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Design and Implementation of a Nonlinear Flight Control Law for the

Yaw Channel of a UAV Helicopter

Guowei Cai, Ben M. Chen, Kemao Peng, Miaobo Dong, Tong H. Lee

AbstractWe present in this paper a flight control system

design for the yaw channel of a UAV helicopter using anewly developed composite nonlinear feedback (CNF) controltechnique. From the actual flight tests on our UAV helicopter,it has been found that the commonly used yaw dynamicalmodel for the UAV helicopter proposed in the literatures isvery rough and inaccurate. This motivates us to first obtaina comprehensive model for the yaw channel of our UAVhelicopter. The CNF control method is then utilized to design anefficient control law, which gives excellent overall performance.In particular, our design has achieved a Level 1 performance

according to the standards set for military rotorcraft. Theresults are verified through actual flight tests.

I. INTRODUCTION

Unmanned aerial vehicles (UAVs) have recently gained

much attention in the academic circle worldwide. They have

potential military and civil applications as well as scientific

significance in the academic research. The UAV, in particular,

the unmanned helicopter, can serve as an excellent platform

for studying plants with maneuverability and versatility. Our

motivation in developing a UAV helicopter is to build a

test bed for implementing some newly developed linear

and nonlinear control techniques, particularly, the composite

nonlinear feedback (CNF) control, which has proven to be

capable of yielding a very fast transient response with no

or very minimal overshoot. We have recently constructed asmall-scale UAV helicopter platform, called HeLion, which

is upgraded from a radio-controlled hobby helicopter, Raptor

90. To realize automatic flight control [2] of the UAV,

we have designed and integrated an avionic system to the

bare helicopter. A linearized model which has a similar

structure as that proposed in [7] for hover or near hover

flight condition is obtained and an automatic control law

using the CNF control method has been implemented on

the actual UAV helicopter [3]. The implementation is quite

successful. However, performance on certain aspects is not

as good as expected. In particular, we have found that the

commonly used dynamical model for the yaw channel is

very rough and inaccurate, especially in the relative highfrequency region (above 4 Hz in HeLion), which is seldom

stimulated by manual control. Such inaccuracy causes the

helicopter to shake severely in many occasions during the

actual flight tests.

This work is supported by the Defence Science and Technology Agency(DSTA) of Singapore.

G. Cai, B. M. Chen, M. Dong and T. H. Lee are with the Departmentof Electrical and Computer Engineering, National University of Singapore,Singapore 117576.

K. Peng is with Temasek Laboratories, National University of Singapore,Singapore 117508.

To improve the overall performance, we carry out in this

work a comprehensive modeling process to obtain a more

accurate dynamical model for the yaw channel. We have

found a mathematical model that gives a much more accurate

response. The CNF control technique is then utilized to

design a high performance flight control law with excellent

performance. In particular, our design has achieved a Level

1 performance according to the standards set for military

rotorcraft. The results have been verified through simulation

and actual flight tests.

I I . COMPREHENSIVE

MODELING OF

UAV YAW

CHANNEL DYNAMICS

Yaw direction control is one of the most challenging jobs

in controlling small-scale UAV helicopters as the torque

associated with the yaw channel is relatively small and highly

sensitive. The dynamics of traditional hobby helicopters is

open-loop unstable, which makes manual hovering of a

helicopter extremely difficult. To overcome such a problem,

modern hobby helicopters are commonly equipped with a so-

called yaw channel governor, which consists of a low-cost

yaw rate gyro and a simple controller to stabilize the yaw

rate and/or heading angle. The block diagram of the control

configuration is depicted in Figure 1. The performance of

such a simple control scheme is generally acceptable for

manual control. It has problems, however, when it comes

to precise automatic control. To enhance the performance of

the yaw channel, a more accurate model characterizing the

input-output relationship of the channel is necessary.

