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Flight Controller Design for anAutonomous MAV
Dissertation
Submitted in partial fulfillment of the requirements
for the Master of Technology Program
by
Gopinadh Sirigineedi
03301012
Under the guidance of
Prof. S.P. Bhat
Department of Aerospace Engineering
Indian Institute of Technology, Bombay
JULY 2005
Dissertation Approval Sheet
Dissertation entitled “Flight Controller Design for an Autonomous MAV”,
submitted by Gopinadh Sirigineedi (Roll No. 03301012) is approved for the
degree of Master of Technology.
Guide
Examiners
(Internal)
(External)
Chairman
Date: July, 2005.
i
Certificate
Certified that this M.Tech Project Report titled as “Flight Controller Design
for an Autonomous MAV”, submitted by Gopinadh Sirigineedi (Roll No.
03301012) is approved by me for submission. Certified further that, to the best
of my knowledge, the report represents work carried out by the student.
Date: 8-7-2005.
Prof. S.P. Bhat
(Project Guide)
ii
Abstract
The report presents a flight controller designed to enable the Mini Air Vehicle (MAV)
Kadet MkII to navigate autonomously from an initial way-point to a final way-point.
A guidance strategy reported in the literature is adopted. Since the guidance strategy
requires the aircraft to fly in one of three trim states, namely, straight level flight, level
coordinated right circular turn, level coordinated left circular turn, linearized models
are generated for these three trim states. Three controllers, each consisting of a state
feedback controller and a reduced-order observer are designed for each of these lin-
earized aircraft models. Each controller stabilizes respective trim state, and also tracks
altitude commands. The performance of linear controllers is demonstrated by simula-
tions performed with a nonlinear aircraft model. Closed-loop guidance is performed
on the nonlinear aircraft model. The guidance strategy is modified to smoothen the
trajectory of the aircraft by eliminating continuous switching between left and right
turns. Simulations for three dimensional way-point navigation are performed. Sim-
ulations are used to study the effect of closed-loop poles on guidance. The effect of
discrete position and heading updates is simulated.
Keywords: Autonomous, State feedback, Reduced-order observer, Guidance Strategy.
iii
Contents
Abstract iii
Nomenclature viii
List of Figures x
1 Introduction 1
2 Controller Design for Full Aircraft Dynamics 4
2.1 Linearized Aircraft Model . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 State Feedback Controller . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Selection of Closed-Loop Poles . . . . . . . . . . . . . . . . . . . 7
2.3 Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Control Law Implementation On Nonlinear Aircraft Model . . . . . . . 9
3 Closed-Loop Guidance 15
3.1 Guidance Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Closed-Loop Guidance . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 Modified Guidance Strategy . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Guidance Under Discrete Position and Heading Updates . . . . . . . . 22
3.5 Effect of Closed-Loop Poles On Guidance . . . . . . . . . . . . . . . . . 25
4 Conclusion 29
Appendices 29
A Nonlinear Aircraft Equations and Aircraft Data 30
A.1 Nonlinear Aircraft Equations . . . . . . . . . . . . . . . . . . . . . . . . 30
A.2 Aircraft Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
B Linear Controller Design for Level Straight Flight 37
iv
C Linear Controller Design for Level Coordinated Right Circular Turn 41
D Linear Controller Design for Level Coordinated Left Circular Turn 45
E Simulink Models 49
References 52
v
Nomenclature
English Symbols
A,B,C State, control and output matrices of the linearized aircraft model
A State matrix of reduced-order observer
Aaa, Aab, Aba, Abb Sub-matrices of the state matrix
Ba, Bb Sub-matrices of the control matrix
B Sub-matrix of control matrix of reduced-order observer
C Output matrix of reduced-order observer
D Direct transmission matrix of reduced-order observer
F Sub-matrix of control matrix of reduced-order observer
g Acceleration due to gravity in metre/sec2
h Perturbation in altitude from trim value in metres
H Altitude at which the aircraft is flying level and trim in metres
hd Desired change in the altitude from trim value in metres
K State feedback gain matrix
Ke Reduced-order observer gain matrix
M Maximum turn rate of the aircraft in radians/sec
n Load factor
p Perturbation in roll rate from trim value in radians/sec
P Cross range error of the aircraft in metres
PT Roll rate of the aircraft at trim state in radians/sec
q Perturbation in pitch rate from trim value in radians/sec
Q Down range error of the aircraft in metres
QT Pitch rate of the aircraft at trim state in radians/sec
r Perturbation in yaw rate from trim value in radians/sec
RT Yaw rate of the aircraft at trim state in radians/sec
u Control vector for the aircraft model
uc Input command to the closed-loop aircraft model
V Velocity of the aircraft
vi
v Perturbation in true air-speed from trim value in metre/sec
VT True air-speed of the aircraft at trim state in metre/sec
x State vector of the linearized aircraft model
∆x Error between aircraft state vector and desired state vector
x Estimated state vector
xa Measured part of the state vector
xb Unmeasured part of the state vector
xb Estimation of unmeasured part of the state vector
xd Vector of desired aircraft states
Greek Symbols
α Perturbation in angle of attack from trim value in radians
αT Angle of attack of the aircraft at trim state in radians
β Perturbation in side-slip angle from trim value in radians
βT Side-slip angle of the aircraft at trim state in radians
φ Perturbation in bank angle from trim value in radians
φT Bank angle of the aircraft at trim state in radians
θ Perturbations of pitch angle from trim value in radians
θT Pitch angle of the aircraft at trim state in radians
δa Perturbation in aileron deflection from trim value in radians
δaT Deflection of the aileron when the aircraft is flying at trim state in radians
δe Perturbation in elevator deflection from trim value in radians
δeT Deflection in the elevator when the aircraft is flying at trim state in radians
δn Perturbation in propeller speed from trim value in rpm
δnT Propeller speed when the aircraft is flying at trim state in rpm
δr Perturbation in rudder deflection from trim value in radians
δrT Deflection of the rudder when the aircraft is flying at trim state in radians
η Reduced-order observer state vector
ψ Function used in guidance strategy
γ Angle made by the line with x-axes in modified logic
ν Turn rate of the aircraft
λ Heading angle of the aircraft
vii
Subscripts
c Command
d Desired
f Final destination
T Trim
viii
List of Figures
1.1 Kadet MkII . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Block diagram of closed-loop aircraft with trim selector . . . . . . . . . 10
2.2 Reduced-order observer based controller . . . . . . . . . . . . . . . . . 11
2.3 Response of the aircraft for trim selector commands. . . . . . . . . . . 12
2.4 True air-speed and altitude responses. . . . . . . . . . . . . . . . . . . . 13
2.5 Angle-of-attack and sideslip responses. . . . . . . . . . . . . . . . . . . 13
2.6 Response of propeller speed to trim selector commands. . . . . . . . . . 14
2.7 Response of elevator deflection to trim selector commands. . . . . . . . 14
2.8 Response of aileron deflection to selector commands. . . . . . . . . . . 14
2.9 Response of rudder deflection to selector commands. . . . . . . . . . . . 14
3.1 Phase portrait under guidance strategy . . . . . . . . . . . . . . . . . . 16
3.2 Schematic diagram of guidance strategy implementation . . . . . . . . 17
3.3 Aircraft path under closed-loop guidance . . . . . . . . . . . . . . . . . 18
3.4 Phase portrait under modified guidance strategy . . . . . . . . . . . . . 19
3.5 Aircraft path under modified guidance strategy . . . . . . . . . . . . . 20
3.6 Trim selector commands with original guidance strategy. . . . . . . . . 21
3.7 Trim selector commands with modified guidance strategy. . . . . . . . . 21
3.8 Trajectory of the aircraft for visiting three way-points. . . . . . . . . . 22
3.9 Variation of altitude for visiting three way-points. . . . . . . . . . . . . 22
3.10 Trim selector commands as the aircraft visits the way-points. . . . . . . 23
3.11 Response of true air-speed of the aircraft. . . . . . . . . . . . . . . . . . 23
3.12 Variation of angle of attack. . . . . . . . . . . . . . . . . . . . . . . . . 23
3.13 Variation of sideslip angle. . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.14 Variation of propeller speed. . . . . . . . . . . . . . . . . . . . . . . . . 24
3.15 Variation of elevator deflection. . . . . . . . . . . . . . . . . . . . . . . 24
3.16 Variation of aileron deflection. . . . . . . . . . . . . . . . . . . . . . . . 24
3.17 Variation of rudder deflection. . . . . . . . . . . . . . . . . . . . . . . . 24
ix
3.18 Guidance strategy implementation with sample and hold block . . . . . 25
3.19 Guidance under discrete updates . . . . . . . . . . . . . . . . . . . . . 26
3.20 Trajectories of aircraft with faster and slower poles . . . . . . . . . . . 27
3.21 Trim selector commands to the closed-loop aircraft with faster poles. . 28
3.22 Trim selector commands to the closed-loop aircraft with slower poles. . 28
A.1 Variation of coefficient of thrust with advance ratio. . . . . . . . . . . . 34
A.2 Variation of coefficient of power with advance ratio. . . . . . . . . . . . 34
E.1 Simulink model to implement closed-loop guidance . . . . . . . . . . . 50
E.2 Simulink model of the controller . . . . . . . . . . . . . . . . . . . . . . 51
x
Chapter 1
Introduction
An aircraft flight control system provides the capability to stabilize and control the
aircraft. The evolution of modern aircraft created a need for automatic-pilot control
systems. In addition, the widening performance envelope created a need to augment
the stability of the aircraft dynamics over some parts of the flight envelope. Because
of the large changes in aircraft dynamics, the dynamic mode that is stable and ade-
quately damped in one flight condition may become unstable, or at least inadequately
damped, in another flight condition. These problems are overcome by using feedback
control to modify the aircraft dynamics. The aircraft motion variables are sensed and
used to generate signals that can be fed into the aircraft control actuators, thus mod-
ifying the dynamic behavior. This feedback must be adjusted according to the flight
conditions. The adjustment process is called gain scheduling because, in its simplest
form, it involves only changing the amount of feedback according to the predetermined
schedule.
