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

Control refinement for DAE systems: A behavioral approach via simulation relations