
Screen shot #11 (Merely a picture to illustrate that our GUI is totally selfexplanatory) Click here to go to our criticism of LQG Optimal Control Strategy and our Endorsement of LQG/LTR. Click here to go to our TKMIP Classified Processing Capability. Click here to go to see how TKMIP handles Rotations and Known Bias Offsets. Click here to go to our discussion of Search and Sweeprate Considerations. Also see Astrom, K. J., “Comments on 'Robustness of ContinuousTime 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. 881889). Also see Rosenbrock, H. H., McMorran, P. D., “Good, Bad, or Optimal?,” IEEE Trans. on Automatic Control, Vol. 16, No. 6, pp. 552554, 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 stepbystep 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 timevarying) 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 backtoback 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 Pitfalls of trying
to use Feedback Linearization on REAL systems If the system were
of the state variable form: (1)
dx/dt = f(x(t),t) + G(t) u(t), where u(t) is the
exogenous control to be specified to achieve some type of objective or goal, then in order to
theoretically employ “feedback linearization”, one would need to use a control of the form: (2)
u(t) = [G(t)]^{1} [  f(x(t),t) + Ax(t) + u_{2}(t)] In the idealization,
for things to work out satisfactorily, it would need to be the case that G(t) be
square, known
perfectly for all time, t, and be invertible
too so that: (3)
dx(t)/dt = f(x(t),t) + G(t) u(t) = f(x(t),t) + G(t) [G(t)]^{1}
[  f(x(t),t) + Ax(t) + u_{2}(t)] = f(x(t),t)  f(x(t),t)
+ Ax(t) + u_{2}(t) = Ax(t) + u_{2}(t) or, now: (4)
dx(t)/dt = Ax(t) + u_{2}(t), which
is linear and, ostensibly, easier to control! Reality
differs significantly from the above described ideal situation for handling the
hard problem of adequately controlling a nonlinear system. Even
for the extremely easy case of a constant Gain Matrix: (5)
G(t) = G for all t and being of full rank, square and invertible, the
actual system would still
exhibit time delay
(i.e., transport delay) through wires or other components and the form of
f((x),t) will likely not match the original perfectly so what one really has is the residual of
an attempted cancellation: (6) dx(t)/dt = f(x(t),t) + G(t) u(t)
= f(x(t),t)  f(x(t+
delta), t+delta) + Ax(t+delta) + u_{2}(t+delta) = f_{2}(x(t), x(t+delta),
t, t+ delta) + A x(t+ delta) + u(t+ delta), being
a differentialdifference equation that is far less manageable in practice than
the "idealization" of Eq. 4, which is "perfect" in
simulation when things are treated as though there were perfectly exact cancellation of the nonlinearity throughout. Real
implementations would behave otherwise. Another
complaint that could be raised is that about needing to have G(t) or G in
the above be square. For systems exhibiting controllability, it is usually not
necessary for the number of control components to match the state size. Usually
the number of control inputs is considerably less than the state size. So
this aspect in the above is unrealistic as well. Finally,
as we discussed in my office when you stopped by last Friday, 1963 book by
Richard Bellman and Cooke, Differential Difference Equations says how to
handle such systems. Differential Difference equations are more
difficult to handle than mere differential equations and are even more
vulnerable to instabilities. Differential Difference equations are
classified as being of the “advanced”, “neutral”, or “retarded” types.
Only the last two types have any hope of tractably being made stable. Recall,
my paper on Neural Networks was not about how to apply Neural Networks per
se but was about what to watch out for lest one be “chumped” by
a lot of the stuff that was being said about NN’s being applicable for control
applications:
Kerr, T. H.,
“Critique of Some Neural Network Architectures and Claims for Control and
Estimation,”
IEEE Transactions on Aerospace and Electronic Systems,
Vol. 34, No. 2, pp. 406419, Apr. 1998.
(an expose)
Kerr, T. H., “Extending Decentralized Kalman Filtering (KF) to 2D for RealTime Multisensor Image Fusion and\or Restoration: Optimality of Some Decentralized KF Architectures,” Proceedings of the International Conference on Signal Processing Applications & Technology (ICSPAT96), 710 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 LMStype Adaptive Parameter Estimates,” IEEE Trans. on Automatic Control, Vol. 28, pp. 5460, Jan. 1983. Bittanti, S., “L^{2} Convergence Analysis of Adaptive RLS Algorithms under Stochastic Excitation,” IEEE Trans. on Automatic Control, Vol. 36, pp. 963967, Aug. 1991. Stateoftheart from TKMIP is always provided in bitesize chunks that are easy to understand and followup on. Callegari, S., Rovatti, R., Setti, G., “Embeddable ADCBased True Random Number Generator for Cryptographic Applications Exploiting Nonlinear Signal Processing and Chaos,” IEEE Trans. on Signal Processing, Vol. 53, No. 2, pp. 793805, 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 crosscorrelations in the random number generators contaminating or degrading output results.] Go to Top When running TKMIP in the classified mode, the following slight variations (among familiar screens routinely encountered) are depicted below: User can set or undo passwordprotected 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 startup 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 TKMIP to their needs by providing their own Basic code to implement rotations or bias offsets (possible timevarying for each). Users may also select rotations to invoke from our wide repertoire of precoded 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 TKMIP code relating to rotations and bias offsets, we prevent our USERs from inadvertently clobbering the portions of TKMIP 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 tradedoff 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:
[6] 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., 911 June 1981. (Selected for reprinting in Navigation: Journal of the Institute of Navigation, Vol. 28, No. 4, pp. 263285, Winter 198182). [7] Kerr, T. H., “Impact of Navigation Accuracy in Optimized StraightLine Surveillance\Detection of Undersea Buried Pipe Valves,” Proceedings of National Marine Meeting of the Institute of Navigation (ION), Cambridge, MA, pp. 8090, 2729 October 1982. [8] Koopman, B. O., Search and Screening: General Principles with Historical Applications, Pergamon Press, NY, 1979. [9] Washburn, A. R., Search and Detection, Military Applications Section, Operations Research Society of America, Arlington, VA, 1981. [10] Stone, L. D., Theory of Optimal Search, 2^{nd} 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.] [11] Stone, L. D., Corwin, T. L., Barlow, C. A., Bayesian Multiple Target Tracking, Artech House Radar Library, Norwood, MA, 1999. [13] Mahdavi, M., “Solving NPComplete Problems by Harmony Search,” on pp. 5370 in MusicInspired Harmony Search Algorithms, Zong Woo Gee (Ed.), SpringerVerlag, NY, 2009. [14]
Kerr, T.H., “GPS\SSN Antenna Detectability,” Intermetrics Report No.
IRMA199, Cambridge, MA, 15 March 1983, for NADC (George Lowenstein). [15] Kerr, T. H., “Further Comments
on ‘Optimal Sensor Selection Strategy for DiscreteTime Estimators’,” IEEE
Trans. on Aerospace and Electronic Systems, Vol. 31, No. 3, pp. 11591166,
June 1995. [17] Kerr, T. H., “Sensor Scheduling in Kalman Filters: varying navaid fixes for tradingoff submarine NAV accuracy vs. ASW exposure,” Proceedings of The Workshop on Estimation, Tracking, and Fusion: A Tribute to Yaakov BarShalom (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. 104122, 17 May 2001. 
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