Sophisticated control configurations are needed to meet the mission requirements
for advanced aircraft. The required vehicle performance during low altitude, low speed
and high angle-of-attack, all-weather, day and night operations must be achieved. One
of the difficulties dealing with flight control are nonlinearities that must be considered.
Lyshevski [1] presents aircraft flight control system design under state and control
bounds. Yang and Kung [2] present the application of nonlinear H∞ state feedback
theory to flight control which solves the aircraft equations without linearization. The
aircraft configuration also impacts control response through variations in centre of
gravity and moment of inertia. Keating, Parchter and Houpis [3] present Quantitative
Feedback Theory (QFT) based robust flight controller for varying flight conditions.
Mini Air Vehicles (MAVs) are miniature airplanes designed to be small, light, and
highly resilient. The purpose of Mini Air Vehicles (MAVs) is to provide inexpensive
1
and expendable platforms for surveillance and data collection in situations where larger
vehicles are not practical. They can be used for battlefield surveillance or mapping the
extent of chemical spills or viral outbreaks. Other applications include use in search
and rescue operations, traffic coverage and crop or wildlife monitoring [4].
Most of the MAVs are fully human piloted and make use of off-the-shelf radio
control systems. These planes are difficult to fly due to their unconventional designs.
Another limitation of human piloted MAVs besides the range of the radio control
transmitter, is the range of the pilot’s sight. Cameras have been used on MAVs to
extend their usable range; however mapping three-dimensional control inputs from a
two-dimensional video is foreign for most pilots. So, there is an urgent need for an
MAV which can fly autonomously, to extend the operational range and to perform
diversified tasks.
Unmanned Aerial Vehicles(UAVs) are susceptible to battle damages and failures
as there is no pilot on-borad. In such a event, the aerodynamics can change rapidly
and deviate significantly from the model used for control design. To stabilize the air-
craft dynamics and achieve accurate command tracking in presence of significant model
errors, Farrel, Sharma and Polycarpou [5] present a on-line approximation based lon-
gitudinal control that is based on ideas from feedback linearization and back stepping.
Andrievsky and Fradkov [6] present combined adaptive control law with forced sliding
motion and parametric approximation for the attitude control of an unmanned aerial
vehicle.
In this report, we outline the design of flight controller for a rigid MAV to enable
it to fly autonomously. The work done is at its rudimentary stage and assumes gust
free and obstacle free environment.
The aim of the project is to design a flight controller for an MAV to enable the
MAV to navigate autonomously from a initial way-point to a final way-point. Figure 1.1
shows Kadet MkII chosen for present work. The feedback guidance strategy proposed
by Bhat and Kumar [7] achieves perfect guidance under the assumption that true
ground speed and altitude are constant and the turn rate of the MAV is perfectly
tracked. So the controller has to be designed such that the velocity, altitude and turn
rate follow the guidance command. In the first stage of the project a controller for
tracking velocity and altitude commands was designed using the root-locus approach.
The controller obtained demanded unrealistically huge propeller speed and elevator
deflection. In the second stage a controller was designed for the longitudinal dynamics
of the aircraft by combining a pole placement using state feedback with observer. In the
2
Figure 1.1: Kadet MkII
final stage we have designed the controller for complete aircraft dynamics combining
the pole placement state feedback using reduced-order observer. Closed-loop guidance
is performed with guidance in outer-loop.
In Chapter-2 the linearized aircraft models are developed for three trim states
required for guidance, that is, straight flight, right turn and left turn. We present
a controller that combines state feedback with a reduced-order observer such that
the aircraft maintains trimmed straight or coordinated turning flight at a specified
speed and turn rate and tracks altitude commands. Three controllers are designed for
stabilizing each of the three trim states required for guidance. The state feedback gain
is obtained using the pole placement technique. A reduced-order observer is used to
estimate the unmeasured states. The linear controllers are then used in simulations
with the nonlinear aircraft model and the results are presented.
In Chapter-3 the guidance strategy proposed by Bhat and Kumar is described.
Simulations of closed-loop guidance are performed with guidance block in the outer-
loop. It was found that, due to the stringent demand for straight flight, the guidance
causes the aircraft to continuously switch between right and left turns without flying
straight. Modified guidance strategy is developed, to eliminate this switching behavior.
The effectiveness of the guidance is demonstrated by simulating a three-dimensional
mission involving visiting of three way-points. Guidance under discrete position and
heading updates is simulated. Effect of closed-loop poles on closed-loop guidance is
studied. Simulations results are presented to show the effect of discrete position and
heading updates.
3
Chapter 2
Controller Design for Full Aircraft
Dynamics
We wish to implement the guidance strategy described in reference [7] on the nonlinear
aircraft model. This guidance strategy achieves perfect guidance under the assumption
that the aircraft flies at a constant speed and altitude and follows turn rate commands
perfectly. Hence, for successful implementation, the guidance strategy requires a con-
troller that keeps the speed and altitude of the aircraft constant and tracks turn rate
commands perfectly. In this chapter we design a controller such that the aircraft main-
tains trimmed straight or coordinated turning flight at a specified speed and turn rate
and tracks constant altitude commands. Three controllers are designed for stabilizing
each of the three trim states required for guidance, that is, straight flight, right turn
and left turn.
Linearized aircraft models are developed for the three trim states required for the
guidance strategy. The controllers, which consist of a state feedback controller and an
observer for estimating unmeasured states, are designed using these linearized aircraft
models. The three controllers are then implemented on a Simulink model of the non-
linear aircraft. Simulation results are presented to demonstrate the performance of the
controllers.
2.1 Linearized Aircraft Model
A nonlinear aircraft model is developed in MATLAB, using the AeroSim aeronautical
simulation block set. Variations in the density of air with altitude are neglected. The
value of air density is fixed at 1.1117 kg/m3, which is the density of air at an altitude
of 1000 metres [8]. Consequently, the dynamics of the aircraft do not depend on the
4
altitude of the aircraft. The moment generated by the propeller is also neglected. The
dynamics of engine are not considered. Instead, the propulsion system is assumed
to consist only of a propeller. The control input to the propulsion system is propeller
speed instead of traditionally used throttle setting. Static propeller data is used to find
thrust for a given advance ratio and propeller speed. Thus, the propeller is assumed to
react instantaneously for given input commands. Mass of the aircraft remains constant
throughout the flight. The aerodynamic data and propeller data used in the model is
given in Appendix A. The dynamics of the actuators deflecting the control surfaces are
not considered. So, the control surfaces deflect instantaneously when input command
is given.
The nonlinear model is linearized around a trim condition. Three trim conditions
are considered — a straight level flight at a true air-speed of 17.44 m/s, coordinated
level turning flight at a true air-speed of 17.44 m/s and a bank angle of 10 degrees, and
coordinated level turning flight at a true air-speed of 17.44 m/s and a bank angle of
−10 degrees. A trim routine available in the AeroSim aeronautical simulation blockset
was modified such that it runs on an aircraft model without engine block. This routine
was used to find the control inputs and aircraft states in each of the three trimmed
flight conditions. The values of trim aircraft states and trim control inputs of the
aircraft for the three trim states are given in appendices B, C and D.
The equations of motion for an aircraft when linearized about a trim condition can
be written in linear state variable form as [9, Chap-2]
x = Ax+ Bu, (2.1)
where the state vector
x =[
v β α φ θ p q r h
]T
(2.2)
consists of perturbations v in true air-speed, β in sideslip, α in angle of attack, φ in
bank angle, θ in pitch angle, p in roll rate, q in pitch rate, r in yaw rate and h in altitude
from the trim values VT, βT, αT, φT, θT, PT, QT, RT, H of the respective quantities. The
control vector u = [δn δe δa δr]T in equation (2.1) consists of perturbations δn in the
propeller speed, δe in the elevator deflection, δa in the aileron deflection and δr in the
rudder deflection from the trim propeller speed δnT, trim elevator deflection δeT, trim
aileron deflection δaT and trim rudder deflection δrT, respectively.
Linearized equations about a general trim condition were derived. These were used
along with the trim states and trim inputs generated by the trim routine to generate
5
linearized aircraft models for each trim state. The equations used to derive linearized
equations are given in Appendix A. The nonlinear aircraft is linearized about each of
the three trim conditions discussed, and three linearized aircraft models are generated.
The A and B matrices of the linearized aircraft models are given in appendices B, C
and D.
