vijaya aeat 2008

8/12/2019 Vijaya Aeat 2008

1/13

Design of a stability augmentation systemfor a helicopter using LQR control and ADS-33

handling qualities specificationsM. Vijaya Kumar and Prasad Sampath

Rotary Wing Research and Design Centre, Hindustan Aeronautics Limited, Bangalore, India, and

S. Suresh, S.N. Omkar and Ranjan Ganguli

Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India

AbstractPurpose This paper aims to present the design of a stability augmentation system (SAS) in the longitudinal and lateral axes for an unstablehelicopter.Design/methodology/approach The feedback controller is designed using linear quadratic regulator (LQR) control with full state feedback and LQRwith output feedback approaches. SAS is designed to meet the handling qualities specification known as Aeronautical Design Standard (ADS-33E-PRF).A helicopter having a soft inplane four-bladed hingeless main rotor and a four-bladed tail rotor with conventional mechanical controls is used for thesimulation studies. In the simulation studies, the helicopter is trimmed at hover, low speeds and forward speeds flight conditions. The performance ofthe helicopter SAS schemes are assessed with respect to the requirements of ADS-33E-PRF.Findings The SAS in the longitudinal axis meets the requirement of the Level 1 handling quality specifications in hover and low speed as well as forforward speed flight conditions. The SAS in the lateral axis meets the requirement of the Level 2 handling quality specifications in both hover and lowspeed as well as for forward speed flight conditions. The requirements of the inter axis coupling is also met and shown for the coupled dynamics case.The SAS in lateral axis may require an additional control augmentation system or adaptive control to meet the Level 1 requirements.Originality/value The study shows that the design of a SAS using LQR control algorithm with full state and output feedbacks can be used to meetADS-33 handling quality specifications.

Keywords Helicopters, Flight dynamics, Control systems

Paper type Research paper

Nomenclature

Ixx, Iyy, Izz

moments of inertia of the helicopterabout x-, y- and z-axes

Kp, Kq, Kr roll, pitch and yaw rates feedback gain,

respectively

Ku, KV, KW longitudinal, lateral and vertical velocity

feedback gain, respectively

L, M, N external aerodynamic moments about

the x-, y- and z-axes

Lv, Mq, Nr, etc. moment derivatives normalized by

moments of inertia

Ma mass of helicopter

p, q, r fuselage angular rates (pitch, roll and

yaw rate)

M, v, w longitudinal, lateral and normal

velocity components of fuselage masscenter

X, Y, Z external aerodynamic forces acting along

x-, y- and z-axes

Xu, XB, etc. X force derivatives normalized aircraft

mass

Yv , Yr, etc. Y force derivatives normalized aircraftmass

Zw , Zq, etc. Z force derivatives normalized aircraft

mass

z, wn damping factor and natural frequencyu, c fus elag e p itch , roll and azimuth

(heading) angle, respectively

u0, u0t collective and p edal p ilot inp ut,

respectively

Q1s, u1c longitudinal and lateral pilot input,

respectively

Tp phase delay between helicopter response

and the control input

1. Introduction

Helicopters are a difficult type of air craft to control (Shaheed,

2005; Szumanski et al., 2002; Kowaleczko and Dzygadlo,

2002; Su and Cao, 2001; Cao et al., 2004; Lin and Meng,

2003). Generally, they exhibit a complex, nonlinear dynamic

behavior and are subject to a high degree of inter axis

coupling. In addition, helicopters are open-loop and most

mathematical models contain a moderate high degree of

uncertainty associated with neglected dynamics and poorly

understood aero-mechanical couplings.

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/1748-8842.htm

Aircraft Engineering and Aerospace Technology: An International Journal

80/2 (2008) 111123

q Emerald Group Publishing Limited [ISSN 1748-8842]

[DOI 10.1108/00022660810859337]

111

8/12/2019 Vijaya Aeat 2008

2/13

8/12/2019 Vijaya Aeat 2008

3/13

are 0.1 and 0.05s for the tail rotor actuator. The actuator

drives the input signals to the control surfaces.From the linear model we can see that the longitudinal and

lateral dynamics are highly coupled. Although the

longitudinal and lateral dynamics are highly coupled, SAS is

designed separately for longitudinal and lateral dynamics. The

performance of the controller is then evaluated using coupled

dynamics.

2.1 Longitudinal dynamics

The linearized model for longitudinal dynamics can be

written in the state space form as:

_x5Ax1Bu

y5Cx1Du 3

where A, B, C and D are system matrices, x5 u; w; q; uT

and u 5 u0; u1sT:

The A, B, C and D matrices for straight and level flight

condition at forward speed 100 Kmph are given below:

A

20:38 0:96

20:24

20:17

0:1 21:2 0:49 20:0121:2 1:3 22:2 0:00

0:00 0:00 0:99 0:00

26643775

B

0:19 10 20:11 1020:28 101 20:62 10

0:12 102 0:25 102

0:00 0:00

2664

3775

C

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2664

3775; D

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

2664

3775

The longitudinal dynamics has pitch subsidence, heave

subsidence and phugoid mode with the poles at 22.66,

20.868 and 0.0347 ^ 0.344 i, respectively, at 100Kmph.

