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APPLICATION OF THE MNA VcSIGN METHODTO A NON-LINEAR TURBOFAN ENGINE

Or. Gary l*ininy*r

tSA -CK- 1ojd:) t)) APPLICATIUN uY THE MNA

DESIGN METt1UD 10 A NUNLISeAb TUUhUtAt+ tNuiNE

rLUal iie pott ( e'ucdut: Univ.) 1J7 p

IfL Alit,/t!r AU1 CSCL 11EUT

\\AA TI .^

CONTROLCERITER

SCHOOL OF MECHANICAL ENGINEERING

January 1981

eL^ v^

Aftwow^a ^^y7r^

^Lw' -11161- IjJU1

PUmDUE UNIVERSITY

WLST LAFAYETTE, INDIANA

1FORWARD

aThis document represents the final report to the

National Aeronautics and Space Administration for work

performed under Grant number NSG-3171. The NASA technical

monitor was Dr. Bruce Lehtinen, NASA, Lewis Research Center,

Cleveland, Ohio.

TABLE OF CONTENTS Ij

Chapter I - T^^troduction . . . . . . . . . 1 r

Chapter II -- M"Itivariable. Nyquist Array Method . . . . . 7

` A -Background and Notation . . . . . . . 8

i B - Perspectives. . . . . . . . . .1a

C - Design Procedure. . . . . . . . . . .22 r

D - Summary . . . . . . . . . . . . . . .33

Chapter III - MNA Applied to QCSEE Engine . . . . . . . .34

A - QCSEE Turbofan Engine'. . . . . . . . .35

B - Non-Linear QCSEE Simulationand Linear Models . . . . . . ..41

C - MNA 'Design . . . . . . . . . 46

References. . . . . . . . . . . .96

Appendix A - Linear operating Point Models . . . . . . .99a

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1 CHAPTER x

INTRODUCTION

To meet the performance demands of future military

and commercial aircraft, propulsion system requirements

will make necessary the design and development of sub-l

stantally more complex engine configurations. These new

propulsion systems will dictate the need for advanced multi-

variable control system design methodologies capable of

' interrelating the design goals with thv: physical :Limitations

of the turbomachiner,y. This means that although steady

state operating line conditionswiTl retain their importance

in the design specifications further emphasis will be placed

on the short term transient performance of the engine. Toi

satisfy these performance requirements, more control vari-

ables will be introduced and more output variables will be }

monitored to evaluate engine performance.

in the early years of jet aircraft propulsion, single-

' spool turbojets were used almost exclusively.. The first

control systems for these engines wre hydromechanical and

used the principle of the flyball governor extensively

for fuel-rotor speed control [1). As tubojet engine

performance demands increased, a second compressor

turbine section was added to drive a fan to remove energy

i from the primary jet airstream, thus reducing engine air-

flow velocity and increasing cycle efficiency. The addition

of this second compressor-turbine section enabled greater

flexibility of compressor performance at high discharge

to intake pressure ratios, but required the control of two

mechanically independent turboshafts. Turbofans now

power rrr.7st modern subsonic aircraft and, with afterburners,

many high performance supersonic military aircrafts as well.

The turbofan engines in use today are basically fixed-Y

geometry designs. The control system modulate: one or two ='

variables and is configured as a set of single loop controllers

which are scheduled in an open loop manner according to

K aerodynamic and environmental conditions. The majority of

` the control systems have been designed using single-input

single-output (SZSO) classical frequency response techniques.

The resulting control configurations have typically been

implemented using hydromechanical or electronic analog

components.

As aircraft turbine engines become more complex,

additional requirements for the control system will result.

operational requirements will become more stringent thus

x ^;j'

necessitating increased control system capability in terms

of computational accuracy and static and dynamic response.

As performance requirements increase, more variables must

be controlled and more engine parameters must be monitored.

SISO oontroi system design methods will no longer be

appropriate as cross-coupled system interactions must be

taken into consideration. Open-loop, trajectory-dependent

control designs will yield to closed loop design procedures

as the now-standard, analog/hydromechanical devices yield

to full authority digital electronic control packages.

Some research and development activity for turbofan

engine control systems using modern control theory has been

reported by Michael and Farrar [2,3], Stone and co-workers

[4]', Weinberg (51, and Merrill and Leininger (6]. The most

comprehensive application of modern control theory to air-

craft propulsion system design was performed by Systems Control

Inc. on the Pratt and Whitney Fl00-PW-100 turbofan engine [7-i

111. This LQR control, design was successfully tested in an

altitude chamber at NASA-LeRC by Lehtinen and co-workers

[12,13]. The results of this test demonstrate the effectiveness`

of modern control theory for highly non-linear "sophi-

sticated turbofan application.,; in addition to providing

an evaluation of digital electronic devices for turbofan

engine controlsystems.

in 1977, the r-100 sea level static condition was used

Ias the theme problem to explore alternatives for linear

multivariablo control in a symposium sponsored by the National

Engineering Consortium, The results reported in [14) demon-

strate the successfil. "pplicatlon of multivariable frequency

domain control system design methods to the F'1Qb design

problem • i, 1I 1 d

I During the period 1974-1979, the General Electric 4

Corporation, under contract to NASA, designed, developed,

and tested an experimental commercial aircraft engine [15,16).

This now Quiet Clean Shorthaul Experimental Engine (QCSEE)A

incorporates performance and structural characteristics

unlike those in any turbofan engine in production today.

' The QCSEE program introduced high bypass ratios for turbo-

machinery noise suppression, reversible pitch fan blades for

rapid thrust response, total digital electronic engine controls,

and extensive use of composites for drag reduction and weight

considerations. in addition to fuel flow and fan blade pitchG

angle, variable nozzle area 'control is also provided. To

incorporate all of -these characteristics into a single

propulsion system, represents a significant breakthrough

in turbofan engine technology.

During the period of development and testing of the

QCSEE engine, NASA developed a highly non-linear, accurate,

real-time digital simulation of the engine at sea-level

static conditions. This non-linear model was used in ex-

tensive tests at the NASA--Ames in-flight simulator facility r

for test pilot evaluations of integrated engine airframe

combinations [17]

Using the non-- linea r, digital simulation as a repro-r

sentative model of the dynamic operation of the QCSBB

turbofan engine, a feedback control system is designed in

I this report by multivariable frequency domain design tech-

niques. Transfer function matrices are generated at each

a of five power lever settings covering the ranee of operation

3 from approach power to full throttle, i. e., 62.5% to 100

full power. These transfer functions are then used by an

interactive control system design synthesis program to

provide a closed loop feedback control using the mu ltivar iable

Nyquist array (MNA) method [18-201 and the recent extensionsi

to multivariable Bode diagrams (MBD) and Nichols Charts

(MNC) [21-22]

The QCSM design is initially performed holding nozzle

area full open with fixed fan blade pitch angle at the power

setting of 62.5% of full power. This SISO design is evalu-

ated at other power settings and tested in the non-linear

simulation to evaluate engine performance during a powern

slam from 62.5% to full power (100%))

{The second phase of the MNA design uses fuel flow and

fan blade pitch angle in realizing a two input two output

feedback design, These results are extended to the three

input three output system with nozzle area as the thirdu $

input. This variable is used top provide additional control

over inlet-Mach number for noise suppression. In each case,

j the MNA design is compared with the Standard QCSEE control,

F The next chapter details the basic concepts associated

with the Multivariable Nyquist Array Method and introduces

several recent extensions to the theory. Chapter 3 then

uses this design philosophy to determine the closed loop

feedback control system for the QCSBU turbofan engine. The

successful application of the MNA method clearly demonstrates

i the effectiveness and utility of the multivariable classi-

cal frequency design procedure.

—7—

CHAPTER 1ICHAPTERr ,

MULTIVARIABLE NYQUIST ARRAY METHOD s

t The Multivariable Nyqui.st Array (MNA) method is com-

posed of two Nycjuist array design rrocedures one pro-

cedure for the direct polar. plane (ANA) and one for the

inverse polar plane '(INA) Roth methods have identical

design objectives and are founded upon a common mathematical

structure. The principal point of departure is the use and

` interpretation of the multivariable Nyquist stability

criterion in achieving the final system design.s

The INA method was first introduced in 195$ [231 based

upon the premise that compensator selection to achieve the

desired diagonal dominance condition was easier to obtain in

the inverse polar plane. In 1974, a more comprehensive j

description of the INA method became available [,291 which

also provided the forum for the introduction of the ANA method.

