November 1989
LIDS-R-1926
LABORATORY FOR INFORMATION AND DECISION SYSTEMS
Massachusetts Institute of Technology
Cambridge, Massachusetts 02139
Annual Technical Reportfor
Grant AFOSR-88-0032
ANALYSIS, ESTIMATION AND CONTROL FOR
PERTURBED AND SINGULAR SYSTEMS AND FOR
SYSTEMS SUBJECT TO DISCRETE EVENTS
for the period
1 October 1988 through 30 September 1989
Submitted to: Lt. Col. James Crowley
Program Advisor
Directorate of Mathematical and Information Sciences
Air Force Office of Scientific Research
Building 410
Bolling Air Force Base
Washington, D.C. 20332
November 15, 1989
I. Summary
In this report we summarize our accomplishments in the research program presently
supported by Grant AFOSR-88-0032 over the period from October 1, 1988 to
September 30, 1989. The basic scope of this program is the analysis, estimation, and
control of complex systems with particular emphasis on (a) multiresolution modeling and
signal processing; (b) the investigation of theoretical questions related to singular
systems; and (c) the analysis of complex systems subject to or characterized by sequences
of discrete events. These three topics are described in the next three sections of this
report. A full list of publications supported by Grant AFOSR-88-0032 is also included.
The principal investigator for this effort is Professor Alan S. Willsky, and
Professor George C. Verghese is co-principal investigator. Professors Willsky and
Verghese were assisted by several graduate research assistants as well as additional thesis
students not requiring stipend or tuition support from this grant.
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II. Multiresolution Modeling and Signal Processing
In the past few years our research efforts in this portion of our project have focused on
systems with dynamics at multiple time scales. We begin, however, with a discussion of
a new aspect of our research which we consider to be one of the most promising and
important directions for our future efforts.
This work is motivated by recent results on multiresolution representations of
signals and the related notion of wavelet transforms. Our objective has been to develop
probabilistic counterparts to this deterministic theory that can then be used as the basis
for signal modeling, optimal estimation, and related algorithms. During the past year we
have obtained some promising initial results in this area [32, 33, 41, 42].
The basic form of a multi-scale representation in 1-D is
+oo00
xm(t) =
i
x(m,n)4(2mt- n)
n=- 00
where ¢(t) is the basic scaling function defining the representation and m denotes the
resolution of the representation. Our work has focused on developing and exploiting
stochastic models for the coefficients x(m,n) as the scale, m, evolves. One natural
structure for the indices (m,n) is a dyadic tree, with a point (m,n) having two
descendants, (m + 1, 2n) and (m + 1, 2n + 1), at the next scale. This has led us to an
investigation of stochastic processes on such trees. In one part of our work [33, 41] we
are in the process of developing a theory of auto-regressive modeling for such processes.
The results here are far more complex than their time series counterparts because of the
tree structure, but we have been able to develop Levinson-like efficient algorithms for
constructing such models. In the other part of our work [32, 42] we have studied a class
of "state" processes evolving from coarse-to-fine scales. We have investigated the
estimation for such processes based on multiresolution, noisy measurements and have
developed three algorithmic structures: a fine-to-coarse algorithm reminiscent of the
Laplacian pyramid and taking great advantage of fast Haar transforms, an iterative
relaxation algorithm reminiscent of pyramidal multigrid methods, and a fine-to-coarse-tofine algorithm reminiscent of Rauch-Tung-Streibel. The last of these introduces a new
class of Riccati equations involving three steps-a measurement update step, a "predict"
step for moving up the tree, and a "merge" step for combining information from different
subtrees.
In our opinion the research area of wavelets, multiresolution stochastic models
and associated identification and estimation algorithms is an extraordinarily rich one,
with potential applications in a wide variety of fields. We expect to intensify our efforts
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in this area during the coming year. Among the topics to be investigated are the detailed
study of the new Riccati equations that we have discovered, the development of sensor
fusion algorithms based on our estimation theory, the formulation of the problem of
choosing the statistically optimal wavelet transform (in terms, for example, of whitening
the data), and the study of processes on more general weighted lattices arising in wavelet
theory.
