Design of a Multivariable Helicopter Flight Control System for Handling Qualities Enhancement

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t~o ;

Design of a Multivariable Helicopter Flight Control System for Handling Qualities Enhancement

I~'r

L. Garrard

aeeretl

wn

Bradic\, S, Liehst

eslpdtrWilliam

IIn'rXsir\ ("t A11juuutcsli Al innutapue AlL1nLI'h

u,

Vi

i Ietu fri Iniaat c PauI1(Aol A? I Fol (

hnow Ct(.01t

Nehandling qualities specifications aecrntbig evodfratckhelicopters. Mlost unautimented helicopters is ill not meet these specifications and feedhack control is nercessari, to improse handling qualities so taf ae (operation close ito the earth in pou~r sseather conditions and or at night is possihle. In this paper a methodolog) for the direct design of helicopter flight control sisterns Ahich meet handling qualities specifications ipresented. t'his methoduilogL uses full state feedback to place closed loop eigensalues to achiese handssidth specificatioas and to shape closed lo0p vigensectors to decouple lateral and longitudinal responses to control inputs, l-ull state feedback requires that all state sariahles he knossn: hosseser. onlN angular rates and normal acceleration are measured hi, sensors. Thus. a state estimator is required in the feedback loop in order to eonsert sensor outputs is) control inputs. 'This estimator is designed using eigenstrueture assignment so as to achiest looip transfer recoseri. D~esign of a ftedhack sYstem for use in precise hosering control for a modern attack helicopter is used to illustrate the method. Control las s syinthesis is accomplished using an eighth order model ,Ahich includes ortli, rigid bod ' modes. Control lais performance is etaluated using a 37th order model oshich includes, rigid hod), actuator. rotor. sensor. and flexure dfsnamics. It is found that a notch filter must he added to the design in order ito eliminate a high frequvnc instabiliti. Once this is accomplished, hoth the tinme and frequenci response characteristics of the augmented helicopter are much improsed conmpared ssith the unaugmented helicopter.

Notation

11 = lateral cyclic pttch. dcg = tail rotor collective pitch. dc,U4 (trAl = mtntimum singular value (if mnatrix A (rAl) maximum sinizuiar value of mnatrtx A = pttch angle. rad Ii =roll atnglc. rad 1P

Scalars

it p q

It

arbtrarix numtber oMcteensalues number of controls number o1 states P.=normal acceleration of c. L. .:' = roll rate. rad sec =pitch rate. rad sec = ya~k rate. rad'scc Laplace operator forward veloct . ft, sec lateral velocit. f't scc downward veloeitl%. tt sec t-th transmtsston Zero) U, collective pitch. deg 1, longitudtnal cyclic pitch. deg

Vectors 14

= control vector. I u. u-

14, A .i V Zp

command control vector from control mixer = state vector. In. z-. wi. p. q. r. t;.fl =estimate of state vector = measurement vector. 11p.q. r, n-11 =pilot input command vector. Icollective. lonugttudtnal. lateral. directional I'

ii

.

u4

Matrices

A

= open loop dynamics matrix =

___8_

This is a reised version of a paper presented at the 43rd Annual Forum (if the American Helicopter Socieno. St Louis. Mo . Mas. 1987

control dtstribution matrix

C

= measurement distribution matrix, state vector

D

=

measurement distribution matrix, control vector

23

Ifi 2

02

24 Es

,\.L GARRARD

= = FMu = H = KA.I = K L

multiplicative error matrix. G, Is) G-,x. open loop transfer matrix. C(l.v - A) 'B full state loop transfer matrix. Ktls - I) 1B control mixer for pilot commands compensator transler matrix. K!.. - A BK LC; 'L = feedback gain matri\ = estimator vain matrix

Superscripts T = transposed - I = inverse Subscripts 8 81thorder model = 37th order model 37 Introduction ecentl\ a number of papers have appeared %%hich discuss the application of various modern feedback control desin techniques to helicopter flight control synthcsis. In the past. classical single-input-sinele-output O frequency t' response tcchniques have been used to design control la,,\ s for helicopter flight control systems. Hiwever. since helicopter responses to control inputs are highly coupled, helicopter dynamics arc characterized b% multi-input-multi-output NIIMOi mathematical models and use of classical SIS() techniques ma. require a creat deal of time. consuming trial and error effort. Modem MIMO technique, are ", ell suited to the design of control la\%.s for hclcopters, and numerous papers have appeared w\hich describe such applications These include linear quadratic regulator theor\ (Refs. 1-2). multi'ariable loop shaping IRet. 3). t defIfollow in- Ref 4). optimal output feedback deswin tRef. i. and 1t' techniques Ref. 6). The advantages and disadsan tage, of some of these techniques arc discussed in Ref.5 The obtlectie of this paper is to present the appliation of another technique. eigenstructurc assignment, to the design of a hclicopter tlight control system. Etgcnstructurc assignment is a technique for synthesis of feedback control las, s which allows the designer to directly place closed loop cigenvalues and ciecnvector. in specified configurations. These configurations are selected so that the closed loop response characteristics of the controlled helicopter satist\ handling qualities specifications. In this paper eigenstructure assignment is used to synthesize control laws for precisc control of forward. side and vertical velocity and yaw rate for an attack helicopter in hover. It is desired" to have pilot longitudinal stick commands correspond to forward velocit. lateral stick to side velocity, collective to vertical velocity. and pedal position to yaw rate. Coupling between longitudinal, lateral, vertical veiocity and vas'. modes is to be minimized. The closed loop band. idth must be large enough to insure crisp response to pilot inputs and the closed loop dynamic response should be stable in the presence of errors in the design model. due to effects such as unmodeled dynamics, nonlincaiities. and variations in parameters, The rest of the paper is divided into five parts. First, the mathematical models used for both control system design and evaluation are discussed. Next. the performance requirements which the closed loop helicopter must meet are described. The design of the control system is outlined in the third major section and the evaluation and modification of the design is given in the fourth section. The last section consists of conclusions and suggestions for additional work. R

