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On Structured-Closed-Loop versus Structured-Controller Design the Case of Relative Measurement Feedback

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Document pages: 15 pages

Abstract: We consider optimal distributed controller design problems with two separatestructural constraints: locality and relative feedback. The locality constraintaccounts for subcontroller interactions restricted to local neighborhoods asspecified by an underlying graph structure. We provide a detailedcharacterization of such locality constraints imposed on an implementation of acontroller (sparsity of state space matrices), in contrast to localityconstraints on the sparsity of the controller input-output mapping (transferfunction), which is the main tool used in the recently developed System LevelSynthesis (SLS) framework. We formally show how the latter problem is a convexrelaxation of the former, original problem. We then formulate and show how toappend relative feedback constraints that are natural in many problems whereonly differential sensors are available, such as in vehicular formation andconsensus-type problems. We show that for a class of plants, these constraintscan be imposed in a convex manner. Our main result demonstrates that whenrelative feedback constraints are imposed in addition to locality, the SLSconvex relaxation can become infeasible. We use these insights to provide acomparative context for the various structured distributed control designmethods such as SLS, funnel causality and quadratic invariance, as well as theopen problems of structured and networked realizations.

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