In a related vein, we have begun to reexamine familiar 2D models (such as partial
differential equation models and ther discretizations into nearest neighbor models [24], as
well as general Markov random fields), in order to represent multiresolution sensing and
sensor fusion possiblitites. For example, [24] examines the solution and linear estimation
(smoothing) of the nearest neighbor model
xi,j = A xil, j +A
+A
i+l,j
3 xi,j-l
+A 4xij+1 +B ui,
j
with an observation equation of the form
Zi, j = Cxi,j
where x's and z's can all be vectors. With a low resolution sensor, one might have
measurements that are weighted local averages of the z's over some region. When can
this situation be transformed to the one above by appropriate augmentation of the local
state vector? The answer evidently depends on the extent and precise nature of the
averaging carried out by the sensor. Also of interest is the dependence of the smoothing
error covariance on the resolution of the sensor. Some of these questions are of
significance even in the context of ID systems. Associated with this are questions of
observability, such as determining the extent of observations around the point ij that will
suffice for xij to be uniquely determined. Similar questions can also be asked for partial
differential equation models and Markov random fields.
We have also been examining questions of multiresolution modeling for the
purposes of control, in the context of power electronic circuits. The switching behavior
of these circuits makes them an excellent vehicle for studying discrete event continuous
time systems. The switching is typically determined by the relationship between time
varying control inputs and periodically varying clocking waveforms.
The main focus of our research here is on averaged models: how to obtain them,
how to do control design with them, and how to translate averaged controls back to the
level of the switched system; see [34-37, 47, 48]. Averaged models are extensively used
already for certain classes of power circuits, but proper justification for what is done is
lacking, and hence the limits of validity are not known. In addition, the literature
contains several examples of incorrect derivations or applictions of averaged models.
Some of these are noted in [34, 47, 48].
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Our recent research has concentrated on understanding the connections between
classical averaging (e.g. Sanders and Verhulst, Averaging Methods in Nonlinear
Dynamical Systems, Springer, 1985) and averaging methods that are natural in the
context of power electronic circuits. While it has recently been demonstrated that
classical averaging can be applied to power electronic circuits (P. Krein et al., "On the
use of averaging for the analysis of power electronic systems," IEEE PESC, Milwaukee,
June 1989), we have begun to understand some of the limitations of this approach.
Classical averaging requires that the dynamical system be periodic or almost
periodic in order that certain infinite-time averages exist. It can thus be used to analyze
time invariant systems whose external drives are periodic or almost periodic; the resulting
averaged model is then essentially modeling the response to initial conditions in the
system. This is typically sufficient for analyzing the behavior of power circuits after a
feedback control loop has been closed around them, because the remaining external
drives are periodic clocking waveforms. The averaged models obtained by the classical
approach can thus permit tuning of parameters in a given feedback controller. However,
to design a feedback controller, it is more satisfactory to first model, simulate and analyze
the response of the open loop system to variations in control variables. Classical
averaging is not able to provide models for this purpose.
By contrast, the sort of averaging that seems more relevent to power electronics is
the running average over some interval T that is related to the period of the clocking
wavforms. This is well defined for non-periodic waveforms. (Almost nowhere in the
power electronics literature on averaging, however, is the definition of an average for
non-periodic waveforms made explicit! Nevertheless, the running average of a switching
variable, which we term the continuous duty ratio, is what the power electronic designer
typically thinks of as a control variable.)
We have begun to systematically study how models involving the running
averages of control variables and state variables may be obtained from a switched model,
what constraints must be observed in analyzing them, what the error is between the
averaged and switched modes, and how controls designed with the averaged models may
be lifted back to the switched models. Our averaged models can also be periodically
varying, as seen in [48], which leads to yet another layer of structure.
The other questions on averaging outlined in our proposal continue to be of
interest, and we shall continue to devote attention to them in the coming months. We
expect that what we are learning here will also be useful in more general discrete event
systems.
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We have also continued our work on singularly perturbed processes, although at a
reduced level compared to previous years. Results during this past year have been in the
development of asymptotic aggregation results for finite-state semi-Markov processes
and the asymptotic analysis of estimation algorithms for finite-state Markov processes.
