LTR Method for Aeroengine Control System

V ol. 17

No. 3

CH INES E JO U RN A L O F AER ON A U TICS

A ugust 2004

Reduced Order H / LTR Method for Aeroengine Control System YANG Gang, SUN Jian guo ( Col lege of Energy & Pow er engineer ing , Nanj ing Uni v . of A eronaut ics & A str onautics , Nanj i ng Abstract:

210016 , China)

T his paper proposes a new loop recov er y method to solve the reduced o rder problem of H /

L T R method. T he result ed lower order controller shares almost t he same performance and robustness as t he orig inal H

/ L T R co ntroller . Further mor e, this paper develops a new order reduction method:

slow fast mode order r eduction ( SFM OR) method. T his order reduction method is particular ly effective for t hose controllers w hose modes can be divided into a slow part and a fast part according to their veloci ties. Applicat ion of these methods to a benchmar k example and a certain tur bofan eng ine is descr ibed. Key words:

aeroeng ine; H / L T R; SF M OR; Schur; or der reduction

航空发动机中的降阶 H / LT R 设 计方法. 杨

刚, 孙健 国. 中国航 空 学报( 英 文版 ) , 2004, 17

( 3) : 129- 135. 摘 要: 提出了一种新的, 可以解决航空发动机 H / LT R 设计方法阶数 过高问题的 目标回路 设计 方法。这种方法 设计出的低 阶控制器 保持了原 H / LT R 方法设 计出的控 制器的性能 和鲁棒性。 此外, 还提出了一种新的降阶方法: 快、 慢模态降阶法( SF M OR) 。这种降阶方法对那些模态可以按 其速度分为快、 慢两部分的控制器特别适用。最后, 给出了设计实例。 关键词: 航空发动机; H / LT R; SF M OR; Schur; 降阶 文章编号: 1000 9361( 2004) 03 0129 07

中图分类号: V233. 7 + 1

Simple linear cont rollers are normally pre f erred to complex linear cont rollers. T here are few er t hings to go wrong in t he hardw are or bugs t o fix in the soft ware; t hey are easier to understand; and t he comput at ional requirement s are less. But

文献标识码: A

many ot her robust m ultivariable design met hods, H / LT R methods can only result in a high order cont roller. T he order of t he cont roller must be re duced before any pract ical implement ation.

the robust mult ivariable met hods oft en result in

A new loop t ransfer recovery met hod is devel oped in t his paper t o solve the reduced order prob

cont rollers w ith relat ively high order. So t he con t rol engineers have to solve t he pract ical problems

lem of H / L T R met hod. T he resulted controller has relat ively low order, but it m aint ains t he per

of how to simplify these desig n procedures and how to effect ively reduce t he orders of t he result ed con

f orm ance and robust ness of t he original syst em. Furt her more, a slow fast mode order reduct ion

t rollers. H / L T R met hod[ 1] uses H

( SFMOR) met hod is developed in this paper t o opt imization t o

recover t he loop transfer f unction. It solves t he problem of L QG/ L T R controllers w hose gain are usually unduly high and w hose bandw idth unduly w ide[ 2, 3] . Wit h H / L T R method, the result ed cont roller not only recovers LQR loop transfer

furt her reduce the order of t he above controller. SFM OR f irst div ides the H / LT R cont roller int o tw o subsyst ems according to t he velocit ies of t he cont roller modes, t hen uses Schur [ 4, 5] method t o reduce t he orders of the t wo subsyst ems separately, and finally recombines t he tw o reduced order sub

funct ion, but also has relat ively low gain and nar row bandw idt h. So t he H / L T R cont roller has

syst ems t o f orm t he ultimat e reduced order con t roller. Examples show that SFMOR achieves t he

superior noise rejection propert ies. How ever, like

expect ed goal.

R eceived dat e: 2003 07 25; R evision received dat e: 2004 06 07 Foundation it em: A eronautic Science Foundat ion of China( 00C52030) ; N at ional D oct oral E ducation Foundation( 2000028701)

% 130 %

YA NG Gang , SU N Jian guo

1

H / L T R M et hod

CJA

- SI ) WS) is sm all enough, to make sure t hat t he

Consider t he unit feedback syst em as show n in F ig 1. G ( s ) is the desig n plant model ( DPM) ,

cont roller can recover the performance of t he target loop; and ( 2) ,

U(

= UWU ) is small enough, t o

make sure t hat the controller has sat isfactory band w idt h.

