Inference-Based Decentralized Prognosis in Discrete Event Systems

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VII. CONCLUSION

In this technical note, a new compensator with adaptive internalmodel for minimum and non-minimum phase linear MIMO systemsis presented. It is shown that the proposed controller ensures globalconvergence of the estimator and preserves the output regulation prop-erties despite variations of the external signal generator parameters.The property of robustness of the closed-loop system in presence ofthe plant parameters variations in a neighborhood of nominal valuesis also guaranteed and simulations results on an illustrative exampleshow the good performance of the regulator.

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[15] H. W. Knobloch, A. Isidori, and D. Flockerzi, Topics in ControlTheory. Berlin, Germany: Birkhäuser Verlag, 1993.

[16] A. Isidori, Nonlinear Control Systems, 3rd ed. New York: Springer-Verlag, 1995.

[17] B. A. Francis and W. M. Wonham, “The internal model principle ofcontrol theory,” Automatica, vol. 12, pp. 457–465, 1976.

[18] E. J. Davison, “The robust control of a servomechanism problem forlinear time-invariant multivariable systems,” IEEE Trans. Autom. Con-trol, vol. AC-21, no. 1, pp. 25–34, Jan. 1976.

[19] B. A. Francis, “The linear multivariable regulator problem,” SIAM. J.Control Optim., vol. 14, pp. 486–505, 1977.

[20] C. I. Byrnes, F. Delli Priscoli, A. Isidori, and W. Kang, “Structurallystable output regulation of nonlinear systems,” Automatica, vol. 33, no.3, pp. 369–385, 1997.

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[24] G. Obregón-Pulido, B. Castillo-Toledo, and A. Loukianov, “Adaptiveinternal model regulation of a PVTOL vehicle,” in Proc. 2nd IFACSymp. Syst., Struct. Control, Oaxaca, Oaxaca, México, 2004, pp.573–578.

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Inference-Based Decentralized Prognosisin Discrete Event Systems

Shigemasa Takai, Member, IEEE, and Ratnesh Kumar, Fellow, IEEE

Abstract—For discrete event systems, we study the problem of predictingfailures prior to their occurrence, also referred to as prognosis, in the infer-ence-based decentralized framework where multiple decision-makers in-teract to make the global prognostic decisions. We characterize the class ofsystems for which there are no missed detections (all failures can be prog-nosed prior to their occurrence) and no false alarms (all prognostic deci-sions are correct) by introducing the notion of -inference-prognosability,where the parameter represents the maximum ambiguity level of anywinning prognostic decision. An algorithm for verifying -inference-prog-nosability is presented. We also show that the notion of coprognosability in-troduced in our prior work is the same as 0-inference-prognosability, andas the parameter is increased, a larger class of prognosable systems isobtained.

Index Terms—Discrete event systems (DES), failure prognosis, inference-based decentralized decision-making.

I. INTRODUCTION

T HE behaviors of a discrete event system (DES) consist of allsequences of events (called traces) it can execute starting from

its initial state. A system specification is used to identify the set oftraces that are desirable. Execution of a trace that violates a specifi-cation constitutes a failure. The task of failure prognosis is to predictan impending failure prior to its occurrence. This helps provide a timefor reacting to an impending failure so that appropriate corrective ac-tions may be initiated prior to its occurrence. Note the contrast with thetask of diagnosis which requires the detection of a failure after its oc-currence. The prediction of a failure based on a statistical analysis was

Manuscript received October 29, 2009; revised May 02, 2010; acceptedSeptember 08, 2010. Date of publication October 11, 2010; date of currentversion January 12, 2011. This work was supported in part by the Na-tional Science Foundation Grants NSF-ECS-0601570, NSF-ECCS-0801763,NSF-CCF-0811541, and NSF-ECCS-0926029, and in part by MEXT underGrant-in-Aid for Scientific Research (C) 21560462. Recommended by Asso-ciate Editor H. Marchand.

S. Takai is with the Division of Electrical, Electronic and Information En-gineering, Osaka University, Osaka 565-0871, Japan (e-mail: [email protected]).

R. Kumar is with the Department of Electrical and Computer Engineering,Iowa State University, Ames, IA 50011-3060 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2010.2085590

0018-9286/$26.00 © 2010 IEEE

166 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 56, NO. 1, JANUARY 2011

considered in [1]. Due to the statistical nature of the analysis, issuing aprediction alert only means a high confidence in a future occurrence of afailure (but not full confidence). To capture the inevitability of a futurefailure, the notion of indicator traces (which indicate that a failure isguaranteed to occur) was introduced in [3], where their bounded delaydetection was also studied.