In Figure 1, ped [1, 1] is the normalized inputto the yaw channel. is the yaw rate in rad/s, which

can be measured either by a yaw rate gyro or an inertial

measurement unit (IMU) and z is the output signal of the

embedded controller. Traditionally, the identification of this

model is based on two important assumptions (see, e.g., [7]):

1) the simplified bare yaw dynamics is a first order system,

and 2) the embedded controller is characterized by a first

order low-pass filter,

z

=

K

s + Kz(1)

where K and Kz are the unknown parameters. For our

UAV helicopter, HeLion, we manage to identify a simplified

model in the state-space form and is given as

z

=

5.5561 36.6740

2.7492 11.1120

z

+

58.4053

0

ped.

(2)

Proceedings of the

46th IEEE Conference on Decision and Control

New Orleans, LA, USA, Dec. 12-14, 2007

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i - -

6

- Bare Yaw Dynamics

Embedded Controller Gyro

ped

z

Fig. 1. Configuration of the yaw channel of the UAV helicopter.

101

100

101

20

15

10

5

0

5

10

15

20

25

Magnitude(dB)

101

100

101

800

700

600500

400

300

200

100

0

Pha

se(Degree)

Frequency (Hz)

Experimental

Simulation

Fig. 2. Frequency-domain verification of the conventional 2nd order modelfor the yaw channel.

This model works pretty well for manual control and

its feasibility has been proved by manual flight tests and

automatic hovering tests in [3]. However, it is found that

the accuracy of such a model is not acceptable for high

bandwidth flight control. Figures 2 shows the frequency-domain verifications of the identified model together with

the actual response of the system. It can be clearly observed

that the traditional model yield a poor performance in the

high frequency region. This motivates us to carry out a

comprehensive modeling process to identify a more accurate

model for the yaw channel of our UAV helicopter.

To ensure the quality of data, flight experiment has to be

carefully designed. In [3], [7], [8], it has been verified that

for small-scale UAV helicopter the yaw channel dynamics

can be physically decoupled from other channels in hover

and near hover flight conditions. Thus, it is reasonable to

assume that the yaw channel dynamics is a SISO system.

The data collection experiment is performed in closedloop, which means the extra feedback controller is included

to ensure the stabilization. To a system like the HeLion

UAV which is inherently unstable, closed loop experiment is

an ideal choice. The feasibility and identifiability of closed

loop identification have been discussed in [6] in detail.

Compared with the open loop experiment which is conducted

under purely manual control, closed loop experiment has the

following two unique advantages: 1) With the extra feedback

controller, the stability of the identified system is guaranteed

and the augmented system can be fully excited by computer-

i - --

6

Yaw Channel withEmbedded Controller

External Control Law

ped ped

Fig. 3. Schematic diagram of data collection in the closed loop setting.

0 20 40 60 80 100 120 140 160 180 200

0.1

0.05

0

0.05

0.1

Time (s)

ped

(1~

1)

a. The resulting input to the yaw channel.

0 20 40 60 80 100 120 140 160 180 200

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Yawr

ate(rad/s)

b. The resulting output of the yaw channel.

Fig. 4. Experimental data for system identification of the yaw channel.

generated input signal over the interested frequency range;

and 2) Since the designed controller is known, the closed

loop identification can be transferred to open loop identifi-

cation. Then all of the identification algorithms which are

suitable for open loop identification can still be used.

The schematic diagram of data collection experiment in

the closed loop setting is shown in Figure 3. The bare

yaw dynamics augmented with the embedded controller

is regarded as a whole system. The additional feedback

controller is part of the automatic flight control law designed

to stabilize the overall helicopter at the hovering flight

condition. When automatic hovering is achieved, we inject aset of sinusoidal signals generated by the onboard computer

system into ped . The resulting signals {ped , } shownrespectively in Figures 4.a and 4.b are then used for system

identification.

Black-box state space model is selected to represent the

SISO yaw channel dynamics. The prediction error method

(PEM), which is commonly used in system identification

area, is adopted as the identification algorithm. In PEM

the unknown parameter set is estimated by minimizing

the sum of squared prediction error at all sampled points.

46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 WePI25.9

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101

100

101

20

15

10

5

0

5

10

15

20

25

Magnitude(dB)

101

100

101

800

700

600

500

400

300

200

100

0

Phase(Degree)

Frequency (Hz)

Experimental

Simulation

Fig. 5. Frequency-domain verification of the 4th order yaw channel model.

