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The Existence of a Discrete-Event Representation of Linear Continuous-Variable Systems with Quantised State Measurements
Copyrighl © 1996 IFAC 13th Triennial World Congress. San Francisco. USA
3d-04 I
THE EXISTENCE OF A DISCRETE-EVENT REPRESENTATION OF LINEAR CONTINUOUS-VARIABLE SYSTEMS WITH QUANTISED STATE MEASUREMENTS J. Lunze Technische Univer'siliit Hamburg-Harburg Arbeitsbereich Regelungstechnik Eissendorfer Strasse 40, D-21071 Hamburg email: Lunze @tu-harburg.d400.de
Abstract. This paper is concerned with the representation of stable linear discrete-time continuous- variable systems with quanti sed state measurements. It is assumed tha t the state space IRn is partitioned into disjoint regions Qx and that the number z of the region Qx(z) to which the state x belongs at time k is used as the qualitative value of x(k): [xJ = z. For a given z(O) [x(O)] the system can generate different qualitative trajectories [XJ [x(O)J, [x(I)J, [x(2)J ... The paper presents a necessary and sufficient condition under which the sequence [XJ is a Markov chain. This condition is rather res trictive and shows that there is, in general, no representation of the qualitative behaviour of linear systems by some stochastic automaton.
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Keywords. Linear dynamical systems, discrete-event systems , representation , Markov models, stochastic automaton, qualitative analysis, quantized states . one discrete state [x( k)J to another discrete state r,,(k + 1)] while generating the qualitative state sequence
I. INTRODUCTION
Problem statement. The subject of this paper is a linear autonomous discrete-time system
x(k
+ 1) = Ax(k),
x(O)
= "'0
(1 )
whose state vector x E IRn can only be qualita. tively measured. The qualitative value [x(k)J E Z n is defined by a partitio n of th e stat o space IRn into disjoint sets Qx( z), (z E Z " )_ [xJ is th e 'number' z E zn of the set Q. to which ., belongs . From the point of view of an observer who only knows the qualitative values [x(k)] (k = 0, 1,2, ... ) the system (1) behaves as a discrete- evell! system with discrete state space Z " _ The sys tem jumps from
[X (xo)] = ([Z(O)J, [x(I)], [x(2)J, ... ). The paper concerns the question whether the relation between the qualitative initial state [x(O)J and the qualitative state sequence [X(xo)J can be re. presented by some automaton with state space zn. The fundamental difficulty of this modelling prDblem resul ts from the fact that the qualitative behaviour [X C"'o)J of the system (1) is, in general, not unique because instead of x(O) only the qualitative value [x(O)J is known (cf. Lemma 2). Consequently, no model can precisely predict the qualitative be. haviour [X(xo)] of the systelll (1)_ Therefore, x(O)
3892
is a.ssumed to be random ly distrib uted over the set Qx([:z:oJ) of all initial states that have the given qualita tive value [x(O)] and a model is to be set up that describ es the random sequen ce [X("o) ]. The central question dealt with in this paper is Is [X(xo) ] a Markov chain? If the answer is in th.e affirma tive, the random se· quence [X(xo) ] can be genera ted by some model
[x(k
+ l)J = f ( [", (k) ],v(k» , [",(O)J = [xoJ (2)
that relates [x(k + 1)] with the predecessor state [o:(k)J and some whi t e random liaise v(k). Hence,
which symbo lic observ ations are eventu ally periodic. Franke (1994) used the formal analog ies of discret e-time system s and determ inistic autom ata to transfe r results from linear control theory to the analysi s and design of discret e-even t system s. Fur· ther results on the discre! e-even t represe ntation of continu ous- variabl e .yst"m s can be found in the literature on qualitative reasoning, which is summarise d in Chapte r 6 of the book (Lunz. 1995b) . Notati ons. IR and 71. are T;he fields of real or integer numbers, respectively. Boldfa ce letters as ", z or A denote vectors or matric ~s , respectively, whereas italics such as Xl, Xi, z, denote scalars .
there exists a model wi t h the state space z n. In case of a negati ve answer [x(k + 1)] depend s on more than one predec essor state
[",(H I)J
= j([x( k )], ["'( k -
1)], [x(k - 2)], ... , v(k))
2. CONT INUOU S-VAR IABLE SYSTE MS WITH QUAN TISED STATE SPACE Consid er an a.symp toticall y stable discret e-time system (1), which genera tes a unique state traject ory
and no represe ntation with state space tl n exists.