Papers on both of these topics are being prepared.
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III. Singular Systems
During the past year we have continued our work on the analysis of two point boundaryvalue descriptor systems (TPBVDS's):
Ex(k + 1) = Ax(k) + Bgp(k)
(3.1)
with boundary conditions
v = Vix(O) + Vfx(N)
(3.2)
y(k) = Cx(k)
(3.3)
and output
Such models are a natural choice for the description of non-causal (e.g. spatial)
phenomena and signal processing tasks.
Our early work [13, 20, 21, 38] dealt with some of the basic system-theoretic
properties of these systems. Specifically, we investigated well-posedness for TPBVDS's
and in the process discovered a very useful normalized form. We also investigated the
two natural "state" processes for these systems, namely the inward process,
corresponding to propagating the boundary condition inward. and the outward process,
which involves propagating the effect of the input outward towrd the boundary. With
these nations in hand we investigated corresponding pairs of notions of reachability and
observability.
Subsequent work of ours provided a realization theory for TPBVDS's [39] and an
analysis of estimation problems for TPBVDS's [13, 40], including the introduction of a
new class of generalized Riccati equations. A number of additional papers are in
progress (e.g. [14, 38, 40] and, in addition, several new problems are under investigation.
In particular, we have begun the development of a corresponding boundary-value system
theory in two-dimensions. In this context we expect to make contact with the theory of
Markov random fields (MRF's). One concept under development is the extension of our
inward-outward 1D recursions to 2D. This should lead to new structures for
implementing 2D processing algorithms as well as new results in multidimensional
system theory.
Another important open question under investigation is concerned with the
development of inward-outward implementations of the optimal estimator. We expect
that such an implementation will be a more natural one than those developed to this point.
In addition such inward/outward structures have significant appeal in the 2D case.
We have previously sketched out an extension to singular systems of our work on
efficiently analyzing selected modes of large, linear, time invariant models. Our
Selective Modal Analysis (SMA) framework proceeds via iterative construction of
reduced models, obtained by appropriately modifying models obtained via partitioning of
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the original system. While numerical tests of a basic SMA algorithm for singular systems
have been successful, [27], there are many more choices and possibilities at each stage of
the algorithm in the singular case, each giving rise to associated research questions. The
options are richer than in the regular case, and discussion of such issues as convergence is
more delicate. We have also not yet obtained satisfactory extensions for some of the
results we obtained earlier for the regular case, such as simultaneous iterations for
multiple modes. These tasks continue to be of interest, and will be pursued in the coming
months.
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IV. Systems Subject to Discrete Events
During the past few years there has been considerable interest in the development of
control concepts and algorithms for complex processes that are characterized more by the
occurrence of discrete events than by differential equations representing the laws of
physics. Such processes are typically man-made-flexible manufacturing systems,
computer networks, etc.-and are often best described in symbolic, rather than numeric
form. Our work in this area is aimed at combining concepts from computer science and
from control in order to develop a meaningful theory of control for such systems. In
particular, the models and formalisms used in such a study come from the field of
computer science (automata, synchronous processes, etc.) while the problems and design
paradigms come from control (stability, regulation, robustness...).
Our research [16,17,29-31,43-46] has dealt with systems modeled as finite-state
automata in which control is effected by enabling or disabling particular events and in
which observations consist either of state measurements or the intermittent information
provided when any of a particular set of observable events occurs. The results described
in [16,17,29-31,43-46], which provide the basis for a servo theory for such discrete event
dynamic systems (DEDS), include the following:
a. The development of a stability theory for DEDS, characterizing the
system's ability to return to normal status after an excursion. The
development of methods for stabilization by state feedback.
b. The development of a theory of observability for DEDS. Given the
fact that in many complex systems our sensed information concerns
events rather than states, we have considered an intermittent
measurement model-i.e. we sense when one of a subset of events has
occurred but do not receive direct information about other events.