These pole locations show that the poles of the pitch and

heave modes are stable since they lie on the left half of the S-

plane. The phugoid mode poles are unstable since they have a

positive real part. The pole locations for the other flight

conditions are given in Table I. From Table I, we can observe

that pitch and heave subsidence is always stable. The phugoid

mode is always unstable except at 50Kmph where it is

marginally stable. The instability increases with increase in

forward speed condition. The helicopter is highly unstable in

phugoid at the high forward speed of 290 Kmph. This clearly

indicates the need for the SAS.

2.2 Lateral dynamics

The linearized model for lateral dynamics can be written in

the state space form as in equation (3) where x v;p; r;fT

and u5 u1c; u0tT: The A, B, C and D matrices for straight

and level flight condition at forward speed 100Kmph are

given below:

A

20:16 0

:022 20

:47 0

:17

27:4 27:4 20:38 20:495:0 21:7 21:3 0:00

0:00 1:00 0:073 0:00

26643775

B

0:15 10 0:19 100:95 102 0:52 101

0:23 102 20:16 102

0:00 0:00

2664

3775

C

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

2664

3775; D

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

2664

3775

The poles of the linear system are listed in Table II. The

lateral dynamics has stable roll subsidence and a marginally

stable spiral mode. The dutch-roll mode is unstable in hover

and is stable at forward speeds as clear from the Table II.

Again, lateral dynamics clearly indicate the need of a stability

augmentation system for the dutch-roll mode in hover.

3. Handling qualities and control law design

The qualities or characteristics of an air craft that govern the

ease and precision with which a pilot is able to perform the

tasks required to support the aircraft mission are known as

handling qualities (Tomczyk, 2006; Tomczyk, 2002).

Handling qualities encompass all aspects of the man-

machine interface. This includes the cockpit ergonomics,the choice of inceptor and display, the presentation and

update rate of the display for the digital systems, the feel or

force feedback from the stick, the field of view, the available

autopilot functions and response types and the vehicle

response to small, moderate and large amplitude inputs.

The automatic fight control system design begins by

determining the design goals from the relevant handling

qualities specification. The handling qualities specifications

widely applied in industry arethe UK Ministry of Defence DEF

STAN 00970, Vol. 2, Part 6, (Pitkin, 1989), MIL-H-8501A

(Anonymous, 1961) and the ADS-33E-PRF (Anonymous,

1999). The ADS-33 standards reported in Anonymous (1999)

clearly indicate the effect of rotor loadings without

compromising the helicopter maneuverability requirements.

The new metricspresented in the ADS-33 provide more insight

in helicopter design from a multidisciplinary optimization point

Table II Lateral mode

Flight condition Dutch-roll Roll subsidence Spiral mode

Hover 0.0000738 ^ j0.544 28.0 21.1

50 Kmph 20.521 ^ j1.29 27.97 20.0901

100Kmph 20.594 ^ j1.74 27.58 20.0535

200Kmph 20.784 ^ j2.34 27.37 20.046

290Kmph 21.01 ^ j3.23 27.04 20.0507

Table I Longitudinal modes

Flight condition Phugoid

Pitch

subsidence

Heave

subsidence

Hover 0.0635 ^ j0.544 22.4 20.419

50 Kmph 2 0.205 ^ j0.342 22.14 21.26

100Kmph 0.0347 ^ j0.344 22.66 20.868

200Kmph 0.362 ^ j0.256 24.24 20.302

290Kmph 1.50, 0.146 25.82 20.213

Design of a stability augmentation system

M. Vijaya Kumaret al.


Volume 80 Number 2 2008 111 123

113

8/12/2019 Vijaya Aeat 2008

4/13

of view. Hence, the work in this paper makes use of ADS-33

metrics for the design of controller.

It is customary to rate handling qualities in terms of Cooper

and Harper (1969) Levels. A system is Level 1 if essentially it

is satisfactory without improvement. Level 2 implies thatimprovement is warranted. With a Level 2 system, adequate

performance is attainable, though the pilot may face very

objectionable deficiencies. A Level 3 system is at leastcontrollable but has major deficiencies, possibly requiring

intense pilot compensation. The three main levels are further

subdivided into handling quality ratings (HQRs), which range

from 1 to 10. HQRs of 1-3.5 equate overall to Level 1. HQRs

of 3.5-6.5 equate to Level 2. HQRs of 6.5-9 equate to Level 3.For details and a discussion of pilot ratings, the reader is

referred to Padfield (1996) and Cooper and Harper (1969).

Handling quality requirements are conceived from the aspectof demonstrating compliance and, therefore, must invariably

be converted into an alternative format appropriate for the

controller synthesis technique.

3.1 ADS-33 handling qualities

The goal of the helicopter control system design is to achieve

Level 1 ADS-33E-PRF (Anonymous, 1999) handling

qualities performance. The ADS-33 specification is divided

into hover/low speed and forward flight regimes. It depicts the

various requirements to achieve Level 1, 2 or 3 handlingqualities ratings of Cooper Harper rating levels. A selection of

flight test maneuvers are provided in the form of precisely

defined mission task elements (MTEs). All the MTEs havebeen listed in the document and the MTEs are generally

classified into target acquisition and tracking and all other

MTEs. Sometimes different requirements are specified for

MTEs in visual meteorological condition and instrument

meteorological condition. The performance requirements for

hover and forward flight regimes are listed with respect to

mission task-elements. ADS-33 mandates some frequency

domain and some time domain criteria for the evaluationperformance. The small amplitude criteria are in the

frequency domain and the moderate and large amplitude

requirements are spelt out in the time domain. The smallamplitude, or short term, pitch, roll, and yaw response are

important because they establish the pilots ability for control

precision. The criteria expressed within a phase-magnitude

plot in ADS-33, the criteria for the moderate amplitude

changes (also known as attitude quickness criteria) and

1large amplitude criteria are detailed below.