Main result . The main result of the paper is gi· ven in Theore m 1 in Section 4, which describ es a necessa ry and sufficient conditi on on the system (I) togeth er with the quanti ser for the sequen ce [X] to be a Marko v chain. This conditi oll turns out to be rather restrictive . Consequently, there is, in general, no represe ntation (2) of the qualita tive behavi our of system (1). Relev ant literat ure. The problem dealt with in this paper occurs in the study of hybrid systems, which consist of continuous-variable and discret e-varia ble compo nents (cf. (Gross man et al. 1993» . Althou gh the theorie s of continu ous and discret e system s have been elabor ated almost independ ently, the re are some results concer ning the relatio ns betwee n the continu ous and the discret e represe ntation of the same system . Lunze (1994) propos ed to use" stocha s tic autom aton a.s approximate represe ntation of [X] . The following re· sults will show that such a relHese ntation of [X] can be precise only under rather reslrkt ive conditions. Ramad ge (J 990) derived conditi ons under
X = (", (0), x(l) . x(2), . . .).
(3)
It is a.ssumed that the sta.te x( k) cannot be measured precisely but is observ ed throug h a directionw ise unifor m quant iser whose output is the qualita tive value z = [x] of the state x where
[xi(k)] =
Z; E Z if
(Zi-~)qxi:5
xi(k) <
(Zi+~)qXi
(4) holds. This quantis ation yields a partiti on of the state space IRn into the dis joint sets
Qx(z)
= {x E IRn : [:cJ = z}
z E zn.
(5)
For any given initial state Xo eqns . (1) and (4) yield the sequen ce of qualita tive states
[X("'o)]
= ([x(O)], ["'( I)], [x(2)], .. .),
(6)
which is also called the qualitative trajectory. This paper concer ns the situati on in which only the qualita tive value [:ro] = Zo of the initial state is known. Tl, en the systel1l can genera te any trajectory of the set X(zo)
3893
= {X(xo ) : "'0 E Qx(zo) }.
(7)
The set
[X(%o)J
= ([xJ : X
E X(zon
(8)
results from a quanti sation of all trajectories belonging to X(zo). An import ant fact is that [X(zo)J is a set with more thar one element unless the system (1) satisfies the restric tive condition given in the following lemmas. Lemm a 1 (Lunze 1994) The system (1) togethe r with the quantiser (4) has for all qualitative initial states Zo E tl n n unique qualitative tmjeetory (6) if and only if for all z\ E tl n there exists some %2 E tl n such that
holds. Then the set [X(zo)J is a singleton for all 7l n •
%0 E
Defini tion 1 If eqn. (9) holds for some pair (ZI,Z2 ) then the pair (ZI,%2) is called deterministic. The pair ( %j, %2) is called nondet ermini stic if {A'" : x E Qx(z, )} n Qx( Z2 ) '" 0 (10)
is valid but eqn. (9) violated. The question under what conditions such nond etermin istic pairs exist is answered by the following lemma . Lemm a 2 (Lunze 1994). Consider the asymptotically stable system (1) and define
A = diag
q;/ A diag qxi'
3. THE MODELLING PROB LEM The modeUing aim is to find a genera tor that determines for any given Zo = ["'oJ the qualita tive trajectory [X("'o)J of the system (1), (4). As Lemm a 2 states, such a model does not exist unless the condition (12 ) is satisfied. Th erefore , the model should predict the 'average qualita tive behavi our' as will be made more precise now. If the initial state "'0 ib assume d to be a random variable (ur, more pr"cisely, a random vector) , which is uniformly distrib uted over the set Qx(zo) , then [",(k )J,(k = 0,1, ... ) "re random variables and [X("'o)J is a random sequence. For any % E 7l"
P(z , k I zo} '" Prob([" ,(k )J = % if ["'oJ = zo) (13) describes the probab ility that the system (1), (4) assumes th.e qualita tive state z at time k if its initial state has the qualita tive value %0. For fixed k , P( z, k I %0) is the conditi onal probab ility distribution descTibing which qualitative states % E Z" the system can assume at time k. The aim is to find a model that genera tes for any given qualita tive initial state %0 E 7l" the sequen ce of conditional probab ility distrib utions P=(P (% , llz(O) ), P{%,2 Iz(0)), ... ).
In thefoUowing consid eration s, %(0), z(l), z(k) etc. denote random varia.bles d'~scribjng the qualitative state [",(O )J, [",(1)]' [x(k)J etc_ whereas %1 , %2 , %k describe speci fic elements of 7l" . In order to describe the modelling proble m in more detail, the following result, on random sequences are recalle cl. In general , the random sequence [X("'o)J is described by the sequence of probability distrib utions
(11)
Po(z(lI)) PI(%( l) , z(O))
The system (1) together with the quantiser (.4) has for all qualitative initial states %0 E tl n a unique qualitative trajecto ry (6) ;f and only if A = diag
1
2ni
+I
P
(14)
Pk(%(k ), %(k - 1), ... , %(0))
(15)
(12)
holds where P is a permutation matrix and n; E 7l , ni > 0 hold for i '" 1,2, ... ,n.
because
P(% , k 1%0)
L
Z(k- l)eZ n
3894
'"
L: z(1 )eZ n
Pk(%, z(k - I) , ... , zo)
holds. In genera l, the whole probab ility distrib ution Pk : 7l. n X 7l." X ... X 7l. n ..... [0, I] has to be known. Only if [X(xa) ] is a Markov chain, the
model to be set up has a concise representatjon.
describes the probab ility that on these traject ories the system assumes at time T the qualita tive state ZT e zn if it has assume d at time T - I the qualitative state Z p E 7l. n .
