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Participation Factors for Linear Systems

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Participation Factors for Linear Systems
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  ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical,heterogeneous and dynamic problems of engineering technology and systems for industry and government.ISR is a permanent institute of the University of Maryland, within the Glenn L. Martin Institute of Technol-ogy/A. James Clark School of Engineering. It is a National Science Foundation Engineering Research Center. Web site http://www.isr.umd.edu  I R INSTITUTE FOR SYSTEMS RESEARCH T ECHNICAL  R ESEARCH  R EPORT On Participation Factors for Linear Systems by E.H. Abed, D. Lindsay and W.A. Hashlamoun T.R. 99-41  Report Documentation Page Form Approved OMB No. 0704-0188  Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.   1. REPORT DATE   1999   2. REPORT TYPE   3. DATES COVERED   - 4. TITLE AND SUBTITLE   On Participation Factors for Linear Systems 5a. CONTRACT NUMBER   5b. GRANT NUMBER   5c. PROGRAM ELEMENT NUMBER   6. AUTHOR(S)   5d. PROJECT NUMBER   5e. TASK NUMBER   5f. WORK UNIT NUMBER   7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)   Air Force Office of Scientific Research,875 North Randolph Street,Arlington,VA,22203-1768   8. PERFORMING ORGANIZATIONREPORT NUMBER   9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)   10. SPONSOR/MONITOR’S ACRONYM(S)   11. SPONSOR/MONITOR’S REPORT NUMBER(S)   12. DISTRIBUTION/AVAILABILITY STATEMENT   Approved for public release; distribution unlimited   13. SUPPLEMENTARY NOTES   14. ABSTRACT   see report   15. SUBJECT TERMS   16. SECURITY CLASSIFICATION OF:   17. LIMITATION OF ABSTRACT   18. NUMBEROF PAGES   18   19a. NAME OFRESPONSIBLE PERSON   a. REPORT   unclassified   b. ABSTRACT   unclassified   c. THIS PAGE   unclassified   Standard Form 298 (Rev. 8-98)  Prescribed by ANSI Std Z39-18  On Participation Factors for Linear Systems Eyad H. Abed David Lindsay Department of Electrical and Computer Engineering Applied Mathematics Programand the Institute for Systems Research and the Institute for Systems ResearchUniversity of Maryland University of MarylandCollege Park, MD 20742 USA College Park, MD 20742 USAE-mail: abed@eng.umd.edu E-mail: lindsay@eng.umd.edu Wael A. Hashlamoun Department of Electrical EngineeringBirzeit UniversityP.O. Box 14Birzeit, West Bank, via IsraelE-mail: hwael@eng.birzeit.edu Revised Manuscript: June 1999 Abstract Participation factors are nondimensional scalars that measure the interaction be-tween the modes and the state variables of a linear system. Since their introductionby Verghese, P´erez-Arriaga and Schweppe, participation factors have been used foranalysis, order reduction and controller design in a variety of fields. In this paper,participation factors are revisited, resulting in new definitions. The aim of these defi-nitions is to achieve a conceptual framework that doesn’t hinge on any particular choiceof initial condition. The initial condition is modeled as an uncertain quantity, whichcan be viewed either in a set-valued or a probabilistic setting. If the initial conditionuncertainty obeys a symmetry condition, the new definitions are found to reduce tothe srcinal definition of participation factors. Keywords:  participation factors, linear systems, modal analysis, stability, dynamics, prob-ability 1 Introduction Since its introduction by Verghese, P´erez-Arriaga and Schweppe [9],[5],[10], Selective ModalAnalysis (SMA) has become a popular tool for system analysis, order reduction and actuatorplacement. In particular, this tool is extensively used in the electric power systems area [4].1  Participation factors, a key element of SMA, provide a mechanism for assessing the level of interaction between system modes and system state variables.In this paper, participation factors are revisited, resulting in new definitions. The pur-pose of these definitions is to achieve a conceptual framework that doesn’t hinge on anyparticular choice of initial condition. The initial condition is modeled as an uncertain quan-tity, which can be viewed either in a set-valued or a probabilistic setting. If the initialcondition uncertainty obeys a symmetry condition, the new definitions are found to reduceto the srcinal definition of [9],[5],[10]. Since the initial condition is viewed as uncertain,the definitions model its effect on the level of mode-state interaction in an average sense.(For simplicity, the term “state” is used here interchangeably with “state variable.”) Thiswork provides a framework for a deeper appreciation of participation factors, as well as anopening to further useful generalizations of the concept in various directions. For instance,generalizations to other system types can be pursued. Also, definitions along the same linescould facilitate analytical treatment of the relation between control inputs and states/modeinteraction, which in turn would have implications for actuator and sensor placement.Fowllowing Verghese, P´erez-Arriaga and Schweppe [9],[5],[10], participation factors areconsidered in two basic senses. In the first sense, a participation factor measures the relativecontribution of a mode to a state. In the second, a participation factor measures the relativecontribution of a state to a mode. It isn’t clear at the outset that these two senses should leadto identical formulas for participation factors. However, the precise definitions in [9],[5],[10]for these two senses of participation factors did indeed result in identical mathematicalexpressions. The same conclusion is found to apply in the present paper, under assumptionsvalid for a large class of problems.It should be noted that there have been other interpretations of participation factorssince the srcinal work of [9],[5],[10]. For example, participation factors are often viewed assensitivities of eigenvalues to changes in the diagonal entries of the state dynamics matrix(see, e.g., [8]). Interpretations in terms of modal energies are also common (see, e.g., [1]).Still another interpretation relates to eigenvalue mobility under state feedback [6].The remainder of the paper is organized as follows. In Section 2, the srcinal definition of participation factors [9],[5],[10] is recalled, and motivation for the work of this paper is given.In Section 3, the new definitions of this paper that address participation of   modes in states  2

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