Fig 1 U nit feedback system Fig. 2 Standard H

and can be described by st at e space form as x = Ax + Bu y = Cx + Du

G( s )

problem

( 1)

w here x , u and y belong to R n , Rm and Rp , re spect ively ; n, m and p represent syst em , input and output dimensions, respectively. Furt her more, it should be assumed that ( A, B) is st abi lizable and ( A, C) det ect able. T hen t he transfer funct ion matrix of t he DPM is A B = C( sI n - A) - 1 B + D G( s) = : C D

F ig. 3 Block diag ram representation

( 2)

w here s is Laplace operat or and I n is the unit ma t rix of dimension n. Consider t he case w hen t he loop is broken at

of performance index J

In t he sequel, discuss how to solve Eq. ( 4) . Fig 2 is t he diagram of standard H problem. Cont roller C( s ) in F ig 2 can be desig ned by st an procedure to satisfy ∀T zr ∀ # , w here

the input of the plant of Fig 1( t he case w hen t he

dard H

breakpoint is at t he output of t he plant is sim ilar) . Assume t hat the sensit ivit y f unction of t he closed

Tzr is the transfer function f rom r to z, and > 0 is const ant. In order to t ranslate Eq. ( 4) into t he

loop is Si , and t he L QR targ et loop sensit ivit y

formula of standard H

funct ion is SI , t hen t he t arget loop t ransf er f unc

SI , WS and W U into F ig 1 and redraw it in

t ion can be recovered by minimizing the f ollowing performance index ( P I) w ith H opt imization

Fig 3. T hen t hose in t he dashed block correspond to P in F ig 2, and the diagram satisfies t hat J=

J

min

Q! R

: =

H

min

Q! R

( S i - SI )

( 3)

H

But it is pointed out by H an and Hsia[ 3] t hat this met hod can not reject measurement noises ef f ect ively. In order t o maintain system performance as w ell as t o reject measurement noises effect ively, Ref . [ 1] proposes t he f ollow ing P I min

Q! R

H

J

=

m in

Q! R

H

problem, let ∃ s int roduce

Tzr . As a result, Eq. ( 4) can now be solved as a standard H problem according to Fig 3. Ref . [ 1] also gives t he guidelines t o how t o choose t he w eight ing matrices WS and W U. With these guidelines, an int egrator is incorporated in the cont rol syst em t o eliminat e st eady st at e error, but t he DPM is not augmented. So t he controller

( Si - SI ) WS UW U

( 4)

w here U = ( I + GC) - 1 C, represent ing the t rans

order has been reduced.

2

Reduced Order T arget L oop Design

f er funct ion from r to u in Fig 1, and WS and

Analy zing t he nominal cont rolled plant P in

W U are w eighting mat rices. T hen Eq. ( 4) can

Fig 3, it can be seen t hat S I is an import ant part

= ( Si

of P . T herefore, if a low order SI can be arrived,

achieve the f ollow ing simultaneously: ( 1)

S(

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R educed Order H

% 131 %

/ LT R M et hod for A eroengine Cont rol System

the order of the nominal cont rolled plant P w ill be reduced accordingly, and so does t he cont roller.

1) Divide t he controller into tw o subsystems according to the mode velocit ies: C ( s ) = C1 ( s)

It is w ell know n in t he LT R procedure t hat

+ C2 ( s ) , where C1 ( s ) and C2 ( s ) cont ain t he

the requirement s f or a certain loop to be a targ et loop are that this loop should have g ood perf or

slow and fast modes of C( s) , respect ively. 2) Reduce t he orders of subsystems C1 ( s )

mance and robustness. Ex amining the design pro

and C2 ( s ) w ith Schur met hod, denoting t he re

cedure of H / LT R method in sect ion 1, it can found that the t arget sensitivity f unction is de

sult ed reduced order subsystems as C 1r ( s) and C2r ( s) , respectively. Because subsystem C1 ( s ) ( C 2

signed using L QR method, but it can also be found

( s) , respectively) contains only t he slow ( fast , re

that none of ot her propert ies of LQR loop is neces

spect ively) modes of the orig inal cont roller C( s ) ,

sary for the design. It is used because and only be cause it is a loop t hat & has good performance and

