Decentralized Supervisory Control With Conditional ... - IEEE Xplore
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ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
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Decentralized Supervisory Control With Conditional Decisions: Supervisor Realization Tae-Sic Yoo and Stéphane Lafortune
REFERENCES [1] C. Antoniades and P. D. Christofides, “Non-linear feedback control of parabolic partial differential difference equation systems,” Int. J. Control, vol. 73, no. 17, pp. 1572–1591, 2000. [2] S. Bourrel and D. Dochain, “Stability analysis of two linear distributed parameter bioprocess models,” Math. Comput. Model. Dyna. Syst., vol. 6, no. 3, pp. 267–281, 2000. [3] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Texts in Applied Mathematics. New York: Springer-Verlag, 1995, vol. 21. [4] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, ser. Graduate Texts in Mathematics. New York: Springer-Verlag, 2000, vol. 194. [5] F. L. Huang, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,” Ann. Diff. Equat., vol. 1, no. 1, pp. 43–56, 1985. [6] S. Julien, J. P. Babary, and M. T. Nihtilä, “On modeling of boundary conditions and estimation for fixed-bed bioreactors,” Math. Model. Syst., vol. 1, no. 4, pp. 233–243, 1995. [7] H. Kanoh, “Control of heat exchangers by placement of closed-loop poles,” in Proc. IFAC Symp. Design Methods of Control Systems, Zürich, Switzerland, 1991, pp. 662–667. [8] N. Kunimatsu and H. Sano, “Stability analysis of heat-exchanger equations with boundary feedbacks,” IMA J. Math. Control Inform., vol. 15, pp. 317–330, 1998. [9] Y. Orlov and D. Dochain, “Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1293–1304, Aug. 2002. [10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, ser. Applied Mathematical Sciences. New York: Springer-Verlag, 1983, vol. 44. [11] M. Renardy, “Spectrally determined growth is generic,” Proc. AMS, vol. 124, no. 8, pp. 2451–2453, 1996. [12] H. Sano, “Boundary control of a linear distributed parameter bioprocess,” J. Franklin Inst., vol. 340, pp. 293–306, 2003. [13] H. Sano, “Exponential stability of a mono-tubular heat exchanger equation with output feedback,” Syst. Control Lett., vol. 50, pp. 363–369, 2003. [14] C. Z. Xu, J. P. Gauthier, and I. Kupka, “Exponential stability of the heat exchanger equation,” in Proc. 2nd Eur. Control Conf., vol. 1, Groningen, The Netherlands, 1993, pp. 303–307.
Abstract—The strategy of decentralized supervisory control of discreteevent systems using so-called “conditional decisions” initiated in prior work is further investigated in this note. Specifically, a constructive methodology for realizing supervisors that employ conditional decisions is developed. This methodology is based on the construction of (deterministic) observers of nondeterministic automata that are built so as to track violations of C&P and D&A coobservability. Index Terms—Conditional decisions, co-observability, decentralized control, discrete-event systems.
I. INTRODUCTION In [10], we consider decentralized supervisory control problems where supervisors are allowed to make unconditional (“enable” and “disable”) as well as conditional (“enable if nobody disables” and “disable if nobody enables”) decisions. This control architecture is referred to as the conditional architecture. In this architecture, the controllable events are partitioned a priori so that some controllable events are disabled by default and the remaining ones are enabled by default when supervisors do not issue unconditional or conditional decisions. Given such a partition of the controllable events, the necessary and sufficient conditions for the existence of supervisors in the context of the conditional architecture are identified in [10]. The notion of conditional coobservability appears in these conditions, together with the familiar controllability and Lm (G)-closure conditions (see, e.g., [1]). Furthermore, [10] also contains the following results: i) a polynomial-time algorithm for verifying conditional coobservability, and ii) a polynomial-time technique to partition the set of controllable events in the conditional architecture so as to satisfy conditional coobservability, if one exists. These polynomial-time verification and controllable event partitioning results build upon the original results in [4] about C&P coobservability. This note addresses supervisor synthesis issues regarding the results in [10]; note that only supervisor existence issues are treated in [10]. More specifically, given a regular language that is achievable in the conditional architecture, we develop in Section III a synthesis procedure for building realizations1 of finite-state automata supervisors that encode the required unconditional and conditional decisions. The key feature of this procedure is the realization of the conditional decisions, which is somewhat intricate and requires building deterministic observers2 of suitably-modified versions of the nondeterministic automata used in [8] to verify the properties of C&P and D&A coobservability. An example illustrating the application of the above synthesis
Manuscript received April 15, 2003; revised April 13, 2004 and March 6, 2005. Recommended by Associate Editor L. E. Holloway. This work was supported in part by the National Science Foundation under Grant CCR-0082784. The major part of this work was done while T.-S. Yoo was a Ph.D. student at the University of Michigan, Ann Arbor. T.-S. Yoo is with Idaho National Laboratory, Idaho Falls, ID 83403 USA (e-mail: [email protected]). S. Lafortune is with Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.852556 1The 2In
word “realization” is used here in the sense of [6] and [1, Ch. 3]. the sense of [1, Ch. 2].
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coobservability for this architecture is called conditional C&P coobservability and it is defined as follows [10]. is said to be conditionally Definition 2: A language K M = M C&P coobservable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K and 8 2 6c = [in=1 6c;i s:t: s 2 L(G) n K
(9i 2 I ())[CD] c
where CD denotes the following condition:
(8s 2 E (s) \ K )[9j 2 I () s:t: E (s ) \ M K ]: i
Fig. 1. Decentralized control architecture.
results is presented in Section IV. Due to space constraints, we assume in the remainder of this note that the reader is familiar with supervisory control theory and with the salient features of [10]. II. CONDITIONAL ARCHITECTURE AND COOBSERVABILITY Let us consider the decentralized control architecture depicted in Fig. 1 where a set of supervisors jointly controls a system G by each observing subsets of 6o (denoted by 6o;i ) and controlling subsets of 6c (denoted by 6c;i ) in order to achieve the desired behavior K L(G) 63 . We denote by 6uo = 6 n 6o and 6uc = 6 n 6c , the unobservable and uncontrollable event sets, respectively. As in [10], we assume at the outset that the controllable event sets 6c;i ; i = 1; . . . ; n, are not mutually disjoint. We briefly review notation, concepts, and results (proved in [10]) that are required for understanding and gaining insight into the synthesis methodology of Section III. Conditional D&A coobservability is a key component of the necessary and sufficient conditions for the existence of a decentralized control system that exactly achieves the desired behavior when supervisors make “disable,” “enable,” “enable if nobody disables” decisions, and a controllable event is disabled by default if there is no decision over that event [7], [10]. The architecture where supervisors are allowed to make the above three decisions is referred to as the conditional-enabling architecture and conditional D&A coobservability is defined as follows [7], [10]. In that definition, 3 ; P 01 (s) := fs0 2 63 : Pi is the projection operation from 63 to 6o;i 0 1 0 , and Ic (i ) := fi : 2 6c;i g. Pi (s ) = sg; Ei (s) := Pi Pi (s) \ K is said to be conditionally Definition 1: A language K M = M 3 D&A coobservable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K and 8 2 6c = [in=1 6c;i s.t. s 2 K
(9i 2 I ())[CE] c
where CE denotes the following condition:
(8s 2 E (s) \ (M n K ))[9j 2 I () s:t: E (s ) \ K = ;]: i
i
c
j
i
The CE condition implies that for each illegal controllable continuation that the ith supervisor estimates, there is a supervisor that can ensure that this continuation with is illegal. That is, the ith supervisor can infer that there is a supervisor (j ) that can disable with certainty. In [10], another decentralized control architecture, the conditionalenabling architecture, is considered. In this architecture, supervisors make three types of decisions: “enable,” “disable,” and “disable if nobody enables.” Moreover, a controllable event is enabled by default if there is no decision over this event. The analogue of conditional D&A 3This
notion is called EDF-partitionability in [7].
