Control of Distributed Parameter Systems as Lumped Input and Distributed Output Systems

Copyright © IFAC System Structure and Control, Nantes, France, 1998

CONTROL OF DISTRIBUTED PARAMETER SYSTEMS AS LUMPED INPUT AND DISTRIBUTED OUTPUT SYSTEMS

G. Hulk6, C. Belavj, J. Belansky, A. Heugerova, M. Lavrinc

Department ofAutomation and Measurement, Faculty of Mechanical Engineering, Slovak University o/Technology, Nam. Slobody 17,81231 Bratislova, Slovak Republic. E-mail:[email protected]

Abstract: In this paper a new decomposition of distributed parameter systems to lumped and distributed parameter subsystems in a framework of a whole structure of lumped input and distributed output system is presented. This new structure offers large possibilities for distributed parameter systems control - practically equivalent to lumped parameter systems control possibilities. Copyright © 1998 IFAC Keywords: distributed parameter systems, lumped input and distributed output systems, transfer functions, control.

I . INTRODUCTION

of a human body have been chosen (Hulk6 and MikuleclcY, 1984).

In general ~istributed £arameter fu'stems (DPS) are systems whose state or output variables/quantities are distributed quantities or fields of quantities.

In general it was found . that fats. sacharides. albumen are transferred in the blood by arteries and/or veins as lumped quantities into the particular human body organs. They are further decomposed by various anatomic structures and distributed input quantities are produced on inputs of organs' parts where processes of the metabolism are realized. State quantities of these processes have the time and space distributed character. e.g. see liver complex. Fig. 2.

In control theory these systems are frequently considered as systems whose dynamics is described by £artial ~ifferential Equations (PDE) , (Lions, 1971). In the input-output relation PDE define ~istributed input and ~istributed output fu'stems (DDS) between distributed input and distributed output quantities at initial and boundary conditions given. Fig. 1.

U(~I _D_DS-----'I ~~) L....-

Fig. 1 Distributed input and distributed output system It is well-known. that formulation and solution of real distributed parameter systems control tasks in engineering practice strike against serious obstacles. This forced us to study of real DPS internal structures in detail. As a research object real distributed systems

Fig. 2. Liver excretion process: H- liver, LC- liver cellular structure, BC- bile canalicus, S-sinusoid,

781

VP- portal vein, VF- gall-bladder, AH- arteria hepatica, VH- hepatic vein, DCH- choledochus,

{UK; (t) LI

n-

2. BASIC CHARACTERlSTICS OF LDS DYNAMICS

contrast material flows, lumped Let us consider LDS distributed on the interval , as shown in Fig. 5. Lumped discrete input quantity Uj(k) enters through the zero-order hold Hj as Uj(t) into the LDS block. The output of the system will be in the form of distributed quantity Yj(x,t). At a point Xi it will be Yj(Xj,t) or Yj(x;,k) respectively assuming unit sampling period. Discrete transfer function between Uj(k) and Yj(xj,k) is denoted as SH;(xj,z).

input quantities, UK{x, t) - contrast material concentration on the input of excretion process, YK(x, t) - contrast material concentration on the output

of

excretion

process,

{GK; };=I.D -

anatomic bodies, distributed input quantities generators, x -{x" X2, X3} Such systems considered in input-output relation are 1,umped input and lListributed output fu'stems (LDS), (Hulk6, 1987, 1989. 1998), Fig.3.

When the unit input quantity is used, in the output of the system distributed transient response -;»I ; (x, k) is got. Its reduced form in the steady-state is expressed as:

(I)

Fig. 3. Lumped input and distributed output system This procedure enables to introduce characteristics for all lumped input quantities of the studied system:

Similar structure of distributed parameter systems can be found also in the engineering practice. Raw materials, energy and other working materials are transmitted to various equipments using pipings, conduits and electrical lines as lumped parameter quantities, further transformed to distributed quantities. These enter into parts of equipments where technological or working processes take place by interaction of fields of quantities. In general such kinds of technical objects - real DPS - can be supposed having the structure according Fig. 4.