This leads to what we think is a useful notion of observability, in
which there are points in time at which we know the state of the
system, but they are intermittent.
c. The synthesis of a and b to address output stabilization. This is
somewhat more complex that in usual control theory, thanks to the
intermittent nature of the observations.
d. The reconstruction of complete event history from the observed
events. This is important for troubleshooting and in estimation
problems and complex inference problems. In this context we have
characterized resiliency-i.e., the ability of the system to contain the
effects of measurement errors without catastrophic error propagation.
e. Tracking and restrictability of DEDS-the control of DEDS to follow
specified strings of particular marked events or to produce strings with
specified properties. This is critical for the servo problem. Again we
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have characterized reliable control, i.e. the ability to recover from one
or more failures or errors.
f.
Task following, aggregation, and higher-level control. Using the
results of e, we have developed a theory for controlling DEDS so that
one of a specified set of tasks is performed, where a task is specified as
the completion of one of a set of strings. This leads directly to the
task-level modeling of a DEDS, which greatly reduces model
complexity and thus our ability to consider higher-level control
questions.
While these results are extremely promising and provide a solid foundation for
our research in this area, they have key limitations that must be addressed. In particular
although finite-state automata can be used to model many processes, the automaton
model is often not a natural framework for describing DEDS, and its use often leads to
cumbersome descriptions having apparent complexity far greater than the real system. In
particular, many algorithms in [16,17,29-31,43-46]] have computational complexity that
is polynomial in the cardinality of the automaton's state space. On the other hand, in
many cases a DEDS consists of an interconnection of many smaller DEDS so that the
Cartesian product state space can be enormous. However in many cases (e.g. in flexible
manufacturing systems), although these subsystems interact strongly, they do so
relatively infrequently--i.e. the systems evolve independently except for intermittent
coordination times. Taking advantage of structure such as this is one of the major
objectives of our future work.
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Publications
The publications listed below represent papers, reports, and theses supported in whole or
in part by the Air Force Office of Scientific Research under Grant AFOSR-88-0032:
[1]
J.R. Rohlicek and A.S. Willsky, "The Reduction of Perturbed Markov Generators:
an Algorithm Exposing the Role of Transient States," Journalof the ACM, Vol. 35,
No. 3, July 1988, pp. 675-696.
[2]
R. Nikoukhah, M.B. Adams, A.S. Willsky, and B.C. Levy, "Estimation for
Boundary-Value Descriptor Systems," Circuits,Systems, and Signal Processing,
Vol. 8, No. 1, April 1989, pp. 25-48.
[3]
X.-C. Lou, G.C. Verghese, A.S. Willsky, and P.G. Coxson, "Conditions for ScaleBased Decompositions in Singularly Perturbed Systems, "LinearAlgebra and Its
Applications.
[4]
H.J. Chizeck, A.S. Willsky, and D.A. Castanon, "Optimal Control of Systems
Subject to Failures with State-Dependent Rates," in preparation for submission to
Int'l J. of Control.
[5]
X.-C. Lou, G.C. Verghese, and A.S. Willsky, "Amplitude Scaling in Multiple Time
Scale Approximations of Perturbed Linear Systems", in preparation for submission
to Int'l J. of Control.
[6]
J.R. Rohlicek and A.S. Willsky, "Multiple Time Scale Approximation of DiscreteTime Finite-State Markov Processes," System and Control Letters, Vol. 11, 1988,
pp. 309-314.
[7]
J.R. Rohlicek and A.S. Willsky, "Hierarchical Aggregation and Approximation of
Singularly Perturbed Semi-Markov Processes", in preparation for submission to
Stochastics.
[8]
J.R. Rohlicek and A.S. Willsky, "Structural Analysis of the Time Scale Structure of
Finite-State Processes with Applications in Reliability Analysis," in preparation for
submission to Automatica.
[9]
M.A. Massoumnia, G.C. Verghese, and A.S. Willsky, "Failure Detection and
Identification in Linear Time-Invariant Systems," IEEE Trans. on Aut. Control,
Vol. 34, No. 3, March 1989, pp. 316-321.
