About Disturbance Decoupling of Timed Event Graphs in Dioids
Mehdi Lhommeau, Laurent Hardouin, Bertrand Cottenceau Laboratoire d’Ing´enierie des Syst`emes Automatis´es
62, avenue Notre-Dame du lac 49000 ANGERS
f
lhommeau,hardouin,cottence
g@istia.univ-angers.fr
Abstract
This paper deals with control of Timed Event Graphs (TEG) when a disturbance acts on transitions. We synthe- size the greatest feedback controller which allows to match the disturbance action. Formally, this problem is very close of the classical problem of disturbance decoupling for lin- ear systems.
1 Introduction
About 20 years ago a linear theory was introduced for a particular class of Discrete Event Dynamic Systems (DEDS) called Timed Event Graphs (TEG). Timed Event Graphs (TEG) constitute a subclass of timed Petri nets whose each place has exactly one upstream and one down- stream transition. It is well known that the timed/event be- havior of a TEG, under the earliest functioning rule1, can be expressed by linear relations over some dioids [1]. Strong analogies then appear between the classical linear system theory and the(max;+)-linear system theory. In particular, the concept of control is well defined in the context of TEG study. In the literature, an optimal control for TEG exists and is proposed in [6], [11]. It is an open-loop control that requires the knowledge of the whole reference input trajec- tory to compute the control law. For a given reference input, this open-loop control yields the latest input firing date in order to obtain the output before the desired date. Moreover, recent works deal with the problem of closed-loop control [7], which consists in synthesizing a controller in a model matching objective. Furthermore the proposed controllers allow delaying, as much as possible, the tokens input inside the TEG. Several recent studies lead to extend the range of systems admitting a linear representation in these algebraic structures (see [10] for unstationary TEG, [12] for nonlinear TEG).
Moreover, these algebraic structures have been appre-
1i.e. a transition is fired as soon as it is enabled
hended since1996under a geometric approach [4], [5]. In the classical linear system theory, the interest of the geo- metric point of view has been shown [13]. The notions of controllability and observability amount to surjectivity, resp. injectivity, of certain linear operators. Hence images and kernels as geometric objects are central. In this paper the disturbance decoupling problem for TEG is discussed.
In section 2, we summarize some theoretical results from the (max;+) literature. In the next parts, modeling and properties of TEG in these algebraic structures are pre- sented [1]. The fourth part introduces the problem of con- trol when a disturbance acts on the system. In particular, by taking into account the nature of these systems, the direc- tion to give to this problem is discussed. Section 5 presents synthesis of an optimal output feedback controller which takes into account the disturbance. Finally, an illustration of these results is given in section6.
2 Algebraic Preliminaries
We first recall in this section some notions from the dioid theory. The reader is invited to consult [1] for a complete presentation2.
2.1 Dioid
A dioid is a set D endowed with two operations de- noted by (addition) and (multiplication), both asso- ciative and both having neutral elements denoted by "and
erespectively, such thatis commutative and idempotent (aa=a). Theoperation is distributive with respect to
, and"is absorbing for the(i.e. "a=a"=").
Dioids are algebraic structures where an idempotent addi- tion has no inverse. This idempotent addition defines a (par- tial) order relation (denoted) asab,a=ab. The upper bound of two elements aandbinDisaband the bottom element of Dis ". A dioidD is complete if it is
2An electronic version is available on http://www- rocq.inria.fr/scilab/cohen/SED/book-online.html.
closed for infinite sums and if the product distributes over infinite sums too. IfDis complete, then the top element of
Dis>. Symbolis often omitted.
Remark 1 If D is a dioid, set Dnn of nn matrices with coefficients inDis also a dioid. Sum and product are defined in the following way:
(AB)
ij
=a
ij b
ij
; (AB)
ij
= n
L
k =1 a
ik b
k j :
Theorem 1 Over a complete dioidD, the implicit equation
x =axbadmitsx=abas least solution, wherea =
L
i2N a
i(Kleene star operator) witha0=e.
