On the stability of qualitative models

Systems & Control Letters 21 (1993) 137-142 North-Holland

137

On the stability of qualitative models Jan Lunze

L

Technische Universitdt Hamburg-Harburg, Arbeitsbereich Regelungstechnik, Eiflendorfer Str. 40, D-W2100 Hamburg 90, Germany

continuous-variable d y n a ~

1

Received 30 December 1991 Revised 14 April 1992 Process control equipment

Abstract: This paper considers a discrete-time continuous-variable autonomous system for which only a quantised state measurement Ix(k)] is available. The qualitative model of this system is a nondeterministic automaton that describes, for a given quantised initial state [x(0)], the sequence of quantised states Ix(k)]. The paper deals with the question for which stable linear systems is the qualitative model also stable. A necessary and sufficient condition on the qualitative model is proved under which the continuous-variable system is stable. This condition refers to the eventual boundedness of th" nondeterministic automaton. An example demonstrates tnat this stability criterion can be used only if a qualitative model of the system is available.

I

[u(k)]l

I [y(k)]

Fig. 1. The human operator observes and controls the system through process control equipment.

~

Keywords: Dynamical systems; linear systems; qualitative model; stability; nondeterministic automata.

x(O)

I

System(1)

~~il~ I

I [x(k)]:;>

Quantiser |.

1. Introduction Qualitative modelling concerns the situation depicted in Fig. 1. A continuous-variable system with input u and output y is observed through process control equipment. The operator does not know the quantitatively precise value y(k) of the output but merely the qualitative value [y(k)], and he does not have access to the quantitative value u(k) of the input but only to the qualitative value [u(k)]. In general, [y(k)] and [u(k)] are related to y(k) or u(k) by a nonlinear mapping. The qualitative model describes the relation between [u] and Iy]. Qualitative modelling has been dealt with in the fields of control theory and artificial intelligence. A comparison of the main lines of research in both fields has been given, for example, in [31. The Correspondence to: Technische Universit~R Hamburg-Harburg, Arbeitsbereich Regelungstechnik, EiBendorfer Str. 40, D-W2100 Hamburg 90, Germany

(a)

[ [x(O)l I

Qualitativemodel

~)l

(b) Fig. 2. The autonomous system under consideration. (a) The system observed through a quantiser. (b) The qualitative model.

following investigations concern the specific situation in which the qualitative values of the system variables are received by a directionwise uniform quantiser that can be defined by (4)-(6) in Section 2 (Fig. 2a). This situation has been the subject of the

0167-6911/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

J. Lunze / On the stability of qualitative models

138

recent papers [1, 2, 4, 5]. In [1] it is investigated how the state x( k) of the system can be reconstructed from the sequence [y(k)] (k = 1, 2 , . . . ) of qualitative measurements. In i2] and 1.5] it is described how the system can be controlled by a feedback that has access to 1.x(k)] rather than x(k). The paper [4] shows that for the autonomous system x ( k + 1)= Ax(k),

x(0)= Xo,

(1)

the relation between the qualitative states Ix(k)] and [ x ( k + 1)] is nondeterministic (Fig. 2b). Hence, nondeterministic automata N are reasonable forms of qualitative models. The relation between system (1) with quantiser and the qualitative model N that has been proved in [4] is summariscd in Lemma I. This paper concerns the question of how to investigate the stability of system (1) if only a qualitative model N is available. The main result is given in Theorem 1, which says that all trajectories of the model n reach a bounded equilibrium set if and only if system (1) is stable. As a basis for these results, the system with quartriser is introduced in Section 2, and relevant results of 1.4] are surveyed in Section 3. Whereas the stability of system (1) can be investigated by well-known methods, as reviewed in Section 4, no widely accepted definition of boundedness of nondeterministic automata N is available. Therefore, this property has to be defined in Section 5 before the main result can be proved in Section 6. An example in Section 7 demonstrates the applicability of the results.

of the state x(k). Since the state variables x~ are quantised independently from each other with resolution qxi, Ix#)]

=

(5)

holds where zi(k) satisfies the relation (6)

(zi(k) - ½) qxi <- xi(k) < (zi(k) + ½)qxi.

