International Journal of Advanced Robotic Systems Predictive Function Control for Communication-based Train Control (cbtc) Systems Regular Paper

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In Communication‐Based Train Control (CBTC) systems, random transmission delays and packet drops are inevitable in the wireless networks, which could result in unnecessary traction, brakes or even emergency brakes of trains, losses of line capacity and passenger dissatisfaction. This paper applies predictive function control technology with a mixed  H H / 2 control approach to improve the control performances. The controller is in the state feedback form and satisfies the requirement of quadratic input and state constraints. A linear matrix inequality (LMI) approach is developed to solve the control problem. The proposed method attenuates disturbances by incorporating  H H / 2 into the control scheme. The control command from the automatic train operation (ATO) is included in the reward function to optimize the train's running profile. The influence of transmission delays and packet drops is alleviated through improving the performances of the controller. Simulation results show that the method is effective to improve the performances and robustness of CBTC systems.

The method of combining the 2 H norm with the  H norm can be a desirable measure to improve the robust performance of the system with uncertainties in its model [24]. Thus, the mixed  H H / 2 performance criterion provides an interesting measure to evaluate controllers. The theoretic motivation for the mixed  H H / 2 control method has been extensively discussed in [25][26][27][28]. Some important results about output feedback control can be found in [29][30][31].
Introduced by Richalet and ADERSA, predictive function control (PFC) is efficiently applied in industrial systems, especially in fast systems [32]. Generalized predicative control (GPC) [33], dynamic matrix control (DMC) [34], PFC [35,36] and model predictive heuristic control (MPHC) [37] follow the same principles, but differ in the model structure and computational complexity. However, DMC and GPC need high computational complexity due to matrix computations and inversions. The control command of PFC does not minimize a cost function. PFC avoids high dimensional matrix computations and inversions, which can reduce the computational time [38], and this is especially important for a fast dynamic system. The robustness in PFC controllers is also important for time varying dynamical systems such as CBTC systems.
A CBTC system is an automatic train control system using bidirectional train-ground communications to transmit train status and control commands to ensure the safe operation of trains. The train-ground data should be transferred accurately, reliably and in real time. However, transmission delays and packet drops are inevitable in train-ground transmission which could result in unnecessary traction, brakes or even emergency brakes of trains, loss of line capacity and passenger dissatisfaction. So we need to develop control algorithms to improve the system performances under transmission delays and packet drops. In this paper, we propose a PFC scheme based on a mixed . We derive sufficient conditions, in the form of LMIs, for the existence of control law to achieve the design specifications. In Section III we present an example of CBTC systems to illustrate our algorithm. Finally, we conclude this study in Section IV.

System Description
The train control model in CBTC systems is presented in Fig.1 is the extra resistance acting on the train, is the controlled output and 1 , , , A B C C and 12 D are matrices of appropriate dimensions based on system dynamics and physical constraints.

as a CBTC Controller
Kuntze and Richalet proposed the PFC in [32]. 1) Referenced profile: the referenced profile of PFC is given by where is the expected output, is the current measured output, is sample time and is the desired closed-loop system response time constant.
2) Base Function: the future control variable of PFC is associated with specific base functions which are set according to the process nature and set points, namely the linear combination of the base functions. The base function may be a step function, ramp function, exponential function, etc. The base function makes the controller's output more regular and improves the rapidity of the system response. Both the base function and its sampled value can be computed offline. A little linear weighting coefficient is optimized to reduce the computational complexity. The control variable of PFC is The parameters of the PFC controller can influence the performance of the system.
3) Feedback correction: in order to overcome the imprecise of the model, we introduce feedback correction. Feedback correction of the predicted output is based on the error between the actual y(k) and the predicted output ˆ( ) p y k at time , the predictive output at time is: where is the ratio for error modification.
If disturbance rejection exists, then where is the transfer matrix from w to z . For given and , the quadratic input constraints are satisfied.
, ; for , For given and , the quadratic state constraints are satisfied.
, , for Adding (2) and (5) , we can get We assume that the disturbance is bounded with where . (4) and (2) gives (10) Consider a quadratic function , of the state of (2). Then Based on (2) and (4), it has Consider the sum of (11) from to , to get (13) Assuming that Adding (9) and (13) gives where is defined in (12). An application of the bounded real lemma [30] shows that is stable and (6)   Condition (14) might be considered as the normalized mixed objective function. This is a more natural combination of the control objectives since it emphasizes the trade-off between the normalized cost and the normalized bound.

Simulations and Discussions
In our simulation, we consider the discrete-time model of a CBTC given by (2) . We select a low disturbance rejection level by setting =2.5. MATLAB is used to simulate the CBTC system, while the PFC based on a mixed 2 / H H  process is implemented. Fig. 2-Fig. 7 show the performances comparison of different prediction lengths and the effect of disturbance. Fig. 2 and Fig. 4 show the velocity tracking performance of the proposed method under different prediction lengths. As we can see from Fig. 2, the tracking speed profile almost coincides with the ATO's speed profile, which means an improved CBTC performance. The prediction length influences the real-time performance of the system. Fig. 4 shows that the system can track the ATO's speed profile after a long time when the prediction length becomes large. In addition, we can also see there is some overshoot under large prediction length which may lead to the train exceeding the protected speed and affecting safety. A properly selected prediction length can ensure that the proposed method provides better realtime and safety performances. The train position tracking performance under different prediction lengths are given in Fig. 3 and Fig. 5, respectively. As it is shown, with an appropriate prediction length, the proposed method can always track the ATO's position profile accurately to meet the requirement of accurate train parking in a station. But as the prediction length increases, the position tracking performance decreases which may affect the accuracy of train parking. In conclusion, the choice of prediction length in the proposed method is important and the appropriate prediction length can optimize the performances of CBTC. For the disturbance consideration, we introduce a disturbance in the form of =2 when the sampling time is between 90s and 100s. Fig. 6 and Fig. 7 give the results of the velocity tracking curve of the train with disturbance under different prediction lengths. The simulation results show that the system can reach stability quickly under the disturbance, meaning that the proposed method has good robustness.

Conclusion and Future Work
In CBTC systems, it is very important to improve the controller of ATC systems to mitigate the impact of transmission delays and packet drops. In this paper, we proposed a novel PFC based on a mixed 2 / H H  control approach for CBTC systems. Different prediction lengths were studied to optimize the performance of the system. The proposed control method showed the capability of achieving a more complicated CBTC control and good balance between different or even conflicting performance requirements. A challenging future work is to design and implement an adaptive PFC based on a mixed 2 / H H  control to increase the dynamic range of the controller.