Until recently these design methods were considered to be

mutually exclusive in the sense that once the open loop_

design conditionswere met in a specific polar plane the

closed loop design analysis was restricted to that polar plane.

Thus an INA design could not at that time be projected into

the more favored direct polar plane. This lod Fisher at al

[251 in a comprehensive application study to indicate a

preference for the DNA method over the INA ;.iethod whereas

the INA method is used most frequently in Burope.

in this chapter the MNA design objective is briefly

reviewed. New theoretical concepts are presented which

eliminate the polar plane dependency and establish a found-

4tion for finite frequency range designs. A new graphical

design procedure is then. introduced which utilizes the

fundamental ideas of Bode and Nichols in multi-loop systems.

A. MNA Background and Rotation

For the system in Figuva la, O(s) represents an mxm

transfer matrix relating the output vector y(s) to the system

input vector U(s). The vector y,, C (s) denotes the reference

output vector. The design objective is to determine the input

vector u(s) such that the system output vector follows the

reference vector with dynam:wcs acceptable to the system

designer. if, for example, y ref(s)

represents a desired set

point operation then y(t) should achieve the desired level

within prespecified rise time, settling time, and percent

overshoot requirements.

To the Oystom of Figure la, three square compensators

are used to close the feedback loop as in Figure lb. For

reasons which will be apparent later in the development

L and F(s) are assumed to be diagonal and K is assumed to be

an arbitrary constant matrix. All matrices Are assumed to

be of rank m. Shifting the constant diagonal matrix L to

the left side of the summing junction Figure lc is obtained.

Re-arranging Figure lc and letting

zref "" L yref(1)

z ( t) = L y (t)

the standard MNA block diagram of Figure 1d is obtained.

Let r(s) equal the identity matrix and define

Q(s) = L G (s)K (2)

Following Rosenbrock (241, K and L must be selected such

that Q(s) and/or

Q(s) K G(s)L (3)

is diagonal, dominant by rows or columns for all s on the

standard Nyquist D contour.

+ e^ c)

.mac G ( s> e^ t

,FIGURE ,IA-W

OPEN Loo p SYSTEM

p K G (s

FIGURE IB; CLOSED LOOP CONFIGURATION I

F[E csa

FIGURE 1C CLOSED Loo p CONFIGURATION

F (s, G cs^ 3 fit) 3,

FIGURE ID; CLOSED LOOP CONF'I'GURATION 3

The concept of diagonal dominant matrices is well

documented (18,23,241 and is defined as follows for the

DNA methods

Definition: Let Q(s) be a complex square matrixof dimension m. Then Q(s) isdiagonal dominant if either or bothof the following conditions exist

f for all seDi

mi e Max Z lgi}(s)l/lgii(s)l < 1.0 (4)` seD j=1

Y seD ^l Iqji(s)I/jgii(s)j < 1.0 (5)

where (4) utilizes the rows of5Q (s) (row dominance) and (5)uses the columns of Q(s) (columndominance)

Clearly if Q(s) in (2) (or Q (s) in (3)) is purelyP '

diagonal, then m feedback controllers (and m loop gains)

tcan be designed independently for the uncoupled system.

To achieve this degree of independence, K and L would of

^. necessity be complicated functions of s and could resultr_

in severe implementation problems. Rosenbrock questioned

the need for a purely diagonal open loop transflr matrix - Q(s)j

i ,i,r. "" ^ •. xaaz-^ttw•+wvwt+atcwVUaw.mne+ rrm.—wv+w+J' rca.wtem amw'+.aveaw° .:^'^vo-ax`.:+J?356a_."n'-r—•..,. A.l.{ : .: lW6^G#^A...—:•-ea.°la.ml..M......sty.3.ow.rrL.r-...A.....«^-._».-.-..v _2.zs.^..„e..e.!r,Y

and developed a procedure wherein the off-diagoial entries 11

in Q(s) were allowed to be non -zero but of sufficiently }I

small moduli to permit design synthesis by single loop9

methods. By relaxing the diagonality condition of Q(s); v ^

to that of a diagonal dominant Q(s), the compensator matrices#

K and L in most cases remain constant thus significantly;

improving the implementation requirements.

Using the above definition for diagonal dominance,g

Roscabrock was able to extend the classical SISO NyquistY

Criterion to the multivariable case. This development led

to a graphical interpretation of the multi-input multi-outputT

(MIMO) Nyquist design objective where the traditional }

Nyquist curve is replaced by a band of Circles. The radiig

i of the circles (called the Gershgorin circles) consist

of the sum of the moduli of the off -diagonal terms in Q(s)

by rows or by columns. For the DNA method the Gershgorin

radii for ( 4) and ( 5) become

i mdi (s) =1 qij (s) (6)

m'di (s)Z I gj( s ) I ()

s.,. ,

Hence, the dominance levels can be written as

e^ Max d^(s) /J q, , (s) l11

8y - Max dy(s)/Igii(s)

To determine the elements of K and L, an optimization

procedure was developed by Leininger [18,19] based upon5

the dominance levels in (8) and (9),, rundamenta.ly, theE

optimization objective is to secure the lowest level o f

dominance in the respective rows (or columns) using a

constant compensator structure. This method of achieving

dominance can be used in conjunction with any of the twelve

MNA forms and thus significantly_ reduces the time required

to obtain dominance from several weeks (or months) using

previous trial and error procedures [24] to several CPU

seconds.'

Once the compensator pairs have been selected in

accordance with the above concepts, the respective control

loops can then be designed independently using classical.'

SISO design procedures. The resulting closed loop design

configuration can be further reviewed on an individual

loop basis through the utilization of Ostrowski's theorem [26]

..-«rv.,Fam}^y„^ -... ,_ :. ..—`_-•-----_• .. ..^^...•:.ny.4=^'..y:x^an _-rn*-s^^'+,.._... .. ...<._ ....+.w.a..` ^SmtK:.iIR dfa.3B$rffi:rs`-a:'**#..*..t::»°wLdwrxda.W?•.4....a-v+.cr.+»...t•an f ....: - __..,.•» _ ..«-,...,.

This theorem provides for the reduction of the Gershgorin

radii in loop i based upon the feedback configurations

in loops jai. Thus a more acourate assessment of the respective3

phase margins and gain margins oin be made.

it is useful at this point in the development to di-

gress from the formalities of the MNA design procedure to

closely examine the closed loop structure that has been

created above. This analysis will place the MNA designr

procedure in proper perspective and provide the foundation 1

for subsequent designs using multivariable Bode and Nichols

methods.

B. MNA Perspectives

In most applications of the MNA design method Q(s)

(or Q(s,)) cannot be made diagonal dominant when L is the 3 f

identity matrix and K is constrained to be a constant

matrix. The alternatives are therefore to either insert

a constant postcompensator matrix or allow K to be a dy-

namic function of s To design a K(s) for open loopj

dominance over a large frequency range would be an ex-

tremely cumbersome and tedious task. It is also questionable A

whether the end result (i.e, dominance of Q(s)) could

justify the means by which it was achieved. That is, if y

^^^'^w.-a• '°. ~ _ .a• Y .o ,.: q:. .zw.G. .s ..x 2x s..° sa ti - s

dynamics are to be introduced into the feedforward path,

it should be to the benefit of the closed loop design

rather than achieving a particular open loop configuration.

The problem of inserting loop dynamics to meet design

specifications will be addressed in the next subsection;

i Consider the utilization of a postcompensator matrix

L as indicated in Figure 1-C. Note that the shift of matrix

L to the left side of the summing junction completely

changes the characteristics of the design problem unless

L is constrained to be diagonal. For a non-diagonal L

the design problem becomes one in which a linear combination

of the system outputs, L y(t), are expected to track a lineard

combination of the reference values, L yref. The error j

vector for this case is thus a linear combination of the

output errors,di

ey(t) L(yref y('t)) (10)

and may become zero for conditions other than

y(t)y (l )ref

However, when L is diagonal the error vector in (10) becomes (

s zero if and only if (11) is satisfied. We therefore, im

pose this constraint on L and utilize the concept of dominance it

sharing [19,201 to achieve the desired dominance condition -

for 4(s) (or Q(s).)0.

in Figure ld, assume r(s) is the identity matrix,

L is diagonal, and K and L are constant matrices which

have been determined such that Q (s) in (2) is column

diagonal dominant. A common misconception associated with

MNA methods concerns an alleged modification of the rela

tionship between y(t) and u(t). Clear,l^`, this relationship

is always dictated,by G(s) and remains intact regardless

of the selection of K and L. What is dependent upon K and

L, however, is the relationship between the corresponding

entries in z (t) and e z (t)

ez (t) Zrefz i (t) (13)

with cross-coupled error signal (i#j) interactions determined

by the degree of dominance achieved and respective loop gains

(inherent within the elements of Q(s)).