i
c
j
i
The CD condition implies that for each legal controllable continuation that the ith supervisor estimates, there is a supervisor that can ensure that this continuation with is legal. That is, the ith supervisor can infer that there is a supervisor (j ) that can enable with certainty. The conditional architecture where individual supervisors are allowed to make “enable,” “disable,” “enable if nobody disables,” and “disable if nobody enables” decisions is also considered in [10]. That is, local supervisors are formally defined as follows:
SP : Pi (63 ) ! 26
2 26 2 26 2 26
where
SP (Pi (s)) = (ei (Pi (s)); di (Pi (s)); eci ((Pi (s)); dci (Pi (s))), for i 2 f1; . . . ; ng. The “enable,” “disable,” “enable if nobody disables,” and “disable if nobody enables” decisions of the ith local supervisor are represented by ei (Pi (s)); di (Pi (s)); eci (Pi (s)), and dci (Pi (s)), respectively. The joint control action of local supervisors SP ; . . . ; SP is denoted by Sfc . For the Sfc supervisor, fc stands for
“fusion of decentralized unconditional and conditional decisions.” Since Sfc is a joint action of local supervisors, the domain of Sfc is P (63 ) and the role of Sfc is to issue joint global “enable” and “disable” decisions. That is
Sfc : P (63 )!26 2 26 Sfc (P (s)) = (e(P (s)); d(P (s))) where e(P (s)) and d(P (s)) are global “enable” and “disable” decisions, respectively, which are defined as follows. For 2 6c ; 2 e(P (s)) iff n
2
=1
_
ei (Pi (s))
n
2
eci (Pi (s)) ^ 2=
=1 Similarly, for 2 6c ; 2 d(P (s)) iff i
2
n
=1
i
di (Pi (s))
_
2
i
n
=1
n
=1
di (Pi (s)) :
i
n
dci (Pi (s)) ^ 2=
i
=1
ei (Pi (s)) :
i
The conditional architecture is more powerful than both the conditional-enabling and the conditional-disabling architectures in the sense that a relaxed version of coobservability appears in the necessary and sufficient conditions for the existence of a set of supervisors that achieves a given desired language [10]. Define the following sets of events: For i 2 f1; . . . ng
:= 6 \ 6 where 6 [_ 6 = 6 . 6 6
c;e;i
c;i
c;e
and 6
c;d;i
:= 6 \ 6 c;i
c;d
c;e c c;e;i is the set of locally controllable events c;d whose default setting is enablement while 6c;d;i is the set of locally controllable events whose default setting is disablement. is said to be conditionally Definition 3: A language K M = M coobservable w.r.t. M; 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n , if the following two conditions hold: i) K is conditionally C&P coobservable w.r.t. M; 6o;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;e;n ; ii) K is conditionally D&A coobservable w.r.t. M; 6o;1 ; 6c;d;1 ; . . . ; 6o;n ; 6c;d;n .
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The existence result of the conditional architecture can now be presented. Theorem 1: [10]: Consider the language K Lm (G) where K 6= ; and consider a fixed partition of 6c such that 6c = 6c;d [_ 6c;e . There exists a nonblocking and control-nonconflicting supervisor Sfc such that Lm (Sfc =G) = K and iff the following three conditions hold: i) K is L(Sfc =G) = K controllable w.r.t. L(G) and 6uc ; ii) K is conditionally coobservable w.r.t. L(G); 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n ; iii) K is Lm (G)-closed. When the previous necessary and sufficient conditions hold, the following local decision rules can be applied to achieve the desired lan i 2 f1; . . . ; ng guage K : For s 2 K; fc
( ( )) =
SP Pi s
fc
( ( ))
fc
( ( ))
fc
( ( ))
fc
( ( ))
ei Pi s ; di Pi s ; eci Pi s ; dci Pi s
where
6 : E (s ) (G ) = ( ) K fc (G ) = d (P (s)) := 6 : E (s) = E (s ) K fc ec (P (s)) := 6 : E (s) (G ) = ) s E (s ) ( (G ) K j I ( ) s:t: E (s ) K = fc dc (P (s)) := 6 : E (s) (G ) = s E (s ) K (G ) j I ( ) s:t: E (s ) fc
( ( )) :=
ei Pi s
f
2
()
i
f
i
2
i
f
i
2
2
8 i
9
i
2
f
i
2
2
6
;
6
;
g
\ L
i
(1)
;g
c;i
\ L
i
\
i
c
2
8 i
9
c;i
\
i
\ L
i
c;i
Ei s \ L G
L
j
c;i
6
;
n
\
i
\ L
i
6
j
c
i
\L
;
K g:
(2)
III. REALIZATION OF SUPERVISORS Suppose that the desired behavior Lm (H ) is a solvable regular language under the conditional architecture and it is desired to realize, in the form of finite-state automata, local supervisors that result in this desired behavior. Given that solvability is assured, local unconditional decision rules are obtained from (1). Without loss of generality, we can assume that H is a strict subautomaton of G [2], [3]. Then, local unconditional decision rules can be realized by constructing for each local site i the observer of H for projection Pi and finding “enable” and “disable” decisions according to (1) for each observer state as is done in [5] and [8]. Details are omitted here. Hereafter, we focus on the realization of the first part of (2), regarding the “enable if nobody disables” decision rule. The treatment of the “disable if nobody enables” decisions is similar, with suitable modifications as will be described later. Equation (2) is the basis for the realization of local conditional decision rules. The “enable if nobody disables” decision rule ecifc (Pi ( 1 )) in (2) implies that a controllable continuation is enabled conditionally by supervisor i only if supervisor i is certain that all illegal controllable continuations with can be disabled by some local supervisor. In other words, based on its observation and inference on observations of other supervisors, if there is some illegal continuation that cannot be disabled by some local supervisor for sure, then is not conditionally enabled. Formally, 62 ecifc (Pi (s)) if Ei (s) \ L(G) = ; or 9si 2 Ei (s) \ (L(G) n L(H )) such that
(
8
j
2
( ))[E (s ) (H ) = ]: (3) (G) = simply means that the estimated
Ic
j
i
\ L
6
=
()
Ei s \ K
;g
\
i
only concentrate on the second part of the condition 62 ecifc (Pi (s)). Let us recall the definition of C&P coobservability from [8] for the sake of further discussion. is said to be C&P coobDefinition 4: A language K M = M and 8 2 servable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K 6c = [in=1 6c;i s.t.s 2 M n K , there exists i 2 Ic () s.t.
