Pitch Control System Using LQR and Fuzzy Logic Controller

Nurbaiti Wahid Faculty of Electrical Engineering

Universiti Teknologi MARA 23000 Dungun, Terengganu, Malaysia

nurbaiti@tganu.uitm.edu.my

Mohd Fua’ad Rahmat Faculty of Electrical Engineering

Universiti Teknologi Malaysia 81310 Skudai, Johor, Malaysia

fuaad@fke.utm.my

Abstract—This paper presents a comparative assessment based on time response specification performance between modern and intelligent controller for a pitch control system of an aircraft system. The dynamic modeling of pitch control system is considered on the design an autopilot that controls the pitch angle of an aircraft. It begins with a derivation of suitable mathematical model to describe the dynamics of an aircraft. To study the effectiveness of the controllers, the Linear Quadratic Controller (LQR) and Fuzzy Logic Controller (FLC) is developed for controlling the pitch angle of an aircraft system. Simulation results for the response of pitch controller are presented in time domain. Finally, the performances of pitch control systems are investigated and analyzed based on common criteria of step’s response in order to identify which control strategy delivers better performance with respect to the desired pitch angle and pitch rate. It is found from simulation, LQR controller give the best performance compared to fuzzy logic controller.

Keywords—Aircraft; Flight control; Autopilot; Longitudinal dynamic; LQR; Fuzzy logic.

I. INTRODUCTION Today’s aircraft designs rely heavily on automatic

control system to monitor and control many of aircraft’s subsystem. The development of automatic control system has played an important role in the growth of civil and military aviation. Modern aircraft include a variety of automatic control system that aids the flight crew in navigation, flight management and augmenting the stability characteristic of the airplane. For this situation an autopilot is designed that control the pitch of aircraft that can be used by the flight crew to lessen their workload during cruising and help them land their aircraft during adverse weather condition in the real situation [1]. The autopilot is an element within the flight control system. It is a pilot relief mechanism that assists in maintaining an attitude, heading, altitude or flying to navigation or landing references [2]. Designing an autopilot requires control system theory background and knowledge of stability derivatives at different altitudes and Mach numbers for a given airplane [3]. Lot of works has been done in the past to control the pitch of an aircraft for the purpose of flight stability and yet this research still remains an open issue in the present and future works [4], [5], [6], [7] and [8].

Pitch is controlled by the rear part of the tail plane's horizontal stabilizer being hinged to create an elevator. By moving the elevator control backwards the pilot moves the elevator up (a position of negative camber) and the downwards force on the horizontal tail is increased. The angle of attack on the wings increased so the nose is pitched up and lift is generally increased. In micro-lights and hang gliders the pitch action is reversed and the pitch control system is much simpler, so when the pilot moves the elevator control backwards it produces a nose-down pitch and the angle of attack on the wing is reduced. The pitch angle of an aircraft is controlled by adjusting the angle and therefore the lift force of the rear elevator. The aerodynamic forces (lift and drug) as well as the aircraft’s inertia are taken into account. This is a third order, nonlinear system which is linearized about the operating point.

This work presents investigation into the development of pitch control schemes for pitch angle and pitch rate of an aircraft systems. Pitch control systems with full state feedback controller are investigated. A modern controller (LQR) and intelligent fuzzy logic controller (FLC) is developed for control the pitch of an aircraft systems. Performance of both control strategy with respect to the pitch angle and pitch rate is examined. Comparison of both control schemes to the system performance of aircraft system is presented and discussed

II. MODELLING OF A PITCH CONTROL Flight control system has been designed using

mathematical models of the aircraft linearized at various flight condition parameters varied with the flight operating conditions [9]. This work is developed to control the pitch angle of an aircraft for pitch control in order to stabilize the system when the airplane is nose up and nose down. The pitch control system considered in this work is shown in Fig. 1 where Xb, Yb and Zb represent the aerodynamics force components. θ, Ф and δe represent the orientation of aircraft (pitch angle), orientation of aircraft (roll angle) in the earth-axis system and elevator deflection angle.