2.2 State Feedback Controller
We first design a state feedback controller to stabilize the trim state and track altitude
commands. Since, density variations have been neglected, the inputs required to trim
the aircraft at a given true air-speed and bank angle as well as the corresponding
trim values of the state variables do not depend on the altitude. Specifically, every
vector of the form xd=[0 0 0 0 0 0 0 0 hd]T is an equilibrium solution of the linearized
equation (2.1). Hence, the stabilization of the trim state as well as the tracking of the
desired constant altitude hd can be simultaneously achieved by stabilizing the desired
equilibrium xd, so that the closed-loop solutions satisfy limt→∞ x(t)=xd.
Let u, the control vector be given by
u = −Kx + uc, (2.3)
where K ∈ <4×9 is the state feedback gain matrix and uc is the input command. On
substituting (2.3) in (2.1) we get
x = (A− BK)x+ Buc. (2.4)
Denoting, ∆x=x − xd and noting that xd is a constant vector, (2.4) yields
∆x = (A−BK)∆x + (A−BK)xd +Buc. (2.5)
Since Axd=0, (2.5) yields
∆x = (A−BK)∆x +B(uc −Kxd). (2.6)
The error ∆x converges to zero if all the eigenvalues of the matrix A − BK have
negative real parts and uc −Kxd = 0. This gives uc = Kxd, so that
u = −K(x− xd). (2.7)
We next use the pole placement technique to find the gain matrix K such that the
matrix A− BK has its eigenvalues at specified locations.
6
2.2.1 Selection of Closed-Loop Poles
The open-loop eigenvalues of the aircraft model linearized about trimmed level flight
are located at −4.22± j7.0651, −0.034± j0.7507, 0, −15.4836, −0.0092 and −0.1377±j2.007. For the aircraft model linearized about level coordinated turn of bank an-
gle 10 degrees the eigenvalues are located at −4.2194 ± j7.0648, −0.0336 ± j0.7574,
0, −15.4806, −0.0113 and −0.1394 ± j2.0062, and for the aircraft model linearized
about level coordinated turn of bank angle −10 degrees the eigenvalues are located at
−4.2194 ± j7.0648, −0.0336 ± j0.7574, 0, −15.4806, −0.0113 and −0.1394 ± j2.0062.
As all the eigenvalues have negative real parts each of the three trim states are stable.
The first two eigenvalues correspond to short-period and phugoid modes, respectively.
The eigenvalue at zero arises because altitude is taken as a state while developing the
aircraft model. The last three eigenvalues correspond to roll, spiral and dutch-roll
modes, respectively.
Flying qualities for level-1 and category-A flight are selected, as flying qualities for
MAVs are not available. The damping ratio of the poles corresponding to the short-
period mode is selected to be between 0.35 and 1.30. The undamped natural frequency
is selected to be between 2.5 rad/sec and 9 rad/sec [10, Chap-3]. Unlike a manned
aircraft, for an MAV the phugoid mode should also be well damped with a damping
ratio around 0.7 [11]. The time constant of the roll mode eigenvalue should be less
than 1 sec. For dutch roll eigenvalue pair, the minimum values of ζ, ζωn and ωn are
given as 0.19, 0.35 and 1 rad/s, respectively [10, Chap-3]. The closed-loop poles, that
is, the eigenvalues of the matrix A−BK are placed at −4± j5, −0.3± j0.2, −0.3, −5,
−0.5± j0.5 and −0.3 for all the three trim conditions. The matrix K which places the
closed-loop poles at these points is found by using the command place in MATLAB.
Choice of poles is made taking into account the settling time of the closed-loop
system and the demands on the control inputs. If the eigenvalues of A−BK are placed
further left, the system response becomes fast, but the demands on the propeller speed
and elevator deflection will be higher than in the case of slower system. So a system
with faster response will make the control inputs reach the saturation limits even for
small commanded increase in the altitude and may cause problems of instability when
used in nonlinear aircraft model.
The state feedback controller is designed for each of the three linearized aircraft
models developed. Three state feedback controllers were designed— one for maintain-
ing the aircraft operate around the straight level flight, one for maintaining the aircraft
operate around the coordinated right turning flight and the other for maintaining the
7
aircraft operate around the coordinated left turning flight. The controller gain ma-
trix K for aircraft models trimmed for level, 10 degree banked coordinated turn and
−10 degree banked coordinated turn, which places the closed-loop poles at the above
mentioned locations are given in appendices B, C and D, respectively.
2.3 Observer Design
The feedback controller described in the previous section uses all the state variables for
feedback. However, we assume that only true air-speed, pitch rate, roll rate, yaw rate
and altitude are available for feedback. Hence we next consider the design of reduced-
order observer for estimating the unmeasured state variables based on the theory given
in reference [12, Chap-12].
The reduced-order observer generates the estimates of the perturbations β, α, φ, θ
based on the measurements of the perturbations v, p, q and r and the control vector
u. A reduced-order observer is designed for each of the three linearized models. The
dynamics of the aircraft are independent of the altitude, and are hence completely
captured in the evolution of the eight state variables v, β, α, φ, θ, p, q, r. The state
variable h is included only to achieve tracking of the commanded altitude. So the A
matrix of the system used for observer design is the 8× 8 matrix corresponding to the
state variables v, β, α, φ, θ, p, q, r.
Consider the linear aircraft model with 8 state variables
x = Ax +Bu,
y = Cx,(2.8)
where the state vector x=[v β α φ θ p q r]T can be partitioned into two parts, the
measured variables xa=[v p q r]T and the unmeasured variables xb=[β α φ θ]T . The
output matrix C is given by[
I4×4 04×4
]
so that the state variable xa is equal to the
output y. Then the partitioned state and output equations become
xa
xb
=
Aaa Aab
Aba Abb
xa
xb
+
Ba
Bb
u, (2.9)
y =[
I 0
]
xa
xb
. (2.10)
8
The equations which define the reduced-order observer are [12, Chap-12]
˙η = Aη +[
B F
]
y
u
,
x = Cη + Dy,
(2.11)
where
A = Abb −KeAab,
B = AKe + Aba −KeAaa,
F = Bb −KeBa,
C =
04×4
I4×4
,
D =
I4×4
Ke
.
(2.12)
The state estimate x and the observer state η are given by
x =
xa
xb
, (2.13)
η = xb −Key. (2.14)
The matrix Ke is found by placing the eigenvalues of A at desired locations. As
a general rule the observer poles must be two to five times faster than the controller
poles to make sure the estimation error converges to zero quickly. Such faster decay
of the observer error compared with the desired dynamics makes the controller poles
dominate the system response [12, Chap-12]. The eigenvalues of A are placed at −16,
−15 ± j1 and −17.
The observer along with the state feedback controller stabilizes the aircraft around
the trim state. The matrix Ke for aircraft models trimmed for level, level coordi-
nated right turn and level coordinated left turn are given in appendices B, C and D,
respectively.
2.4 Control Law Implementation On Nonlinear Air-
craft Model
The linear controllers described in Section 2.2 and 2.3 for the three trim states were
used in simulations with the nonlinear aircraft model. Each controller consists of a
9
SaturationBlock
Controllerxd
YTrimUTrim
YU
Trim selector
Model
linearNon
Commandedtrim
Aircraft
Figure 2.1: Block diagram of closed-loop aircraft with trim selector
state feedback controller and a reduced-order observer. Each controller stabilizes the
aircraft around the respective trim state, for which it is designed. At any given time
only one of the three controllers is used. With the help of the three controllers we can
operate the aircraft in either of the three trim states. Moreover, the aircraft can be
made to switch between any two of the three trim states by switching the corresponding
controller. This is necessary because the aircraft should be able to switch from one
trim state to other trim state for the guidance strategy to be implemented. The block
diagram depicting the selection of controller by the trim selector is shown in Figure
2.1.
The inputs to the controller are perturbations from the trim operating point, as
the linear controller operates on the perturbations in the states. The block diagram in
Figure 2.2 shows the reduced-order observer based controller employed for the nonlinear
aircraft model. Inputs to the observer are perturbations from the trim values. As
shown in Figure 2.2 the inputs to the reduced-order observer are u and y which are
the perturbations in inputs and outputs, respectively, from the trim values UTrim and
YTrim. The nonlinear aircraft model works on the true inputs. So the controller output
u is added to the trim input vector UTrim and passed to the nonlinear aircraft model.
To prevent the control inputs from going out of operating range, a saturation block
has been placed before the aircraft model. The operating speed of propeller is main-
tained between 3,000 rpm and 10,000 rpm. The deflections of elevator, aileron and
rudder are kept between −10 degrees and 10 degrees.
10
UTrim
SaturationBlock
UTrim
MatrixGain
State feedback
K
YTrim
xd
Reduced−OrderObserver
Air−speed,True
Model
Nonlinear
Altitude
AOA, bank angle Pitch angle&
Side−slip angleEstimates of
u
x
Y
y
angular rates
Aircraft
Figure 2.2: Reduced-order observer based controller
The load factor for 10 degrees bank angle turn is 1.0154. The expression for turn
rate is given by
M =g√n2 − 1
V, (2.15)
where n is load factor and V is the velocity of the aircraft. For a circular turn at a
bank angle of 10 degrees and a true air-speed of 17.44 m/s, the turn rate is −0.0992
rad/s and it takes 63.33 seconds to complete one full circular turn. Figure 2.3 shows
the response of the aircraft for commands from the trim selector. The turn rate that
has to be tracked by the aircraft is shown in dashed lines. At t = 0 the aircraft is
commanded to do a level coordinated turn at a bank angle 10 degrees. At t = 100 sec
it is commanded to fly level and straight. At t = 200 sec is commanded to do a level
coordinated turn at a bank angle −10 degrees. Figure 2.4 shows the responses of true
air-speed and altitude for commands of the trim selector. The maximum variation of
true air-speed from trim value of 17.44 m/s is 0.017 m/s. The maximum variation of
altitude from the trim altitude of 1000 m is 0.11 m. Figure 2.5 shows the responses
of angle of attack and sideslip angle to trim selector commands. Both the angle of
attack and sideslip angle vary rapidly when the aircraft switches from one trim state
to the other. Figure 2.6 shows the response of propeller speed. The propeller speed
falls sharply when the aircraft switches from one trim state to the other and then
11
0 50 100 150 200 250 300−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
t (s)
Tur
n ra
te (
rad/
s)
Figure 2.3: Response of the aircraft for trim selector commands.
settles to a constant value. Figure 2.7 shows the response of elevator deflections. The
elevator deflects sharply when the aircraft switches from one trim state to the other.
