CDS Lecture Series


Andrew Alleyne
Ralph and Catherine Fisher Professor of Engineering
University of Illinois at Urbana-Champaign

Control of Systems in a Dimensionless Framework

The use of dimensional analysis is prevalent in several fields of the physical and life sciences. In Mechanical Engineering, the common concepts of Fluid Mechanics, Heat Transfer, and Thermodynamics are all represented by dimensionless variables. These include the well known dimensionless numbers such as the Reynolds Number, Froude Number, Nusselt Number, etc. The question we raise and hope to answer in this talk is whether or not the field of Systems and Control can benefit from dimensionless analysis as other fields have done.
This talk will begin by detailing the initiation of our study into dimensional analysis for control systems. Vehicle systems were the primary motivation and several types of planar vehicle systems will be examined. The class of systems under study is LTI systems. We demonstrate the process of taking a dimensional system representation and transforming it into a dimensionless one. It can be shown that the dimensionless form for this linear system can be thought of as a very convenient similarity transformation of the original dimensional system.
Subsequent to presenting a dimensionless form for the dynamics of particular systems, we illustrate several key benefits that we have found from working in a dimensionless framework. First, it is possible to uncover underlying dynamical relationships that do not seem clear when studying the dimensional system dynamics. Secondly, the parametric uncertainty associated with nominal vehicle representations is greatly reduced in a dimensionless framework, thereby leading to less conservative controller constraints for robustness requirements. Finally, parametric interdependence uncovered and can be used to greatly reduce system excitation requirements for identification or adaptation mechanisms.

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