# Research on Real-Time Monitoring and Performance Optimization of Suspension System in Maglev Train
## 5 Control Performance Optimization Architecture  
### Quadratic Performance Index  
Although the nominal controller, based on the simplified model, can realize the stable suspension of the suspension system, when the suspension system is medium degraded, the nominal controller alone cannot realize the performance optimization. According to the Youla, parameterized realization form of the controller based on the residual generator, \(Q_{c}(z)\) can be regarded as a dynamic compensator based on the residual signals. When \(Q_{c}(z)\) itself is stable, the robustness of the system can be further improved, and the performance optimization of the system can be achieved [39; 40]. The design of the dynamic compensator \(Q_{c}(z)\) and the iterative implementation based on the gradient descent are given below.  
Considering the input standard type parameter [18] of the dynamic compensator \(Q_{c}(z)\) as shown in (13):  
\[\theta_{r}=\left[\begin{array}{c}\theta_{AB,r}\\ \theta_{C,r}\\ \theta_{D,r}\end{array}\right] \tag{13}\]  
Among them, \(\theta_{AB,r}\) is the parameterization of \(A_{r}\) and \(B_{r}\); \(\theta_{C,r}\) and \(\theta_{D,r}\) are the parameterization of \(C_{r}\) and \(D_{r}\), respectively.  
The quadratic performance indicator is defined as [32]:  
\[J=\frac{1}{2N_{r}}\sum_{k=k_{0}}^{k_{0}+N_{r}-1}\left[e(k)^{T}W_{e}(k)e(k)+u( k)^{T}W_{u}(k)u(k)\right] \tag{14}\]  
\[\frac{\partial x_{0}(k+1)}{\partial\theta_{AB,r}(i)}=\Big{(}\mathcal{A}- \mathcal{B}\mathcal{D}D_{c}g^{-1}C_{o}\Big{)}\frac{\partial x_{o}(k)}{\partial \theta_{AB,r}(i)}+\mathcal{B}\mathcal{D}C_{c}\frac{\partial x_{c}(k)}{\partial \theta_{AB,r}(i)}+\mathcal{B}\mathcal{D}C_{r}\frac{\partial x_{r}(k)}{\partial \theta_{AB,r}(i)} \tag{23}\] \[\frac{\partial x_{r}(k+1)}{\partial\theta_{AB,r}(i)}=\frac{ \partial A_{r}}{\partial\theta_{AB,r}(i)}x_{r}(k)+A_{r}\frac{\partial x_{r}(k) }{\partial\theta_{AB,r}(i)}+\frac{\partial B_{r}}{\partial\theta_{AB,r}(i)}r(k)\] (24) \[\frac{\partial u(k)}{\partial\theta_{AB,r}(i)}=\mathcal{D}C_{c} \frac{\partial x_{c}(k)}{\partial\theta_{AB,r}(i)}-\mathcal{D}D_{c}g^{-1}C_{ o}\frac{\partial x_{o}(k)}{\partial\theta_{AB,r}(i)}+\mathcal{D}C_{r}\frac{ \partial x_{r}(k)}{\partial\theta_{AB,r}(i)}\] (25) \[\frac{\partial e(k)}{\partial\theta_{AB,r}(i)}=-D_{o}\mathcal{D}C _{c}\frac{\partial x_{c}(k)}{\partial\theta_{AB,r}(i)}+\Big{(}D_{o}\mathcal{D }D_{c}g^{-1}-I\Big{)}C_{o}\frac{\partial x_{o}(k)}{\partial\theta_{AB,r}(i)}- D_{o}\mathcal{D}C_{r}\frac{\partial x_{r}(k)}{\partial\theta_{AB,r}(i)}\] (26) \[\theta_{AB,r}(i+1)=\theta_{AB,r}(i)-\frac{\lambda}{N}\sum_{k=k_{ 0}}^{k_{0}+N-1}\Big{(}e(k)^{T}W_{e}(k)\frac{\partial e(k)}{\partial\theta_{AB, r}(i)}+u(k)^{T}W_{u}(k)\frac{\partial u(k)}{\partial\theta_{AB,r}(i)}\Big{)} \tag{27}\]  
Through (22)-(26), \(\frac{\partial u(k)}{\partial\theta_{AB,r}(i)}\) and \(\frac{\partial e(k)}{\partial\theta_{AB,r}(i)}\) can be obtained, then parameters \(\theta_{AB,r}\) can be updated online iteratively according to (27).  
The update equation of \(\theta_{C,r}\) and \(\theta_{D,r}\) can be obtained in the same way. Through the above equation, \(\theta_{r}\) is continuously updated iteratively during the operation of the system. When the performance index \(J\) meets a certain condition or reaches the maximum number of iterations, the online update stops and outputs the dynamic compensator \(Q_{c}(z)\).