Marius Schmitt

Flow-maximizing equilibria of the Cell Transmission Model

We consider the freeway ramp metering problem, based on the Cell Transmission Model and address the question of how well decentralized control strategies, e.g. local feedback controllers at every onramp, can maximize the traffic flow asymptotically under time-invariant boundary conditions. We extend previous results on the structure of steady-state solutions of the Cell Transmission Model and use them to optimize over the set of equilibria. By using duality arguments, we derive optimality conditions and show that closed-loop equilibria of certain decentralized feedback controllers, in particular the practically successful “ALINEA” method, are in fact globally optimal.

An approach for model predictive control of mixed integer-input linear systems based on convex relaxations

Model predictive control of systems with mixed discrete and continuous inputs usually requires the online solution of a mixed integer optimization problem. Optimal solutions of such problems require methods whose worst-case complexity is exponential in the number of binary variables. In this paper we propose an approximate approach in which the integer input constraints are initially relaxed. A projection is then applied to the relaxed solution in order to obtain inputs satisfying the integer constraints. Satisfaction of state constraints under the projected input sequence is to be guaranteed by applying a robust reformulation to the original relaxed problem. We restrict our approach here to the practically important class of Pulse-Width Modulated power electronic systems, and present a suitable projection function for such systems. We demonstrate an attractive trade-off between performance and computational cost, using the examples of a DC-DC buck converter and a single-phase AC-DC grid inverter.

Convex, monotone systems are optimally operated at steady-state

We consider a special class of monotone systems for which the system equations are also convex in both the state and the input. For such systems we study optimal infinite horizon operation with respect to an objective function that is also monotone and convex. The main results state that, under some technical assumptions, these systems are optimally operated at steady state, i.e. there does not exist any time-varying trajectory over an infinite horizon that outperforms stabilizing the system in the optimal equilibrium. We draw a connection to recent results on dissipative systems in the context of Economic Model Predictive Control, where systems that are optimally operated at steady state have already been studied. Finally, we apply the main result to a problem in traffic control, where we are able to disprove the existence of improving periodic trajectories involving the alternation of congestion and free flow for freeway ramp metering.

Sufficient optimality conditions for distributed, non-predictive ramp metering in the monotonic cell transmission model

We consider the ramp metering problem for a freeway stretch modeled by the Cell Transmission Model. Assuming perfect model knowledge and perfect traffic demand prediction, the ramp metering problem can be cast as a finite horizon optimal control problem with the objective of minimizing the Total Time Spent, i.e., the sum of the travel times of all drivers. For this reason, the application of Model Predictive Control (MPC) to the ramp metering problem has been proposed. However, practical tests on freeways show that MPC may not outperform simple, distributed feedback policies. Until now, a theoretical justification for this empirical observation was lacking. This work compares the performance of distributed, non-predictive policies to the optimal solution in an idealised setting, specifically, for monotonic traffic dynamics and assuming perfect model knowledge. To do so, we suggest a distributed, non-predictive policy and derive sufficient optimality conditions for the minimization of the Total Time Spent via monotonicity arguments. In a case study based on real-world traffic data, we demonstrate that these optimality conditions are only rarely violated. Moreover, we observe that the suboptimality resulting from such infrequent violations appears to be negligible. We complement this analysis with simulations in non-ideal settings, in particular allowing for model mismatch, and argue that Alinea, a successful, distributed ramp metering policy, comes close to the ideal controller both in terms of control behavior and in performance.

Distributed learning in the presence of disturbances

We consider a problem where multiple agents must learn an action profile that maximises the sum of their utilities in a distributed manner. The agents are assumed to have no knowledge of either the utility functions or the actions and payoffs of other agents. These assumptions arise when modelling the interactions in a complex system and communicating between various components of the system are both difficult. In [1], a distributed algorithm was proposed, which learnt Pareto-efficient solutions in this problem setting. However, the approach assumes that all agents can choose their actions, which precludes disturbances. In this paper, we show that a modified version of this distributed learning algorithm can learn Pareto-efficient solutions, even in the presence of disturbances from a finite set. We apply our approach to the problem of ramp coordination in traffic control for different demand profiles.