Schur method works eff ect ively in t he mode recom binat ion procedure. 3) Recombine C1r ( s) and C 2r ( s ) t o get t he

robustness∋. T herefore, if t here ex ists a loop t hat is simple and has t he good performance and robust ness of LQR loop, it can be doubtlessly used as our targ et loop.

final reduced order cont roller Cr ( s) Cr ( s ) = C1r ( s) + C2r ( s)

T herefore, t his paper considers reducing t he

From t he above procedure, it is clear t hat SF M OR is particularly ef fect ive if t he modes of t he

order of LQR target loop first . Based on Edmunds

cont roller can be div ided int o t wo parts according t o

[ 6, 7]

model m at ching design met hod , the L QR loop can be replaced w ith a lower order loop. Ex amples

their velocit ies. On t he ot her hand, H / LT R cont roller cont ains integrat ors t o eliminate steady

in sect ion 4 w ill show t hat when t he t arget L QR

state errors. T he velocities of t he modes corre

loop is replaced w it h a simplified one, t he result ed

sponding t o t he integrat ors w ould def init ely be

cont roller w ill have relat ively lower order, but it w ill have almost t he same good performance and

much slower t han ot her modes. According t o t his correspondence, t he modes of H / L T R controller

robustness as the controller w hose target loop is not

are obviously divided int o tw o parts. As a result,

simplif ied.

SFM OR should be ef fect ive for H / LT R con t roller, and the examples in sect ion 4 will show

3

Controller Order Reduction

Although t he orders can be reduced by careful ly choosing weig ht ing mat rices and simplif ying t he targ et loop, the orders of H / L T R controller are st ill very hig h. T he cont roller st ill needs furt her order reduct ion before im plementat ion. In t his paper, a simple yet effect ive met hod,

this.

4

Exam ples

In t his sect ion, t wo ex am ples will show t he procedure and t he ef fectiveness of the met hods dis cussed in sect ion 2 and section 3. Example 1 First let∃ s consider t he benchmark ex

called slow f ast mode order reduct ion ( SFM OR) met hod, is developed. Assume that t he controller

ample as in Ref. [ 3] .

C( s ) has been derived. T hen the modes of t he

follow ing st at e space represent at ion 0 1 0 x= x+ e - 3 - 4 1

cont roller can be easily calculat ed. If t hese modes have w idely separat ed velocit ies, especially when including int eg rators, normal order reduct ion met hods may result in great errors in t he modes re

Consider the DPM in F ig 1. G ( s ) has t he

y= 2 1 x First, tw o controllers are designed. With

combinat ion procedure. So, in this paper, SF MOR is used to reduce t he controller order. T his method

LQR loop as t he t arget loop, H

/ LT R method

div ides t he order reduction procedure into 3 st eps:

plif ied t arget loop, this m et hod t urns out controller

yields cont roller ( C ( ( s ) ; w hen using t he sim

% 132 %

YA NG Gang , SU N Jian guo

) C ) ( s)

CJA

comparing t he orders of cont roller ( and ) , it is

C ( ( s) = ( 1195s + 1. 349 ∗ 10 s + 1. 615 ∗ 6

5 5

6 4

6 3

7 2

10 s + 7. 476 ∗ 10 s + 1. 592 ∗ 10 s + 7

6

7

1. 521 ∗ 10 s + 5. 283 ∗ 10 ) / ( s + 6

5 5

clear that cont roller order is reduced. T he order of cont roller ) is reduced compar ing to cont roller ( , but it is still t oo hig h for t he

6887s + 1. 218 ∗ 10 s + 8. 161 ∗ 105 s 4 + 2. 47 ∗ 106 s 3 + 3. 312 ∗ 106 s 2 +

cont roller t o be pract ically im plemented. T here f ore, t he order of controller ) is furt her reduced

1. 543 ∗ 10 6 s )

by SFM OR.