The deflections are well within the operating range. Figure 2.8 shows the response of
aileron deflections. Initially the aileron deflects away from its final steady value. Figure
2.9 shows the response of rudder deflection to trim selector commands.
12
0 50 100 150 200 250 30017.438
17.44
17.442
17.444
17.446
17.448
17.45
17.452
17.454
17.456
17.458
t (s)
Tru
e a
ir−
sp
ee
d (
m/s
)
0 50 100 150 200 250 300999.88
999.9
999.92
999.94
999.96
999.98
1000
1000.02
t (s)
Altitude (
m)
Figure 2.4: True air-speed and altitude responses to trim selector commands.
0 50 100 150 200 250 3002.1
2.15
2.2
2.25
2.3
2.35
2.4
2.45
t (s)
Angle
of attack (
deg)
0 50 100 150 200 250 300−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t (s)
Sid
e−
slip
an
gle
(d
eg
)
Figure 2.5: Angle of attack and sideslip responses to trim selector commands.
13
0 50 100 150 200 250 3005900
5920
5940
5960
5980
6000
6020
6040
6060
t (s)
Pro
pelle
r speed (
rpm
)
Figure 2.6: Response of propeller speed
to trim selector commands.
0 50 100 150 200 250 3004
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
t (s)
Ele
vato
r deflection (
deg)
Figure 2.7: Response of elevator deflec-
tion to trim selector commands.
0 50 100 150 200 250 300−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
t (s)
Aile
ron
de
fle
ctio
n (
de
g)
Figure 2.8: Response of aileron deflection
to selector commands.
0 50 100 150 200 250 300−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
t (s)
Ru
dd
er
de
fle
ctio
n (
de
g)
Figure 2.9: Response of rudder deflection
to selector commands.
14
Chapter 3
Closed-Loop Guidance
3.1 Guidance Strategy
Bhat and Kumar [7] present a guidance strategy to steer a MAV from a given initial
position and heading to a specified destination way-point in an obstacle-free environ-
ment. The strategy achieves perfect guidance at constant altitude and speed under
continuous and perfect position and heading updates, and perfect tracking of turn rate
commands for small as well as large inter way-point distances.
Reference [7] considers a MAV flying at a constant altitude and speed in a two-
dimensional plane. A kinematic model for such an aircraft is given by
x = V cos λ,
y = V sin λ,
λ = ν,
(3.1)
where x and y are the position co-ordinates of the aircraft, λ is the heading angle of
the aircraft, ν is the turn rate and V is the speed of the aircraft. If the maximum
permissible load factor of the MAV is n, then the maximum permissible rate at which
the aircraft can turn is given by equation (2.15).
The turn rate constraint |ν| ≤ M leads to a lower bound V/M on the turn radius.
Under the assumption of constant velocity, time-optimal trajectories of (3.1) between
specified initial and final positions and initial headings under the turn rate constraint
|ν| ≤ M consist of arcs of circles of minimum turn radius V/M and straight lines. In
other words, a time-optimal turn rate time history takes the values ±M and 0.
Let ~r and ~rf be the position vectors of the instantaneous location and destination
way-point respectively. The cross range error P and the down range error Q to the
15
R
R
L
L
Cross range
Down range
Figure 3.1: Phase portrait under guidance strategy
destination are defined as
P = − 1
V(~V × ~D).k, (3.2)
Q =1
V( ~D · ~V ), (3.3)
where ~D = ~rf − ~r is the relative displacement between the instantaneous location and
the destination way-point, ~V is the instantaneous velocity vector of the MAV, and ~k
is a unit vector orthogonal to the plane of motion of the MAV. The magnitudes of
~D and ~V are√P 2 +Q2 and V , respectively. Letting ψ(P,Q) = (|P | − V/M)2 + Q2,
the feedback strategy that steers the MAV from a initial position and heading to a
specified destination way-point along the shortest path is given by
ν(t) = λ(P (t), Q(t)), (3.4)
where
λ(P,Q) = −Msign(P ), ψ(P,Q) ≥ V 2/M2, P 6= 0,
= Msign(P ), ψ(P,Q) < V 2/M2,
= 0, P = 0, Q > 0,
= −M, P = 0, Q < 0.
(3.5)
Figure 3.1 shows the phase portrait of the closed-loop system obtained by applying
the guidance strategy (3.5) to the kinematic equations (3.1). It demonstrates that the
guidance strategy (3.4) - (3.5) steers the vehicle to the destination way-point by driving
cross range and down range to zero in a finite time.
16
SaturationBlock
Controllerxd
Trim selectorGuidanceLatitudeLongitudeHeading
Model
Nonlinear
UTrimYTrim
YU
Commandedtrim
Aircraft
Figure 3.2: Schematic diagram of guidance strategy implementation
3.2 Closed-Loop Guidance
The guidance strategy described in Section 3.1 is used for guiding the aircraft from the
initial way-point to the final way-point. A schematic diagram of the nonlinear aircraft
model with guidance in the outer-loop is shown in the Figure 3.2. The guidance strategy
given in Section 3.1 is implemented in the guidance block. The guidance block takes
position and heading updates of the aircraft and gives commands to the trim selector
in order to select one of the three trim states described in Section 2.1. The guidance
block then commands the aircraft to either fly straight, turn right, or turn left and
selects the correct inner-loop controller to ensure that the aircraft operation converges
to that trim state. As the distances between the way-points considered by us are small
compared to the radius of earth, x and y co-ordinates of the position of the aircraft
are obtained by multiplying longitude and latitude by the radius of earth, respectively.
The radius of earth is taken as 6371.3 km [13].
Results of simulations performed on closed-loop aircraft with guidance in the outer-
loop are presented. Figure 3.3 shows the four paths followed by the aircraft to reach
each of the way-points (0,200,1000), (0,800,1000) (0,-800,1000) and (1000,0,1000) start-
ing from (0,0,1000) with a initial heading along the x-axes. For inter way-point dis-
tances greater than minimum turn radius given by VT/M=175.84 m, the aircraft first
takes a turn until the heading is aligned with destination way-point. For inter way-
point distances less than the minimum turn radius, the aircraft first moves away from
destination way-point and then turns towards it. The simulation is stopped as soon as
the trajectories reach within 20 m of the destination way-point. The aircraft can not
come very close to the destination way-point because the aircraft takes time to settle
17
−400 −200 0 200 400 600 800 1000 1200
−800
−600
−400
−200
0
200
400
600
800
X (m)
Y (
m)
Figure 3.3: Aircraft path under closed-loop guidance
down in the trim state while switching between the trim states. The guidance strat-
egy assumes that the guidance commands are obeyed instantaneously, but the aircraft
takes a finite time to switch from one trim state to another. This sluggishness of the
aircraft to implement guidance commands results in the aircraft missing the target by
20 metres.
As shown in Figure 3.3, instead of flying straight to reach the way-point (1000,0,1000),
which is aligned with the initial heading of the aircraft, the aircraft reaches the way-
point by performing a series of right and left turns. This is because of the stringent
demand of P = 0 by the guidance strategy for straight flight. When the aircraft head-
ing is aligned with the destination way-point the guidance block should command the
aircraft to fly straight, but by the time command is issued the aircraft over-shoots and
the heading of the aircraft is no longer aligned with the destination way-point. As the
condition P = 0 is satisfied only at discrete instants, the guidance block commands the
aircraft either to turn right or left. So, the aircraft reaches the destination way-point
by performing a sequence of right or left turns and the path of the aircraft is wavy,
instead of smooth circular turns and straight lines.
3.3 Modified Guidance Strategy
The guidance strategy described in Section 3.1 is modified so that the aircraft reaches
the destination way-point smoothly without continuously switching between right and
18
R
R
L
L
S
Down range
Cross range
Figure 3.4: Phase portrait under modified guidance strategy
left turns. Conditions for straight flight are relaxed such that the aircraft flies straight
even if the destination way-point is not aligned with the heading of the aircraft. The
guidance strategy is modified such that the aircraft flies straight when the destination
way-point is within a cone of angle 180-2γ containing the current heading. The strategy
is thus given by
ν(t) = λ(P (t), Q(t)), (3.6)
where
λ(P,Q) = −Msign(P ), ψ(P,Q) ≥ V 2/M2, P 6= 0, Q− |P | tanγ < 0,
= Msign(P ), ψ(P,Q) < V 2/M2, Q− |P | tanγ < 0,
= 0, Q− |P | tanγ ≥ 0, Q > 0,
= −M, P = 0, Q < 0.
