Self-Tuning Control of Lumped Input and Distributed Output Systems

Copyright © IFAC Adaptive Systems in Control and Signal Processing, Budapest, Hungary, 1995

SELF-TUNING CONTROL OF LUMPED INPUT AND DISTRIBUTED OUTPUT SYSTEMS Hulk6, G., Belansky, J., Belary, C., Kovalcik, J., Antoniova, M., Szuda, J., Vegh, P.

Department ofAutomation and Measurement. Faculty ofMechanical Engineering. Slovak Technical University. NiJm. Slobody 17.83112 Bratislava, Slovak Republic

Abstract: Distributed parameter systems can be very often found in the engineering practice in the form of lumped input and distributed output systems. In the paper possibilities of self-tuning control of uncertain lumped input and distributed output systems will be presented. Keywords: Distributed parameter systems, lumped input and distributed output systems, lumped input and distributed output predictor, self-tuning control.

I. INTRODUCTION

In the paper first some typical examples of real LDS will be presented. Further the lwnped input and distributed output predictor will be proposed and distributed parameter self-tuning system of control will be designed. The perfonnance of such a system will be then demonstrated by self-tuning control of a thermal process physical model.

Real Qistributed farameter SYstems (DPS) can be very often found in the engineering practice in the fonn of L,wnped Input and Qistributed Output SYstems (LDS). On the other hand by the design or construction of new technical distributed parameter systems it is often possible to fonn them as LDS. In this way LDS fonn a wide and important class of DPS.

Y(X,t)

The LDS dynamics is decomposed to time and space components. According to this fact also problems of identification and control synthesis are decomposed to problems in the time and space domains solutions as well. The time domain problems are then solved using methods and algorithms of L,wnped farameter SYstems (LPS) identification and control, space problems being solved using functions approximation methods and algorithms. Thus, on an algorithmic level, there are possibilities offered practically the same as possibilities of LPS control are. See Fig. 1., Hu1k6, et al. (1990-94).

11 (k)

The main goal of the paper will be to show similar possibilities also for uncertain LDS self-tuning control.

Fig. 1. Distributed parameter system of control HLDS - LDS and fonning unit H; IDJSD - time/space components of dynamic characteristics; TSJSS - time/space parts of control synthesis; K - time/space sampling; Y(x,t) - controlled variable; W(x,k) - control variable; V(x,t) - disturbance quantity, E( x,lc) - control deviation; i(k) - vector of actuating variables.

In this case problems of self-tuning control will be decomposed to problems in the time and space domains solutions too. By solving problems in the time domain we will methodically start from the predictor based selftuning control of uncertain LPS scheme. Individual identification and control synthesis steps in the time domain, at choosen points of controlled distributed systems, are to be taken in these relations, as, due to the limited extend of the paper, that cannot be pointed out particularly, Peterka (1984), Karny et aJ. (1989). 437

2. REAL LUMPED INPUT AND DISTRIBUTED OlITPUT SYSTEMS Technical processes input variables as raw materials, various other materials, energy, etc. are transmitted to process (using pipings, conduits and electrical lines) in the fonn of lumped parameter variables. Technological processes however mostly operate in the interactions of fields of variables form, state variables and/or output variables being distributed variables. Considering inputoutput relations there are real LDS shown e.g. in Figs. 2. t04.

of input and output variables the matrix of linear finite memory predictors can be specified in the block IDENI1FlCATION and PREDICTION: Tab. 1. See also Fig. 8. YT(x,t)

U1 It) 's

7'-

Fig. 4 Control ofhot-house's temperature field in agriculture Ui(t) - steam flow, lumped input quantity, Yf(x,t) temperature field, distributed output quantity, {Gi}i=l,n - heating units

.2

~t)

ftt

tM.l(tl

Y

DPS

ftt

(x, t )

UA1Ctl

LDS

licu

Fig. 5. Qistributed ~ararneter fu'stem (DPS) as Lumped Input and Qistributed Output fu'stem (LDS) {ui(t)}i=l,n - lumped input quantities, Y(x,t) distributed output quantity

Fig. 2 Fluidised fireplace U~(t) - supplied quantity of fuel and additives, lumped input quantity, FV - fluidised layer; P high-pressure parts of steam generator, {Si}i servo-drives; {Ki}i - throttle valves; Y(X ,t) temperature field, distributed output quantity

Now, we have vectors of input and output variables in the fonn: ui(Kf ={u~k),... ,ui(k-nb)} for i=l,n; Y(xj,kf = {Y(xj,k)}

j=O,m;

Y(xj,k-lf = {Y(xj,k-l), ... , Y( xj,k-na)}

j=O,m;

Further

p(x:} i) = {p(x:} i, k), ..., p(x:} i, k - nb)} p(x:} xi) = {p(x:} xi. k - I), ..., p(x:} xi. k - na)}

.v

being vectors of predictors of specified structures parameters, for i=l,n; j=(),m. These predict distributed output variables at the time k in points xO, ... , xi, ... , xj, ... , xm of the interval <0,1>. The integers na, nb defme the memory of the predictors. e(xi,k), e(xj,k) for arbitrary xi .. xj; xi, xj E (0, L) being not correlated random variables.

