Optimal Model Predictive Control of Timed Continuous Petri Nets

Abstract

This paper addresses the optimal control problem of timed continuous Petri nets under infinite servers semantics. In particular, our goal is to find a control input optimizing a certain cost function that permits the evolution from an initial marking (state) to a desired steady-state. The solution we propose is based on a particular discrete-time representation of the controlled continuous Petri net system, as a certain linear constrained system. An upper bound on the sample period is given in order to preserve important information of the timed continuous net, in particular the positiveness of the markings. The reachability space of the sampled system in relation to autonomous continuous Petri nets is also studied.Based on the resulting linear constrained model, the optimal control problem is studied through Model Predictive Control (MPC). Implicit and explicit procedures are presented together with a comparison between the two schemes. Stability of the closed-loop system is also studied.

Piecewise-linear constrained control for timed continuous Petri nets

“…Therefore, for a small enough value of (e − γe 0 ) = e ′ , i.e. for m close enough to m ′ d , γ is a solution for (12). A particular case occurs at m 0 , in which (e − γe 0 ) = (1 − γ)e 0 = e ′ (i.e.…”

“…A particular case occurs at m 0 , in which (e − γe 0 ) = (1 − γ)e 0 = e ′ (i.e. a value of γ close enough to 1 makes e ′ be small enough, and then such γ is a solution for (12) and the LPP). Now, suppose that at time τ 1 a solution β 1 is computed.…”

“…Now, suppose that at time τ 1 a solution β 1 is computed. Substituting γ = β 1 − ∆γ into (12) and reordering the terms we obtain…”

“…In [12] a solution based on Model Predictive Control was proposed, finding computational complexity problems for the explicit approach that make its application prohibitive. In [13] a tracking control approach was introduced, considering step and ramp references and low-andhigh gain controllers.…”

An Efficient Heuristics for Minimum Time Control of Continuous Petri nets

“…In Mahulea et al (2008b) it is shown that, the optimal steady state control problem of timed contPN system can be solved by means of Linear Programming Problem (LPP) when all transitions are assumed to be controllable. The transitory control problem is also solved by means of implicit and explicit MPC control strategy (Mahulea et al (2008a)). A Lyapunov-function-based dynamic control algorithm was proposed in (Xu et al (2008)), which can ensure global convergence of both system states and input signals.…”