Notation 1 The Kleene star operator, defined on a com- plete dioidD, will be represented by the application
K : D!D
x7!
M
i2N x
i
:
Property 1 ([9]) LetDbe a complete dioid.
8a;b2D
(a
)
= a
(1)
a
a
= a
(2)
(ab)
= (a
b)
a
(3)
Definition 1 (Kernel [4],[5]) Let C : X ! Y be a map- ping. We call kernel ofC (denoted bykerC), the equiva- lence relation overX :
x kerC
y,C(x)=C(y): (4)
Remark 2 The usual kernel definitionfx2XjC(x)="g becomes meaningless in dioid algebra. Relation (4) cor- responds to the kernel definition of a mapping defined on lattices [8].
Definition 2 (Isotone mapping) A mappingfdefined over ordered sets is isotone ifab)f(a)f(b).
Definition 3 (Closure mapping) An isotone mapping f :
E !Edefined on an ordered setE is a closure mapping if
f Id
E andfÆf =f.
Example 1 The Kleene Star mapping (see Theorem 1) is a closure mapping sincea =
L
i2N a
i
aand(a) =a (see Property 1).
Definition 4 (Restricted mapping) Let f : E ! F be a mapping and A E. We will denote fjA
: A ! F the mapping defined byfjA
=f ÆId
jA whereIdjA
: A! E,
x 7! x is the canonical injection. Identically, let B F withImf B. MappingB j
f : E ! Bis defined byf =
Id
jB Æ
B j
f, whereIdjB
: B ! F,x 7! xis the canonical injection.
2.2 Residuation theory
The residuation theory provides, under some assump- tions, optimal solutions to inequalities such as f(x) b, wheref is an isotone mapping defined over ordered sets.
One can note that a complete presentation of this theory is given in [2], and see [1] for a specialization to dioids.
Definition 5 (Residuated mapping) An isotone mapping
f :E!F, where(E;E
)and(F;F
)are ordered sets, is a residuated mapping if for ally2Fthe least upper bound of the subsetfx2Ejf(x)F
ygexists and belongs to this subset.
Theorem 2 Letf :E!Fbe an isotone mapping from the complete dioid (E;E
)into the complete dioid (F;F ). The following two statements are equivalent :
(i) f is residuated.
(ii) There exists an isotone mappingf]:F!Esuch that
fÆf ]
F Id
Fandf]ÆfE Id
E.
Consequently,f]is unique. Whenf satisfies these proper- ties, it is said to be residuated andf]is called its residual.
Proposition 1 ([7]) Letf :E ! Ebe a closure mapping.
MappingImfj
f is a residuated mapping of which residual is the canonical injectionIdjImf
:Imf !E,x7!x. It means that a closure mapping restricted to its image is a residuated mapping, with the canonical injection as residual.
2.3 Residuation theory and dioid
Theorem 3 ([1]) Letf : D ! E be a mapping where D and E are complete dioids of which bottom elements are respectively denoted by"Dand"E. Then,fis residuated iff
f("
D )="
E and8AD,f(
L
x2A x)=
L
x2A f(x). Example 2 ([1]) MappingsLa
:x7!axandRa
:x7!xa
defined over a complete dioidDare both residuated. Their residuals are usually denoted by respectively L]
a (x) =
aÆ
nx= x
a
andR]
a (x)=
xÆ
= a= x
a
.
Theorem 4 ([3]) LetA2Dnn, are equivalent :
(i) A=A (ii) A=AÆ=A
We recall that A belongs to the image of K (denoted by
ImK).
Proposition 2 ([7]) MappingImKj
K(Kleene star operator) is a residuated mapping of which residual is (ImKj
K ) ]
=
Id
jImK.
Proof: The proof is a direct application of Proposition 1, sinceKis a closure mapping.