To achieve the optimized identified result, the state space

models with the order from 2 to 7 are identified and then

compared with the real measured frequency response. We

find that a fourth order model given in (3) yields the best

agreement in frequency domain with the fitness of 65.48%and is finally selected as the identified model. The frequency-

domain verification between the identified 4th order model

and real measured data is presented in Figure 5. It is clear

that the identified model is very accurate for frequencies up

to 10 Hz.

x =

2.6571 21.9350 3.8290 6.049731.0290 3.5154 17.0990 3.0897

6.1059 6.9623 9.7553 96.375017.1690 25.7330 37.1760 33.0820

x

+

0.6258

6.217529.199014.6430

ped,

= [15.3190 10.3210 0.7307 4.7274] x,(3)

Lastly the fidelity of the identified model is evaluated by

using flight data which is not used in identification procedure.

Two types of input signals are used. The first one is the

automatic chirp signal which covers the frequency range

from 0.01 Hz to 10 Hz and the second one is manual step-like signals with different input amplitudes. The verification

result presented in Figure 6 shows a perfect matching be-

tween the simulated yaw rate and the real measured data.Such agreement indicates the identified model is accurate

enough for controller design.

III. CONTROL OF YAW CHANNEL USING CN F

TECHNIQUE

By examining the identified model of yaw channel dynam-

ics, we find that the system is internally stable because of the

embedded controller. The closed loop poles of the system

with the embedded controller are located at 12.2507 j27.0777 and 12.2541 j57.4222. However, the actual

373 374 375 376 377 378 379 380 381 382 3830.1

0.05

0

0.05

0.1

ped

(1~1)

Time (s)

373 374 375 376 377 378 379 380 381 382 383

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Yaw

rate(rad/s)

Experimental

Simulation

Fig. 6. Model validation using automatic chirp input signal.

performance is pretty bad. For this reason designing an

extra control law to improve the yaw channel performance

is necessary. We propose in this section to design a high

performance controller using a newly developed composite

nonlinear feedback (CNF) technique. The CNF control tech-

nique was first introduced by Lin et al. [5] to improve the

tracking performance under state feedback laws for a class of

second-order systems subject to actuator saturation. Recently,

it has been fully developed to handle general systems with

input constraints and with measurement feedback (see, e.g.,

Chen et al. [4]).

To be more specific, we consider a linear continuous-time

system with an amplitude-constrained actuator character-ized by

x = A x + B sat(u), x(0) = x0y = C1 x

h = C2 x

(4)

where x Rn, u R, y Rp and h R are, respectively,the state, control input, measurement output and controlled

output of . A, B, C1 and C2 are appropriate dimensionalconstant matrices, and sat: R R represents the actuatorsaturation defined as

sat(u) = sgn(u)min{ umax, | u | } (5)

with umax being the saturation level of the input. The

following assumptions on the system matrices are required:

i) (A, B) is stabilizable, ii) (A, C1) is detectable, and iii)(A ,B ,C2) is invertible and has no invariant zeros at s = 0.The objective is to design a CNF control law that causes

the output to track a high-amplitude step input rapidly

without experiencing large overshoot and without the adverse

actuator saturation effects. This is done through the design

of a linear feedback law with a small closed loop damping

ratio and a nonlinear feedback law through an appropriate

Lyapunov function to cause the closed loop system to be

highly damped as the system output approaches the com-

mand input to reduce the overshoot.


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In what follows, we recall from [4] the step-by-step proce-

dure of the CNF control design with full order measurement

feedback:

STE P 1: Design a linear feedback law,

uL = F x + G r (6)

where F is chosen such that 1) A + BF is an asymptotically

stable matrix, and 2) the closed loop system C2(sI A BF)1B has certain desired properties, e.g., having a smalldamping ratio. We note that such an F can be designed using

methods such as the H2 and H optimization approaches.

Furthermore, G is a scalar and is given by

G =

C2(A + BF)1B

1

(7)

and r is a command input. Here we note that G is well

defined because A + BF is stable, and the triple (A ,B ,C2)is invertible and has no invariant zeros at s = 0.