3.2 Short-term response to control inputs (bandwidth)

A key parameter related to the tracking performance of a

control system is bandwidth. For a flight control system givingan attitude response, ADS-33E-PRF defines the handling

qualities bandwidth vbw to be that frequency at which the

phase has fallen to 21358 (Figure 1). This is also known asthe phase limited bandwidth vBWPHASE : ADS-33 als odefines the gain-limited bandwidth vBWGAIN as the frequencyat which the gain is 6 dB greater than the gain at phase

crossoverv180. For a rate (as opposed to attitude) response,vbw is defined as the lesser of the phase and the gain limitedbandwidths. ADS-33 states that if the gain-limited bandwidthis less than the phase limited bandwidth, the rotorcraft may be

PIO prone, i.e. prone to pilot induced oscillations. The sameis true if the gain bandwidth is undefined; that is, if at no

frequency below phase crossover does the gain exceed its

value at phase crossover by 6 dB or more. Another key

parameter is phase delay tp. Phase delay is a measure of therate of change of phase with frequency beyond the bandwidth,and is defined as the average slope of the phase response (in

radians) in the range v180-2v180 rad/s: i.e. from the phasecrossover frequency to twice that frequency. It gives a measureof the systems effective dead-time. To determine bandwidth

and phase delay, frequency response data are required. Whenlinear models are available, these can be determined from the

definitions.Boundaries consistent with Cooper Harper Levels 1-3

handling qualities are defined in terms of bandwidth andphase delay. Different sets of boundaries apply to different

types of operation but broadly speaking, one seeks to achieve

high bandwidth and low phase delay. To compute the phasedelay (Figure 1), the bode plot is developed for the parameter

of interest. From this phase and magnitude plot, thefrequency v180 at which the phase cross over takes place isobtained. The phase difference DF2*v180 ) to go from v180 to2v180 is computed from the bode plot. If the phase isnonlinear between v180 and 2v180, then DF2*v180 can bedetermined from a linear least squares fit to the phase curve

betweenv180 and 2v180. Now, tp is given by:

tpDF2*v180

57:3*2*v1804

3.3 Moderate amplitude attitude changes (attitude

quickness)

Moderate amplitude change is also known as attitudequickness and is expressed as the ratio of peak angular rate

to peak change in attitude angle. The attitude quickness

parameter, for example, in pitch axis Quin ADS-33 is definedas the ratio of the maximum pitch rate qpkto the peak attitude

angle change Dupk, that is:

Qudef q

pk

Dupks21 5

ADS-33 defines handling quality boundaries for the attitudequickness parameter as a function of the minimum attitude

change Dumin. However, this criterion and these boundariesapply only to hover and low speed manoeuvres (,45kn) forpitch, roll and yaw. In forward speeds (.45kn) quantitativerequirements are defined only for roll.

In forward flight (.45kn) in case of pitch axis response,

ADS-33 is more qualitative in terms of flight path handlingqualities, and no quantitative levels are defined. Bearing in

mind that the present investigation also addressees forwardflight, it was decided to extend the definition of the minimum

pitch change Dumin in the ADS-33 attitude quicknesscriterion. The criterion of the hover and low speed forpitch is extended to the forward speed case as in Pavel and

Padfield (2002).The attitude changes required for compliance with this

requirement vary from 58 in pitch (108 in roll) to the limits of

the operational flight envelope or 308 in pitch (608 in roll),whichever is less. To compute the attitude quickness, the

required attitude change is made as rapidly as possible fromone steady attitude to another without significant reversals in

the sign of the cockpit control input relative to the trim

position. The peak angular rate is measured after giving theinput (Dqpk). The definition of the peak attitude and the




Volume 80 Number 2 2008 111 123

114

8/12/2019 Vijaya Aeat 2008

5/13

minimum attitude is shown in the Figure 2. The ratio for

example in the pitch axis of the peak pitch rate to the peak

pitch angle (Dqpk/Dupk) is plotted against Dumin.

3.4 Large amplitude attitude changes

The achievable angular rate (for rate response-types) or

attitude change from trim (for attitude response-types) are

prescribed in ADS-33. The specified rates or attitudes shall be

achieved in each axis while limiting excursions in the other

axes with the appropriate control inputs. For the moderate

agility requirements in low-speed case, the achievable pitch

and roll attitudes for Level 1 are ^20 to ^308 and ^608

respectively. For the moderate agility requirements in forward

flight case, the achievable roll attitudes for Level 1 is ^258.

The achievable pitch attitudes are not specified for the

forward flight case.

4. Control law design

Flight control design is an important problem for any aircraft

(Nobahariet al., 2006; Guglieri et al., 2006). In this paper, a

LQR with full SF and OF is considered. Therefore, the LQR

design with full SF for longitudinal and lateral SAS design is

described.