2010 IEEE Symposium on Industrial Electronics and Applications (ISIEA 2010), October 3-5, 2010, Penang, Malaysia

978-1-4244-7647-3/10/$26.00 ©2010 IEEE 389

Fig. 2 shows the forces, moments and velocity

components in the body fixed coordinate of aircraft system. The aerodynamics moment components for roll, pitch and yaw axis are represent as L, M and N. The term p, q, r represent the angular rates about roll, pitch and yaw axis while term u, v, w represent the velocity components of roll, pitch and yaw axis. and β are represents as the angle of attack and sideslip. In this study the data from General Aviation Airplane [1] is used in system analysis and modeling. The parameter include in dimensional derivatives Q = 36.8lb/ft2, QS = 6771lb, QS c = 38596ft.lb and ( 02/ uc ) = 0.016s is considered.

The longitudinal stability derivatives parameter used

are denoted in Table I.

A few assumption need to be considered before

continuing with the modeling process. First, the aircraft is steady state cruise at constant altitude and velocity, thus the thrust and drag are cancel out and the lift and weight balance out each other. Second, the change in pitch angle does not change the speed of an aircraft under any circumstance.

The section provides a brief description on the modeling of the pitch control system. In order to analyze the dynamics of aircraft in flight, the aircraft dynamic system must be model in terms of mathematical equations. Mathematical modeling involves process to describe dynamics of a system in a set of differential equations.

Referring to the Fig. 1 and Fig. 2, the following dynamic equations include force and moment equations are determined.

)( rvqvummgSX −+=− θ (1)

)( qupvwmCmgCZ −+=+ Φθ (2)

)()( 22 rpIIIrqqIM xzzxy −+−+= (3)

It is required to completely solved the aircraft problem

with considering the following assumption: (1) rolling rate, θψSp −Φ= , (2) yawing rate, ΦΦ += SCCq θψθ , (3) pitching rate, ΦΦ −= SCCr θψ θ , (4) Pitch Angle,

ΦΦ −= rSqCθ , (5) roll Angle, θθ TrCTqSp ΦΦ ++=Φ , and (6) Yaw Angle, θψ sec)( ΦΦ += rCqS .

Equation (1), (2) and (3) should be linearized using small disturbance theory. The equations are replaced by a variables or reference value plus a perturbation or disturbance, as shown below.

wwwvvvuuu ooo Δ+=Δ+=Δ+=

rrrqqqppp ooo Δ+=Δ+=Δ+= ZZZMYMMXXX ooo Δ+=+=Δ+=

δδδ Δ+= o

Figure 1. Description of pitch control system.

Figure 2. Definition of force, moments and velocity in body fixed

coordinate.

TABLE I. LONGITUDINAL STABILITY DERIVATIVE PARAMETERS

Longitudinal Derivatives

Components X-Force

(S-1) Z-Force

(F-1) Pitching

Moment (FT-1) Rolling

velocities Xu = -0.045 Zu = -0.369 Mu = 0

Yawing velocities

Xw = 0.036

X w = 0 Zw = -2.02

Z w = 0

Mw = -0.05

M w = -0.051

Angle of attack

X α = 0

X α = 0

Z α = -355.42

Z α = 0

M α = -8.8

M α = -0.8976

Pitching rate Xq = 0 Zq = 0 Mq = -2.05

Elevator deflection Xδe = 0 Zδe = -28.15 Mδe = -11.874

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For convenience, the reference flight condition is assumed to be symmetric and the propulsive forces are assumed to remain constant. This implies that,

00000000 ===Φ==== wrqpv ψ . After linearization the (4), (5) and (6) are obtained.

eeowu XgwXuXdtd δδθθ Δ=Δ+Δ−Δ⎟

⎠⎞

⎜⎝⎛ − )cos( (4)

eeqowwu ZgdtdZuwZ

dtdZuZ δθθ δ Δ=Δ⎥⎦

⎤⎢⎣⎡ −+−Δ⎥⎦

⎤⎢⎣⎡ −−+Δ− 0sin)()1( (5)

eeqwwu MdtdM

dtdwM

dtdMuM δθ δ Δ=Δ⎟

⎟⎠

⎞⎜⎜⎝

⎛−+Δ⎟

⎠⎞

⎜⎝⎛ +−Δ− 2

2 (6)

By manipulating the (4), (5), (6) and substituting the

parameters values of the longitudinal stability derivatives, the following transfer function for the change in the pitch rate to the change in elevator deflection angle is shown as (7) obtained.