2.4 Applications in Geometry and System Theory
We briefly recall a few algebraic results here, more de- tails can be found in [4, 5].Theorem 5 ([4, 5]) For two mappingsA:X !DandB:
Y ! DwithAresiduated, the following three statements are equivalent :
1: ImBIm A;
2: B=AÆA ]
ÆB;
3: there existsL:Y!X such thatB=AÆL.
2.5 Projections on the Image of a Mapping Paral- lel to the Kernel of Another Mapping
We considerB :U ! X andC : X ! Yand we call projection ofx2X onImBparallel tokerCanyx0which belongs toIm Band is equivalent tox modulokerC, that is,
x 0
=B(u) and C(x0)=C(x)
The conditions of existence and uniqueness of such a pro- jection are discussed in [4] for the case of residuated map- pings B andC, and in [5] for the case of -morphisms.
When the projector on ImB parallel to kerC exists and is well-defined, it is denotedC
B
and it is given byCB
=
B ÆB ]
ÆC ]
Æ C in the residuated case, and by CB
=
(BÆ
=(CB))C=B((CB) Æ
n Cin the case of morphisms.
3 TEG representation 3.1 Transfer function
Timed Event Graphs (TEG) are well adapted to model synchronization phenomena; moreover, they can be seen as linear dynamic systems in dioid algebra [1]. TEG behav- ior can be expressed over many dioids, for instance in the dioid of formal power series in one variable and coeffi- cients inZmax. This dioid is usually denoted by Zmax
[[]]
in literature.
Consequently, for a given TEG we can obtain the follow- ing representation over dioidZmax
[[]]
x = AxBu
y = CxDu
(5)
where,
x() 2 X represents the state vector ; u() 2 Urepre- sents the control vector ;y()2Yrepresents the output vector.
MappingsAnn : X ! X,Bnr : U ! X,Cmn :
X !YetDmr
:U!Yare represented by constants matrices whose terms are inZmax
[[]].
For instante considering the TEG drawn in solid black lines in Fig. 1 (without taking the dotted arcs into account).
It models an elementary production workshop composed of three machines (M1 to M3
). Machine M1 can process
2parts simultaneously, each processing lasts 6times units.
MachineM3processes the parts released by machines M1
andM2. For this TEG, a state representation is
0
@ x
1
x
2
x
3 1
A
= 0
@ 6
2
" "
" 6 2
"
7 8 6
2 1
A 0
@ x
1
x
2
x
3 1
A
0
@ 11 "
" 9
" "
1
A
u
1
u
2
(6)
y
= " " 1
0
@ x
1
x
2
x
3 1
A (7)
3 6
5 1 2 6 1
u1u 2 6 y
x1x2 x3
q 1q 2 q 3
M 1M 2 M 3
65
v2v1
Figure 1. A TEG endowed with a controller (dotted lines)
4 About disturbance decoupling in dioid
In this chapter we first discuss if the disturbance decou- pling problem have a sense for the TEG. Examination of this problem allows naturally giving a practical sense to the kernel definition of a mapping in dioid. Let us consider the system :
x = AxBuSq (8)
y = Cx (9)
The termqin (8) represents a disturbance which is assumed to be not directly measurable by the controller. Let us as- sume thatqbelongs to a setQand that mappingS:Q!X is a mapping which binds the perturbationqto the statex. In the conventional linear system theory [13], the distur- bance decoupling problem consists in finding a control u such that disturbance qhas no influence on the controlled output y. A particular problem is to find (if possible) an output feedbackF, i.e.u=Fy, which allows reaching this objective. The disturbance decoupling problem can also be solved by the mean of a state feedback, y =Fx. From an algebraic point of view, it amounts to finding F such that the state trajectory remains in a subspace of the kernel of mappingC, i.e. a state which leads to a null output8q.
