Design of a Feedback Controller for a Cryogenic Windtunnel

Copn'ighl © IFAC IlI lh Trielllli,t1 Irorld Con g rcss , 1'.lullich. FRG. 19H7

DESIGN OF A FEEDBACK CONTROLLER FOR A CRYOGENIC WINDTUNNEL R. Steinhauser f)F\ ' U~ - /I/, ! i ! lI ! j li l /)\'11 (11111/;

r/I'I F!lIg ",.I!I' III1'. O/}(·rjJfall i- lI/wji- lI . D-80]l lL',.,/ill g . FRG

Abstract: A controller design is presented for a cryogenic windtunnel. The nonlinear mUlti-input mUlti-output model with state dependent time delays in input variables is first simplified: For the simplified model a suitable choice of controller structure is straightforward. Values for the controller coefficients are first obtained by pole assignment and are then taken as initial values for several controller designs using optimization of a vector performance index. The problem of obtaining satisfactory performance with a simplest possible controller structure (e.g. using fixed gain PI-control) is investigated. All control designs are evaluated by nonlinear simulations. Keywords: Multivariable control system, robust control, multicriteria design, parameter optimization based design, cryogenic windtunnel control.

INTRODUCTION

The controller design is based on a linearized multivariable model where two input variables have dead times. Since the control specifications have to be performed over a wide operating region several linearized system data sets are included in the design: The control problem is treated as a so called (linear) multimodel problem. For it the robustnes s aspect is important. In a first stage of the design a suitable controller structure is chosen and initial values for the controller coefficients are determined. The parameter values are then change d by means of a design procedure based on parameter optimization of a vector valued performance index (Kreisselmeier, Steinhauser, 1979). The method allows many different design specifications to be included directly in the design while proceeding it e ratively in a very systematic way. In three designs controller simplifications are pursued (structure reduction, no gain scheduling=robust controller). All design results are checked and compared in nonlinear system simulations.

Aerodynamic simulations in a windtunnel for subsonic speed give correct real world behaviour if the similarity law of Reynolds holds. That means the Reynoldsnumber in the windtunnel must be the Reynolds-number in reality. The Reynolds-number Re (w'l) / v is the product of flow-velocity wand a geometric measure 1 divided by the dynamic viscosity of the flow v. The current me thod of increas ing the Reynolds-number in the wind tunnel involve increasing w by using a higher number of fan revolutions. Obviously the techn ical 1 imi t is imposed by the fan-performance. This limit prevent the Reynolds-number from being increased to a sufficiently high value. However the dynamic viscosity of the gas flow medium depends on its temperature. For gaseous nitrogen, v decreases with decreasing temperature. So -for maximal poss ible wand 1- it is poss ible to f urther increase the Reynolds-number by injecting liquid nitrogen into the tunnel. This new method of increasing the Reynolds-number is being used in a DFVLR windtunnel at Cologne (Vieweger, 1983).

CONTROL PROBLEM

According to a higher complexity of the cryogenic windtunnel type the installation of a control system is necessary for guaranteing accurate a n d fast tunnel operation. This paper deals with the design of a feedback controller for the cryogenic wind tunnel of Cologne. After some background information about the cryogenic windtunnel realization the windtunnel model is introduced and the control problem is formulated.

The new method of increasing the Reynolds number (by using nitrogen as the gas flow medium), is being used in a DFVLR windtunnel at Cologne. For this purpose 2 tanks have been installed with liquid nitrogen which is injected into the tunnel by several valves. However if mass is injected into the tunnel the static pressure of the flow changes. This creates the danger of damaging the insulating wall. In order to allow the pressure deviation from the environmental level to be kept small additionally a gas injection system and a gas

39

40

R. Sleinhallse r

exhaust system fig. 1).

have

been

installed

(s.