;
The condition Ei (s) \ L ; behaviors (Ei (s)) cannot be continued with event within the system behavior. Therefore, is excluded from the decision since decisions over are irrelevant to the resulting controlled language. Therefore, we
;
:
From Definition 4 and (3), when the system executes trace s, the following happens: i) there exists si such that Pi (si ) = Pi (s) and si violates C&P coobservability, and ii) is excluded from the “enable if nobody disables” decision of supervisor i. In order to identify traces violating C&P coobservability, we will modify the automaton Mc (6c ) of [4]. In [4], Mc (6c ) is denoted by M . The notation Mc (6c ) here emphasizes that the M of [4] pertains to C&P coobservability and that it is parameterized by the set of controllable events. The modification involves changing the labels of some of the (unobservable) transitions to in order to identify the precise trace executed by the system. More precisely, we have mod
Mc
(6 ) =
M
Q
c
;
6
;
[f g
M
M
M
; q0
; Qm
where Q
:= (Q Q Q := q0 ; q0 ; q0 ; q0 := vc :
M
H
M
H
q0
M
Qm
f
H
2
H
2
H
H
2
G
Q
)
vcg
[ f
G
g
Hereafter, only the accessible part of the state space QM is consid. Let us recall from [4] the set of conered when we refer to QM ditions implying the violation of C&P coobservability; we retain these conditions in Mcmod (6c ). For 2 6c
( ) ( ) ( ) (q4 ; )
H q1 ; H q2 ; H q3 ; G
is defined if is defined if is not defined is defined
The transition relation M brevity, given that q1 ; q2 ; q3 the following:
2
0
H
0
G
i
2
6 6
c;1
( ):
c;2
3
is defined as follows. For the sake of G 2 Q , and 2 6, let us define
QH ; q4
:= (q ; ) for i q4 := (q4 ; ) and q := (q1 ; q2 ; q3 ; q4 ): ~ qi
2
2 f
1; 2; 3
g
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Fig. 2. Automata
and
.
Note the use of the label in three instances that follow. For and 62 6o;2
For
For
26
0
0
o;2
0
0
o;1
and
62 6
0
0
0
0
0
o;2
(q~; ) = (q1 ; q2 ; q3 ; q4 ) if q2 exists (q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q1 ; q3 ; and q4 exist 0
0
0
M
M
and
0
0
M
26
0
0
(q~; ) = (q1 ; q2 ; q3 ; q4 ) if q1 exists (q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q2 ; q3 ; and q4 exist
M
26
o;1
M
For
M
62 6
0
0
o;1
((~q); ) = ((qq1 ;; qq2;; qq3 ;; qq4 )) ifif qq1 exists 1 2 3 4 2 exists ( q1 ; q2 ; q3 ; q4 ) if q3 and q4 exist (q~; ) = vc if(3): 0
M
62 6
o;1
and
26
0
0
0
0
0
o;2
(q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q1 ; q2 ; q3 ; and q4 exist 0
0
0
0
0
0
0
0
For 2 6; M (vc; ) is undefined. The state–space QH 2 QH 2 QH 2 QG tracks all traces s; s0 ; s00 2 L(H ) such thatmodP1 (s) = P1 (s0 )andP2 (s) = P2 (s00 ). The transition relation of Mc (6c ) is defined to track the traces in the following manner: Q
H
s 0
00
2 Q 2Q 2 Q H
s
H
G
:
s
Two traces s and s are tracked by the first two H ’s and the last pair H and G tracks s. The violation of C&P coobservability is characterized by traces s; s0 ; s00 2 L(H ) and event such that P1 (s) = P1 (s0 ); P2 (s) =
( ) 2 L(H ); s 2 L(H ); and s 2 L(G) n L(H ), where 2 6 \ 6 2 . The characterization of the violation of C&P
P2 s00 , s0 c;1
00
c;
coobservability demands to track legal traces (this is done by the first two H ’s) and one illegal trace (the last pair H and G tracks this). The occurrence of this violation that is captured by condition (3) causes a transition into marked state vc. In order to identify the illegal traces [denoted by s in (3)] violating C&P coobservability, we need to distinguish the transitions tracking trace s from the transitions tracking traces s0 and s00 (traces that have the same local projections as s at local sites 1 and 2, respectively). Hence, when we consider a transition tracking s (that is, when the last pair of H and G involved), we attach the label to the transition. Otherwise (that is, when the last pair H and G is not involved), the label is attached to the transition. With this modification, if s reaches state vc, then we know for sure that s 2 L(G) nL(H ) and that s violates C&P coobservability. In order to realize the local “enable if nobody disables” decision rule of supervisor i from (3), we need to identify every si 2 Ei (s) \ (L(G) nL(H )) violating C&P coobservability and exclude such 2 6c;i from the local “enable if nobody disables” decisions. Let us remove marked state vc and its attached transitions from Mcmod (6c ) and denote the result by Mc0 (6c ). We construct the (deterministic) observer of Mc0 (6c ) with respect to 6o;i , denoted by Obsi (Mc0 ). The purpose of constructing the observer of Mc0 (6c ) with respect to 6o;i is to identify the reachable states of Mc0 (6c ) with Ei (s) when s is the trace executed by the system and the observed trace is Pi (s). Moreover, Mcmod (6c ) is constructed to explicitly identify traces violating C&P coobservability. Therefore, if a state of Obsi (Mc0 ) that is reached by trace Pi (s) contains states of Mcmod (6c ) that reach state vc with some transition in automaton Mcmod (6c ), then we know that there exists si 2 Ei (s) \ (L(G) nL(H )) violating C&P coobservability and event is excluded from the local “enable if nobody disables” decision of supervisor i after observing Pi (s). In this manner, the automaton Obsi (Mc0 ), together with Mcmod (6c ), can be used as a realization of the local “enable if nobody disables” decision rule of supervisor i. Algorithm 1 formalizes the procedure described previously. The correctness of Algorithm 1, namely the realization of the local “enable if nobody disables decision,” is shown in Theorem 2.
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Fig. 3. Supervisor 1: unconditional decisions. (a)
( ( )). (b) ( ( )).
Theorem 2: For all s 2 L(H ) and i 2 f1; . . . ng; eci (Pi (s)) = eci (Pi (s)). Proof: () Suppose that 62 ecifc (Pi (s)). This implies that for 2 6c;i (i) Ei (s) \ L(G) = ; or (ii) there exists si 2 Ei (s) \ (L(G) n L(H )) s.t. fc
(8j 2 I ())[E (s ) \ L(H ) 6= ;]: c
j
i
(5)
If i) holds, it is clear that 62 eci (Pi (s)). Therefore, we focus on condition ii) hereafter. Since si 2 Ei (s) \ (L(G) nL(H )), we have that si 2 L(H ); si 2 L(G) n L(H ), and Pi (s) = Pi (si ). With (5) indicating the violation of C&P coobservability and from the construction of Mcmod (6c ), we know that there exists si 2 L(Mcmod (6c )) 63 4 such that
M
M
q0
; si
(6 )
= vc:
4Note that the automaton is nondeterministic and the transition represents the silent transition. label in
(6 )
1209
Let q M
:=
(q0
M
M
Since Pi (si )
M
; si
)= (q
M
M
;
(q0
M
)
; si . Then, we have that
) = vc:
= P (s), we have i
q
M
2 q := obs
Obs (M
)
obs
()
q0 ; Pi s
:
Therefore, we have 2 neci (qobs ). Consecutively, we have 62 eci (Pi (s)). () The proof of this case follows similar arguments to the preceding case (in reverse), starting from the assumption that 62 eci (Pi (s)). The realization of the “disable if nobody enables” decision rule in (2) can be performed analogously. The key difference is centered on the use of a suitably modified version of the nondeterministic automaton introduced in [8] for testing D&A coobservability. Namely, we modify Md (6c ) of [8] to get Mdmod (6c ) by using labels, in the same spirit as was done when obtaining Mcmod (6c ) in Algorithm 1. Then, state
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Fig. 4.