Our problem must be stated in a different way since trajec- toriesu;x;yandqare monotonous and no decreasing (date
x
i
(k+1)is later than datexi
(k)). The output cancellation is consequently meaningless in this context. Nevertheless finding a controluwhich keeps the statexin the kernel of
Cfor all disturbanceqis relevant, provided that the notion of kernel be redefined. In dioid (see Definition 1), the ker- nel ofCis an equivalence relation3, i.e. the space splits up in equivalence classes (each class contains all the elements which map to the same image, in [4], the term ”fibration”
is used). Obviously the set of these controls may contain many elements, hence we are interested in computing the greatest one, since it is the one which satisfies the just-in- time criterion. Formally this problem can be established in the following way. The explicit solution of (8) is :
x=A
BuA
Sq; (10)
which leads to the output
y=CA
BuCA
Sq:
This equation allows establishing that all controlsusuch as
CABu CA
Sq keep unchanged the output generated by q. In agreement with the objective stated previously, this consists in establishing the greatest control usatisfy- ing this inequation. This greatest control uallows delay- ing as much as possible the tokens input inside the TEG by taking into account the disturbance action. Actually, it is useless that tokens be inserted too soon in the TEG since the uncontrollable disturbanceqdelays the output fir- ing. The residuation of mappingLCA
B(Theorem 3) yields
u =CA
B Æ
n CA
Sq:It is the greatest control which keeps statexin the equivalence class ofASqmodulokerC, i.e.
in the class of states that yields the output CASq. This problem is very close with the disturbance decoupling prob- lem of the classical system theory, since the control objec- tive is to keep the statexin the kernel ofC.
Remark 3 Another way of interpreting this result is the fol- lowing one. If we replaceuin Equation (10) by its formal valueu=CABnCAÆ Sq, Equation (10) becomes :
x = A
B(CA
B Æ
nCA
Sq)A
Sq;
thanks to (2), Equation (10) is also equal to :
x = A
B(CA
A
B Æ
nCA
A
Sq)A
Sq:
Let AB : U ! X, CA : X ! Y be mappings. The termAB(CAABÆnCAASq)is the projection ofASqon
ImA
Bparallel tokerCA.
IndeedAB(CAABÆn CAASq)can be also written as
CA
A
B (A
Sq)=A
BÆ(A
B) ]
Æ(CA
) ]
ÆCA
(A
Sq);
In other wordsCA
A
B (A
Sq)is the maximal elementof the image of ABsuch thatCA CASq(but equality does hold true ifASqis already inImAB).
3i.e. the kernel ofCis not a ’subspace’ ofX
Practically this control computation requires the distur- bance4knowledge. Our problem is then to find a feedback
F which allows avoiding this assumption.
5 Output feedback for disturbance decou- pling
In this part we discuss the existence and the compu- tation of two output feedback controllers which lead to a closed-loop system making the disturbance decoupling (in the sense defined previously). The objective of the first con- troller (denoted by F1) is to keep unchanged state xwhat- ever be disturbance q. The second controller (denoted by
F
2) keeps unchanged outputywhatever be disturbanceq.
Figure 2. The output feedback control
A system provided with a controllerFis represented Fig.
1. Its behavior is described by the following equations:
x = AxBFySq (11)
y = Cx (12)
5.1 Output Feedback which keeps the state
In this section we look for a controller which achieves disturbance decoupling (always in the sense defined before) by keeping statex. Equation (11) is written also :x = AxBFCxSq=(ABFC)xSq; (13)
thanks to Theorem 1, we establish :
x=(ABFC)
Sq:
This equation is to be compared with the expression of the transfer between the state and the disturbance without con- troller :x=ASq.
The search for a closed loop control which kept state x, can then be formally expressed as the controller synthesis F1
such that :
(ABF
1 C)
S=A
S: (14)
Obviously this controller leads to an output unchanged with respect toq, formally it means that :
C((ABF
1 C)
S)=C(A
S),(ABF
1 C)
S kerC
A
S
4In a manufacturing system,qmay represent the supply of raw material which is a priori known. The problem is then very similar to the problem introduced in [11] which establishes an optimal open-loop control in pres- ence of known uncontrollable inputs.