The Marko v prope rty says t1,at
P(Zk I Zk-\, Zk_2, ... , za)
= P(Zk I zk-ll.
(16)
holds. Then, the sequence (14) can be obtaine d according to the Chapm an- Kolmogorov equatio n
P(z,k lz a) =
L
P,(zlz )P(z , k-lI zal
( 17) for given P(z, I za) = Pa(zo) . Eqn. (17) is a con· eise represe ntation of the sequence [X(",o)]. It can be interpr eted as a stocha stic autom aton s(zn, Pt) with state space 7l." and transit ion probab ility dis· tributi on Pt : zn X 7l.n ..... [0, I].
denotes the set of qualita tive states from which there exists a traject ory that after k time steps re· aches the qualita tive state ZT. Obviously, min . P(ZT, T I zp , zo) :,; P(ZT' T I Zp, %0) ZOE!!(ZT,7 )
°
The main proble m to be dealt with in the fol· lowing section s is the question whether the ran· dom sequence IX(",o) ] possesses the Markov pro· perty (16).
4. PROPE RTIES OF THE QUAL ITATI VE
hold.
Lemm a 4 Consider the system (1) together with the quantiser (4) and assume that the initial state :eo is uniformly distribu ted oiler some gillen set Qx(zo). Consider po;'.' (ZP,ZT ) with ZT E 7l. n , %p e 8(ZT,I ). For sufficiently large T the follo wing relations hold:
(aJ For all pairs (ZP,ZT ):
TRAJE CTOR IES Thls section presen ts proper ties of the qualita tive traject ories that will be used in the derivation of the main results in Section 5.
max P(zy, Tlzp, zo) = 1 ZoEI3(ZT.T)
(h) For deterministic pairs (ZP,ZT ): min P(zT, Tlzp,z o) = ZoEI3(ZT,T) P(ZT , T I zp, Zo) = = Zo EI3( max P(Z·i" , T I Zp, Zo) = 1 z" ,T)
Lemm a 3 Consider stable linear time- invariant system s (I). For any given set X C IRn there exist some finite T E Z awl E zn such that
z
(18)
implies
:e(T) E X.
P(ZT, T I Zp, zo) = Prob([",(T)] = ZT if [",(T - I)l = zp , [:>:(O)J = zo)
(22)
(c) For nondeterministic pairs (zp, ZT): (23)
(19)
The proof has to be omitte d due to space limitat i· ons. Consid er now the set of qualita tive traject ories that the system (1 ) genera tes for uniformly distributed initial state "'a E QA za). The conditional prohab ility
(21)
The Proof is omitte d due
10
space limitat ions.
Rema rk 1 The results of Ramage (19g0) concerning the periodicity of tht qualitative tmjectories apply to the modelling problem considered here because the system (I) is assumed to be asymptotically stable. [n fact, the qualit"tive tmjectory becomes eventually periodic beenus< for aU "'0 E IRn there
3895
exists a sufficiently large time l' such that the relation P(o,k I zo) = 1 for all k ~ l' holds where 0 denotes the qualitative zero state. I'hat is, any qualitative tmjectory eventually becomes deterministic: [",(T)J
=0 ,
[.,(T
+ l)J = 0 ,
[x(T
and, thus,
P(z, I'-lli)
=
{
P(ZT, l' I i)
=
1
Theor em 1 Consider the system (1) together with the quantiser (4) and assume that the initial state Xo is uniform ly distributed over some set Qx( zo) . For all Zo E tl n the qualitative trajectory [X(xo)J defined by eqn. (6) is a Markov chain if and only if the condition (12) is satisfied. That is, the qualita tive traject ory possesses the Marko v proper ty (16) if and only if the system (1) has for all qualita tive initial states a unique qualitative traject ory.
0< P,(ZT, zp) < 1 0< P,(zjV ,zp) < 1,
P(zT, l'lz)
-f 0 n Q.(ZN ) -f 0
=1>
= zp [x(I')J = %,-
<
1
(30)
P,(ZT 1%) P(i,I' -11 z)
I zp)
P(zp, T
-11 z)
which comple tes the proof.