[10] C.M. Ozveren, G.C. Verghese, and A.S. Willsky, "Asymptotic Orders of
Reachability in Perturbed Linear Systems," in the 1987 IEEE Conf. on Dec. and
Control and the IEEE Trans. on Automaic Control, Vol. 33, No. 10, Oct. 1988,
pp. 915-923.
[11] J.R. Rohlicek and A.S. Willsky, "Aggregation for Compartmental Models", in
preparation for submission to IEEE Trans. on Circuits and Systems.
[12] T.M. Chin and A.S. Willsky, "Stochastic Petri Net Modeling of Wave Sequences in
Cardiac Arrythmias," Computers and Biomedical Research, Vol. 22, 1989, pp.
136-159.
[13] R. Nikoukhah, "A Deterministic and Stochastic Theory for Boundary Value
Systems," Ph.D. thesis, September 1988.
[14] R. Nikoukhah, A.S. Willsky, and B.C. Levy "Generalized Riccati Equations for
Two-Point Boundary Value Descriptor Systems," 1987 IEEE Conf. on Dec. and
Control; journal version in preparation.
[15] J.R. Rohlicek and A.S. Willsky, "Structural Decomposition of Multiple Time Scale
Markov Processes," Allerton Conference, Urbana, Illinois, October 1987.
[16] C.M. Ozveren, A.S. Willsky, and P.J. Antsaklis, "Some Stability and Stabilization
Concepts for Discrete-Event Dynamic Systems," 1989 IEEE Conf. on Decision and
Control, Tampa, Fl., Dec. 1989.
[17] C.M. Ozveren, "Analysis and Control of Discrete-Event Dynamic Systems," Ph.D.
thesis, August 1989.
[18] C.A. Caromicoli, "TimeScale Analysis Techniques for Flexible Manufacturing
Systems," S.M. Thesis, January 1988.
[19] C.A. Caromicoli, A.S. Willsky, and S.B. Gershwin, "Multiple Time Scale Analysis
of Manufacturing Systems," 8th INRIA International Conference on Analysis and
Optimization of Systems, Antibes, France, June 1988.
[20] R. Nikoukhah, A.S. Willsky, and B.C. Levy, "Reachability, Observability, and
Minimality for Shift-Invariant, Two-Point Boundary-Value Descriptor Systems," to
appear in the special issue of Circuits, Systems and Signal Processingon Singular
Systems.
[21] R. Nikoukhah, B.C. Levy, and A.S. Willsky, "Stability, Stochastic Stationarity, and
Generalized Lyapunov Equations for Two-Point Boundary-Value Descriptor
Systems," IEEE Trans. on Aut. Control, Vol. 34, No. 11, Nov. 1989, pp. 11411152.
[22] P.C. Doerschuk, R.R. Tenney, and A.S. Willsky, "Modeling Electrocardiograms
Using Interactive Markov Chains," to appear in Int'l J. of Systems Science.
[23] P.C. Doerschuk, R.R. Tenney, and A.S. Willsky, "Event-Based Estimation of
Interacting Markov Chains with Applications to Electrocardiogram Analysis," to
appear in Int' IlJ. of Systems Science.
[24] B.C. Levy, M.B. Adams, and A.S. Willsky, "Solutions and Linear Estimation of
2-D Nearest-Neighbor Models," to appear in May 1990 special issue of
Proceedings of the IEEE on multidimensional signal processing.
[25] X.Z. Liu, "Transformations and Stability of Electrical Machines in Nearly Ideal
Operation," S.M. Thesis, Mech. Engr. Dept., M.I.T., May 1988.
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[26] X.Z. Liu and G.C. Verghese, "Stabilty of Electrical Machines in Nearly Ideal
Operation," to be submitted.
[27] G.C. Verghese, I. Perez-Arriaga, L. Rouco, and F.L. Pagola, "Selective Modal
Analysis for Large Singular Systems," invited paper for session on Singular
Systems, SIAM Conf. on Control in the 90's, San Francisco, May 1989.
[28] G.C. Verghese and X.Z. Liu, "Averaged Models for Resonant Converters," in
preparation.
[29] C.M. Ozveren and A.S. Willsky, "Analysis and Control of Discrete-Event Dynamic
Systems: A State Space Approach," SIAM Conf. on Control in the 90's,
San Francisco, May 1989.