Among the controllers satisfying the objective (Eq. 14), we seek the greatest one, i.e. the one which will generate the greatest control : u = F1
y. Thanks to Property (3), our objective becomes:
(A
BF
1 C)
A
S = A
S: (15)
First we look for the greatest controller such that
(A
BF1C)
A
SA
S; say
^
F
1
=
L
fF
1 j(A
BF
1 C)
A
SA
Sg F
1 :
Thanks to the results of section 2we have the following equivalences :
(A
BF
1 C)
A
SA
S, (A
BF
1 C)
A
SÆ
=A
S (16)
, A
BF1CA
SÆ
=A
S (17)
, F
1 A
B Æ
nA
SÆ
=CA
S=
^
F
1(18)
(16) since mappingRA
S is residuated (see Example 2)
(17) sinceASÆ=A2ImK(see Theorem 4)
(18) since mappingsLA
BandRCare residuated We now prove that F^1 is a solution, i.e(ABF^1
C)
A
S =
A
S. It is obvious sinceF1
="is a solution.
Then the greatest controller satisfying (14) is :
^
F1=A
B Æ
nA
SÆ
=CA
S: (19)
It is the greatest output feedback which preserves state x generated by q. From a practical point of view, it is the controller which will generate the greatest controlu=F^1
y, i.e. the controller leading to the control trajectory (u =
^
F
1
y) which delays as much as possible the input of tokens.
Since the disturbances will delay the firing of some internal transitions, delaying the inputu(by the way ofF^1) avoids a useless accumulation in the upstream places of the disturbed transitions.
5.2 Output feedback which keeps the output
In this section we are interested in synthesizing the great- est controller which preserves outputy.Let us solve (Theorem 1) the implicit Equation (11) :
x = A
BF2yA
Sq;
hence outputyis expressed by :
y=CA
BF
2 yCA
Sq: (20)
Equation (20) is also an implicit equation, by using Theo- rem 1 again we obtain
y = (CA
BF
2 )
CA
Sq:
The objective of this controller is to leave the output un- changed whatever be disturbanceq, i.e formally :
(CA
BF2)
CA
S = CA
S: (21)
First we look for the greatest controller :
^
F
2
=
L
fF
2 j(CA
BF
2 )
CA
SCA
Sg F
2 :
Thanks to the results of section 2we have the following equivalences :
(CA
BF2)
CA
SCA
S, (CA
BF2)
CA
SÆ
=CA
S (22)
, CA
BF2CA
SÆ
=CA
S (23)
, F
2 CA
BÆnCA
SÆ=CA
S=
^
F
2(24)
(22) since mappingRCA
Sis residuated (see Example 2)
(23) sinceCASÆ=CAS2ImK(see Theorem 4)
(24) since mappingLCA
Bis residuated We now prove that F^2 is solution, i.e(CABF^2
)
CA
S =
CA
S. It is obvious sinceF2 = "is a solution. Then the greatest controller is :
^
F
2
=CA
B Æ
n CA
SÆ
=CA
S: (25)
This new controllerF2is the one which keeps unchanged the output. It leads to the transfer relation between pertur- bationqand statex:
^ x=(A
B(CA
B Æ
nCA
SÆ
=CA
S)C)
A
Sq
It belongs to the equivalence class ofxmodulokerC:
(A
B
^
F
2 C)
A
S k erC
A
S,C((A
B
^
F
2 C)
A
S)=C(A
S):
6 Application
Let us consider the TEG of Fig. 1 Equations (6) and (7) provide the entries of matrices A;B andC. Matrix S is given below :
S = 0
@ e " "
" e "
" " e 1
A
;
Trajectoriesq1
;q
2andq3represent inputs delaying the re- lease of parts from machinesM1
;M
2andM3. It may rep- resent the effect of exogenous events on the machines be- havior. Our objective is thus to synthesize the greatest state feedback controllerF^=
^
F11
^
F
21
such that controlu=F^yis the greatest one that leaves the output unchanged. Relation (25) provides the formal expression of this controller. Its computation leads to :
^
F=
19(6 2
)
18(6 2
)
:
The matrix entries are periodical, but not causal. According to ([7]), we use the maximum realizable controller F^+that is equal toF^without terms with negative exponents, i.e.