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(6 ).
of Mdmod (6c ) [which is analogous to state vc of Mcmod (6c )] is deleted and Md0 (6c ) is obtained. The realization of the conditional “disable if nobody enables” decisions of supervisor i is then encoded in the observer of Md0 (6c ) for event set 6o;i , using Mdmod (6c ). (For complete details, see [9].)
vd
IV. EXAMPLE Consider the uncontrolled behavior generated by automaton G in Fig. 2(a) and the desired behavior generated by automaton H in Fig. 2(b). We define that Lm (H ) = L(H ) = K and Lm (G) = L(G) (i.e., marking is omitted for all states). We set 6o;1 = fa; a0 ; c; dg; 6o;2 = fb; b0 ; c; dg; 6c;1 = 6c;2 = fc; dg; 6c;e = fcg, and 6c;d = fdg. In [10], we showed that L(H ) is conditionally coobservable w.r.t. L(G); 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n . We focus here on the realization of the supervisors. Finite-state automata realization of the unconditional decisions of supervisor 1 are shown in Fig. 3. In Fig. 3(a), the event information appearing in each state of the realization (rectangle in figure) represents the enabled events; in the case of Fig. 3(b) it represents the disabled events. The automata in Fig. 3 are obtained using the techniques presented in [5] and [8], which are not repeated here. Following the procedure discussed in Section III, we construct mod Mc (6c ), shown in Fig. 4, to identify the traces violating C&P coobservability (Step 1). By removing state vc from Mcmod (6c ), we get Mc0 (6c ) (Step 2). In order to realize the local “enable if nobody disables” decision rules, we build observers of Mc0 (6c ) w.r.t. 6o;1 and 6o;2 , respectively (Step 3). The observer for supervisor 1 is shown in Fig. 5 where, for the sake of readability, the states of Mcmod (6c ) are renamed as described
in the table in the figure. For Mcmod (6c ), states (1; 5; 2; 2) state d in Fig. 5) and (8; 1; 4; 4) state o in Fig. 5) reach vc with event c and state (12; 14; 11; 11) state z in Fig. 5) reaches vc with event d. Therefore, the observer states of Mc0 (6c ) containing state d (observer state (a, b, c, d, e, f, g, h) in Fig. 5) or state o (observer state (j, k, l, o, p, q) in Fig. 5) exclude c from the “enable if nobody disables” decisions, and the observer states of Mc0 (6c ) containing z (observer state (t, u, v, w, x, y, z, aa) in Fig. 5) exclude d from the “enable if nobody disables” decisions (Step 4). The realization of the “enable if nobody disables” decision rule of supervisor 1 is therefore as shown in Fig. 5 (set of events next to observer state components). Note that c = nec1 ((a; b; c; d; e; f ; g; h)); c = nec1 ((j; k; l; o; p; q)), and d = nec1 ((t; u; v; w; x; y; z; aa)). States (a; b; c; d; e; f ; g; h); (j; k; l; o; p; q), and (t; u; v; w; x; y; z; aa) are reached with observed traces ; a, and cd, respectively. Therefore, we have that c
62 ec1 ()
c
62 ec1 (a)
and d 62 ec1 (cd):
Finally, following (4), we have
( ) = ec1 (a) = ec1 (cd) = ;:
ec1
Note that trace cd remains feasible because event c is enabled by default initially. We omit the realization of supervisor 2, which can be performed similarly; see [9]. The realization of the local “disable if nobody enables” decision rules is omitted here due to space constraints; again, we refer the reader to [9] in this regard.
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[8]
, “A general architecture for decentralized supervisory control of discrete-event systems,” Discrete Event Dyna. Syst.: Theory Appl., vol. 12, no. 3, pp. 335–377, 2002. [9] , “Decentralized supervisory control with conditional decisions—Part II: Verification and synthesis,” Univ. Michigan, Ann Arbor, MI, Tech. Rep. CGR-03-18, 2003. [10] , “Decentralized supervisory control with conditional decisions: Supervisor existence,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 1886–1904, Nov. 2004.
A Fast Nonlinear Model Identification Method Kang Li, Jian-Xun Peng, and George W. Irwin Abstract—The identification of nonlinear dynamic systems using linear-in-the-parameters models is studied. A fast recursive algorithm (FRA) is proposed to select both the model structure and to estimate the model parameters. Unlike orthogonal least squares (OLS) method, FRA solves the least-squares problem recursively over the model order without requiring matrix decomposition. The computational complexity of both algorithms is analyzed, along with their numerical stability. The new method is shown to require much less computational effort and is also numerically more stable than OLS. Index Terms—Computational complexity, fast recursive algorithm, nonlinear system identification, numerical stability.
I. INTRODUCTION
Fig. 5. Supervisor 1: conditional decision
(
( )).
V. CONCLUSION The procedure presented to synthesize supervisors that implement conditional decisions is novel and relies on specially constructed nondeterministic automata that track violations of C&P and D&A coobservability, respectively. This realization procedure gives insight into the nature of conditional decisions, especially regarding the inferencing process that is at the heart of the conditional architecture.
REFERENCES [1] C. G. Cassandras and S. Lafortune, Introduction to Discrete Event Systems. Norwell, MA: Kluwer, 1999. [2] E. Chen and S. Lafortune, “On the infirnal closed and controllable superlanguage of a given language,” IEEE Trans. Autom. Control, vol. 35, no. 4, pp. 398–404, Apr. 1990. [3] H. Cho and S. I. Marcus, “On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation,” Math. Control Signals Syst., vol. 2, pp. 47–69, 1989. [4] K. Rudie and J. C. Willems, “The computational complexity of decentralized discrete-event control problems,” IEEE Trans. Autom. Control, vol. 40, no. 7, pp. 1313–1318, Jul. 1995. [5] K. Rudie and W. M. Wonham, “Think globally, act locally: Decentralized supervisory control,” IEEE Trans. Autom. Control, vol. 37, no. 11, pp. 1692–1708, Nov. 1992. [6] W. M. Wonham, “Notes on control of discrete-event systems,” Univ. Toronto, Toronto, ON, Canada, Tech. Rep., Jul. 2003. [7] T. Yoo and S. Lafortune, “Decentralized supervisory control: A new architecture with a dynamic decision fusion rule,” in Proc. 6th Int. Workshop Discrete Event Systems, Zaragoza, Spain, 2002, pp. 11–17.
Some widely used nonlinear regression models and neural networks constitute linear-in-the-parameters models, for example the nonlinear autoregressive model with exogenous inputs (NARX) and radial basis function (RBF) networks [1]–[8]. Such models form a linear combination of model terms, or basis functions, which are nonlinear functions of the system variables. Depending on the nonlinear functions employed, linear-in-the-parameters models possess broad approximation capabilities, and have wide applications [1]–[8]. One problem with such models is that an excessive number of candidate model terms or basis functions usually have to be considered initially [2]–[8]. From these, a useful model is then generated based on the parsimonious principle [9], [10], of selecting the smallest possible model, in terms of size, which explains the data. Given a model selection criterion, this can be achieved by an exhaustive search of all possible combinations of candidates using a least-squares method. This is computationally very expensive. To reduce the computational complexity, efficient suboptimal search algorithms have been proposed, among which orthogonal least-squares (OLS) method is perhaps among the most popular [2]–[9]. OLS was first applied to nonlinear dynamic system identification [2], [3] and is now widely used in many other areas [4]–[8]. In general, OLS approaches are derived from an orthogonal (or QR) decomposition of the regression matrix [2]–[9]. The elegance of the OLS approach lies in that, the net decrease in the cost function can be Manuscript received September 18, 2003; revised January 10, 2005. Recommended by Associate Editor E. Bai. This work was supported by the U.K. EPSRC under Grant GR/S85191/01 to K. Li. The authors are with the School of Electrical and Electronic Engineering, Queen’s University of Belfast, Belfast BT9 5AH, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2005.852557
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ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments and suggestions.
