Abstract— This paper presents, a state observer based

controller for the twin rotor multiple-input-multiple-output

(MIMO) system. The twin rotor MIMO system (TRMS)

belongs to a class of nonlinear system having high coupling

effect between two propellers, unstable and nonlinear

dynamics. A state observer is designed using coordinate change

which transforms the TRMS into an approximate normal form.

Based on the proposed observer, a feedback linearization

controller is designed for TRMS. The control effort is further

compensated using a compensator based on Chebyshev neural

network (CNN) to ensure good tracking performance and

bounded control effort. Finally simulation results are presented

to illustrate the effectiveness of the proposed observer based

controller. Keywords— Approximate normal form, Chebyshev neural

network, nonlinear system, state observer, twin rotor MIMO

system.

I. INTRODUCTION

The nonlinear control problems are tackled easily using well known feedback linearization control technique. Feedback linearization uses a nonlinear transformation to transform an original nonlinear dynamic model into a linear model by diffeomorphism mapping [1]. Feedback lineari-zation approach has been applied successfully to address many control applications [2]-[5]. Neural network (NN) based feedback linearization has been proposed in [6], [7].

The TRMS is a laboratory set up designed for control experiments [8]. The modeling and control of the TRMS has gained a lot of attention because the dynamics of the TRMS and a helicopter are similar in certain aspects [9]-[12]. Due to the high coupling effect between two propellers, unstable and nonlinear dynamics, the control problem of the TRMS has been considered as a challenging research topic. Several classical as well as intelligent controllers for TRMS have been proposed in the literature [13]-[16]. Feedback linearization controller has been proposed for TRMS in [17]-[18]. However the simplified model of TRMS has been utilized for designing the controller. Moreover all state variables are assumed to be measurable which is generally not practically feasible. One solution is design of observer.

The problem of designing an observer for nonlinear systems has been recognized as an important and challenging one. One way to solve the observer synthesis problem is based on the transformation of a system into the simpler form via change of coordinates [19], [20]. In particular, it is shown in [19] that, under some Lipschitz conditions, there exists a state observer for a nonlinear system if it has relative degree n . However, if the relative

Bhanu Pratap and Shubhi Purwar are with the Department of Electrical

Engineering, M. N. National Institute of Technology, Allahabad, India (e-

mail: bhanumnnit@gmail.com, spurwar@rediffmail.com).

degree of the system is not well defined, such as the ball and beam system [21], then the technique of [19] is no longer adequate. As far as the control problem is concerned, when the system has failed to have well-defined relative degree, the notion of “robust relative degree” is introduced in [22] to solve the local control problem.

A Chebyshev neural network (CNN) proposed in [23] can be used for function approximation and pattern classification with faster convergence and lesser computational complexity than an MLP network. A dynamic nonlinear system identification methodology using CNN is reported in [24].

In this paper, a state observer based feedback linearization controller for TRMS having robust relative degree is presented. The proposed observer based controller utilizes the coordinate change which transforms a given system into an approximate normal form [21] and guarantees the local exponential convergence of the state estimates into the true states. Feedback linearization control is applied to design a controller based on the proposed observer. A CNN-based scheme for saturation control for TRMS is also presented. The approach is applied to the nonlinear plant with a general model of actuator saturation which does not necessitate exact knowledge of the actuator output. The simulation results obtained reveal that the proposed control strategy gives good tracking performance.

The remainder of the paper is arranged as follows. In Section II, the preliminaries comprising of TRMS and feedback linearization control is introduced. The problem statement is introduced in the section III. Robust feedback linearization controller with compensator using CNN is designed in Section IV. The local state observer is demonstrated in Section V. Section VI validates the performance of the proposed observer based controller through simulation results. Finally conclusions are given in the Section VII.

II. PRELIMINARIES

A. Twin Rotor MIMO System (TRMS):

The TRMS mechanical unit has two rotors placed on a beam together with a counterbalance whose arm with a weight at its end is fixed to the beam at the pivot and it determines a stable equilibrium position as shown in the Fig. 1 [1]. The beam is pivoted on its base in such a way that it can rotate freely both in the horizontal and vertical planes. At both ends of the beam there are rotors (the main and tail rotors) driven by dc motors. The main rotor produces a lifting force allowing the beam to rise vertically making a rotation around the pitch axis. While, the tail rotor is used to make the beam turn left or right around the yaw axis. The whole unit is attached to the tower allowing for safe helicopter control experiments. The system parameters of

State Observer Based Robust Feedback Linearization Controller for

Twin Rotor MIMO System

Bhanu Pratap and Shubhi Purwar

2012 IEEE International Conference on Control Applications (CCA)Part of 2012 IEEE Multi-Conference on Systems and ControlOctober 3-5, 2012. Dubrovnik, Croatia

978-1-4673-4505-7/12/$31.00 ©2012 IEEE 1074

the TRMS are taken from [12].