In [2], a notion of uniformly-bounded predictability of failures wasformulated: Each failure trace must possess a nonfailure prefix suchthat any indistinguishable trace has the property that a failure is in-evitable within a uniformly bounded number of steps. Note while theexistence of a uniform bound for the delay of failure detection is es-sential for defining diagnosability (otherwise a diagnoser may end upwaiting for an arbitrarily long period before diagnosing a failure), theexistence of a uniform bound within which a failure is guaranteed tooccur is not essential for defining prognosability. This observation ledus to weaken the definition of prognosability in [7]: Each failure tracemust possess a nonfailure prefix such that each indistinguishable traceis an indicator trace. (Recall an indicator trace is one for which a failureis inevitable.)

In our work [7], a local prognoser issues a prognostic decision onlywhen it is fully unambiguous about it. It is known through earlier worksreported in [8], [11] that adding inferencing can help arrive at a cor-rect decision even when the local decision-makers are not fully unam-biguous. While these prior inferencing-based approaches relied on a“single-level” of inferencing, a framework allowing multilevel infer-encing was later presented in [5] (for supervisory control), [6] (for di-agnosis of failures), and [10] (for diagnosis of nonfailures).

In this note, we introduce the inference-based decentralized deci-sion-making framework for the prognosis of failures. The frameworksupports multiple levels of inferencing over the ambiguities of the selfand the others. Each local prognoser uses its observations of the systembehavior to come up with its prognostic decision together with a gradeor level of ambiguity for that decision. A minimum (level-zero) ambi-guity decision is issued by a local prognoser when following all traces,producing the same observation as the one received, either a failure isinevitable (so a positive decision is issued) or it is not inevitable (soa negative decision is issued). In general a local prognoser will issuea positive (respectively, negative) decision with an ambiguity level �following a certain observation if for each ambiguous trace, there existsanother local prognoser that can issue a negative (respectively, positive)decision with an ambiguity level at most � � �. Note in certain situa-tions it is possible that a local prognostic decision is neither “positive”nor “negative,” but “unsure.” The global prognostic decision is taken tobe the same as a local prognostic decision whose ambiguity level is theminimum. (Such a local decision can be considered to be a “winning”local decision.)

We characterize the class of systems for which there are no misseddetections (all failures can be prognosed prior to their occurrence) andno false alarms (all prognostic decisions are correct) by introducingthe notion of � -inference-prognosability, where the parameter � rep-resents the maximum ambiguity level of any winning prognostic de-cision. An algorithm for verifying � -inference-prognosability is pre-sented. We also show that as the parameter � is increased, a largerclass of prognosable systems is obtained. Further the notion of coprog-nosability introduced in [7] is the same as 0-inference-prognosability,implying that even the class of 1-inference-prognosable systems sub-sumes the class of coprognosable systems introduced in [7].

The results presented in this note were first reported at a conference[9] but without proofs. This note contains their proofs.

II. NOTATION AND PRELIMINARIES

We consider a DES modeled by a nondeterministic automaton � �� , where � is the set of states, � is the finite set of

events, a function � � � � �� is the transition function,�� is the set of initial states, and�� is the set of marked oraccepting states. � is said to be deterministic if the transition functioncan be written as a partial function � � � � � � � and �� .Let �� be the set of all finite sequences of events including the emptysequence �. Elements of �� are called traces, and subsets of �� arecalled languages. For each trace � ��, �� denotes its length. For any� , where denotes the set of all nonnegative integers, �� denotes the set of all traces with� or more events.The transition function � can be generalized to � � ��

in a natural way. The generated and marked (or accepted) languages of� are respectively defined as, �� , and�� .

For a trace � ��, the set of all prefixes of � is denoted by ��.The notation � � � denotes that � is a prefix of �. For a language , theset of all prefixes of traces in is defined as ��

�� . is said to be (prefix-)closed if � �� . A language is saidto be deadlock-free if for any � �� , there exists a trace � ��

such that �� ; otherwise � is called a deadlocking traceof . The language after � ��, denoted by ��, is defined as �� .