)/()/()//()/(

)()(

002

000

αααα

αδδαδαδ

δ MuMZsuZMMsuZMuZMsuZMM

ssq

qq

eeee

e −+++−−−+−=

ΔΔ (7)

The transfer function of the change in pitch angle to the

change in elevator angle can be obtained from the change in pitch rates to the change in elevator angle in the following way.

θΔ=Δq (8)

)()( sssq θΔ=Δ (9)

)()(1

)()(

ssq

sss

e θδθ

ΔΔ⋅=

ΔΔ (10)

Therefore the transfer function of the pitch control

system is obtained in (11) and (12) respectively.

)/()/()//()/(1

)()(

002

000

αααα

αδδαδαδδθ

MuMZsuZMMsuZMuZMsuZMM

sss

qq

eeee

e −+++−−−+−⋅=

ΔΔ (11)

ssss

ss

e 941.129676.4578.227304.11

)()(

23 +++=

ΔΔδθ (12)

The transfer function can be represented in state-space form and output equation as state by (13) and (14).

[ ]eqq δ

θ

α

θ

αΔ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−−

−=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

07304.1116.0

01009476.29868.60102.2

(13)

[ ] [ ]0100 +⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔΔ

=θ

αqy (14)

III. METHODOLOGY In this section, two control schemes are proposed and

describe in detail which is Linear Quadratic Regulator

(LQR) and Fuzzy Logic Controller (FLC). Furthermore, a few of design specification have to be set to investigate the performance of both control strategies. In this work, four considerations have to be met which are rising time less than 3 second, settling time less than 5 second, percentage of overshot less than 10% and steady state error less than 2% for controlling the pitch angle of 0.2 radian (11.5 degree).

A. LQR Controller LQR is a method in modern control theory that used

state-space approach to analyze such a system. Using state space methods it is relatively simple to work with a multi-output system. The system can be stabilized using full-state feedback system. The configuration of this control system is shown in Fig. 3.

In designing LQR controller, lqr function in Matlab can be used to determine the value of the vector K which determined the feedback control law. This is done by choosing two parameter values, input R = 1 and Q = CT*C where CT is the matrix transpose of C from state equation (14). The controller can be tuned by changing the nonzero elements in q matrix which is done in m-file code as obtained.

R = 1; Q = [0 0 0;0 0 0;0 0 x]; K = lqr [A, B, Q, R]; Consequently, by tuning the value of x = 500, the

following values of matrix K are obtained. If x is increased even higher, improvement to the response should be obtained even more. But for this case, the values of x = 500 is chosen because it satisfied the design requirements while keep x as small as possible.

Matrix K = [-0.5704 1.6929 22.3607] In order to reduce steady state error of the system

output, a value of constant gain Nbar should be added after the reference. With a full-state feedback controller all the states are feedback. The steady-state value of the states should be computed, multiply that by the chosen gain K, and used a new value as the reference for computing the input. Nbar can be found using the user-defined function which can be used in m-file code. The method used in simulation work is done by exported both value of matrix K and constant gain, Nbar as shown in Fig. 4. For this controller design, the value of constant gain, Nbar are found to be, Nbar = 22.360.

Figure 3. Full-state feedback controller with reference input.

Figure 4. Block diagram of the system with matrix, K and gain

Nbar.

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B. Fuzzy Logic Controller In this work, fuzzy logic controller has been applied for

stabilization of the pitch control system. FLC is conceived as a better method for sorting and handling data but has proven to be an excellent choice for many control system applications because of non-linearity, complex mathematical computation and real-time computation need. It can be built into anything from small, hand-held products to large computerized process control systems. It uses an imprecise but very descriptive language to deal with input data more like a human operator. It is very robust and forgiving of operator and data input and often works when first implemented with little or no tuning. Based on these properties, fuzzy logic controller plays the best to fit the requirements in such cases. FLC incorporates a simple rule-based “If X and Y then Z” approach to solving control problem rather than attempting to model a system mathematically.

