Decentralized Nonlinear Control of a LaboratoryModel Helicopter

Reza Banaei KhosroushahiDepartment of Electrical Engineering

Sahand University of TechnologyP.O.Box 51335-1996, Tabriz, Iran

Email: rbanaei@sut.ac.ir

Mohammad Javad KhosrowjerdiDepartment of Electrical Engineering

Sahand University of TechnologyP.O.Box 51335-1996, Tabriz, Iran

Email: khosrowjerdi@sut.ac.ir

Abstract— In this paper, design, simulation and implementa-tion of a decentralized nonlinear feedback linearization controllerare addressed. The experimental results are presented, and theperformance of the proposed controller is compared with a linearstate feedback controller. It has been shown that the nonlinearcontroller has fairly better results in comparison with linearcontroller. Some practical tips which have been used to achievebetter implementation results are discussed.

I. INTRODUCTION

Helicopters are the most versatile flying machines that havedemonstrated their merit because of their ability to take off andto land in the out-of-the-way places and their feature of hov-ering flight. A Helicopter consists of numerous complicatedmechanical parts besides that it is faced with various aerody-namic forces which result in a sophisticated dynamical model.Because of its unstable behavior it needs to apply a dynamiccontrol law. However, it is well-known that it exhibits a signif-icant nonlinear behavior which makes the controller design achallenging task. A Laboratory Helicopter, the so-called Twin-Rotor setup prepares a suitable workbench for implementationof various ideas for the controlling of an actual helicopter.Being unstable, being Multiple-Input Multiple-Output (MIMO)and having intense interactions, make the Twin-Rotor systeman attracting subject for research so that numerous controllaboratories have employed it for educational and researchpurposes [1]–[3]. In recent years many attempts system havebeen done to design an effective controller for the Twin-Rotor.Many of these works just remain at the simulation step andhave not presented practical results (for example, see [4]). Thismay be done due to the fact that there are various types ofdisturbances besides the channel interactions in this system.Others tried to use a more simple setup for achieving practicalresults by adding a mass with a pendulum structure to thecenter of the main body so that it lowers the center of thegravity of the system and balances it [5]. The presented workhas been done on the main mentioned model and attempts havebeen done to get reasonable practical results. A schematic ofTwin-Rotor is shown in Fig. 1, where ψ is elevation and ϕ isazimuth directions of setup body movements and ωm and ωrare main motor-propeller and side motor-propeller rotationalspeeds, respectively. The laboratory helicopter has 2 Degree-of-Freedom (DOF) in body motion in the elevation and in the

ψ

ϕωr

ωm

Fig. 1. Schematic of the Laboratory Helicopter

azimuth orientations. For the simulation of the hovering flightof an actual helicopter, the body elevation of the laboratorymodel helicopter should be regulated in 90 degree and theazimuth should be regulated in spite of the disturbances causedby the variations of the setup body in elevation, variations ofthe rotational speed of the main motor-propeller and changesin the center of gravity . Also there must be the ability to trackthe desired azimuth.

In this paper, two different controller design schemes wereexperimentally evaluated. A decentralized nonlinear feedbacklinearization controller and a linear state feedback controller.Feedback linearization is an approach to nonlinear controldesign which has attracted a great deal of research interest inrecent years. The basis of feedback linearization was suggestedby [6] and fully derived in [7], [8]. The proposed conditionsin these works related the possibility of feedback linearizationof a nonlinear system to involutiveness of vector fields.

Over the last decade, geometric nonlinear control theoryhas provided powerful tools for systematic design of nonlinearfeedback system [9], [10]. Most of the available methodsfor nonlinear tracking control system design are based onlinearizing the input-output response of a nonlinear systemusing state feedback, or exact state linearization using acoordinate change and a state feedback [8]–[10]. Majorlimitations to these approaches come from the fact that theyrequire certain regularity conditions such as involutivity,existence of a (vector) relative degree, minimum phaseproperty, and that they rely on exact cancellation of nonlinear

1–4244–0342–1/06/$20.00 c© 2006 IEEE ICARCV 2006

terms.

An idea to overcome the design complexities of a nonlinearfeedback linearization controller design for a MIMO systemwhen there is not a well-defined relative degree vector isto simplify the model to Single-Input Single-Output (SISO)systems by considering the interactions as disturbances and toeliminate them for controller design. It needs a modificationfor implementation, wherein offsets are added to the controlinputs to compensate the value of interactions in the operatingpoint. By applying this technique in many cases the resultingSISO models will have well-defined relative degree and cansatisfy other conditions needed to design a feedback lineariza-tion controller.