6. CONS EQUE NCES FOR MODE LLING THE QUAL ITATI VE BEHA VIOUR O}' SYSTE M (1) In this section several conseq uences for modell ing the qualita tive behavi our of the system (1) with quantis er (4) are given. Since %0 = [x01 is assume d to be known ,
= {l o
for Z else
=Zo
(31)
describes the 'initial state' of the m odel. Corol lary 1 If the conditi on (12) is satisfied n(
(24) (25)
I:
= P,(ZT
I )=
{1o
if [AxJ else
= ZI for 80me[.,] = %2
holds. Hence, eqns. (17) and (31) yields
For sufficiently large time l' there exists some i such that the relatio ns
[x(1' - I)J
(29)
ZEZ n
rt %1 %2
Qx(ZT )
(28)
which can be seen by conside ring the set X(zp) of trajectories startin g in zp. Hence,
P(z,O I z) Proof. The sufficiency of the conditi on (12) for the Markov proper ly follows directl y from eqn. (22) because eqn. (12) implies that there does not exist any nondet ermini stic pair (ZI, Z2). In order la prove the necessi ty assume tltat there exist nondet ermi· nistic pairs (ZP , ZT) and (ZP,Z N) satisfy ing
n
(26)
holds. Eqns. (26), (27) and (30) violate the eqn. (17) becaus e
The following theorem represe nt.s the main result of the paper.
{Ax ::z: E Q.(zp )} {Ax : x E Q. (zp)}
= Zp
(27)
P,(ZT I zp) 5. MARK OV PROP ERTY OF THE QUAL ITATI VE BEHA VIOUR
%
bold. On the other hand, eqns. (24) and (25) imply
+ 2)J = 0, ...
However, the madel to be set up has also to describe the possibly nondeterministic part ([.,(O)J, [.,(l)J, ... , [x(1' - 1)]) of these tmjectories.
1 for 0 else
z = [:r(k)J P( Z, k I Zo ) -_ {I0 for I e se
(32)
(33)
and, thus, d..cribes the unique qualitative trajectory (6).
3896
This coroUary foU ows from Theorem 1 and Part (b) of Lemma 4. The automaton s(zn, Pt) with Pt given by eqn. (32) is a deterministic automaton.
ries [X(zoJ can be represented by a stochastic automaton s(zn, Pt). Theorem 1 shows that such a representation is possible only under restrictive conditions.
Corollary 2 If the condition (12) is not satisfied there does not exist a stochastic automaton s(zn, Pt) that for any given initial state Zo = ["'0) describes the set [X(zo») of qualitative trajectories of the system (1),
In the light of this result the question occur why deterministic automata have often been successfully used for describing discrete-event systems. The answer lies obviously in the fact that in all these cases the system has not been considered with equidistant clock time, but the system under consideration itself determines the clock time of the next event or the system has specific properties comparable with that given by eqn. (12).
m.
It has been proposed by Lunze (1994) to use the stochastic automaton s(Zn, Pt) with
= Prob([A"'J = ZI
for", E QxCZ2)) (34) as qualitative model of the system (1) with quanP(z,Olz) is given as in tiser (4). If P(z,Olz) eqn. (31) this model generates the sequence
Pt(Zl [Z2)
REFERENCES
=
P(z, k 1 zo)
= L:
Pt(z 1z) p(z, k -
11 zo).
zeZ n
(35) Since no stochastic automaton can generate the sequence P described in eqn. (14) precisely, p(z, k [ zo) can only be an approximation of P(z, k 1 zo). Nonetheless, the automaton with transition probability (34) has been used successfuUy used for solving different control problems like the stabilisation of unstable systems by means of qualitative controUers or the ohservation of unmeasurable qualitative states (cf. (Lunze 1995a) or (Lichtenberg and Lunze 1995), respectively). Theorem 1 shows that although the methods reported in these references rely only on an approximate evaluation of the qualitative state, they use the 'best' model in the sense that no stochastic automaton can represent the qualitative behaviour of the system (1) precisely.
7. CONCLUSIONS The paper presents a necessary and sufftcient condition under whcih the set of qualitative trajecto-
Franke, D. (1994): Sequelltie/le Systeme, Vieweg, Braunschweig. Grossman, R.L.; Nerode, A.; Ravn, A.P.; illschel, H. (Eds.) (1993) Hybrid Systems, Springer-Verlag, Berlin. Lichtenberg, G .; Lunze, J. (1995): Observation of qualitative states by means of a qUalitative model, Intern. Journal of Control (submitted for publication). Lunze, J. (1994): Qualitative modelling of linear systems with quantised state measurements, Automatica 33, 417- 432. Lunze, J. (1995a): 'Stabilisation of nonlinear systems by qualitative feedback controllers', Intern. J. Control 62, pp. 101-128. Lunze, J. (1 995b): Kunstliche InteUigenz fUr Ingenieure. Band 2: Technische Anwendungen, Oldenbourg Verlag, Munchen, Wien. Ramage, P.J. (1990): On the periodicity of symbolic observations of piecewise smooth discrete-time systems, IEEE Trans. Autom. Control, AC-35, 807-813. . Wunsch, G., Schreiber, H. (1989): Stochastische Systeme, Springer Verlag, Berlin.
