  The controller design of the so-called "difference algebraic equation" (DAE)
systems that are frequently shown in industrial processes, tend to be
challenging because of the combination of algebraic equations and high state
dimensions. In this paper, we tackle this problem by developing control
refinement approaches for DAE systems via the notions of (bi)simulation
relations and approximate simulation relations from computer science. The
quantified refinement accuracy is achieved by defining observation metrics over
a general system framework named transition systems. We employ the behavioral
theory to tackle dynamical systems and control problems in a more general
framework. Due to the difficulty in dealing with a DAE system directly, we
derive another system, which is behaviorally equivalent to the related DAE
system and in standard state space form, to provide ease for further control
refinement. Consequently, well-developed model reduction approaches can be
applied to obtain an abstract simplified system, which can be rewritten into a
DAE system again. Based on the (bi)simulation relations, approximate simulation
relations and the initialization conditions, we show that for any given
well-posed controller of the abstract model, we can always refine it to a
controller for the concrete model such that the two systems have the same
controlled output behavior or the distance between their output behavior is
bounded.