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Decentralized Supervisory Control With Conditional Decisions: Supervisor Realization Tae-Sic Yoo and Stéphane Lafortune
REFERENCES [1] C. Antoniades and P. D. Christofides, “Non-linear feedback control of parabolic partial differential difference equation systems,” Int. J. Control, vol. 73, no. 17, pp. 1572–1591, 2000. [2] S. Bourrel and D. Dochain, “Stability analysis of two linear distributed parameter bioprocess models,” Math. Comput. Model. Dyna. Syst., vol. 6, no. 3, pp. 267–281, 2000. [3] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, ser. Texts in Applied Mathematics. New York: Springer-Verlag, 1995, vol. 21. [4] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, ser. Graduate Texts in Mathematics. New York: Springer-Verlag, 2000, vol. 194. [5] F. L. Huang, “Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,” Ann. Diff. Equat., vol. 1, no. 1, pp. 43–56, 1985. [6] S. Julien, J. P. Babary, and M. T. Nihtilä, “On modeling of boundary conditions and estimation for fixed-bed bioreactors,” Math. Model. Syst., vol. 1, no. 4, pp. 233–243, 1995. [7] H. Kanoh, “Control of heat exchangers by placement of closed-loop poles,” in Proc. IFAC Symp. Design Methods of Control Systems, Zürich, Switzerland, 1991, pp. 662–667. [8] N. Kunimatsu and H. Sano, “Stability analysis of heat-exchanger equations with boundary feedbacks,” IMA J. Math. Control Inform., vol. 15, pp. 317–330, 1998. [9] Y. Orlov and D. Dochain, “Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor,” IEEE Trans. Autom. Control, vol. 47, no. 8, pp. 1293–1304, Aug. 2002. [10] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, ser. Applied Mathematical Sciences. New York: Springer-Verlag, 1983, vol. 44. [11] M. Renardy, “Spectrally determined growth is generic,” Proc. AMS, vol. 124, no. 8, pp. 2451–2453, 1996. [12] H. Sano, “Boundary control of a linear distributed parameter bioprocess,” J. Franklin Inst., vol. 340, pp. 293–306, 2003. [13] H. Sano, “Exponential stability of a mono-tubular heat exchanger equation with output feedback,” Syst. Control Lett., vol. 50, pp. 363–369, 2003. [14] C. Z. Xu, J. P. Gauthier, and I. Kupka, “Exponential stability of the heat exchanger equation,” in Proc. 2nd Eur. Control Conf., vol. 1, Groningen, The Netherlands, 1993, pp. 303–307.
Abstract—The strategy of decentralized supervisory control of discreteevent systems using so-called “conditional decisions” initiated in prior work is further investigated in this note. Specifically, a constructive methodology for realizing supervisors that employ conditional decisions is developed. This methodology is based on the construction of (deterministic) observers of nondeterministic automata that are built so as to track violations of C&P and D&A coobservability. Index Terms—Conditional decisions, co-observability, decentralized control, discrete-event systems.
I. INTRODUCTION In [10], we consider decentralized supervisory control problems where supervisors are allowed to make unconditional (“enable” and “disable”) as well as conditional (“enable if nobody disables” and “disable if nobody enables”) decisions. This control architecture is referred to as the conditional architecture. In this architecture, the controllable events are partitioned a priori so that some controllable events are disabled by default and the remaining ones are enabled by default when supervisors do not issue unconditional or conditional decisions. Given such a partition of the controllable events, the necessary and sufficient conditions for the existence of supervisors in the context of the conditional architecture are identified in [10]. The notion of conditional coobservability appears in these conditions, together with the familiar controllability and Lm (G)-closure conditions (see, e.g., [1]). Furthermore, [10] also contains the following results: i) a polynomial-time algorithm for verifying conditional coobservability, and ii) a polynomial-time technique to partition the set of controllable events in the conditional architecture so as to satisfy conditional coobservability, if one exists. These polynomial-time verification and controllable event partitioning results build upon the original results in [4] about C&P coobservability. This note addresses supervisor synthesis issues regarding the results in [10]; note that only supervisor existence issues are treated in [10]. More specifically, given a regular language that is achievable in the conditional architecture, we develop in Section III a synthesis procedure for building realizations1 of finite-state automata supervisors that encode the required unconditional and conditional decisions. The key feature of this procedure is the realization of the conditional decisions, which is somewhat intricate and requires building deterministic observers2 of suitably-modified versions of the nondeterministic automata used in [8] to verify the properties of C&P and D&A coobservability. An example illustrating the application of the above synthesis
Manuscript received April 15, 2003; revised April 13, 2004 and March 6, 2005. Recommended by Associate Editor L. E. Holloway. This work was supported in part by the National Science Foundation under Grant CCR-0082784. The major part of this work was done while T.-S. Yoo was a Ph.D. student at the University of Michigan, Ann Arbor. T.-S. Yoo is with Idaho National Laboratory, Idaho Falls, ID 83403 USA (e-mail: [email protected]). S. Lafortune is with Department of Electrical Engineering and Computer Science, The University of Michigan, Ann Arbor, MI 48109-2122 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.852556 1The 2In
word “realization” is used here in the sense of [6] and [1, Ch. 3]. the sense of [1, Ch. 2].
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coobservability for this architecture is called conditional C&P coobservability and it is defined as follows [10]. is said to be conditionally Definition 2: A language K M = M C&P coobservable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K and 8 2 6c = [in=1 6c;i s:t: s 2 L(G) n K
(9i 2 I ())[CD] c
where CD denotes the following condition:
(8s 2 E (s) \ K )[9j 2 I () s:t: E (s ) \ M K ]: i
Fig. 1. Decentralized control architecture.
results is presented in Section IV. Due to space constraints, we assume in the remainder of this note that the reader is familiar with supervisory control theory and with the salient features of [10]. II. CONDITIONAL ARCHITECTURE AND COOBSERVABILITY Let us consider the decentralized control architecture depicted in Fig. 1 where a set of supervisors jointly controls a system G by each observing subsets of 6o (denoted by 6o;i ) and controlling subsets of 6c (denoted by 6c;i ) in order to achieve the desired behavior K L(G) 63 . We denote by 6uo = 6 n 6o and 6uc = 6 n 6c , the unobservable and uncontrollable event sets, respectively. As in [10], we assume at the outset that the controllable event sets 6c;i ; i = 1; . . . ; n, are not mutually disjoint. We briefly review notation, concepts, and results (proved in [10]) that are required for understanding and gaining insight into the synthesis methodology of Section III. Conditional D&A coobservability is a key component of the necessary and sufficient conditions for the existence of a decentralized control system that exactly achieves the desired behavior when supervisors make “disable,” “enable,” “enable if nobody disables” decisions, and a controllable event is disabled by default if there is no decision over that event [7], [10]. The architecture where supervisors are allowed to make the above three decisions is referred to as the conditional-enabling architecture and conditional D&A coobservability is defined as follows [7], [10]. In that definition, 3 ; P 01 (s) := fs0 2 63 : Pi is the projection operation from 63 to 6o;i 0 1 0 , and Ic (i ) := fi : 2 6c;i g. Pi (s ) = sg; Ei (s) := Pi Pi (s) \ K is said to be conditionally Definition 1: A language K M = M 3 D&A coobservable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K and 8 2 6c = [in=1 6c;i s.t. s 2 K
(9i 2 I ())[CE] c
where CE denotes the following condition:
(8s 2 E (s) \ (M n K ))[9j 2 I () s:t: E (s ) \ K = ;]: i
i
c
j
i
The CE condition implies that for each illegal controllable continuation that the ith supervisor estimates, there is a supervisor that can ensure that this continuation with is illegal. That is, the ith supervisor can infer that there is a supervisor (j ) that can disable with certainty. In [10], another decentralized control architecture, the conditionalenabling architecture, is considered. In this architecture, supervisors make three types of decisions: “enable,” “disable,” and “disable if nobody enables.” Moreover, a controllable event is enabled by default if there is no decision over this event. The analogue of conditional D&A 3This
notion is called EDF-partitionability in [7].