(2) { -;:m:R. 1 (x, co)}.l=l ,n

(3) y i( ....,

y

Fig. 4. Lumped input and distributed output system structure: {SAb)}j=l,n - transfer functions of actuating gears of lumped quantities {SAj};=I,n , {SGj(~,S)};=l,n - transfer functions of generators of distributed input quantities {GUj}i=l,n, S(x,~,s)- transfer function of distributed input and distributed output system

Fig. 5. Lumped input and distributed output system with zero-order hold Hj on the input

3. DISTRlBUTED PARAMETER SYSTEM OF CONTROL For LDS control synthesis, let us introduce distributed parameter feedback control loop, Fig.6. Let the goal of control is to ensure the steady-state control deviation to be minimal, i.e. :

LDS in this structure offers large possibilities for solving tasks of modelling, control and design of distributed parameter systems in the engineering practice. In this paper first some basic characteristics of LDS dynamics will be presented, further possibilities of distributed parameter systems control will be described. 782

minllE( x, 00)11 = minltw( x, 00) - Y( x, 00)11 =

(4)

= Itw( x, 00) - Y( x, 00)11 =IIE( x, 00)11

(5)

controllers is done for one-parameter control loops

where 11.11 is a nonn appropriately chosen.

{SH;(x;,z), R;(z)};, Fig.9.

U(lt)

U(k)

cs

R.(z)

~

~

Il((z)

~

~ TS

R.(z)

F.(It)

,. E,,(It)

Fig. 8. Time synthesis block TS

Fig. 6. Distributed parameter feedback control loop: HLDS- LDS system with zero-order holds {HiL on .the input, CS- control system, TS- control synthesis in time domain, SS- control synthesis in space domain, K- sampling, Y(x,t)distributed controlled quantity, W(x,k)- control quantity, V(x,t)- disturbance quantity, E(x,k)control deviation

SHi(Xi,z)

Ui(k) R;(z)

Fig. 9. i-th one-parameter control loop

During the control process an approximation problem (6) in the block of the ~pace fu'nthesis (SS) is solved in the set of reduced steady distributed transient ~esponses {-:»IR; (x, co i=l,n, Fig. 7.

During the control process for k -+ 00 we get the relation

H;,

~~IIE{ x, k) - ~ E;{k)~HR; (x, 00 = IIE{X,k)-

~

il

~~IIE( x, 00) - ~ E j (00)-;mRj (x, 00 )11 = IIE( x, 00 )11 (7)

= thus the control task (4), (5) is accomplished.

(6)

~E; {k)-:»IR; (x, co)11

Control synthesis technique shows, that for the attainment of required control quality in space domain it is possible to use results of approximation theory.

E(x,k)

E

To design of one-parameter control loops for controllers {R;(z)}; tuning there are results of the control theory of lumped systems at disposal. Thus, for achievement control quality requested in time domain it is necessary to select appropriate control synthesis methods. x

o

4. DISTRIBUTED PARAMETER SYSTEMS CONTROL PROCESS

Fig. 7. Approximation problem solution

input of controllers {R;(z)}; and the sequence of

In previous part we have analyzed the basic functions of distributed parameter control system. For solving various control tasks of distributed parameter systems it is possible to use similar schemes. Further control processes for some typical tasks will be presented, where the notation of figures represents: a) distributed control quantity W(x,t), b) - distributed output quantity Y(x,t), c) - actuating quantities Ui(k), d) - quadratic norm of distributed control deviation

actuating quantities D(k) is generated. Tuning of

IIE{k}ll·

Further the control deviations vector {E;(k)}; enters into the lime Synthesis block (TS), Fig. 8. Here vector components

{E; (k)};

are fed through the

783

4.1 Control of smart material structure string deviation The string with smart material structure is considered as real LDS, Fig. 10.