The i th element in (12) can he written as

z. (s) q.. (s) ` a (s) + qj (s) a (s) (14)11

...Ej{

at.lira... .:..e ... _ .n7_.. .. n., .., ^...._d-#^, ' z.. .... [ ^x.^o. .4, _Y,.z:e•.yn.P_.. ri.l. T.>».,e.^ LL,...a _...,__ ...... ,_. ..

z (s) Q(s) ez (5) (12)

When Q(s) is diagonal dominant, z i (s) is primarily dependent

upon the i th error signa l

where i 1 1 2 1 ..., m. Note that although we assumed

Q (s) column dominant, it is the row elements of Q (s) which

contribute to the interaction terms in (14). To examine

this anomaly further, let m 2 and postmultiply (2) by

r as in figure ld

I Q(s)r. L G (s) K F (15)

where F its a, diagonal constant gain matrix, since Q(s)

is assumed column dominant, the dominance levels of Q(s)

are unaffected by the moduli of the elements of F. Thus

(12) becomes a4

z (s) gll (s) fl e z l (s) + q12 (s) f2 ez2 (s) (16)

i z2 (s) = q21 () £l ez (S) + q22 (S) f 2 ez (s) (17)i

r With J `iii (s) ( > J g j i (s) I for all seD it is clear that the

error term e z (s) has proportionally more influence on z(s)X

than on z i ( ), iy^j But note what happens as the gain f2

is increased with f l fixed. The interaction of ez (s) on2_

z (s) increases while the effect of e z (s) on z2 (s) decreases.l

This interaction influence of the gain matrix F is independent

of the respective dominance levels.

Thus low levels of dominance associated with each

diagonal entry in Q(s) are not sufficient to insure low inter-

action levels unless the impact of the respective loop

gains has been properly addressed. This analysis explains '+

why the DNA (or INA) bands may be very narrow (low dominance

levels) yet the closed loop time domain responses indicate

high levels of cross-coupled interactions.'

To .pursue this analysis further, consider the closed }'j j

loop system of Figure ld and assume that K and L have been jidetermined for column dominance of Q(s). Shift the diagonal {

L matrix around the loop as in Figure 2 recognizing than this

is the most probable implementation configuration since aour interest is in the y(t) vector and not necessarily z(t)

where i

i z (t) - ,L y(t) (18)

From Figure ld, the closed loop relationship isr

Z (s) = (l + LGKF) -1 LGKF Zref (s) (19)

Z (s) P (s) Zref (s) (20)'

P (s) (I + LGKF) -1 LGKF(21)

(I + Q(s)F)-1 Q(s)F

LA LLY ..

'ujF i

C Wcn 1

T)wOOLL

^-^,3x

L....... a . b ^...,. n a .. .. ^ _ .... ,..., .^.u?. a ....»^.. x ^ ..a ^ ^........n....^, .i ..t ^ ^.^^" k.> . ...5 ,...^... r..+. w.y ,.. ,r ...... ... .,.. .. ._

y(s) (I + G(s) KFL) G (s) KFL xref (s) (22)

y(s) H(s) YreE (s) (23)

H(s) _ (I * Lr 1 Q(s) FL)-1

L-1 Q (s) FL (24)

Comparing (21) with (24)t

if follows that

H(s) -- L (s) L (25)

P (s)17^?1

p12 (s) . m P (s)lm

P (s)21 p22 (s`)

Z Pm l ( S)

. . . . . .p

(s). . mmm

...21w

Hence, the diagonal elements of the closed loop transfer

function matrix are unaffected by the compensator shift.

Further, if each control loop is designed for a small

steady state error condition (i.e., high loop gains),

then the of-diagonal terms in (26) tend to zero and y(t)

tracks yref' If the gain in loop i decreases, then the

degree of interaction in loop i from the remaining high

gain loops will increase. This interaction effect is not

easily detected from the Nyquist array diagrams as it is

independent of the dominance levels when using a DNA

column or an INA row dominant design objective

The concerns associated with closed loop interaction

levels and system dominance can be resolved simultaneously

when using a DNA row and/or an INA column dominanceobjective.

For these N.NA design configuration the m2 elements of K.

! m elements of L, and the m dynamic compensators of F(s)

must be simultaneous) selected. In this wa , system domi-Y Y Y

trance, loop gains, and response shaping for all loops are

addressed simultaneously since they are highly interconnected.

As a result of the compl,uxity of these design formats;

they have never been seriously considered as viable design

procedures.

in the following section recent extensions to the MNA'

1 design technique are briefly described,.

G. New Design procedures

The MNA method as originally conceived by Rosenbrock

was composed of two design procedures. Each design pro-

cedure was ,relegated to the specific complex plane in

which the diagonal dominance conditions was obtained.

Stability theorems based upon the extended Nyquist cri-

terion were then developed for each design procedure:

In a typical application of the MNA design methods,

Q (s) and Q (s) are not, in general, simultaneously diagonal

dominant in the Gershgorin sense for all s on the Nyqust

D contour. This is due to the fact that

qij (s) ^ qij (s) (27)

for i t j, and s on D. Hence it was thought that the MNA

forms were not compatible, i.e., DNA compensation methods

could not be used when INA dominance was obtained and visa

versa. This limitation of the ,MNA design philosophies

is !removed by introducing a conformally mapped image process.

The details, of the imaging process are contained in

[201 and will b4it 'oriefly reviewed,

Assume Q(s) is Gershgorin dominant in the inverse

polar plane Z where the origin is exterior to the respective

Gershgorin bands. LetF j

W 1/Z (28)

represent the conformal mapping from the inverse polar jI

plane to the direct polar plane W. Thus under (28) there

4 exists a one-to-one correspondence between the paints

z(s) - x(s) +j y(s) in the Z plane and the points w(s)

IU(s) + J y(s) in the W plane. Since (28) is a conformal i

mapping circles in the Z plane map to circles in the W !

plane,

Let d(s) represent the radii of the circles in the Z

plane and r (s) represent the radii of the image circles

in the W plane. Then the image center point and image

radii are given bX

U(s) X(s) (z9)x (s) + y (s) - d (s)

V(s) Y(s) (30)r x2 (s) y, (s2^) — d ^

d( s)r (s) = - 2 ------^------ 2 (31)x (s) + Y (s) - d (s)

Since the origin is not an interior point of the Z plane

circles, the interior of these circles map to the interior

of the image circles. Furthermore, the relationship be

tween respective circles in the W plane will be maintained

since. (28) is conformal. Hence, any gain and/or phase

margins measured with respect to the Gershgorin circles I

(or envelope) in the Z plane will be retained in the imagei

plane.

This mapping procedure works equally as well when

Q(s) is diagonal dominant and the Gershgorin bands are

projected into the inverse plane. Note that the imaging

process is independent, of Gershgorin dominance levels in

the image mode as the imaging process retains the dominance

levels from the projected domain.

Stability theorems for the image bands as well asifurther insights into the imaging process 'are contained in

Appendix A. The significance of this theoretical development

will become more apparent when the Bode and Nichols methods

are discussed for M1MO systems later in this section.

A second limitation of the MNA design methods as pro-

posed by Rosenbrock was the requirement of diagonal dominance

for Q(s) or Q(s) by rows or by columns over the entiref

Nyquist n contour. This condition was imposed to ensure that

the stability theorems of Rosenbrock would remain valid. For

if the system lost diagonal dominance, then the theorems

neither confirm nor deny stability.t

Since all zeal systems are band limited (i.e , strictly um

proper in Rosenbrock's terms), the search for dominance

only needs to be checked as w proceeds up the imaginary

axis on the D contour. Although this simplifies the search r

to the customary ranee of 0 < w < - it imposes an excessive`

burden on the system designer if he knows that frequencies

beyond some finite value, wo, are not likely to contribute

to the system response.