Let � � �� denote the index set of local prognosers thatperform the task of prognosis without sharing their observations. Weassume that the limited sensing capabilities of the �th local prognoser�� (� �) can be represented as the local observation mask, �� , where � is the set of locally observed symbols,and �� . The map �� is generalized to ��

� � �� and

�� in a natural way

��

��

��

III. INFERENCE-BASED DECENTRALIZED PROGNOSIS FRAMEWORK

Let �� be a closed language representing the generated languageof a plant, and � be a nonempty closed language representinga nonfailure specification language. Traces in � are consideredfailure traces and the task of prognosis is to predict the execution ofany trace in � . Without loss of generality, the plant language can be taken to be deadlock-free. Otherwise we can extend each dead-locking trace by an unbounded sequence of a newly added event thatis unobservable to all prognosers. This will make the language dead-lock-free without altering any of the prognosability properties since thenewly added event does not produce any observation to any of the prog-nosers.

We introduce the notion of an inference-based decentralized prog-noser that consists of a set of local prognosers and a central decisionfusion unit. Let the set � � �� be the set of prognostic deci-sions, where “1” means a failure is guaranteed to occur in future (i.e.,all extensions are eventually faulty), “0” means a failure is not guaran-teed to occur in future (i.e., some extensions are never faulty), and “�”means a prognoser is unsure whether or not a failure is guaranteed tooccur (i.e., unsure whether all extensions are eventually faulty or someextensions are never faulty). Each inference-based local prognoser ��is defined as a map �� , where for each �

��

Here �� denotes the prognostic decision of �� followingan observation�� , and�� denotes the am-biguity level of the prognostic decision of��. Let �� be the minimumambiguity level of local decisions, i.e., �� .


Fig. 1. Automata and of Example 1.

The decentralized prognoser �� that consists of local prog-nosers �� (� � �) issues global prognostic decisions. Formally,�� is defined as a map �� . For each � � � , theprognostic decision �� is given as follows:

��

�

�� if ��

�� if ��

�� otherwise.

In other words, the global prognostic decision is taken to be the same asa local prognostic decision possessing the minimum level of ambiguity.

A useful notion of a decentralized prognoser is the largest ambiguitylevel � � of any sure decision, and the preservation of certainty of adecision with a decrease in the ambiguity level (if a certain ambiguitylevel decision is “sure,” then all lower ambiguity level decisions arealso “sure”). We refer to such a prognoser to be “ -inferring.”

Definition 1: A decentralized prognoser �� is saidto be -inferring if the following two conditions hold:

1) �� ;2) ��

�� .

IV. EXISTENCE/SYNTHESIS OF INFERENCE-BASED

DECENTRALIZED PROGNOSERS

We introduce the following notions of boundary traces (for which afailure can occur in a next step), indicator traces (for which a failurein future is guaranteed), and nonindicator traces (that are not indicatortraces).

Definition 2: [7] Given a pair �� of closed languages with� ��, we define the set of

• boundary traces of � with respect to � as, �� ;

• indicator traces of � with respect to � as, �� ;

• nonindicator traces of � with respect to � as, �� .

Note �� .In this section we introduce the notion of -inference-prognos-

ability as a necessary and sufficient condition for the existence of an -inferring prognoser with the following properties.

• There are no missed detections, i.e., each failure is prognosed priorto its occurrence

��

�� (1)

• There are no false alarms, i.e., an incorrect prognostic decision isnever issued

��

�� (2)

Condition (1) requires the existence of a nonfailure prefix of eachfailure trace where a prognostic decision “1” is issued, and which iscontinued to be held in future (i.e., the prognoser does not change itsmind), whereas (2) requires that for each indicator (respectively, non-indicator) trace a prognostic decision “0” (respectively, “1”) is not is-sued. Since � � � is a nonfailure prefix of a failure trace � � �� ,�� automatically holds in (1).

In order to introduce the notion of -inference-prognosability,we inductively define a monotonically decreasing sequence�� of language pairs as follows:

• Base step

��

• Induction step

��

��

��

��

The computation of the sequence �� of languagepairs starts with �� , the set of indicator traces, and�� , the set of nonindicator traces. Note that �� is asublanguage of �� consisting of those traces for which for each � � �there exists an �-indistinguishable trace in �� . As a result, whenthe plant executes a trace in ��, all the local prognosers will beambiguous as to whether the executed trace is in �� or in �� . Thesublanguage �� of �� can be understood in a similar fashion.