Fig. 3 The rolling-mill heat furnace UMi- gas flow, lumped input quantity; Gi - i-th generator of the infmite-dimensional input quantity, T - thermometers of the vault temperature; Q dimensions; V - composition; v - speed of semiproduct shift; D - transport to a rolling-mill train; T/f0 - input/output temperature of the semiproduct; YK(x,t),YR(x,t),YP(x,t) - temperature fields, distributed output quantities In general DPS can be defmed as systems having state and/or output variables in the fonn of fields of variables distributed variables - i.e. LDS fonn a class ofDPS. Fig.5.

3. LUMPED INPUT AND DISTRIBUTED OlITPUT PREDICTOR

KT...

o

Assume that input-output variables of an uncertain distributed system being considered are measured in some arbitrary points of real HLDS. See Fig. 6. Using sequences

>11...

_...•

, .....

L

Fig 6. Input-output variables of an uncertain RiDS

438

Tab. I. Matrix of linear ftnite-memory predictors

ul(}0,

... ,

..

ui(}0,

"

-!.

-!. P(xO,l),

...

P(xO.i}.

P(xl,l),

... ,

P(xl,i),

~

...• .. "

... ,

... , P(~i,l),

... ,

... ,

..

P(xm,l),

... ,

... ,

P(~i,i), "

... ,

P(xm,i),

un(}0,

Y(xO,lcl),

-!.

-!.

P(xO.n).

P(xO.xO}.

P(xl,n),

P(xl,xO),

... ,

... ,

P(~i,n),

P(~i,xO),

... ,

... ,

P(xm,n),

P(xm,xO),

Tab. 1. let us defme the vector Yz(xj, k) = {Yz(xj, k)} j = 0, In as a response to

Using

vectors: Y(xO,k-l),...,Y(XIll.k-I). Further the vector Yui(xj,k) = {P(xj,i). ui(~} j = O,m as a response to individually acting vector ui~). The selected components lYui(xi,k)=P(xi,i).ui(~1 for i=l,n are then responses in points {xi}i=l,n to individually acting input variables. Here •.• means the scalar product. Let us introduce the relation Yu(xj, k) = {Yu(xj, k)} ,- 0 ,a = =

{i Yui(xj, k) = i 1-1

p(xj, i) . Ui(f)}

1_1

,-0,.

where YU ( xj , k ) is a vector generated by simultaneous actions of input variables sequences U(f) = {ui(k)}i = I, n. Hence the total vector of output variables predicted is: YP(xj, k) = Yu(xj, k) + Yz(xj, k). Let us analyse now the part of DPS dynamics on a ftnite-dimensional level generated by a vector of actuating values u(t): Yu(xj, t); t E (k, kl). The vector {Yui(xi,f)} i = I,n forms cmves leading discretes Yui(xj, f). The sum of discretes gives then the total output variable, directly generated by actuating values U(f):Yu(xj,f)=i:Yui(xj,f) on the interval -

Y(xm,1cl),

1

-!.

-!.

....

P(xO.xm},

e(xO.k},

-+ Y(xO.k)

... ,

P(xl,xm),

e(xl,k),

-+ Y(xl.k)

... ,

... ,

... ,

P(~i,xm),

e(~i,k),

-+ Y(xLk)

... ,

.0.,

... ,

P(xm,xm),

e(xm,k),

-+Y(xm,k)

..