In response to this concern, a new theoretical develop- fment was introduced wherein the contribution to the MNA or

image bands for all frequencies beyond wo could be properly

accounted for. The specific mechanism for finite frequency

domain design evaluation and related stability theorems are

detailed in - [201a

Fundamentally, the contribution to the dynamic response

for frequencies beyond some wo are contained within a circle o

centered about: the origin in the complex plane. The radius

of this circle is determined by the moduli of the elements

in Q (s) or Q (s) at s = jwo . As wo increases the corresponding

circle radii de , rease for the DNA method and increase for the

INA method. The feedback gains are then selected in accordance

with the appropriate array bands and origin centered circles.

The origin-centered circles associated with a finite

frequency range evaluation can be conformally mapped toN

the image plane. In this case the origin is contained within ►the circle, hence under the mapping principal the interior

of the circle in one domain maps to the exterior of the

circle in the image domain.

The two now theoretical developments discussed abover

provide the basis for the equivalence between the multi-

variable Nyquist array design procedures. As a result,

closed loop compensator design analysis and synthesis is

no longer restricted to the polar plane in which the desired

diagonal dominance condition was obtained. Thus, DNA

designs can be used for INA dominance conditions and the

desirable INA minor loop synthesis procedures can be used

when DNA dominance has been achieved. In addit$.oe, the

theory is now complete for finite frequency dominance.I

With this equivalence principal we are now in a position

to extend the design synthesis procedure to the magnitude/

frequency and phase/frequency planes. The resulting graphi-

cal display will allow for design analysis and compensator

synthesis using standard Bode techniques. Using the

magnitude/phase angle plane, a graphical display for each

control loop can be reviewed by the familiar Nichols chart

methods.

IX 1ViaK

• EoI o ^•

p allww

t uOO Q W W.................... ^„ • ZLYa:... .. ..: A 1r.^ _.7

Ox^ ••^^y

I ^^•:^ sst.NZ . K N it sar ss sts^ iR r Illiii• ^►K

p 1! W

d ; I N^ a, : I _NQ'Coo

^--• ; i 1-nm 2 1 1-,^a a

Q. ^`^ r • ^^ ¢ j • SAS. a

:'i^ I • Z S

roa _ vw

The fundamental concept of multivariable Bode Plots

and Nichols Charts to be presented below are described in

more detail in [21). Reference [221 describes a computer

aided design suite for closed loop system design and includes

all of the graphical. procedures referred to in this chapter.

The multivariable Bode diagrams are constructedidirectly from the data used to calculate a direct polar

plane Nyquist diagram. If the OLTF matrix is DNA dominant,

the envelope curves [27] of the corresponding Gershgorin

radii are each converted to Polar form and independently

drafted as a function of frequency. When system dominance

is obtained in the inverse polar plane, it is necessary

to first form the equivalent set of image bands in the r

direct polar plane. The envelope of the image bands are i

then used to construct the multivariable Bode diagrams.

Figure 3a represents an_INA diagram which is dominant

over the finite frequency range 0 < w < 3 0. Figure 3b

contains the corresponding image band for Figure 3a with

} Figures 3c and 3d displaying the Bode information.

Since the mapping between the respective complex polar

planes is conformal, it is not necessary to actually compute

and/or display the image circles. The envelope curves for

the image domain can be computed directly from the envelope,

curves in the domain in which dominance was achieved.

aw+w^+ww yrrac'<aew.w.n++^•++sa._ i ,..^.,-.-.-..w_. -.. .,.. .:.

Let c1(w) and c 2 (w) ,represent the envelope curves

for an MA diagram. The envelope curves, C 1 (w)and C

2 (w),

to the corresponding image circles in the image plane are

g iven by,

Cl ( w) = Lcl (w) ] -1 and C2 (w) = [c 2 (w) ] 1

Proof •

The relationships in (32) follow directly from the

conformality of the mapping W l/Z between the respective

polar planes,

Q.E.D.`

Thus for an INA system dominant condition, the Bode

diagrams can be calculated based upon the inverse of the

INA Gershgorin bands. The multivariable Nichols chart

is then obtained directly from the data used to construct

the Bode diagrams and is presented in Figure ,4 for the

conditions detailed in Figure 3.

With -the graphical analysis procedures described above,

system output response shaping can be considered by any of

the standard SISO methods. Since INA and DNA design methods

are described elsewhere, the remainder of this section will

be :..restricted to the Bode: design form.

0pP 24

C GA is^I ND

CI' B-^E

v S-18

Consider a DNA column dominant condition for Q(s)

E in (2), and assume the system is type o Since the

loop gain determined at w = 0 may be low, a reasonably

large steady state error could be expected. If the

design specification will accept an error no greater

than 10%, we may compute the required gain increase

te5S = 1 + f i P -ToT (33

where P i (0) is the value of the i th Gershgorin envelope

(or image DNA envelope) closest to the origin at w = 0.

usina this value in (33) ensures that the steady state

error will always be less than the required minimum level.

_ 1 - e

f i e SS) P i (0) i l l 2, ..., m (34)

t Similar relationships can be obtained for type 1 or

higher systems.

Inserting (34) into the system will only shift the

magnitude curve of the Bode diagram. As the curve is )

shifted, the closed loop bandwidth of the system (-3 dby

point on magnitude curve) will be simultaneously affected.

It is also possible the system dominance may be lost

as a result of the gain change. This is apparent, forthose cases in which the 180 0 phase angle informationcorresponding to the new zero db crossover point is

contained within the phase envelope curves.

This condition, however, does not present anyserious complications since we can easily synthesize anappropriate compensator to shift the -180 0 point outside

the phase angle bands at the crossover frequency. This

can be accomplished using standard lead, lag, or lead-lag

methods. The details of the compensationprocedure

y ' are described in [21, 221. Similar procedures apply for

type n systems.i It is now apparent that the multivariable Bode diagrams

and Nichols chart can play an important role in the effective xdesign of multivariabl.e closed loop systems. In, addition, f

it provides an effective measure of the frequency range '.over which dominance-was obtained. If the desired closed

loop bandwidth is beyond the range in which dominance wasobtained, new efforts are required to compensate for this

9 fitideficiency. At the other end of the spectrum, if the zero

z db crossover frequency is well within the dynamic range,_

this could reflect too conservative of a frequency range

(i.e., frequency range too large) for dominance acquisi-

tion. New design efforts over a reduced range may provide

less interaction levels in the bandwidth regions of interest

and could improve the overall design significantly.

D. Summary.

In this chapter the MNA design procedure was brieflyi

reviewed and new design tactics were introduced. The Imagef

bands play an important role in establishing the equivalence

of the INA and DNA design methods and provide the theoreticali

foundation for viewing the INA Gershgorin envelope curves

in the Bode and Nichols domain. The finite frequency de-

velopment provides the theory needed to reduce the com-

putational burden for the diagonal dominance search.. The

multivariable Bode diagrams provide a new view of the system

design where SISV compensator design procedures can be

used effectively to secure dominance and simultaneously4

achieve the desired closed loop performance. The correspond-

ing multivariable Nichols chart displays the Bode information

in a familiar form which again is suitable for design

analysis and control system synthesis.

The next chapter utilizes the design procedures and

theoretical developments of this chapter to design a closed

loop control for the General Electric QCSEE`turbofan engine.

CHAPTER III

MNA APPLIED TO QCSEE ENGINE

The turbofan engine selected for the MNA control

design study was the quiet clean shorthaul experimental i

engine ( QCSEE) [16). The QCSEE engine program was a NASA

sponsored effort to design, build, and test two experimentalengines which would serve as test beds for advancsd

propulsion technology evaluation. one engine was designed 3

for under--the-wing (UTw) mounting while the second designwas developed for an over-the-wing (OTW) mounting. Thisstudy was limited to the UTW design.

iTurbofan engines of the QCSEE design are under con-

sideration to power shorthaul aircraft of the STOI con-r figuration_. These aircraft would operate from small,hir

ports located near major urban communities. In this wayi

large transports could land at metropolitan airports many

miles from the urban center with passengers and cargo

transported to the civic centers via low _noise and high

'k reliability STOL aircraft.fl

To provide these services, primary propulsion re-

quirements for the aircraft are those of low noise levels

and exhaust emissions at all operating levels, rapid

thrust response capability including reverse thrust and

low weight. The design challenges were met by the

General ;Electric Company.