For the cryogenic wind-tunnel of Cologne we have now four input variables to influence three state variables in the test-section, namely temperature T, static pressure P and flow velocity w. The control variables are the fan angular velocity NR which was previo~sl¥ u~ed alone and tlle three mass flows mL , mc;+' mG _ (liquid nitrogen-, gaseous nitrogen Injection, gaseous nitrogen exhaustion). Since there is now the possibility of changing the flow velocity and the gas temperature the cryogenic windtunnel of Cologne has a two dimensional operating region. The temperature can be cooled down to 100 Kelvin and the flow velocity can be increased up to 100 m/so Several operating phases are distinguished. One is the cool-down phase where the fan is idle and the temperature is decreased to a desired value. In the run-up phase the flow veloc i ty is increased to a desired value and decreased in the rundown phase. In the testing phase the aerodynamic measurements for the aeroplane model in the test-section are performed (s. fig. 2). As already mentioned the static pressure must always be kept near the environmental level. But also for the two remaining variables T and w hard tolerance demands are specified for all operating phases. The multivariable control problem is now to keep T, P and w always within prescribed tolerances without exceeding physically given limits for the control variables and their derivatives (s. table 1). CONTROLLER DESIGN The starting point for the controller design is a multivariable strongly coupled nonlinear wind tunnel model which was derived essentially by thermo-fluidmechanical balance principles. (This model will not be discussed here, but for more details see (Palancz, Kronen, 1982; Kraft, 1986). The mass flow control variables have flow velocity dependent dead times ('l(w), '2(w)) which are caused by the time delay of the mass transport from the valves to the test section. For a nonlinear mUlti-input multi-output system with state dependent dead times in input variables no suitable control design procedures exist. In such situations it is first necessary to obtain a simplified model before an existing design procedure can be applied.

problem was reformulated as a so called linear multimodel problem: That is a controller is sought which gives satisfactory system performance at several (here five) operating points which are indicated in fig. 2. An analysis of the linearized models led to a model decomposition. The model was decomposed into two parts, a simple SISO 'flow-velocity plant' without dead time and a MIMO 'temperature-pressure-plant' with operating point dependent dead time in all input variables. Compared to the 'temperature-pressure plant' the controller design for the 'flow-velocity plant' is a relatively easy task and will not pursued here (but see (Steinhauser, 1986). One of the main objects for the controller design was to avoid complexity of the control system. For instance switching to different controllers for different operating phases was not desirable. The identification of the run-up phase as the most critical operating phase at all for the controller design led to the following multimodell regulator-problem formulati o n for the 'temperature-pressure-plant' (details are to be found in (Ste inhauser, 1986)): For the system

~(t)

x(

t)

a multimodel regulator is sought. That me ans for arbitrary initial conditions ~eo' y(t) must go to zero (y(t + "') + 0) und e r the restriction closed loop system at ing poi!}ts. In (1) vector x and the ~e =

-e

.•

.

T

(mL,mG+,m G_),

-Ae

all

a 12

0

0

0

a 21

a 22

0

0

a 24

a

a 32

0

34

a 44 '

31

0

0

b

b

0

0

0

b

0

b32 b 33 '

22 0

Q

MODEL SIMPLIFICATIONS

According to the wide operating region of the cryogenic windtunnel it is not sufficient to do the controllpr design only for a single operating point and so the

,~=

where

" the d1sturbance vector, ~ = [ NR,w )T s 1S and Tm is the temperature o f the metal p arts in the windtunnel. Th e matrices are as follows (s. table 2 for data) :

~ 2e

The model simplifications were done in several steps an overview of which is given in fig. 3. In a first step the model was linearized.

T -

(Tm,T,P,~)

of stability of the the 5 chosen operatthe extended _state input vector ~ are

0

0

44

a 44

:e1e

0

0

0

b

21

0

0

b

31

0

0

~.(

: : 1

0

0

1

0

0

)

Furthermore the 'temperature-pressureplants' were discretized. The discretization inter v al TD was chos e n flow-velocity dependent TD(W) = const / w. By means of this variable discretization interval the operating point dependent time delays

,~i)

and

,~i)

of

the continous model des-

cription are transformed to an number of shift operators (6 for for '2) for all operating points.

equal '1' 3

Design

ur a Feedback COIlI mller

Physical and control theoretical considerations led to a simple order reduction - for the controller design the very slow dynamics of the metal temperature T could be neglected. The final resul t 0'£ all the model simplifications is represented by the system structure of fig. 4 with the respective data of table 3.