STE P 2: Given a positive definite matrix W Rnn, wesolve the following Lyapunov equation:

(A + BF)P + P(A + BF) = W (8)

for P > 0. Such a solution is always existent as A + BF isasymptotically stable. The nonlinear feedback portion of the

enhanced CNF control law, uN, is given by

uN = (e)BP(x xe) (9)

where (e), with e = h r being the tracking error, is asmooth and nonpositive function of |e|. It is used to graduallychange the system closed loop damping ratio to yield a better

tracking performance. The choices of the design parameters,

(e) and W, will be discussed later. Next, we define

Ge := (A + BF)

1

BG and xe := Ge r (10)If all the state variables of the system are available for

feedback, the CNF control law is given by

u = uL + uN = F x + G r + (e)BP(x xe) (11)

STE P 3: For the case when there is only a partial measure-

ment available, the state feedback CNF control law of (11)

should be replaced by the following measurement feedback

controller:xv = (A + KC1)xv Ky + Bsat(u)

u = F(xv xe) + Hr + (e)BP(xv xe)(12)

where K is the full order observer gain matrix such that

A + KC1 is stable and

H = [1 F(A + BF)1B]G. (13)

The following result is due to [4].

Theorem 1: Given a positive definite matrix W Rnn,let P > 0 be the solution to the Lyapunov equation

(A + BF)P + P(A + BF) = W (14)

Given another positive definite matrix WQ Rnn with

WQ > FBP W1P BF (15)

let Q > 0 be the solution to the Lyapunov equation

(A + KC1)Q + Q(A + KC1) = WQ (16)

Note that such P and Q exist as A + BF and A + KC1are asymptotically stable. For any (0, 1), let c

be the

largest positive scalar such that for all

xxv

X

F := x

xv

: x

xvP 0

0 Q x

xv

c

(17)

we have [ F F ]

x

xv

umax(1 ) (18)Then, there exists a scalar > 0 such that for any nonpos-itive function (e), locally Lipschitz in e and |(e)| ,the full order measurement CNF control law of (12) drives

the system controlled output h(t) to track asymptotically astep command input of amplitude r from an initial state x0,

provided that x0, xv0 = xv(0) and r satisfy:

|Hr | umax and x0 xe

xv0 x0

X

F (19)We note that the freedom to choose the function in

the CNF design is used to tune the control laws so as

to improve the performance of the closed loop system as

the controlled output, h, approaches the set point, r. Since

the main purpose of adding the nonlinear part to the CNF

controller is to shorten the settling time, or equivalently to

contribute a significant value to the control input when the

tracking error, e, is small. The nonlinear part, in general, is

set in action when the control signal is far away from its

saturation level, and thus it does not cause the control input

to hit its limits. The following nonlinear function proposed

in [4] meets such a requirement:

(e) = exp(|e|) exp(|e(0)|), (20)

where and are tuning parameters that can be adjusted

to yield a desired performance.

In what follows, we adopt the above mentioned CNF

control technique to design the controller for yaw channel

dynamics that yields a top level performance specified by

the United States Army Aviation and Missile Command in

[1]. It follows from the previous section that the identified

fourth order yaw dynamical model can be written as that in

(4) with

A =

2.6571 21.9350 3.8290 6.0497

31.0290 3.5154 17.0990 3.08976.1059 6.9623 9.7553 96.3750

17.1690 25.7330 37.1760 33.0820

,

B =

0.62586.2175

29.199014.6430

,

(21)

and y = h = = C1x = C2x with

C1 = C2 = [15.3190 10.3210 0.7307 4.7274] .(22)


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For safety consideration, the maximal amplitude of the pedal

input is kept within 0.4. It is straightforward to verifythat (A, B) is controllable and (A, C1) is observable. Fur-thermore, the triple (A ,B ,C2) is invertible with 3 unstableinvariant zeros respectively at 29.013 j29.572 and 990.68.This nonminimum-phase nature of the yaw channel gives lots

of troubles in designing a high performance controller.