C ) ( s) = ( 3353s 4 + 3. 655 ∗ 105 s 3 + 3. 101 ∗ 6 2

6

10 s + 7. 787 ∗ 10 s + 5. 048 ∗ 6

5

4 4

10 ) / ( s + 1. 855 + 10 s + 2. 58 ∗ 5 3

6 2

6

10 s + 1. 155 ∗ 10 s + 1. 426 ∗ 10 s) Next , cont roller ( and ) are connect ed to G ( s) according t o Fig 1 t o do some simulat ions, respec

Analy zing controller ) , its modes can be easi ly got: 0, - 1 9989, - 5 9583 + 1 728i, 5 9583 - 1 728i, - 18536. Clearly, t he f irst mode is much slow er t han t he ot hers. Cont roller ) is t herefore easily div ided into t wo subsyst ems as follow ing C ) ( s) = C ) , 1 ( s ) + C ) , 2 ( s)

t ively. T he result s are plot ted in Figs 4 and 5. In one of t he simulations for each cont roller, t he unit st ep input is cont am inat ed by a zero mean white noise of variance 2n = 0 1.

3 54 s 3 C ) , 2 ( s) = ( 3349s + 2. 998 ∗ 105 s 2 + 2. 188 ∗

where

C ) , 1 ( s) = 6

6

4

10 s + 3. 698 ∗ 10 ) / ( s + 1. 855 ∗ 4 3

5 2

6

10 s + 2. 58 ∗ 10 s + 1. 155 ∗ 10 s + 6

1. 426 ∗ 10 ) Since t he f irst subsyst em cont ains only one state, its order doesn∃ t need reducing anymore. But the second subsystem has 4 st at es, and w ith Schur m et hod t he reduced syst em can be got as fol low ing F ig 4 U nit step respo nses of controller (

5

3291s + 2. 638 ∗ 10 s + 1. 817 ∗ 105 s + 1. 076 ∗ 105

C ) , 2r ( s ) =

2

which contains the follow ing t wo modes: - 5. 922, - 18167. It∃ s very clear now how Schur method recombines the modes of C ) , 2 ( s ) . F inally, the re duced order cont roller + C + ( s ) result s by recom bining the t w o reduced order subsystems 2

5

5

1152s + 1. 144 ∗ 10 s + 1. 323 ∗ 10 3 2 4 s + 6585s + 3. 865 ∗ 10 s which cont ains modes: 0, - 5. 922, - 18167. C + ( s) =

F ig 5 U nit step respo nses of controller )

As show n in Figs 4 and 5, t he cont roller w ith

F or comparison, C ) ( s ) is direct ly reduced wit h Schur met hod. If C ) ( s) is reduced to a 3 or

reduced order t arget loop works almost as w ell as cont roller ( does. It also rejects noise eff icient ly.

der syst em, controller ,C ,( s ) w ill be got

T he variance of t he noise aft er controller ) be

C ,( s) = ( 3349s 2 + 3. 067 ∗ 105 s + 2. 823 ∗

comes 0. 0013443 ( 0. 0012977 for cont roller ( ) .

10 ) / ( s + 1. 855 ∗ 10 s + 3. 287 ∗

It∃ s much sm aller t han 0. 1, t he variance of the o

105 s + 1. 909 ∗ 106 )

6

3

4 2

riginal w hite no ise. T heref ore, a conclusion can be draw n: H / L T R wit h reduced order t arget loop

wit h modes: - 8 624 + 4 943i, - 8 624 4 943i, - 18536. Clearly, in t he modes recombi

w ould not deg rade cont roller performance. But by

nat ion procedure, t he int egrator disappears.

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R educed Order H

% 133 %

/ LT R M et hod for A eroengine Cont rol System

F ig 6 shows singular values of t he errors be

achieves almost the same performance as t he un re

t ween C ) ( s ) and the t wo reduced order con

duced controller does. Cont roller , fails t o elimi

t roller. T he solid line st ands for the error bet ween C ) ( s ) and C + ( s ) : E + ( s ) = C ) ( s ) - C + ( s ) ,

nat e st eady state error. T his is because Schur met hod has incorrectly eliminated t he int egrat or

and t he dash dot ted line st ands for t he error be

from controller ) . T he variance of the noise af ter

t ween C ) ( s ) and C ,( s ) : E ,( s ) = C ) ( s ) -

cont roller + is 0. 0013312. It is also close t o that of t he un reduced cont roller. T he f ollow ing conclu