Z denotes the set of integers, Ix(k)] is called the qualitative state and Z" the qualitative state space. Obviously, the introduction of qualitative values Ix] imposes a partition of the quantitative state space _R" R" -- ~

Q~(z),

Qx(zl) n Qx(z2) = 0

for zt # z2,

• 6Z n

where ~(_Z) is given by = =

Ix] = z} =

(7) i=l

for (zl • •. z,)' = zz_Z". The vector e~ is defined as ex~ --(O, . . . , O, qx,, O, . . . , O)',

where q,,~is the ith elermnt. 3~e qualitative trajectory of system (l) is given by [X(x(0))] = ([x(0)], Ix(l)], Ix(2)],... ).

(8)

3. Qualitative models of system (1) 2. Continuous-variable systems with quantised state measurement The continuous-variable system (1) has, for any given initial state x(0), a unique state trajectory X(x(O)) = (x(O),x(1),x(2), . . . ),

(2)

with x(k) = Akx(O)

(k = 0, 1, 2 , . . . ).

(3)

R" is called the quantitative state space. In the following, it is assumed that the state

= {X(xo): xo

Qx(z(O))}.

Hence, for any given qualitative initial state z(O), the qualitative model has to describe the set IX1 of qualitative trajectories:

x(k) = (xl (k), x 2 ( k ) , . . . , xn(k))'

cannot be measured quantitatively but is observed through a directionwise uniform quantiser that yields the qualitative value 1.x(k)] = ([xt(k)], [x2(k)] , . . . , [ x , ( k ) ] ) '

Qualitative models have to describe the qualitative trajectory [X(x(O))] that system (1), (4)-(6) generates for a given qualitative initial state z(O) = [x(O)l. Since only the qualitative description z(O) of the initial state Xo is available, Xo is only known to lie within the set Qx(z(O)). Consequently, the system trajectory cannot be determined unambiguously, but is known to belong to the set

(4)

IX(z(0))] = {IX]" X~£(z(0))}.

(9)

It has been proved in 1"4] that the set 1.~] is, in general, not a singleton but includes more than one qualitative trajectory. As a consequence, a

139

J. Lunze / On the stability of qualitative models

nondeterministic automaton N(_Z", H, Zo) is used as a qualitative model of system (1). The transition relation H: _Z"

--~

x2

2 z" ~x 1

describes for every given state z e_Z" a set H(z) of states that the automaton can assume at the next time instant. For every given initial state Zo the automaton yields the sets of states

/ Ox(,

Qx(Zl)

M (0) = {Zo},

,~z,v]

M=(k) = {zeH(~):~.eM=(k- 1)} (k = 1 , 2 , . . . ) and, hence, the set of trajectories Z(zo) = {(z(O),z(1),z(2),. . . ): z(k)eH(z(k - 1))}.

(10) The question of how to determine the transition relation H for given system (1),(4)-(6) has been answered in [4], as summarised in the following lemma. Lemma 3.1 (Lunze [41). Consider the sets of qualitative trajectories [X(zo)] and Z(zo) of system (1), (4)-(6) or the nondeterministic automaton N(_Z", H, Zo), respectively (cf. (9) or (10)). The relation

__ 2,(Zo)

(11)

holds for every oiven qualitative state zo~Z_" if and only if H satisfies the relation H(z)D_H*(z)={[x]:x~Do(z)}

for all z~Z_", (12)

where Do(z) is defined by Do(z) = {x = Ag: gcQx(z)}.

(13)

That is, if the transition relation H is chosen to satisfy (12) (cf. Fig. 3), the automaton N yields all qualitative trajectories of system (1), (4)-(6). If (12) holds with the equality sign, the corresponding automaton and the sets M:(k) and 7~(Zo) are marked by an asterisk: N*(Z_",H*,zo), M*(k), Z*(zo). The model N* yields the smallest set Z,*(Zo) of trajectories and is the "best" of all qualitative models for which (11) holds. Therefore, further investigations will deal with N*.