Next, we proceed to design a CNF control law for this

system. Our main goal is to reduce the overshoot of the

time-domain response. Since the identified model of the yaw

channel with the embedded controller is stable with two pairs

of pole having very small damping ratios (0.47 and 0.22,

respectively), we can safely choose the state feedback gain

for the linear part of the CNF control law as F = 0, whichyields

G = [C2(A + BF)1B]1 = 0.2675 (23)

and

Ge = (A + BF)1BG =

0.05600.0217

0.00540.0785

. (24)

Solution to Lyapunov equation AP+P A = W is obtainedwith W = I. For the full order, we choose an observer gainmatrix

K =

1.20164.0081

2.90735.4800

, (25)

which places the observer poles, i.e., the eigenvalues of A +KC1, at 24 j14.6 and 26 j14.6, respectively. Finally,we obtain a full order measurement feedback CNF control

law,

xv =

15.7502 9.5333 4.7070 0.369330.3711 44.8830 20.0277 22.0376

38.4310 23.0439 11.8797 82.6310101.1171 30.8261 41.1802 58.9882

xv

1.20164.0081

2.90735.4800

+

0.62586.2175

29.199014.6430

sat(ped)

(26)

and

ped = (e)

[ 0.5745 0.1570 0.5716 0.6469] xv

+0.0183 r+ 0.2675 r (27)with

(e) = 9.6exp(1.05|e|) 0.3679. (28)

IV. SIMULATION AND IMPLEMENTATION RESULTS

We evaluate its performance through an intensive simu-

lation process. In order to compare the overall performance

of the yaw channel with and without the additional CNF

controller, we choose a step reference of 0.5 rad/s. Thesimulation result is shown in Figure 7. It is clear that the

CNF control yields a better performance. In what follows, we

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.2

0

0.2

0.4

0.6

0.8

Yaw

rate(rad/s)

Without CNFCNF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.06

0.08

0.1

0.12

0.14

0.16

Time (s)

ped

(1~1)

Fig. 7. Step responses of the yaw channel with and without CNF control.

proceed to examine a comprehensive performance analysis

for the overall system with the CNF controller on the actual

UAV helicopter. It is done by replacing the control law for

the yaw channel as given in [3] with the one given in (26) and

(27). The rest of the control system in [3] remains unchanged

throughout the whole experimental test.

We first present the automatic hovering turn tests for

the yaw channel on the UAV system. Figure 8 shows the

results of actual responses of the yaw channel with and

without the additional CNF controller. In the actual flight

test, the setpoint is chosen to be 0.3 rad/s. We note thatthe overshoot levels are slight different with the simulation

results. This is practically acceptable and it is mainly due

to mechanical friction and external environmental factors

such as the precision of the GPS measurement signals.

Nonetheless, the system with the additional CNF controlonce again gives a much better performance compared to

that of the system with only the embedded controller. In

particular, the system with the CNF controller is capable of

tracking the setpoint in 0.4 s and keeping it there within 0.1rad/s. On the other hand, the system with only the embedded

controller needs about 5 s to reach the steady state and is

only able to maintain in the target with a 0.2 rad/s accuracy.The advantage of using an additional CNF control is obvious

in this regard.

Next, we carry out the performance evaluation of the

overall system. More specifically, we follow the standard set

by the United States Army Aviation and Missile Command

in [1] to examine 1) the stabilization, and 2) the agility ofthe overall flight control system.

1) Stabilization Test: The stabilization test examines the

hovering stability and the accuracy of heading-hold.

The performance is categorized by two levels [1],

namely, the desired performance and adequate perfor-

mance. The result of the actual automatic hovering

test for a duration of 35 s is shown in Figure 9. The

stabilization test results together with the standards are

summarized in Table I. It is once again clear that our

design is very successful in this category. We would


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451 452 453 454 455 456 457 458 459

0

0.1

0.2

0.3

0.4

0.5

Time(s)

Yaw

rate(rad/s)

585 586 587 588 589 590 591 592 593 594

0

0.1

0.2

0.3

0.4

Time(s)

Yaw

rate(rad/s)

Embedded controller

CNF controller

Fig. 8. System responses with and without the CNF controller.