"

... , ... ,

Here Yui(xi,f) represents the time component of the output variable dynamics in the point xi and (Yui(xj,kl )}j=O,m represents the space component of the output variable dynamics in the point k I being generated by individually acting input sequence ui. Hence the predictor P(xi,i), represents the time component of the output variable dynamics and {P(xj,i)}j=O,m represents the space component of the distributed output variable dynamics (on a fmite-dimensional level) generated by individually acting input variable ui. Till now our scope was focused on DPS dynamics study on a ftnite-dimensional level only, but our goal is the formulation of an infinite-dimensional model predicting the distributed output variables course with an accuracy known or given. Let us assume now an infinite-dimensional number of wriform displaced predictors in the interval <0, 1>. The outputs of these predictors give us the infinite-dimensional variable: YP(x,k) - the distributed conditional mean value of the distributed output variable. Our task now consists of fmding approximation: YPa(x,k) the measuring points layout - discretes: y p ( x j, k) given

and approximation methods choosen, and to determine an approximations accuracy x(k) in the discrete time: 11 yP (

x, k) - YPa (x, k) 11 ~ X(k)

(I)

iEl

Or, the accuracy X given, to find such measuring points layout and such approximation method, that the deviation between modelled YP(x,Je) and model output YPa(x,k) variables holds the relation

Fig. 7.

Tui(Xi,!)

yw.(xj,kl)

11

YP (x, k) - YPa (x, k ) 11 ~ X

(2)

where 11.11 is an appropriate norm chosen, X(k) and X are positive real numbers.

Y

These approximation problems can be solved using the approximation theory results. The differential properties of YP(x,k) will be analysed using continuity modules and derivatives norms in dependence to x. Fig. 7. Discretes responding to distributed output variables changes in the time and space domains - on the interval X

439

The more detailed analysis of LDS dynamics proves that differential properties of YP(x,k) can be analysed using discretes {Yui(xj, k)} i = 1, n. E.g. the lumped

- a space of continuous functions V(... ) in <0,1> with the norm

input variables range given from an interval using discrete quantity Yu(xj, kl) =

Iy( ... )1Ic'{O,L)

±

1-1

Differential properties got using discretes can be then generalised on the whole interval <0,1> and function YP(x,f) respectively. (This is a standard approach to infinite dimensional systems modelling. E.g. by mathematical description of continuum dynamics distributed in the interval <0,1> an arbitrary infinite-dimensional element of continuum is taken. Using this element the differential relation is formulated and further generalised on the whole interval <0,1> in the form of partial differential equation.) By uniform measuring points layout and approximation using cubic spline functions the maximal deviations values between YP(x,f) and YPa(x,f) are shown in Tab. 2., Zavjalov et al. (1980).

v[yp"(...)]:s; v[YPw"(·.·)]

VP(. ..) eW;(O.L}

~:

hIllY PM" ( ... )

In general we get the lumped input and distributed output predictor in the form:

11..

YPa(x,f) :

(3)

Using results of the approximation theory it is possible to formulate and to solve many similar problems also in the case of a not uniform measuring points displacement and for further appropriate classes of approximations functions.

2.8 h v[v PM '( ...)]

v[Y P'C ..)]:s; v[Y pw'(...)]

YP(...) eC I (O,L)

8X(k,kl) h C '(o.L}(k,kl) = 9V[ YP M'(x,kl)]

Ilk.kl)

YP(...) eC1(O.L)

IYP"(..·l ;s;lvp.. "(.··l

)1

The relations shown in Tab. 2 can be used, the accuracy X (k,k 1) given, to define the minimal distance of measuring points h(k,kl) by uniform measuring points layout. E.g. using the first row of Tab. 2., the value X(k,kl) given, the minimal distance between measuring points is:

Tab. 2. Deviations of sets M={yp(x,m and Ma={YPa(x,m

YP(. ..)eW;(O,L)

XE(O,L}

- a class of functions with continuous derivatives ofk-th degree - a class of functions V(... ) on interval <0,1>. His elements have continuous derivatives of (I-l)-th degree and their l-th derivatives are from the space Wp <0,1>, l:S;p:s;oo. -l>k, a class of functions V( ... ) holding V( .. )eCk«),1>, Y( ..)eCIk, 1SP:S;OO, a class of functions V(... ) holding V(.. )eCk«),1>, V(.. )e Wl
p(xj, i)· ui max(kl).

M={Yp(x.f)}

= maxIY( •.•

hI v[V PM "( ... )]

=

SA{t.,-I

P(xj,i). ui(f) + YZ(Xj,f)}

(4)

where SA {.} is the spline approximation using discretes appropriately choosen.

8: h11lY PM

IVP·"(..·l ;s; IVp.. "'(.··l

Then

"'(···)11..

YPa(x,f) =tYuia(x,f) + yza(x,f) =yua(x,f) + yza(x,f) ;'1

(5)

YP(...) eCIC~(O,L)

~YP'''(...)]:s; ~YPw "'(...)]