CONTROLLER STRUCTURE After the model simplifications of the last section the choice of a suitable controller structure was straightforward (for the following compare fig. 5). Using gain scheduling in all three input var iables (VT,Vp+'V _) allowed the parameter variations of 'lhe multimodel problem to be drastically reduced. Gaseous nitrogen should not be injected and exhausted at the same time. This is why it is switched when the input signal m +_ changes sign. G Using simple feedforward decoupling (kKl,kK2,kK3,VKP) the MIMO system was ap-

41

design spec ifications, free controller coefficients in a given control structure are determined in such a way that a systematic improvement of the result in each iteration step is guaranteed, even if a great many of design specifications have to be considered. Here only an overview of the iteration scheme is presented. (More details about the design procedure and its application in different complex control problems can be found e.g. in (Kreisselmeier, Steinhauser, 1979; Kreisselmeier, Steinhauser, 1983; Sander, Steinhauser, 1984; GrUbel, Joos, Kaesbauer , Hillgren, 1984».

For' a yiven mathemat ical description of the plant

r----

-" .:1)

ch oose a CO lltr ol ler struc tur e and initial

l:ontl-oller coefficient s ~j =

I

I .

1 III

proximately decomposed into two SISO systems, a 'tempera tur e -plant' and a 'pres sure p lan t'. For each of these two systems a state feedback structure with additional PID resp. PI controller dynamics was incorporated (kTl-kT9,kpl-kp5)'

I-- - -. bl

1::. j

1

I J;j +l

(ol."mulate the de s ired system behaviour mathe-

I

m.J.tic ally by a vector pe rformance index

I

C;

=

(G I , · · · , GLl ;

I II

v

I =

Gi

=

1

par.m~ter

, -_ _ _ Cj

choose the

Gi
vec tor SOj v

in a

systcmath: ma nn e r

I

CONTRO LLER SYNTHESIS An analysis of the system parameter variations subject to the operating point (i. e . wand T) led to the following ana lytical interpolation formulas for gain scheduling:

uj

01

.j. optimizat i on ( m~no. :

Vp+(w)

Vp_(w) .

The coefficients for the decoupling structure were determined heuristically. Essentially an exact feedforward decoupling from the 'temperature-' to the 'pressurep lant' (where now the above gain scheduled coeffic i ents where included in the model) at operating point 5 was chosen. According to the different time delays T , T2 only a '(T I -T 2 ) - retarted' gain I scheduled decoupl i ng from the 'press ure'to the 't emperature -plant' (V KP = 0.0005' (300. O-T) ) was in troduced. The sys tern could now be regarded as perfectly decoupled at operating point 5. For it a pole placement routine was applied for determination of the remaining controller coefficients.

CONTROL DESIGN BY OPT1M IZING A VECTOR PERFORMANCE INDEX By optimizing a vector performance index, the components of which rate diffe r en t

0.

=- ffi1x

(Gi/cij>j

I

jutJye system behaviour by system a nalys is and - s illlulcl t ion

VT(w,T)= VT(w) • VT(T) , VT(w) = 1+0.105(w-lO.0), VT(T) =1.5«(300.0-T) /100 .0-0.2)2+ 0 • 94 ), Vp _(w,T)= v p_ ( w) • V p_ ( T) , Vp_(w) I + 0 • I ( w-l 0 • 0) , 300 .0/ T, Vp_(T)

< Ej v < ~j v -I

~(1:j v-I)

v.:tv +l

1

yes

result ok ?