Fig. 1. The twin rotor MIMO system

B. Feedback linearization Control:

Feedback linearization has been used successfully to address some practical control problems. It includes the control of helicopters, high performance aircraft, industrial robots, and biomedical devices. The basic idea of feedback linearization approach is to use a control consisting two components: one component cancels out the plant nonlinearities and other controls the resulting linear systems. The method is limited to a class of dynamical systems, plants, whose models are sufficiently smooth; i.e. the plant whose right hand sides of the modeling differential equations are sufficiently many times differentiable.

Consider a MIMO nonlinear system described by the state equation

( ) ( )

( )1

, 1, 2, ,

m

i i

i

i i

x f x g x u

y h x i m

=

= +

= =

∑ɺ

…

(1)

where nx ∈ℜ is state vector, m

u ∈ ℜ is input vector and m

y ∈ℜ is output vector. The objective of feedback

linearization is to create a linear differential relation between the output y and a newly defined input v .

The system is said to have a vector relative degree

[ ]1 1, , ,

mr r r⋯ at a point

0x if [1]

1) ( ) 0j

k

g f iL L h x = for all 1 j m≤ ≤ , all 1 i m≤ ≤ , 1

ik r< − ,

and for all x in the neighborhood of 0

x ;

2) m m× matrix

( )

( ) ( )

( ) ( )

1 1

1

1

1 1

1

1 1

m

m m

m

r r

g f g f i

r r

g f m g f m

L L h x L L h x

A x

L L h x L L h x

− −

− −

=

⋯

⋮ ⋮

⋯

(2)

is nonsingular at 0

x x= , which is called as a decoupling

matrix. Based on the defined relative degree, the control law of a

MIMO nonlinear system is defined as

( ) ( ) ( )1u A x b x v x

− = − + (3)

where ( ) ( ) ( )1

1m

Trr

f f mb x L h x L h x = ⋯ (4)

( ) [ ] ( ) ( )1

1 1m

TT rr

m mv x v v y y = = ⋯ ⋯ (5)

Note that the control law in (3) transforms the nonlinear system into a linear one in which the aforementioned input–

output relation is linearized and decoupled.

III. PROBLEM STATEMENT

The complete dynamics of TRMS [8] represented in state space form is as follows

( ) dis

x f x gu

y hx

τ= + +

=

ɺ (6)

where the state vector [ ]1 2

Tx ψ ψ ϕ ϕ τ τ= ɺ ɺ , input

vector [ ]1 2

Tu u u= , output vector [ ]

Ty ψ ϕ= , 6

disτ ∈ ℜ

is disturbances in TRMS and ( )f x , g , C are given by

( )

( ) ( )

( ) ( )

( )

2 21 1

1 1

1 1 1 1

1 2

1 1 1 1

1 1

12 22 2

2 2 1 1 1 1

2 2 2 2

10

1

11

20

2

21

20.0326

sin sin2

cos

1.75

g

gy

c

Ma b

I I I I

kBa b

I I

f xBa b

k a bI I I I

T

T

T

T

ψ

ϕ

ψ

τ τ ψ ψ ϕ

ψ ψ ϕ τ τ

ϕ

τ τ ϕ τ τ

τ

τ

+ − + − − +

=

+ − − +

− −

ɺ

ɺ

ɺ ɺ

ɺ

ɺ

1

11

2

21

0 0 0 0 0

0 0 0 0 0

Tk

Tg

k

T

=

, 0 0 0 0 0

0 0 0 0 0

1

1h

=

.

where ψ = pitch (elevation) angle, ϕ = yaw (azimuth)

angle, 1

τ = momentum of main rotor, 2

τ = momentum of

tail rotor, 1

u = voltage applied to main rotor, 2

u = voltage

applied to tail rotor, . The objective of this paper is to design a state observer

based feedback linearization controller for TRMS, that

forces the plant output [ ]T

y ψ ϕ= to track a specified

smooth reference trajectory 1 2

T

d d dy y y = i.e.,

( )lim 0d

ty y

→∞− = , subjected to the constraint 2.5

iu ≤ for

1, 2i = .