Mathematical Models The helicopter modeled in this stud\ is a modem attack helicopter similar to the YAH-64 (Ref. 3). The control laws are designed using an eighth-ordcr rigid body model and eval-

JOURNAL Of THE AMERICAN HELICOPTER SOCIET uated using a thirt.-scventh-order model which includes actuator. rotor, sensor, and flexure d. narnics. Main rotor collectisc pitch. longitudinal cvclic pitch. lateral cyclic pitch. and tail rotor collective pitch are the control inputs. Threc bod\ rate gyroscopes and an accelerometer which measures nornial acceleration are used as sensors. The mathematical model used is semi-empirical and wkas developed for the hover flight condition. The linearized rigid body equations of motion are expressed in standard state variable form as = ,'.

-

Bi

I)

The A. B. C. and 1) matrices \,ere obtained by numerical Itnearization of a nonlinear analytical mod,.l of the helicopter and are Liven in Table I . The actuator, rotor and Sensor flcxural dynamics models used are given in FiL. I and wcre obtained from flight test data. Sensor outputs 'Acrc measured and recorded for pilot input, at various frequencies. The transfer functions shown in Fiv. I v,ere obtained b ittine assumed forms of the input and output transfer functions )ouplCd Ls th the model given in Eqs. (I) and 121 to the measured data and numerically adjustinp time constants. damping factor,. and natural frequencies until the trequenc. responsc of the modcl matched that of the helicopter. The \alidit (f this t. pc o model for flight control design o,as verified in Ref. 3 in s hich a flight control s\sten \.as desiened using a nodel similar t, that Liven in Table I and Fie. 1. This control \,.sticni '.as successfully fli,,ht tested. The open loop cigcn\alucs and non-dimensional cigcnectoirs of the design model are gicn in Table 2. Non-diniensionalization of the state yector was achieved bs dividine the lincar velocities b, 25 It se.. thc aneular rates b', 20; dce sec. and the angular displacements b\ 20 dce, These arc the Iaxinui values of the state variables expected during hover maneuscr, The control variables %kere non-dinensionalied b di\ idine by their maximum \alues of 9 de collectise. lI dee longptudinal cyclic. 8.75 dee lateral cyclic, and I8.5 deg tail rotor collective pitch. Examination of he cicenvc%.tors and elienvalues in I able 2 indicates significant coupling betf.cen lateral and longitudinal modes. Both forward and side %clocities exhibit lo . frequcnc. instabilities and as is discussed in the next section. the bandwidth in the pitch, vertical velocit\,. and yav directions is not large enough to guarantee level I handling qualities. In addition to the modal coupling, examination of the control distribution matrix B in Table I reveals strong control coupling bctween lateral cyclic. longitudinal cyclic, and tail rotor collective Onl\ main rotor collective Is relativel\ uncoupled. Time histories of open loop responses to a lateral cyclic step input are shown in Fig. 2. The response of the unaugmented helicopter is indicated by solid lines. It can be seen that the responses about all axes become very large due to the unstable cigcenvalues and significant coupling which exists between modes. The dynamic responses for the helicopter modeled in this study are typical of most high performance helicopters. Simulations and flight tests have shown that even experienced helicopter pilots are unable to accomplish relativel\ simple hover tasks in conditions of degraded visual cueing and or divided attentions tasks with such typical helicopter dynamics (Refs. 7. 8). However flight tests with variable stabilit. helicopters have shown that stabilit. augmentation is an eftectise method for compensating for missing visual cues and for use in situations in which the pilot must devote a significant amount of time to tasks other than piloting. Performance Requirements Three response types have been proposed to quantify mission oriented rotorcratt handling qualit. requirements fRef'.. 7-91.

