Minimum sensitivity adaptive systems

mINIm um SE NSITIVITY

AD~PT IVE

SYSTEmS

S. Bingul ac In s titute of Nuclea r Sciences "Bori s Bel grade, Yugo s lavi a

K idri ~ ",

Introduction Thi s

p a ~3r

di scusses the proble m of reducing system sens itivity

cau se d by s ma ll v a ri a ti o ns of process pardffi9ters . p a r~m e t e r

I n the caS 8 of

~ ma ll

v 3 ri d tions it i s possible to desig n ada ptive s ys te ms in

w ~ich

the a ppropri." te adjustment of control l e r p21rdmeters i s dchi9ved by on-line solution o f line a r a lg eb r a i c e qua tions .

The numbe r of

e ~ uations

to be

so lved i n the on-lin e p a r3meter computer i s equal to the nu mber of adjustable controller pa r ameters .

One of th8 ma j or ad vant a ges of such a

system is t ha t coefficients of th e equations may be calculated off-line. This t ype o f adaptatio~ avoids the widely applied principle of canc e llation of the process and controller tr ans fer functions /1,2/, and thus ceases to be rastricted to lin ear systems only. re ~ uirem en t

Besides the

imposed on system line ar ity, adaptive systems based on

tr a nsfer functions cancellation (Fig. 1 ) suffer from the additional defficiency that the number of adjustable controller parameters b j (j = 1, 2 , ••• J) must e q ual the number of variable process parameters a i (i = 1,2, ••. 1). Furtherm o re each c o ntroller parameter b j must be chosen in such a way that its influence on the syste'm output x(t ) is e~u a l

to th e influen ce of one variabl e process parameter a i •

Th es e two

requirements may be expre sse d as: (1) i, j

1, 2 , ••••• I

Another type of adaptation

/3,4/

J.

through controller pa rameter

adjustment is characterized by continious minimization of the given cost Although condition (1) is not necess a ry function F [e (t)), (Fig. 2) . here, practical app licatio~ of this type of aoaption i s limited by stability considerations and th e need for extensive on-line calcul a tions. With the gereral aim of devising more effective means for compensating the affect of small parameter variations, this paper deals

- 249 -

with the problem of synthesising adaptive systems which (i)

(ill

require large on-line computers,

do not

are not subject to limi tations

Jue to process nonlinearities and (iii)

du not reguire the number of

adjustable controller parameters to be eyual to the number of variable process parameters.

Description of the system The proposed method of adaptation is applicabele to processes described by equations of the form

(2)

where

=(

Y(t)

ys(t)}, s

= 1,2, ••• 5

is an 5-dimensional state vectol

of the process, UI(t) = {wr(t)}, r = 1,2, ••• R, vector of the controller,

A = {aJ

i

an R-dimensional state

= 1,2, ••• 1,

an I-dimensional vector

whose coordinates are the variable process parameters.

The coordinates

gs of the vector G are scalar functions of the S+R+I independent variables i • e. s

(3j It is assumed that environmental influences on the process dynamice will

be reflected in slow variations of the parameters a i •

The mathematical model of the controller is as follows

=

H [UI(t),

where, similarly to equation (2),

8

Y(t),

I

(4)

B, r(t)]

j = 1,2, •• J, is a

{b j} ,

J-dimsnsional vector whose coordinates are adjustable controller parameters. r (t) is the input Signal, and H

{ hrl

{ hr

[iU(t), ¥Tt),

8, r(t)] }

With ths substitution

x n (t)

for

1

Ys (t)

for

R+l

{

wr (t)

~

n

:!i

R

~

n

~

R+S

(5)

- 250-

N

e~uations

(2)

and (4)

may be combined into

F

[X(t), A,

B,

(6)

r(t) ]

which is the usual mathematical model of a feedback control system. According to equation (5)

the output coordinate of the process YS(t)

now becomes

(7) It should be noted that there are no restrictions concerning the linearity of the process and the controller .

The only requirement,

as

will be cl ea r later on, is the ex ister.ce of sensitivity functions of the output variable xN(t)

with respect to proces s and controller param 8 ters,

i.e.

(8)

and

Conse~uently,

in the parametric spaces of the process and controller tne

following sensitivity vectors may be defined

u (t)

{ u i (t)

grad I xN(t)

{ axN(t) } aai (9)

V (t)

grad J xN(t)

{vj(t)\

where indices I and J indicate that xN(t) respect to I parameters

ai

J

and

{

N(t) } ab . J

8X

s h ould be differentiated with

parameters

0j'

respectively.

Parameter

values b jo ' which ensure optimal system performance for nominal values a io of process p a r ame ters, should be determined by one of the existing methods /5,6/.

