Control of a deterministic system es a fuzzy environment over infinite planning horizon

Fuzzy Sets North-Holland

and

Systems

10 (1983)

291-298

291

CONTROL OF A DETERMINISTIC ENVIRONMENT OVER INFINITE Janusz KACPRZYK’” Sys~tns

Resectrch

Received

Institute.

January

Revised

May

and Piotr STANIEWSKI P&s/l

Academy

of Scimces.

0 I-147

Warstrw.

Pohd

I979

control infinite

of a time-invariant finite-state planning horizon is considered.

fuzzy constraints imposed on controls assumed to he state-dependent. The constraints and politics relating For the problem.

and fuzzy

deterministic system The fuzzy environment

fuzzy goals to he attained. The decision is assumed to he the

in a fuzzy is meant as

fuzzy constraints are intersection of fuzzy

fuzzy goals. The problem is to find an optimal strategy. i.e. a sequence of the control to the state, maximizing the mcmhership function of fuzzy decision. a functional equation is formulated. An algorithm of policy iteration type is

existence of a time-invariant optimal A numerical example is shown.

Keywords: Policy

IN A FUZZY G HORIZON

1979

The multistage environment over

given. The arc proved.

SYSTEM PLANNIN

Control

(ci~cision-making)

strategy

in a fuzzy

and

the convergence

cnvironmsnt.

Infinite

of the algorithm

planning

horizon.

iteration.

1. Introduction Many control processes in the real world proceed in a fuzzy environment, which-due to Bellman & Zadeh Cl]- may be formalized as fuzzy constraints imposed on controls and fuzzy goals to be attained. The problem is to find an optimal decision, i.e. a sequence of controls maximizing the membership function of fuzzy decision. Following the above source paper, a number of contributions appeared. see e.g. [2-91, discussing various aspects. However. in all of them the termination time of the control process is finite, i.e. fixed and specified (e.g. [l, 2, 4, 7, 9]), implicitly given by entering a specified termination set of states (e.g. [l]) or fuzzy (e.g. [3, 6, 81). There arc, however, some long, low-varying control processes in which the goal is just to maintain some level of activity, to avoid decay, etc. and the planning horizon is very long. Possible examples may concern many social processes, agriculture in numerous regions and countries. industries producing commodities for which the demand is saturated, etc. In such cases the use of conventional techniques, i.e. to iterate over subsequent control stages, is evidently inefficient and a different approach should be employed. The purpose of this paper is to present such a different approach. The ” On Rochelle,

Icavc NY

0165~0114/83/$03.00

at

Machine

10801.

Intelligence

Institute.

Hagan

School

of

Business.

USA (Q 1983 Elsevier

Science

Publishers

B.V.

(North-Holland)

Iona

College,

New

292

J. Kacprzyk.

P. Staniewski

termination time is assumed to be infinite. The system under control is a time-invariant finite-state deterministic one. The set of controls is finite as well. An optimal time-invariant strategy is sought. Its existence is proved and a converging algorithm of policy iteration type for determining it is given. A numerical example and some concluding remarks are presented. What concerns the notation used, the fuzzy sets will be denoted by capital Latin letters, e.a. A E X and - assumed to be defined in finite spaces X = {s,, , s,,,) given as A = p+,(x,)/s,f. .+ pa(s,)/s,,,, where ~~ is the membership function and F~(. 1y) means pLA conditioned on y. Moreover, ar\b=min(a.b), /\~,a,=a,r\.~.r\a,. ~~~,~=a,r\a,r\~~~ and i/J.) = max, (.). 2. Decision-making

in a fuzzy environment

over finite planning

Let us assume that the system under control deterministic one given by a state equation X,+I=f(X,,U,),

t=0,

1..

horizon

is a time-invariant

finite-state

.,

(1)

where: x,,,+, is the state (output) at the control stage r/t+ 1, +1+1 E X= {s,, . , s,,,}, u, is the control (input) at t, U,E U={c,, . . . ,ck} and f:Xx U-*X. The initial state is x,). At each control stage t the control is subjected to a state-dependent fuzzy constraint C(x,,)c U characterized by pc(u, 1x,). The rationale for the statedependency of fuzzy constraints is that, first, it exists in most real cases (e.g. in economic systems what we can invest depends usually upon what we have produced) and, second, the termination time is infinite, hence the timedependency is extremely inconvenient. The fuzzy goal G cX, characterized by am, is assumed to be the same at each control stage, which evidently stems from the very nature of the problem. The fuzzy decision D c U x U x . . . , characterized by kr,( uo, ui, . . . , x,,), is given by the intersection of fuzzy constraints and fuzzy goal at subsequent control stages with a discount factor b > 1, which reflects an obvious fact that what happens at earlier control stages should have more influence on the value of in than that what happens at later stages, i.e. D(x,,)

= C(x,,) n G n b(C(x,)

n G) n b’(C(x,)

n G) n . .