OCTOBER 1990

MULTIVARIABLE FLIGHI CONTROLS

25

lable I State and (ain Matrices A= - .0286

.04)47 .0039 .0000 .0000

- .0637 -. 2311 - .0257 - .0500 .0118 - .0049 .0(00 .0011

.0205 .0059 - .2610 .0095 .0002 .0008 .0000 .0000

.4350

.5760

-. 1140

.136) .0575 - .06(10 -. 31111 .0097 .400(0 .00(1

.4910 - .02504 .6470

.00001

.1)00

.0O0

.0(011 .0008

.0000 .0081

.00(00

.0000

.0001) ,0000H

.0000

.00(0 .1327

.0000t(1 .(0018

.0000) .0008

.0(00( 0.0000

.0883

- .AMU4

.0779

.0046 .0079

.2290 -8.2900) - .3790 - 2.70001 - .0092 - 1.0500 1.011(1 .1100

7.9700 - 1.03001 2.2500 - .1340 - .7500 .4130 - .(0)51 .9990

- .2570 - 1.6400 2.1900 - .6620 .0244 - .400) .1031 .0499

.0000 32.0001 1.6000 .0000 .0000 .0000 .0000 .0000

- 32.0000 -

.1641 3.2811

0000

.0(00 .0000 .010 .(1101

B= -. 1581 -4.27(0 -

.0438 .11172

.08111 .o00 .00001o

.0009 .2820 .0012 .08110 - .11019 -0 .455 .0oo .0000 -

- .09(11

.20001 .0000 .0)00

1: = .0000O

.1t000 .0(00 -. 0001

1.00((

)1

00.000

.0000 .0118

.(XflO 1.1)1

.0000 - .0699

.0000

.(100

I .L3(1(1() - .0681

.0(00 .01011

.0000

.0)O00 .0000

.(0(0 .0000(t0

.00110

1) = .(f)O0

.0000X -

.0000)

H= .6735 -

.1258 .40897 .7622

- A751.1041) .11532 .3554

- .2381 .2533 .8386

-

.0255

.1103 - .8169

I= 1.14211) + 5 -

1.06571) + 05 9.74191) + 05 3.53811) + 03 1.61011)+03 1.67271)+144 3.9652D+ OI

1.06901) + 05

- 1.93421) + 04

1.00881) + 0

- 1.83881)

- 9.49971) + 05

6.09631) + 04

14

5.82621) + ()4

1.65481) + 05 - 3.32721) + 02

- 5.14871) + 05 1.47361) + 03

2.7094D + 03 -6.45451)+02 - 1.6531114 3.65221) + 0

4.88611) + 01

2.88231) +02 3.06051)+03 - 7.38530 + 00

6.13941)+00

- 8.36411D- 01

2.82941) + 01

-. 0541

-. 1145 -. 1784 - .00146 .0013

-. 1802 .0396 .0335 - .1782

-. 1214 -. 4143 .3469 24.1517

.5974 .0628 .2.46

- 1.11251)+03 -9,50081)+03 2.32231) +01

These arc as follows.

.9174 -32.4650 - .6961 - 18.3387

-. 5795 -3.7509 4.1961 -41.6498

-1.9272 -6.2656 6.2384 21.6354

4.(179

54.0280( - 2.9799 - 20.3557

I. Rate Command (RC) Attitude Command with Attitude Hold (ACAH 3. Translation Rate Command stth Position Hold (TRCPH)

considerable concentration is required for the pilot to perceive pitch or roll attitude and lateral. longitudinal, or vertical translation rates (Ret. 7). Th. use of cigenstructure a',sItmmcnt for the design of RC and RCAH has been discussed tn a preious paper (Ref. 10); therefore, this paper will concentrate on svn-

In RC systems. attitude must diverge from trim for at least 4 seconds following a step input command. In ACAH. a constant control input must produce a proportional angular dis. placement and must maintain this attitude in the presence of external disturbances. In TRCPH, constant control input must result in constant translational rate and the rotorcralt must hold position if the force on the cockpit controller is zero. TRCPH systems are preferred in nap of the earth maneuvers in fair to poor usable cue environments, In lair usable cue environments

thesis of a TRCPH using eigenstructure assignment Both clasical (Ref. I I) and modern handlhng qualities. Itterature (Refs. 7-9) indicate minimum bandwidths of 2 rad'sec in pitch. roll. and yaw rate and/or attitude and 0.25 to 0.75 rad/sec in vertical velocity arc required for level I handling qualities. In addition. coupling between lateral. longitudinal. vertical velocity, and yaw modes should be mintmized. Bandwidth requirements of 0.2 to I rad,sec have been postulated for TRCPH (Ref. 12).

26

W \.L. GARRARD)

JOURNAL iF THE AMIERICAN' I-LLIC( )TEfR

Roior[Actuator Dynamics

Flexure/Sensor Dynamics

3

1

2

5010

-

§846

Dynamics

itae qJ

R.r-

--

~~Rigid Body

-

SOCIFTh

_____________________________

i

4

(sec)

A.Time

22

Time (sec)

6

_____.C)

5s

Fig. I C.ain, pole, and zero, locations for rotoractuators and Ilexihiei modes: AII for real eigen%alues. iw,; EI for complex eigensalues.

