Analysis, estimation and control for perturbed and singular systems and for systems subject to discrete events.

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

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.

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-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

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 3xi,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].

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.

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.

III. Singular Systems

During 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

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.

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

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

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," 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 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 Scale-Based Decompositions in Singularly Perturbed Systems, "Linear Algebra 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 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 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 Processing on 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. 1141-1152.

[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' Il J. 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.

[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, 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 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-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 Weighting Patterns with Boundary-Value Descriptor Systems," submitted to SIAM J. Contr. and Opt. (also Rapport INRIA, No. 977, Feb. 1989).

[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.