Also see Astrom, K. J., “Comments on 'Robustness of Continuous-Time Adaptive Control Algorithms in the Presence of Unmodeled Dynamics’ by C. E. Rohrs et al,” IEEE Trans. on Automatic Control, Vol. 30, No. 9, pp. 889, Sep. 1985 (original paper, Sep. 1985, pp. 881-889).
Also see Rosenbrock, H. H., McMorran, P. D., “Good, Bad, or Optimal?,” IEEE Trans. on Automatic Control, Vol. 16, No. 6, pp. 552-554, Dec. 1971.
There is no stability problem associated with Linear Quadratic (LQ) Optimal Feedback Control. The issue raised above concerns the tenuous stability of LQG optimal control laws involving, first, a Kalman filter then followed by an LQ feedback regulator in providing a LQG control implementation. The classic textbook by E. B. Lee and L. Markus, Foundations of Optimal Control, (published 1966) contains accessible, easy to understand, detailed step-by-step proofs of the stability of just the LQ portion, as rigorously established within this textbook using the two familiar supporting regularity conditions of observability and controllability along with Matrix Positive Definiteness verification within the context of a completely specified associated Lyapunov function (using the aforementioned parameters). The associated Matrix Riccati equation is also encountered within this context in specifying the appropriate (possibly time-varying) feedback gain matrix to be implemented. Prof. David Kleinman (UCONN and later Naval Postgraduate School in Monterey, CA) has several published papers in the IEEE Transactions of Automatic Control in the 1970’s on aspects relating to the computational calculation of the solution to the Matrix Riccati equation that arises within this same context. Similar rigorous proofs of the stability of the Kalman filter alone have been available since the inception of the Kalman filter methodology. The problem identified more recently (and summarized in the figure above) is with the two being used together back-to-back as LQG. The remedy to avoid LQG instability is to add an additional step involving a slight variation, denoted as Loop Transfer Recovery (LTR), abbreviated as LQG/LTR. Go to Top
Kerr, T. H., “Extending Decentralized Kalman Filtering (KF) to 2D for Real-Time Multisensor Image Fusion and\or Restoration: Optimality of Some Decentralized KF Architectures,” Proceedings of the International Conference on Signal Processing Applications & Technology (ICSPAT96), 7-10 October 1996.
Question: Who would ever need the knowledge conveyed in the above diagram? Answer: Competent researchers, of course, like:
Bitmead, Robert, R., “Convergence in Distribution of LMS-type Adaptive Parameter Estimates,” IEEE Trans. on Automatic Control, Vol. 28, pp. 54-60, Jan. 1983.
Bittanti, S., “L2 Convergence Analysis of Adaptive RLS Algorithms under Stochastic Excitation,” IEEE Trans. on Automatic Control, Vol. 36, pp. 963-967, Aug. 1991.
State-of-the-art from TK-MIP is always provided in bite-size chunks that are easy to understand and follow-up on.
Callegari, S., Rovatti, R., Setti, G., “Embeddable ADC-Based True Random Number Generator for Cryptographic Applications Exploiting Nonlinear Signal Processing and Chaos,” IEEE Trans. on Signal Processing, Vol. 53, No. 2, pp. 793-805, Feb. 2005 [TeK Comment: this approach to generation may be too strong for easy decryption but results still germane to excellent computational simulation of systems without subtle cross-correlations in the random number generators contaminating or degrading output results.] Go to Top
User can set or undo password-protected secure encrypted processing (with secure deletes) enforced by invoking it from the Main Menu.
After invoking secure classified processing, the Main screen is modified (with additional accoutrements activated on top RHS) as below:
Upon subsequent start-up with secure processing enforced, Banner screen is modified (with additional accoutrements activated) as depicted:
If USER successfully enters correct password (within the allotted three tries), only then does the Banner screen change to allow USER through:
Upon subsequently entering the Main screen, any important security breach attempts are reported to the user (as required):
Then USER may resume classified processing.
To turn off classified processing, user needs to return to the Activate/Deactivate Password Protection, and enter the correct password...
Afterwards, USER must enter the correct password or else the following will occur:
USER may customize TK-MIP to their needs by providing their own Basic code to implement rotations or bias offsets (possible time-varying for each). Users may also select rotations to invoke from our wide repertoire of pre-coded and validated standard familiar rotational transformations.