(3.7)
Figure 3.4 shows the phase portrait of the closed-loop system obtained by apply-
ing the strategy (3.6) to the kinematic equations (3.1). The phase portrait in the
figure demonstrates that the guidance strategy (3.6) - (3.7) steers the vehicle to the
destination way-point.
Simulations are done with different values of γ, and γ = 80 degrees is found to give
better guidance in terms of the termination proximity to the destination way-point.
With γ = 80 the aircraft reaches within 20 m to the destination way-point. The path
followed by the aircraft with this modified guidance strategy is shown in the Figure 3.5
for the same destination way-points considered in Section 3.2. The aircraft flies straight
for a considerable amount of time. The aircraft reaches the way-point (1000,0,0) flying
19
−400 −200 0 200 400 600 800 1000 1200
−800
−600
−400
−200
0
200
400
600
800
X (m)
Y (
m)
Figure 3.5: Aircraft path under modified guidance strategy
straight. Even for other way-points the path followed by aircraft is smooth.
Figures 3.6 and 3.7 compare the sequence of trim states commanded by the origi-
nal guidance strategy to reach the destination way-point (0,800,1000) with that com-
manded by the modified guidance strategy. Trim selector command of 1 represents
right turn, 2 represents straight flight and 3 represents left turn. The aircraft with the
original guidance strategy switches between left and right turns and never flies straight,
whereas the aircraft with the modified guidance strategy flies straight switching only
occasionally to right and left turning trim states.
Three dimensional way-point navigation is performed, as the controllers are capable
of tracking altitude commands. Figure 3.8 shows the trajectory followed by the aircraft
for visiting the three way-points (1500,1000,1005), (4000,0,1030) and (1000,-3000,995)
in sequence starting from (0,0,1000) with a initial heading along the x-axes under the
modified guidance strategy. The co-ordinates of the next way-point are loaded as soon
as the aircraft is within 20 m from the current destination way-point. The first way-
point (1500,1000,1005) is reached at t = 104.2 secs, second way-point at t = 262.9 secs
and the final way-point at t = 522.4 secs. The simulation is stopped when the aircraft
reaches within 20 m from the third way-point.
To reach the way-point the aircraft first climbs to the altitude of the way-point,
and then moves at constant altitude. Figure 3.9 shows the altitude variation of the
aircraft. As the three way-points are not in the same plane, the altitude of the aircraft
changes while visiting the way-points.
20
0 10 20 30 40 50 60
1
2
3
t (s)
Co
mm
an
ds t
o t
rim
se
lecto
r
Figure 3.6: Trim selector commands with
original guidance strategy.
0 10 20 30 40 50 60
1
2
3
t (s)
Co
mm
an
ds t
o t
rim
se
lecto
r
Figure 3.7: Trim selector commands with
modified guidance strategy.
Figure 3.10 shows the trim selector commands as the aircraft visits the three way-
points. The commands change rapidly as the aircraft comes close to the way-point.
Figure 3.11 gives true air-speed response of the aircraft. There is sudden variation in
the true air-speed just after each way-point is visited. As soon as the aircraft reaches
a way-point, the aircraft is commanded to change altitude to match the altitude of
the next way-point. So the true air-speed of the aircraft either increases or decreases
depending on whether the aircraft is descending or ascending. Figure 3.12 shows the
variation of angle of attack. The angle of attack momentarily reaches 10.3 degrees when
the aircraft is climbing from an altitude of 1005 metres to 1030 metres. Figure 3.13
shows the variation of sideslip angle. The sideslip angle reaches a maximum of 11.5
degrees. Figure 3.14 shows the variation of engine speed. The engine speed is pushed to
the limits of its operating range. The propeller speed reaches the upper limit of 10,000
rpm when the aircraft is climbing and reaches the lower limit of 3,000 rpm when the
aircraft is descending. Figure 3.15 shows the variation of elevator deflection. The
elevator deflection reaches its upper limit of 10 degrees when the aircraft is descending
and reaches the lower limit of −10 degrees when the aircraft is climbing. Figure 3.16
shows the variation of aileron deflection. The variation in aileron deflection is well
within its operating range. The aileron deflection reaches a maximum of 0.86 degrees
and a minimum of −0.77 degrees. The variation in aileron deflection is so small that
it may be difficult for the actuator to deflect the aileron by such small angle. The
21
0 1000 2000 3000 4000−3000
−2000−10000
1000
995
1000
1005
1010
1015
1020
1025
1030
x (m) y (m)
z (
m)
Figure 3.8: Trajectory of the aircraft for
visiting three way-points.
0 100 200 300 400 500 600990
995
1000
1005
1010
1015
1020
1025
1030
1035
t (s)
Altitu
de
(m
)
Figure 3.9: Variation of altitude for visit-
ing three way-points.
resolution of the actuator should be high and the mechanical couplings should not
have dead zones for the aileron to be deflected to such small angle. Figure 3.17 shows
the variation in rudder deflection as the aircraft visits the way-points. The rudder
deflection reaches a maximum of 4.45 degrees and a minimum of −3.85 degrees.
3.4 Guidance Under Discrete Position and Heading
Updates
One of the main assumptions for the guidance strategy is that the position and heading
updates are continuously available. This assumption is not valid in practice. In order
to sense the position and heading, we anticipate use of the Global Positioning System
(GPS) which receives updates typically at an update frequency of 1Hz. In this case,
only discrete position and heading updates are available. In this section we present the
effect of discrete updates on the guidance strategy.
In between the updates, no data is available to the guidance. To simplify the
situation, we assume that the guidance command in between the updates is based only
on the last updates of position and heading. To study the effect of discrete position
and heading updates on the guidance strategy the GPS is simulated by a sample and
hold block with a sampling period of 1 sec. The closed-loop guidance with sample and
hold block is shown in Figure 3.18.
22
0 100 200 300 400 500 600
1
2
3
t (s)
Trim
se
lecto
r co
mm
an
ds
Figure 3.10: Trim selector commands as
the aircraft visits the way-points.
0 100 200 300 400 500 60015
16
17
18
19
20
21
22
23
24
25
t (s)
Tru
e a
ir−
sp
ee
d (
m/s
)
Figure 3.11: Response of true air-speed of
the aircraft.
0 100 200 300 400 500 600−2
0
2
4
6
8
10
12
t (s)
An
gle
of a
tta
ck (
de
g)
Figure 3.12: Variation of angle of attack.
0 100 200 300 400 500 600−10
−5
0
5
10
15
t (s)
Sid
e s
lip a
ng
le (
de
g)
Figure 3.13: Variation of sideslip angle.
23
0 100 200 300 400 500 6002000
3000
4000
5000
6000
7000
8000
9000
10000
11000
t (s)
Pro
pelle
r speed (
rpm
)
Figure 3.14: Variation of propeller speed.
0 100 200 300 400 500 600−10
−8
−6
−4
−2
0
2
4
6
8
10
t (s)
Ele
va
tor
de
fle
ctio
n (
de
g)
Figure 3.15: Variation of elevator deflec-
tion.
0 100 200 300 400 500 600−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t (s)
Aile
ron
de
fle
ctio
n (
de
g)
Figure 3.16: Variation of aileron deflec-
tion.
0 100 200 300 400 500 600−4
−3
−2
−1
0
1
2
3
4
5
t (s)
Ru
dd
er
de
fle
ctio
n (
de
g)
Figure 3.17: Variation of rudder deflec-
tion.
24
SaturationBlock
Controllerxd
Hold
Guidance
Model
linearNon
UTrim YTrim
YU
LatitudeLongitudeHeadingCommanded
Trim selector
trim
Sample &
Aircraft
Figure 3.18: Schematic diagram of guidance strategy implementation with sample and
hold block
Figure 3.19 presents two trajectories — the one in red generated under discrete po-
sition and heading updates, and the one in blue generated under continuous position
and heading updates, the modified guidance strategy being used in both the cases.
The aircraft starts at (0,0,1000) with a initial heading along the x-axes and reaches
the destination way-point (600,500,1000). The simulation is stopped when the aircraft
is within 20 m from the destination way-point. The crosses on the trajectory represent
the locations where the updates are made available from the GPS block. While the
closed-loop aircraft follows straight path when position and heading updates are con-
tinuously available, the aircraft deviates from the straight path under discrete position
and heading updates. This is expected, because the guidance strategy demands contin-
uous position and heading updates for perfect guidance. As the position and heading
are updated at a interval of 1 second, the aircraft deviates from the path under discrete
updates.
3.5 Effect of Closed-Loop Poles On Guidance
The guidance strategy assumes that the guidance commands are implemented instan-
taneously, but the closed-loop aircraft takes a nonzero amount of time to settle in a
trim state after switching from either of the other two trim states. Clearly faster set-
tling yields better guidance. To study the improvement provided by faster settling,
25
−100 0 100 200 300 400 500 600 700−100
0
100
200
300
400
500
600
x (m)
y (
m)
Figure 3.19: Guidance under discrete updates
three controllers are designed to place the closed-loop poles for each of the three linear
models at −4 ± j5, −1 ± j1, −1, −5, −1.5 ± j1 and −1 by selecting a suitable K
matrix. Now, the closed-loop aircraft will be able to react quickly to the guidance
commands, as the poles are placed further left in the complex plane than those given
in subsection 2.2.1. The effect of closed-loop poles on guidance is studied by comparing
the trajectory and trim selector commands for the closed-loop aircraft with faster poles
with those of the aircraft with slower poles.
