Control of continuum models of production systems

Michael La Marca, Hans Armbruster, Michael Herty, Christian Ringhofer

Research output: Contribution to journalArticlepeer-review

55 Scopus citations


A production system which produces a large number of items in many steps can be modelled as a continuous flow problem. The resulting hyperbolic partial differential equation (PDE) typically is nonlinear and nonlocal, modeling a factory whose cycle time depends nonlinearly on the work in progress. One of the few ways to influence the output of such a factory is by adjusting the start rate in a time dependent manner. We study two prototypical control problems for this case: i) demand tracking where we determine the start rate that generates an output rate which optimally tracks a given time dependent demand rate and ii) backlog tracking which optimally tracks the cumulative demand. The method is based on the formal adjoint method for constrained optimization, incorporating the hyperbolic PDE as a constraint of a nonlinear optimization problem. We show numerical results on optimal start rate profiles for steps in the demand rate and for periodically varying demand rates and discuss the influence of the nonlinearity of the cycle time on the limits of the reactivity of the production system. Differences between perishable and non-perishable demand (demand versus backlog tracking) are highlighted.

Original languageEnglish (US)
Article number5443732
Pages (from-to)2511-2526
Number of pages16
JournalIEEE Transactions on Automatic Control
Issue number11
StatePublished - Nov 2010


  • Adjoint calculus
  • output control
  • partial differential equation (PDE)
  • production lines

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Computer Science Applications
  • Electrical and Electronic Engineering


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