+

END

Remarks : The des ign procedure is not constrained to a linear plant and/or to a linear controller . - Each performance criterion has to be chosen such th at its value is the less the better the respective specification is satisfied . - In the inn e r l oop (I) it is a lways guaranteed:

v

~(.J<:j)

v

~ .£j

v -I

< ::j

<

1

... <::j

That means the des ign cr iter ia a re improved s t ep by step very systematica ll y . After an opt imizati on run no criteria value G· does exceed its given parameter value c t j ! This quality allows the design e r t o proceed in his desired design direction. - The outer loop (11) means that the last formulation of the des ir ed system behaviour by the respective perfor mance criteria can be changed, for instance by remo v ing some criteria which were id e ntified as conflicting ones or by adding c rit e ria for a more precise formulation of the desired sys tem behav i our. In the outer loop (Ill) the controller s truc ture ca n be changed, for ins tance by an ext e nsi on of th e last chosen struc ture. Note that entering loop (11) or (Ill) does n o t necessarily mean that improve me n ts ach ieved in loop (I) are los t.

42

R. Steinhauser

The design can be continued with the last design result! Four different controller designs have been performed, which should lead to answers of the following questions: Up to what controller simplifications a satisfying system behaviour can be guarenteed? Is it possible to avoid gain scheduling? To what extent can the control performance be improved? Investigation of such problems can not be done (systematically) by any synthesis procedures as for instance by pole placement. Here the application of a controller design procedure based on parameter optimization of a vector performance index is a powerful tool. For the wind-tunnel controller design step a) was described in the two sections before. For step b) the following types of performance criteria were formulated: type 1: [V (3)

-

x

V ( i )]2

x

,

2

i=!, 2,4,5,

x =T, P+, p-

type 2: k xj

j £{ 1,2,3,4,5,6,9 } , x=T,P

typ e 3: m~x Ak i

i =1,4,S,

l =l, ... ,n

i =1 ,2 ,S ,

y =T,P

1

type 4:

.

L y1(k k= m

o

T ) / (1-m)T A

A

(i is the number of the operating point, j is the number of the controller coeffic ient, n is the system oder, A k represents an eigenvalue, TA is the discretization interval). The criterion type 1 is for formulating the robustness demand. Reduction of all criteria values to zero (for x=T,P+,P- and i=1-5) means that no gain scheduling furthermore is necessary. Reducing a criterion of typ 2 to a very small value means that the respective controller coefficient can be neglected. Keeping the values of the criteria of type 3 less than 1 guarantees stab il i ty of the closed loop system. (For the stability over the complete operating region it proved to be sufficient if the linear system was stable at operating point 1,4 and 5). Concerning a criterion of type 4 it is clear that the less its value the better the respective variable (T or P) is regulated in the time interval [mT A , ITA]. Remember that a multimodel problem was given and therefore any of the above criterion had to be included for several operating points (as indicated above). So up to 18 performance criteria were used during the iteration steps. A detailed description of the proceeding for all 4 controller designs can be found in (Steinhauser, 1986).

RESULTS The controller simplifications (controller 1,2,3 of table 4) produce a more or less performance reduction in comparison with the initial controller 0 of section 'controller synthesis'. The linear simulations without gain scheduling (controller 1 resp. 2) especially show a sluggish temperature behaviour (s. fig. 6a,6b). But note that the closed loop system is stable at all 5 operating points (robust

control). Simulations with a PI-controller structure but where additional gain scheduling is used (controller 3) proves that control without gain scheduling here detoriates the control performance seriously more than an essential simplification of only the controller structure does (compare fig. 6c with 6a,6b). This is confirmed by nonlinear simulations of the operating phases run-up, testing, run-down, especially at the most critical temperature of 100 K (s. fig. 7). For the robust controller 1 and 2 the system behaviour is satisfactory at most in the region of ambient temperature. However controller 3 satisfies all tolerance specifications given in table 1. (Results of improving the system performance using the (unreduced) controller structure of fig. 5 can be found in (Steinhauser, 1986) ) •

CONCLUSIONS For a multivariable cryogenic windtunnel with flow velocity dependent dead times in input variables the design of several feedback control schemes was described. After several model simplifications (linearization, multimodel problem, discretization, order reduction) the choice for a suitable controller structure was straightforward (gain scheduling, approximate decoupling, state feedback, integral action). The controller coefficients first were determined essentially by pole placement and than used for four controller designs based on optimizing a vector performance index. The possibility of controller simplifications (robust control, simple PI-control) were investigated. The resulting controllers were checked by nonlinear simulations.

