DESIGNING AND IMPLEMENTATION OF AN … Optimized PID Controller for Longitudinal Autopilot., ... Root Locus, Bode Plot, Nyquist ... Technique has its advantages and Disadvantages so

DESIGNING AND IMPLEMENTATION OF AN OPTIMIZED PID

CONTROLLER FOR LONGITUDINAL AUTOPILOT

ABSTRACT

This paper presents the design of an Aircraft model for

Longitudinal Dynamics. We formulated a design of inner

and outer loop of an aircraft with and without optimization

(PID controller) for continuous and discrete controller by

controlling the variables of an aircraft using a simulation

loop of a MATLAB. The different flight conditions were

arrived using Orthogonal Array (OA) based on different

Aircraft weight, Altitude, Mach number configurations.

This attempts to span the aircrafts across the regimes in

aircrafts flight envelope. Performance against uncertainties

also included like turbulence and variation of atmospheric

conditions.

Index Terms :

Attitude control,

Aircraft navigation,

PID controller,

Discrete optimization,

Pitch Attitude Hold Mode

Name of the Author:

Princy Randhawa

Assistant Professor, Department of Mechatronics Engineering

School of Automobile,Mechanical and Mechatronics

Engineering, Manipal University

Jaipur,Rajasthan (INDIA)

Advance Research Journal of Multi-Disciplinary Discoveries ISSN NO : 2456-1045

ISSN CODE: 2456-1045 (Online)

(ICV-EE/Impact Value): 3.08

(GIF) Impact Factor: 2.174

Copyright@IJF 2017

Journal Code: ARJMD/EE/V-11.0/I-1/C-5/MARCH-2017

Category : ELECTRICAL ENGINEERING

Volume : 11.0 / Chapter- V / Issue -1 (MARCH)

Website: www.journalresearchijf.com

Received: 06.03.2017

Accepted: 25.03.2017

Date of Publication: 05-04-2017

Page: 21-26

Citation of the Article

Original Research Article

Randhawa P. (2017, March). Designing and Implementation of

an Optimized PID Controller for Longitudinal Autopilot., Advance

Research Journal of Multidisciplinary Discoveries.11.0,C-

5(2017):21-26 ISSN-2456-1045.

http://www.journalresearchijf.come;

www.journalresearchijf.com

I 07

An open access journal of International Journal Foundation Page I 21



I. INTRODUCTION

An autopilot is a system used to control the trajectory

of an aircraft without constant 'hands-on' control by a human

operator being required. The first aircraft autopilot was

developed by Sperry Corporation in 1912.The main aim of the

autopilot is to track the desired goal. There are several various

control techniques available for the design of an autopilot like

Frequency domain techniques: Root Locus, Bode Plot, Nyquist

Plot, PID Design and for Time domain Techniques: Pole Place

–ment Technique, Eigen Structure Assignment, Optimal

Control (LQR) Design and Advance Techniques like Robust

Control, Sliding Mode Control, Adaptive Control etc. [2] Every

Technique has its advantages and Disadvantages so as to

improve its performance new techniques has proposed. In this

paper presents the simplest technique using optimized PID

controller for determining the stability of the modes of the

autopilot. . In control law design there are thousands of

different sets and combinations of altitude, velocity, etc. In this

we discuss about the method known as orthogonal array to

reduce the no. of sets.

II. CONTROL LAW DESIGN OF AN AIRCRAFT

To design a plant model through, we need a longitudinal

equation. For the dynamics of a longitudinal aircraft, we need

variables which are (small) deviations from operating point or

trim conditions state (components)

u : velocity of aircraft along body axis

α : angle of attack (the angle between the velocity

vector and the x-axis of the aircraft

: Angle between body axis and horizontal (up is

positive)

q =

.