3897
3d-04 I
THE EXISTENCE OF A DISCRETE-EVENT REPRESENTATION OF LINEAR CONTINUOUS-VARIABLE SYSTEMS WITH QUANTISED STATE MEASUREMENTS J. Lunze Technische Univer'siliit Hamburg-Harburg Arbeitsbereich Regelungstechnik Eissendorfer Strasse 40, D-21071 Hamburg email: Lunze @tu-harburg.d400.de
Abstract. This paper is concerned with the representation of stable linear discrete-time continuous- variable systems with quanti sed state measurements. It is assumed tha t the state space IRn is partitioned into disjoint regions Qx and that the number z of the region Qx(z) to which the state x belongs at time k is used as the qualitative value of x(k): [xJ = z. For a given z(O) [x(O)] the system can generate different qualitative trajectories [XJ [x(O)J, [x(I)J, [x(2)J ... The paper presents a necessary and sufficient condition under which the sequence [XJ is a Markov chain. This condition is rather res trictive and shows that there is, in general, no representation of the qualitative behaviour of linear systems by some stochastic automaton.
=
=
Keywords. Linear dynamical systems, discrete-event systems , representation , Markov models, stochastic automaton, qualitative analysis, quantized states . one discrete state [x( k)J to another discrete state r,,(k + 1)] while generating the qualitative state sequence
I. INTRODUCTION
Problem statement. The subject of this paper is a linear autonomous discrete-time system
x(k
+ 1) = Ax(k),
x(O)
= "'0
(1 )
whose state vector x E IRn can only be qualita. tively measured. The qualitative value [x(k)J E Z n is defined by a partitio n of th e stat o space IRn into disjoint sets Qx( z), (z E Z " )_ [xJ is th e 'number' z E zn of the set Q. to which ., belongs . From the point of view of an observer who only knows the qualitative values [x(k)] (k = 0, 1,2, ... ) the system (1) behaves as a discrete- evell! system with discrete state space Z " _ The sys tem jumps from
[X (xo)] = ([Z(O)J, [x(I)], [x(2)J, ... ). The paper concerns the question whether the relation between the qualitative initial state [x(O)J and the qualitative state sequence [X(xo)J can be re. presented by some automaton with state space zn. The fundamental difficulty of this modelling prDblem resul ts from the fact that the qualitative behaviour [X C"'o)J of the system (1) is, in general, not unique because instead of x(O) only the qualitative value [x(O)J is known (cf. Lemma 2). Consequently, no model can precisely predict the qualitative be. haviour [X(xo)] of the systelll (1)_ Therefore, x(O)
3892
is a.ssumed to be random ly distrib uted over the set Qx([:z:oJ) of all initial states that have the given qualita tive value [x(O)] and a model is to be set up that describ es the random sequen ce [X("o) ]. The central question dealt with in this paper is Is [X(xo) ] a Markov chain? If the answer is in th.e affirma tive, the random se· quence [X(xo) ] can be genera ted by some model
[x(k
+ l)J = f ( [", (k) ],v(k» , [",(O)J = [xoJ (2)
that relates [x(k + 1)] with the predecessor state [o:(k)J and some whi t e random liaise v(k). Hence,
which symbo lic observ ations are eventu ally periodic. Franke (1994) used the formal analog ies of discret e-time system s and determ inistic autom ata to transfe r results from linear control theory to the analysi s and design of discret e-even t system s. Fur· ther results on the discre! e-even t represe ntation of continu ous- variabl e .yst"m s can be found in the literature on qualitative reasoning, which is summarise d in Chapte r 6 of the book (Lunz. 1995b) . Notati ons. IR and 71. are T;he fields of real or integer numbers, respectively. Boldfa ce letters as ", z or A denote vectors or matric ~s , respectively, whereas italics such as Xl, Xi, z, denote scalars .
there exists a model wi t h the state space z n. In case of a negati ve answer [x(k + 1)] depend s on more than one predec essor state
[",(H I)J
= j([x( k )], ["'( k -
1)], [x(k - 2)], ... , v(k))
2. CONT INUOU S-VAR IABLE SYSTE MS WITH QUAN TISED STATE SPACE Consid er an a.symp toticall y stable discret e-time system (1), which genera tes a unique state traject ory
and no represe ntation with state space tl n exists.