The utilized Twin-Rotor setup has a nonlinear MIMO modelwith interactions between its channels. Its operating point isnot an equilibrium point, and another feature of this systemis its slow motor-propellers dynamic versus its fast bodydynamic. For designing of a feedback linearization MIMOcontroller for this system, the relative degree of the systemshould be defined. It is shown that the MIMO system doesn’thave a well-defined relative degree, so this system is notsuitable to design a MIMO feedback linearization controller.Dynamic extensions was not used because it increases theorder of the system and produces a highly complicate feedbacklaw. As mentioned above, we simplify the problem to design oftwo SISO controllers. This technique also leads to get simplercontrol laws that can be executed much faster in the real-time implementation. In the experiment, an integral actionhas been employed to regulate steady state errors caused byvarious disturbance sources. Performance of the both linearand nonlinear controllers depend on accuracy of the estimationof the state variables. The Luenberger observer is used toestimate the state variables in real-time implementation of theexperiment. Feedback linearization has been successfully usedto address some practical control problems of the aerospacevehicles, including the control of helicopters [11]–[13].

The outline of this paper is as follows. In section II, theexperimental setup and the control goals are described. InSection III we have focused on modeling and have presented adynamic model for the system. In section IV, the problem hasbeen briefly formulated and the designing of the controllershas been described. Experimental results have been presentedin Section V. The conclusion besides some offers has beenpresented in Section VI.

II. LABORATORY HELICOPTER SETUP

The utilized setup in the experiment is a laboratory-scalebench-top simplified model of an actual helicopter, producedby TQ Inc. A photograph of this model is shown in Fig. 2. Thesetup has 2 DOF and those are body movement in elevation(ψ) and azimuth (ϕ) directions.

The laboratory helicopter consists of the five importantparts:

• Main body of the model• Power Supply and PWM driver units

Fig. 2. Laboratory Helicopter Setup

• Computer interface card• Computer that runs controller program• Driver software that link computer and interface card

In this system, signals from sensors (Incremental RotaryEncoders) are collected at the Power-PWM unit and by con-ditioning at the desired level come to interface card. Thesesignals are read by driver software from the interface cardand by applying some post-processes, are taken to the mainprogram. Controller design has been done in Matlab and thesystem output variables (sensors output) have been read by RTtoolbox using driver software. After applying the control law,the output variables are sent to the card using RT toolbox.These signals are changed to PWM pulses by the interfacecard, and by amplifying in the power unit they are appliedto the motors which are mounted on the model. At thismodel we have not been facing with changes in the cyclicor collective mean for creating thrust variations [14]. Thecontrolled thrust forces are achieved by making changes inthe rotational speed of the DC motors which are coupled topropellers. In this experiment, the main goal of the controlis the body stabilization of the Twin-Rotor in hovering thatmeans elevator and azimuth regulation. The tracking of theazimuth is another goal in this work.

III. MODELING

In this section, a mathematical model for the laboratoryhelicopter is presented. The simplified block diagram of themodel is shown in Fig. 3. This MIMO nonlinear model isderived from physical data which can be described as:

x = f(x) + g(x)uy = h(x)

(1)

where x ∈ Ω ⊂ Rn is system state vector and is defined

x = [ψ ψ α α ϕ ϕ ζ ζ (η −KrTorTpr

uψ)]T and u, y ∈ Rm are

input and output vectors and are defined u = [uψ uϕ]T and

f(x) =

x21Iψ

(−τg sin (x1) −Bψx2 + x3(a1x3 + b1))x4

−1T 21(x3 + 2T1x4)

x61Iϕ

(−Bϕx6 − x9 + x7(a2x7 + b2))x8

−1T 22(x7 + 2T2x8)

−1Tpr

(x9)

, g(x) =

0 0kgyroI (x6 cos (x1)) 0

0 01T1

2 00 0

−1Iϕ

(kr TorTpr) 0

0 00 1

T22

1Tpr

(kr(1 − TorTpr

)) 0

, h(x) =[x1

x5

](2)