C ,( s) . From Fig 6, it is clear t hat error E + ( s ) is much smaller than error E ,( s) . T his also t est i f ies t o the declaration: SFM OR is part icularly fit t ed for H / LT R cont roller, w hile direct ly apply ing Schur method w ould result in bigger error.

sions can now be draw n: SF MOR is part icularly fitt ed for H / L T R cont roller; t he effect on t he performance is almost neg ligible; SF MOR is more eff ect ive t han Schur method for H / L T R con t roller. Example 2 Nex t , a low er order controller w ill be designed for a certain t urbofan eng ine. F irst derive a linear model from a non linear model of t he t urbof an engine model, at H = 0, M = 0 and int ermediat e pow er level( DPM) . Incorpo rating actuat ors in t he linear model, its A, B, C, D matrices can be got as

Fig 6

Singular value plot of er ror functio n

Nex t the reduced order cont rollers are con nect ed to G( s ) to do sim ilar sim ulat ions as F igs 4 and 5. T he simulat ion results of controller + and cont roller , are plot ted in Figs 7 and 8, respec t ively.

- 4. 6855 1. 7323 A = - 0. 5565 - 2. 7219 0

0

0 1. 1791 1. 9258 , B = 0 10

- 10

C = [ 1 0 0] , D = 0 Once the DPM is got , it is possible to design H / LT R controllers wit h f ull order and its lower order compartment s. Because the design procedure is almost t he same as Ex ample 1, only t he result s of tw o controllers will be g iven: ( 1) the f ull order cont roller C lqr ( s) ; ( 2) the cont roller C r ( s) w hose order is reduced by SFM OR. T he first cont roller, i . e. , Clqr ( s ) , has t he follow ing representat ion 8

7

4 6

Clqr = (107 7s + 4947s + 9 447 ∗ 10 s + 9 75 ∗ F ig 7 U nit step respo nses of controller +

105 s 5 + 5 93 ∗ 106 s 4 + 2 173 ∗ 107 s 3 + 4 668 ∗ 107 s 2 + 5 331 ∗ 107 s + 2 43 ∗ 107 ) / ( s 9 + 1186s 8 + 4 228 ∗ 104 s 7+ 6 141 ∗ 105 s 6 + 4 663 ∗ 106 s 5 + 2 ∗ 10 7 s 4 + 4 831 ∗ 7 3

7 2

7

10 s + 6 02 ∗ 10 s + 2 898 ∗ 10 s ) When it is used to cont rol t he non linear turbof an eng ine model, it g ives result s as show n in Fig 9 and 10. T he aeroengine model is contaminated by a F ig 8 U nit step respo nses of controller ,

Simulation result s show that only cont roller +

zero mean w hite noise w it h variance 2

2 n=

641 3493

( rev/ min) , and t he data are obtained from t he statistics of a real semi physical simulat ion t est

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YA NG Gang , SU N Jian guo

CJA

st and of aeroengine control system . Fig 9 is the re sult of a step response and F ig 10 is the result of an accelerat ing process. T he control schedule f or t he turbof an engine is t o maintain t he fan speed t o be const ant .

F ig 12 A ccelerating line of aeroengine controlled by Cr ( s )

First of all, t he order of Cr ( s ) is 3 w hile that of Clqr ( s) is 9. T his means that t he cont roller or der has been reduced to an accept able ex tend. F ig 9

Step response of aeroeng ine

T herefore, t he order reduct ion procedure proposed

controlled by C lqr( s )

in t his paper is effect ive. Next , f rom the simulation results, it is clear that t he reduced order controller Cr ( s ) performs almost as w ell as t he un reduced cont roller C lqr ( s) does. Cr ( s ) also reject s noise effect ively. And t he simulation results show that C r ( s) is a robust con t roller as Clqr ( s ) . T his can be easily demonstrated by the f ollow ing t hree reasons: ( the controller is

F ig 10 A cceler at ing line of aeroengine controlled by Clqr ( s )

T he second cont roller, i . e. , C r ( s) , has t he follow ing f orm 326. 8s 2 + 4359s + 1. 064 ∗ 104 C r ( s) = s 3 + 3102s 2 + 1. 171 ∗ 104 s Cr ( s ) is also connect ed to the aeroengine model t o do sim ilar simulations as Clqr ( s ) . T he result s are plot ted in F ig 11 and 12.