11~~11~~

t

i u,

rrr!,[Jo, "

~~

a)

IIIIIII/'

t

"

I : I In

I

u,

u.

b)

, .......: ....... : ..............................

': :::::::t:::: ::t::::::::::::::l:::::~:

~~:~~~

Fig. 10. Oscillating string with smart structure

..

.

....... ~ ...... ~ ..

Exciting subsystem Ti includes both blocks SAj and GUj, Fig. 4. Dynamics of these blocks represent transfer functions {SAj(s) L, {SGj(s) L and shaping units of distributed input quantities {Tj(x) L. Thus, the dynamics T j is expressed as transfer function SA j(s) SG j(s) Tj(x) . The forms of shaping units of

..... i

.

.~ .. .

c)

d)

Fig. 12. Control process of string deviation

distributed input quantities in space domain are shown in Fig. 11 .

4.2 Control of smart material structure string deviation

T(x) ..

I

J

T,(x) T, (x)

1/\

The metal plate with heating units is real LDS , Fig. 13. Dynamics of heating units is given by transfer

Cfs::)

functions {SAj(s) SGj(s) Tj(x,y)}j' where shaping

x

L

units of distributed input quantities have following forms : TJ(x,y) - constant form, T 2(x,y) - conic form, T3(X,y) - sinusoidal form, T4 (x,y) - constant form, Ts(x,y) - conic form in specified subareas {nJ j'

Fig. 11 . Layouts and forms of shaping units The string dynamics is described by distributed transfer function (Butkovskij, 1982): . n1tx . n1t~

2 -

SlO--SlO--

S(x,~,s)=- L L

n=1

S2

y

L 2 ~ 2 n 1t a +2as+ 2 L

Distributed transfer function

S(x , ~, s)

(8)

x

due to a

boundary value problem:

U( x, t)

(9)

Fig. 13. Metal plate with heating units (10)

y(o, t)=gJ(t), 0::; x ::; L,

t 2!

Y(L,t)=g2(t) 0,

a :;t 0

Metal plate dynamics as DDS is given in the form of boundary value problem due to parabolic type PDE in an arean E E2 :

(11)

ay at

d - - V(cVY)+aY= f Mathematical model and dynamic characteristics necessary are determined from lumped and distributed transfer functions . Control synthesis in time domain is realized using results of lumped parameter systems algebraic theory.

(12)

with constants d, c, a and Neumann type boundary conditions ii(cVY) +qY = g

Typical results of string deviation control process are presented in Fig. 12.

where 784

(13)

ii is the outward unit normal and q = 0, g = O.

Mathematical model and dynamic characteristics necessary are determined using the Einite ~lement Method (FEM). Control synthesis in time domain is based on results of the state-space theory of lumped parameter systems. Typical results of the control process are presented in Fig. 14.

(14)

with parameters d, c, a. In fixation points of the membrane Dirichlet boundary conditions, otherwise Neumann ones are to be considered. Mathematical model and dynamic characteristics necessary are determined using the FEM. Control synthesis in time domain is realized using PSD controllers.

"I····

;'..0 .

t ;c .....

Typical results of the control processes are shown in Fig. 16. a)

b)

,:1

.

.. .

~ ~:t j.~ .·.~.;:~:~ ;. ·.;:~J\,,~~...... ~t~ ~.:l

.:j~~:!:=:~::q:::~ .1 -2

c)

d)

, Fig. 14. Control process of metal plate temperature field

b) . .... •...

........ .:[jj[]

2.5 2 ··

l:;~

4.3 Control of smart material structure string deviation

.0.50

'0

20

,.,. 30

c)

Fixed membrane having smart material structure is real LDS , Fig. 15. The left side the membrane is fixed . Dynamics of excitation subsystems including distributed input quantities shaping units in specified subareas {n; are represented by transfer

..

<&0

50

: eo

d)

Fig. 16. Control process of fixed membrane deviation

l

functions : {SA;(s) SG j(s) T;(x,y)};.

a) ..