-

Time (sec) Feedback control ss stems must maintain stability in the piesence of1both uncertainties and errors in the mathematical models, used tisr desien and in %ariations in sssteni parameters during actual operations. This is termed stahilits%robustness. Stabilit% robustness is usuall specified indirectly in terms ot ,ain and phase margin. Minimium Lain margins ot 6 db and minimum phase margins of 45 arc t\spica] stability criteria used for control la%% desi.-n.3

(sec)

-Time

or

'a

*

C -

Time (sec)

(sec)

Cnrl aASiihssTime The control lass design process is performed in isso staces tor this studs. First, a lull state reizulator is des eloped us"ing cicenstructure assignment. Even though good deeoupling of the closed loop cigenvectors is aehievcd. control coupling ol the helicopter is so g~reat that control command misine is required Both the feedback control la\s\ and the mixer arc desiened assunmne lull state feedback. The feedback control lass cannot be implemented dlireetlh since all system states cannot be measured. Thus it is necessar\ to realize the control lass b means, of a state estimator in the feedback loop. The statec estimator is synthesized using an cigenstructure assignment technique w&hich results in recovery oif the loop transfer properties of the lull state regulator.

Fig. 2 Open loop and full %lateclosed loop responses to a side! Neliciis\ open loop.--full state feedback.

command:

Full State Regulator The feedback control lass is a linear tunctiotn ot the state As

it

(%

The feedback L'ain matrix K is selected to Live a desired Closed loop eigenvalue'eicens'ecior configuration. The thcsrs for cigenstrueture assignment b\ feedback control is goisen inl

Table 2Open-Loop Eigensalues. Eighth-Order Model Open-Loop Eigen' alues 1. -3.2610 + .0000i roll 2. - .9760) + .0000i pitch 3. .08120 ± .62%i1 side veloclit. 4. .101)±.547i forward sel4,cils 5. - .2588*±. .0428i1 sass 'ertical Open-Loop Eillenvectors 1. U

r0

.3605 .0102

-

w P q

4'

.0024 -. 07814 - .0012 - .8921 .02681 .3407 .2814 - .00311

2.

.0525 .0373 - .6131 -

.3338 .0767 .6104

3. .0710) .4-3103i -. 2171 + .1269i .0040 + .0241i .1334 - .2352i - .0234 + .281721 .33812+ .4070i - .26"9 - .3026i .47815 + .0727i -

4. .0462 + -. 0979 + .0082 + - .00166 .0666 + .4026 + - .19812.58162 -

5 .4723i

.1297i AMU41 .1424i .2829i1 .2892i .1093i .0431i

.0244 .0616 + -. 0949 + .08150 + .41)1W+ - .7740 - .0231 + - .41257 -

.1983i .0423i .54681 .00814i .0270i .14849i .02314i .07"i1

27

ML'LTI\'ARIABLE FILIGHT CONTROLS

OCTOBER 1990

elements equal to one and arbitrar\ non-zero values associated with roll rate and roll angle. All other elements are zero. Thus the lateral modes. roll and side slip. are dlecoupled frornt the long~itudinal, vertical velocity and ,,a\%rate modes. The desired and cig~envectors associated w&ith the pitch cigenvalue. -2.9. the forward velocity cig~envectors associated with -0.801 0.3871. are selected in a similar manner to decouple the iongitudinal modes from the lateral, vertical velocity and .axk rate modes. The cigenvector associated with the vertical velocit\ cigenvalue. -I .0. is selected so that all components are zero except the vertical velocity., which is unity. Similarly,. all elemenits of the cicenveetor associated wkith the yaAs rate mode. - 3.0. arc selected as zeros. except for the element associated with the vawk rate which is chosen to be unity. The attainable eicenvectors, for a unity weighting on the squared error betwveen all elements of the desired and attainable een vectors are shown in Table 1. Examination of the cieensectors associated with roll indicates excellent decoupling bct\ecn this~ and all modes. except the side velocity.. The pitch mode is also decoupled from all modes except the forward vcloeii '1ask and vertical velocity modes are also dlecoupled from the other modes. There is mild va" coupling in the side \clocit\ cigenvecter and a more severe vecrtical velocity coupling, in the torssard velocit\ eigenvector. Responses wecre significanil'

Refs. 13- 17. The details of the application of the theory to the problem described in this paper is given in Ref. 17: therefore. this section will be limited ito a discussion of the philosoph\ of design and a presentaton of the results. The desired cigenvalues and eigenvectors for the final design are shown in Table 3. The desired eigenvalues were selected to satisf\ the handling qualities specifications described above. The roll eieenvalue was selected to give a roll bandwidth of' 3.5 rad sec. a pitch bandwidth of 2.9 rad see. a yaw%bandw,%idth of' 3 rad sec. and a vertieal veloity bandwidth of I rad see. The cigenval ues associated with the forward and side veloeities were selected to be complex with a natural frequencN of" 0.9 rad sec and a damping factor of 0.9. This resulted in bandwidths of about I rad sec for the forward and side velocities as \kell as the vertical velocits. The desired eiensectors are shown in 'Fable 3. The first desired ciicenvector associated %kithroll rate. 1p. is made equal ito unit\. Since the roll aneLe. Lr. is the integral of the roll rate. the element associated \%ith roll anele is the inverse of the roll cigenvalue. Also. since some side slip is inevitable when the helicopter is rolled, an arbitrary non-zero element is associated with side velocity. Ali other element,, of this ci-en\ ector arc zero~ The tw'o comples. eio-enveectors, associated w ith the side (i .1881;, ha~ e side velocit\ velocnN complex roots. - 0.802

_Table 3

Closed toop Desired and A-tainahle Eigcnsalues and Eigenscctnrs

Closed-Loop Desired Eigensalues 1. 2. 3. 4. 5. 6.