Main result . The main result of the paper is gi· ven in Theore m 1 in Section 4, which describ es a necessa ry and sufficient conditi on on the system (I) togeth er with the quanti ser for the sequen ce [X] to be a Marko v chain. This conditi oll turns out to be rather restrictive . Consequently, there is, in general, no represe ntation (2) of the qualita tive behavi our of system (1). Relev ant literat ure. The problem dealt with in this paper occurs in the study of hybrid systems, which consist of continuous-variable and discret e-varia ble compo nents (cf. (Gross man et al. 1993» . Althou gh the theorie s of continu ous and discret e system s have been elabor ated almost independ ently, the re are some results concer ning the relatio ns betwee n the continu ous and the discret e represe ntation of the same system . Lunze (1994) propos ed to use" stocha s tic autom aton a.s approximate represe ntation of [X] . The following re· sults will show that such a relHese ntation of [X] can be precise only under rather reslrkt ive conditions. Ramad ge (J 990) derived conditi ons under
X = (", (0), x(l) . x(2), . . .).
(3)
It is a.ssumed that the sta.te x( k) cannot be measured precisely but is observ ed throug h a directionw ise unifor m quant iser whose output is the qualita tive value z = [x] of the state x where
[xi(k)] =
Z; E Z if
(Zi-~)qxi:5
xi(k) <
(Zi+~)qXi
(4) holds. This quantis ation yields a partiti on of the state space IRn into the dis joint sets
Qx(z)
= {x E IRn : [:cJ = z}
z E zn.
(5)
For any given initial state Xo eqns . (1) and (4) yield the sequen ce of qualita tive states
[X("'o)]
= ([x(O)], ["'( I)], [x(2)], .. .),
(6)
which is also called the qualitative trajectory. This paper concer ns the situati on in which only the qualita tive value [:ro] = Zo of the initial state is known. Tl, en the systel1l can genera te any trajectory of the set X(zo)
3893
= {X(xo ) : "'0 E Qx(zo) }.
(7)
The set
[X(%o)J
= ([xJ : X
E X(zon
(8)
results from a quanti sation of all trajectories belonging to X(zo). An import ant fact is that [X(zo)J is a set with more thar one element unless the system (1) satisfies the restric tive condition given in the following lemmas. Lemm a 1 (Lunze 1994) The system (1) togethe r with the quantiser (4) has for all qualitative initial states Zo E tl n n unique qualitative tmjeetory (6) if and only if for all z\ E tl n there exists some %2 E tl n such that
holds. Then the set [X(zo)J is a singleton for all 7l n •
%0 E
Defini tion 1 If eqn. (9) holds for some pair (ZI,Z2 ) then the pair (ZI,%2) is called deterministic. The pair ( %j, %2) is called nondet ermini stic if {A'" : x E Qx(z, )} n Qx( Z2 ) '" 0 (10)
is valid but eqn. (9) violated. The question under what conditions such nond etermin istic pairs exist is answered by the following lemma . Lemm a 2 (Lunze 1994). Consider the asymptotically stable system (1) and define
A = diag
q;/ A diag qxi'
3. THE MODELLING PROB LEM The modeUing aim is to find a genera tor that determines for any given Zo = ["'oJ the qualita tive trajectory [X("'o)J of the system (1), (4). As Lemm a 2 states, such a model does not exist unless the condition (12 ) is satisfied. Th erefore , the model should predict the 'average qualita tive behavi our' as will be made more precise now. If the initial state "'0 ib assume d to be a random variable (ur, more pr"cisely, a random vector) , which is uniformly distrib uted over the set Qx(zo) , then [",(k )J,(k = 0,1, ... ) "re random variables and [X("'o)J is a random sequence. For any % E 7l"
P(z , k I zo} '" Prob([" ,(k )J = % if ["'oJ = zo) (13) describes the probab ility that the system (1), (4) assumes th.e qualita tive state z at time k if its initial state has the qualita tive value %0. For fixed k , P( z, k I %0) is the conditi onal probab ility distribution descTibing which qualitative states % E Z" the system can assume at time k. The aim is to find a model that genera tes for any given qualita tive initial state %0 E 7l" the sequen ce of conditional probab ility distrib utions P=(P (% , llz(O) ), P{%,2 Iz(0)), ... ).
In thefoUowing consid eration s, %(0), z(l), z(k) etc. denote random varia.bles d'~scribjng the qualitative state [",(O )J, [",(1)]' [x(k)J etc_ whereas %1 , %2 , %k describe speci fic elements of 7l" . In order to describe the modelling proble m in more detail, the following result, on random sequences are recalle cl. In general , the random sequence [X("'o)J is described by the sequence of probability distrib utions
(11)
Po(z(lI)) PI(%( l) , z(O))
The system (1) together with the quantiser (.4) has for all qualitative initial states %0 E tl n a unique qualitative trajecto ry (6) ;f and only if A = diag
1
2ni
+I
P
(14)
Pk(%(k ), %(k - 1), ... , %(0))
(15)
(12)
holds where P is a permutation matrix and n; E 7l , ni > 0 hold for i '" 1,2, ... ,n.
because
P(% , k 1%0)
L
Z(k- l)eZ n
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'"
L: z(1 )eZ n
Pk(%, z(k - I) , ... , zo)
holds. In genera l, the whole probab ility distrib ution Pk : 7l. n X 7l." X ... X 7l. n ..... [0, I] has to be known. Only if [X(xa) ] is a Markov chain, the
model to be set up has a concise representatjon.
describes the probab ility that on these traject ories the system assumes at time T the qualita tive state ZT e zn if it has assume d at time T - I the qualitative state Z p E 7l. n .