~hl v[VP "'(. .. )] 1728 M

This model - with accuracy known: X(k,kl)

Iyp(x, f) - YPa(x, f)1I S X(k, kl)

(6)

or given X: (by the choose of approximation methods) Vp(...) eC1W:.(O,L) ~Vp'·,,(.··t :s;IVp.. ''''(..·l

3~

4

h 1/V PM

lIyp(x,f)-YPa(x,nl;s;x (7) predicts distributed conditional mean values of distributed output variables in the set <0,1> x .

""(···)11..

4. DISTRIBUTED PARAMETER SELF-TIJNING SYSTEM OF CONTROL

M - sets of approximated variables in the classes of functions choosen in the interval <0,1>, Ma - sets of approximants; X(k,kl) - deviations between M and Ma on the interval s - net of approximation nods in the interval <0,1>, s: O=xl
Self-tuning control of distributed parameter uncertain system can be realised using the scheme as shown in Fig.8. Let the problem of control synthesis consist of minimising

440

DPSTC distributed parameter self-tuning controller, TSISS - time/space parts of control synthesis; K - time/space sampling; {iili( f)}i / {Yuia(x,k1 )}i - time/ space components for control synthesis; Y(x,t) - controlled variable; W(x,k) - control variable; V(x,t) - disturbance quantity, U(k) - vector of actuating variables.

• - in the time domain - the functionals: {Ji}i=l,n of the type

Ji = tim

.!.-E{ '.k.. ~[Yi(xi.f)- Wi(f)]' +OO(6Ui(f))1} (7)

T.... T

where Wi(t) is the control variable, CJ)~.T is the control horizon, 6ui(t) are increments of input quantities, Peterk.a (1984), KArny et al. (1989). • - in the space domain - the functional (8)

JT = Tlim IIw (x, T) - Y(x, T)II ... c

4.3 Space pan ofcontrol synthesis

1.1

where is an appropriately choosen nonn. In the space domain it is possible to solve more real, engineering problem too. Namely to assure the relation

rrsB,

In the k-th step let us solve the approximation problem

Itw (x, f) -

where

8 is a positive real number given, to be valid

Y ( x , k I )11 :S; 0

(11)

where W(x,t) is the control variable in the interval <0,1>, with constant value in time direction in the interval , fe and 11./1 is an appropriately choosen nann.

At the self-tuning control these problems in the step k are to be solved in an appropriately choosen finite horizon h. First, in the block IDENTIFICATION and PREDICTION - Fig. 8, components for DPSTC synthesis, using the matrix of predictors and lumped input and distributed output predictor designed, are prepared.

Then, using the relations (4), (5), we get IIW(x. f) - [yza (x, kl)+

L aiYuia (x, kl) + T](x, kl)]11 ~

~ 0 - x(k,kl)

4.1 Time domain control synthesis components

(12) The time domain control synthesis component for i-th input quantity will be found using the predictor P(xi,i). Let us assume unifonn step function of the control variable. Specifying further conditions of the solution using well known methods and algorithms we can determine the optimal sequence of actuating values ii li(f) ,. Peterka (1984), KArny et al. (1989).

4.2 Space domain control synthesis components

Yza(x. k I) +

L aiYuia(x. k I) + T](x. kl) =

approximates the distributed controlled quantity: Y(x,k1), with the deviation less than X(k,k1) in the interval <0,1>.

L aiYuia (x, kI) + T](x, k I) ]11:s; :s; IW(x. r) - [yza (x. k I) + L aiYuia (x, kl) 11 + IITl(X, k 1)11

IIw(x, f) - [yza (x, k I) +

(14)

(10)

Therefore the approximations problem ( 12) can be rewritten as follows:

where k 1=k+h. k-th

step of control

(13)

YPa(x,kl) + T](x.kl)

Using triangle inequality we get:

The i-th space domain control synthesis component can be found using the variable Yuia(x,t). This variable approximates with deviation less than X(k,k1) the distributed variable Yui(x,t); the i-th space domain control synthesis component is then given in the fann: Yuia(x,k1) = SA!Yui(x,k1)j = SA!~xj,i). iili(!!)!

where 11(x,kI) is the random component of the distributed output variable.The distributed variable

IIw(x,f) - [yza (x,k1)+

synthesis

L aiYuia (x,k1)]I:s;

:s; ok1

(15)

where ok1 = 0- X(k,k1) -llrJ(x,k1)1. Let the solution of this approximation problem be a vector aokl = {iiiokI}i_I.•. Then after the solution of space domain control valid:

synthesi~

problem the following relation is

!w(x,f) - [Yza(x, k1) + LCiiok1Yuia(x, k1)]II:s;

(16)

:s; ok1

4.4 Time part ofcontrol synthesis By the time domain control synthesis individual solutions components from the block SS: {a i 0 k 1 } 1 _ 1.. are present in the block 1'8 in the fann of lumped step control variables: {W i( f) = aiok I}.,-I,D - see Fig.8.