: Angular velocity of aircraft (pitch rate)

Inputs

Control or actuator inputs:

e : Elevator angle ( e >0 is down)

If we introduce the longitudinal state variable vector

x = [u α q ] (1)

& the longitudinal control vector

)(tu = [ e ] (2)

These equations are equivalent to the system of first-

order equations

)(txIn

= )()( tButAx (3)

)()()( tDutCty

x represents the time derivative of the state vector x, and the

matrices appearing in this equation are

An =

0100

0

sin

cos0

0

00

0

00

00

00

qu

u

MMM

u

g

u

uZ

u

Z

u

Z

gXXu

(4)

In =

1000

010

00)1(0

0001

.0

.

M

u

Z (5)

Bn=

0

0

e

e

e

M

u

Z

X

(6)

x = uBIxAI nnnn

11 (7)

On solving (4), (5), (6) and substitute in (7) then we get the

approximation form of the linearized equations for longitudinal

motions. The various dimensional stability derivatives and

control derivatives are related to their dimensionless

aerodynamic coefficient. The selected aircraft Boeing 747-400

is flying in straight level flight at some altitude with a velocity

and the compressibility effects are neglected. On the basis of

Boeing 747-400 structures and its aerodynamic configuration,

the longitudinal state space model established. For this aircraft

the values are given below. After calculating all the coefficients

in matrix A and matrix B, we substitute the values for stability

and control coefficient derivatives as shown in table 2

Table 1: Longitudinal Derivatives

S.No.

Aerodynamic Coefficients

Stability Derivatives

1. Xu )**2(**

*0

0

DMCMCDuM

SQ

2. X )(**

*0

0

CLCDuM

SQ

3. Zu

)*))1(

/(*2(**

*

0

2

2

0

0

CLMach

MachCLuM

SQ

4. .

M

CMuIy

CSQ*

**2

2^**

0

5. Z )(**

*0

0

LCCDuM

SQ

6.

Z

CzuM

cSQ*

**2

**2

0

.

7. Zq LqC

uM

CSQ*

**2

**

0

8. Mu mMCMach

uIy

cSQ**

*

**

0

9. M M

CuIy

cSQ*

*

**

0

10. qM

mq

y

CuI

cSQ*

**2

**

0

2

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When the aircraft is in status (H=300000Kg, Mach=0.35,

Velocity=114.97m/s, Altitude= 7000m).The altitude (Height)

and velocity will be varying but all the coefficient derivatives

will remain constant.

A. Modes of typical aircraft

The natural response of most aircraft to longitudinal

perturbations typically consists of two under- damped oscillatory

modes having rather different time scales. One of the modes has

a relatively short period and is usually quite heavily damped; this

is called the short period mode. The other mode has a much

longer period and is rather lightly damped; this is called the

phugoid mode.

Aircraft Boeing 747-400

Parameters

Altitude-H(metre) 7000

Flight Mach no.(Mach) 0.35

True Speed-u0(metre/sec) 116.84

Density(Newton/m^3) 0.5895

Static pressure(Newton/m^2) 41061

Dynamic Pressure-Q(Newton/m^2) 4024

Weight(Kg) 300000

Wing area-S(metre^2) 510.96

Wing chord-c(metre) 8.324

I y (slug/.ft^2) 32.3e+6

Pitch angle(θ)(degree) 0

Longitudinal derivatives

CD (Drag variations along x axis)rad 0.66

0CD (Reference Drag coefficient) 0.102

0CL (Reference lift coefficient) 1.11

DMC 0

CL (Airplane lift curve slope)rad 5.70

qCL (Effect of pitch rate on lift)rad 5.40

eCL (Lift force due to elevator deflection)rad 0.338

mqC (Damping in pitch)rad -20.8

mC (Static longitudinal stability)rad -1.26

Cm (Downwash lag on moment)rad -3.2

CL (Downwash lag on lift)rad 6.7

CmM (Effect of slipstream and flexibility

and thrust) 0

eCm (Moment force due to elevator

deflection)rad -1.38

Error! Reference source not found.