Statement of the problem It is assumed that the deterioration of system response is caused by variations of parameters

around the values

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In accordance

with the usu"l practice in control ~n

variation s in parameters a i pard me ters b j _

by

In other words,

is to minimise the error

theory,

compensat~

ad6ptive system wh ich will

oOj ~ ctive

t ne

is t o o8si<;n

t h E undesir8 c e ffect o f

dppr op ridte dOjustm an t of the obj ec tive of t h is

controll~r

~8~p tive

syste~

~uantity

(lU) which repr 3 sents

the deviation of t he system r espo n s e

Aa, Ba )

response xN(t,

atte mpt t o minimise th e error small vdriations

of proces s


by adj usting bjo

4x(t)

x (t)

L i=l

(11)

1

may be written as

J

I
In the c u se o f

i. e .

pdfdmeters,

« the error

from the optimal

Un o er such circu ms tdnce s i t is customary to

0

L J=l

LI a.

1

LI b

(12 )

4 b. J

(13 )

or: J

L j=l Using vector notation equation


where


(t)

A

=

Aa,

defined by equation (9)

(13)


B

(t)

becomes

v' (t)

+

vj

Ba


U 'it)

(14) and V'(t)

are row vectors

One possiole method of minimising error

0

is to apply condition (1),

x

(t)

which giv8s

I .:\X

(t)

L i,j=l

(15 )

Thus, if the controller parameters are adjusteo so that

(16)

h . J

the error

LlX(t)

will be always equal to Zero.

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It should be noted that equation (16) requires the effect of each particular process parameter a i t o be c ompensated by the a d jus tm en t of only one controller parameter b j , i.e. that the condition J = I be satisfied

/ 1/.

We shall demonstrate now th at th e abo ve method Can be ~xt end e d also to th e case J < I, wh ich is of particular importance when the system contains a larg e numb~r of variable parame tars a i • The adaptation scheme wh ich we propose i s not restricted by condition (1) and therefore permits greater freedom in the selection of adjustable parameters b j • Inste a d of considering the effects of a particular pa r a meter a i' it is more convenient to consider the total error I

L i=l

e (t) p

(17)

caused by variation of all process parameters together. The total error ep(t) of the process should be compensated by simultaneous adjustment of the J chosen controller parameters, whose effect on the output x(t) may be represented as J

L j=l Thus, the 6rror

~

x(t) of eyuation (13) becomes:

In order to a voio on-line computation of time-varying functions which may often Cause instabilities /3,7,8/, we Shall approximate the sensiti vity functions u. (t) and v. (t) by linear combinations of 1. J appropriately chosen functions gh(t), (h = 1,2, ... H). The number H is determined by the desired accuracy. This transformation may be represented as G(t) (19)

11

0 where IICI = {c ih } , {d· h } and G(t) Generaly speaking the choice of gh1t) may be arbitrary /9/.

- 253-

(gh(t)}. More details

concerning the functions 9 h (t) ana th e calculation of constants c ih and d jh will be present ed in d subse~uent section. Th e re are many analog computer metnoDS availdble f or the ge neration of s e ns iti v it y f u ncti o ns in both linear a na nonlin8Qr s yst em s Introdu cing transfor mdt i o ns

/10-13/.

( 1 9)

int o

€~uat i on

(14)

\!I6

obt ain

H

L h =l

.4 x (t)

E

where

E' ::; ( t)

d A

" ell

dB

D

1

Here

11 ell

,

or

J

L i= 1

eh

(20)

c ih

~

and

minimisation of

d di

0

L J =l

+

I

.4 b j '

h

(21)

1,2, ... H

ara trdns p os6d matrices.

(t)

.1x

It is obvious that the error coefficients eh are equal to zero. the system of algrebraic

o

E

d jh

IIx(t) "'ill be zero if all error tor the Case J = H one can solve

e~uations

or

11

dB

0 11

11 ell

(22)

and determine the corrections to controller parameters B wnich will mi nimi ze the error

11 x (t) •

(In this Case tne mini mu m value of

d x (t)

is zero). In the case the general Case.

J

< H, however, equation (22) cdnnot be solved for

Therefore, it is only possible to minimize all error

coeFficients eh which will in an appropriate manner also minimize the error

Ax(t).

To this end, let us consider the quantity m

which is a

scalar Function of the J independent variables b j , i.e. H

m(11 B)

L h=l

(23 )

where kh ara weiQhtinQ factors. The quantity m will be minimum at those values of 11 b j , "'hich satisfy equation

- 2S4-

o From equations

(23) and (24) we obtain /14/

~ 0 I I K 11 where

11 KII

(24)

{II e ~

,

11 0 I '

AA +

AB}

= 0 ,

(25)

is a diagonal matrix whose elements in the main diagonal

are the weighing factors

kh •

On-line solution of these

J

equations with

J

unknowns

yields the appropriate values of controller parameters which, if

(25) A ai

of process parameters are available, will always ensure minimum value of

Ax(t).