Let � � be a given nonnegative integer. ( represents anindex of prognosability to be elaborated later.) Using the sequence�� of language pairs, a local prognoser computes itsprognostic decision and associates a level of ambiguity with sucha decision as follows. For each � � � , the �th local prognoser ��computes

�� (3)

�� (4)

Note that �� and �� are bounded above by � �.Here �� represents the ambiguity level of a failure prognosticdecision “contemplated” by the �th prognoser following the observa-tion��. (When�� , it denotes the minimum index� such that the observation�� does not match with the observationsof any of the traces in �� .) Similarly, the notation �� repre-sents the ambiguity level of a nonfailure prognostic decision “contem-plated” by the �th prognoser following the observation ��. Whichof the two contemplated decisions is ultimately issued is decided bycomparing the two ambiguity levels, �� vs. �� , and


favoring the smaller one. That is, the prognostic decision and ambiguitylevel �� is determined as follows:

��

�� if ��

�� if ��

�� if ��

(5)

and

�� (6)

Example 1: We consider a plant modeled by the finite automaton shown in Fig. 1(a). Let ��

�� if � � � � �� otherwise,

�� if � � �� otherwise.

Also, let � � � be a language generated by the finite automaton �

shown in Fig. 1(b).We synthesize the decentralized prognoser using (3)–(6) for � � .

We first need to compute the language pairs ��. Ini-tially, we have

��

��

��

��

��

Since

��

��

��

��

we have �� and �� . Also,since

��

��

we have �� and �� .The local decisions of �� and �� computed using (3)–(6) are shown

in Table I. For example, �� is computed as follows. Since � �� , we have by (3) that �� . Also, since � ��, we have by (4) that �� . It followsthat � � �� . By (5) and (6), we have �� and �� , which implies that �� issues a prognostic decision“1” following the observation � � �� with the ambiguity level1.

Then, the global prognostic decisions of the decentralized prognoser�� are computed as shown in Table II. For example, �� is computed as follows. Since � � �� and �� , we have �� and �� .

The following lemma shows that the decentralized prognoser givenby (3)–(6) is an � -inferring one with no false alarms. The proof issimilar to Lemma 3 in [6].

Lemma 1: Consider the decentralized prognoser �� consisting of local prognosers �� (� � �),and defined by (3)–(6). Then, �� is an � -inferring decentralizedprognoser satisfying (2).

TABLE ILOCAL DECISIONS OF AND

TABLE IIGLOBAL DECISIONS OF

We next introduce the notion of � -inference-prognosability andshow that under this condition, the decentralized prognoser, as definedby (3)–(6), also has no missed detections. In fact, this conditionserves as a necessary and sufficient condition for the existence of an� -inferring decentralized prognoser with no missed detections andfalse alarms. The condition requires that each boundary trace is alsoan indicator trace and can be inferred to be so within � levels ofinferencing.

Definition 3: The pair �� of closed languages with � � � issaid to be � -inference-prognosable if �� .

Remark 1: The definition of � -inference-prognosability dependson local observation masks �� since the language �� does.In this technical note, we make a standard assumption that the set�� of local observation masks is given. The mask-designproblem, namely, how to find �� for the property of � -in-ference-prognosability, is certainly interesting but beyond the scopeof this technical note. This may be performed by a top-down or abottom-up approach as presented in [4].


We need the following lemma, the proof of which is similar toLemma 1 in [6].

Lemma 2: Consider the decentralized prognoser �� consisting of local prognosers �� (� � �), anddefined by (3)–(6). Then for any � � �

� � ��

The following lemma states that if �� is � -inference-prognos-able, then there are no missed detections under the decentralized prog-nosers given by (3)–(6).

Lemma 3: Consider the decentralized prognoser �� consisting of local prognosers �� (� � �), anddefined by (3)–(6). If the pair �� of closed languages is � -infer-ence-prognosable, then (1) holds.

Proof: For any � � � , there exist � � �� and� � � such that �� . It suffices to prove that �� . Suppose for contradiction that �� . Since �� is� -inference-prognosable, we have � � �� .Since � � �� , there exists � � � suchthat � � � , � � ��, and � �� . So, there exists � � � suchthat � ��

� ��. It follows that �� . We have

��

Since �� , there exists � � � such that �� and �� . Then we have

��

which together with (4) implies that

��

We have �� , which implies that � �� . Since� � ��, this contradicts Lemma 2.

