Chance-Constrained Model Predictive Controller Synthesis For Stochastic Max-Plus Linear Systems.pdf

## Chance-Constrained Model Predictive Controller Synthesis for Stochastic Max-Plus Linear Systems

This technical report by Delft University of Technology's Center for Systems and Control presents a novel approach for controlling stochastic max-plus linear (MPL) systems using model predictive control (MPC). MPL systems are used to model discrete event systems with continuous state spaces, often found in infrastructure networks like communication and railway systems. The added complexity of randomness in processing or transportation times makes stochastic MPL systems more realistic but also poses challenges for control design.

Key Contributions:

1. Convexity Analysis: The authors demonstrate the convexity of a specific class of functions (max-plus-nonnegative-scaling functions) used in MPC for stochastic MPL systems, simplifying previous proofs and enabling efficient optimization.

2. Chance-Constrained MPC: They propose a randomized technique to handle chance constraints, which are necessary to account for probabilistic uncertainty in system parameters. This approach avoids assumptions about the distribution of uncertainties and allows for high-dimensional problems (relevant for real industrial applications).

3. Practical Applications: The framework is validated on two case studies: a production system and a subset of the Dutch railway network.

Problem Setup:

The research focuses on minimizing a cost function over a receding horizon while satisfying constraints on states, inputs, and outputs of a stochastic MPL system. Unlike robust MPC which assumes bounded uncertainty, stochastic MPC interprets constraints probabilistically, allowing for a specified violation probability level.

Methodology:

1. Scenario-Based Approach: The control problem is approximated by optimizing control inputs over a receding horizon considering a finite number of scenarios (samples) of uncertain parameters.

2. Convex Feasible Set: The authors show that the approximate optimization problem is convex with respect to decision variables, providing probabilistic guarantees for desired constraint fulfillment.

Advantages:

No assumptions on uncertainty distribution are needed.
Suitable for high-dimensional problems (realistic industrial scenarios).