The Marko v prope rty says t1,at
P(Zk I Zk-\, Zk_2, ... , za)
= P(Zk I zk-ll.
(16)
holds. Then, the sequence (14) can be obtaine d according to the Chapm an- Kolmogorov equatio n
P(z,k lz a) =
L
P,(zlz )P(z , k-lI zal
( 17) for given P(z, I za) = Pa(zo) . Eqn. (17) is a con· eise represe ntation of the sequence [X(",o)]. It can be interpr eted as a stocha stic autom aton s(zn, Pt) with state space 7l." and transit ion probab ility dis· tributi on Pt : zn X 7l.n ..... [0, I].
denotes the set of qualita tive states from which there exists a traject ory that after k time steps re· aches the qualita tive state ZT. Obviously, min . P(ZT, T I zp , zo) :,; P(ZT' T I Zp, %0) ZOE!!(ZT,7 )
°
The main proble m to be dealt with in the fol· lowing section s is the question whether the ran· dom sequence IX(",o) ] possesses the Markov pro· perty (16).
4. PROPE RTIES OF THE QUAL ITATI VE
hold.
Lemm a 4 Consider the system (1) together with the quantiser (4) and assume that the initial state :eo is uniformly distribu ted oiler some gillen set Qx(zo). Consider po;'.' (ZP,ZT ) with ZT E 7l. n , %p e 8(ZT,I ). For sufficiently large T the follo wing relations hold:
(aJ For all pairs (ZP,ZT ):
TRAJE CTOR IES Thls section presen ts proper ties of the qualita tive traject ories that will be used in the derivation of the main results in Section 5.
max P(zy, Tlzp, zo) = 1 ZoEI3(ZT.T)
(h) For deterministic pairs (ZP,ZT ): min P(zT, Tlzp,z o) = ZoEI3(ZT,T) P(ZT , T I zp, Zo) = = Zo EI3( max P(Z·i" , T I Zp, Zo) = 1 z" ,T)
Lemm a 3 Consider stable linear time- invariant system s (I). For any given set X C IRn there exist some finite T E Z awl E zn such that
z
(18)
implies
:e(T) E X.
P(ZT, T I Zp, zo) = Prob([",(T)] = ZT if [",(T - I)l = zp , [:>:(O)J = zo)
(22)
(c) For nondeterministic pairs (zp, ZT): (23)
(19)
The proof has to be omitte d due to space limitat i· ons. Consid er now the set of qualita tive traject ories that the system (1 ) genera tes for uniformly distributed initial state "'a E QA za). The conditional prohab ility
(21)
The Proof is omitte d due
10
space limitat ions.
Rema rk 1 The results of Ramage (19g0) concerning the periodicity of tht qualitative tmjectories apply to the modelling problem considered here because the system (I) is assumed to be asymptotically stable. [n fact, the qualit"tive tmjectory becomes eventually periodic beenus< for aU "'0 E IRn there
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exists a sufficiently large time l' such that the relation P(o,k I zo) = 1 for all k ~ l' holds where 0 denotes the qualitative zero state. I'hat is, any qualitative tmjectory eventually becomes deterministic: [",(T)J
=0 ,
[.,(T
+ l)J = 0 ,
[x(T
and, thus,
P(z, I'-lli)
=
{
P(ZT, l' I i)
=
1
Theor em 1 Consider the system (1) together with the quantiser (4) and assume that the initial state Xo is uniform ly distributed over some set Qx( zo) . For all Zo E tl n the qualitative trajectory [X(xo)J defined by eqn. (6) is a Markov chain if and only if the condition (12) is satisfied. That is, the qualita tive traject ory possesses the Marko v proper ty (16) if and only if the system (1) has for all qualita tive initial states a unique qualitative traject ory.
0< P,(ZT, zp) < 1 0< P,(zjV ,zp) < 1,
P(zT, l'lz)
-f 0 n Q.(ZN ) -f 0
=1>
= zp [x(I')J = %,-
<
1
(30)
P,(ZT 1%) P(i,I' -11 z)
I zp)
P(zp, T
-11 z)
which comple tes the proof.