REFERENCES GrUbel, G., and others (1984) Robust Back-up Stabilization for Artificial Aircraft. Proceedings of 14th ICAS Congress, 1085-1095. Kraft, D., (1986) Optimalsteuerungen ein systematisches Hilfsmittel zur rechnergestUtzten Erfot:'schung der dynamischen Moglichkeiten eines Tieftemperaturwindkanals. DFVLR-FB 86-23, or: Optimal Control a Systematic Device for the Computer-Aided Explot:'ation of the Dynamical Feasibilities of a Cryogenic Wind Tunnel, ESA-TT1016. Kreisselmeier, G., Steinhauset:', R., (1979) Systematische Auslegung von Reglern durch Optimierung eines vektoriellen GUtekritet:'iums. Regelungstechnik 27, 3, 76-79, or: Systematic Control Design by Optimizing a Vector Performance Index. Proceedings of IFAC Symposium on Computer Aided Design of Control Systems, ZUt:'ich, 113-117.

43

Design of a Feedback COlllrolle r

1 ---- - - - - - - - - - ;

Kreisselmeier, G., Steinhauser, R., (1984) Application of Vector Performance Optimization to a Robust Control Loop Design for a Fighter Aircraft. Int. J. Contr., 37 251-284.

I

palancz, B., Kronen, R., (1982) Modeling, Simulation and Control of a Cryogenic Windtunnel, Part I: Model and Simulation of the Dynamic Behaviour (in German). DFVLR-Institutsbericht, WKT 21/82.

I I

1,

I I I I IL _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ JI

Sander, N., Steinhauser, R., (1984) Design of a Robust Missile-Controller by Vector Performance Optimization (in German). DFVLR-FB 84-05. Steinhauser, R., (1986) Reglerentwurf fiir einen Tieftempertur-Windkanal mittels Giitevektoroptimierung. DFVLR-FB 8537, or: Design of a Control System for a Cryogenic Wind Tunnel Optimimizing a Vector Performance Index, ESATT-965.

0',

:

Fig. 4

simplified windtunnel system structure

Fig. 5

controller structure

Viehweger, G., (1983) The Cryogenic Wind Tunnel Cologne. AGARD CP 348, 4-1 4-8.

~eous

nitrogen

Fig. 1 the DFVLR c r yo g e nic windtunnel of Cologne T 1

1

300 K

•

"-

2 00

3

" -, .. ..

,

100

so

10

Fig. 2

/

r un-up - - r un-down x I esling

r 70

rn ls

100

I

cool-down

!

warm-up w

o perating re g i o n and operating phas e s

I 5 ''f\pj'!,~ot'ons I

y

~~

Fig. 6a

~ .= 1, ... •5

ord~r

Fig. 3

l eoucllon

model simplifications

1 = controller structure 0, no gain scheduling (12 = results after 12 iteration steps)

44

R. Steillhauscr ,"," , ,",,J tol~,·.,! ",J ,I."" i a l ions ( r Ofll refe r e nce va l ue3 ('".,_", v .. ltJ" s ) Ol>" ~ ~ ling

p h ..

"c

&T{ II )

,

s t a nd- b y

, .. ,

t el tlng

I o.~ )

i' ]

"

.. ctuat o t

l im Lt

,

u. ~9 /~ u. kg/"

, ""G-

,,

(O.O'»

, '- ,

~U"-\'P

,"un -
I -I

"

,, ,,

'-'

, ,

(0.5)

,, ,,

,.

..,

"'C • '

tate

~oO

.,-

k
.,.

, ,

,

1<.9/ &

,.

"

-,

H "Ht.

Ch i n.,. "

O. lH kg /II

"

,,

kg / a

• l'Iin

!:!..~-1 / .