Fig. 5 shows the overall closed-loop system for FLC with the pitch control of an aircraft. The inputs to the fuzzy controller are the error (e) which measures the system performance and the rate at which the error changes (Δe), whereas the output is the change of the control signal (Δu). From the Fig. 5, the error (e) is computed by comparing the reference point (desired angle) with the plant output. The change of error (Δe) is generated by the derivation of the error. The error and change of error is fed to the fuzzy controller through a multiplexer.

Fuzzification involves the conversion of the input and

output signal into a number of fuzzy represented values (fuzzy set). Each fuzzy set consists of three types membership function, which is negative (N), zero (Z) and positive (P). The appropriate membership function to represent each fuzzy set need to be defined and each fuzzy set must have the appropriate universe of discourse. In addition, the membership functions are evenly distributed so that the tuning process of the controller can be easily done. In designing FLC, the standard fuzzy rules generated from the under damped response curve. This response is transform into fuzzy rules using the formula obtained below.

)()()( kykrke −= and )1()()( −−=Δ kekeke In this work, the triangular and trapezoidal membership

functions are chosen for each fuzzy set. The universe of discourse is set between -0.4 to 0.4 that implies the range of pitch angle (±0.4 radian). A two input, one output fuzzy pitch control can be designed by defining error as the reference angle minus the measured angle, and implementing the expert knowledge in a form of IF-THEN rule structure. These are nine rules that have been utilized in designing the controller and the rule is defined in Table II.

Since there are a total of three fuzzy variables (two

inputs and one output), and each fuzzy variable has three membership functions, the fuzzy controller for pitch control of an aircraft has a total of nine membership functions. Each membership function is constrained to be triangular so each membership function has three parameters (a modal point and two half-width).

In this study, the triangular and trap membership function are chosen for each fuzzy set. Fig. 6, Fig. 7 and Fig. 8 shows the fuzzy set of the input ‘error’ and ‘Δerror’ and the output ‘output1’.

Figure 5. Fuzzy logic controller in feedback loop of pitch control

system.

TABLE II. RULES FOR THE FUZZY CONTROLLER

Error, e Delta error. de Delta u, du 1 N N N 2 N Z N 3 N P N 4 Z N N 5 Z Z Z 6 Z P P 7 P N P 8 P Z P 9 P P P

Figure 6. Fuzzy set of the input ‘error’.

Figure 7. Fuzzy set of the input ‘∆error’

Figure 8. Fuzzy set of the input ‘outout1’.

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IV. RESULTS AND DISCUSSION In this section, the proposed of control schemes are

implemented and the corresponding results are presented. A unit step command is required in order for pitch angle to follow the reference value of 0.2 radian = 11.5 degree.

The pitch control system with both LQR and fuzzy logic controller produced the response of pitch angle, θ. The system response with the linear quadratic regulator (LQR) is shown in Fig. 9. The summary for the performance characteristics of the step response for the pitch angle between LQR and fuzzy logic controller is shown in Table III quantitatively. By referring to the Fig. 9 and Table III, the results clearly demonstrate that LQR controller has the fastest response with the settling time of 0.3655 second and rising time of 0.1335 second. For the percent of overshoot (%OS), LQR has 4.35% which is met the desired requirement of controller design. Furthermore the LQR controller tends to produce very small steady state error (Ess) and it is within the limit that is 0.01%. This can be indicating that LQR controller can handle the effect of disturbances in the system.

Fig. 10 shows the closed loop system response of the pitch angle, θ with fuzzy logic controller. Two inputs have been applied to fuzzy logic controller which is the error (e) that computed by comparing the reference point (desired angle) with the plant output and the change of error (Δe) which generated by the derivation of the error. The fuzzy logic controller provides good performance in term of percent overshoot that is 0%. As depicted from Fig. 10, it can be observed that the pitch angle follows the reference value respectively. This controller is able to give a good response without produce any overshoot. The response is comparatively fast that give the settling time (Ts) about 2.002 second and rise time (Tr) about 1.03 second. The results also demonstrated that the artificial fuzzy logic controller can eliminate the effect of disturbances in the system up to 0.5%.