TABLE I

PARAMETER DEFINITION

Elevation ψ

Azimuth ϕ

Gravitation torque of the body mass τg

Parameters of the main motor a1, b1, T1

Parameters of the side motor a2, b2, T2

Viscous friction coefficient in elevator movement Bψ

Viscous friction coefficient in azimuth movement Bϕ

Model Body’s moment of inertia around horizontal axis Iψ

Model Body’s moment of inertia around vertical axis Iϕ

Parameters of interaction Kr, Tor , Tpr

Gyroscopic gain Kgyro

y = [yψ yϕ]T and f and g are nonlinear mappings as shown in(2). The definition of parameters are presented in Table I. Theparameters α , ζ and η are defined according to Fig. 3. Forthe laboratory helicopter, m and n are 2 and 9 respectivelywhich means that the model has two channels in input andtwo channels in output and the resulting model is of 9th order.The model which is presented in (1) is a more general class ofthe control-affine systems that typically employed in feedbacklinearization studies.

IV. NONLINEAR CONTROLLER DESIGN

Assume x0 is an operating point of the system. For MIMOfeedback linearization design, at first the relative degree vector r1, · · · , rm must be defined. A multivariable nonlinear sys-tem of the form (1) has a (vector) relative degree r1, · · · , rmat a point x0 if(i)

LgjLkfhi(x) = 0

for all 1 ≤ i, j ≤ m, for all 0 ≤ k < ri − 1 and for all x in aneighborhood of x0

(ii) the m×m matrix, so called decoupling matrix,

E(x) =

Lg1Lr1−1f h1(x) · · · LgmL

r1−1f h1(x)

Lg1Lr2−1f h2(x) · · · LgmL

r2−1f h2(x)

· · · · · · · · ·Lg1L

rm−1f hm(x) · · · LgmL

rm−1f hm(x)

Fig. 3. Simplified Block Diagram of System

be nonsingular at x = x0 [9].Using definition (i) and having calculation with laboratory

helicopter model, r1 = 4 and r2 = 2 is achieved becauseLg1L

3fh1(x) and Lg1L

1fh2(x) aren’t zero in a neighborhood

of x0. By constructing the decoupling matrix

E(x0) =

Lg1L

3fh1(x0) Lg2L

3fh1(x0)

Lg1L1fh2(x0) Lg2L

1fh2(x0)

since Lg2L3fh1(x) = 0 and Lg2L

1fh2(x) = 0, it is obvious

that E(x) is singular at x = x0. Thus the relative degreevector for MIMO model can not be defined so input-outputlinearization can not be applied to this system. We convertthe mentioned MIMO system in (1) to two SISO subsystemsby ignoring the interactions. For this purpose, interactions inthe simplified block diagram of the system (Fig. 3) have beeneliminated and the nonlinear differential equations have beenrewritten. In this simplification one of the system states, x9

, which belongs to M(s) eliminated because η considered asinteraction. So the MIMO system is converted to two 4th orderSISO system. By using of this technique both of the SISOsubsystems do have a well-defined relative degree rψ = 4 and

rϕ = 4. If the elevation subsystem is indexed with ψ and theazimuth subsystem is indexed with ϕ, the general equation ofa multivariable nonlinear system in (1) is converted to a SISOsystem for ψ subsystem:

xψ = fψ(xψ) + gψ(xψ)uψyψ = hψ(xψ)

(3)

where xψ ∈ R4, xψ = [x1 x2 x3 x4]T . fψ(xψ), gψ(xψ) and

hψ(xψ) are achieved as follow:

fψ(xψ) =

x21Iψ

(−τg sin (x1) −Bψx2 + x3(a1x3 + b1))x4

−1T 21(x3 + 2T1x4)

gψ(xψ) =

0001T 21

, hψ(xψ) =

[x1

]

and a SISO system for ϕ subsystem:xϕ = fϕ(xϕ) + gϕ(xϕ)uϕyϕ = hϕ(xϕ)

(4)

where xϕ ∈ R4 and xϕ = [x5 x6 x7 x8]T . fϕ(xϕ), gϕ(xϕ)

and hϕ(xϕ) are achieved as follow:

fϕ(xϕ) =

x61Iϕ

(−Bϕx6 + x7(a2x7 + b2))x8

−1T 22(x7 + 2T2x8)

gϕ(xϕ) =

0001T 22

, hϕ(xϕ) =

[x5

]

It can be easily shown that the required conditions for exactfeedback linearization are satisfied for both of these SISOsubsystems and they are separately linearizable by using thefollowing SISO feedback linearization design approach foreach one in the form:

uψ = ( −L4fψhψ(xψ) +

rψ−1∑i=0

kiψ ( yd(i)ψ − yψ

(i))

