Abstract
Statistical model validation is treated for a class of parametric uncertainty models and also for a more general class of nonparametric uncertainty models. We show that, in many cases of interest, this problem reduces to computing relative weighted volumes of convex sets in R N (where N is the number of uncertain parameters) for parametric uncertainty models, and to computing the limit of a sequence (V k ) ā1 of relative weighted volumes of convex sets in R k for nonparametric uncertainty models. We then present and discuss a randomized algorithm based on gas kinetics for probable approximate computation of these volumes. We also review the existing Hit-and-Run family of algorithms for this purpose.
Finally, we introduce the notion of testability to describe uncertainty models that can be statistically validated with arbitrary reliability using input-output data records of sufficient (finite) length. It is then shown that some common nonparametric uncertainty models, such as those involving ā1 or H ā norms, do not possess this property.
Supported in part by the National Science Foundation under Grant ECS 89-57461, by gifts from Rockwell International, and by the Air Force Office of Scientific Research under a National Defense Science and Engineering Graduate Fellowship.
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References
D. Applegate and R. Kannan, āSample and Integration of Near Logconcave Functions,ā Comp. Science report CMU, 1990.
I. Barany and Z. Furedi, āComputing the Volume Is Difficult,ā Proc. of the 18th Annual Symp. on Theory of Computing, pp. 442ā447, 1986.
C. J. P. BĆ©lisle, H. E. Romejin, and R. L. Smith, āHit-and-Run Algorithms for Generating Multivariate Distributions,ā Mathematics of Operations Research, 18, No. 2, 255ā266, 1993.
M. Dahleh and J. B. Pearson, āā1-Optimal controllers for MIMO discrete-time Systems,ā IEEE Trans. Automatic Control, vol. AC-32, pp. 314ā323, 1987.
M. E. Dyer and A. M. Frieze, āOn the Complexity of Computing the Volume of a Polyhedron,ā SIAM J. Computing, vol. 17, pp. 967ā974, 1988.
B. A. Francis, A Course in H ā Control Theory, Spring-Verlag, 1987.
B. K. Ghosh, Sequential Tests of Statistical Hypotheses, Addison-Wesley, Reading, 1970.
G. C. Goodwin, B. Ninness, and M. E. Salgado, āQuantification of Uncertainty in Estimation,ā Proc. of the 1990 ACC, pp. 2400ā2405, 1990.
G. Gu and P. P. Khargonekar, āLinear and Nonlinear Algorithms for Identification in H ā with Error Bounds,ā to appear in IEEE TAC, 1994.
A. J. Helmicki, C. A. Jacobson, and C. N. Nett, āIdentification in H ā: a Robustly Convergent Nonlinear Algorithm,ā Proc. of the 1990 ACC, pp. 386ā391, 1990.
M. H. Karwan, ed., Redundancy in Mathematical Programming, Springer-Verlag, New York, 1983.
M. J. Kearns and U. V. Vazirani, Topics in Computational Learning Theory, preprint, 1993.
L. G. Khachiyan, āOn the Complexity of Computing the Volume of a Polytope,ā Izvestia Acad. Nauk. SSSR Engr. Cybernetics, vol. 3, pp. 216ā217, 1988. (in Russian)
R. L. Kosut, M. Lau and S. Boyd, āIdentification of Systems with Parametric and Non-parametric Uncertainty,ā Proc. of the 1990 ACC, pp. 2412ā2417, 1990.
J. M. Krause and P. P. Khargonekar, āParameter Identification in the Presence of Non-parametric Dynamic Uncertainty,ā Automatica, vol. 26, pp. 113ā124, Jan. 1990.
L. Lee and K. Poolla, āStatistical Testability of Uncertainty Models,ā Proc. of the 1994 IFAC Symposium on System Identification, Copenhagen, 1994.
L. H. Lee and K. Poolla, āOn Statistical Model Validation,ā submitted to ASME Journal of Dynamic Systems, Measurement and Control.
L. H. Lee and K. Poolla, āStatistical Validation and Testability of Uncertainty Models,ā in preparation.
E. L. Lehmann, Testing Statistical Hypotheses, Wiley, New York, 1986.
L. Lin, L. Y. Wang and G. Zames, āUncertainty Principles and Identification n-widths for LTI and Slowly Varying Systems,ā Proc. of the 1992 ACC, pp. 296ā300, 1992.
L. Ljung, System Identification, Theory for the User, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.
L. Ljung and Z-D Yuan, āAsymptotic Properties of Black-box Identification of Transfer Functions,ā IEEE TAC, vol. 30, pp. 514ā530, June 1985.
P. M. MƤkilƤ and J. R. Partington, āRobust Approximation and Identification in H ā,ā Proc. of the 1991 ACC, pp. 70ā76, 1991.
S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag, New York, 1993.
A. Packard and J.C. Doyle, āThe complex structured singular value,ā Automatica, vol. 29, no. 9, pp:71ā110, 1993.
J. R. Partington, āRobust Identification and Interpolation in H ā,ā Int. J. Contr., 54, pp. 1281ā1290, 1991.
K. Poolla, P. P. Khargonekar, A. Tikku, J. Krause, and K. M. Nagpal, āA Time-Domain Approach to Model Validation,ā IEEE TAC, vol. 39, no. 5, pp. 951ā959, 1994.
K. Popper, Conjectures and Refutations, Harper and Row, 1963.
M.G. Safonov, āStability margins of diagonally perturbed multivariable feedback systems,ā IEE Proceedings, Part D, vol. 129, no. 6, pp:251ā256, 1982.
R. L. Smith, āEfficient Monte Carlo Procedures for Generating Points Uniformly Distributed over Bounded Regions,ā Operations Research, 32, No. 6, pp. 1296ā1308, 1984.
R. Smith and J. C. Doyle, āTowards a Methodology for Robust Parameter Identification,ā Proc. of the 1990 ACC, pp. 2394ā2399, 1990.
R. Tempo and G. Wasilkowski, āMaximum Likelihood Estimators and Worst Case Optimal Algorithms for System Identification,ā Systems and Control Letters, vol. 10, pp. 265ā270, 1988.
D. N. C. Tse, M. A. Dahleh, and J. N. Tsitsiklis, āOptimal Asymptotic Identification under Bounded Disturbancesā, Proc. of the 1991 ACC, pp. 1786ā1787, 1991.
L. G. Valiant, āA Theory of the Learnable,ā Communications of the ACM, vol. 27, pp. 1134ā1142, 1984.
R. C. Younce and C. E. Rohrs, āIdentification with Non-parametric Uncertainty,ā Proc. of the 1990 CDC, pp. 3154ā3161, 1990.
G. Zames, āOn the metric complexity of causal linear systems: Īµ-entropy and Īµ-dimension for continuous timeā IEEE Trans. on Auto. Contr., vol. AC-24, pp. 222ā230, 1979.
T. Zhou and H. Kimura, āTime domain identification for robust control,ā Systems and Control Letters, vol. 20, pp. 167ā178, 1993.
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Dedicated to Professor George Zames on the occasion of his sixtieth birthday
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Lee, L.H., Poolla, K. (1995). Statistical validation for uncertainty models. In: Francis, B.A., Tannenbaum, A.R. (eds) Feedback Control, Nonlinear Systems, and Complexity. Lecture Notes in Control and Information Sciences, vol 202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027675
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DOI: https://doi.org/10.1007/BFb0027675
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