- 3.5000 +s - 2.90(H) +- .8020 t - .8010 =

.00001i .0000i .3880i .3870i -1.LOW0 + .00001i - 3.0000 + .0000i

roll pitch side %elocit.% forward selociis hease sass

(losed-tLoop D~esired Eigen'.ector% I. .00)0H)

U

.0004) 1.040M) .04)0 AM(00 -. 3448

1.0000 .000 .011) .2857 .000W

44)

*~~~~~~

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

+ + + + X .0000 + .0001) +

1.00041 .0000 .000 .1)00

.000) + O(0H0i LO0W) + .04)0 .100M1+ .0488)i

.00)00 .0000

is.00

p q r

4.

3.

2.

.000 + .0K00i .00)00 + .0000i x .0000 + .4000i *

*

*

I

*

*

.10001 .010001 .0011.000)0 .00001i .00001 .00001

**. .

* *

6.

5.

***

.0000 .0000 .0000 .00(K) .0048) .04)18) . .

*

.00181 .1H)0 .0000 .0W8) .00001 1.0(9)(1 M041 .4888K)

.*

ILNITV WEIGHTING Altainable FEigenvectors u

1.0890 is0.0MI8

.0024 .9577 .0001 - .0002 - .27216 0.0000

p q r 4

o

-1? - .0235 - .0293 .0041 .9357 -. 0036 .41003 - .3223

.0118 .4745 - .0375 + .3724 +.00)92 + -. 0282 + -. 5925 - .0.%68-

.0107i .00100i .0192i .443 1 .0564i .0555i .2726i .0559i

.4254 .0131 - .2279 .0389 - .4185 - .0183 -. 0571 .6197

-.

000)0i

+ .0.143i + .0045i

+ .0346i - .39"i1 - .00)82i - .0171i + .1999i

-. 1003 - .0181 ."939 .0089 - .0290 - .0314 - .0089 .0290

-. 0112 - .0366 _ 0028 16; - .01373 .9986 - .0321 - .0142

-. 1003 - .0181 ."939

-. 0112 - .0.6 _ 0028 - .)65 - .0073 .9986 -. 0321 - .0142

FINAL. WEIGHTING Attainable Eigenvectors .0024 .0890 0.0000 .9577 .0001

U

1. Ws

p

q r-.0002 4-.2736 0

0.0000

- .1381

.0235 - .0293 .004) .9357 -. 0036 .000.1 - .3223 -

.0118 -. 0103i .4737 + .0000i - .0376 + .0193i .37201+ .4473i .0084 + .058 1 -. 0028 +.0056i -. 5943 -. 27051 - .0M19 - .0549i

.4143 + .0093 + - .0024 + .0408 + - .4350 -. 0192 -. -. 0606 -. .6440 +

00001 .0332i 00001~

.0377i

.0089

.4154i 00851 0 194i

- .0290

.2075i

.0290

-. 0014 -. (X)89

JOURNAL OF THE AMERICAN HELICOPTER SOCIETY

W.L. GARRARD

28

35

dcecoupled. except in the two cases mentioned above. This was corrected by weighting the error between the desired and attainable vertical velocity element in the forward velocit. cigenvector by a factor of l0. The vaw coupling. while not serious, was also reduced by increasing the weihtint on the error in the yaw, direction between the desired and attainable side velocity eigcenvectors by a factor of 10. The attainable eigenvectors for this set of weighting terms are shown in Table 3. The resultine time histories for lateral step input for this design are shown in Fig. 2 by dashed lines. It can be seen that excellent decoupling has been attained. The forward velocity is nearly zero. as is the vertical velocity. Yav, and pitch ancular velocities and pitch angle are also nearlN zero. The roll anglc overshoots its steady state value enough to produce a ncarl\ constant side acceleration. which exists until the commanded side velocity is achieved. The roll anglc then decreases until the side force is sufficient to counteract the steadx state drac in the lateral direction. thercb maintainine constant side yCSimilar results were obtained for longitudinal. collcclocit%. tive. and directional commands. 1 he gain matrix. K. for this design is gven in Table I. In single-input-single-output (SISOI sN'stems. stahilit\ robusiness is measured b\ pain and phase margnns obtained from Bode or N.quist diagrams. For ,ariations in the input gain and margins phase. the MIMO equivalent of these classical stabilit\ is the minimum sincular value (,MS\'j of the return difference matrix Il/ KIs)(10i for s = iw (Refs 18-2 11 The MIM() gain and phase margins can be expressed as

o 2 > M "5 0 E

i 25

..._

20 I

15 ................................ 10

0

2

t0-

;0

10110

Frequency (rad/s) Fig. 4 Minimum singular values of closed loop transfer function and maximum singular salues of multiplicatihe error matrix s.Irequenc .