4.1 LQR design with full state feedback

We describe the LQR approach in this section (Stevens and

Lewis, 1942). Consider a linear system whose dynamics is

described by:

_x Ax Bu

y Cx 6

where matrices x [ Rn is the state, u [ Rm is the control

input, and y [ Rp is the measured output. The matrices A is

of the order (n n),B is of the order (n m) and C is of the

order (p n). We look at a control law with full SF which can

be expressed as:

u 2Kx 7

whereK is the SF gain matrix and results in the closed loop

system:

_x A 2 BKx 8

The problem is to determine an optimal control u which

minimizes the performance index, J, is given by:

J1

2

Z 1

0

{xTQx uTRu}dt 9

Figure 2Definition of moderate-amplitude criterion parameters

Time -->

(

)()

pk(pk)[pk]

min(min)[min]

Figure 1Definitions of bandwidth and phase delay

10

0

10

20

[X/Xi](dB)(X=,,)

(Xi=F

SorS)

235

180

135

90

(

deg)

BWphase 180

M= 45 deg

2180

2 180

GM = 6 dB

BWgain




Volume 80 Number 2 2008 111 123

115

8/12/2019 Vijaya Aeat 2008

6/13

where Q(n n) and R(m m) are symmetric positive

semidefinite matrices. The following algebraic Ricatti

equation is solved for P:

0 ATcfP PAcf Q PBR21BTP 10

and the Kalman gain K R21BTP is calculated. If the system

is controllable and observable (Kailath, 1980),the optimal gainK for the state-feedback law u 2Kx can be found. The

algorithm we used to arrive at optimal set of gains is as follows:

1 Select the design parameters Q and R, determine the

optimal gainK, and simulate the closed loop response and

the frequency domain characteristics.

2 If the results are not suitable, choose a different matricesQ and Rand repeat the procedure.

This approach involves the notion of tuning the design

parameters and for good performance.

4.2 LQR design with output feedback

The LQR approach described above guarantees asymptotic

stability for the optimized gains and is well treated

theoretically (Stevens and Lewis, 1942). Unfortunately, inhelicopter control systems it is usually not possible or

economically feasible to measure all the states accurately. It is

possible to design a dynamic observer or Kalman filter to

provide estimates xtof the states x(t), and then use feedbackof the estimates (LQG approach). We use a method where we

feed back not the entire state x(t), but only the measurable

out puts y(t). The OF control law is:

u 2Ky 2KCx 11

whereKis an (m n) feed back gain matrix to be determined.

On substitution into state space equation (equation (3)), this

yields:

_x A 2 BKCx ; Acx 12

where:

Ac A 2 BKC 13

We now briefly state the method used from Stevens and Lewis

(1942) and Moerder and Calise (1985). The performance

index J is given by:

J1

2

Z 1

0

{xTQx uTRu}dt 14

where Q of the order (n n) and Rof the order (m m) are

symmetric positive semidefinite matrices. The design problem

now is to select the gain Kso that J is minimized subject to

the dynamical constraint stated above. Suppose that we canfind a constant, symmetric, positive semi definite matrix P so

that:

d

dtxTPx 2xTQ CTKTRKCx 15

Assuming that the closed loop system is asymptotically stable,

the performance index becomes:

J1

2xT0Px0 16

We may write this equation as:

J1

2trPx 17

where the (n n) symmetric matrix X is defined by:

X x0xT0 18

The necessary conditions for the solution of the LQR

problem with OF are given by:

0 ATc P PAc CTKTRKC Q 19

0 A cS SATc x 20

0 RKCSCT 2 BTPSCT 21

The equations (19) and (20) are Lyapunov equations and

equation (20) is an equation for thegain K. Theequation(19) is

solved for P and the equation (20) is solved for S. IfRis positive

definite andCSCT is non-singular, then equation (21) may be

solved forKto obtain:

K R21BTPSCTCSCT21 22

Unfortunately, the dependence of X on the initial condition

makes the optimal gain dependent on the initial state. The

numerical technique described in Stevens and Lewis (1942)

and Moerder and Calise (1985) is used to solve for K and

the detailed algorithm is provided in Appendix 2. The

conditions for convergence is given in Moerder and Calise

(1985).The algorithm used to arrive at an optimal set of gains is as

follows:1 Select the design parameters Q and R, determine the

optimal gainK, and simulate the closed loop response and

the frequency domain characteristics.2 If the results are not suitable, choose different matricesQ

and Rand repeat the procedure.

This approach involves the notion of tuning the design

parameters Q and R for good performance. Since,

computation of K involves solution of the three coupled

equation, the method is iterative. In this method, the first step

is to find an initialK0. The LQR approach with full state back

described in the previous section is used to arrive at an initial

guess of K0.

5. Simulation results and discussion

The SAS are designed to follow rate command and attitude

hold for an unstable helicopter. The MTE considered for this

study is target acquisition and tracking. The procedure for

obtaining optimal feedback gains are described.

1 Select the design parametersQ and R.2 Determine the optimal gainK.

3 Study the closed loop response in time and the frequency

domain characteristics.4 Use ADS-33 performance metrics described above to

decide on the suit ability of the gains.5 If the results are not suitable, choose different matricesQ

and R repeat the procedure.6 Repeat this process for all the speeds and arrive at an

optimal set of gains at all speeds.7 Using the linear interpolation, gains for intermediate

speeds are obtained. This gain scheduling approach is




Volume 80 Number 2 2008 111 123

116

8/12/2019 Vijaya Aeat 2008

7/13

8/12/2019 Vijaya Aeat 2008

8/13

8/12/2019 Vijaya Aeat 2008

9/13

rate in hover is high in out put feedback approach as

compared to full SF approach. This is due to the fact that the

lateral component feed back is not available in OF approach.

