CONTROL PITCH, ROLL AND YAW
AXIS OF AIRCRAFT USING RELAY
FEEDBACK METHOD
ABSTRACT
Conventional PID (Proportional-Integral-Derivative) controller is widely used to control
system because of its simple structure and robustness with many commonly met process
control problems as for instance disturbances and nonlinearities. Relay feedback
methods have been widely used in tuning proportional-integral-derivative controllers
due to its closed loop nature. We used relay feedback method to control pitch, roll and
yaw axis of aircraft. This methodology is used to derive PID parameter and
implemented on transfer function of pitch, roll and yaw. The performance of derived
parameters has been checked using simulation results. The entire simulation has been
carried out in MATLAB environment. The results will show the effectiveness of derived
PID parameters.
INTRODUCTION
Modern flight dynamics is concerned with the provision of safe controllable air
vehicles which are easy to fly and capable of delivering an enhanced performance. As
there is a pilot in the cockpit, which flies an aircraft; however there can be a provision
to control the aircraft dynamics without the interventions of pilot, which is known as
Autopilot and which can also flies an aircraft. This work describes the application of
tuning three term parameters of controller and thereafter using those values as
parameters for the autopilot program which is fed into an aircraft as per requirements.
FLIGHT DYNAMICS
There are four primary forces acting on an aircraft: Thrust, Drag, Weight and Lift. An aircraft has six
degree of freedom as shown in above figure. They are linear motions about x, y, z axes and angular
motions about x, y, z axis.
Longitudinal equations of motion of an aircraft
Lateral equations of motion of an aircraft
For yaw control heading variable is added to lateral equation;
PID (PROPORTIONAL-INTEGRAL-DERIVATIVE) CONTROLLER
P-I-D controller has the optimum control dynamics including zero steady state error, fast response
(short rise time), no oscillations and higher stability.
PID controller
Block diagram of PID controller
• Here where
and all non-negative, denote the coefficients for the proportional, integral,
and derivative terms respectively.
RELAY FEEDBACK METHOD
• The Relay allows its output to switch between two specified values. When the relay is on, it
remains on until the input falls below the value of the Switch off point parameter. When the relay
is off, it remains off until the input exceeds the value of the Switch on point parameter.
• The relay feedback method is also called as the method “ATV” (auto tune variation).
Block diagram for relay feedback
Input and output of system
•• The further procedure is the following:
1.
Record the plant output amplitude and period from output graph.
2.
Calculate ultimate gain from below equation
3.
Find PID Ziegler–Nichols tuning constants using Ziegler–Nichols table as below.
P
-
PI
PID
Ziegler–Nichols tuning rules
4.
-
Use this tuning constant to find PID constants , and as below
FLOW CHART
FOCUSING AREA
Design pitch, roll and yaw control system in MATLAB Simulink
Analyse this model using PID control with relay feedback method.
SOFTWARE
Control model and analysis: MATLAB Simulink
PITCH CONTROL OF AIRCRAFT
• Aircraft pitch is governed by the longitudinal dynamics.
Assumptions
• aircraft is in steady-cruise at constant altitude and velocity
• change in pitch angle will not change the speed of the aircraft under any circumstance
• the longitudinal equations of motion for the aircraft can be written as modelling
equation in following form;
TRANSFER FUNCTION OF AIRCRAFT PITCH AXIS
SYSTEM
• To find the transfer function of the system, we need to take the Laplace transform
of the above modeling equations. Recall that when finding a transfer function,
zero initial conditions must be assumed. The Laplace transform of the above
equations are shown below.
• After few steps of algebra, we will obtain the following transfer function.
OUTPUT WITHOUT CONTROLLER
First we find behaviour of system without controller to find settling time of system
Block diagram of Step input to system without controller
• When we direct apply step input without controller the output will be;
Output of system without controller
DESIGN OF PID PARAMETER
• We have to use trial and error method to find switch on point and switch off point until we get
sustain oscillations. The block diagram of relay feedback system is shown below;
Block diagram of relay feedback for pitch control
• After trial and error method we find switch on point 2 and switch off point -2 and oscillations are
shown as below;
Oscillations of relay output
•• From graph we can find ultimate period () which is 5.3132 second, amplitude is 2 and relay
amplitude d is equal to 1 from this data we can find ultimate gain so;
So, after calculation = 0.6366
From table 1 we can find PID Ziegler–Nichols tuning constants;
0.381
2.6566
0.6641
Ziegler–Nichols tuning constants for
pitch
0.381
0.3768
0.6641
PID constants for pitch control
IMPLEMENT OF PID CONTROLLER ON
SYSTEM
• As we find PID constants we can now implement on out system. The block diagram of System is
shown below;
Block diagram of system with PID controller
• Deflection v/s time graph of system is shown below;
Deflection v/s time graph of system with controller
• Here system can automatically tune PID parameter we can compare both result.
Relay feedback tune and system tune
• From the graph following results are obtained;
System tuned
Rise time (sec)
Peak time (sec)
Peak overshoot (%)
Settling time(sec)
12.6
1.07
6.78
145
Block
tuned
3.3
1.12
12.5
20.8
• Amount of deflection (δ) in elevators is the input and the pitch angle (θ) will be the output. So to
reach at particular pitch angle (reference step) it will take in total 20.8 seconds. But before this at
7.8 seconds aircraft’s pitch angle will overshoot by 0.87 degree from the required pitch degree
then it will reach to desired pitch angle in next 13 seconds. Hence the settling time (with 2%
tolerance band) is 20.8 seconds and overshoot is 12.5 %.
