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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]
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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
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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
Design of a stability augmentation system
M. Vijaya Kumaret al.
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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
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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
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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)
Design of a stability augmentation system
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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
Design of a stability augmentation system
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Aircraft Engineering and Aerospace Technology: An International Journal
Volume 80 Number 2 2008 111 123
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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
Design of a stability augmentation system
M. Vijaya Kumaret al.
Aircraft Engineering and Aerospace Technology: An International Journal
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.
Design of a stability augmentation system
M. Vijaya Kumaret al.
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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]
Design of a stability augmentation system
M. Vijaya Kumaret al.
Aircraft Engineering and Aerospace Technology: An International Journal
Volume 80 Number 2 2008 111 123
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