III. DESIGN OF PITCH ATTITUDE HOLD MODE

There are basically two loops in the design: one inner loop

(control loop) and one outer loop which controls the top level

guidance parameters such as heading or altitude known as

(guidance loop). The Pitch Attitude Hold mode (PAH) controls

the pitch angle by applying appropriate deflections of the

elevator if the actual pitch angle differs from the desired

reference value. The pitch angle θ is fed back to damp the

phugoid mode and to ensure that the desired pitch angle is

maintained. A proportional and integrating controller is applied

in order to make sure that no steady-state errors in the pitch angle

will remain. As long as the error signal θ−θ ref is not equal to

zero, the signal from the integrator will increase, which leads to

an increasing elevator deflection which eliminates the error.

Figure 1: Block diagram of the discrete pitch attitude hold mode

Outer loop constitutes a feedback loop of pitch angle. It can

improve the damping of aircrafts long period motion. Parameter

selection of control law for the pitch angle control system

consists of two parts: the first part is the feedback gain K of the

damping circuit (inner loop), the second part is the PID

parameters in the pitch angle control signal of pitch angle 1

degree. Table 1 shows the particular PID values of pitch attitude

hold mode for different nine plant models without optimisation

which uses trial and error method .It is a time consuming method.

Table 3: Different PID values for different 9 Plant models

(Without optimization)

Plant model

Parameters (W=Weight) (A=Altitude) (M=Mach. No.)

Proportional Gain

Integral Gain

Derivative Gain

1. W=200000 A=1000 M=0.25

2 0.05 3

2. W=200000 A=6000 M=0.45

1.6 0.03 2

3. W=200000 A=12000 M=0.65

2 0.04 2.5

4. W=300000 A=1000 M=0.45

1.6 0.03 2

5. W=300000 A=6000 M=0.65

1.6 0.03 2

6. W=300000 A=12000 M=0.25

17 0.3 23

7. W=400000 A=1000 M=0.65

3 0.03 2

8. W=400000 A=6000 M=0.25

10 0.3 8

9. W=400000 A=12000 M=0.45

16 0.5 11

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Figure 2: Longitudinal Aircraft closed loop step Response

for pitch attitude hold mode (without optimization) to 1-deg

Elevator step Input

The figure 2 shows the response for pitch attitude hold mode

using PID technique (without optimisation) .In that graph the

settling time and overshoot was more in some plant models. To

reduce that, we use a technique called PID optimisation. It is

also known as automatic tuning.

IV. DISCRETE PID CONTROLLER (With Optimization)

For optimisation, we obtain a required response using the

second order equation, we choose such a, and ξ value where

we get the good step response with minimum overshoot and

less settling time.

2^22^

2^

s Where ɷ= 1.5 and ξ=0.9 (8)

Now take this as a transfer function, to optimize the PID

controller parameters such as K p , iK , dK by using f-min

search.This is chosen so that we get the best response. Using

these values we obtained a required graph against which all the

other plant models will be compared so as we get optimised

values of K p , iK , dK .It minimises the error between the

required and obtained graph to match with the required one.

Error! Reference source not found.

After optimisation we get the optimised values of K p , iK ,

dK for all nine plant models. The settling time thus obtained

is very less and less overshoot the one we got from the un-

optimised controller. Figure 5 shows the aircraft closed loop step

response for pitch attitude hold mode (with optimisation) for-1

degree elevator step input.

From figure 5 and table 2 shows that after optimisation, the

response improves in respect of settling time, and overshoot of a

system.

Figure 4: Block diagram of pitch attitude hold mode using

optimization method.

V. RESULTS

P

l

a

n

t

N

o

.