Figure 3.20 shows the trajectory of the closed-loop aircraft with faster poles in red,
and that with slower poles in blue. The original guidance strategy is implemented and
the aircraft starts at (0,0,1000) with a initial heading along the x-axes in both the
cases. The closed-loop aircraft with faster poles goes as close as 2 m to the destination
way-point (-300,800,1000) where as the closed-loop aircraft with slower poles can go
only up to 20 m to the destination way-point. Because of the small settling times
of its states, the closed-loop aircraft with faster poles is quick to settle to one of the
three trim states and achieves better guidance. This is evident from red plot which
shows that the the closed-loop aircraft with faster poles reaches the way-point along
a straight line path. The trajectory is smooth even if the original guidance strategy
is used, because the aircraft is quick enough to respond to guidance commands. So,
better guidance can be achieved with original guidance strategy, if the aircraft settles
faster in the trim state commanded.
Figure 3.21 shows the trim selector commands of the closed-loop aircraft with faster
26
−500 −400 −300 −200 −100 0 100 200 300 400
−100
0
100
200
300
400
500
600
700
800
900
x (m)
y (
m)
Figure 3.20: Trajectories of aircraft with faster and slower poles
poles and Figure 3.22 shows the trim selector commands of the closed-loop aircraft
with slower poles. The closed-loop aircraft with faster poles switches between the trim
states more frequently than the closed-loop aircraft with slower poles. So, more energy
is expended to deflect the control surfaces. The closed-loop aircraft with faster poles
reaches the destination way-point in 62.96 secs, where as the closed-loop aircraft with
slower poles takes 66.84 secs. So, the closed-loop aircraft with slower poles has to fly
for longer time as it takes longer path to reach to the way-point.
27
0 10 20 30 40 50 60 70
1
2
3
t (s)
Trim
se
lecto
r co
mm
an
ds
Figure 3.21: Trim selector commands to
the closed-loop aircraft with faster poles.
0 10 20 30 40 50 60 70
1
2
3
t (s)
Trim
se
lecto
r co
mm
an
ds
Figure 3.22: Trim selector commands to
the closed-loop aircraft with slower poles.
28
Chapter 4
Conclusion
A controller combining state feedback and observer is designed for nonlinear aircraft
model of MAV, Kadet Mk-II, such that the aircraft maintains trimmed straight or co-
ordinated turning flight at a true air-speed of 17.44 m/s and tracks altitude commands.
Closed-loop guidance is performed to steer the aircraft from the initial way-point to
within 20 m of the destination way-point when the minimum turn radius is 175.84
m. The propeller speed and control surface deflections are well with in the operating
range, when the aircraft is visiting way-points in a plane.
The guidance strategy is modified to smoothen the trajectory of the aircraft by
eliminating continuous switching between right and left turns. Simulation of three-
dimensional way-point navigation is performed. When the aircraft visits way-points
at different altitudes in three-dimensional space, the propeller speed and elevator de-
flection are pushed to the limits of operating range. Altitude commands of 35 metres
are tracked simultaneously following the guidance commands. The aircraft with faster
closed-loop poles performs better guidance reaching as close as 2 metres to the desti-
nation way-point. Under discrete position and heading updates the aircraft deviates
from straight line path.
29
Appendix A
Nonlinear Aircraft Equations and
Aircraft Data
The wind axes force equations and body axes moment equations are used for generating
linearized aircraft models.
A.1 Nonlinear Aircraft Equations
The wind axes force equation in vector form is [9, Chap-2]
FW
m= VW + ΩRVW + ωW × VW − TWBTBIg (A.1)
where m is the mass of the aircraft, FW is the force vector expressed in wind axes,
ωW is the angular velocity vector expressed in wind axes, TWB is the transformation
matrix from body to wind axes, TBI transformation matrix from inertial to body axes
and g is the gravitational vector expressed in inertial frame.
ΩR =
0 −β −α cos β
β 0 α sin β
α cos β −α sin β 0
TWB =
cosα cos β sin β sinα cos β
− cosα sin β cos β − sinα sin β
− sinα 0 cosα
30
TBI =
cos θ cosψ cos θ sinψ − sin θ
− cosφ sinψ + sinφ sin θ cosψ cosφ cosψ + sinφ sin θ sinψ sinφ cos θ
sinφ sinψ + cosφ sin θ cosψ − sin φ cosψ + cos φ sin θ sinψ cos φ cos θ
g =
0
0
9.81
The force vector expressed in wind axes FW in (A.1) can be replaced by
FW = FWA+ FWT
= FWA+ TWBFBT
(A.2)
where FWA=
−DY
−L
is the aerodynamic force vector expressed in wind axes, FWT
is the thrust vector expressed in wind axes, FBT=
FT
0
0
thrust vector expressed in
body axes. The angular velocity vector ωW expressed in wind axes can be replaced
by TWBωB. FT is the thrust force generated by the propeller. It is assumed that the
thrust vector passes along the body x-axes. After the two replacements the three force
equations reduce to
mV = −D + FT cosα cos β +mg1,
mβV = Y − FT cosα sin β −mV (−P sinα +R cosα) +mg2,
mαV cos β = −L− FT sinα +mV (−P cosα sin β +Q cos β −R sinα sin β) +mg3,
(A.3)
where
g1 = g(− cosα cos β sin θ + sin β sinφ cos θ + sinα cos β cosφ cos θ),
g2 = g(cosα sin β sin θ + cos β sinφ cos θ − sinα sin β cosφ cos θ),
g3 = g(sinα sin θ + cosα cos θ cosφ).
g=9.81 m/s2 is the acceleration due to gravity. V is the true air-speed, while P,Q,R
are the body axes roll, pitch and yaw rates, respectively.
The body axes moment equation in vector form is
TB = JωB + ωB × JωB (A.4)
where TB the moment vector expressed in body axes and J is the moment of inertia
31
matrix given by
J =
Jx 0 −Jxz
0 Jy 0
−Jxz 0 Jz
.
The three moment equations are
L = JxP − JxzR +Q(−JxzP + JzR) − JyQR,
M = JyQ+R(JxP − JxzR) − P (−JxzP + JzR),
N = −JxzP + JzR + JyPQ−Q(JxP − JxzR).
(A.5)
L,M,N are the body axes rolling, pitching and yawing moments, respectively.
The rotational kinematic equations are
φ = P +Q tan θ sinφ+R tan θ cosφ,
θ = Q cosφ− R sin φ,
ψ = Q sinφ sec θ +R cos φ sec θ.
(A.6)
The altitude equation is obtained by transforming the wind axes velocity vector
into inertial frame. The velocity vector[
Xi Yi −H]T
in inertial frame is given by
Xi
Yi
−H
= T TBIT
TWB
V
0
0
. (A.7)
Expanding (A.7), the equation for −H is given by
− H = V (− sin θ cosα cos β + sin φ cos θ sin β + cos φ cos θ sinα cos β). (A.8)
The nine nonlinear aircraft equations (A.3), (A.5), (A.6) and (A.8) are linearized.
The equation for ψ is not considered because ψ is not a state in the aircraft model.
Solving the implicit state equation f(X,X, U) = 0 with the nine nonlinear equations
at equilibrium point (Xe, Ue), we obtain linearized equations of motion, which can be
expressed in linear state variable form as
Ex = Ax +Bu, (A.9)
where x and u are perturbations from the equilibrium values of the state and control
vectors. The state vector x is given by
x =[
v β α φ θ p q r h
]T
(A.10)
32
and the control vector u is given by
u =[
δn δe δa δr
]T
. (A.11)
The coefficient matrices
E = −
∆Xf1
...
∆Xf9
X = Xe
U = Ue
, A = −
∆Xf1
...
∆Xf9
X = Xe
U = Ue
and B = −
∆Uf1
...
∆Uf9
X = Xe
U = Ue
.
(A.12)
are calculated at the equilibrium point (Xe, Ue).
A.2 Aircraft Data
The aircraft data for Kadet Mk-II is listed below
Moment reference point =
0.056
0
0
m
Mean Aerodynamic Chord c = 0.29 m
Wing span b = 1.7 m
Wing area S = 0.493 m2
Mass of the aircraft = 3.7 kg
Location of center of gravity =
0
0
0
m
Moment and products of inertia
Jx = 0.22584 kgm2
Jy = 0.80374 kgm2
Jz = 1.00505 kgm2
Jxz = 0.0491527 kgm2
33
CT
J 0.8
0.1
Figure A.1: Variation of coefficient of
thrust with advance ratio.
CP
0.1
0.45 0.8J
Figure A.2: Variation of coefficient of
power with advance ratio.