The following theorem establishes the main result of the note.Theorem 1: Given a pair �� of closed languages with � � ,

there exists an� -inferring decentralized prognoser �� satisfying (1) and (2) if and only if �� is� -inference-prognosable.

Proof: (�) Consider the decentralized prognoser �� consisting of local prognosers �� (� � �), and defined by (3)–(6). By Lemmas 1 and 3, �� is an� -inferring decentralized prognoser satisfying (1) and (2).

(�) Let �� be an � -inferring decentralized prog-noser satisfying (1) and (2). We show that

�� (7)

For any � � ��, there exists � � � such that �� .Since �� satisfies (1), there exists � � �� such that�� for any � � � �� . Since � � and� � �� , we have �� . Further, it follows from(2) that � � ��. Thus, �� holds.

It remains to prove that �� . Suppose for con-tradiction that �� . We first show that there exist�� and �� such that

• �� if � is an even number�� if � is an odd number

(� � �� );

• �� (� � �� );

• �� ;• �� (� � �� ).We can pick up �� . It follows from (7) that

�� . Since �� is � -inferring, we have �� .Then, we have �� . Also, there exists �� suchthat �� and �� .

Since �� , we have �� . Then,

we can pick up �� such that �� . Since�� , we have by (2) that �� . Also,since �� , it followsfrom the second condition of Definition 1 that �� . Thereexists �� such that �� and �� . Moreover, since �� , we have�� . Thus, we have�� .

Since �� , we have �� . Then,

we can pick up �� such that �� . Since�� , we have by (2) that �� . Also, since�� , it follows fromthe second condition of Definition 1 that �� . There ex-ists �� such that �� and �� . Moreover, since �� , we have�� . Thus, we have�� .

By repeating this procedure, we can obtain �� and�� which satisfy the above four conditions. Thenwe have the following two cases for �� with �� that�� and �� .

We first consider the case that �� . Since �� ,we have by (2) that �� . Also, since � � �� and �� , it follows from the second conditionof Definition 1 that �� . There exists � � � such that� � �� and �� . Since ��

� ��, there exists �� such that�� . Further, since � � �� , we have�� , which implies together with(2) and the second condition of Definition 1 that �� . However, since �� and �� , we have �� , which contradicts�� .

We next consider the case that �� . Since �� , we have by (2) that �� . Also, since �� , it follows from the second condition of Definition 1 that�� . There exists � � � such that � � �� and �� . Since ��

� ��, thereexists �� such that �� . Fur-ther, since � � �� , we have �� , which implies together with (2) and thesecond condition of Definition 1 that �� . However,since �� and �� , we have �� , which contradicts �� .

In the following, we show that the system of Example 1 is 2-in-ference-prognosable but it is not 1-inference-prognosable. (It will beshown in a theorem below that in general, the class of prognosable sys-tems grows as the parameter � is increased.)

Example 2: We revisit the setting of Example 1. We have

��

��

��


It follows that �� , which implies that �� isnot 1-inference-prognosable. Since

��

we have�� . Thus, we have �� , whichimplies that �� is 2-inference-prognosable.

By Table II, we can verify that �� is a 2-inferring decentralizedprognoser satisfying (1) and (2).

The following remark discusses how to verify -inference-prognos-ability of a pair �� of closed regular languages with � � �.

Remark 2: The verification of -inference-prognosability is equiv-alent to checking the emptiness of the languages �� and �� . Since checking the emptiness of a lan-guage can be done linearly in the size of an acceptor of the language, weonly discuss the computation of certain acceptors for �� and �� . A point of our construction is to estab-lish that the step of “determinization” (which is exponential in the sizeof the automaton being determinized) is never required as long as thegenerator for the nonfailure specification is a deterministic automaton.

Let � �� be the finite plant model with �� , and � �� be a finite deterministic gen-erator of the nonfailure specification language, i.e.,�� . For computing ��, we construct the synchronous composi-tion �� of and , where � � � � ,�� , and � � � �� is defined in theusual manner. Then �� holds. Let�� .Then, for the finite automaton � ��

��, we have��

�� .Next, we discuss the computation of�� and �� (� � �) inductively

over �. For the base step (� � �), let �� be the set of indi-cator states of from which no cycle in can be reached [7]. Also,let �� be the set of nonindicator states of from which acycle in can be reached [7]. The identification of �� and �� can be performed in complexity linear in the size of . Then the lan-guages�� and�� are computed by replacingthe marked state set � of with �� and �� , respectively. Noteto check the emptiness of �� , the complement�� of�� can be computed in the usual manner (a deter-minization step is avoided since is already deterministic), whereasthe intersection �� can be computed using thesynchronous composition operator.