The results are studied separately for:. hover and low speed; and. forward speeds.

5.2.1 Hover and low speed. We have considered hover and 50 Kmph for this study.

The limits on roll oscillations are plotted along with ADS-

33 boundaries in Figure 15. The ADS-33 boundaries are

shown by the dotted line. The poles lie on the left of the

ADS33 lines and meet the Level 1 requirements.. The short-term response to control inputs are studied and

shown in the Figure 16. It can be seen from the figure

that the Level 1 criteria is satisfied in hover. However,

at 50Kmp h, s hort-term res pons e meets Level 2

requirements.. Attitude quickness is studied by giving a suitable pulse

input to generate a roll angle of 10-608 and is shown in

Figure 17. It can be seen from the figure that the Level 1

criteria is satisfied in hover. However, at 50Kmph,

attitude quickness criteria meets the Level 2 criteria.. To study the requirement of large amplitude attitude

changes criteria, the test is carried out by giving

the appropriate lateral cyclic control input. The

achievable attitude change from trim is shown to be

608 and meets the Level 1 requirement for the moderate

agility requirements (Figure 17).

Figure 13 Controller structure with output feedback lateraldynamics

+

Helicopter

Dynamics

command

Kr

rKp

p

Figure 12Controller structure with SF lateral dynamics

+

Helicopter

Dynamicscommand

K

Kr

rKp

p

Kv v

Figure 14 Feedback gains scheduling with respect to airspeed lateral

0 50 100 150 200 250 3000.5

0

0.5

1

1.5

2

2.5

3

Speed (Kmph)

StateFeedbackGains

Kv SF

Kp SF

Kr SF

K SF

Kp OF

Kr OF

K OF

Figure 15Limits on roll oscillations hover and low speed

1.5 1 0.5 0 0.50

0.5

1

1.5

n

n

*sqrt(12) ADS 33

ADS 33

ADS 33

Hover SF

Hover SF

Hover OF

Hover OF

50 Kmph SF

50 Kmph SF

50 Kmph OF

50 Kmph OF

Level 1 Level 2Level 3

Figure 11 Helicopter response to pulse longitudinal cyclic input:290Kmph, 2 s pulse

0 2 4 6 8 1040

20

0

20

40

pitchrateq(deg/s)

0 2 4 6 8 1010

0

10

20

30

pitchattitude(deg)

time (s)




Volume 80 Number 2 2008 111 123

119

8/12/2019 Vijaya Aeat 2008

10/13

5.2.2 Forward speed. Straight and level flight conditions at forward speeds of

100, 200 and 290 Kmph are considered. In case of

forward flight speeds of 100, 200 and 290Kmph, the

poles are within the left of the S-plane for both OF and SF

methods and are stable.. Bandwidth and phase delay are calculated at different

forward speed conditions. The bandwidth and phase

delays for SF and OF methods are shown in Figure 18.

From the figure, we can see that both SF and OF methods

satisfy the Level 2 requirement for short-term response

except at 290 Kmph for OF method. In case of 290 Kmph

with OF method, the Level 1 criteria is satisfied.. The attitude quickness criteria for forward flight

conditions are calculated for SF and OF methods and

are shown in Figure 19. From the figure, we can see that

both SF and OF satisfy the Level 1 requirement when the

minimum attitude change is on the higher side (608).

When the attitude change is near 308 Level 2 criteria

is met.. To study the requirement of large amplitude attitude changes

criteria, the test is carried out by simulating the appropriate

lateral cyclic control input. The achievable attitude change

from trim is shown to be 608 and meets the Level 1

requirement forthemoderateagility requirements(Figure19).

5.2.3 Coupled dynamics

In thecoupled dynamics both longitudinaland lateral dynamics

are considered in addition to the coupled state as in equation

(2). Theresponse ofthe helicopter to a longitudinalpulse(5 s)is

provided forall thestates in Figure20 for thecoupled dynamics.

ADS-33 does not provide any quantitative measures for the

inter axis coupling for the moderate agility that is of interest to

us. It can be seen that the response of the helicopter in the roll

axis is negligible for an input in the longitudinal cyclic and this

meets the requirement of the inter axis coupling for the

moderate agility helicopter.

Figure 17Moderate amplitude roll attitude change hover and lowspeed

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Minimum attitude change min(deg)

ppk/pk(1/sec) ADS33

ADS33ADS33ADS33HoverSFHoverOFHoverSFHoverOF50 KmphSF50 KmphOF50 KmphSF50 KmphSF

Level 3

Level 2

Level 1

Figure 18Lateral axis bandwidth and phase delay forward speed

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

BandwidthBW (rad/sec)

PhaseDelayp(

sec)

100 KmphSF100 KmphOF200 KmphSF200 KmphOF290 KmphSF290 KmphOF

Level 3

Level 2

Level 1

Figure 19Moderate amplitude roll attitude change forward speed

0 10 20 30 40 50 600

0.5

1

1.5

2

2.5

Minimum attitude change min(deg)

ppk/pk(1/sec)

ADS33

ADS33

ADS33

ADS33

100 KmphSF

100 KmphOF

100 KmphSF

100 KmphOF

200 KmphSF

200 KmphOF

200 KmphSF

200 KmphOF

290 KmphSF

290 KmphOF

290 KmphSF

290 KmphOF

Level 1

Level 2

Level 3

Figure 16 Lateral axis bandwidth and phase delay hover and lowspeed

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Bandwidth BW(rad/sec)