6. CONS EQUE NCES FOR MODE LLING THE QUAL ITATI VE BEHA VIOUR O}' SYSTE M (1) In this section several conseq uences for modell ing the qualita tive behavi our of the system (1) with quantis er (4) are given. Since %0 = [x01 is assume d to be known ,
= {l o
for Z else
=Zo
(31)
describes the 'initial state' of the m odel. Corol lary 1 If the conditi on (12) is satisfied n(
(24) (25)
I:
= P,(ZT
I )=
{1o
if [AxJ else
= ZI for 80me[.,] = %2
holds. Hence, eqns. (17) and (31) yields
For sufficiently large time l' there exists some i such that the relatio ns
[x(1' - I)J
(29)
ZEZ n
rt %1 %2
Qx(ZT )
(28)
which can be seen by conside ring the set X(zp) of trajectories startin g in zp. Hence,
P(z,O I z) Proof. The sufficiency of the conditi on (12) for the Markov proper ly follows directl y from eqn. (22) because eqn. (12) implies that there does not exist any nondet ermini stic pair (ZI, Z2). In order la prove the necessi ty assume tltat there exist nondet ermi· nistic pairs (ZP , ZT) and (ZP,Z N) satisfy ing
n
(26)
holds. Eqns. (26), (27) and (30) violate the eqn. (17) becaus e
The following theorem represe nt.s the main result of the paper.
{Ax ::z: E Q.(zp )} {Ax : x E Q. (zp)}
= Zp
(27)
P,(ZT I zp) 5. MARK OV PROP ERTY OF THE QUAL ITATI VE BEHA VIOUR
%
bold. On the other hand, eqns. (24) and (25) imply
+ 2)J = 0, ...
However, the madel to be set up has also to describe the possibly nondeterministic part ([.,(O)J, [.,(l)J, ... , [x(1' - 1)]) of these tmjectories.
1 for 0 else
z = [:r(k)J P( Z, k I Zo ) -_ {I0 for I e se
(32)
(33)
and, thus, d..cribes the unique qualitative trajectory (6).
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This coroUary foU ows from Theorem 1 and Part (b) of Lemma 4. The automaton s(zn, Pt) with Pt given by eqn. (32) is a deterministic automaton.
ries [X(zoJ can be represented by a stochastic automaton s(zn, Pt). Theorem 1 shows that such a representation is possible only under restrictive conditions.
Corollary 2 If the condition (12) is not satisfied there does not exist a stochastic automaton s(zn, Pt) that for any given initial state Zo = ["'0) describes the set [X(zo») of qualitative trajectories of the system (1),
In the light of this result the question occur why deterministic automata have often been successfully used for describing discrete-event systems. The answer lies obviously in the fact that in all these cases the system has not been considered with equidistant clock time, but the system under consideration itself determines the clock time of the next event or the system has specific properties comparable with that given by eqn. (12).
m.
It has been proposed by Lunze (1994) to use the stochastic automaton s(Zn, Pt) with
= Prob([A"'J = ZI
for", E QxCZ2)) (34) as qualitative model of the system (1) with quanP(z,Olz) is given as in tiser (4). If P(z,Olz) eqn. (31) this model generates the sequence
Pt(Zl [Z2)
REFERENCES
=
P(z, k 1 zo)
= L:
Pt(z 1z) p(z, k -
11 zo).
zeZ n
(35) Since no stochastic automaton can generate the sequence P described in eqn. (14) precisely, p(z, k [ zo) can only be an approximation of P(z, k 1 zo). Nonetheless, the automaton with transition probability (34) has been used successfuUy used for solving different control problems like the stabilisation of unstable systems by means of qualitative controUers or the ohservation of unmeasurable qualitative states (cf. (Lunze 1995a) or (Lichtenberg and Lunze 1995), respectively). Theorem 1 shows that although the methods reported in these references rely only on an approximate evaluation of the qualitative state, they use the 'best' model in the sense that no stochastic automaton can represent the qualitative behaviour of the system (1) precisely.
7. CONCLUSIONS The paper presents a necessary and sufftcient condition under whcih the set of qualitative trajecto-
Franke, D. (1994): Sequelltie/le Systeme, Vieweg, Braunschweig. Grossman, R.L.; Nerode, A.; Ravn, A.P.; illschel, H. (Eds.) (1993) Hybrid Systems, Springer-Verlag, Berlin. Lichtenberg, G .; Lunze, J. (1995): Observation of qualitative states by means of a qUalitative model, Intern. Journal of Control (submitted for publication). Lunze, J. (1994): Qualitative modelling of linear systems with quantised state measurements, Automatica 33, 417- 432. Lunze, J. (1995a): 'Stabilisation of nonlinear systems by qualitative feedback controllers', Intern. J. Control 62, pp. 101-128. Lunze, J. (1 995b): Kunstliche InteUigenz fUr Ingenieure. Band 2: Technische Anwendungen, Oldenbourg Verlag, Munchen, Wien. Ramage, P.J. (1990): On the periodicity of symbolic observations of piecewise smooth discrete-time systems, IEEE Trans. Autom. Control, AC-35, 807-813. . Wunsch, G., Schreiber, H. (1989): Stochastische Systeme, Springer Verlag, Berlin.
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