P1111128655

1CSE-11 ICGST International Conference on Computer Science and Engineering, 19 - 21 December 2011, Istanbul, Turkey
Multivariable PID Controller
for Lateral Motion of Aircraft
Dr. Abdelkader MADDI
Electronics Department
University of Blida, Algeria.
Tel: +213773150235
Fax: +21325433633
Email:a_maddi@hotmail.com

Table of Contents
1. Introduction
2. Problem Formulation
3. Application to CESSNA-182 Aircraft System
4. Simulations and Results
5. Conclusion
Introduction
Problem Formulation
Application to CESSNA-182 Aircraft System
Simulations and Results
Conclusion

Introduction
Mathematical modelling for analysis of non-linear aircraft system can be represented by
the differential equations that describe the evolution of the basic states of an aircraft
[1]. These equations of motion are all nonlinear first order ordinary differential
equations. In addition they are highly coupled, i.e., each differential equation depends
upon variables [2].
The synthesis of MRAC is done using the concept of hyper-stability [4]. To eliminate some
of the undesirable phenomena, we suggested the use of proportional integrator law
control based on hyper-stability approach that is dominantly rich to eliminate the SSE.
This law control is shown to guaranty a good convergence to the following model.
The LQG gives a very good following to the outputs of plant with a steady shift error
limited and the Kalman filter is an optimal estimator when dealing with Gaussian white
noise [3].
The MV control strategy has been applied to a specific case of aircraft system [5]. If a
statically independent noise acts directly on the controlled variable, minimum
variance controllers cannot decrease the variance of the controlled variable; only for
colored noise, can the variance of the controlled variable be reduced.
Hence, in this work, we develop a strategy of PID controller to determine the desired
sideslip angle and roll angle with respect of the characteristics control surfaces of
aircraft.
Introduction
Problem Formulation
Conclusion

Problem Formulation
Introduction
Problem Formulation
Conclusion
The transfer function of the PID controller is given by the following equation:
1. PID controller synthesis method
Figure 1. Closed loop process with PID controller
s
ksksk
sk
s
k
ksH dpd
d
i
p
++
=++=
2
)(
)(ty
)(tr
The PID controller compares the measured process value y(t) with a reference set-point value r(t) as
represented in Figure 1.

Introduction
Problem Formulation
Conclusion
The error is then processed to calculate a new process input. This input will try to adjust the
measured process value back to the desired set-point. The control signal is now equal to the
following expression:
dt
ted
KdtteKteKtu dip
)(
)()()( ++= ∫
2. Control signal of PID controller
where kp, ki and kd are the proportional, integral and derivative gain respectively.
dt
tdE
KdEKtEKtU d
t
ip
)(
)()()(
0
++= ∫ ττ
2. Multivariable PID controller
In multivariable case, the control input system is generated by the following expression:
where Kp, Ki and Kd are the PID controller gain matrices of appropriate dimensions.

Application to CESSNA-182 Aircraft
-The CESSNA-182 was introduced in
1956, as a tricycle gear variant of the
180.
- In 1957, the name was changed to
the 182A and the name Skylane was
introduced.
- The characteristics of CESSNA-182
are presented on Table 2 and having
the following lateral factors of stability
represented on Table 3.
Figure 2 : CESSNA-182 Aircraft
1. Introduction
Introduction
Problem Formulation
Conclusion

Rolling. The ailerons control the
roll or lateral motion and are
therefore often called the lateral
controls.
Pitching. The elevator controls pitch
or the longitudinal motion and thus is
often called the longitudinal control.
Yawing. The rudder controls yaw
or the directional motion and thus
is called the directional control.
Figure 3: Definition of the aircraft’s axis system
2. Body Axis
The Wright Brothers were the first to develop a
fully controllable airplane. They understood that
flying machines needed to be controlled on three
axis :
- Pitch
- Roll
- Yaw
Introduction
Problem Formulation
Conclusion























∆
+
∆
+
∆
+
∆
+
∆
+
∆
+






−





+
=
0)(10
0
0
)cos()cos()sin(
0
0
0
0
0
0
00
θ
θθθ
ββ
ββ
β
tg
aLNaLNaLN
aNLaNLaNL
V
g
V
Y
V
Y
V
Y
A
rrpp
rrpp
rp






















∆
+
∆
+
∆
+
∆
+
=
00
00
aarr
aarr
ar
bLNbLN
aNLaNL
V
Y
V
Y
B
δδδδ
δδδδ
δδ
3. Aircraft Lateral Dynamics
UBXAX +=
We can develop the equations of motion for the lateral dynamics as:
[ ]φβ rpX T
=
[ ]ra
T
U δδ=
Where
Introduction
Problem Formulation
Conclusion

4. Geometric characteristics of CESSNA-182
Description Values for CESSNA-182
Wing area 174.00 sq. ft
Wight 2645.00 Ibs
Wing span 35.80 ft
Mean aerody. chord 4.90 ft
Air speed 219.00 ft/sec
Air density 0.00205 slugs/ cu. ft
Initial theta 0.00 rad
High 5000 ft
Xcg 0.25
Iyy 1346 slugs. sq. ft
Ixx 948 slugs. sq. ft
Izz 1967 slugs. sq. ft
Ixz 0.00 slugs. sq. ft
Table 1: Geometric characteristics of CESSNA-182
Introduction
Problem Formulation
Conclusion

