Robust control of constrained max‐plus‐linear systems

Abstract

Max-plus-linear (MPL) systems are a class of nonlinear systems that can be described by models that are 'linear' in the max-plus algebra. We provide here solutions to the three types of finite-horizon min-max control problems for uncertain MPL systems, depending on the nature of the control input over which we optimize: open-loop input sequences, disturbances feedback policies, and state feedback policies. We assume that the uncertainty lies in a bounded polytope and that the closed-loop input and state sequence should satisfy a given set of linear inequality constraints for all admissible disturbance realizations. Despite the fact that the controlled system is nonlinear, we provide sufficient conditions that allow one to preserve convexity of the optimal value function and its domain. As a consequence, the min-max control problems can be either recast as a linear program or solved via N parametric linear programs, where N is the prediction horizon. In some particular cases of the uncertainty description (e.g. interval matrices), by employing results from dynamic programming, we show that a min-max control problem can be recast as a deterministic optimal control problem.

Optimal input design for uncertain max‐plus linear systems

“…For an uncertain max-plus linear system described as interval system (4), a bounded interval vectorũ is called an approximate interval input if for any output y ∈ y and any transition matrix H ∈ H, there exists an input u ∈ũ such thatũ is the optimal solution of problem (8) for p = ∞, ie, (Hũ, ) = min u∈Q(H, ) (Hu, ).…”

“…Max-plus linear systems, [1][2][3] which have maximization and addition as their basic operations, describe a class of dynamical systems with synchronization but no concurrency, although such systems are generally nonlinear descriptions in conventional algebra. Many effective methods are available for assessing the characteristics and performances of max-plus linear systems such as periodicity and stability, [4][5][6][7][8][9] controllability and observability, [10][11][12] and optimization problems. [13][14][15][16][17] Applications of max-plus linear systems often arise in the context of manufacturing plants, traffic scheduling, queuing systems, etc.…”

“…The transfer of the system may fluctuate due to the uncertain or variable parameters caused by noises, time delays, or system malfunctions, and such fluctuations can be framed by an interval describing the transfer's upper and lower behaviors. To take parameter perturbations into account, a class of uncertain max-plus linear systems whose parameters are not exactly known but belong to intervals is drawing increasing attention from researchers, and several analysis and control problems have been studied, including predictive control, 19,20 optimal control, 21-23 solvability, [24][25][26][27][28][29] robustness, 8,30,31 disturbance decoupling, 32 and reachability. 33 One problem inherent in control systems is how to design inputs so that the outputs can meet the predetermined requirements.…”

Max-plus algebra and max-plus linear discrete event systems: An introduction

“…Worked examples of MPC for maxplus linear systems and related results can be found in [35], [38]- [41].…”

Modeling and control of switching max-plus-linear systems with random and deterministic switching

“…The MPC approach is essentially based on the minimization of the error between the actual output times and the due dates, possibly subject to additional constraints on the inputs and the outputs. The MPC approach for MPL systems has been developed in De Schutter and van den Boom (2001), Masuda (2006), Masuda and Goto (2007), Necoara et al (2009) and van den Boom and De Schutter (2002). Another approach, based on invariant subspaces, is proposed in Katz (2007) and Maia et al (2011).…”