One of the possible configurations of the parameter computer

is given in Fig. 3. farm in Fig. 4. Fig. 4

The proposed adaptive system is shown in general

Advantages of this system may be visualized by comparing

with Figs. 1 and 2. It is obvious from Fig. 3 that the adaptive system of thts type

does not require on-line use of either the sensitivity functions ui(t), v j (t) or the functions

I

gh(t).

It is necessary only to dispose with

I

matrices ~ e~ o~ and K~ moreover, matrices and may be calculated off-line. The calculation of these matrices should be

,

lie 11

UolI

performed using functions u i (t), v j(t) and gh(t).

The choice of functions gh (t) Since the sensitivity functions u i (t) and vj(t) are dependent an si~nal applied, the coefficients c ih and d jh will be

the type pf input

also input dependent in the general case. undesired dependance, selected.

In order to avoid this

the arbitrary functions ~h(t) should be properly

Therefore, since functions ui(t) and vj(t) characterize the

given system and the applied si~nal, the system and the input

si~nal

functions 9 h (t) must also represent in an appropriate way.

It is shown in the Appendix that for linear systems the

followin~

relations hold without any approximation

u (t)

Id

G(t)

(26) G(t)

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=

Here H

~

+

m+

1.

m the

N is the order of the system and

zeros of the overall syst e m transfer function.

[j',,,trices

number of

11 ell

and

11 D 11

are in this Case independent of the input signal and adaptation Can De successfully achieved. In the case of nonlinear

syst~m5,

the functions gh (t) may be Then JJe get

u (t) or V (t) •

obtained by ortnonormalisdtion of U (t)

1 ell

G(t)

D 11

G(t)

(L?)

-

V( t)

11

If the vector U(t) is orthonormalised, only the second equation is approximate and vice versa.

Due to system nonlinearity and the

approximations involved in relations matrices

ell

and

(27),

th9 input d3pendence of

can not be completely eliminated. Analog computer examination /14/ has shown, however, that satisfactory reduction 11

11

DII

of system sensitivity with fixed

11 ell

and

IIll 11

cun be dchieved even

with changing input signal. If all functions U(t), Vet)

and

G(t) are aVailable for Bach

particular case, analog computer calculation of matrices

le 11

and

is straightforward for both linear and nonlinear systems /14/.

Example As a simple example we have taken a nonlinear system with saturation between the controller and the process. of this system together with the model

The block diagram

for the generation of sensitivity

ell

functions is shown in Fig. 5. The matrices 11 and 11 D 11 were calculated using the above procedure and the whole system waS simulated on an

analo~

computer.

It waS assumed that the variation of parameters

a i could be obtained by process identification. The

tr~j8ctory

of the two-dimensional vector

~

together with the corresponding responses of the system from JJhich the sensitivity to parameter variations may be obtained is

sho~n

Using equation (25) and taking into account the trajectory of

in

Fi~.

~,

6.

the

trajectory of vector 8" {b l , b 2 } waS calculated (Fig. 7). RespDns~s indicated in the figure are obtained by appropriate adjustment of controller parameters

B,

which make the system practically insensitive

to process·parametars variations.

- 256-

Conclusion

The basic advan td ges of

t~e

propose d method of adap tation

m~y

be summer i sed dS follows :

(i ) ve r y

The

stron~ly

( ii)

sele~tion

of adjustaole contro ll er paramete rs is not

i nfluenced by process parameters . The computat i on speed of the

made as h i gh as des ire u ,

t hus

prov i o in ~

p~ra~eter

co m~ute r

advanta~eous

may be

stabi l i t y

condi t i ons . ( iii ) The method o f adaptat i on i s applicab l e to both linear and nonlinsar processes .

( iv )

As f ilr as prdc ti cal realis at i ons i s concerned,

t he re is no

need to use a l arge number of computing co mp one n t s , most of the calculations be i ng perfo r med off-li ne .

(v)

By on- line compu t at io n of mat ri ces

11 ell

the

and

metho d may be app lie d to l a r ge Pdrameter v a ri ations. The systhesis of th e proposed type of o f the f oll owing steps : t he controlled pr ocess ,

(a) (b)

ad~ptive

sys tem consists

f o r mu l a tio n of the mathemat ic a l model o f choice of the ma t hemat i ca l mode l

dnd

structure of th e controller,

(c)

a ll process pa r amete rs

(d) calculation of the op ti mu m set ti ngs

a io '

de t e r mi nation of the nomi na l

v"lues of

of controller pa r ame ter s

b jO corresp on d in g to a io ' (e) me asu r emen t o r estimation of process parameter v ar i d tions, (r) prov i s i on for the

a djustment of

~Cl

J

controller parameters,"

computation of matrices

I

( h) introdu c tion o f a nd DI and the choice of ma trix a parameter computer wh ic h will solve the s yst em of algebraic e qua ti o ns (25) and compute the app ropriate values of controller parameters.

