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Journal of Optimization Theory and Applications

, Volume 146, Issue 2, pp 511–537 | Cite as

New Risk-Averse Control Paradigm for Stochastic Two-Time-Scale Systems and Performance Robustness

  • K. D. Pham
Article

Abstract

This work is concerned with the optimal control of stochastic two-time-scale linear systems with performance measure in a finite-horizon integral-quadratic form. Nature, modeled by stationary Wiener processes whose mean and covariance statistics are known, malevolently affects the state dynamics and output observations of the control problem class. With particular focus on the system performance robustness, the use of higher-order statistics or cumulants associated with the performance measure of chi-squared random variable type makes it possible to restate the stochastic control problem as the solution of a deterministic one, which subsequently allows disregarding all sample-path realizations by Nature acting on the original problem.

The distinguishing feature of the risk-averse control paradigm is that the performance index is multiobjective in nature, being composed of both risk-neutral integrals and risk-sensitive costs associated with the ubiquitous linear-quadratic-Gaussian (LQG) and rather recent risk-sensitive control problems. Another issue that makes this class of control particularly interesting is the fact that Nature has the ability to exercise all the higher-order characteristics of the uncertain chi-squared performance measure. The efficient controller, having access to Nature’s apriori statistical knowledge and employing dynamic output feedback, seeks to minimize the performance uncertainty that Nature can do over the set of mixed random realizations.

Furthermore, the results herein potentially generalize the existing results for the single-objective H 2, H , and risk-sensitive control problems to a substantially larger class of systems, wherein Nature selects mixed sample-path realizations that need not be Gaussian. That is, the entire probability density function of Nature’s choices is not necessarily known except for its first two statistics. Finally, the numerical simulations for a two-time-scale longitudinal dynamics of the F-8 jet aircraft demonstrate that the proposed control paradigm has competitive performance in the closed-loop system responses and offers multiple levels of robustness for the system performance.

Keywords

Two-time-scale stochastic system Multiobjective cumulant-based control Risk-averse controllers Mayer form optimization 

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References

  1. 1.
    Pham, K.D., Sain, M.K., Liberty, S.R.: Robust cost-cumulants based algorithm for second and third generation structural control benchmarks. In: Proceedings of American Control Conference, pp. 3070–3075 (2002) Google Scholar
  2. 2.
    Pham, K.D., Sain, M.K., Liberty, S.R.: Cost cumulant control: state-feedback, finite-horizon paradigm with application to seismic protection. J. Optim. Theory Appl. 115(3), 685–710 (2002) CrossRefMathSciNetMATHGoogle Scholar
  3. 3.
    Pham, K.D., Sain, M.K., Liberty, S.R.: Infinite horizon robustly stable seismic protection of cable-stayed bridges using cost cumulants. In: Proceedings of American Control Conference, pp. 691–696 (2004) Google Scholar
  4. 4.
    Pham, K.D., Sain, M.K., Liberty, S.R.: Statistical control for smart base-isolated buildings via cost cumulants and output feedback paradigm. In: Proceedings of American Control Conference, pp. 3090–3095 (2005) Google Scholar
  5. 5.
    Kokotovic, P.V., Yackel, R.A.: Singular perturbation of linear regulators: basic theorems. IEEE Trans. Automat. Contr. 17, 19–37 (1972) MathSciNetGoogle Scholar
  6. 6.
    Haddad, A.H., Kokotovic, P.V.: On a singular perturbation problem in linear filtering theory. In: Proceedings of 5th Annual Princeton Conference on Information Sciences and Systems (1971) Google Scholar
  7. 7.
    Teneketzis, D., Sandell, Jr. N.R.: Linear regulator design for stochastic systems by a multiple time-scales method. IEEE Trans. Automat. Contr. 22(4), 615–621 (1977) CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Pham, K.D.: Cooperative solutions in multi-person quadratic decision problems: finite-horizon and state-feedback cost-cumulant control paradigm. In: The 46th IEEE Conference on Decision and Control, pp. 2484–2490 (2007) Google Scholar
  9. 9.
    Brockett, R.W.: Finite Dimensional Linear Systems. Wiley, New York (1970) MATHGoogle Scholar
  10. 10.
    Fleming, W.H., Rishel, R.W.: Deterministic and Stochastic Optimal Control. Springer, Berlin (1975) MATHGoogle Scholar
  11. 11.
    Pham, K.D.: Statistical control paradigms for structural vibration suppression. Ph.D. Thesis, University of Notre Dame, Notre Dame, Indiana, 2004. Available via http://etd.nd.edu/ETD-db/theses/available/etd-04152004-121926/unrestricted/PhamKD052004.pdf. Cited 14 August 2009
  12. 12.
    Etkin, B.: Dynamics of Atmospheric Flight. Wiley, New York (1972) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Aerospace Engineer, Air Force Research LaboratoryKirtland Air Force BaseNew MexicoUSA

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