Screen shot #11 (Merely a picture to illustrate that our GUI is totally self-explanatory) Click here to go to our criticism of LQG Optimal Control Strategy and our Endorsement of LQG/LTR. Click here to go to our TK-MIP Classified Processing Capability. Click here to go to see how TK-MIP handles Rotations and Known Bias Offsets. Click here to go to our discussion of Search and Sweeprate Considerations. Astrom, K. J., "Computer Control of a Paper Machine--an Application of Linear Stocastic Control Theory," IBM Journal, pp. 389-405, July 1967. Mitter, S. K., and Moro, A. (Editors), Nonlinear Filtering and Stochastic Control, No. 972, Springer-Verlag, NY, 1982. 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 in 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 by IEEE Fellows Karl J. Astrom, H. H. Rosenbrock, and F. L. Lewis (and summarized in the image 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. However, LQG/LTR is only available for the time-invariant systems case. It utilizes an LTR inequality (all discussed in Lewis's textbook cited above). MIT Emeritus Prof. Sanjoy K. Mitter muddies the water somewhat by claiming success for LQG in:
Takashi Tanaka, Peyman Mohajerin Esfahani, and Sanjoy K. Mitter, "LQG Control With Minimum Directed Information: Semidefinite Programming Approach,"
IEEE Transactions on Automatic Control, Vol. 83, No. 1, pp. 37-52, Jan. 2018.
http://web.mit.edu/~mitter/www/publications/129_LQG%20Control_IEEE.pdf,
which proclaims the benefits of LQG in a signal processing context (that is much different from the
classical, well known, 50 year old methodology of the LQG problem that had
originally been addressed for years at MIT in Stochastic
Control applications and had already been widely rebuffed, as discussed
above. In the above paper discussed in this paragraph, they attempt to reinstate
LQG as a valid solution approach yet their Fig. 8 depicts a pre-Kalman filter
block and a
post-Kalman filter block but NO Kalman filter
block appears in between! However, the pre-Kalman filter
block is itself a Kalman Filter and likewise for the post-Kalman filter block
being itself a Kalman Filter.
Historically, MIT LQG utilized only one Kalman filter and, according to the
authors and demonstrated in the appendix of this paper, this current LQG resurrection
requires two Kalman filters in order to calculate the proper control to exert.
The original description of LQG may be found in Athans, M., Falb, P. L., Optimal
Control, McGraw-Hill Book Company, Inc., NY, 1966 and is elaborated on much
more in the Special LQG Issue of the IEEE Transactions on Automatic
Control, Vol. 16, No. 6, December 1971. 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 differential-difference 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, the 1963 book by Richard Bellman and Kenneth L. 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. 406-419, Apr. 1998.
(an expose)
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., “L^{2} 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 When running TK-MIP in the classified mode, the following slight variations (among familiar screens routinely encountered) are depicted below: 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:
Classified processing is sometimes required for navigation and target tracking where sensitive accuracies may be revealed or inferred
even from input data, such as gyro drift-rates, or from final processing outputs if encryption associated with classified processing were not invoked. In some sensitive situations, attention may be focused exclusively on output residuals instead since they can be used to investigate adequate performance without revealing any whole values that, otherwise, would be classified.
TK-MIP can be used for either unclassified or classified (CONFIDENTIAL) Kalman Filter-like estimator processing tasks since it incorporates PASSWORD protection, which can be enabled or disabled (i.e.,
"turned off" or "turned on"). When PASSWORD protection is invoked, intermediate computed files and results are encrypted except for output tables and plots, which would, otherwise, be useless
and indecipherable if they were encrypted too. Instead, USER should properly protect
OUTPUTTED tables and plots/graphs by the USER storing these in an approved
containers such as a safe or locking in a file cabinet that is equipped with a
metal bar using a combination lock (as is standard procedure). When intermediate processing results are encrypted, it slightly increases the signal processing burden, since
encryption operations and corresponding decryption operations must be performed at the transition boundary between each major step resulting in an output file to be further operated on before the entire process is complete. USER may customize TK-MIP to their needs by providing their own Basic code to implement rotations or bias offsets (possibly 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:
[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., 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). [7] 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. [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 NP-Complete Problems by Harmony Search,” on pp. 53-70 in Music-Inspired Harmony Search Algorithms, Zong Woo Gee (Ed.), Springer-Verlag, NY, 2009. [14]
Kerr, T.H., “GPS\SSN Antenna Detectability,” Intermetrics Report No.
IR-MA-199, Cambridge, MA, 15 March 1983, for NADC (George Lowenstein). [15] 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,
June 1995. [17] 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. |
TeK Associates’ Motto: “We work hard to make your job easier!” |