Fig. 3. Interpretation of the transition relation H. The state z at time k has four successors: z(k + 1) ~ {zl, z,, z3, z4}.

the stability analyses of the qualitative model in the sections to follow. In order to simplify the investigations, the transformation = (diag q~l)x

(14)

is introduced. Obviously, system (1) is asymptotically stable if and only if the system ~(k + 1)= A'~(k),

~(0) = (diag q~l)Xo,

(15)

with ,4 = (diag q~ ~) A(diag qx~), is asymptotically stable. It is well known that system (15) is asymptotically stable if and only if there is some symmetric positive-definite matrix Q such that the equation ,4'PA- P = - Q

(16)

has a unique symmetric positive-definite solution P (cf. [5 pp. 120-121]). With the vector norm II~ llp = x / ~ P ~ = x / ~ ) ,

(17)

every trajectory of system (15) or, equivalently, system (1), (14) satisfies the relation II~(k + l)llp < II~(k)llP for x # 0.

4. Stability of system (1)

The vector norm (17) induces the matrix norm

This section briefly reviews those well-known resuits on the stability of system (1) that are used in

IIAIIp- max [12~[le. '~ p= 1

(18)

(19)

J. Lunze / On the stability of qualitative models

140

Since (18) is satisfied for arbitrary ~(k) with ~(k + 1) from (! 5), the inequality (20)

II 2 lip < 1

Theorem 6.1. The qualitative model N*(Z_", H*, Zo) is

holds if system (15) is asymptotically stable.

eventually bounded if and only if system (1) is asymptotically stable.

5. Stability definition for the qualitative model The stability analysis of nondeterministic automata has not received much attention. For a discussion of this problem, the reader is referred to [7]. In the following definition, the notion of boundedness, as defined below, is used for characterising the kind of stability of the automaton which is necessary for proving the stability of system (1). N o t e that for the state space _Z" of the automaton N, a norm !1' II can be introduced. Equation (12) implies that the automaton N is live, that is, H(z) ~ 0 holds for all zeZ". A state zeeZ" is called the equilibrium state of N(_Z", H, Zo)if {ze } = H(ze) holds. Nondeterministic automata have, in general, no such single equilibrium state but merely one or more equilibrium sets Z~ in which all trajectories Z(z(O))=(z(O), z(l) . . . . )eZ(z(0)) remain for a sufficiently large time k. A set Z~ c _Z" is called the equilibrium set of N~Z_",H, Zo) if for all trajectories the relation z([)eZ, for some ume k implies the relation

z(k)eZ~

model N* since, due to Lemma 3.1, no qualitative model N can eventually be bounded or asymptotically stable unless N* has these properties.

for all k >/~.

H*(z) = { [~]" ~e/~o(z)},

(23)

with

(24) and (~x(z)-- {~

--

z + I" I -- (I,,..., I,)',-½ <_ l,< ½}. (25)

First, it will be shown that max Ilzlle>_mimplies :e Mz(k)

max zeMz(k

Ilzlle< max Ilzlle

+ I)

ze Mz(k)

for some m > 0. Equations (23)-(25) yield

(22)

The equilibrium set is called bounded if there is a norm II'll defined in Z" such that IIz II < m < oo holds for all z e Z,.

Definition 5.1. The automaton N(Z_", H, zo) is called eventually bounded if it has a bounded equilibrium set Z~ and if for all initial states z(0) = zo there exists some k-such that (22) holds for all trajectories (z(O), z ( l ) , z ( 2 ) . . . .

Proof. If system (1) is not stable, IIx(k)II exceeds all bounds and so do the trajectories of the automaton, because (11) holds. Therefore, it remains to be shown that the qualitative models N* of all stable systems are eventually bounded. The vector norm I1" II~, defined in (17) is now used for R" and Z_". The following relations follow from (7) and (12)-(14):

)eZ(Zo).