i
c
j
i
The CD condition implies that for each legal controllable continuation that the ith supervisor estimates, there is a supervisor that can ensure that this continuation with is legal. That is, the ith supervisor can infer that there is a supervisor (j ) that can enable with certainty. The conditional architecture where individual supervisors are allowed to make “enable,” “disable,” “enable if nobody disables,” and “disable if nobody enables” decisions is also considered in [10]. That is, local supervisors are formally defined as follows:
SP : Pi (63 ) ! 26
2 26 2 26 2 26
where
SP (Pi (s)) = (ei (Pi (s)); di (Pi (s)); eci ((Pi (s)); dci (Pi (s))), for i 2 f1; . . . ; ng. The “enable,” “disable,” “enable if nobody disables,” and “disable if nobody enables” decisions of the ith local supervisor are represented by ei (Pi (s)); di (Pi (s)); eci (Pi (s)), and dci (Pi (s)), respectively. The joint control action of local supervisors SP ; . . . ; SP is denoted by Sfc . For the Sfc supervisor, fc stands for
“fusion of decentralized unconditional and conditional decisions.” Since Sfc is a joint action of local supervisors, the domain of Sfc is P (63 ) and the role of Sfc is to issue joint global “enable” and “disable” decisions. That is
Sfc : P (63 )!26 2 26 Sfc (P (s)) = (e(P (s)); d(P (s))) where e(P (s)) and d(P (s)) are global “enable” and “disable” decisions, respectively, which are defined as follows. For 2 6c ; 2 e(P (s)) iff n
2
=1
_
ei (Pi (s))
n
2
eci (Pi (s)) ^ 2=
=1 Similarly, for 2 6c ; 2 d(P (s)) iff i
2
n
=1
i
di (Pi (s))
_
2
i
n
=1
n
=1
di (Pi (s)) :
i
n
dci (Pi (s)) ^ 2=
i
=1
ei (Pi (s)) :
i
The conditional architecture is more powerful than both the conditional-enabling and the conditional-disabling architectures in the sense that a relaxed version of coobservability appears in the necessary and sufficient conditions for the existence of a set of supervisors that achieves a given desired language [10]. Define the following sets of events: For i 2 f1; . . . ng
:= 6 \ 6 where 6 [_ 6 = 6 . 6 6
c;e;i
c;i
c;e
and 6
c;d;i
:= 6 \ 6 c;i
c;d
c;e c c;e;i is the set of locally controllable events c;d whose default setting is enablement while 6c;d;i is the set of locally controllable events whose default setting is disablement. is said to be conditionally Definition 3: A language K M = M coobservable w.r.t. M; 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n , if the following two conditions hold: i) K is conditionally C&P coobservable w.r.t. M; 6o;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;e;n ; ii) K is conditionally D&A coobservable w.r.t. M; 6o;1 ; 6c;d;1 ; . . . ; 6o;n ; 6c;d;n .
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The existence result of the conditional architecture can now be presented. Theorem 1: [10]: Consider the language K Lm (G) where K 6= ; and consider a fixed partition of 6c such that 6c = 6c;d [_ 6c;e . There exists a nonblocking and control-nonconflicting supervisor Sfc such that Lm (Sfc =G) = K and iff the following three conditions hold: i) K is L(Sfc =G) = K controllable w.r.t. L(G) and 6uc ; ii) K is conditionally coobservable w.r.t. L(G); 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n ; iii) K is Lm (G)-closed. When the previous necessary and sufficient conditions hold, the following local decision rules can be applied to achieve the desired lan i 2 f1; . . . ; ng guage K : For s 2 K; fc
( ( )) =
SP Pi s
fc
( ( ))
fc
( ( ))
fc
( ( ))
fc
( ( ))
ei Pi s ; di Pi s ; eci Pi s ; dci Pi s
where
6 : E (s ) (G ) = ( ) K fc (G ) = d (P (s)) := 6 : E (s) = E (s ) K fc ec (P (s)) := 6 : E (s) (G ) = ) s E (s ) ( (G ) K j I ( ) s:t: E (s ) K = fc dc (P (s)) := 6 : E (s) (G ) = s E (s ) K (G ) j I ( ) s:t: E (s ) fc
( ( )) :=
ei Pi s
f
2
()
i
f
i
2
i
f
i
2
2
8 i
9
i
2
f
i
2
2
6
;
6
;
g
\ L
i
(1)
;g
c;i
\ L
i
\
i
c
2
8 i
9
c;i
\
i
\ L
i
c;i
Ei s \ L G
L
j
c;i
6
;
n
\
i
\ L
i
6
j
c
i
\L
;
K g:
(2)
III. REALIZATION OF SUPERVISORS Suppose that the desired behavior Lm (H ) is a solvable regular language under the conditional architecture and it is desired to realize, in the form of finite-state automata, local supervisors that result in this desired behavior. Given that solvability is assured, local unconditional decision rules are obtained from (1). Without loss of generality, we can assume that H is a strict subautomaton of G [2], [3]. Then, local unconditional decision rules can be realized by constructing for each local site i the observer of H for projection Pi and finding “enable” and “disable” decisions according to (1) for each observer state as is done in [5] and [8]. Details are omitted here. Hereafter, we focus on the realization of the first part of (2), regarding the “enable if nobody disables” decision rule. The treatment of the “disable if nobody enables” decisions is similar, with suitable modifications as will be described later. Equation (2) is the basis for the realization of local conditional decision rules. The “enable if nobody disables” decision rule ecifc (Pi ( 1 )) in (2) implies that a controllable continuation is enabled conditionally by supervisor i only if supervisor i is certain that all illegal controllable continuations with can be disabled by some local supervisor. In other words, based on its observation and inference on observations of other supervisors, if there is some illegal continuation that cannot be disabled by some local supervisor for sure, then is not conditionally enabled. Formally, 62 ecifc (Pi (s)) if Ei (s) \ L(G) = ; or 9si 2 Ei (s) \ (L(G) n L(H )) such that
(
8
j
2
( ))[E (s ) (H ) = ]: (3) (G) = simply means that the estimated
Ic
j
i
\ L
6
=
()
Ei s \ K
;g
\
i
only concentrate on the second part of the condition 62 ecifc (Pi (s)). Let us recall the definition of C&P coobservability from [8] for the sake of further discussion. is said to be C&P coobDefinition 4: A language K M = M and 8 2 servable w.r.t. M; 6o;1 ; 6c;1 ; . . . ; 6o;n ; 6c;n , if 8s 2 K 6c = [in=1 6c;i s.t.s 2 M n K , there exists i 2 Ic () s.t.
