Abstract
Real-time failure detection for systems having linear stochastic dynamical truth models is posed in terms of two confidence region sheaths. One confidence region sheath is about the expected no-failure trajectory; the other is about the Kalman estimate. If these two confidence regions of ellipsoidal cross section are disjoint at any time instant, a failure is declared.
A test for two-ellipsoid overlap is developed which involves finding a single pointx* whose presence in both ellipsoids is necessary and sufficient for overlap. Thus, the overlap test is contorted into a search forx*, shown to be the solution of a nonlinear optimization problem that is easily solved once an associated scalar Lagrange multiplier is known. A successive approximations iteration equation for λ is obtained and is shown to converge as a contraction mapping. The method was developed to detect failures in an inertial navigation system that appear as uncompensated gyroscopic drift rate. For simulated gyroscopic failures, the iterations converged very quickly, easily allowing real-time failure detection.
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Kerr, T. H.,A Two Ellipsoid Overlap Test for Real Time Failure Detection and Isolation by Confidence Regions, Proceedings of the Conference on Decision and Control, Phoenix, Arizona, 1974.
Mehra, R. K., andPeschon, J.,An Innovations Approach to Fault Detection and Diagnosis in Dynamic Systems, Automatica, Vol. 7, No. 5, 1971.
Jones, H. L.,Failure Detection in Linear Systems, MIT, Department of Aeronautics and Astronautics, PhD Thesis, 1973.
Willsky, A. S., andJones, H. L.,A Generalized Likelihood Ratio Approach to State Estimation in Linear Systems Subject to Abrupt Changes, Proceedings of the Conference on Decision and Control, Phoenix, Arizona, 1974.
Harrison, J. V., andChien, T. T.,Failure Isolation for a Minimally Redundant Inertial Sensor System, IEEE Transactions on Aerospace and Electronics Systems, Vol. AES-11, No. 3, 1975.
Boozer, D. D., andMcDaniel, W. L.,On Innovation Sequence Testing of the Kalman Filter, IEEE Transactions on Automatic Control, Vol. AC-17, No. 1, 1972.
Martin, W. C., andStubberud, A. R.,An Additional Requirement for Innovations Testing in System Identification, IEEE Transactions on Automatic Control, Vol. AC-19, No. 5, 1974.
Athans, M.,On the Elimination of Mean Steady-State Errors in Kalman Filters, Proceedings of the Symposium on Nonlinear Estimation Theory, San Diego, California, pp. 212–214, 1970.
D'Appolito, J. A., andRoy, K. J.,Reduced Order Filtering with Applications in Hybrid Navigation, Proceedings of the IEEE Electronics and Aerospace Systems Conference, Washington, D.C., 1973.
Kerr, T. H.,Implementing a Two Ellipsoid Overlap Test for Real Time Failure Detection, IEEE Transactions on Automatic Control (to appear).
Nash, R. A., Jr., Kasper, J. F., Jr., Crawford, B. S., andLevine, S. A.,Application of Optimal Smoothing to the Testing and Evaluation of Inertial Navigation Systems and Components, IEEE Transactions on Automatic Control, Vol. AC-16, No. 6, 1971.
Jazwinski, A. H.,Stochastic Processes and Filtering Theory, Academic Press, New York, New York, 1970.
Sage, A. P.,Optimum Systems Control, Prentice-Hall, Englewood Cliffs, New Jersey, 1968.
Anderson, T. W.,An Introduction to Multivariate Statistical Analysis, John Wiley and Sons, New York, New York, 1958.
Kerr, T. H.,Applying Stochastic Integral Equations to Solve a Particular Stochastic Modeling Problem, University of Iowa, PhD Thesis, 1971.
Luenberger, D. G.,Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, Reading, Massachusetts, 1973.
Fletcher, R., andPowell, M. J. D.,A Rapidly Convergent Descent Method for Minimization, Computer Journal, Vol. 6, No. 2, 1963.
Holtzman, J. M.,Nonlinear System Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1970.
Blum, E. K.,Numerical Analysis and Computation Theory and Practice, Addison-Wesley Publishing Company, Reading, Massachusetts, 1972.
Gallager, R. G., andHelstrom, C. W.,A Bound on the Probability that a Gaussian Process Exceeds a Given Function, IEEE Transactions on Information Theory, Vol. IT-15, No. 1, 1969.
Kerr, T. H.,False Alarm and Correct Detection Probabilities Over a Time Interval for Failure Detection Algorithms (to appear).
Kleinman, D. L., andPerkins, T. R.,Modeling Human Performance in a Time-Varying Anti-Aircraft Tracking Loop, IEEE Transactions on Automatic Control, Vol. AC-19, No. 4, 1974.
Athans, M., andSchweppe, F. C.,Gradient Matrices and Matrix Calculations, Lincoln Laboratory, Lexington, Massachusetts, 1965.
Lee, E. B., andMarkus, L.,Foundations of Optimal Control Theory, John Wiley and Sons, New York, New York, 1967.
Kwakernaak, H., andSivan, R.,Linear Optimal Control Systems, Wiley-Interscience, New York, New York, 1972.
Van Valkenburg, M. E.,Modern Network Synthesis, John Wiley and Sons, New York, New York, 1964.
Bose, N. K.,Test for Two-Variables Local Positivity with Applications, Proceedings of the IEEE, Vol. 64, No. 9, 1976.
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Communicated by A. V. Balakrishnan
This work was supported by the Department of the Navy, Strategic Systems Project Office, SP-24.
This paper is based on an earlier paper (Ref. 1) which was presented at the IEEE Conference on Decision and Control, Phoenix, Arizona, 1974. A stronger convergence proof is presented in the present paper.
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Kerr, T.H. Real-time failure detection: A nonlinear optimization problem that yields a two-ellipsoid overlap test. J Optim Theory Appl 22, 509–536 (1977). https://doi.org/10.1007/BF01268172
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DOI: https://doi.org/10.1007/BF01268172