Max-Plus Algebra and Control of Discrete Event Systems

Max-plus algebra is a mathematical framework in which the usual addition and multiplication operations are replaced by maximisation and addition, respectively. This tropical semiring offers a natural language for modelling systems governed by synchronization constraints rather than by continuous interactions. Discrete event systems (DES) describe processes in which state changes occur at discrete points in time, triggered by events. When concurrency is absent or negligible, such systems can be represented as max-plus linear DES, where event dates evolve linearly in the max-plus sense. The max-plus representation captures timing dependencies and enables algebraic analysis of system behavior, including throughput, stability and periodicity. Control objectives—such as enforcing deadlines, sequencing tasks or matching a reference trajectory—can be formulated in this setting through residuation theory and model predictive strategies. Extensions of the basic theory accommodate switching structures, semi-cyclic schedules and mixed-integer formulations, broadening applicability to manufacturing lines, public transport timetabling and communication protocols. The interplay between algebraic spectral properties and system-theoretic constructs underpins both analysis and controller synthesis, yielding explicit conditions for reachability, observability and performance optimisation. This synergy has driven global interest in applying max-plus methods to real-world planning, scheduling and synchronization challenges.

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Max-Plus Algebra and Control of Discrete Event Systems publication trend

The graph below shows the total number of articles in max-plus algebra and control of discrete event systems across all publications each year (not limited to Nature Index journals).

Technical terms

Max-plus algebra: A semiring in which “addition” is replaced by maximisation and “multiplication” by arithmetic addition, used to model timing relations.

Discrete event system (DES): A dynamical system where state transitions occur at discrete event occurrences rather than continuously.

Residuation: A mathematical operation in ordered semirings that computes the greatest input satisfying a given inequality, enabling feedback controller synthesis.

Model predictive control (MPC): A control strategy that optimises future control actions over a moving horizon, here formulated with max-plus dynamics.

Switching max-plus linear model: A representation combining several max-plus linear subsystems with state-dependent selection rules.

Dynamic graph: A time-indexed graph structure used to visualise and analyse event ordering and dependencies in semi-cyclic DES.

References

1. Analysis and control of max-plus linear discrete-event systems: An introduction. Discrete Event Dynamic Systems (2019).
2. The Model Matching Problem for Max-Plus Linear Systems: A Geometric Approach. IEEE Transactions on Automatic Control (2022).
3. Model predictive scheduling of semi-cyclic discrete-event systems using switching max-plus linear models and dynamic graphs. Discrete Event Dynamic Systems (2020).