We discussed above the computation of �� and �� (the base step).For the induction step, let � and � be finite acceptors of �� and�� , respectively. For each � � , a finite acceptor of ��

��

is constructed as follows: Replace each transition that exists in �by a set of transitions on all ��-indistinguishable events (including�). Note that since an �-transition is implicitly defined at each stateas a self-loop, unobservable events will get added as self-loops ateach state of � . Then, the resulting, possibly nondeterministic,automaton accepts ��

��. It should be noted that this re-

sulting automaton, denoted by ��

�� , has the same stateset as � . In the same way, we can construct a finite automatonaccepting ��

��, denoted by ��

�� . Then, the

synchronous compositions � ��

�� and � ��

��

�� accept �� and ��,respectively. Let �� and �� be the state sets of � and � ,respectively. Then the size of the state sets of � and � are�� and �� , respectively. Once anacceptor for �� has been computed (using the above inductive

procedure), the emptiness of �� can be checked by areachability analysis over � �� .

It can be concluded that the complexity of checking -inference-prognosability is polynomial in the sizes of and , and ��-foldexponential in ��, the number of sites.

V. PROPERTIES OF -INFERENCE-PROGNOSABILITY

In this section, we study properties of -inference-prognosable sys-tems. First, we show that the class of coprognosable systems studied in[7] is equivalent to the class of 0-inference-prognosable systems. (From[7], coprognosability serves as a necessary and sufficient condition forpredicting all failures without any errors, when no inferencing over mu-tual ambiguities is performed.) The following definition and lemma aretaken from [7].

Definition 4: [7] A pair �� of closed languages with � � � issaid to be coprognosable if

��

where ��

�� .Lemma 4: [7] A pair �� of closed languages with � � � is

coprognosable if and only if

��

�� (8)

Theorem 2: A pair �� of closed languages with � � � is0-inference-prognosable if and only if it is coprognosable.

Proof: (�) Assume that �� is coprognosable. By Lemma 4,(8) holds. First, we show that �� . For any � ��,there exists � � such that � ��

��. Since ��

��, we have � ��, which implies that � ��.It follows that �� . Further, by (8), we have

��

��

� ��

��

� ��

��

Thus, we have �� .(�) Assume that �� is 0-inference-prognosable. Consider any

� ��. We have � �� and � ��. Then there exists� � such that � ��

��. Thus, (8) holds. By Lemma 4,

�� is coprognosable.We also establish that the classes of -inference-prognosable sys-

tems form a monotonically increasing sequence as a function of .Since the sequence �� of language pairs is monotonicallydecreasing, the following result is easily obtained (the proof is omitted).

Theorem 3: For any � , if a pair �� of closed languageswith � � � is -inference-prognosable, then it is ��-inference-prognosable.

The converse relation of Theorem 3 need not hold. For example, thesystem of Example 1 is 2-inference-prognosable, but not 1-inference-prognosable.

VI. CONCLUSION

It is desirable to have systems in which it is possible to prognose(predict) failures prior to their occurrence. We studied the prognosis offailures in a decentralized framework where multiple prognosers, basedon their observations of the executed behavior, infer the inevitability ofan impending failure. Each local prognostic decision (of whether or not


failure is inevitable) is tagged with an ambiguity level (zero being theminimum), and the global prognostic decision is taken to be the win-ning local one (i.e., one with the minimum level of ambiguity). We in-troduced the notion of � -inference-prognosability to characterize theclass of systems for which any failure can be predicted prior to itsoccurrence in a manner that the maximum ambiguity level of a win-ning decision does not exceed � . An algorithm for verifying � -in-ference-prognosability was presented. We also showed that a bigger� corresponds to a larger class of prognosable systems. Further theclass of 0-inference-prognosable systems coincides with the class ofcoprognosable systems presented in [7], implying that even the classof 1-inference-prognosable systems subsumes the class of coprognos-able ones studied in [7].

REFERENCES

[1] H. K. Fadel and L. E. Holloway, “Using SPC and template monitoringmethod for fault detection and prediction in discrete event manufac-turing systems,” in Proc. IEEE Int. Symp. Intell. Control/Intell. Syst.Semiot., Cambridge, MA, Sep. 1999, pp. 150–155.