Fig. 8. Distributed parameter self-tuning system of control HLDS - uncertain controlled LDS and forming units H;

441

a

Then we can get individual components of optimal input sequences in the fonn:

b

c

d

~(f) = {iiiBklUli(f)},.1.n Therefore in the control action we can get the optimal input sequence in the block TS in an interval simply by the multiplication of components {iiiBU1,'l.n and {uli(f)},.,.n'

UJ.(t)

Ui (t)

•••

Un (t)

Fig. 10. Physical model of a thermal distributed parameter system, a - air, b - insulation, c - infared lamps: generators of distributed input variables {GUi}i, d measuring points, {ui(t)}i - lumped input variables (voltage 0-5 V), Y(x,t) - temperature field, distributed output quantity

By the influence of the sequence ~ (f ), the values of functionals {Ji}i=l,n are minimised, that means the given condition of a time domain control is valid. By the influence of the sequence i1 (f ), the relations (16), (11) are valid, that proves (in accordance with chapter 4.3) the space domain control synthesis solution to be got.

5. CONCLUSION By solving individual problems of distributed parameter self-tuning control

For the control in the k-th period the first element of the sequence fi ( f) is asswned to be an actuating value.

• - in the time domain - methods and algorithms of on line identification and self-tuning control synthesis were used,

4.5 Distributed Parameter Self-Tuning Control of Thermal Process Physical Model

• - in the space domain - methods and algorithms of functions approximation in the design of lumped input and distributed output predictor and self-tuning control synthesis were used.

The perfonnance of DPSTC designed was verified successfully by the control of physical model of distributed parameter thermal process - Fig. 10.

Using the scheme of LDS self-tuning control proposed, there are possibilities to solve distributed parameter selftuning control problems practically the same as for lumped parameter systems self-tuning control.

Typical courses of lumped input and distributed output variables are shown in Fig.9.

6. REFERENCES

lseepaSOISloJ V(M,k)

•

lOCI,':'

HULKO, G.: Control of Lumped Input and Distributed Output Systems. Preprints of 5-th IFACIIMACSIIFIP Symposium on Control of Distributed Parameters Systems. Perpignan, 1989.

!Jl

60 30

•

HULKO, G. et al.: Computer Aided Design of Distributed Parameter Systems of Control. II-th World Congress of IFAC, Tallinn 1990. HULKO, G.: Control of Lumped Input and Distributed Output Systems at the Control of Distributed Parameter Systems. Problems of Control and Information Theory. Vol. 20, no. 2., Pergamon Press, Budapest-Oxford 1991.

• •••.••. , .100 , ••••.••. , .••••••, .•••••. IVI~

: ~ ~ 1~ 1~ ~

HULKO, G.: Control and Design of Lumped Input and Distributed Output Systems. D.Sc. thesis, Bratislava, I 99 I. (In slovak).

J=== 00

k 2!lD

l'l.~ : : . : 1 •

· ·•

o o

:

:

:

5lD

llDl

15lD

2

,

•• .;. ••••••• :

.'. -

.; . ._. -; -_. -_.

.; .

HULKO, G. et al.: Design of Distributed Parameter Systems of Control. Preprints of IFAC Workshop NTDCS'94, Srnolenice, 1994.

.

--~_. -_._-

:

DD

k

2!lD

KARNY,M., KULHA vY,

HALOUSKOVA, A., BOHM, 1., R., NEDOMA, P.,: Design of a Linear Quadratic Control. Theory and Algorithms for Practice. Kybernetika, VoI.21., Acadernia, Prague, 1989. (A supplement).

IVI~ +.....+..... : ' . .........,

a3(l<) , ••••••.• ,

--

.

3 .. -- ..

· . . 2~ 1 -

00

.:

~

•• - _. ~- _.

,~

• ••• ~ ••• - -

1~

- - •• -

~.

2!lD

Fig. 9. Lumped input and distributed output quantities of the self-tuning control of thermal distributed parameter system physical model; ul(k), u2(k), u3(k) - sequences of actuating quantities; Y(x,k) controlled temperature field, distributed output quantity, W(xJc) - control quantity

PETERKA, V.: Predictor-based Automatica, Vo1.20, No.1., 1984.

Self-tuning

control.

ZAVJALOV, Y., S., KVASOV, P., I., MIROSPICENKO, v., L.: Methods of spline functions. Nauka, Moscow, 1980. (In russian).

442