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Advances in Statistical Control, Algebraic Systems Theory, and Dynamic Systems Characteristics pp 99–125Cite as

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A Multiobjective Cumulant Control Problem

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Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

Summary

The \(H_2/H_{\infty}\) control problem is well known in the control community. It mixes the results of two powerful control techniques; to balance two objectives: minimizing the \(H_2\) norm of the system, while constraining the system’ \(H_{\infty}\) norm. In the presence of random noise, this is akin to solving a Nash game with the players’ objectives to minimize the mean of their costs. In this chapter, recent trends in minimizing further cumulants will be analyzed, in particular one in which wishes to minimize the variance and other cumulants of a cost, while constraining the system’ \(H_{\infty}\) norm. This problem formulation will begin for a class of nonlinear systems with nonquadratic costs. Sufficient conditions for a Nash equilibrium for a two player game in which the control wishes to minimize the variance of its costs and the disturbance wishes to minimize the mean of its cost are found. The case of linear systems and quadratic costs is applied and equilibrium solutions are determined. Further cumulants are also examined. The results of the control formulation are applied to a problem in structural control, namely, the third generation structural benchmark for tall buildings subject to high winds.

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Author information

Authors and Affiliations

  1. Department of Engineering, University of SouthernIndiana, Evansville, IN 47712, USA

    Ronald W. Diersing

  2. Freimann Professor of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

    Michael K. Sain

  3. Department of Electrical and Computer Engineering, Temple University, Philadelphia, PA 19122, USA

    Chang-Hee Won

Authors
  1. Ronald W. Diersing
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  2. Michael K. Sain
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  3. Chang-Hee Won
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Corresponding author

Correspondence to Ronald W. Diersing .

Editor information

Editors and Affiliations

  1. Dept. Electrical &, Computer Engineering, Temple University , North 12th Street 1947, Philadelphia, 19122, U.S.A.

    Chang-Hee Won

  2. College of Engineering, Boise State University, University Drive 1910, Boise, 83725-2100, U.S.A.

    Cheryl B. Schrader

  3. Department of Electrical Engineering, University of Notre Dame, Fitzpatrick Hall 275, Notre Dame, 46556-5637, U.S.A.

    Anthony N. Michel

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© 2008 Birkhäuser Boston, a part of Springer Science+Business Media, LLC

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Diersing, R.W., Sain, M.K., Won, CH. (2008). A Multiobjective Cumulant Control Problem. In: Won, CH., Schrader, C., Michel, A. (eds) Advances in Statistical Control, Algebraic Systems Theory, and Dynamic Systems Characteristics. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4795-7_5

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