^
F
+
=
5 8
(6 2
)
6
(6 2
)
: (26)
A realization of this controller is given on Fig. 1 (dotted arcs). It leads to the following control laws inZmax:
u1(k) = 6u1(k 2)5y(k 8)
u2(k) = 6u1(k 2)y(k 6)
6.1 Numerical application
Let us consider the following trajectories for inputs :
v1 = e5 3
13 9
16 12
+1 13
;
v2 = 210 5
12 8
13 9
17 12
+1 13
:
The states are delayed by disturbances whose trajectories are as follows :
q
1
= 19 5
12 8
16 10
18 12
+1 13
;
q
2
= 18 2
9 5
13 7
14 9
15 10
+1 13
;
q
3
= 610 4
13 6
15 11
19 12
+1 13
:
StateX of the TEG (Fig. 1) without controller (open loop behavior) is equal to :
X = 0
@ x1
x
2
x3 1
A
=A
B
v
1
v2
A
S 0
@ q1
q
2
q3 1
A
= 0
B
B
@ 1117
2
23 4
29 6
35 8
41 10
47 12
+1 13
11117 3
23 5
29 7
35 9
41 11
+1 13
181924 2
25 3
30 4
31 5
36 6
37 7
42 8
43 9
48 10
49 11
54 12
+1 13
1
C
C
A
OutputY =CX(without controller) leads to the trajectory given below :
Y = 192025 2
26 3
31 4
32 5
37 6
38 7
43 8
44 9
49 10
50 11
55 12
+1 13
:
By considering the TEG with controllerF^+(Fig. 1), output
Y is given by
Y = (CA
B
^
F
+ )
CA
BV(CA
B
^
F
+ )
CA
SQ:
Obviously the outputs with and without controller are equals.
ControlU =
u
1
u
2
is equal to :
U =
^
F
+ YV
=
^
F+(CA
B
^
F+)
CA
BV
^
F+(CA
B
^
F+)CA
SQV:
=
e5 3
24 8
25 9
30 10
31 11
36 12
210 5
19 6
20 7
25 8
26 9
31 10
+1 13
37 12
+1 13
32 11
Clearly these control trajectories are greater thanv1andv2, i.e the input tokens are delayed as much as possible by pre- serving the outputY.
And the state vector in closed-loop is given by:
X = (A
B
^
F+C)
A
BV(A
B
^
F+C)
A
SQ
= 0
@
1117 2
23 4
29 6
35 8
36 9
41 10
11117 3
23 5
28 6
29 7
34 8
35 9
181924 2
25 3
30 4
31 5
36 6
37 7
42 11
47 12
+1 13
40 10
41 11
46 12
+1 13
42 8
43 9
48 10
49 11
54 12
+1 13
1
A
This state is also greater than the one obtained without con- troller and is the greatest which can be obtained with a con- trol lawU =FY which keep the outputY.
7 Conclusion
In this paper is discussed the problem of disturbance de- coupling in dioids. The objective is to synthesize a control law keeping statexin the kernel ofC. It presents a strong analogy with the disturbance decoupling of the traditional control systems. However it must be noted that the reached objective does not lead to an output cancellation, indeed the specific kernel definition of a mapping on a lattice and the nature of the considered systems allow obtaining the great- est control such that the output remains unchanged what- ever be the disturbance. It seems interesting now to con- sider state feedback controller synthesis in order to achieve disturbance decoupling.
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