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[5] R. Kumar and S. Takai, “Inference-based ambiguity management indecentralized decision-making: Decentralized control of discrete eventsystems,” IEEE Trans. Autom. Control, vol. 52, no. 10, pp. 1783–1794,Oct. 2007.

[6] R. Kumar and S. Takai, “Inference-based ambiguity management in de-centralized decision-making: Decentralized diagnosis of discrete-eventsystems,” IEEE Trans. Autom. Sci. Eng., vol. 6, no. 3, pp. 479–491, Jul.2009.

[7] R. Kumar and S. Takai, “Decentralized prognosis of failures in discreteevent systems,” IEEE Trans. Autom. Control, vol. 55, no. 1, pp. 48–59,Jan. 2010.

[8] S. L. Ricker and K. Rudie, “Knowledge is a terrible thing to waste:Using inference in discrete-event control problems,” IEEE Trans.Autom. Control, vol. 52, no. 3, pp. 428–441, Mar. 2007.

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Adaptive Control for Plants in the Presence ofActuator and Sensor Uncertain Hysteresis

Xinkai Chen and Toshikuni Ozaki

Abstract—This note discusses the output tracking control for a linearplant containing uncertain hysteresis nonlinearities in actuator and sensordevices simultaneously, where the hysteresis is described by Prandtl–Ishlin-skii model. A new adaptive control scheme is developed to compensate theplant, the actuator and the sensor uncertainties and to generate an adap-tive estimate of the plant output. The proposed control law ensures the uni-form boundedness of all signals in the closed-loop system. Furthermore, thetracking error between the estimated plant output and the desired outputis guaranteed to converge to zero asymptotically.

Index Terms—Adaptive control, hysteresis nonlinearity, Prandtl–Ishlin-skii model.

I. INTRODUCTION

The hysteresis phenomenon occurs in all the smart material-basedactuators and sensors [4], [6], [10], [11], [16]. When the hysteresis non-linearity exists in the controlled system, the system usually exhibits un-desirable inaccuracies or oscillations and even instability due to the un-differentiable and nonmemoryless character of the hysteresis. For theplants preceded by hysteresis which means that the system is driven byactuators with hysteresis, the control problem has received consider-able attention recently. The common control approach is to constructan inverse hysteresis model to compensate the effect of the hysteresis[6], [7], [15], [16]. Essentially, the inversion problem depends on hys-teresis modeling methods. Some hysteretic nonlinearities are very com-plicated with multivalues and non-smooth features, such as those inpiezoelectric actuators and magnetostrictive actuators, where the op-erator-based hysteresis models are generally applied. The hysteresiscancellation by the direct inversion will result in compensation errors,which may cause difficulties in stability analysis for the closed-loopsystem. To avoid such difficulties, some new approaches have beenproposed in the literature [3], [4], [13], [14]. Instead of directly con-structing the inversion from the operator-based hysteresis model, animplicit inversion was introduced in [3], [4] for the convenience of sta-bility analysis of the closed-loop systems.

For sensor failure detection and identification research, tremendouseffort has been devoted recently. One typical design method for con-trol of systems with sensor failures is based on the neural networks andsensor redundancy, which is rendered by the measurements from mul-tiple sensors [9], [12]. Another typical method is reported recently in[8], where sensor characteristics are modeled as parametrizable uncer-tain functions and a compensator is constructed to adaptively cancel the

Manuscript received November 13, 2009; revised November 17, 2009; ac-cepted June 29, 2010. Date of publication October 07, 2010; date of current ver-sion January 12, 2011. This work was supported in part by the Grants-in-Aidfor Scientific Research of Japan Society for the Promotion of Science (JSPS)21560474, and by the Key Laboratory of Integrated Automation for the ProcessIndustry (Northeastern University), Ministry of Education of China. Recom-mended by Associate Editor F. Wu.

X. Chen is with the Department of Electronic and Information Systems,Shibaura Institute of Technology, Saitama-city, Saitama 337-8570, Japan(e-mail: [email protected]; [email protected]).

T. Ozaki is with the Graduate School of Engineering, Shibaura In-stitute of Technology, Saitama-city, Saitama 337-8570, Japan (e-mail:[email protected]).

Color versions of one or more of the figures in this technical note are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TAC.2010.2084151

0018-9286/$26.00 © 2010 IEEE

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Inference-Based Decentralized Prognosis in Discrete Event Systems