Discrete-Controller

(Before optimisation)

Discrete Controller (After

optimisation)

pK

iK

dK

Set

tlin

g

Ti

me

(se

c)

pK iK

dK

Settli

ng

time(

sec)

1. 2 0.05 3 7 4.3999 0.06806 4.747 4

2. 2.6 0.03 2 8 2.6087 0.03996 2.774 4

3. 2 0.04 2.5 11 3.0203 0.04489 3.333 5

4. 1.6 0.03 2 6 1.4491 0.03131 1.368 4

5. 1.6 0.03 2 6 1.3259 0.02813 1.292 4

6. 17 0.3 23 15 20.812 -0.3628 24.99 5

7. 3 0.03 2 5 0.74934 0.01988 0.611 4

8. 10 0.3 8 12 9.0119 -0.22401 10.44 4

9. 16 0.5 11 14 6.6983 0.15927 7.664 4

Table 4: Comparison between Discrete and optimized values

of PID


for pitch attitude hold mode for 1-deg Elevator step Input

(after optimization)

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VI. DESIGN OF FORWARD SPEED HOLD MODE

It is used during cruise flight as a Mach hold mode. Aircraft

flies at constant Mach through automatic control of pitch angle

by the elevator. As the aircraft flies the fuel is burned and

weight decreases, and speed tends to increase. Speed increase

detected by control system and corrected by elevator. There are

basically three loops in the design: two inner loops (control

loop) and one outer loop which controls the top level guidance

parameters such as velocity (guidance loop). In the block

diagram, the initial speed is given as a unit step input

(1m/sec).The output is also coming as a step input which

indicates the holding of forward speed mode. Figure 7 shows

the response of forward speed hold mode for unit step input

velocity. Table 3 shows the gain values for speed hold mode.

Figure 6: Block diagram of the discrete

Forward speed hold mode


for Forward speed hold mode for 1m/sec -step Input

Plant

model

Proportional

controller

( pK )

Integral

Controller

( iK )

Derivative

Controller

( dK )

1. -0.004 -0.0064 0.009

2. -0.008 -0.0004 0.0098

3. -0.004 -0.003 0.005

4. -0.003 -0.007 0.0098

5. -0.004 -0.005 0.007

7. -0.003 -0.007 0.0098

Table 5: Different PID values for Forward

Speed Hold Mode

For this mode also we can use optimization technique to get

better responses for all the models. From the above section, we

concluded that optimization have more advantages over without

optimization.

Advantages of PID optimisation

1. The frequency response specifications (settling time,

overshoot etc.) are improved as compared to

conventional PID controller.

2. It is an automatic tuning method.

3. The elevator angle deflection is within the limits. (-28

degrees to 28 degrees).

Trade-offs for PID optimisation

1. The error computation.

2. Noise degradation in the derivative control.

3. Over simplification and the loss of performance in the

control law in the form of linear weighted sum.

4. Complications brought by the integral control.

5. No active Disturbance Rejection.

To overcome the trade-offs in PID optimisation, there are

various techniques available for plant uncertainty, un-modelled

dynamics and disturbance using Active Disturbance Rejection

Control (ADRC) scheme that involves an observer design,

Kalman filtering,various estimation techniques etc. But still PID

controller is more prevalent nowadays because it is the simplest

design to develop which caters the problem of dynamics.

VII. CONCLUSION

We established the longitudinal equation based on small

perturbation and designed the vertical control law of autopilot

system of flight simulator using classical control technique.

More modern control design techniques will be involved in the

full six degree of freedom linear autopilot design.

Make the design of controller based on the states. These will be

developed using the aircraft plant model. Since the model used

(Boeing 747) is a general model, and can construct the rnodel of

many other aircraft in the same structure. Robust autopilot

control laws are designed for pitch attitude hold mode and

Forwards Speed Hold mode in MATLAB Simulink such as PID

(without optimisation), PID (with optimisation). More models

can be designed for different phases of flight which will define

the different set/subsets of experiments with different values of

Kp, Ki, and Kd.

REFERENCES

[1] Gao jian-shu, Li-Jing, “Design and optimisation of

autopilot system ofFlight simulator” 2010

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Documents

DESIGNING AND IMPLEMENTATION OF AN … Optimized PID Controller for Longitudinal Autopilot., ... Root Locus, Bode Plot, Nyquist ... Technique has its advantages and Disadvantages so