[30] C.M. Ozveren, A.S. Willsky, and P.J. Antsaklis "Stability and Stabilizability of
Discrete Event Dynamic System," submitted to J. of the ACM.
[31] C.M. Ozveren and A.S. Willsky, "Observability of Discrete Event Dynamic
Systems," accepted for publication in IEEE Trans. on Aut. Control.
[32] K.C. Chou, A.S. Willsky, A. Benventiste, and M. Basseville, "Recursive and
Iterative Estimation Algorithms for Multi-Resolution Stochastic Processes,"
28th IEEE Conf. on Dec. and Control, Dec. 1989.
[33] A. Benveniste, M. Basseville, A.S. Willsky, and K.C. Chou, "Multiresolution
Signal Processing and Modeling of Isotropic Processes on Homogeneous Trees,"
Int'l Symp. on Math.Theory of Networks and Systems, Amsterdam, June 1989.
[34] G. Verghese, C. Bruzos, and K. Mahabir, "Averaged and sampled-data models for
current mode control," IEEE Power Electronics Specialists Conf., Milwaukee, June
1989.
[35] S.R. Sanders, NonlinearControl of Switching Power Converters,Ph.D. Thesis,
EECS Department, M.I.T., January 1989.
[36] S.R. Sanders and G.C. Verghese, "Synthesis of averaged circuit models for
switched power converters," prepared for submission to IEEE Trans. Circuits and
Systems, and to IEEE ISCAS '90.
[37] S.R. Sanders, G.C. Verghese, and D.E. Cameron, "Nonlinear control of switching
power converters," to appear in Control-Theoryand Advanced Technology.
[38] R. Nikoukhah, A.S. Willsky, and B.C. Levy, "Reachability, Observability,
Minimality, and Extendability for Two-Point Boundary-Value Descriptor Systems,"
Rapport INRIA, No. 920, Oct. 1988; to be submitted to Int'l. J. Control.
[39] R. Nikoukhah, B.C. Levy, and A.S. Willsky, "Realization of Acausal Weighting
Patterns with Boundary-Value Descriptor Systems," submitted to SIAM J. Contr.
and Opt. (also Rapport INRIA, No. 977, Feb. 1989).
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[40] R. Nikoukhah, A.S. Willsky, and B.C. Levy, "An Estimation Theory for Two-Point
Boundary-Value Descriptor Systems," IFAC Workshop, Prague, Czechoslovakia,
Sept. 1989; journal version in preparation.
[41] M. Basseville, A. Benveniste, and A.S. Willsky, "Multiscale Autoregressive
Processes," M.I.T. Lab. for Inf. and Dec. Systems Report LIDS-P-1880,
June 1989; journal version in preparation.
[42] A.S. Willsky, K.C. Chou, A. Benveniste, and M. Basseville, " Wavelet Transforms,
Multiresolution Dynamical Models, and Multigrid Estimation Algorithms,"
submitted to the 1990 IFAC World Congress, Tallinn, USSR, Aug. 1990.
[43] C.M. Ozveren and A.S. Willsky, "Output Stabilizablity of Discrete Event Dynamic
Systems," submitted to IEEE Trans. on Aut. Control.
[44] C.M. Ozveren, "Invertibility of Discrete Event Dynamic Systems," submitted to
Mathematics of Control, Signals, and Systems.
[45] C.M. Ozveren and A.S. Willsky, "Tracking and Restrictablity in Discrete Event
Dynamic Systems," submitted to SIAM J. on Cont. and Opt..
[46] C.M. Ozveren and A.S. Willsky, "Aggregation and Multi-Level Control in Discrete
Event Systems," submitted to Automatica.
[47] G.C. Verghese, "Averaged models in power electronics," invited paper under
preparation for special issue of J. Inst. of Electr. and Tel. Engineers, India.
[48] K. Mahabir, G.C. Verghese, J. Thottuvelil, and A. Heyman, "Linear models for
large signal control of high power factor ac-dc converters." submitted to IEEE
PESC, San Antonio, June 1990.
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