In the next section the specific design for the QCSEE

UTW engine is detailed with principal emphasis placed on

the control system design requirements. The following

section describes the non-linear digital simulation program

developed by NASA from which the linear operating point

models for the MNA design are obtained. The last section

discusses the results of the control system design study

performed using the multivariable frequency domain design

procedure.

A. QCSEE Turbofan Engine

A cross-section of the under-the-wing QCSEE turbofan

engine design is detailed in Figure 5. The design in-ir corporates performance and structural characteristics

iunlike those in any engine in production today and includes;

i1. An extremely high by-pass ratio (12:1)

and a high throat Mach number inlet~ fornoise suppression.

2. Reversible pitch fan blades for rapid 1`thrust response.

"36—

4A LiJ

Y 5 `n

k t ^y nm m^ /

1 z r u

m U-. G^

>v C7 ';

f © ^ aW

.-..z.^ ._. • u., _. .an a v. + .. ..N.. ^^c. siw^F.ww^F , .,., «- .w ...va.w. a.n .,,..wvw .,. >^,.v.-^a.m3 x.. ^ ^^..,,... .».. >~...+..« q,.i _ .. ^l .^

3. Geared turbine/fan combination for lowfan tip speeds with a high thrust rating.

4. Digital electronic engine controls.

S. Extensive use of composif;es for dragreduction and weight conziderati,ons.

The engine also incorporates a variable area fan duct

exhaust and variable compressor stator blades. The fan

is driven by the low pressure turbine through reduction

gears to achieve an efficient low pressure ratio, low

tip speed operation. The variable pitch mechanism is

capable of moving the blades from a forward thrust position

to a full, reverse thrust position in less than one second.

In the reverse thrust position, the hydraulically actuated

fan duct exhaust nozzle is opened to a flare position

to serve as an efficient inlet,.

The power requirements of a shorthaul aircraft system

dictated the design specifications which had to be met by

the QCSEE control, The primary specifications were:

1. to allow the pilot to command percentrated thrust,

2i to assure engine operation at all timeswithin safe engine limits,

3. to control inlet Mach number to meetnoise suppression levels,

4. to achieve rapid thrust response (inboth forward and reverse modes):

kLLe.3a ._ via >:^KSYNi+ air ^ wS? rY "ms n v+e saa,cuvUy,p f. xxx crcn.ie. a-+w.x«.a

These requirements all relate mainly to safety and noise

emission considerations. The first requirement resulted

from the desire to decrease pilot work load in a STOL

environment. This can be achieved by closing the feedback

loops using a parameter closely related to thrust (such

as engine pressure ratio) instead of using a parameter

indirectly related to thrust (such as spool speed). This

reduces the time the pilot must spend adjusting the

throttle to achieve the desired thrust level.

The second requirement meant that logic must be.

incorporated to prevent compressor and fan overspeqds

and turbine overtemperature. To meet noise suppression

specifications f a high inlet throat Mach number is required

to significantly reduce noise radiation from the inlet

duct. The requirement for rapid thrust response arises

from the need to insure safe STOL operation. in particular,

to have a safe go-around capability, thrust changes from

62% (approach power) to 95% of full power must be achieved

within one second. This is two to four times as fast as

is possible with current production engines.

The above control system requirements were met by the

engine manufacturer by using an engine mounted digital

control system shown in Figure 5. All primary control

-7 1 9-

functions are programmed on the digital control computer.

Digital command signals are sent directly to the three

input actuators:I

1. fuel flow actuator,ri

2. fan pitch actuator,

3. nozzle area actuator.

A hydromechanical control provides fuel flow limiting and

is also used to provide the command to the compressor

variable stator actuators.

An important consideration in the manufacturer's

design process was the control mode selection. This is

the process of pairing the four control variables with

four output variables. The mode selection was made such

that at key, selected steady-state operating conditions,

primary (unmeasurable) ;performance variables such as thrust, :p

specific fuel consumption, and stall margins are relatively

insensitive to small sensor inaccuracies. The result of3

the mode selection process for the QCSEE UTW engine control

was to pair fan speed with fan pitch angle, engine pressure

ratio (P /P ) with fuel flow and inlet Mach number withS3 TO Y

t fan duct nozzle area. The core stators were controlled

essentially independently, being scheduled (according to

standard practice) as a function of corrected core compressor

'speed.

The steady state operation of the QCSEE engine is

determined by a set of demand schedules. These schedules

command optimum values of engine pressure ratio (EPR),

inlet Mach number and fan speed as functions of the pilot's

power request and .ambient conditions. In particular, the {

EPR schedule is a monotonic increasing function of the

power request. The inlet Mach number is scheduled to be

a constant 0.79; however, at low power (less than 70t of

maximum), the nozzle is full open and the Mach number then

varies with power request. The fan speed is scheduled1

as a constant value for power requests above 555. Below

that value, fan pitch is at its minimum allowable setting j

and hence fan speed varies with power, request. Keeping

fan speed constant jeans that thrust can be rapidly mod-

ulated without the necessity of waiting for the rotor

speeds to change. This is especially important when

attempting to meet the stringent thrust response requirement r

of one second between a 62% to 95% power request change.c

in addition to the steady _state schedules, a number

of limit schedules are incorporated; for engine protection.

.Fuel flow is limited by maximum fan speed, core speed, and .r

fan turbine inlet temperature schedules. Also fan pitch and F

nozzle area controls have maximum and minimum limit schedules.

And finally, closed loop proportional plus integral control

,_ _.. .....'. L..»^:a-x z^< e_„vsv __ ..zvm+ttn`a..+x sc -gym ry ^t.._

was implemented on EPRI fan speed and inlet static pressure

(inlet Mach number) to ,insure that these three outputs

follow their respective demand schedules.

The QCSEE control designed using the Multivariable

Nyquist Array (MNA) method (described in Chapter 2) makes

use of much of the control logic described above. Ini

particular, demand schedules and limit logic designed by

the engine manufacturer are used with the MNA design as

are the max and min limit logic. hfowever? the MNA design

replaces the three independently designed uncoupled servo

loops on EPR, fan speed, and Mach number with an MNA

designed pre-compensator matrix, which provides cross-

coupling between the command schedules for optimum overall a

control. performance.

In the next section the non-linear digital simulation

for the QCSEE engine is described. This simulation is then

used to obtain linear models for the MNA design method,e1

B. Non-Linear QCSEE Simulation and Linear Models

During the hardware test of the UTW engine at the NASA

Lewis Research Center, NASA developed a highly non-linear

accurate real-time digital simulation of the QCSEE engine

at sea-level static conditions. This non-linear model Was y

used in extensive tests at the NASA-Ames in-flight simulator l

facility for test pilot evaluations of integrated engine air-

frame STOL configurations (17)

a _ _ y __._.

The digital simulation of the QCSEE UTW engine, shown

in Figure 7, was based on both steady state, flow-pressure-

temperature relationships (maps) for each component (fan, 'Icompressor, ,combustor, etc.) and on dynamic relationships.

These relationships are based upon continuity considerations

for mass, momentum, and energy.

Maps of pressure and temperature ratio as functions

of corrected rotor speed and weight flow are used to model

fan tip and also high pressure compressor performance.

Compressor variable stators were assumed to be on schedule

at all times. Fan hub performance is a polynomial function

of fan tip performance. The inlet throat and duct are

modeled by static pressure loss relationships. The combustor

is represented by a heat addition and a pressure drop, with

combustor volume dynamics neglected. Both high and low

pressure turbines are represented by maps of flow and

enthalpy drop which are functions of pressure ratios across

i the turbines and corrected speeds. Compressor speed is

computed by forming a torque balance between torque generated

by the turbine and absorbed by the high pressure compressor.

The same procedure is followed to compute fan speed.4

The complete non-linear simulation is programmed in

Fortran and uses a modified Euler method for integration.

It runs much faster than real-time, having a problem time to

CPU time ratio of approximately 25 to one on a UNIVAC 1100-40.