;
The condition Ei (s) \ L ; behaviors (Ei (s)) cannot be continued with event within the system behavior. Therefore, is excluded from the decision since decisions over are irrelevant to the resulting controlled language. Therefore, we
;
:
From Definition 4 and (3), when the system executes trace s, the following happens: i) there exists si such that Pi (si ) = Pi (s) and si violates C&P coobservability, and ii) is excluded from the “enable if nobody disables” decision of supervisor i. In order to identify traces violating C&P coobservability, we will modify the automaton Mc (6c ) of [4]. In [4], Mc (6c ) is denoted by M . The notation Mc (6c ) here emphasizes that the M of [4] pertains to C&P coobservability and that it is parameterized by the set of controllable events. The modification involves changing the labels of some of the (unobservable) transitions to in order to identify the precise trace executed by the system. More precisely, we have mod
Mc
(6 ) =
M
Q
c
;
6
;
[f g
M
M
M
; q0
; Qm
where Q
:= (Q Q Q := q0 ; q0 ; q0 ; q0 := vc :
M
H
M
H
q0
M
Qm
f
H
2
H
2
H
H
2
G
Q
)
vcg
[ f
G
g
Hereafter, only the accessible part of the state space QM is consid. Let us recall from [4] the set of conered when we refer to QM ditions implying the violation of C&P coobservability; we retain these conditions in Mcmod (6c ). For 2 6c
( ) ( ) ( ) (q4 ; )
H q1 ; H q2 ; H q3 ; G
is defined if is defined if is not defined is defined
The transition relation M brevity, given that q1 ; q2 ; q3 the following:
2
0
H
0
G
i
2
6 6
c;1
( ):
c;2
3
is defined as follows. For the sake of G 2 Q , and 2 6, let us define
QH ; q4
:= (q ; ) for i q4 := (q4 ; ) and q := (q1 ; q2 ; q3 ; q4 ): ~ qi
2
2 f
1; 2; 3
g
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Fig. 2. Automata
and
.
Note the use of the label in three instances that follow. For and 62 6o;2
For
For
26
0
0
o;2
0
0
o;1
and
62 6
0
0
0
0
0
o;2
(q~; ) = (q1 ; q2 ; q3 ; q4 ) if q2 exists (q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q1 ; q3 ; and q4 exist 0
0
0
M
M
and
0
0
M
26
0
0
(q~; ) = (q1 ; q2 ; q3 ; q4 ) if q1 exists (q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q2 ; q3 ; and q4 exist
M
26
o;1
M
For
M
62 6
0
0
o;1
((~q); ) = ((qq1 ;; qq2;; qq3 ;; qq4 )) ifif qq1 exists 1 2 3 4 2 exists ( q1 ; q2 ; q3 ; q4 ) if q3 and q4 exist (q~; ) = vc if(3): 0
M
62 6
o;1
and
26
0
0
0
0
0
o;2
(q~; ) = (vcq1 ; qif2(;3q)3:; q4 ) if q1 ; q2 ; q3 ; and q4 exist 0
0
0
0
0
0
0
0
For 2 6; M (vc; ) is undefined. The state–space QH 2 QH 2 QH 2 QG tracks all traces s; s0 ; s00 2 L(H ) such thatmodP1 (s) = P1 (s0 )andP2 (s) = P2 (s00 ). The transition relation of Mc (6c ) is defined to track the traces in the following manner: Q
H
s 0
00
2 Q 2Q 2 Q H
s
H
G
:
s
Two traces s and s are tracked by the first two H ’s and the last pair H and G tracks s. The violation of C&P coobservability is characterized by traces s; s0 ; s00 2 L(H ) and event such that P1 (s) = P1 (s0 ); P2 (s) =
( ) 2 L(H ); s 2 L(H ); and s 2 L(G) n L(H ), where 2 6 \ 6 2 . The characterization of the violation of C&P
P2 s00 , s0 c;1
00
c;
coobservability demands to track legal traces (this is done by the first two H ’s) and one illegal trace (the last pair H and G tracks this). The occurrence of this violation that is captured by condition (3) causes a transition into marked state vc. In order to identify the illegal traces [denoted by s in (3)] violating C&P coobservability, we need to distinguish the transitions tracking trace s from the transitions tracking traces s0 and s00 (traces that have the same local projections as s at local sites 1 and 2, respectively). Hence, when we consider a transition tracking s (that is, when the last pair of H and G involved), we attach the label to the transition. Otherwise (that is, when the last pair H and G is not involved), the label is attached to the transition. With this modification, if s reaches state vc, then we know for sure that s 2 L(G) nL(H ) and that s violates C&P coobservability. In order to realize the local “enable if nobody disables” decision rule of supervisor i from (3), we need to identify every si 2 Ei (s) \ (L(G) nL(H )) violating C&P coobservability and exclude such 2 6c;i from the local “enable if nobody disables” decisions. Let us remove marked state vc and its attached transitions from Mcmod (6c ) and denote the result by Mc0 (6c ). We construct the (deterministic) observer of Mc0 (6c ) with respect to 6o;i , denoted by Obsi (Mc0 ). The purpose of constructing the observer of Mc0 (6c ) with respect to 6o;i is to identify the reachable states of Mc0 (6c ) with Ei (s) when s is the trace executed by the system and the observed trace is Pi (s). Moreover, Mcmod (6c ) is constructed to explicitly identify traces violating C&P coobservability. Therefore, if a state of Obsi (Mc0 ) that is reached by trace Pi (s) contains states of Mcmod (6c ) that reach state vc with some transition in automaton Mcmod (6c ), then we know that there exists si 2 Ei (s) \ (L(G) nL(H )) violating C&P coobservability and event is excluded from the local “enable if nobody disables” decision of supervisor i after observing Pi (s). In this manner, the automaton Obsi (Mc0 ), together with Mcmod (6c ), can be used as a realization of the local “enable if nobody disables” decision rule of supervisor i. Algorithm 1 formalizes the procedure described previously. The correctness of Algorithm 1, namely the realization of the local “enable if nobody disables decision,” is shown in Theorem 2.
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Fig. 3. Supervisor 1: unconditional decisions. (a)
( ( )). (b) ( ( )).
Theorem 2: For all s 2 L(H ) and i 2 f1; . . . ng; eci (Pi (s)) = eci (Pi (s)). Proof: () Suppose that 62 ecifc (Pi (s)). This implies that for 2 6c;i (i) Ei (s) \ L(G) = ; or (ii) there exists si 2 Ei (s) \ (L(G) n L(H )) s.t. fc
(8j 2 I ())[E (s ) \ L(H ) 6= ;]: c
j
i
(5)
If i) holds, it is clear that 62 eci (Pi (s)). Therefore, we focus on condition ii) hereafter. Since si 2 Ei (s) \ (L(G) nL(H )), we have that si 2 L(H ); si 2 L(G) n L(H ), and Pi (s) = Pi (si ). With (5) indicating the violation of C&P coobservability and from the construction of Mcmod (6c ), we know that there exists si 2 L(Mcmod (6c )) 63 4 such that
M
M
q0
; si
(6 )
= vc:
4Note that the automaton is nondeterministic and the transition represents the silent transition. label in
(6 )
1209
Let q M
:=
(q0
M
M
Since Pi (si )
M
; si
)= (q
M
M
;
(q0
M
)
; si . Then, we have that
) = vc:
= P (s), we have i
q
M
2 q := obs
Obs (M
)
obs
()
q0 ; Pi s
:
Therefore, we have 2 neci (qobs ). Consecutively, we have 62 eci (Pi (s)). () The proof of this case follows similar arguments to the preceding case (in reverse), starting from the assumption that 62 eci (Pi (s)). The realization of the “disable if nobody enables” decision rule in (2) can be performed analogously. The key difference is centered on the use of a suitably modified version of the nondeterministic automaton introduced in [8] for testing D&A coobservability. Namely, we modify Md (6c ) of [8] to get Mdmod (6c ) by using labels, in the same spirit as was done when obtaining Mcmod (6c ) in Algorithm 1. Then, state
1210
Fig. 4.