IIz II~ -

max

II ~ IIP = II z IIP -

e H* (z)

max

II [,4

"" IlzllP-

II1"A(z +/)]lie

max -½ < 1~<½

> IIZHe-Ill/i" z]lle>__ min

(llzll~ -

max

111"2/]lie

IIEh zllla) - m

ilzllp~ m

6. Relation between the asymptotic stability of system (I) and the boundedness of the qualitative model N* In this section the relation between the stability of system (1) and the existence and boundedness of an equilibrium set of the qualitative model N* is investigated and the main result of this paper proved. Attention is restricted to the qualitative

~] lie

.ie~xlz)

> min ( l l x l l p - Ilia X'IlIp)-x~a_ n

][XlIp > m

> m

min (llxlIP - IIU/T x]llp) ilxl!e_> 1

= m(l -

>0,

II/iflp) - .a

J. Lunze / On the stability of qualitative models

with r~ =

141

Therefore, the relations max

-½
H(z) = {entier (0.5z), entier (0.5z) + 1} for z # O, (29)

II I-2 1-1lip > 0

where entier(z) denotes the greatest integer that does not exceed z, and

if m satisfies the relation n~

m >

.

(26)

1 -11,4lie

Note that according to (20) the right-hand side of (26) is positive. Hence, for every initial state Zo, the automaton N* has a set of trajectories Z(zo) for which (22) holds with Z, = {z: IIzlle
H, Zo) of system (1) which satisfies (12). I f N is eventually bounded then system (1) is asymptotically stable. If N is known to be the best model N* then the eventual boundedness of N is also necessary for the stability of system (I).

7. Example

The following example demonstrates how the stability of system (1) can be proved only if a qualitative model N of this system is available. This can be done only if a one-dimensional system

x(k + 1) = 0.Sx(k), x(0) = Xo

H(0)-

{0}

(3O)

satisfy (12) and, thus, represent the transition relation of a qualitative model N of system (27). Note that the "best model" N* has the transition relation H* given in (28). Now it is assumed that only the qualitative model N is available for the stability analysis. First, it can be shown that Z, = {0, 1, 2} is an equilibrium set of the model N because H(z) c_ Ze holds for all zeZe. This set is obviously bounded. Second, it can be shown that for all initial states Zo there exists some /? such that (22) holds for all trajectories in the set Z(zo), because max I:~l = max (lentier(0.5z)l, lentier(0.5z) + l I) ~¢H(z)

2 . Hence, Iz(k+ l)l 2 holds. All trajectories reach and remain within the set Z~. Owing to Theorem 6.1, system (1) is asymptotically stable. Note that the stability analysis exploits merely information that is provided by the qualitative model N. The model N has not the property that all trajectories reach and remain in the origin z=0.

(27) 8. Conclusions

is considered. For q,, = 1 a qualitative model can be derived according to (12) and (13). Equations (7) and (13) yield

Do(Z) =

{o.5~: ~ e Q x ( z ) }

= {o.stz -

½) _< x

< o.5(z +

½)}.

The paper shows that the stability of system (1) can be checked only if a qualitative model N is available. Theorem 6.1 says that the boundedness of the equilibrium set of the qualitative model N* is a necessary and sufficient indicator of the stability of system (1).

Hence, (12) reads as

H(z) ~_ H*(z) = {[x]: 0.5(z - ½) .<_x < 0.5(z + ½)}

{0.Sz} =

{0.Sz - ½, 0.Sz +

number, for z = odd number. (28) for z = e v e n

½}

References [1] D.F. Delchamps, Extracting state information from quan.. tized output records, Systems Control Lett. 13 (!989) 365-372.

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2. Lunze / On the stability of qualitative models

[2] D.F. Delchamps, Stabilizing a linear system with quantized state feedback, IEEE Trans. Automat. Control 35 (1990) 916-924. [3] J. Lunze, Qualitative analysis of dynamical systems, Prec. Workshop on Decision Support Systems and Qualitative Reasoning, Toulouse (1991). [4] J. Lunze, Qualitative modelling of linear dynamical systems with quantised outputs, Automatica (submitted).

[5] J. Lunze, Qualitative modelling of continuous-variable systems by means of non-deterministic automata, Intelligent Systems Engng. 1 (1992) 22-30. [6] P.C. M(iller, Stabilitdt und Matrizen (Springer, Berlin, 1977). [7] C.M. Ozveren and A.S. Willsky, Stability and stabilizability of discrete event dynamic systems, J. ACM 38 (1991) 720-752.