PhaseDelayp(sec)

hover SF

hover OF

50 KmphSF

50 KmphOFLevel 3

Level 2

Level 1




Volume 80 Number 2 2008 111 123

120

8/12/2019 Vijaya Aeat 2008

11/13

6. Conclusions

The performance of a SAS designed for an unstable

helicopter in longitudinal and lateral axes is assessed with

respect to the ADS-33 requirements. The design of the SAS isbased on LQR with full SF and LQR with OF controllers and

uses the ADS-33 metrics. A helicopter having a soft inplane

four-bladed hinge-less main rotor and a four-bladed tail rotor

with conventional mechanical controls is used for the

simulation studies. For the simulation study, a linearized

helicopter model at straight and level flight condition is

considered. The equilibrium conditions of hover, low speeds

and for forward speeds are used for this study. The simulation

results shows the following:. The SAS in longitudinal axis meets the requirement of the

Level 1 handling qualities in hover and low speed as well

as forward speed flight conditions..

The SAS in lateral axis meets the requirement of the Level2 handling qualities in both hover and low speed as well as

forward speed flight conditions.. The SAS in lateral axis may require an additional control

augmentation sys tem to meet the Level 1 requirement. It

may also call for an adaptive control to meet the Level 1

requirement.. The requirement of the inter axis coupling is met as shown

for the coupled dynamics case.. The study shows that the design of a SAS using LQR

control algorithm using full SF and OF can be used to

meet ADS-33 handling quality specifications.

References

Anonymous (1961), General requirements for helicopter

flying and ground handling qualities, MilH-8501 A.

Anonymous (1999), Performance specification, handlingqualities requirements for military rotorcraft, Aeronautical

Design Standard-33E-PRF, US Army Aviation System

Command, St Louis, MO.

Cao, Y.H., Zhang, G.L. and Su, Y. (2004), Mathematical

modeling of helicopter aerobatic maneuvers, Aircraft

Engineering & Aerospace Technology, Vol. 76 No. 2,

pp. 170-8.

Cooper, G.E. and Harper, R.P. (1969), The use of pilot

rating in the evaluation of aircraft handling qualities,

Technical Note TN D-5153, National Aeronautics and

Space Administration, NASA, Washington, DC.Ekblad, M. (1990), Reduced-order modeling and controller

design for a high-performance helicopter, AIAA Journal of

Guidance, Control and Dynamics, Vol. 13 No. 3, pp. 439-49.Gribble, J.J. (1993), Linear quadratic Gaussian/loop transfer

recovery design for a helicopter in lowspeed flight, AIAA

Journa l of Guidance, Control and Dynamics, Vol. 16,

pp. 754-61.Guglieri, G., Pralio, B. and Quagliotti, F. (2006), Flight

control system design for a micro aerial vehicle, Aircraft

Engineering & Aerospace Technology, Vol. 78 No. 2,

pp. 87-97.Horn, J.F., Tolani, D.K., Lagoa, C.M., Wang, Q. and Ray, A.

(2005), Probabilistic robust control of rotorcraft, Control

Engineering Practice, Vol. 13, pp. 1037-46.

Figure 20Helicopter response to pulse longitudinal cyclic input (200 Kmph, 5 s pulse)

10

5

0

u(m/s)

0 5 100.5

0

0.5

v(m/s)

5

0

5

w(m/s)

0 5 100.1

0

0.1

p(deg/s)

20

0

20

q(deg/s)

0 5 101

0

1

r(deg/s)

0.2

0

0.2

(deg)

0 5 10

10

0

10

(deg)

time (s)

0 5 10

0 5 10

0 5 10

0 5 10

0 5 10

0

5

10

longcyc(deg)

time (s)

SF

OF

200 Kmph level flight response to longitudinal cyclic input




Volume 80 Number 2 2008 111 123

121

8/12/2019 Vijaya Aeat 2008

12/13

Ingle, S.J. and Celi, R. (1994), Effects of higher order

dynamics on helicopter flight control law design, Journal of

the American Helicopter Society, Vol. 39 No. 3, pp. 12-23.

Kailath, T. (1980), Linear Systems, Prentice-Hall, Englewood

Cliffs, NJ.Kowaleczko, G. and Dzygadlo, Z. (2002), Effect of free play

in rotor blades hinges on the non-linear ground resonance

of a helicopter,Aircraft Engineering & Aerospace Technology,Vol. 74 No. 6, pp. 508-16.

Krishnakumar, K., Sawhney, S. and Wai, R. (1994), Neuro-

controllers for adaptive helicopter hover training, IEEE

Transactions on Systems Man and Cybernetcs, Vol. 24 No. 8,pp. 1142-52.

Krupadanam, A.S., Annaswamy, A.M. and Mangoubi, R.S.

(2002), Multivariable adaptive control design with

applications to autonomous helicopters, AIAA Journal of

Guidance, Control and Dynamics, Vol. 25 No. 5, pp. 843-51.

Lee,S., Ha,C. and Kim,B.S.(2005),Adaptive nonlinearcontrolsystem design for helicopter robust command augmentation,

Aerospace Science and Technology, Vol. 9, pp. 241-51.