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(6 ).
of Mdmod (6c ) [which is analogous to state vc of Mcmod (6c )] is deleted and Md0 (6c ) is obtained. The realization of the conditional “disable if nobody enables” decisions of supervisor i is then encoded in the observer of Md0 (6c ) for event set 6o;i , using Mdmod (6c ). (For complete details, see [9].)
vd
IV. EXAMPLE Consider the uncontrolled behavior generated by automaton G in Fig. 2(a) and the desired behavior generated by automaton H in Fig. 2(b). We define that Lm (H ) = L(H ) = K and Lm (G) = L(G) (i.e., marking is omitted for all states). We set 6o;1 = fa; a0 ; c; dg; 6o;2 = fb; b0 ; c; dg; 6c;1 = 6c;2 = fc; dg; 6c;e = fcg, and 6c;d = fdg. In [10], we showed that L(H ) is conditionally coobservable w.r.t. L(G); 6o;1 ; 6c;d;1 ; 6c;e;1 ; . . . ; 6o;n ; 6c;d;n ; 6c;e;n . We focus here on the realization of the supervisors. Finite-state automata realization of the unconditional decisions of supervisor 1 are shown in Fig. 3. In Fig. 3(a), the event information appearing in each state of the realization (rectangle in figure) represents the enabled events; in the case of Fig. 3(b) it represents the disabled events. The automata in Fig. 3 are obtained using the techniques presented in [5] and [8], which are not repeated here. Following the procedure discussed in Section III, we construct mod Mc (6c ), shown in Fig. 4, to identify the traces violating C&P coobservability (Step 1). By removing state vc from Mcmod (6c ), we get Mc0 (6c ) (Step 2). In order to realize the local “enable if nobody disables” decision rules, we build observers of Mc0 (6c ) w.r.t. 6o;1 and 6o;2 , respectively (Step 3). The observer for supervisor 1 is shown in Fig. 5 where, for the sake of readability, the states of Mcmod (6c ) are renamed as described
in the table in the figure. For Mcmod (6c ), states (1; 5; 2; 2) state d in Fig. 5) and (8; 1; 4; 4) state o in Fig. 5) reach vc with event c and state (12; 14; 11; 11) state z in Fig. 5) reaches vc with event d. Therefore, the observer states of Mc0 (6c ) containing state d (observer state (a, b, c, d, e, f, g, h) in Fig. 5) or state o (observer state (j, k, l, o, p, q) in Fig. 5) exclude c from the “enable if nobody disables” decisions, and the observer states of Mc0 (6c ) containing z (observer state (t, u, v, w, x, y, z, aa) in Fig. 5) exclude d from the “enable if nobody disables” decisions (Step 4). The realization of the “enable if nobody disables” decision rule of supervisor 1 is therefore as shown in Fig. 5 (set of events next to observer state components). Note that c = nec1 ((a; b; c; d; e; f ; g; h)); c = nec1 ((j; k; l; o; p; q)), and d = nec1 ((t; u; v; w; x; y; z; aa)). States (a; b; c; d; e; f ; g; h); (j; k; l; o; p; q), and (t; u; v; w; x; y; z; aa) are reached with observed traces ; a, and cd, respectively. Therefore, we have that c
62 ec1 ()
c
62 ec1 (a)
and d 62 ec1 (cd):
Finally, following (4), we have
( ) = ec1 (a) = ec1 (cd) = ;:
ec1
Note that trace cd remains feasible because event c is enabled by default initially. We omit the realization of supervisor 2, which can be performed similarly; see [9]. The realization of the local “disable if nobody enables” decision rules is omitted here due to space constraints; again, we refer the reader to [9] in this regard.
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[8]
, “A general architecture for decentralized supervisory control of discrete-event systems,” Discrete Event Dyna. Syst.: Theory Appl., vol. 12, no. 3, pp. 335–377, 2002. [9] , “Decentralized supervisory control with conditional decisions—Part II: Verification and synthesis,” Univ. Michigan, Ann Arbor, MI, Tech. Rep. CGR-03-18, 2003. [10] , “Decentralized supervisory control with conditional decisions: Supervisor existence,” IEEE Trans. Autom. Control, vol. 49, no. 11, pp. 1886–1904, Nov. 2004.
A Fast Nonlinear Model Identification Method Kang Li, Jian-Xun Peng, and George W. Irwin Abstract—The identification of nonlinear dynamic systems using linear-in-the-parameters models is studied. A fast recursive algorithm (FRA) is proposed to select both the model structure and to estimate the model parameters. Unlike orthogonal least squares (OLS) method, FRA solves the least-squares problem recursively over the model order without requiring matrix decomposition. The computational complexity of both algorithms is analyzed, along with their numerical stability. The new method is shown to require much less computational effort and is also numerically more stable than OLS. Index Terms—Computational complexity, fast recursive algorithm, nonlinear system identification, numerical stability.
I. INTRODUCTION
Fig. 5. Supervisor 1: conditional decision
(
( )).
V. CONCLUSION The procedure presented to synthesize supervisors that implement conditional decisions is novel and relies on specially constructed nondeterministic automata that track violations of C&P and D&A coobservability, respectively. This realization procedure gives insight into the nature of conditional decisions, especially regarding the inferencing process that is at the heart of the conditional architecture.
REFERENCES [1] C. G. Cassandras and S. Lafortune, Introduction to Discrete Event Systems. Norwell, MA: Kluwer, 1999. [2] E. Chen and S. Lafortune, “On the infirnal closed and controllable superlanguage of a given language,” IEEE Trans. Autom. Control, vol. 35, no. 4, pp. 398–404, Apr. 1990. [3] H. Cho and S. I. Marcus, “On supremal languages of classes of sublanguages that arise in supervisor synthesis problems with partial observation,” Math. Control Signals Syst., vol. 2, pp. 47–69, 1989. [4] K. Rudie and J. C. Willems, “The computational complexity of decentralized discrete-event control problems,” IEEE Trans. Autom. Control, vol. 40, no. 7, pp. 1313–1318, Jul. 1995. [5] K. Rudie and W. M. Wonham, “Think globally, act locally: Decentralized supervisory control,” IEEE Trans. Autom. Control, vol. 37, no. 11, pp. 1692–1708, Nov. 1992. [6] W. M. Wonham, “Notes on control of discrete-event systems,” Univ. Toronto, Toronto, ON, Canada, Tech. Rep., Jul. 2003. [7] T. Yoo and S. Lafortune, “Decentralized supervisory control: A new architecture with a dynamic decision fusion rule,” in Proc. 6th Int. Workshop Discrete Event Systems, Zaragoza, Spain, 2002, pp. 11–17.
Some widely used nonlinear regression models and neural networks constitute linear-in-the-parameters models, for example the nonlinear autoregressive model with exogenous inputs (NARX) and radial basis function (RBF) networks [1]–[8]. Such models form a linear combination of model terms, or basis functions, which are nonlinear functions of the system variables. Depending on the nonlinear functions employed, linear-in-the-parameters models possess broad approximation capabilities, and have wide applications [1]–[8]. One problem with such models is that an excessive number of candidate model terms or basis functions usually have to be considered initially [2]–[8]. From these, a useful model is then generated based on the parsimonious principle [9], [10], of selecting the smallest possible model, in terms of size, which explains the data. Given a model selection criterion, this can be achieved by an exhaustive search of all possible combinations of candidates using a least-squares method. This is computationally very expensive. To reduce the computational complexity, efficient suboptimal search algorithms have been proposed, among which orthogonal least-squares (OLS) method is perhaps among the most popular [2]–[9]. OLS was first applied to nonlinear dynamic system identification [2], [3] and is now widely used in many other areas [4]–[8]. In general, OLS approaches are derived from an orthogonal (or QR) decomposition of the regression matrix [2]–[9]. The elegance of the OLS approach lies in that, the net decrease in the cost function can be Manuscript received September 18, 2003; revised January 10, 2005. Recommended by Associate Editor E. Bai. This work was supported by the U.K. EPSRC under Grant GR/S85191/01 to K. Li. The authors are with the School of Electrical and Electronic Engineering, Queen’s University of Belfast, Belfast BT9 5AH, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2005.852557
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