Identification of Nonlinear Objects of Management on the Basis of Critical Phenomena Theory

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Control St ra tegies. Tbilisi . USSR. 1989

IDENTIFICATION OF NONLINEAR OBJECTS OF MANAGEMENT ON THE BASIS OF CRITICAL PHENOMENA THEORY A. Sh. Gugushvili D fpartllll'llt

of

A utolll atic Control, G fo lg ian P olytechnic Institute. 38 007 5. Tbilisi. Gro'g ia , L'SSR

Abstract. The variety and heterogeneity of the specific features of ~ndustr~al objects needs the solution of some actual ~roblems of identification: q~alitative . and quant~tative systematizat~on of its elements, format~on of opt~mum adapt~ve systems of mana~ement w~th an ident~fic ator, elaboration of methods of identificat~on for t h e conditions when measurement of variables and data excerption and a priori information on signals, hindrances and parameters are limited. The study gives the theoretical basis and practical solution of the. p roblem of identification in nonlinear objects of management accord~ng to the three main ~rinciples of physics: the princip les of symmet ry, the variation princ~ple and laws of conservation. The research of identification is based on the critical phenomena (catastrophe theory, Hopf's bifurcation, renormalization groups theory). Ke~words: critic~l p~enomena

ca

~on.

renormal~zat~on

theory, catastrophe theory, Hopf's bifurgroups theory.

While identifying nonlinear objects two main problems are solved: 1.Determination of structure of nonlinear objects of manag ement. 2.Determination of parameters of nonlinear objects for a given structure.

functions c an be got from their linearization form at deg ener at ed critical point. When this has b e en don e we c an consider that behaviour of this system is determined pretty well i n n ei ghbours of such points.

There is vast literature concerninb the second problem but we have only few works dedic ated to the first one.And the latter needs serious elaboration for it is very i mportant (Tsipkin,1 968 ; Gugushvili, 1979, 1982; Kortmann, 1988 ; Mehr a , 1977).

Let us consider us e of t he method based on critical phenomena theory in particular catastroph e t heor y for so me ex amples.

We can solve the bo t h problems fro m one point of view on the basis of critical phenomena theory (catastrophe theory, Hopf's bifurcation and renormalization groups theory ) The aspects of the use of catastrophe theory we can form as following ( Gilmore, 1931 ). 1. System describing equations are unknown.If space variation or evolution in time domain is slow, existance of multiple locally stable modes can signify that behaviour of the system can be described qualitably or even semiqualitably by means of one of the func t ~ons of catastrophe theory • . This considerably widens chances of an ~nvest~ bator up to gett~ng sUbstantiation for the guess of system describing equation. 2. Relative im~ortance of variables are unknown. In th~s case we also do not know the system describin~ equations. If there is an existance of l~ited model only a few (one, two or three) system describing functions of condition variables can change condition of the system. These

Recircul ating ob je cts. Th es e ar e the obJ ects wh en t e 'p art of t h e initi al materi a l not h avin6 b e en worked up completely during t h eir ao ino throuGh th e workinb part of the ob ject return t o the entrance of the latter making the so-called recirculating current. The obj ects with recircul atill6 process es are met in biolo5Y , in chemical, mine-metallurgi c al, wo od-pulp and pap er industry, etc. Mathematical models of single-stage recircul ating nonlinear obj ects can be of four kinds ( Gugushvili, 1973 )

c.~=X~+C!X+C2. dt

T

1

s!.!-+I-Cr2:X4 f dt

£/:0

(~+ cp2')

,-----"

r:Jl+Tz TT

1 2

4+(Tl+T2)d~ +(~-2C!X4)Zdt d"C

_CZ 2

(1)

=0- -€)X 4 + CX~

(2 )

A. Sh. Gugushvili 2

d V ( ) dV T1T2dt.2 + Ti+ T2 ~ + V - CV

2

= (4)

=T2 dd tx

+X + (

4

4

(!.,

J~

+ C13 2)

Z is recYcle, '1. 4 is the initial signal, T1 ,T2 ,-e, C are coefficients. The problem is which equation describes the recirculatin~ object really. The equation (1) after some transformation ~s

(5)

The potential function is / Ci C X2+3'C

ci

v=e'

2

3 3 )(

+C4 )(

The condition for stable or

t

)

po~se ~s

V:() (10)

where

V(X) is Tom's potential function

Vee;, c2 )(X)

= x\

2

C/ + C2 X.

(6)

The critical points of the function are met where \%(C ,C , (x)=Q Vcc1, C)'2 (X) ) 1 or (7 ) 3 + 2C.1 X + C2 0 . The critical points coincide when

=

x'2

or

6 X + 2 C.1 :: 0

from equations (6) and (7) Elccluding we have .1..C 2 _£..(2 +C =0 3 1 3 1 2 or

I

(8)

2 Cl

C2 ="3 Catastrophes of multitudes K for a t5~ven function VCc c )(X) occur where

t' C 2

C2 (T

where

3 Cl +

-C C2 =r! X4 (- ~ + 3CY. ~ ) + X~ +

From equation (3) we have

'" 3

+ 2+

ct=~) C3

-jf

[,2

Z

:-

0

C;:-

Cr

(11)

C3

Equation (11 ) describes peak-type catastrophe. In case of the equation (4) we have peaktype equation X'+ Cl X + C2 =0 • Inserting X LA + l5 and transforming the p eak we will get

=

U3+lJ 3+(UfU)(3UIJ+C.1)+C2 =0

(12 )

Now we have two variables U and U instead of one X and t hat is why one of them is arbitrary or we can find out arbitrary dependence between the two. Let us assume 3 lA l5 + Cl = C3

(9 )

~

Ci ='-1X..,

~s

Lf Cl

1.