= 6 b’ ‘(C(q-,)T\G), ,=I or in terms of membership

Hence, the problem IA:, UT,. . . given by

(2)

functions

is to find an optimal

ELD(U$, UT>. . 1XO)=

decision, i.e. a sequence of controls

V (i bi-‘(/+(Ui-l U,).U I.... ,=I

1x,-,)Apa(f(xi-l,

.I)),). (4)

Control in a fuzzy

Evidently,

it is more convenient

a, :X+

environment

293

to express the solution by a policy function

U,

(5)

a, E A, relating at each control stage t the control u, E U to the state x, E X. The sequence of policy functions will be termed the strategy and denoted by S = policy function and strategy are denoted by a and (a,,, a,, . .). The time-invariant a Q= (a, a, . .), respectively. Thus, the problem (4) -expressed in terms of strategy S - is to find an optima1 strategy S* (time-invariant, if possible), such that cLD(s* I x,,) = v ILLJ(S 1x,J s with the following

preordering

S, 2 S2 e

(6)

defined in the space of strategies:

I+,(S, 1xc,)3 pD(S2 ( x0) for each X,,E X,

(7)

and, evidently, S*s Si for each Si. Let us now proceed to the determination of functional equation. Due to the time-invariance of the system under control and the time-independence of fuzzy constraints and fuzzy goal, (2) may be written as D(x,,) = Ctx,,) n G C-IbD(x,)

(8)

where D(x,)=(C(x,)nG)nb(C(x,)nG)n..

”

(9)

Thus, in general

D(x,) = c(x,) n G n bDb,+,). This is the desired functional used in the following form:

(10)

equation

for our problem.

fiD(S ) x,,) = pc(a,,(xo) 1x0) A k(f(xc,,

In the sequel it will be

a,,(x,,))) A bh(TS

1ftx,,, a,,(x,))) (11)

where T is a time shift operator, i.e. TS = (a ,, a2, .I. Evidently, (11) should be meant as the following set of simultaneous p..,(S) s,)= Fc(a,,(s,)) pD(S 1s2) = pc(a,,(sd

s,)~k(f(s,, I s2)~pG(f(sZ,

a,,(s,)))AbpD(TS

equations

jf(s,, ads,))),

a,(s2)))r\ bpD(TS ( f(s,, a,,(%))).

c12j

PD(S1s,)= f-da&,,) I s,,)AcLGV(S,,,, a,ds,,,)))A b(TS lfts,,, a,,(s,,,))). Let us now remark that since X and U are finite, then CL~(TS 1f(x,,, a,(x,))) = H(a,,)b~D(S

1x,,)

where H(a,,) is an m x m transition matrix and the matrix multiplication usual sense. Let us now introduce an operator L : [0, 11”’ -+ [0, l]“‘, such that L(a)w

= &da

1x~AI-%(~(xo,

a))r\H(a)bw.

(13) is in the

(14)

294

.I. Kacprzyk,

Thus, denoting

(a, S) = (a, a,,, aI, . _ .) we obtain

/JtJ((LI, S) I x,,) = L(a)kl(S We have now an important Theorem Proof.

P. Sraniewski

(15)

Ix.,,).

theorem:

1. The operator L(a) is monotone, i.e. if w, 3 w2, then L(aJw,

2 L(a)wz.

Evident.

As the consequences of Theorem Theorem

1 we obtain the following

three theorems.

2. If (16)

cLD(SX ( x,,) ?=/L”((U, S’“) 1x,,) v E A, tlren the strntegy

for each

Proof.

S” is optimal.

From the relation (15) and the assumptions L(U)~~(S”

/ X,,)~/L”W

for each a E A. Taking we obtain L(a,)/.b(S*

we have

/ x0).

an arbitrary

strategy

s = (a,,, a,, . . . ) and assuming u = a,,

1x,Jc pD(s* I x,,).

Using the superposition operator, we find L(a,,)L(a,) Applying

of the theorem

L(a,,)L(a,)

. . . t(a,, ~, ), which is obviously a monotone

. L(n,,)/LL,(S”

subsequently

1x,,)~L(n,,)L(a,).

the above relation,

!.$cs* 1x,,)~L(aJL(a,)

. L(a,,-,)CLD(S*

1x,,).

we finally get

. . U4,)p,W

1XCJ

= pD((a,,, a,, . . , a,,, S”) 1x,,) for each n. For n + = we obtain pIJs*

1x,,)> CLJJS1x,,)

for each S. Hence S* is optimal. Thorem

3. If pD((a, /&(a”

Proof.

q

S) 1 x,,) > ~~(s 1 x,,). then

I x,,) > /-b(S

(17)

1 x,,).

From (15) and the assumption

of the theorem we have

Uu)/-b(S I Xl,)>/-b(S Ix,,). Applying

a monotone

L”(a)pD(S

operator

L”-‘(a)

(X’J~L”-‘(a)CLD(S

we obtain 1x,,)

Conrrol

for each II >O. Applying following inequality:

in a

the principle

fuzzy

295

environtnenl

of mathematical

induction

we arrive at the

L”(a)kl(S 1Xo)=/.Lo((a,S) (X,,)>k(S I x0) for each

II

> 0. For II + 7, we find

p.,(tr” 1x,,)> /-LD(S ( x,,).

cl

Thus, if we only can improve our strategy, then a better time-invariant strategy may be found. Let us remark here that for a time-invariant strategy TS = S. Now WC formulate the following main theorem, which will form a basis of a procedure for the determination of an optimal strategy. Theorem

4. Let a E A arld for each i E { 1. rn} let us denote

u(s, 1E A. such

by B( i, a) the set of all

that

(18)

CLD(~~“Isi) pD(a”

(19)

( x,,).

Proof. The ith element of the vector pD((z, a’) ( x,,) is equal to j.h((z.

a”)

I s,) =

cLtax!c

control Decision

with control

fuzzy

of a nonfuxzy

termination control

termination

time,

of 21 fuzzy

time.

system Fuzzy

system

in 2,

Sets and in a fury