ *&*&-\lambda I\end{array} \right]\). Based on the non-negativity of the Lyapunov function (24), for \(J<0\) to hold, \(\Xi<0\) must satisfy. Applying the Schur complement and congruence transformations to the matrix inequality and defining \(H_{1}=P_{1}^{-1}\), \(H_{2}=P_{2}\), and \(H_{3}=KP_{1}^{-1}\) yields \(\Xi<0\), which in turn yields \(J<0\). That is, with zero initial condition and any \(d_{L}(t)\
eq 0\), \(\left\|Z(t)\right\|_{2}\leq\lambda\left\|d_{L}(t)\right\|_{2}\) is always satisfied when \(\tau\rightarrow\infty\). **Remark 5**. : _The URTs system estimates and attenuates the unknown lumped disturbances online under the IDOBC strategy. The baseline feedback controller approach near nominal control performance in the absence of disturbances. If subjected to external disturbances, the disturbance observer is activated for disturbance compensation and attenuation, providing robustness to disturbances without involving too much control energy._  
**Remark 6**.