2t (22 =0 2

K

2

There ~s one max~mum of V((' C ) (X) • There . . . . . ',?c2» '3I C12 Mule ~s no cr~t~cal po~nt when (c;) C2 ) movine; on a plane, intersects K in such a way, that C2 diminishes and the minimum has influence with the maximum and disappears. The system beinb described by the function V(C1 ,c,/X) and being in poise corresponding the minimum of VC( C,) ()() will. make. the catastrophical jump(to V=-O::> d~rect~on or X:=- ), as soon as the pointed out situation is realized. The catastrophe of such kind is called a pleat. In equation (2) after some simple transformations we have peak-type equations Lf

We'll have jumps in the system if the maximum and the minimum of the system are the same v=v:O Equa~ion for the catastroph es of multitude

-f- (/- 8) X

4

from (12) we have U\ /}3-+ C3U -+ C3 (f + C2 =0 which corresponds to hyperbolycal ombilic point, the equation of which is

f(X:)Ij)u,I5,W)=-)(3+~3.. 2>,1j +~ Id-f

C3 '::1

~

Cj =0

when

The derivative of (13) 2

~s

2

3u +C3 = 31J .. C3 =0

The second derivative

(15 ) ~s

equal to zero.

F4uation (14) represents the reflection

{~{ (X,'::I)-- (C) ,C) = pi . . 3!/) and intersection of ~f multitude with the plane {w=o} . The reflection resembles a complex handkerchief (Fi 6 .1). The graph is given for W:O and U=U=C3 either for positive or negative values. We see according to the graph that there is at least one finite local regime which is on the other side of the plane (Fig.2) which contains quarter W=O) C'3 70

Identification of Nonlinear Objects of Management

Insertin6

Cl:::;

lP=

~n

(1) and (2)

, C ::: z-Jr-""i-+-W2-Z"':"T-Z:-S-L-n-(l..)-t-+-r.p-"""1) 2

Cl'lctj WT

(16 ) As we see from Fig.3, catastrophes will take place when the following conditions are realized -Co()
q

(17)

is the amplitude of variation of

(2

If condition (17) is not realized there will be no catastrophe or only one will occur. Amplitude A is selected in such a way that condition (17) is reali~ed. The value of C2 when catastrophes take place ~s selected from equation Lt C"3 i

+

2i

2 (2 :::

(18)

0

Considering (16) and (19) we have

4

J

f '- A- j 4+ w 2/

27

'2

IS l, n (w T+ If!)

- r!

( 20)

The equations (20) and (21) are necessary and sufficient conditions for existance of catastrophes in the system. We can determine a moment when a catast,... rophe is beginning in the system. At the same time we can also determine initial values X4 when the system goes from one poise into another at correspondiD6 values of .g, r: , 7' (Fil;5.4). Basic c atastrophes let us to classify the structures (1),(2),(3),(4). In order to make parameter identification we act in the following way:we feed a harmonic signal to the input of the object with such an amplitude that a catastrophe will take place in the system and observe recycle. We fix X4 and Z • Value of X is determined from X Z + X4 Then the expressions

=

X\ t) + CL X (t ) + C2 = 0

Cl

=;

4C}

Cz =~ j1+(,,}T 2

S~n(VJ-/; -Hp)

2

+27 C2 =0

are valid. The equation system, taking into conside-

ration values of X, X4 and Z , allows to build up the system of algebraic equations in order to determine ~1' C2 and ~ which can be solved by means of well-known methods, for example, by Chebishev approximation, which leads to the linear programminc; method. Dynamic points of bifurcation are apparently less known. There is no complete classification in dynamic system theory correspondil16 to that of the basic catastrophe theory. But it has been already found out that Hopf's bifurcation and bifurcation saddle-knot (or pleat catastro~he) are the only structure-stable bifurcat~ons observed during variation of one managing parameter. REFERENCES Gilmore R. (1981). Cat astrophe theory. Gu~ushvili A.Sh. and V.G.Shanshiashvili. (1 973) .Determination of static and dynamic characteristics of nonlinear objects with recycle. Trans. of the GPI, Instrument engineerlng, pp.65-69. Gugushv~li A.Sh. (1979). Principle of symmet ry in identification of nonlinear control objects. In: Trans. of AllUnion seminar on adaptive systems. Alma-Ata, pp.65-69. Gugushvili A.Sh. (1 982) . ldentification at large of nonlinear cont~ol objectl?. In: Mater lals of All-Un~on sC lentlflc conference on Dynamic modell illG of comp lex systems. Tbilisi, pp.178-179. Kortmann i'i . and H.Unberhauer. (1988) . A model structure selection algorithm in the identification of multivariable nonlinear systems with application to a turboGenerator set. lilAC S world conbress on scientific computation. Paris. Mehra R. (1977). Catastrophe theory , nonlinear system identification anc.i. bifurcation control. In: Scientific system. Tsipkin Ya.Z. (1963) . Adaptation and learninb in automatic systems. Nauka , Moscow, 339 p . ---

A. Sh. Gugushvi li

150

!I

~

1.I

-0

y

ZJ.r

I?

(W::O)

Fig. 1

u

Cl

v

Fig. 3

Fig. 2

(I

I I

+-______________________________~------------------------~w :f Fig.

4

!