convert pilot and a fixed Lain control mixer x\as required ito inputs to control inputs (note the control mixer dcecribed helokN was used in obtaining the results in Fig. 2). The control mixer was designed b\ calculating the control i required to produce a given steado, state response. The details ot this desien arc given in Ref. 17. The resulting control nixer Lain matni,. tt. is given in Table I. The closed loop s stem equation, arc rto.k A = 1. -

BK!

.

+ Btt

,5

Gain Mnarcin G.M) = 2(1 lot( I (I- MS\I ') where (5, represents pilot command. =

Phase Nlargin (PMI

! cos

!t

--(%lS\

C 2i

(4) Compensator Design The control la" dceeloped above require, knov,ledge ot the conplete state vector. Since onlx measurements of roll. pitch. and vas rates and norml acceleration are aviilahlc. a stale estimator is required in the feedback loop. The control i,

For a minimuni singular value of units - the gain margins arc infinit, and 6 db, with no change in phasce at the input, and the phasc margins are _ 60 dcL with no change in gain at the input, The mnum singular value ol the return diflerence matrix tor the full state design is plotted solid line) versus frquenc in Fig 4. The minimum singular value of 0.67 occurs at 2 5 rad sec. From Eq. 4 this corresponds to gain niargins of9+63 db and 4 45 db and phase margins of -39 dcc. Since the math model of the helicopter is felt to he reasonably accurate in this frequenc range. it was felt that these margins \xerc sufficientl.o large to cuarantee stability. This proved not to be the case'. hoe\ver.

i=

K.t

(0

where . is the estimate ol the state civen bs the slate estmtor A = A. - Bit

-

li N - C' - I)u 1

)

a\ Itiswell known that an estimator such as piken b\ Eq not recover the stabilitx margins of the lull state controller (Refs. 18-211. In fact, the stability margins ma\ be vcr\ poor. (ontrol Mixer Design even though those of the lull state controller max be excellent an estimator hch recoxr the lull Even with the modal decoupling inthe closed loop system.22. apstate loop translcr stabilit\ properties can be designed the control coupling resulted i excessively coupled responses propriate selection o1 the estimator gain matrix L such that: (I)j of the closed loop eigenvalucs of A - LC approach the finite transmission zeros of the plant. G.). (21 the remainin 1 - j closed loop cigenvalues approach infinit\, and (3 1 the 10t Z 10 left closed loop eigenvectors of A - LC associated ,xith tfhe . finite cigenvalues approaches the left zero direction ot the finite .. I transmission zeros. This results in a controller in which the lull 8 I\ C is approximated bo the coinloop transfer matrix F(.-) \" -state i pensator loop translr matrix KI.oGI., up to the frcqucnc\ of 6 E the infinity poles. -.ES ......... -- ulls, The finite transmission zeros of G(.%) ,.ere calculated as Wn E ! Ole, r"1 % --i

fI -- ,- r, ift:

I

0 L" .. ................ .. ....... t '101,

10

.es

o~.,

0.0

,o

0.0

1000

... +

10

--.......... 102

2

Frequency (radls) Fit. 3 Minimum singular values vs. frequency for full state feedback and various estimator pole locations.

0 .0 0.02147

:4

O 11224

29

NIULTIVARIABLE FLIGHT CONTROLS

OCTOBER 1990 In designing the estimator, the gain matrix L was selected to place the two of the estimator pole , at :, and :, and two of the estimator poles at -0.01 and -0.012 (these approximate the transmission zeros at zero). Initially the iemainine estimator poles were placed at - 10. - 12. and -5 t8.666i (these approximate the transmission zeios at infinity). The resulting singular value plot is shown in Fig. 4. Stability margins are almost nonexistent. The "infinity" pole locations were increased by a factor of ten to - 100. - 120. -50 ! 86.666i. The singular value improves substantially but is still unsatisfactor\ at high frequencies. Finally. the "infinity" pole locations were increased by another factor of eight to - 800. -960. -400 = 693.3i. This pole configuration resulted in essentially full recovery of the full state singular values over the entire frequency range. The resulting estimator gain matrix. L. isgiven in Table 3. Evaluation The design developed above using the eighth-order model as evaluated using the 37th order model described in Fig. 1. The MIIO generalized gain and phase margins of 9.63 and - 4.45 db and = 39 degrees were felt to be adequate to provide stability in the presenceofunmodeled dynamics. However. the cloed loop 37th order system exhibitet an instability near 30 rad s due to coupling between the main rotor collective.actuator dynamics and the sensor flexural dynamics as measured by the normal accelerometer. Instabilities resulting from rotor and sensor dynamics have been noted by Chen and Hindson (Ref. 23) and Hall and Brvson (Ref. I ). To better understand the source of the instability the multiplicative error matrix. E(s). between the 8th order model and the 37th order model was computed at various frequencies. This matrix is dcefincd (Ref. 18) as

