# Optimal scheduling on trains equipped with different braking systems

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Scheduling

An open loop controller is used to calculate the inputs when a train is running in its steady state with the reference velocity and acceleration maintained.
In [11], the off-line schedule for the throttling and braking inputs is chosen in such a way that the train is in its steady state with the reference velocity maintained. The settings do not contribute to additional accelerations/decelerations of the train. The schedule determines the sequencing and the amplitudes of the inputs in case there are continuous input variations and no power limits. The applied inputs ou(t) are nonlinear functions of the schedule parameters p (grade of the track, velocity profile and train data) and the travelled distance z of the train: ou = fu(z, p). The inputs are approximated by step functions of variable amplitudes. Such an optimal problem can be solved with MISER developed Toe in [74]. The sequence of the steps is predetermined and tuned, and the time instants of the step functions at which the steps are applied are decided on line. It is obvious that this off-line schedule is heuristic and subject to the pre-determined control sequence, so it will not be discussed further in this study.

Simulation of heuristic scheduling vs. optimal schedul-ing

There is only one open loop operational strategy for heuristic scheduling, as shown in Fig. 3.7. Figs. 3.8, 3.9 and 3.10 are the optimal scheduling of 1-1 strategy, 2-1 strategy and 2-2 strategy respectively, with Ke = 1, Kf = 1.
In these figures, the first subplot is the front locomotive group speed, rear locomotive group speed and the mean speed of all the cars with respect to the distance from the starting point. The second subplot is maximum and minimum in-train forces and the mean value of the absolute values of all the in-train forces in a specific time with respect to the distance.The third is the steady in-train forces, which are calculated by applying the efforts of the cars to the train model with the reference speed (and the acceleration) maintained and the dynamic process ignored. As can be seen there are dips in the third subplots of these figures when the reference speed changes. This is because the steadystate in-train forces in the third subplots are the calculation results of the algebraic equations. When the reference speed has a step-type change in the algebraic equations, the other variables, such as in-train forces, unavoidably have step-type changes, which results in the dips.

Anti-windup technique

Within a closed-loop controller in this thesis, open loop scheduling is used to calculate the inputs when a train is running in its steady state with the reference velocity maintained and the input constraints are not considered in open loop scheduling. Since the throttle of the locomotives takes discrete values and the braking capacities of the wagons are constrained, when the control inputs u of a closed-loop controller are applied to the train, an anti-windup technique is employed. For the wagons, the application of the anti-windup technique is very simple. For the locomotives, the inputs are discrete with some operation constraints. Similar methods as described in [29] are used to smooth continuous control inputs.

Output regulation with measured output feed-back

The output regulation problem in linear systems has been studied in [30, 31, 32]. The internal model principle is proposed in [30], enabling the conversion of output regulation problems into stabilization problems. The details on the solvability of the problem can be found in [31, 32].
The internal model principle is extended to nonlinear systems in [33], which shows that the error-driven controller of the output tracking necessarily incorporates the internal model of the exosystem. The conditions of the existence of regulators for nonlinear systems are detailed for different kinds of exosystems with bounded signals in [34, 35, 36]. The necessary and sufficient conditions are given in [37] for the local output regulation problem of nonlinear systems, which is the solvability of regulator equations. With an assumption added to the conditions in [37], the results in [37] have been improved in [38]. A differential vector space approach is used in [39] to develop solutions of state feedback for nonlinear systems with both bounded and unbounded exogenous signals. An approach for robust local output regulation problems is presented in [40] in a geometric insight. In [41], an output regulation problem of a class of singleinput single-output (SISO) nonlinear systems is reformulated into an output feedback stabilization problem.
The robust version of output regulation problem of nonlinear systems with uncertain parameters is studied in [42]. Furthermore, the output regulation problem of nonlinear systems driven by linear, neutrally stable exosystems with uncertain parameters is presented in [72], in terms of the parallel connection of a robust stabilizer and an internal model, which has recently been in [73]. Recently, the concept of the steadystate generator has been advanced in [43] as well as that of the internal model candidate. Based on these dynamic systems, a framework for global output regulation of nonlinear systems with autonomous exosystems is proposed in [43] for bounded signals, in [44] for unbounded signals, and in [47] for nonlinear exosystems. The frameworks are in the form of output feedback or plus (partial) state feedback. All the controller design approaches in these papers cannot be extended directly to the form of measurement (measured output) feedback. Measurement feedback is considered in this chapter instead of the output feedback, because generally the measurable output is different from the output to regulate. For example in this study, in the handling of heavy-haul trains, the outputs to be regulated are all the cars’ speeds; however, only part of the speeds (for example, the first and last locomotives’ speeds) can be practically measured. On the other hand, the measurable output covers the form of the output or output plus (partial) state, as considered in [44]. In the above papers, an important idea is to design an internal model to eliminate the effect of the unknown states of the exosystem. In the controllers of [37] for the local version of output regulation problem, the internal model is given together with the stabilizer directly. In the global version, it is proposed in [43] and [44] firstly to design an internal model candidate, and thus the solvability of the output regulation problem is transformed into the solvability of the stabilization problem. This is a very smart technique to deal with the output regulation problem. The internal model candidates incorporate the output (or plus [partial] state, dependent on the measurability of the state) and the input to design. It does not incorporate the information of the stabilizer, which will be known in the controller. In this chapter, another approach is proposed to solve output regulation problems. Similar to [43] and [44], an output regulation problem is transformed into a stabilization problem of a simplified system with the assumption that the states of the exosystem are known, and a stabilizer is designed for it. It will be shown that the existence of the stabilizer is sometimes necessary for the solvability of the output regulation problem. Then an internal model with respect to the stabilizer for the original system is constructed. Specifically, the internal model can incorporate the information of the stabilizer. This approach is more natural than the ones in [43] and [44], where the internal model is a prerequisite and first designed. It can be seen that the existence of the internal model is sometimes not necessary.
However, the existence of the stabilizer is necessary when the output zeroing manifold is unique, which is the case in all the examples given in [43, 44] and examples 1, 2 and 4 in this chapter.
The definition of the output regulation problem of nonlinear systems in [44] is borrowed, but the feedback is in the form of measurement feedback. A stabilizer for the simplified system is firstly designed. Then with respect to the stabilizer, an internal model is constructed to estimate the exosystem states. If successful, the output regulation problem is solved. In this study, the exosystem may be linear or nonlinear, the signals of which may be bounded or unbounded. The results for both the global version and the local version of dynamic measurement feedback output regulation problem (DMFORP) are reported.

1 Introduction
1.1 Background
1.2 Literature review of train handling
1.3 Motivation
1.4 Contributions of thesis
1.5 Layout of thesis
2 Train model
2.1 Introduction
2.3 Stop distance calculation – a model-comparison
2.4 Conclusion
3 Optimal scheduling
3.1 Introduction
3.2 Control strategies
3.3 Optimal scheduling on trains equipped with different braking systems
3.4 Scheduling
3.5 LQR controller
3.6 Conclusion
4 Speed regulation
4.1 Introduction
4.2 Output regulation with measured output feedback
4.3 Speed regulation
4.4 Conclusion
5 Fault-tolerant control
5.1 Introduction
5.2 Fault detectability
5.3 Fault modes of trains
5.4 Fault detection and isolation
5.5 Fault-tolerant control (FTC)
5.6 Simulation
5.7 Conclusion
6 Conclusions
6.1 Summary
6.2 Assessment
6.3 Future work

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