Leitner, J., Calise, A.J. and Prasad, J.V.R. (1997), Analysis of

adaptive neural networks for helicopter flight control,AIAA Journal of Guidance, Control and Dynamics, Vol. 20

No. 5, pp. 972-9.Lin, F.S. and Meng, G. (2003), Study on the dynamics of a

rotor in a maneuvering aircraft, Journal of Vibration and

Acoustics-Transactions of the SME, Vol. 125 No. 3, pp. 324-7.

Low, E. and Garrard, W.L. (1993), Design of flight control

s ys te ms to me et r ot or cr af t ha nd li ng qu al it ie s

specifications, AIAA Journal of Guidance, Control andDynamics, Vol. 16 No. 1, pp. 69-78.

Manness, M.A. and Murray-Smith, D.J. (1992), Aspects of

multivariable flight control law design for helicopters using

eigenstructure assignment, Journal of the American

Helicopter Society, Vol. 37 No. 3, pp. 18-32.

Moerder, D.D. and Calise, A.J. (1985), Convergence of a

numerical algorithm for calculating optimal outputfeedback gain, IEEE Transactions on Automatic Control,

Vol. AC30 No. 9, pp. 900-3.

Nobahari, H., Alasty, A. and Pourtakdoust, S.H. (2006),

Design of a supervisory controller for CLOS guidance

with lead angle, Aircraft Engineering & Aerospace

Technology, Vol. 78 No. 5, pp. 395-406.Ockier, C.J. (1996), Flight estimation of the new handling

qualities criteria using the Bol05, Journal of American

Helicopter Society, Vol. 41 No. 1, pp. 67-76.

Padfield, G.D. (1981), A theoretical model of helicopter

flight mechanics for application to piloted simulation,Technical Report RAE TR 81048, Royal Aircraft

Establishment, Farnborough.Padfield, G.D. (1996), Helicopter Flight Dynamics, Blackwell

Science Limited, Oxford.

Pavel, D.P. and Padfield, G.D. (2002), Defining consistent

ADS-33-metrics for agility enhancement and structuralloads alleviation, Proceedings of the American Helicopter

Society 58th Annual Forum, Montreal.

Phillips, C., Karr, C.L. and Walker, G. (1996), Helicopter

flight control with fuzzy-logic and genetic algorithms,Engineering Applications of Artificial Intelligence, Vol. 9 No. 2,

pp. 175-84.Pitkin, B. (1989), Flight and ground handling qualities,

military defence specification DEF-STAN 00-970,

Technical Note TN D-5153, Ministry of Defence.

Postlethwaite, I., Smerlas, A., Walker, D.J., Gubbels, A.W.,

Baillie, S.W., Strange, M.E. and Howitt, J. (1999), H1control of the NRC bell 205 fly-by-wire helicopter, Journal

of the American Helicopter society, Vol. 44 No. 4, pp. 276-84.Prouty, R.W. (1995), Helicopter Performance, Stability and

Control Krieger, New York, NY.Rozak, J.N. and Ray, A. (1997), Robust multi-variable

control of rotorcraft in forward flight, Journal of theAmerican Helicopter Society, Vol. 43 No. 3, pp. 149-60.

Rysdyk, R.T. and Calise, A.J. (1998), Adaptive model

inversion flight control for tiltrotor aircraft, Proceedings of

the American Helicopter Society 54th Annual Forum,

Washington, DC.Shaheed, M.H. (2005), Feedforward neural network based

non-linear dynamic modelling of a TRMS using RPROP

algorithm, Aircraft Engineering & Aerospace Technology,

Vol. 77 No. 1, pp. 13-22.

Shim, H., Koo, T.J., Hoffman, F. and Sastry, S. (1998), A

comprehensive study of control design for an autonomous

helicopter, paper presented at 37th IEEE Conference on

Decon and Control, Tampa, FL.

Snell, S.A. and Stout, P.W. (1997), Robust longitudinalcontrol design using dynamic inversion and quantitative

feedback theory, AIAA Journal of Guidance, Control and

Dynamics, Vol. 20 No. 5, pp. 933-40.Stevens, B.L. and Lewis, F.L. (1942), Aircraft Control and

Simulation, Wiley, New York, NY.

Su, Y. and Cao, Y. (2001), Studies of helicopter dynamic

stability and control laws, Aircraft Engineering & Aerospace

Technology, Vol. 73 No. 2, pp. 132-7.Suresh, S., Sundararajan, N. and Saratchandran, P. (2006),

Neural adaptive flight controllers for helicopters,

Proceedings of 44th AIAA Aerospace Science Meeting and

Exhibit, Paper No. 2006-1476, AIAA, Reno, pp. 9-12.Szumanski, K., Berezanski, J. and Szumanski, A. (2002),

Elements of the helicopter supermanoeuverability at low

flight velocities, Aircraft Engineering & Aerospace

Technology, Vol. 74 No. 6, pp. 517-24.

Tischler, M.B. (1989), Assessment of digital flightcontrol

technology for advanced combat rotorcraft, Journal of the

American Helicopter Society, Vol. 34 No. 4, p. 66.

Tischler, M.B., Fletcher, J.W., Morris, P.M. and Tuckerg,

G.E. (1991), Flying quality analysis and flight evaluation

of a highly augmented combat rotorcraft, AIAA Journal of

Guidance, Control and Dynamics, Vol. 14 No. 5, pp. 954-63.

Tomczyk, A. (2002), A proposal of handling qualities

shaping for general aviation aircraft, Aircraft Engineering &

Aerospace Technology, Vol. 74 No. 6, pp. 534-49.