Propeller data
Diameter of propeller D = 0.28 m
Advance ratio J = 60V/(δnD)
Coefficient of thrust CT = 0.1 − J/8
Coefficient of power CP = 0.1, J < 0.45
= 8/35 − 2J/7, 0.45 < J < 0.8
Propeller Thrust FT = ρδn2D4CT/3600 N
Propeller Moment LT = −ρδn2D5CP/7200π N − m
Lift coefficient
CL = CL0+ CLα
α + CLδfδf + CLδe
δe + c2V
(CLαα + CLq
q) + CLMM
Lift derivatives
CL0= 0.2
CLα= 4.8116
CLδf= 0
CLδe= 0.51695
CLα= 0
CLq= 10.0727
CLM= 0
34
Drag coefficient
CD = CDmin+
(CL−CLminD)2
πeAR+ CDδf
δf + CDδeδe+ CDδa
δa + CDδrδr + CDM
M
Drag derivatives
CDmin= 0.015
CL at minimum drag CLminD= 0.2
CDδf= 0
CDδe= 0
CDδa= 0
CDδr= 0
CDM= 0
Side force coefficient
CY = CYββ + CYδa
δa + CYδrδr + b
2V(CYp
p+ CYrr)
Side force derivatives
CYβ= −0.06695
CYδa= −0.00933
CYδr= −0.11866
CYp= 0.029337
CYr= −0.072043
Pitching moment coefficient
Cm = Cm0+ Cmα
α + Cmδfδf + Cmδe
δe + c2V
(Cmαα + Cmq
q) + CmM.M
Pitch moment derivatives
Cm0= 0.135
Cmα= −2.95
Cmδf= 0
Cmδe= −1.3104
Cmα= 0
Cmq= −10.4977
CmM= 0
35
Rolling moment coefficient
Cl = Clββ + Clδaδa + Clδr
δr + b2V
(Clpp+ Clrr)
Roll moment derivatives
Clβ = −0.0281
Clδa= 0.2945
Clδr= 0.00204
Clp = −0.49926
Clr = 0.0083
Yawing moment coefficient
Cn = Cnββ + Cnδa
δa + Cnδrδr + b
2V(Cnp
+ Cnr)
Yaw moment derivatives
Cnβ= 0.0265
Cnδa= 0.01165
Cnδr= −0.05734
Cnp= −0.01075
Cnr= −0.03044
Aerodynamic forces and moments
D = 12ρV 2SCD
Y = 12ρV 2SCY
L = 12ρV 2SCL
LMR = 12ρV 2SbCl
MMR = 12ρV 2ScCm
NMR = 12ρV 2SbCn
LMR, MMR and NMR are rolling, pitching and yawing moments about the moment
reference point, respectively. The moments about the centre of gravity L, M and N
are obtained as follows
L
M
N
=
LMR
MMR
NMR
+
0.056
0
0
× TBW
−DY
−L
.
36
Appendix B
Linear Controller Design for Level
Straight Flight
The trim values of the aircraft states for steady straight level flight of the aircraft Kadet
MkII flying with a velocity of VT =17.44 m/s at an altitude of 1000 metres where the
density of air ρ is 1.1117 kg/m3 are
VT = 17.44 m/sec,
βT = 0 radians,
αT = 0.04066 radians,
φT = 0 radians,
θT = 0.04066 radians,
PT = 0 radians/sec,
QT = 0 radians/sec,
RT = 0 radians/sec,
H = 1000 metres.
The control inputs which enable the aircraft to fly in this trimmed condition are
δnT = 6011.1858 rpm,
δeT = 0.0756 radians,
δaT = 0 radians,
δrT = 0 radians.
37
For these aircraft states and control inputs the matrices A and B of the linearized
aircraft model (2.1) are given by
A =
−0.1304 0 4.1078 0 −9.81 0 0 0 0
0 −0.1101 0 0.562 0 0.0425 0 −1.0037 0
−0.0642 0 −6.2386 0 0 0 0.8918 0 0
0 0 0 0 0 1 0 0.0407 0
0 0 0 0 0 0 1 0 0
0 −17.0851 0 0 0 −15.4428 0 0.2069 0
0 0 −60.7325 0 0 −2.1389 0 0
0 2.5035 0 0 0 −0.899 0 −0.2153 0
0 0 −17.44 0 17.44 0 0 0 0
,
B =
0.0226 0 0 0
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
0 0 0 0
.
The state feedback gain matrix required to place the eigenvalues of the closed-loop
system at −4 ± j5, −0.3 ± j0.2, −0.3, −5, −0.5 ± j0.5 and −0.3 is given by
K =
8.7696 −0.0634 145.6218 −0.1451 −294.8725 −0.28 4.8324 0.1878 2.2645
−0.0138 0 1.7131 0 −1.2192 0 −0.166 0 −0.0187
0 −0.0926 0 0.0023 0 −0.0779 0 −0.0003 0
0.0002 −0.3879 0 0.0303 0.0003 0.0184 0 −0.0411 0
.
The sub-matrices of the A and B matrices, used in (2.9)-(2.12) for design of reduced-
38
order observer are given by
Aaa =
−0.1304 0 0 0
0 −15.4428 0 0.2069
0 0 −2.1389 0
0 −0.809 0 −0.2153
,
Aab =
0 4.1078 0 −9.81
−17.0851 0 0 0
0 −60.7325 0 0
2.5035 0 0 0
,
Aba =
0 0.0425 0 −1.0037
−0.0642 0 0.8918 0
0 1 0 0.0407
0 0 1 0
,
Abb =
−0.1101 0 0.562 0
0 −6.2386 0 0
0 0 0 0
0 0 0 0
,
Ba =
0.0226 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
,
Bb =
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
.
The observer gain matrixKe is selected such that the observer poles, that is, eigenvalues
of A are located at −16, −15 ± j1 and −17. The observer gain matrix Ke is given as
Ke =
0 −1.7127 0 0.251
0 0 −0.1607 0
0 −23.041 0 3.3762
−1.7329 0 −0.1172 0
.
The matrix Ke is found using command place in MATLAB.
The matrices A, B, F , C and D which define the reduced-order observer in (2.11)
39
are given by
A =
−30 0 0.562 0
0 −16 0 0
−402.1102 0 0 0
0 0 0 −17
,
B =
0 12.2278 0 −6.2266
−0.0642 0 3.1197 0
0 336.6067 0 −95.3788
29.2338 0 2.7419 0
,
F =
0 0 317.8 1
0 −6.5 0 0
0 0 4274.9 15.3
0 −4.3 0 0
,
C =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,
D =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0 −1.7127 0 0.251
0 0 −0.1607 0
0 −23.041 0 3.3762
−1.7329 0 −0.1172 0
.
40
Appendix C
Linear Controller Design for Level
Coordinated Right Circular Turn
The trim values of the aircraft states for level coordinated right circular turn at a bank
angle of 10 degrees of the aircraft Kadet MkII flying with a true air-speed of VT =17.44
m/s at an altitude of 1000 metres where the density of air ρ is 1.1117 kg/m3 are
VT = 17.44 m/sec,
βT = 0 radians,
αT = 0.04212 radians,
φT = 0.1745 radians,
θT = 0.04148 radians,
PT = −0.004106 radians/sec,
QT = 0.01718 radians/sec,
RT = 0.09746 radians/sec,
H = 1000 metres.
The control inputs which enable the aircraft to fly in this trimmed condition are
δnT = 6022.646 rpm,
δeT = 0.072167 radians,
δaT = −0.000447 radians,
δrT = −0.002601 radians.
41
For these aircraft states and control inputs the matrices A and B of the linearized
aircraft model (2.1) are given by
A =
−0.1305 1.7020 3.8706 −0.0717 −9.8097 0 0 0 0
−0.0056 −0.1104 0.0082 0.5535 −0.0041 0.044 0 −1.0036 0
−0.0642 0 −6.2389 −0.0975 0.0007 0 0.8918 0 0
0 0 0 0 0.0991 1 0.0072 0.0409 0
0 0 0 −0.099 0 0 0.9848 −0.1736 0
−0.0048 −17.0851 0 0 0 −15.4412 −0.0892 0.1913 0
0.0026 0 −60.7365 0 0 0.095 −2.1389 0.0079 0
0.001 2.5035 0 0 0 −0.8188 −0.0068 −0.217 0
0 −3.0258 −17.1755 0.1274 17.4395 0 0 0 0
,
B =
0.0227 0 0 0
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
0 0 0 0
.
The state feedback gain matrix required to place the eigenvalues of the closed-loop
system at −4 ± j5, −0.3 ± j0.2, −0.3, −5, −0.5 ± j0.5 and −0.3 is given by
K =
8.7612 51.5647 135.2113 −3.9086 −290.0693 2.3402 4.9131 −0.949 2.3207
−0.0143 0.2019 1.7134 0.0542 −1.2373 0.0059 −0.1659 0.0012 −0.019
0 −0.0932 −0.0001 0.0024 0.004 −0.0778 −0.0008 −0.0005 0.0001
−0.0009 −0.3947 −0.0053 0.0294 0.0411 0.0186 −0.0007 −0.0408 0.0006
.
The sub-matrices of the A and B matrices, used in (2.9)-(2.12) for design of reduced-
order observer are given by
42
Aaa =
−0.1305 0 0 0
−0.0048 −15.4412 −0.0892 0.1913
0.0026 0.095 −2.1389 0.0079
0.001 −0.8188 −0.0068 −0.217
,
Aab =
1.702 3.8706 −0.0717 −9.8097
−17.0851 0 0 0
0 −60.7365 0 0
2.5035 0 0 0
,
Aba =
−0.0056 0.044 0 −1.0036
−0.0642 0 0.8918 0
0 1 0.0072 0.0409
0 0 0.9848 −0.1736
,
Abb =
−0.1104 0.0082 0.5535 −0.0041
0 −6.2389 −0.0975 0.0007
0 0 0 0.0991
0 0 −0.099 0
,
Ba =
0.0227 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
,
Bb =
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
.