+ kIψ

∫(ydψ − yψ) dt )/( LgψL

3fψhψ(xψ) )

(5)

and

uϕ = ( −L4fϕhϕ(xϕ) +

rϕ−1∑j =0

kjϕ ( yd(j)ϕ − yϕ

(j))

+ kIϕ

∫(ydϕ − yϕ) dt )/( LgϕL

3fϕhϕ(xϕ) )

(6)

where ydψ and ydϕ are the reference signals for elevation andazimuth channels. If the reference signals ydψ and ydϕ are

step inputs then y(i)dψ

and y(j)dϕ

will be zero for i = 1, · · · , rψand j = 1, · · · , rϕ. By using the relative degree definition, itis easy to show that:

y(i)ψ = Lifψhψ(x) and y(j)

ϕ = Ljfϕhϕ(x)

for i = 0, · · · , rψ−1 and j = 0, · · · , rϕ−1. These definitionslead us to the simplified equations in (5) and (6) that are usedfor implementation.

After the calculation of the Lee derivatives the followingrelations are achieved for ψ subsystem:

L0fψhψ = x1

L1fψhψ = x2

L2fψhψ =

1Iψγ(x1, x2, x3)

L3fψhψ =

1Iψ

(−τg cos (x1)x2) − BψI2ψ

γ(x1, x2, x3)

+x4

Iψ(2a1x3 + b1)

L4fψhψ =

τgx2

Iψ(sin (x1)x2 +

BψIψ

cos (x1))

+1I2ψ

(−τg cos (x1) +B2ψ)γ(x1, x2, x3)

+x4

Iψ(−Bψ

Iψ(2a1x3 + b1) + 2a1x4)

+1

IψT 21

(2a1x3 + b1)(−x3 − 2T1x4)

LgψL3fψhψ =

1IψT 2

1

(2a1x3 + b1)

and the following relations are achieved for ϕ subsystem:

L0fϕhϕ = x5

L1fϕhϕ = x6

L2fϕhϕ =

1Iϕδ(x6, x7)

L3fϕhϕ = −Bϕ

I2ϕ

δ(x6, x7) +x8

Iϕ(2a2x7 + b2)

L4fϕhϕ =

B2ϕ

I3ϕ

δ(x6, x7)

+x8

Iϕ(−Bϕ

Iϕ(2a2x7 + b2) + 2a2x8)

+1

IϕT 22

(2a2x7 + b2)(−x7 − 2T2x8)

LgϕL3fϕhϕ =

1IϕT 2

2

(2a2x7 + b2)

where

γ(x1, x2, x3) = −τg sin (x1) −Bψx2 + a1x23 + b1x3

δ(x6, x7) = −Bϕx6 + a2x27 + b2x7

The control law,uψ and uϕ, can be created by using aboverelations.

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 3040

60

80

100

120

140

Time ( s ), Sampling = 0.01

Ele

vato

r B

ody

Ang

le(

Deg

ree

)

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 300

0.2

0.4

0.6

0.8

1

Time ( s ), Sampling = 0.01

Con

trol

Effo

rt(

MU

)

Fig. 4. State Feedback Controller applied to Elevator Channel of ModelHelicopter

V. EXPERIMENTAL RESULTS

This section deals with the Experimental results of thelinear state feedback controller and the nonlinear feedbacklinearization controller. at first Implementation of the linearstate feedback controller is discussed. States estimation havebeen done by minimum order observer. The sampling timeis set to the smallest possible value according to hardwarelimitation.

As shown in Fig. 4, regulation at this channel is achievedin about 7.5s. Because the tracking is important for azimuthchannel and the system must have greater operating zone inthis channel, the following tracking scenario has been definedfor the azimuth channel. As shown in Fig. 5 after regulationa step has been applied to change the azimuth from ϕr = 0

to ϕr = 90 where ϕr is the set-point for azimuth channel.After 40s the set-point has been changed to ϕr = −90 ( a180 rotation ). Results imply that the tracking is performedwith a good performance and the disturbances caused byinteractions are fairly damped at elevator channel. In thepractical implementation of the nonlinear controller, there existstochastic disturbances mainly caused by turbulent flow ofair around the laboratory helicopter. Other uncertainties arepresent because of using simplified model instead of exactone for controller design, using linear observer for nonlinearstate estimation and ignoring the interactions. For dealing withthese types of disturbances and uncertainties an integrator isadded to the control law.