designed using t he linear DPM , but of fers good performance on the nonlinear model; ) T he accel erat ing simulat ions show t hat t he controller offers good performance in a w ide range of t he pow er lev el. It should be pointed out t hat in t he accelerat ing process, t here are tw o kinds of controllers that work: one w orks when n 2 rotates slow er than 85% of its design speed, the ot her w orks w hen n 2 ro t at es faster than 85% of it s design speed. T he con t rollers developed in t his paper t ake in force when n 2 rotates faster. + F urt her sim ulat ions at ot her points in t he f ull envelope ( Simulat ions in t his pa per are at sea level, st at ic point in t he envelope. ) of t he turbofan engine are also done. T he simula t ion result s show t hat cont roller Cr ( s) does offer good performance at t hose points. But because of the lengt h lim it at ion, t hese results are omitt ed.

Fig 11 Step response of aeroeng ine controlled by Cr( s )

Aft er t he controllers are designed and simula

Now t he conclusions can be drawn: t he order reduction procedure proposed in t his paper can ef f ect ively simplify the H / L T R controller, but it

t ions are done, the tw o cont rollers can now be

would not deg rade controller performance and ro

compared.

bustness.

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R educed Order H

5

LQ G / LTR control: an H

Conclusions

( 1 ) T his paper develops a new H / LT R met hod. It can result in a low er order controller, but does not degrade performance and robustness.

[ 4]

A C 34( 7) : 729- 733. [ 5]

T rans on A ut omat Cont r, 1981, A C 26( 1) : 17- 31.

part and a f ast part according to t heir velocit ies.

[ 7]

that t he reduced order controller designed in t his paper has good performance and robust ness for t he turbof an engine. References 杨刚, 孙健国. 一种新的控制系统 H / LTR 设计方法 [ J ] . 航空学报, 2004, 25( 2) : 104- 107. Y ang G, Sun J G . A new H

/ LTR m ethod for control sys

t em design [ J] . Chinese Journal of A eronaut ics, 2004, 25( 2) : 104- 107. ( in Chinese) . [ 2]

Chen B M , Saberi A , Sannut i P. A new st able compensat or design for exact and approximat e loop t ransf er recovery [ J ] . A utomat ica, 1991, 27( 2) : 257- 280.

[ 3]

Han K C, Hsia T C. On reducing compensator bandw idt h of

M oore B C. Prin cipal com ponent analysis in linear systems: cont rollabilit y, observabilit y, and model reduct ion [ J] . IE EE

[ 6]

SFM OR show s its superiority to Schur met hod. ( 3) T he second example in sect ion 4 show s

Safonov M G , Chiang R Y . A S chur met hod for balanced model reduct ion [ J] . IEEE T rans on A ut omat Cont r, 1989,

met hod: SFMOR. SF MOR is very eff ect ive for cont rollers w hose modes can be divided int o a slow Part icularly, in the ex amples ment ioned above,

optimizat ion approach [ A] . A CC

[ C] . 1990. 924- 929.

( 2) T his paper develops a new order reduct ion

[ 1]

% 135 %

/ LT R M et hod for A eroengine Cont rol System

S hrider A , G eroger J C, X in Z, et al . H

control design for

a jet engine [ R] . A IAA 98 3753, 1998. Edmunds J M .

Cont rol system design and analysis using

closed loop N yquist and Bode arrays [ J ] . IJC, 1979, 30 ( 5 ) : 773- 802.

Biographies: YANG Gang Bo rn in 1976, receiv ed his Ph. D. degree from Nanjing U niver sity of Aeronautics and Astronautics in 2004. He is specialized in aer oengine modeling and robust contr ol. Presently he is do ing post doctorate researches in China Av iat ion M otor Contr ol System Institute. E mail: yg 203@ yahoo. com SUN Jian guo ( 1939 ) is a professor and dir ector for doc tors in Colleg e of Energy and Po wer Engineering, N anjing University of A eronautics and Astr onautics during 1982 1984, he was do ing resear ch wor ks in Columbia U niv. in A mer ica. He is specialized in modeling, contr ol and diagnosis o f aeroengine, robust control and integrated flight/ pr opulsion Control. E mail: jgspe@ nuaa. edu. cn