485 490 495 500 505 510 515

1.5

2

2.5

3

Xposition(m)

485 490 495 500 505 510 515

2.5

3

3.5

4

Yposition(m)

485 490 495 500 505 510 515

13

13.5

14

14.5

Height(m)

485 490 495 500 505 510 515

122

120

118

116

Headingangle(deg)

Time (s)

Experimental

Reference

Fig. 9. Flight test: Automatic hovering.

like to further note that the position errors in our actual

test results are mainly due to the inaccuracy of the GPS

signals received. The positioning accuracy of the GPS

receiver is 3 m.

TABLE I

HOVERING PERFORMANCE SPEC IFICATIONS AND ACTUAL TEST RES ULTS

DesiredPerformance

AdequatePerformance

HeLionsPerformance

Stabilized Hover

Duration> 30 s > 30 s > 35 s

HorizontalTolerance

3 ft 6 ft 3 ft

Altitude

Tolerance 2 ft 4 ft 3 ft

Heading AngleTolerance

5 10 3.7

2) Agility Test: In the agility test, the UAV helicopter

is required to perform a 360 self-rotation around itsmain shaft with a required yaw rate. The performance

is categorized into three levels. The actual test results

are given in Figure 10, in which the yaw rate is

maintained at 0.5 rad/s, and summarized in Table IIwith the specifications set in [1]. Our HeLion has again

512 514 516 518 520 522 524

0.05

0

0.05

x

(rad/s)

512 514 516 518 520 522 524

0.2

0.1

0

0.1

0.2

y

(rad/s)

512 514 516 518 520 522 524

0.4

0.2

0

0.2

0.4

0.6

0.8

z

(rad/s)

Time (s)

Fig. 10. Agility test: Angular rates.

achieved a Level 1 performance in this test.

TABLE II

AGILITY PERFORMANCE SP ECIFICATIONS AND ACTUAL TEST RES ULT

Level 1Performance

Level 2 and 3Performance

HeLionsPerformance

Required Yaw Rate > 22 deg/s > 9.5 deg/s > 31 deg/s

V. CONCLUSION

We have carried out a comprehensive study on the yaw

channel of a UAV helicopter. Particularly, we have a fairly

accurate model for the channel with an embedded controller

and improved its performance by augmenting an additional

control law using a nonlinear control technique.

REFERENCES

[1] ADS-33E-PRF, Aeronautical Design Standard Performance Specifica-tion Handling Qualities Requirements for Military Rotorcraft, UnitedStates Army Aviation and Missile Command, 2000.

[2] G. Cai, K. Peng, B. M. Chen and T. H. Lee, Design and assembling

of a UAV helicopter system, Proceedings of the 5th International Con-ference on Control and Automation , pp. 697-702, Budapest, Hungary,2005.

[3] G. Cai, B. M. Chen, K. Peng, M. Dong and T. H. Lee, Modeling andcontrol system design for a UAV helicopter, Proceedings of the 14th

Mediterranean Conference on Control and Automation, pp. 600-606,Ancona, Italy, 2006.

[4] B. M. Chen, T. H. Lee, K. Peng and V. Venkataramanan, Compositenonlinear feedback control for linear systems with input saturation:theory and an application, IEEE Transactions on Automatic Control ,vol. 48, No. 3, pp. 427-439, 2003.

[5] Z. Lin, M. Pachter, and S. Banda, Toward improvement of trackingperformance nonlinear feedback for linear systems, International

Journal of Control, vol. 70, pp. 111, 1998.

[6] L. Ljung, System Identification Theory For the User, 2nd Edn.,Prentice Hall, Upper Saddle River, N.J., 1999.

[7] B. Mettler, Identification Modeling and Characteristics of MiniatureRotorcraft, Kluwer Academic Publishers, Norwell, MA, USA, 2003.

[8] D. H. Shim, H. J. Kim and S. Sastry, Control system design forrotorcraft-based unmanned aerial vehicle using time-domain systemidentification, Proceedings of the 2000 IEEE Conference on Control

Applications, pp. 808-813, Anchorage, Alaska, USA, 2000.


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