Table 1 Design Specifications (L Mode, R=Reference Variable)

'"' r?

!ope .... tin g

a ll

1

Fig. 6b

2 = PI controller, no gain scheduling ,,~-. ",

1

1---0. 00 012

a l2

0 . 000 12

:::

0. 0 0 088

-::::::: 1-::::::::

o . l)~n I -O . 1~b9

'H

I

'"

i

6.bO~& 1

,::1:'y:.~jf~Ft- L~•.~ ,

U . OS 79 1 H

! -u.Jl27 U O.~UbU l

2.)1 4 14

1 _0

1--_'-",, +-_C-+-.:.'-,S7 627

-; l H'! i

O.OO O ~ U

(1.00 2 6 1

'"

1

Idl ,

l -l .~ 80H 9.H4I ~

I

O O~ , BH I - o~

f

211Jl

0 ll 'J-;~- l----o--;;;;-;'J

bH

I O . ~659 1

----;-;;-~ro . 07 H~ I

I--_''-''',-+C-=,-+_-':...:.jj~~~ 0~~7_H~\~~7_~

~ ' S :i:~ · l:.r-rr:[)~ !

1,) 11

,_.-

i>J j

"u i> ]2

I>JJ

oaSu7S - 0 717~7 0

-0 61219

0

0001 2

O OlH ijU

~O_1~b i>_7

1-0 "llblb6

1001.1661

00

-0

O!J ~ :

!

~~J_

OIlI.1} 1 2l CIl L/ UH

1~~BH6l~HH--;:--1~~ so~-I-l-1~~6~lb-ll~ O~J711 2

-2 I , U36 11 J

'1

I _ I UlD

-0

7I q "l~

- ' - - - - --

I

71.1

I

' 2

2

I, UU ~

-2

}.H

141&

I

' . 41

-u IJlUS

-1-

I

71J

10 H

31 . 1

Table 2 System Data of the Continous Linear Model, Operating Point 1-5

' ::~~~Fnn+-~~~

:·[J-.~_· t-l-f:~_ ; _~~ ~L;;~~~J,

,

oper
""

""

,---

'~It,t~}!O,-~-~ij , ~J

c..<

:.,

"'I

!l' , ~ ..

6c

Fig. 6

3 = PI controller, gain scheduling linear simulations at operating point 1-5 o = controller by pole assignment

~5

I 1

~1. l2~

I

. 0 '1 0 -

1"" ":1,

0 . 09 4

:2 05 65

I

I

ibj2

! _ 7 . 0 '1 0 - 5

_2.4.1 0 - 4

_ 2. 0 .1 0 -'>

L l ' l O- '

~ . 1 . 10- 5

i

!1 -2. 0 '1 0 - 5

b~l I

Ib~

5

0.':163

I

'H

I "~1 \

Fig.

_7. 0 . .\. 0 - 5

2

0 . 9&4

-2.5 6 5

I I

).4 0 )

5 . 070 - ],42 0

Table 3 System Data for the Discretized 'Temperature-Pressure-Plant'

~ II ::~

I

k T2

!

1 . 3276

i

- 2 . 9062. 0670

'n

- 1.0 6 44 1

II

i

-0 . 051

(1.1 9 5) 0 . 1)152

I kT 8

1-0.00 2

k T9

! -0 . 00 1

I

!i---"----I-I kpI

!,

1 -0 . 0010 0.01.l4

-r--:--

o . 9 5 00ao

I

o . sa

---I---~------.j

~9

-0 . 6TI~

i kpl

i 0 . 500000 I I . i - 0 . 025000

1 ~N

1 - 0.00 14 00

- 0 . 00 4 6

1

- 0.0063

I " ps

-0.000122

- 0 . 000 1

I

- 0 . 000 4

k pl

!'" Ik

:1-

! '"

i

KZ

0 ,0 4 2 1

O.

I

I

IS~ ~

7

nonlinear simulations

I

!l

Table 4 Controller Coefficients o with Gain Scheduling o = no Gain Scheduling Fig.

!