• In order to check the feasibility of PID parameters using relay feedback, multiple deflection angles
to the system are applied as shown in below figure. It shows the response for the different input
values to the system and its corresponding output. This concludes that the system follows in the
same manner as it was following for the step input. This means that for every different input
values, the system responds correctly and settling time, peak overshoot and other time domain
remains the same.
Series of input v/s time taken to settle the system
ROLL CONTROL OF AIRCRAFT
•
• Lateral dynamics of aircraft defines pitch moment of aircraft. Lateral equation of motions are shown in
below equation. Let us consider that the aircraft is in steady-cruise at constant altitude and velocity thus, the
thrust, drag, weight and lift forces balance each other in the y and z directions. From lateral equation of
motion we can derive roll dynamics for an aircraft. We directly take data of Boeing 747-400 and derive roll
to aileron deflection transfer function.
• Roll Dynamics;
OUTPUT WITHOUT CONTROLLER
• First we find behaviour of system without controller to find settling time of system. The block
diagram of system is shown below. Here we input one step signal to the system and output graph
is as shown next.
Block diagram of roll system without controller
Deflection v/s time graph for roll without controller
DESIGN OF PID PARAMETER
• We have to use trial and error method to find switch on point and switch off point until we get
sustain oscillations. The block diagram of relay feedback system is shown below;
Block diagram of relay feedback for roll control
• After trial and error method we find switch on point 1 and switch off point -1 and oscillations are
shown as below;
Oscillation for roll output
•• From oscillation we can find ultimate period () which is 19.00 second, amplitude is 1.175 and
relay amplitude d is equal to 1 from this data we can find ultimate gain so;
So, after calculation = 1.083
From Ziegler–Nichols table we can find PID Ziegler–Nichols tuning constants;
0.6498
9.505
2.3750
Ziegler–Nichols tuning constants for roll
0.6498
0.1052
2.3750
PID constants for roll control
IMPLEMENT OF PID CONTROLLER ON SYSTEM
• As we find PID constants we can now implement on out system. The block diagram of System is
shown below;
Block diagram of system with PID controller
• Deflection v/s time graph of system is shown below;
Deflection v/s time graph of system with controller
• Here system can automatically tune PID parameter we can compare both result.
Relay feedback tune and system tune
• From the graph following results are obtained;
System tuned
Block
tuned
Rise time (sec)
54.09
7.03
Peak time (sec)
1.09
1.23
Peak overshoot (%)
8.71
23
Settling time(sec)
243
51.3
• Amount of deflection (δ) in ailerons is the input and the roll angle (ϕ) will be the output. So to
reach at particular roll angle it will take in total 51.3 seconds. But before this at 17 seconds
aircraft’s pitch angle will overshoot by 1.8 degree from the required pitch degree then it will reach
to desired pitch angle in next 33.3 seconds. Hence the settling time (with 2% tolerance band) is
51.3 seconds and overshoot is 23 %.
YAW CONTROL OF AIRCRAFT
Aircraft yaw angle definition
•• Let us consider that the aircraft is in steady-cruise at constant altitude and velocity thus, the thrust,
drag, weight and lift forces balance each other in the x and z directions. From lateral equation of
motion we can derive yaw dynamics for an aircraft. We directly take data of NAVION aircraft and
derive roll to aileron deflection transfer function.
• Yaw dynamics;
OUTPUT WITHOUT CONTROLLER
• First we find behaviour of system without controller to find settling time of system. The block
diagram of system is shown below. Here we input one step signal to the system and output graph
is as shown in next slide.
Block diagram of roll system without controller
Deflection v/s time graph for yaw without controller
DESIGN OF PID PARAMETER
• We have to use trial and error method to find switch on point and switch off point until we get
sustain oscillations. The block diagram of relay feedback system is shown below;
Block diagram of relay feedback for yaw control
Oscillation for yaw output
•• From oscillation we can find ultimate period () which is 2.8 second, amplitude is 7.1 and relay
amplitude d is equal to 1 from this data we can find ultimate gain so;
• So, after calculation = 0.1793
• From Ziegler–Nichols table we can find PID Ziegler–Nichols tuning constants;
0.1075
1.4000
0.3000
Ziegler–Nichols tuning constants for yaw
0.1075
0.7142
0.3000
PID constants for yaw control
IMPLEMENT OF PID CONTROLLER ON SYSTEM
• As we find PID constants we can now implement on out system. The block diagram of System is
shown below;
Block diagram of system with PID controller
• When we simulate this system we face singularity problem for this system at 0.00014130 sec so
we cannot apply this PID parameter on this system which is calculated by relay feedback method.
From this we can say that relay feedback method is not applicable for all system this is its
limitation.
Singularity error with this system
CONCLUSION
PITCH CONTROL
• System take 60 second to stable without controller and when we apply PID
controller system takes only 20.8 second to stable so controller must be used to
reduce settling time.
• Tuned parameter is more effective than which system tuned because settling time
for system tuned parameter is 145 second where in our case it is only 20.8
second.
ROLL CONTROL
• System never reach to our input deflection without controller but with PID
controller it reaches to our input deflection as well take only 51.3 second to stable
system.
• When system automatically tuned system it takes 243 second to stable system
where with our gain inputs it takes only 51.3 second to stable
YAW CONTROL
• Relay feedback method is very effective way to find PID gains which is very easy
to understand and less time consuming but sometimes relay feedback method is
not working for some system. In case of yaw control system calculated PID gain
are not capable to stable system and we face singularity error in our result.
IMPROVEMENTS AND FURTHER RESEARCH
• Improve stability by decrease overshoot
• Decrease settling time
• Control yaw axis by different technique