The observer gain matrixKe is selected such that the observer poles, that is, eigenvalues
of A are located at −16, −15 ± j1 and −17. The observer gain matrix Ke is given as
Ke =
0.1095 −1.7599 0.0001 0.2579
−0.0138 0.145 −0.1585 −0.0212
4.268 −24.0722 0.4814 3.5273
−1.6802 0.1661 −0.1069 −0.0243
.
The matrix Ke is found using command place in MATLAB.
The matrices A, B, F , C and D which define the reduced-order observer in (2.11)
are given by
43
A =
−31.0107 −0.4073 0.5613 1.0698
2.5536 −15.8132 −0.0985 −0.1342
−427.3718 12.7172 0.3059 41.9675
5.7576 0.0127 −0.2194 −16.482
,
B =
−2.7907 14.2625 0.0616 −6.6456
0.2373 −2.2018 3.0395 0.6192
−115.728 385.728 −7.4989 −105.0173
27.168 −5.0328 2.4254 0.9019
,
F =
0 0 326.5 1
0 −6.4 −26.9 −0.1
−0.1 17.5 4466.2 16
0 −3.9 −30.8 −0.1
,
C =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,
D =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
0.1095 −1.7599 0.0001 0.2579
−0.0138 0.145 −0.1585 −0.0212
4.268 −24.0722 0.4814 3.5273
−1.6802 0.1661 −0.1069 −0.0243
.
44
Appendix D
Linear Controller Design for Level
Coordinated Left Circular Turn
The trim values of the aircraft states for level coordinated circular turn at a bank angle
-10 degrees of the aircraft Kadet MkII flying with a true air-speed of VT =17.44 m/s at
an altitude of 1000 metres where the density of air ρ is 1.1117 kg/m3 are
VT = 17.44 m/sec,
βT = 0 radians,
αT = 0.04212 radians,
φT = −0.1745 radians,
θT = 0.04148 radians,
PT = 0.004106 radians/sec,
QT = 0.01718 radians/sec,
RT = −0.09746 radians/sec,
H = 1000 metres.
The control inputs which enable the aircraft to fly in this trimmed condition are
δnT = 6022.646 rpm,
δeT = 0.072167 radians,
δaT = 0.000447 radians,
δrT = 0.002601 radians.
45
For trim inputs and state variables given in equation (D) the matrices A and B of the
equation (2.1) are given by
A =
−0.1305 −1.7020 3.8706 0.0717 −9.8097 0 0 0 0
0.0056 −0.1104 −0.0082 0.5535 0.0041 0.044 0 −1.0036 0
−0.0642 0 −6.2389 0.0975 0.0007 0 0.8918 0 0
0 0 0 0 −0.0991 1 −0.0072 0.0409 0
0 0 0 0.099 0 0 0.9848 0.1736 0
0.0048 −17.0851 0 0 0 −15.4412 0.0892 0.1913 0
0.0026 0 −60.7365 0 0 −0.095 −2.1389 −0.0079 0
−0.001 2.5035 0 0 0 −0.8188 0.0068 −0.217 0
0 3.0258 −17.1755 −0.1274 17.4395 0 0 0 0
,
B =
0.0227 0 0 0
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
0 0 0 0
.
The state feedback gain matrix required to place the eigenvalues of the closed-loop
system at −4 ± j5, −0.3 ± j0.2, −0.3, −5, −0.5 ± j0.5 and −0.3 is given by
K =
8.7612 −51.6086 135.2124 3.8087 −290.1035 −2.478 4.9098 1.125 2.3202
−0.0143 −0.2019 1.7134 −0.0542 −1.2373 −0.0059 −0.1659 −0.0012 −0.019
0 −0.0932 0.0001 0.0024 −0.004 −0.0778 0.0008 −0.0005 −0.0001
0.0011 −0.3947 0.0054 0.0294 −0.0413 0.0185 0.0007 −0.0408 −0.0006
.
The sub-matrices of the A and B matrices, used in (2.9)-(2.12) for design of reduced-
order observer are given by
46
Aaa =
−0.1305 0 0 0
0.0048 −15.4412 0.0892 0.1913
0.0026 −0.095 −2.1389 −0.0079
−0.001 −0.8188 0.0068 −0.217
,
Aab =
−1.702 3.8706 0.0717 −9.8097
−17.0851 0 0 0
0 −60.7365 0 0
2.5035 0 0 0
,
Aba =
0.0056 0.044 0 −1.0036
−0.0642 0 0.8918 0
0 1 −0.0072 0.0409
0 0 0.9848 0.1736
,
Abb =
−0.1104 −0.0082 0.5535 0.0041
0 −6.2389 0.0975 0.0007
0 0 0 −0.0991
0 0 0.099 0
,
Ba =
0.0227 0 0 0
0 0 187.1101 −0.6059
0 −36.4084 0 0
0 0 10.7534 −8.6645
,
Bb =
0 0 −0.0121 −0.1533
−0.0001 −0.6677 0 0
0 0 0 0
0 0 0 0
.
The observer gain matrix Ke is selected such that the observer poles, that is, eigen-
values of A are located at −16, −15 ± j1 and −17. The observer gain matrix Ke is
given as
Ke =
−0.1095 −1.7599 −0.0001 0.2579
−0.0138 −0.145 −0.1585 0.0212
−4.268 −24.0722 −0.4814 3.5273
−1.6802 −0.1661 −0.1069 0.0243
.
The matrix Ke is found using command place in MATLAB.
The matrices A, B, F , C and D which define the reduced-order observer in (2.11)
47
are given by
A =
−31.0107 0.4073 0.5613 −1.0698
−2.5536 −15.8132 0.0985 −0.1342
−427.3718 −12.7172 0.3059 −41.9675
−5.7576 0.0127 0.2194 −16.482
,
B =
2.7907 14.2625 −0.0616 −6.6456
0.2373 2.2018 3.0395 −0.6192
115.728 385.728 7.4989 −105.0173
27.168 5.0328 2.4254 −0.9019
,
F =
0 0 326.5 1
0 −6.4 26.9 0.1
0.1 −17.5 4466.2 16
0 −3.9 30.8 0.1
,
C =
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,
D =
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
−0.1095 −1.7599 −0.0001 0.2579
−0.0138 −0.145 −0.1585 0.0212
−4.268 −24.0722 −0.4814 3.5273
−1.6802 −0.1661 −0.1069 0.0243
.
48
Appendix E
Simulink Models
The simulink model used to simulate closed-loop guidance is shown in Figure (E.1).
Multiport switch is used as trim selector, which selects one of the three controllers
based on the guidance command. Desired change in altitude should be multiplied by[
0 0 0 0 0 0 0 0 1
]T
so that it represents desired state vector.
Figure (E.2) shows the implementation of controller designed for stabilizing coor-
dinated right turning flight. Altitude does not enter the reduced-observer because it
is not used to estimate other state variables. Inputs to the reduced-order observer are
perturbations from trim operating point.
49
flap
elevator
aileron
rudder
True air−speed
Angular Rates
Altitude
Guidance Command
Controller to stabilize coordinated left turn
rpsTrim Selector
0.01348
SideslipSaturation
block
0
Reset
R2D
R2D
0.0409
Pitch angle
0
Mix
K*u
States
Input
Desired State Vector
Inputs
Controls
Winds
RST
States
Sensors
VelW
Mach
Ang Acc
Euler
AeroCoeff
PropCoeff
Mass
ECEF
MSL
AGL
REarth
AConGnd
Kadet MKII
0
Ignition
0.7192
Heading
Longitude
Latitude
Heading
Selector
Guidance Block
0
Flap0
Desiderd change in altitude
Demux
Demux
Demux
Demux
States
Input
Desired State Vector
Inputs
Controller to stabilize level straight flight
States
Inputs
Desired State Vector
True control inputs
Controller to stabilizecoordinated right turn
−0.07889
Bank angle
17.46
Airspeed
2.345
AOA
Figu
reE
.1:Sim
ulin
km
odel
toim
plem
ent
closed-lo
opgu
idan
ceon
non
linear
aircraft
model
50
Perturbation inaltitude
Perturbations inmeasurable
states
Perturbations incontrol inputs
Estimated states
1
True control inputs
K*u
State feedbackgain matrix
x’ = Ax+Bu y = Cx+Du
Reduced−orderobserver u+0.072167
Fcn9
u−5.5841*pi/180
Fcn8
u−(−0.2353*pi/180)
Fcn7
u−100.377436
Fcn6
u+0.002601
Fcn5
u+0.000447
Fcn4
u−0.9846*pi/180
Fcn3
u−1000
Fcn2
u+100.377436
Fcn12
u−0.002601
Fcn11
u−0.000447
Fcn10
u−0.072167
Fcn1
u−17.44
Fcn
Demux
Demux
Demux
Demux
3
Desired State Vector
2
Inputs
1
States
Figu
reE
.2:Sim
ulin
km
odel
ofth
econ
troller
51
References
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Acknowledgment
Date: 8-7-2005
I would like to express my sincere gratitude towards Prof. S.P. Bhat, for his invalu-
able guidance and constant encouragement during the course of M.Tech. Project. I
would also like to express my sincere gratitude towards Prof. H. Arya for his valuable
support.
S. Gopinadh
03301012
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