4.4 Robust control of hot-air model temperature field

{1;(x, y)};=1.3

have constant forms .

Physical model of continual technological process hot-air model - can be considered as LDS , Fig. 17. This model consist of cast iron tube having internal diameter 20 mm and active length 600 mm. In this tube four halogen lamps serving as heat sources (voltage supply {UJiZl-4 = 0-10 V dc, input power

1• 11.•' •

:: ...:...:..: .•.: ..1···1···[····1.··•.••

25 W) are displaced. For the measurement of the thermal field temperatures eleven thermistors are displaced in the model. Hot air flow is ensured by electric ventilator V. All elements of the model are realized in the form of exchangeable modules allowing easy displacement in the model or an exchange as well (Rohal'-Il'kiv, et al., 1984).

02

.02

n with smart

Dynamic characteristics necessary for control synthesis are obtained directly from distributed hotair model. Control synthesis in time domain is based on the !MC structure control loop of robust control.

Fixed membrane dynamics is given as a boundary value problem due to hyperbolic PDE in the area n .

Typical results of control process are presented in Fig. 18.

Fig. 15. Fixed membrane of given form material structure

785

5. CONCLUSION This paper is dedicated to the original method of modelling and control of distributed parameter systems. DPS are interpreted as lumped input and distributed output systems. Dynamics of such systems is decomposed to space and time components. This representation enables to divide of control task to the time domain and space domain. Control synthesis in space domain is solved as an approximation problem. In time domain actuating quantities are generated in one-parameter control loops and there is possible to use control synthesis methods of lumped systems. Examples of control of DPS are given to illustrate the using approach.

Fig. 17. Scheme of hot-air system

ACKNOWLEDGEMENT

a)

This work has been carried out under the financial support of the VEGA, Slovak Republic project "Development of Modern Automatic Control Methods" (grant 95/5196/198) and the European Union granted COPERNICUS project ,,Adaptive and Predictive Control with Physical Constraints" (grant CP 941174).

b)

", ~-~-~--

'"

i~ lt~ eo:

f_

t200

tIDC

c)

o

eoo

T_:lOO

'tIOt

REFERENCES

d)

Butkovskij, A. G. (1982). Green 's Functions and Transfer Functions Handbook. Ellis Horwood Limited, Publishers Chichester. Hulk6, G., MikuleckY, M. (1984). Distrtibuted parameter model of liver dye excretion.

Fig. 18. Robust control of temperature field

4.5 Adaptive control of hot-air model temperature field

Preprints of the 1st International Symposium on Mathematical Modelling of Liver Dye Excretion. Bratislava - Smolenice.

Dynamic characteristics necessary for control synthesis are determined directly on the real distributed system during the adaptive control process. In time domain lumped parameter system LQG synthesis method are used. Typical results of the control process are shown in Fig. 19.

Hulk6, G. (1987). Distributed Parameter Systems Control by means of Multi - Input and Multi Distributed - Output Systems I., 11. Preprints of

the IMACS I IFAC Symposium "Distributed Parameter Systems '87, Hiroshima. Hulk6, G. (1989). Control of Lumped Input and Distributed Output Systems. Preprints of the 5th

IFAClIMACSIIFIP Symposium on Control of Distributed Parameter Systems, Perpignan. Hulk6, G. et al. (1998). Modeling, Control and Design of Distributed Parameter Systems with Demonstration in MATLAB Enviroment. a)

Publishing House ofSTU, Bratislava. Lions, J.L. (1971). Optimal control of systems governed by partial differential equations. Springer - Verlag. Rohaf - Ifkiv, B. et al. (1994). Experimental Systems for Distributed Process Control Education,

b)

"r - - - - - - - - - ,

o~

5CO

1000

1500

lOCO

Preprints of the 3rd International symposium on Advances in education of automatic control.

T~

c)

d)

Tokyo. Fig. 19. Adaptive control of temperature field using LQG synthesis 786