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 FORPERTURBED 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 CrowleyProgram AdvisorDirectorate of Mathematical and Information SciencesAir Force Office of Scientific ResearchBuilding 410Bolling Air Force BaseWashington, D.C. 20332

November 15, 1989

I. SummaryIn 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 andsignal processing; (b) the investigation of theoretical questions related to singular

systems; and (c) the analysis of complex systems subject to or characterized by sequencesof 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 ProcessingIn 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 naturalstructure 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 aninvestigation 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-to-

fine 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 modelxi,j = A xil, j +A i+l,j +A 3xi,j-l +A 4xij+1 +B ui,j

with an observation equation of the formZi, 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 SystemsDuring the past year we have continued our work on the analysis of two point boundary-

value descriptor systems (TPBVDS's):Ex(k + 1) = Ax(k) + Bgp(k) (3.1)

with boundary conditions

v = Vix(O) + Vfx(N) (3.2)

and output

y(k) = Cx(k) (3.3)

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 EventsDuring 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 thesystem's ability to return to normal status after an excursion. Thedevelopment of methods for stabilization by state feedback.

b. The development of a theory of observability for DEDS. Given thefact that in many complex systems our sensed information concernsevents rather than states, we have considered an intermittentmeasurement model-i.e. we sense when one of a subset of events hasoccurred but do not receive direct information about other events.This leads to what we think is a useful notion of observability, inwhich there are points in time at which we know the state of thesystem, but they are intermittent.

c. The synthesis of a and b to address output stabilization. This issomewhat more complex that in usual control theory, thanks to theintermittent nature of the observations.

d. The reconstruction of complete event history from the observedevents. This is important for troubleshooting and in estimationproblems and complex inference problems. In this context we havecharacterized resiliency-i.e., the ability of the system to contain theeffects of measurement errors without catastrophic error propagation.

e. Tracking and restrictability of DEDS-the control of DEDS to followspecified strings of particular marked events or to produce strings withspecified 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 oneor more failures or errors.

f. Task following, aggregation, and higher-level control. Using theresults of e, we have developed a theory for controlling DEDS so thatone of a specified set of tasks is performed, where a task is specified asthe completion of one of a set of strings. This leads directly to thetask-level modeling of a DEDS, which greatly reduces modelcomplexity and thus our ability to consider higher-level controlquestions.

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 orin 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," Journal of 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 forBoundary-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 Scale-Based Decompositions in Singularly Perturbed Systems, "Linear Algebra and ItsApplications.

[4] H.J. Chizeck, A.S. Willsky, and D.A. Castanon, "Optimal Control of SystemsSubject to Failures with State-Dependent Rates," in preparation for submission toInt'l J. of Control.

[5] X.-C. Lou, G.C. Verghese, and A.S. Willsky, "Amplitude Scaling in Multiple TimeScale Approximations of Perturbed Linear Systems", in preparation for submissionto Int'l J. of Control.

[6] J.R. Rohlicek and A.S. Willsky, "Multiple Time Scale Approximation of Discrete-Time 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 ofSingularly Perturbed Semi-Markov Processes", in preparation for submission toStochastics.

[8] J.R. Rohlicek and A.S. Willsky, "Structural Analysis of the Time Scale Structure ofFinite-State Processes with Applications in Reliability Analysis," in preparation forsubmission to Automatica.

[9] M.A. Massoumnia, G.C. Verghese, and A.S. Willsky, "Failure Detection andIdentification 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 ofReachability in Perturbed Linear Systems," in the 1987 IEEE Conf. on Dec. andControl 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", inpreparation for submission to IEEE Trans. on Circuits and Systems.

[12] T.M. Chin and A.S. Willsky, "Stochastic Petri Net Modeling of Wave Sequences inCardiac Arrythmias," Computers and Biomedical Research, Vol. 22, 1989, pp.136-159.

[13] R. Nikoukhah, "A Deterministic and Stochastic Theory for Boundary ValueSystems," Ph.D. thesis, September 1988.

[14] R. Nikoukhah, A.S. Willsky, and B.C. Levy "Generalized Riccati Equations forTwo-Point Boundary Value Descriptor Systems," 1987 IEEE Conf. on Dec. andControl; journal version in preparation.