5. Lateral factors of stability
Table 2: Lateral factors of stability for CESSNA-182
19.4730 ft / rad. sec2
0.0000 ft / rad. sec2
1.7859 ft/ rad. sec
-0.3147 ft / rad. sec
-32.2554 ft / rad. sec2
-10.2284 / rad. sec2
-8.2512 / rad. sec2
-1.2597 / rad. sec
-0.3817 / rad .sec
10.1194 / rad. sec2
4.7485 / rad. sec2
57.4984 / rad. sec2
2.5346 / rad. sec
-12.4092 / rad. sec
-28.7492 / rad. sec2
Values for CESSNA-182Quantity
Introduction
Problem Formulation
Conclusion













0010
01.2597−0.3817−10.119
02.534612.409−28.749−
=
0.14980.9918-0.0014-0.1473-
A












00
8.2512−10.228
57.4984.7485
=
00.0886
B






=
1000
0001
C
,
If we assume that the measurable outputs are the sideslip angle and roll angle, the matrix A, B and C are:
2. Modes of the system
0112.01 −=λ
4341.122 −=λ
i3073.36855.04,3 ±−=λ
Spiral Mode
Roll Damping
Dutch Roll
Introduction
Problem Formulation
Conclusion
1. State space system

Introduction
Problem Formulation
Conclusion
Time (sec.)
Step Response from: U(1)
0 5 10 15 20 25
-6
-4
-2
0
2
Beta(deg.)
Time (sec.)
0 1.5 3 4.5 6 7.5 9
0
0.5
1
1.5
Phi(deg.)
Figure 3. P controller with kp1 = 1; kp2 = 1
Figure 3. P controller with kp1 = 1; kp2 = 1

Introduction
Problem Formulation
Conclusion
Phi(deg.)
Time (sec.)
0 1.5 3 4.5 6 7.5 9
0
0.5
1
1.5
Time (sec.)
Beta(deg.)
0 5 10 15 20 25 30 35
-10
-5
0
5
Figure 4. PD controller with kd1 = 0.05; kd2 = 0.05; kp1 = 1; kp2 = 1

Introduction
Problem Formulation
Conclusion
Time (sec.)
Beta(deg.)
0 3 6 9 12 15 18
-6
-4
-2
0
2
Time (sec.)
Phi(deg.)
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
Figure 5. PI controller with ki1 = 0.01; ki2 = 0.01; kp1 = 1; kp2 = 1

Introduction
Problem Formulation
Conclusion
Time (sec.)
Beta(deg.)
0 3 6 9 12 15 18
-6
-4
-2
0
2
Time (sec.)
Phi(deg.)
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
Figure 6. PID controller with kd1= 0.01; kd2 = 0.01; ki1 = 0.01;
ki2 = 0.01; kp1 = 1; kp2 = 1

Conclusion
 Applying a PID controller for lateral motion of aircraft is considered as a practical
example. The controller should work fine, For example to eliminate some of the
undesirable phenomena, we suggested a good choosing of the PID parameters
that is dominantly rich to eliminate the steady state error.
 Using only P control gives a stationary error in all cases except when the system
control input is zero and the system process value equals the desired value. But
the system became unstable for too large P term.
 Using only the I term gives a slow response and often an oscillating system. Note
that the PI controller response is very slow with no stationary error.
 The derivative term D is typically used with the P or PI as a PD or PID controller. A
too large D term usually gives an unstable system. The response of the PD
controller gives a faster rising system process value than the PI controller.
 Results have been quite satisfactory compared to the LQG control [3] or GMV
controller [5]. For more information, you can see the following references:
Introduction
Problem Formulation
Conclusion
0=r

1. A. Gonzalez Blazquez, “Mathematical modelling for analysis of non-linear aircraft
dynamics”, Computers and structures, Vol. 37, No. 2, pp. 193 -197, July 1990.
2. A. Maddi, “Modélisation et contrôle du vol latéral d’un avion”, First International
Conference on Electrical Engineering, ICEE’2000, University of Boumerdes, Algeria,
November 04-06, 2000.
3. A. Maddi, A. Guessoum and D. Berkani, “Using Linear Quadratic Gaussian Optimal
Control for Lateral Motion of Aircraft”, World Congress on Science, Engineering and
Technology, WCSET 2009, Dubai, United Arab Emirates, pp. 285-289, January 28-30,
2009,
4. A. Maddi, A. Guessoum and D. Berkani, “Applying Model Reference Adaptive
Controller for Lateral Motion of Aircraft”, American Journal of Applied Sciences, Issue 7,
Vol. 2, ISSN 1546-9239, pp. 235-240, 2010.
5. A. Maddi, A. Guessoum and D. Berkani, “Generalized minimum variance control for
lateral motion of aircraft”, 4th International Conference from Scientific Computing to
Computational Engineering, 4th IC-SCCE, Athens, Greece, July 7-10, 2010.
References

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