Tomczyk, A. (2006), Simple virtual attitude sensors for

general aviation aircraft, Aircraft Engineering & Aerospace

Technology, Vol. 78 No. 4, pp. 310-4.Vijaya Kumar, M., Suresh, S., Omkar, S.N., Ganguli, R. and

Sampath, P. (2005), A direct adaptive neural command

controller design for an unstable helicopter, Proceedings of

the American Helicopter Society 61st Annual Forum,

Grapevine, TX.

Walker, D.J. (2003), Multivariable control of the longitudinal

and lateral dynamics of a fly-by-wire helicopter, Control

Engineering Practices, Vol. 11, pp. 781-95.Yue, A. and Postlethwaite, I. (1990), Improvement of

helicopter handling qualities using H1 Optimisation, IEE

Proceedings, Vol. 137, pp. 115-29.




Volume 80 Number 2 2008 111 123

122

8/12/2019 Vijaya Aeat 2008

13/13

Appendix 1. Linearized matrices F and G

Matrix F

Xu=m Xw=m Xw=m2we 2gcosue Xv=m Xp=m 0 Xr=m ve

Zu=m Zw=m Zq=m ue 2gcosFesinQe Zv=m Zp=m ve 2gsinFecosQe Zr=m

Mu=Iy Mw=Iy Mq=Iy 0 Mv=Iy Mp=Iy 0 Mr=Iy

0 0 cosF

e 0 0 0 0 2

sinF

e

Yu=m Yw=m Yq=m 2gcosFesinQe Yv=m Yp=m we gsinFecosQe Yr=m2ue

IxLu IxxNu=Ic IxLw IxxNw=Ic IxLq IxxNq=Ic 0 IxLv IxxNv=Ic IxLp IxxNp=Ic 0 IxLrIxxNr=Ic

0 0 SinFetanQe 0 0 1 0 CosFetanQe

IxzLu IxNu=Ic IxzLw IxNw=Ic IxzLq IxNq=Ic 0 IxxLv IxNv=Ic IxzLp IxNp=Ic 0 IxzLrIxNr=Ic

where Ic IxIz 2 I2xz

Matrix G

Xu0 =m Xu1s=m Xu1c=m Xu0T=m

Zu0 =m Zu1s=m Zu1c=m Zu0T=m

Mu0=Iy Mu1s=Iy Mu1c=Iy Mu0T=Iy

0 0 0 0

Yu0=m Yu1s=m Yu1c=m Yu0T=m

IzLu0 IxzNu0 =Ic IzLu1s IxzNu1s =Ic IzLu1c IxzNu1c =Ic IzLu0TIxzNu0T=Ic

0 0 0 0

IxzLu0 IxNu0 =Ic IxzLu1s IxNu1s =Ic IxzLu1c IxNu1c =Ic IxzLu0TIxNu0T=Ic

23

Appendix 2. Optimal output feedback solutionalgorithm

1. Initialize:

Set k 0.

Determine a gain K0 so that A-BK0C is asymptoticallystable

2. kth iteration:Set Ak A-BKkCSolve for Pk and Sk in:

0 ATck Pk PkAk CTKTkRKkC Q

0 A kc Sk SkATck X

Set Jk 12 tracePkX

Evaluate the gain update direction:DK R21BTPSCTCSCT21 2 Kk

Update the gain by:Kk1 Kk aDK

wherea is chosen so that:A-BKk1C is asymptotically stable

Jk 1;

trace(PkX),

JkOtherwise, set k k 1 and go to 2.

3. Terminate:Set K Kk1, J Jk 1Stop.

About the authors

M. Vijaya Kumaris a Chief Manager (Design) at the RotaryWing Research and Design Centre, HAL Bangalore and heads

the auto pilotgroup involved in the development, design, testingand certification of helicopters. He completed his MSc in

Physics from the University of Mysore in 1979 and his MTechfrom the Indian Institute of Science in 1982. He is currently

doing his PhD degree at the Department of AerospaceEngineering, Indian Institute of Science, Bangalore.

Prasad Sampath is the Head of Design Department at the

Rotary Wing Research and Design Centre, HAL Bangalore.He completed his MS and PhD in Aerospace Engineering

from the University of Maryland, College Park, USA, in 1976and 1980, respectively.

S. Suresh did his PhD at the Indian Institute of Science,Bangalore, in 2005. He is currently a postdoctoral researcher

at the National Technical University in Singapore.

S.N. Omkar is a Principal Research Scientist in theAerospace Engineering department of the Indian Institute of

Science, Bangalore, India. He received his MSc and PhDdegree in Aerospace Engineering from the Indian Institute of

Science in 1992 and 1999, respectively.

Ranjan Ganguli is an Associate Professor in the AerospaceEngineering Department of the Indian Institute of Science,

Bangalore, India. He received his PhD and MS degrees fromthe Alfred Gessow Rotorcraft Center at the University of

Maryland, College Park, USA in 1994 and 1991, respectively,and his BTech (Honors) degree from the Indian Institute ofTechnology, Kharagpur, India in 1989. Ranjan Ganguli is the

corresponding author and can be contacted at: [email protected]




Volume 80 Number 2 2008 111 123

To purchase reprints of this article please e-mail: [email protected]

Or visit our web site for further details: www.emeraldinsight.com/reprints

vijaya aeat 2008

Documents