1 s3 10

2

,

61

5s

I >

0,. 0

30

2

Time (sec) 1

(9)

A plot of these tv o functions is shown in Fig. 5. The large error peak near 30 rad:s crosses the minimum singular value curve verifying the source of the instability. To alleviate this problem one might increase the order of the design model to include dynamics in the 30 rad/s range and completely redesign the compensator. However. a filter on the accelerometer signal which notched out the set of complex poles at 30 rad's and rolled off at 200 rad/s was implemented and was found to eliminate the instability. Transient responses to pilot commands for the 37th order model with the notch filter are given in Fig. 5. Figure 5 illustrates a pure side velocity maneuver resulting from step pilot lateral input, of I ft/sec. In this case longitudinal, vertical velocity, and yaw responses are minimal. In general, the responses in Fig. 6 are very close to the full state responses of Fig. 2 with the exception of small lags due to actuator dynamics. The control deflections required to achieve these responses were not large (Ref. 17). Closed loop transfer functions between pilot commands and system outputs are shown in Fig. 6. These figures demonstrate (as the transient responses did) that the closed loop system is now characterized by simple decoupled first order responses over the desired bandwidth range. The vertical velocity. forward, and sideslip transfer functions are flat until about 0.8 rad/sec and then begin to roll off. The yaw transfer function is flat until about I rad/sec then rolls off. If necessary the bandwidth on these transfer functions could have been increased by increasing the natural frequency of the complex closed loop eigenvalues.

4

2

4

5

o 5oM, 0 0

1

2

3

C

1

3

Time (sec)

Time (see)

5

,

a, 10 5 -

Time (see)

To be assured stability in the face of modeling errors E(s). it is known (Ref. 18) that at all frequencies

3 1 2 Time (sec)

o

Gs = G. (.(I -4 E(s)((8

(T(I + (Ks G8 )- ii (E)

0

Time (see)

Fig. 5 Closed loop time responses to a side velocit command-37th order model.

Conclusions The eigenstructure design techniques described in this paper provide a useful method for the design of control laws for helicopter flight control systems for precisin hover tasks. Use

M 10 -o' °

"

1 10

0

100

-

o - 10' > Z

10

-

:02

,-* 10

0o

:0'

2

m0

.

U. 1

Frequency (radis)

0'bo? Frequency (tad s)

-0 :o0 0 , - :0 ' a =_ "' :03 . 10014 1(2 0 1 :0' '0 : ih Frequency (radle)

. "

I,

C1 130

:'

Frequency (reds)

Fin. 6 Bode ploLs or clo ed loop transfer functions between input commands end system outputs.

JOURNAL OF THE AMERICAN HELICOPTER SOCIFA

W.L. GARRARD)

310

of MIMO gain and phase margins based on minimum singular values of the loop gtain did not predict an instabilit% due to actuator/rotor and sensortllexural dynamics in the c'ollective control loop. This instabilitv "~as eliminated bx filtering the accelerometer output through a notch filter. The 'Use of an error model derivsed from an estimate of the dvnamics neelcted in the design model did predict the instabiity and it is recontmended that this approach to evaluating stabilIitv marginms be used in the future. The state estimator in the feedback loop required high gains. This might cause problems in actual iniplementation unless the sensor outputs are filtered. Also, the effects of ditzital implementation of the control laws nmight cause difficulties if the sampling and computational rates are sufficientl\ sloss . but this is dependent on hard%%are considorations ansnot addressed in til, paper. Acknosslidgment This research \%as supported b\ the Arm\ Research Office under contract DAAL03-856-K-1f056. Mir. Miark Eksblad and Mr., Joseph Anderson. Research A\ssistants at the Universits of MNinnesota. provided invaluable assistance. All computini- and graphics presented here "sere performed on Apple Macintosh in AControl" , a control svstent destn packcompuers u Mrf. Jerr\ Farm age developed bsProfessor Bradle\ Liebsi and * of the U.nis ersits oit Minnesota. References Hill.\ Controller

Brisin-\ I3. Inlso W. -lit, att. \,ol li4,. I~,' t h pl icat in oIi Quai i. hlil theis I ipsr lla CI-1 'mun!ori/i H

1. Jr arid tesi,-ti

55 knsend.

If

i

ittsrii

n

Apt11173 ( li

pt i a I Clin tr~l Siti /l/i

111

55icr.