For comparison of controller performance, the

response for pitch control of an aircraft system using LQR and fuzzy logic controller are shown with overall response of both controllers in Fig. 11 and the bar graphs in Fig. 12. The results clearly shows that LQR controller has the best performance as compared to fuzzy logic controller in term of rising time (Tr) that is 0.1335 second, settling time (Ts) that is 0.3655 second and percent of steady state error (Ess) about 0.01%. The results also show the LQR controller is good in eliminating the error from the system which is almost tend to the zero value. However, for the percent of overshoot (%OS) fuzzy logic controller have the best range which is 0%. Therefore, from the result obtained in Fig. 11 and Fig. 12, it can be concluded that the LQR controller provide higher ability in controlling the pitch angle as compared to the fuzzy logic controller.

0 1 2 3 4 5 6-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (sec)

Pitch

angle

(ra

dia

n)

ReferencePitch angle-LQR

Figure 9. Pitch angle response with LQR controller.

0 1 2 3 4 5 6 7 8-0.05

0

0.05

0.1

0.15

0.2

0.25

Time (sec)

Pitch

angle

(ra

dia

n)

ReferencePitch angle-FLC

Figure 10. Pitch angle response with fuzzy logic controller.

TABLE III. SUMMARY OF PERFORMANCE CHARACTERISTIC FOR PITCH ANGLE

Response characteristic

Pitch angle LQR Fuzzy Logic

Rising Time (Tr)

0.1335s 1.03s

Settling Time (Ts)

0.3655s 2.002s

Percent Overshoot (%OS) 4.35 0

Steady-state Error (ess) (%) 0.01 0.5

0 1 2 3 4 5 6 7 8

0

0.05

0.1

0.15

0.2

0.25

Time (sec)

Pitc

h a

ngle

(ra

d)

Reference

LQR

FLC

Figure 11. Pitch angle response with LQR and fuzzy logic controller.

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V. CONCLUSION The validated model of pitch control of an aircraft is

very helpful in developing the control strategy for actual system. Pitch control of an aircraft is a system which requires a pitch controller to maintain the angle at it desired value. This can be achieved by reducing the error signal which is the difference between the output angle the desired angle.

Two controllers, LQR and fuzzy logic are successfully designed and presented. Based on the result and the analysis, a conclusion has been made that, the control approach of LQR and fuzzy logic is capable of controlling the pitch angle of the aircraft system for value of 0.2 radian (11.5degree). Simulation and analysis results show that, LQR controller relatively give the better performance compared to fuzzy logic controller in controlling the pitch angle of an aircraft system. For further research, effort can be devoted through adding another element that make up the control system, following by develop more advanced and robustness control techniques. Beside, the proposed control algorithm can be implements to real plant for validating of theoretical result.

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[4] Pavle Boskoski, Biljana Mileva, Stojche Deskoski, “Auto Landing Using Fuzzy Logic”, 6th International PhD Workshop on Systems and Control, Slovenia, 2005.

[5] D. Choe, Y. Lee, S. Cho, “Nonlinear Pitch Autopilot Design with Local Lines Linear System Analysis”, International Conference on Automatic Control and Systems Engineering, Cairo, Egypt, 2005.

[6] Robert S. Eick, A Reconfiguration Scheme for Flight Control Adaptation to Fixed Position Actuator Failures, Ph. D Theses, University of Florida, 2003.

[7] D. G. Bates, R. Kureemun and I. Postlethwaite, “Quantifying the Robustness of Flight Control Systems Using Nichols Exclusion Regions and the Structured Singular Value”, Proc InstnMech Engrs, Vol 215, Part I, 2001.

[8] Hamid R. Berenji, Sujit Saraf, Ping-Wei Chang, and Steven R. Swanson, “Pitch Control of the Space Shuttle Training Aircraft”, IEEE Transaction on Control System Technology, Vol 9, No 3, 2001.

[9] Ekprasit Promtun, Sridhar Seshagiri, “Sliding Mode Control of Pitch Rate of an F-16 Aircraft”, International Journal on Applied Science, Engineering and Technology, Vol 5, No 5.

Figure 12. Performance comparison between LQR and fuzzy logic controller.

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