Implementation results of the elevator channel regulationare presented in Fig. 6. This figure shows that the regulationis achieved in 2s and has much better performance than thelinear controller presented in Fig. 4. The overshoot is reducedto smaller values. Another characteristic of the nonlinearcontroller is smoothness of the control efforts applied to themotor in real-time implementation versus the result of thelinear controller.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 18040

60

80

100

120

140

160

Time ( s ), Sampling = 0.02

Ele

vato

r A

ngle

( D

egre

e )

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180−150

−100

−50

0

50

100

Time ( s ), Sampling = 0.02

Azi

mut

h A

ngle

( D

egre

e )

Fig. 5. State Feedback Controller applied to Model Helicopter in MIMOMode with Tracking of Azimuth Channel

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2060

65

70

75

80

85

90

95

Time ( s ), Sampling =0.011s

Ele

vato

r A

ngle

( D

egre

e )

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

0.2

0.4

0.6

0.8

1

Time ( s ), Sampling =0.011s

Mai

n M

otor

Con

trol

Effo

rt(

MU

)

Fig. 6. Feedback Linearization Controller Applied to Elevator Channel ofModel Helicopter

In the next step, both SISO feedback linearization con-trollers are simultaneously applied in the real-time implemen-tation to form a decentralized MIMO control system. Theresults are depicted in Fig. 7. This figure shows that it takes10s for error to damp and finally the system has been reg-ulated.Next the tracking performance of the azimuth channelin MIMO implementation is evaluated. This is done using areference signal that has ±90 variation. As shown in Fig. 8,acceptable tracking has been performed and interaction effectsin elevator channel have been eliminated by the controller.The non-homogeneous result at t = 20s has been caused byimpact of the system body with the mechanical stop limit. Thepresence of higher order variables (ie. results of multiplicationof some states) in nonlinear differential state equations leadto extend the bandwidth that is needed by the controller. If

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 3040

50

60

70

80

90

100

110

Time ( s ), Sampling =0.016s

Ele

vato

r A

ngle

( D

egre

e )

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30−150

−100

−50

0

50

100

150

Time ( s ), Sampling =0.016s

Azi

mut

h A

ngle

( D

egre

e )

Fig. 7. MIMO Implementation of Decentralized Feedback LinearizationController

0 10 20 30 40 50 60 70 80 90 100 110 12060

80

100

120

140

Time ( s ), Sampling =0.017s

Ele

vato

r A

ngle

( D

egre

e )

0 10 20 30 40 50 60 70 80 90 100 110 120−150

−100

−50

0

50

100

150

Time ( s ), Sampling =0.017s

Azi

mut

h A

ngle

( D

egre

e )

Fig. 8. Azimuth Channel Tracking in MIMO Implementation of Decentral-ized Feedback Linearization Controller

the sampling time has not been such a small that it cannot proved the required bandwidth, the performance of thecontroller has been drastically decreased and it can lead systemto instability. So it is needed to decrease the sampling timein the real-time implementation as much as possible. This isaccessible by using a faster hardware or by optimizing thesoftware routines. In this experiment, if the gain are increased,better performance of the controller can be achieved (thismeans choosing poles for faster results) but the chatteringwill be increasing in the control action. For achieving goodperformance besides of good control signals, a compromisemust be taken with proper gain selection.

VI. CONCLUSION

This paper presents experimental evaluation of two SISOnonlinear controllers applied to a laboratory model helicopter.

The method is based on feedback Linearization. Also a linearstate feedback controller has been applied for studying of thenonlinear controller performance. As shown in this paper, thenonlinear controllers in SISO mode have remarkable betterperformance than the linear controller beside a much bettercontrol action. The applied method of breaking the MIMOmodel to SISO subsystems by ignoring the interaction hasgood results on the laboratory helicopter model. Nonlinearsystem state variables are used in nonlinear controller con-struction, therefore the better estimation of states, the betterperformance in the control of the system. By using the linearobservers for state estimation, system encounters with stateobservation errors in the operating conditions that are far fromthe main operating point. These errors can be damped byadding an integrator to the feedback linearization control law.The presented results are comparable with relevant works onthis setup. This work can be followed and the results canbe improved by using a Nonlinear observer to estimate thenonlinear states.

ACKNOWLEDGMENT

The authors would like to thank Dr. Safari-shad and Dr.Abedi for their helps and attempts to prepare the laboratoryhelicopter setup for the digital control lab at the K.N.ToosiUniversity of Technology where this research has been per-formed.

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