\ol 32'. 1I1i. Jiln 11 t,' allii is Hicopici . Enrir. tDale. -'stultisriabic t ilir (Ilitrl Poedi, It 1w I 515 Auitiatn, (oiiro!i Coflctrence:. 'sa Juiil) firle [)esir ota 'sodelimr -l llli ing 'ffltcbn K 13 . and Btilller, . r iHelicopters.' Paper No X-41941. Al-S's (idanc Colrlll 'S\,i ste and Contl C.litteren c . Seattle:. 55asi . -Sue I 1)Kotl cirg D 1)P . arid C Alie J .Coifiion 'Praad . J \ R Internrational Coii lnrot Ssstein Ikesigit Techniques.' Hicoprer Ilielti( terenc on Helciiopter H andiij LrQualities aiid Control. Londion. N, 1N "t uc . Andress . and PolsiIsI ati , tar. -Rllusr Hel icopter Colntrol Lass 5 to~r Handling QualIi es F ihancetrient . Inriernat litial (iliiercnc o il Hel, icopter Hanidioln Qualities and Control. Lonrdonri.No\s 1)h Hf. I' 'knato i Requ iremrerir s i tire N"~ Handilinc Qualities -Holt . k Speci ticatwi for I S5.1Itr Rlrrcratt.' International Cuunterence ol Hr~ilclrpner Handlinni Qualities and Cointroll. Lorndorn.Nirs 1915

~l

'Ke%. David 13.A Milirar

Rotorcrati.'

N"s Handling Qualilic' Spcit cain for I S International Conference oil Hisopicr Handliirii

Qualities and Control. London. No% 1988 d 'Hoh. Roger. et ail.. "Proposed A ir"sortiIness )esi en SiandarsiHn

~I

aIte RI ureei lltr ooeai ehRp xo 4 ashre ai S~in ehou~ n. tein ot A\iiiiudc and "Garrard. %VL.. oss. E . and Prouts_ S. Rate Commirand Systems, tor He licopter, I'sn, Licnsirrciurc Assignmiii.-*Ji'iinal o, Giiidamer ioi ) iiii \o!r~r II2. 1h'. No,

~c15

Dec 1989. "Seckel. Edssard. Srihitix nri co'ntrol (of Airplciu id Moon Academnic Press. NessYork. 164i. Ch,. XXtlt arid XXI\ 2 tDcveloprrien lit .t)(xw (l 1 Landis. K. H.. and (ilustitan. S I . troller, and Control La"s.\ ol I and 2. A'sARt)COS. N \S2 111550. uprsit sn cie lte 'Carr.W 13.adtesi Eiirenspace and Linear Quadratic IDesign lcchnisnes.- 1 w iin Control andl Dsniurnti. \siol S. Ma'- Junll) '-Liebst. HI S . Ciarraid. %\ 13..-arid Adinis.- \ M t IDcsl,_ii o! if] 'lit l'tlot Actise Flutter Suppression S~seie.- J ''t (;sitiI ,sirnil .ol S. jan- Feb lIQw* "Moore. B. C . "Oii the Flexibilit~ Offered hs Ilull Star,- I cedbaci, ii aluc .Asseriiiicr %lutliariable Ss[tems ttewnd Closed Loop l-1ieCnsJII oirtioL \ ll 21. 00i lIW6 l~l ulo IfLL 5 lranl o (lIAnltljr hnL JC ' Andrs . A N..SiroEY.an .Aineni for Linear S1sieiis

Sep 14853. '-Giarrard.

/L;. 1

\k 13.. ;Aid Liebsi. B S

A\nnual Forun & Tech nlles [Displ. t\a lI)S7 S, tar)ui. \tIol.. 'Ilisle. J C .an d Seiti. Gi .

(w'il'1vlfl

511

o;~sei i a Stultis riail (I ot tile %\riicri-attH clI i pic S, I

ultisariable teedhback lkcsicr:T Coi. A.I.'(it lia>oil o cepis ilor a Classical Mo1 dern Ss nthesi,. H,11. ol 26, Feb I181 i[Q(t 1WT Prolcdui-C tlo; sni '-'Steii. Gi. aiid \[hall,. Mt .. -iiIti. /Erri LL. 71,'l table Feedback Control t 1eb 17 F5ilaisue. Nc >Letiiirraki. N A.. Sanrdell . N R . and, Ailians. Mt .Rhu-timHii I I suits in Linreat Quadrati tBased Mu t iiarirblc Cl wiii '11r L)Csltj on Auto' Control. \oil 2(,. Ich 1S! Iln Tests I ii hmn_ t hc StIih in I. lot ' Lehiotiaki . N , It ai! . 'Robusttess alid Modelig Error.' Priiceedinrrrs ofi thc: 1N511.1[A ott ott tlcs in Control. Dec 151SI Ka/eriroi. H , and Hllupt . P K . "(In the Loolp ritister Re 5 115cm .4,. -S.I lInt Jiiirral (if Cwnl~trl. 1, 'nl m al and th W\ill air S " Cheni. Robert T N . and Hinrdsolrt. Ins estiat Inn of the Influenc ill Rotosrand tOther H igtr-(rder Ihs iil(in Helicopter Flig~ht -Counirll S5'sten Hand ssidlit. 'Proceed ire', Inirin urinal Conference ott Rllrrrat Research. Research Triarne- pcIark. oN 15