Again, USERs can either write their own code to effect a particular rotation or they can select from our existing repertoire of standard rotations, as already validated by TeK Associates.
After a rotation or a bias shift is specified, its invocation is reflected as a spinning top that appears on pertinent screens near the model status indicator as a mnemonic device to remind the USER that it is invoked.
By only giving our USERs access to modify TK-MIP code relating to rotations and bias offsets, we prevent our USERs from inadvertently clobbering the portions of TK-MIP that have already been validated by TeK Associates as performing correctly, as originally implemented. Go to Top
Historically, within submarine navigational use of a Kalman filter as a navigation filter, the need for occasional external position fixes to compensate for the inherent deleterious gyro drift of its Inertial Navigation System (INS) has to be traded-off against sweeprate exposure to enemy surveillance during such fix taking that involves exposure of an antenna mast.
In the above diagram, R = Detection Range of Surveillance Sensor (definite range law) within which detection of target occurs with probability 0.50; and v = constant velocity of the Surveillance Platform (i.e., vehicle) upon which the surveillance sensor is mounted.
Historical Account of our experience with balancing surveillance with navigation requirements:
 Kerr, T. H., “Modeling and Evaluating an Empirical INS Difference Monitoring Procedure Used to Sequence SSBN Navaid Fixes,” Proceedings of the Annual Meeting of the Institute of Navigation, U.S. Naval Academy, Annapolis, Md., 9-11 June 1981. (Selected for reprinting in Navigation: Journal of the Institute of Navigation, Vol. 28, No. 4, pp. 263-285, Winter 1981-82).
 Kerr, T. H., “Impact of Navigation Accuracy in Optimized Straight-Line Surveillance\Detection of Undersea Buried Pipe Valves,” Proceedings of National Marine Meeting of the Institute of Navigation (ION), Cambridge, MA, pp. 80-90, 27-29 October 1982.
 Koopman, B. O., Search and Screening: General Principles with Historical Applications, Pergamon Press, NY, 1979.
 Washburn, A. R., Search and Detection, Military Applications Section, Operations Research Society of America, Arlington, VA, 1981.
 Stone, L. D., Theory of Optimal Search, 2nd Edition, Operations Research Society of America, Arlington, VA, 2007. [It was wonderful to see Bernard Koopman's critically constructive review of this book, when it first came out (and when Koopman was at Arthur D. Little at age ~79). I gave him a call as an admirer when I was six blocks away at Intermetrics but his manager interceded. I was also aware of Koopman's first classified publications of his material for the Secretary of the Navy during WWII and later in 1956 in the Journal of Operations Research, while he was a professor at Columbia University, where Eli Brookner and Lofti A. Zadeh were matriculating at the time. Eli before 1950.]
 Stone, L. D., Corwin, T. L., Barlow, C. A., Bayesian Multiple Target Tracking, Artech House Radar Library, Norwood, MA, 1999.
 Mahdavi, M., “Solving NP-Complete Problems by Harmony Search,” on pp. 53-70 in Music-Inspired Harmony Search Algorithms, Zong Woo Gee (Ed.), Springer-Verlag, NY, 2009.
Kerr, T.H., “GPS\SSN Antenna Detectability,” Intermetrics Report No.
IR-MA-199, Cambridge, MA, 15 March 1983, for NADC (George Lowenstein).
 Kerr, T. H., “Further Comments
on ‘Optimal Sensor Selection Strategy for Discrete-Time Estimators’,” IEEE
Trans. on Aerospace and Electronic Systems, Vol. 31, No. 3, pp. 1159-1166,
T. H., “Sensor Scheduling in Kalman Filters: Evaluating a Procedure for
Varying Submarine Navaids,” Proceedings
of 57th Annual Meeting of the Institute of Navigation, pp.
310-324, Albuquerque, NM, 9-13 June 2001.
 Kerr, T. H., “Sensor Scheduling in Kalman Filters: varying navaid fixes for trading-off submarine NAV accuracy vs. ASW exposure,” Proceedings of The Workshop on Estimation, Tracking, and Fusion: A Tribute to Yaakov Bar-Shalom (on the occasion of his 60th Birthday) following the Fourth ONR/GTRI Workshop on Target Tracking and Sensor Fusion, Naval Postgraduate School, Monterey, CA, pp. 104-122, 17 May 2001.