[15] J.R. Rohlicek and A.S. Willsky, "Structural Decomposition of Multiple Time ScaleMarkov Processes," Allerton Conference, Urbana, Illinois, October 1987.

[16] C.M. Ozveren, A.S. Willsky, and P.J. Antsaklis, "Some Stability and StabilizationConcepts for Discrete-Event Dynamic Systems," 1989 IEEE Conf. on Decision andControl, 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 ManufacturingSystems," S.M. Thesis, January 1988.

[19] C.A. Caromicoli, A.S. Willsky, and S.B. Gershwin, "Multiple Time Scale Analysisof Manufacturing Systems," 8th INRIA International Conference on Analysis andOptimization of Systems, Antibes, France, June 1988.

[20] R. Nikoukhah, A.S. Willsky, and B.C. Levy, "Reachability, Observability, andMinimality for Shift-Invariant, Two-Point Boundary-Value Descriptor Systems," toappear in the special issue of Circuits, Systems and Signal Processing on SingularSystems.

[21] R. Nikoukhah, B.C. Levy, and A.S. Willsky, "Stability, Stochastic Stationarity, andGeneralized Lyapunov Equations for Two-Point Boundary-Value DescriptorSystems," IEEE Trans. on Aut. Control, Vol. 34, No. 11, Nov. 1989, pp. 1141-1152.

[22] P.C. Doerschuk, R.R. Tenney, and A.S. Willsky, "Modeling ElectrocardiogramsUsing 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 ofInteracting Markov Chains with Applications to Electrocardiogram Analysis," toappear in Int' Il J. of Systems Science.

[24] B.C. Levy, M.B. Adams, and A.S. Willsky, "Solutions and Linear Estimation of2-D Nearest-Neighbor Models," to appear in May 1990 special issue ofProceedings of the IEEE on multidimensional signal processing.

[25] X.Z. Liu, "Transformations and Stability of Electrical Machines in Nearly IdealOperation," 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 IdealOperation," to be submitted.

[27] G.C. Verghese, I. Perez-Arriaga, L. Rouco, and F.L. Pagola, "Selective ModalAnalysis for Large Singular Systems," invited paper for session on SingularSystems, 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," inpreparation.

[29] C.M. Ozveren and A.S. Willsky, "Analysis and Control of Discrete-Event DynamicSystems: 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 ofDiscrete Event Dynamic System," submitted to J. of the ACM.

[31] C.M. Ozveren and A.S. Willsky, "Observability of Discrete Event DynamicSystems," accepted for publication in IEEE Trans. on Aut. Control.

[32] K.C. Chou, A.S. Willsky, A. Benventiste, and M. Basseville, "Recursive andIterative 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, "MultiresolutionSignal 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 forcurrent mode control," IEEE Power Electronics Specialists Conf., Milwaukee, June1989.

[35] S.R. Sanders, Nonlinear Control 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 forswitched power converters," prepared for submission to IEEE Trans. Circuits andSystems, and to IEEE ISCAS '90.

[37] S.R. Sanders, G.C. Verghese, and D.E. Cameron, "Nonlinear control of switchingpower converters," to appear in Control-Theory and 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 WeightingPatterns 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-PointBoundary-Value Descriptor Systems," IFAC Workshop, Prague, Czechoslovakia,Sept. 1989; journal version in preparation.

[41] M. Basseville, A. Benveniste, and A.S. Willsky, "Multiscale AutoregressiveProcesses," 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 DynamicSystems," submitted to IEEE Trans. on Aut. Control.

[44] C.M. Ozveren, "Invertibility of Discrete Event Dynamic Systems," submitted toMathematics of Control, Signals, and Systems.

[45] C.M. Ozveren and A.S. Willsky, "Tracking and Restrictablity in Discrete EventDynamic Systems," submitted to SIAM J. on Cont. and Opt..

[46] C.M. Ozveren and A.S. Willsky, "Aggregation and Multi-Level Control in DiscreteEvent Systems," submitted to Automatica.

[47] G.C. Verghese, "Averaged models in power electronics," invited paper underpreparation 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 forlarge signal control of high power factor ac-dc converters." submitted to IEEEPESC, San Antonio, June 1990.

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