- 257-

I\ppendix The LaplClce transform X(s) 8)

of the output vdriable

xN(t)

(Fig.

is

x (s)

R (s)

pes)

R (s)

(11-1 )

D (s) Here R(s)

is the Ldpldce transform of the input singdl, dnd pes)

u(s) ,He polynomials of fiJ-th Clnd N-th functions

order

respectively.

and

The transfer

of the process and controller dre

P (s)/D (s)

p

Di fferantiClting X (s) p

X (s) = R (s) a.

Ui (s)

p

with respect to

c [lc

QTsT

1

Q

P

~

-

~i

p ~

d.

1

and

b. J

Q

~i

R (s)

Q (s)

we get. Ci (s)

QTs)7 (1\- 2)

X (s) -bJ

V j (s)

R (s)

~

-

Qc P cj

Q (s)

Pc Qcj

D. (s)

R (s)

Q (s)

~

where. p

C i (s)

and

pi

ClP p (s)/ aa i

Qpi

aQp(s)/ aa i

p cj

aPc(s)/ ab. J

QCj =

aQc(s)/ ab. J

Dj(s)

in equa tions

(1\- 2) are polynomiClls of (H-l) s t order

where some of the coeffici ents c ih

and

d jh

may be zero.

Usi ng sUbstitution R (s) Q (s)2

Gh

(s) = s Gh _ l (9) = ••• = 9 h-l Gl (9) I

equations

(A-l) may be written aSI

- 258 -

h

1,2, . . . H

H

Ui

(s)

L h=l

H

c ih Gh(s)

Vj(s)

L h= l

d jh Gh(s)

(A- 3)

App lyin g the inverse Lap l a ce tr a ns formation we o btai n H

u i (t)

L h=l

H c ih gh (t)

vj ( t )

2:::

h=l

d jh gh (t)

(A-4)

whi ch evidently corres ponds to equation (2 6).

Acknowledgement The author wishes to express his gratitude to P. Kokotovi6 a nd L. Ra danovi6 for their support a nd many stimul a ting discussions.

REFERENCES 1. N.N. Puri, C. N. Weyg a ndt, "multil/ariable Adaptil/e Control Systems", Proc. Joint Autom a tic Control Conf., New York, 1960. 2 . H.P. Whitaker, "Oesign Capabilities of model Reference Adaptive Systems", Proc. NEC, Chicago, 196 2 . 3. I.E. Kaz ak ov, L.G. Evl a nol/, "On the Theory of Self-tuning Sy stems with a Sea rch of Gradient by the method of AUXiliary Operator", lI-nd IFAC Congress, Basle, 1963. 4. I.E. Kazakov, "Contribution to the Statistical Theory of Continuous Self-adjusting Systems", Izv. Akad. Na uk SSSR, No. 6, 1962. 5. S. Bingulac, P. Kokotovi6, "AutomatiC Optimization of Control Systems by Analog Computers", Proc. AICA, Bruxelles, 1964. 6. D. mitrovi6, "Graphical Analysis and SyntheSiS of FeedbaCk Control Systems", Trans AIEE, Vol. 77, 1958. 7. m. margolis, C.J. Leondes, "On the Theory of Adaptil/e Control Systems - the Learnig model Approach", I-st IFAC Congress, 1960. 8. H. F. mei ss inger, "Parameter Optimi zation by an Au toma tic Oper.-L oap Computing method", IV-th AICA Congress, Brighton, 1964. 9. R. Courant, D. Hilbert, "method of mathematical Physics", 1953. la. P. Kokotovi6, "Structural Approach to the Analysis of Parameter Influences in Automatic Control Systems", m.s. thesis, Belgrade, 1963.

- 259 -

11. H.F. ~laissinger, "The us e of P a rameter Influence Coefficients a n d Weightin g Functions Applied to Pertubation rinalysis of Dynamic Systems", Hughes Co., California, 1 9 61. 12. R. Tomo v ic, "Sensitivity Analysis of Dynamic Systems", 1963.

Belgrade,

13. J.D. Roberts, "A me thod of O~timizin g Adj usta ble Parameters in Control S ys tems", Pr oc . lEE, Vol. 1 09 , Part B, 1962. 14. S. Bingulac, "Sensitivity Approach to the Synthesis of multipardmeter Adaptive Systems", Ph. D. thesis, Belgrade, 1964.

- 260-

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- 261 -

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CONTROLLR

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10ENTIFICATION

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- 264 -

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- 265 -

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