The MNA control system design procedure requires

linear operating point models. These models can be

generated directly from the non-linear digital simulation

through the use of a perturbation technique. This procedure

(also used to obtain the F100 linear models) (28] producesmatrices A, B, C, and D for the standard state variable

form model:

x Ax + Bu (35)

y = Cx + Du (36)

where x is the system state vector of dimension eight, u

is a third order control vector, and y is an eighth order

output vector.

For the QCSEE UTW engine the physical variables in-

dicated in Table S were used to define the linear state .ymodels. Note that several of the variables in the y(t)

vector are not measurable. {

To obtain 'linear models, the following procedure was

followed;

1. Run the simulation to steady state

y operating point conditions.

2. Disconnect integrator inputs and 1perturb each state variable x j (t) in

turn (j = 1 ..., 8,) by a small amountAx(t) from its steady state level'.

3. Note the corresponding change in thestate derivatives (Ax j (t)) and outputs

Ayj (t). Compute elements of A and C

Aij = Axi/Axe

Cij = Ayi/Axe

4. Compute steady state gain relationshipsbetween u and x i and yi by perturbing

uj in turn (j = 1, 2, 3) and allowing

the simulation to reach steady state.Note the steady state relationships as

Ax O = K Au(-)

Ay(-) = K Au(-)

5. Determine the B and D matrices as

B _ -AKX

D _ K CKy X

Following this procedure linear models were obtained

for five power demand values, i.e., 62.5, 70, 80, 90, and r

100 percent power. The matrices obtained are indicated

in Appendix A

` -45-

' VARIABLES QCSEE-UTW LINEAR MODELS

State Variables

P12 - Fan inlet duct total pressure

P13 - Fan discharge total pressure

P 4 - Compressor discharge pressure

P8- Core nozzle total pressure

NL - Fan speed

NH - Compressor speed

T3 - Compressor discharge temperature

T4 - Combustor discharge temperature

Input°Variables

xMV - Fuel metering valve position

X18 - Fan nozzle area actuator position

Theta I - Fan pitch mechanism drive motor positiont

Output Variables

PS1 1 -

Engine inlet static pressure-I

-P13 - Fan discharge total pressure f

P4 - Combustor pressure

P 8 - Core nozzle total pressure

NL - Fan rotor speed

NH - Compressor speedi

T41C - Calculated turbine temperature

FN - Net thrusti

_tYu..«... \.... .. w a. ... ... ..l... .. .. .s.....n..._..YM.,.., r2EE'v i:tt _x s..,...., ..... ....t_. _.. ..wvG..^: .J.. .. r. ... . ......... ... ? ..^... ..._.. . ., .... < .r.

C. MNA Design

The MNA control systems designs to be discussed in this

section utilize the state equations of Appendix _A aug-

mented by one state variable for each integrated input variable.

This integral control action eliminates steady state offset

and conforms to the standard QCSEE control, form. Since all

control loops utilize integral control action, the entries

in the transfer function matrix, G(s), were modified appro-

priately. Thus, the resulting MNA designs could be mple-mented directly in the non-linear simulation for design

evaluation.

The First MNA design for the QCSEE-UTW engine assumes

nozzle area to be fixed at the full open position and fan

pitch angle fixed at the 62.5% of full power position. With

these two control inputs fixed at their respective values,

the only input variable to be determined then is fuel

metering valve position. For this SISO condition, combustor

exit pressure (P 4;) was selected as the output variable to

be controlled as it is most representative of cjystem thrust

response.

From the linear models of Appendix A, transfer functions

augmented with integral control action for P 4 /xmv at each

operating point are obtained. For the 62.5% condition, the

Nyquist, inverse,Nyquist, and Bode diagrams are indicated

in Figure 6 for a feedback gain of 0.04. The phase margin

is 29.90 degrees with an infinite gain margin. The closed

loop bandwidth at this design point is approximately

10. radians with the feedback control

Xmv,,,04 P4 (37)

To examine the effect of this control at each of the

fother four operating points (70%, 80%, 90%, and 100%) the

composite Nichols chart of Figure 7 may be used. Here

each ct.lrve represents the respective Nichols plot for each

P4/Xm transfer function with a gain of 0.04. The 62.5%w

power curve and the 70% power curve coincide. As the power

level increases, the corresponding curves move to the right.

This signifies an improvement in phase margin (to 50 0 at

100% power) with a corresponding decrease in loop bandwidth.

The transient responses of the nonlinear simulation to

the SISO design discussed above for a power change from 62%

to 70% are indicated in Figures 8A-8D. For referenceI

purposes the transient responses of the simulation using the full

GE QCSEE control are also indicated in the figures.

Over the power range 62.5-80% power, the full GE control

consists of a SISO Xmv control and sufficient control logic1

(i.e., constraints) to impede movement of X1 8 (nozzle area)

-gg.. •

I ^ v ( ^ y

n ~ E': CL

I ` = aN ^ ^ o ^^^ 3

".. r..

_ I tIUWWI. •.......................

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SF4+70 ••"•••_.....AF449i ..«...

^'N R r9swG6FT1144 s. +.-•

zel000,

-2+o -2!• --180 >150 -120 -90 -i* -30 0

PHASE ANGLE IN DEGREES

FIGURE 7: NICHOLS CHART FOR SISO DESIGN AS AFUNCTION OF POWER SETTING

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I I^ ^I 3 I!;L.

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i t S W"..^.... ^ I p S• 1 J

;............................................... . .4 ...a ....... ....... ....... ..N

w .. •- • N A N ^ • F

and Theta I (fan pitch angle). When the environmental

conditions near 80% power are realized, the.GE control 3

logic permits Theta x to change, at or near 90% power

conditions X18 is permitted to vary. It is not surprising', {

therefore, that the steady state conditions for the SISO

MNA and full QCSHE control responses are identical over

62.5-70% full power,

what is surprising, however, is the improved responses

for Xmv , thrust (FN), and fan speed (NL) of Figure 8G. Since

total fuel consumption is the area under the X mv curve,

it appears that the MNA control of (37) is more fuel

efficient at this operating condition.

To examine the impact of the constant gain control of

(37`), in light of the Nichols chart of Figure 7, a non-linear

simulation run was made a power slam situation, i.e.,

r 62.5% to 100% full power. Here the engine command setting

of 100% full power was requested from the 62.5% operating

point condition. The corresponding transient responses for

the MNA-SISO and full GE controls are indicated in Figures

T 9A- 9D.-

From these responses it can be seen that fan speed (NL)

and inlet duct pressure (P12 ) are significantly higher for

the MNA-SISO design than for the full GE QCSEE control.I

`However, engine thrust (FN) and total fuel consumption (area

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f^= w. ............................................. • = c.^..lei

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............................. . .

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under XmV curve) for the MNA SISO could be consideredt

an improvement over the corresponding full GE control'

responses. it appears, th%refore, that both X 18 and

Theta I (fan pitch angle are used for noise suppressionI

in the full GE control and not (as stated in (15)) j

"primarily to provide rapid thrust response." Except i

for the obvious reduction in NL the impact of Theta I

of Figure 9C is unclear.

For a more meaningful comparison, the GE controlsi

on nozzle area (X1 8 ) and fan pitch angle (Theta I) were

held fixed at the 62.5% full power position. This condition

is then equivalent to the MNA STSO design and,is presepted

in figures 1OA-10D for a transient power slam to 100

power. The two designs are comparable in transient and

steady state performance. The control logic to implement

_(37) is considerably simpler than the logic in [15, 161.

As indicated in Figures 8A-8D, good system performance

is obtained at the 62.5% power point by a SISO control design.4

The design point at which two controls are simultaneously used

by the GE QCSEE control occurs at the 80'% power point. Using,

the linearized model corresponding to 80t power, a two input

two output condition was configured for an MNA design

attempt. The inputs selected were xm v and Theta I with

^I^^-z w

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................. .................. to

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: Y^YYY...^•^^.•Y.•Y. • •.YYYY•.YYYYY•Y •Y1•Y^ ^. -

outputs P and NLf4

NL Gl,(s) G (S) X12 Vmv

P G (S) Theta I4 21(s) G2200

liere nozzle area (X was fixed at its full open18 position

(62.5% power)

Application of the MNA design suite described in [223

to the corresponding transfer function matrix at 80% power

yielded system dominance over the range 0.1 to 50 radians.

The resulting constant compensators are

1.1 80163.5K (38)723361.1 -2.1263(10

0.5163. 0.0L (39)

0.0 13.8743

3.0(10 5 0.0F

-80.0 i.i(10(40)

The DNA f INA, Bode, and Nichols plots appears as Figures

11-14. Transient responses to a power step from 80% to 90%

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.. ........... ..... ...................... ....IRS.t I : _. • ^.

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of full power are indicated in Figures 15A-15D with the

two input GE control superimposed. Nozzle area was

held fixed at the full open position which corresponds

to the 62.5 power setting. The peak overshoots are signi-

ficantly lower for the MNA design with no degradation

in performance. Also total fuel consumption (area under

Xmv) is much less for the MNA design.

For a full poorer slam of 62.5% to 100%, the 2x2 MNA

design is compared with the two input GE control in

Figures 16A-16D. In every figure, the MNA design performs

very well with respect to the two input QCSEE control,

A comparison of this 2x2 MNA design with the full GE QCSEE

design is indicated in Figures 17A-17D. The only signi-

ficant deviation is, as expected, in P 12 and XM11. These

variables are directly effected by nozzle area, X18.

There is, however; a significant difference in imple-

mentation Logic. This is apparent when considering the im-

plementation of the constants in(38)-(40) versus. the lbgic

outlined in [15, 16).

To regulate 'inlet Mach number, a third controlled variable,

inlet duct pressure (P 12 ), is introduced. Using nozzle

area as the third input, a three input three output design

condition exists„ Although time did not permit -a full scale );s

three input three output MNA design over the full operating

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range, some analysis and simulation were performed. The

remainder of this section is directed to this condition.

Since the GE nozzle area control was designed using

SISO methods, the SISO MNA and the 2X2 MNA designs were

independently augmented with the GE nozzle area control.

The first case represented in Figures 18A-18D consists

of the SISO MNA with the GE X 1 control while Theta I

is fixed aL the 62.5 power position. Note the rapid

thrust response (same as in Figure 11C) and the respective

decreases in P12 and XMll from their counterparts in Figures

9A and 9D. Execept for fan speed, the Theta I curve for the

full GE control has little influence on the remaining

variables.1

Augmenting the GE nozzle area control to the 2x2 MNAa

control design, results in the transient performancei

indicated in 'Figures 19A-19D. As expected, the dynamics

are unchanged from the previous case except for the improved

performance of P12 and XMll. The MNA control performs very

well with the GE nozzle area control fornoise suppression.

Preliminary design using MNA concepts for the three

input three output case was performed using the 90% power

point. The input vector u(t) and output vector y(t) were

e4eie n .... e. ... . .._..ve, i .. : r .. ..__ .xd r ._ :.-- v rrhxaa F .._ a+m^ _fPf4 ^u..4 _+J'--•++.. :sx ti ra..r e. vif xu+cC2e .f ,., krx .:. a. vz, ar . ir:^. rr 4... .. .., . _ .. _

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L; ^► rn` ; I • ^^., ^

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r.4 n LLnn .,, ..... .. . .«....... . ........,iqu

CLf4p,

4, N w G co*

N YI '1f IY r

ii t,` „4 r

NW S -(AWW ^''N S.^

• Lt

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♦ t^

4.. ..^ r

WIL 1 j

tog 4z

O^ ^ n

tZ ^`:

.as M a^

(a w ' ( H

L:Q ► ^N 3 W $Z Z O a Q dff ^.

C!1_ _ x

j ; ( r

a ^^ ( 4 N

i I •

r :1 d -,W S .^ W :

N ...«.......... .......................«..... ♦ _ r ..^ ..................... .......... IL r

yr v • ^ s a • r; r y

aria T

-91—

chosen as }

a.Xmv NL

U(t) Xl$ y(t) P12

Theta I P4

Ixwhere Xla is nozzle area and P12 is inlet duct pressure.iFor this case, system dominance was easily obtained

lfor each linearized operating point in Appendix A. The

CAD procedure of [18, 22) resulted in the following

compensators when 01 t W C 40 radians at the 90% power

i.2032E-4 -.01856 0.00157

K .0300 7.9359 -.02023 1

0289 103,, 67 .001195

L Diag{0.0415, 71.455, 0.45231

F Biag{10.0, 0.1, 20.0}

The corresponding DNA, Image, and Bode diagrams for this

operating point are presented in Figures 20-22.

R I ^ I ^u

I p^ I a

f I wY^ I ^-

I ' /I Q^^ I

^ fz' .T r•.:e:•_;to• kl« Ito :^^:^:^s:tt^sss^:^t^st^t.•^s::ts^:rsrJ- ^ J

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ia I CIDz

.. s I w

^.1 ..•........•.. R ..•...••.••.•...r..r•rr.•

pS HM Z i NtlOUM

tn 1°w ^ i h_n^ap

rr1 1 1

^^ war f;

' ^ 7i c...k..l .....;.:.-....,v .... ....-.„r^,^.. _..w .0 s....44n.e+,r^.r.^^ti... ..w «,.._.,_.. .... .,..

I A - +^M

thE Bg

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CNf rN

. w••r•r• ' .. • •.}real. ^ ..^ ^ {{ W

i ...» ....................................... •e... ^.^ H J

I N^^a ' ^ Nor►

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t WOW ,.imp +U1.4r ^, ( W..

• ppCl N

.. o ^+ a r v ^ h •

Usu ^ ' •

W^ 0 V

off Q-/ u '^ } W^ J...V : (

,, z::^ s ss: s. f s cis :r .h ^ Atm.w

I _ •

O •rrr ... ..... I

• fnn

ro MOMI r.^o1DO ~ wadi

I• Wes• I W.sW.,

QWW `' ~rte Y

•►+ >>

I ua^a*w

r jr--.itI

swratAg

MW a a sON

F The MNA designs discussed above demonstrate the utility

of the design concept in achieving good turbofan engine

performance. The :Full potential of the Bode and Nichols

concepts for multivariable plants were not fully utilized

in this design application, however, as dynamic compensators

were not required. in each case, appropriately determined

constant compensators provided adequate closed loop system ^f

performance and feedback gain scheduling was not needed. 1

This desirable condition is due principally to the well be-

haved and easily controlled dynamic system structure.f

_ YPvSlf^n r,vwrffiYw+neMww k..wr--ifl Wa. ^xa.. i4rr w. 4Y-,as.x-h,.a,an°#Utdd^:au.^s4rw.ei^..f s. anFsLzthe....ucQa Y.x.a :.iJi3r,:ate+.u.,G.•,v..:i_..ks..w v.t.nx> .. .. W .. _.... _

REFERENCES

1. Sobey, A. and Suggs, A., Control of Aircraft andMissile Powerplants, Wiley and Son, Inc., 1963.

2 Michael, G. and Farrar, F., "Development of OptimalControl Modes for Advanced Technology PropulsionSystems,,"' UARL Report M9-11620-1, 1973.

3. Michael, G. and Farrar, F., "Identification of Multi-variable Gas Turbine Dynamics from StochasticInput-Output Data," UARL Report R 941620-3.

4. Stone, C., Miller, N., Ward, M., and Schmidt, R.,"Turbine Engine Control Synthesis," AFAPL ReportTR75, March 1975.

5. Weinberg, M., "A Multivariable Control for the F100Engine Operating at Sea Level Static,"ASD-QTR-75-28 November 1975.

6. Merrill., W. and Leinginer, G., "Identification and y

Dual Adaptive Control of a Turbojet Engine,"IFAC Symposium on Identification and SystemParameter Estimation, Darmstadt, September 1979.

{7. DeHoff, R. and Hall, W., "Multivariable Control Design {

Principles With Application to the F100 TurbofanEngine," JACC, Purdue, 1976.

84 DeHoff, R. and Hall, W., "Design of MultivariableCon roller for an Advanced Turbofan Engine,'!1976 CDC, Clearwater Beach, Florida.

9. DeHoff, R. and Hall, W., "Multivariable Design Proceduresfor the F100 Turbofan Engine," Final ReportContract Number F33615-75-C-2053, Systems Control, Inc.,Palo Alto, California, January 1977.

10. DeHoff, R. and Hall, W., "A Turbofan Engine ControllerUtilizing Multivariable Feedback,"_IFAC MVTSSymposium, Fredericton, Canada, 1977

11. DeHoff, R. and Hall, W., "System IdentificationPrinciples Applied to the Control and FaultDiagnosis of the F100 Engine," 1977 JACC,

FSan....rancLSCO.

12. Lehtinen, B., Soeder, J, Costakis, W., and Sel ,dner, K."Altitude Tests of a Multivariable Control forthe F100 Turbofan Engine," NASA Technical Reportin preparation.

13. Lehtinen, B., DeBoff, R., and Hackney, R., "Multi-variable Control Altitude Demonstration on theF100 Turbofan Engine," Joint Propulsion Control

I Conference, AIAA 79-1204, June 1979.i

14. Sain, M., Peczkowsk , J., and Melsa, J., Alternativesfor Linear Multivariable Control, NEC. Inc., 1977.

i 15. QCSEE-UTW Engine Simulation Report, General. ElectricCompany, NASA CR-134914 1 July 1977.

16. QCSEE-UTW Digital Control. System Design F.eport, GeneralElectric Company, NASA CR-134920; January 1978.

17 Mihaloew, J. and Hart, C., "Rea; Time Digital PropulsionSystem Simulation for Manned Flight Simulators,"NASA TM-78958, July 1978,

18.. Leininger, G;., "Diagonal Dominance for MultivariableNyquist Array Methods Using Function Minimization,"Automatica, Vol.. 15, pp. 339-345, 1979.

19. Leininger, G., "The Multivariable Nyquist Array - TheConcept of Dominance Sharing," in Alternatives forLinear Multivariable Control, Sain et a1 eds., 1977.

20., Leininger, G., "New Dominance Characteristics for theMultivariable Nyquist Array Method," Int. J. Control,Vol. 30, pp. 459-475, 1979.

21. Leinginer, G. "Multivariable Compensator Design UsingBode Diagrams and Nichols Charts," IFAC CAD ofControl Systems Symposium, Zurich, 1979.

x,.yk y

^..tt.. 't{.iSf"Ys1^,5^+i'^ -- 3-•.S.LNYtYu.tiJ.v"fa[_.vhd[e....cL a m..y,i.11.>Jlm _w.i"W^rc •N

22: Leininger, C., "An Interactive Design Suite for the 1Multivariable Nyqua._st Array Method," IFACCADCS Symposium, Zurich, 1979,

23. Rosenbrock., H, "Design of Multivariable ControlSystems Using the Inverse N uist Array,"Y 9 Y4Proc. IEF`, Vol. 116, 1968.

24. Rosenbrock, H., Computer Aided Design o f Control Systems,{ Academics Press, 19 74 .

25. Fisher, D. G., and Kuon J. and Ex Per mental^ ^ " ComparisonEvaluation of Multivariable Frequency Domain jDesign Techniques," IFAC MVTS Symposium, FFredericton, Canada, 1977.

26. Ostrowski, A., "Note on Bounds for Determinants withDominant Principal Diagonal," Proc. Am Math. Soc. ,Vol. 3, pp. 26-30, 1952.

27. Crossley, T. R., "Envelope Curves to Inverse NyquistArray Diagrams," Int. J. Control,, Vol. 72,, 1975.

t 28 Daniele, C. and Krosel, S., "Generation of LinearDynamic Models from a Digital Nonlinear Simulation,"NASA Technical Paper 1388 0 February 1979.

APPENDIX A

LINEAR MODELS FOR} THE QCSEE ENGINE

1. 62.5% power level - Model includes actuator states.

G 2. 70% power without actuator state s.i

3. 80% power without actuator states.

4. 90t power without actuator states.

5. 100% power without actuator states.3

-100—

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m m m m m m m m m m m Z /- ..1 I ^ ►•1 Q

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I 1 I 1 1 Im@(Um -^ M-44) dm@@mmm mm @ mmmmmm @m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m+ + + + + I I I .......... . ... ♦++♦ ♦+♦ ++♦♦W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WN^DAm00000000AAAf'1f-m ♦ Ammmmmm mmmmmmmmmmm

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lAmr^mm-»N('1ll1111 mdlA00Nmmmmmm mmmmmmm@mmm

H @@mmLAmm00Am f'1r-Mr- @@m@mdl- mmmmmmmmmmmO —m00V-VWWr- r-N AmmmmmmDmmmm@@mm mm

IL . a

Z m m m m m m m m m m m m m m m m m Qie9 m m m Z m ..........

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m fn m N m m v v'o m m m m m m m

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m .-4 .4 m r- N NCD O1 — N 00 -+ 00 Qt

m 01 v- 00 N N U)

m m m m m d m mI I

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W W W W W W W WN ^ m m m m m m

m N m m mm N m m m

m m m m m m m m1 I

m m m — m N m mm m m m m m m mW W W W W W W Li,m m m V — f- m m(D m m m LA U) m mm m m M .-4 .4 m mm m m m m m m m

m m N .-^ .-. N m -+

OW(Dm m m w m m m m m m m m m m m m+ + + I + + + + I + 1 I + 1 + +W W W W W W W W W W W W W W W Wm m W m Ln m m m VM^M WU) MV

m m N v LA m N N m lD lJl 00 m l!1(D V1

m (p + v N (D U) (D NvNNm rl mNm m m m m m m m m m m m m m m (v

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X m m m m m m m m m m m m m m m m X m m m m m m N m•--^ I 1 1 ^--^ 1 1 1 1 1 IMNm-+mN c M<SoW (S)NmQ` M N vHmmmmmmmmmmmmmmS ► mf-mmmmmmmmQ + + + + + 1 + I + + + + + + + + Q I + + + + + + +E:WWWWWWWWWWWWWWWLi1EWWWWWWWW

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^--+ I I I 1 1 ►-+ I

+ 1 + ( + + I I

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m CD N ^ .1 Cu m + .^@Nf NCD@@m m m m m m m m mCflmm(DmmCa+ + + 1 + + + + I + I I ++++ W W W !s! W W W W WWLAJWI►JWWW@ m (D m v VI N IV M •-+ In r P) 00 (D 0)C'v m N N to 1- N m (n 1` V N v--q V (D V9 m qr •-• (D In (D N v m v m w4 In v. .m 0

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Q ..... 1 + 1 ........ Q I .......11 E W W W W W W W W W W W W W W W WX: W W W W W W W W

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m m m m m m m m+ 1 + I + + 1 1

0 MN ^m 111 r M m

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N ^ Nm m m m m m

N -^ m m --4 m m mW m m m m m mmW W W W W W W W(D (^ m m N m (4 9O1 U1 m S1 ^ m m m-+ N 9 m -+ m m mm m m m m m

Im m m -4 m N m mm m m m m m m m

m m m m N U) m m^ -r f^U mm m m 4T

m m m m m 91

m mm m N

--^ --^ N (S> + r-mN^Nmmmm m m m m m m m mmmmmCDsDM+ ♦ + I + + + + 1 + I I +++W W W W W W W W WWWWWWWWm m V -+ 00 v w N M y r- N N w mmm m N m — W 1~ N MMmNm 1 m00CD (P U) N f- W M v M In N N N U). .m m m

. .m m

. . . . . . .m CD mmmmmmmm

1 1 1mmNmN--^mf`1^mMm.-

I 1 1^m.-+m Mw<DNMwNNm m m m m m m m m m m m m m m m m m m m m m m m+ + ♦ + + 1 + I + + + + + + + + I + + I + + + +

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o X m m m m m m m m d m m.

m <D m m m X m m m m m m m mO^ r-^ 1 I I ►-^ I I 1 1

^Nm--+mN--^--+M^'mM-.NmNm^-•m^fNU)U)M^I^-mmmmmmmmmma^a^mmmml--mmmmmmmm

Q ♦ ♦ t ♦ t I + I ♦ + ♦K: W W W W W W W W W W W

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• M • '. • Q t w• • m • m 4+ V •♦ I ♦ 1 ♦ • I I

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♦ A I ♦ ♦ ♦ ♦ ♦ ♦ I I

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H NmNmm^Of^ ^ IAm+mf'1111SON1~- oNb('1('1+f'1(O••m—(9--'!.+1-fTfnNN-4(nN•^700--^N(7)--.M--111

IL . . . . . . a . . . . . .CD m m m m m m m m m m m m m m .m Z m N m m m m m mZ " 1 I I I I ";

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