IV. FEEDBACK LINEARIZATION CONTROLLER FOR TRMS

WITH COMPENSATOR

A. Saturation Nonlinearity:

In control engineering, the most commonly used actuators are continuous drive devices [25]. Saturation nonlinearity with its maximum and minimum operation limits is unavoidable. In this section a feedback linearization controller is designed with a CNN based compensator.

( )6nequ u y

Fig. 2. TRMS with actuator saturation

1075

Assuming ideal saturation {see Fig. 2}, mathematically,

the output of the actuator ( )u t is given by

( )( )

( ) ( )( )

2.5 2.5

2.5 2.5

2.5 2.5

if u t

u t u t if u t

if u t

>

= − ≤ ≤− < −

(7)

If the control signal falls outside the range of the actuator, actuator saturation occurs and the control input can’t be fully implemented by the device. The control that can’t be

implemented by the actuator, denoted as ( )tδ is given by

( ) ( ) ( )

( ) ( )

( )

( ) ( )

2.5 2.5

0 2.5 2.5

2.5 2.5

u t if u t

t u t u t if u t

u t if u t

δ

− >

= − = − ≤ ≤

− − < −

(8)

In this paper, ( )tδ is assumed to be unknown and is

estimated by CNN.

B. Controller Design:

Consider [ ]2T T

f fz h L h L h y y y = = ɺ ɺɺ , the system

(6) is defined in normal form as [1]

( ) ( ){ }T

z y y b x A x u = + ɺ ɺɺɺ (9)

and the desired state vector [ ]T

d d d dz y y y= ɺ ɺɺ , the

tracking error vector given as

T

dz z z y y y = − =

ɺ ɺɺɶ ɶ ɶɶ (10)

where d

y y y= −ɶ .

Now the filtered tracking error vector 2 1e

×∈ ℜ is given by [25]

Te K z= ɶ (11)

where [ ]1 2 2

TK K K I= is appropriately chosen

coefficient vector with 2 2

iK

×∈ℜ , for 1, 2i = and 2

I is

identity matrix, so that 0z →ɶ exponentially as 0e → . The time derivative of (11) in terms of the filtered

tracking error given as

( ) ( ) de b x A x u Y= + +ɺ (12)

where 1 2d d

Y y K y K y= − + +ɺ ɺɺɺɺɺ ɶ ɶ is a known function.

Using (8), (12) becomes,

( ) ( )( ) de b x A x u Yδ= + + +ɺ (13)

As the relative degree vector of the plant given by (6) is

{ }3, 3 , the feedback linearization control law is given as [1]

( ) ( )1

3eq d ru A x b x Y u e K e

− = − − + − (14)

where 2 2

3K

×∈ ℜ . A robust term r

u is added for the

rejection of plant uncertainty [26] is given by

( )( )

/ ,

/ ,r

e eu

e

γ µ

γ µ µ

− ≥=

− < (15)

where 0γ > and 0 1µ< < . Since the control variables

show up after the third derivative of outputs, the relative

degree vector is { }3, 3 , thus the decoupling matrix ( )A x is

defined as

( ) 1 2

1 2

2 2

1 1

2 2

2 2

g f g f

g f g f

L L h L L hA x

L L h L L h

=

(16)

where

( ) ( )1

2 1 1 11 5 1 4 1 5 1

11 1 1 1

2 2cosgy

g f

kk a bL L h x x x a x b

T I I I

= + − +

(17)

2

2

10

g fL L h = (18)

( )1

2 12 1 5 1

11 2

21.75 c

g f

kkL L h a x b

T I

= − +

(19)

2

2 2 2 22 6

21 2 2

2g f

k a bL L h x

T I I

= +

(20)

Additionally, the matrices ( )b x is given by

( ) 3 3

1 2

T

f fb x L h L h = (21)

where ( ) ( ) ( )3 2

1 1 4 1 5 1 5 1

1 1

sin cosgy g

f

k ML h x x a x b x x

I I

= + −

12 21 11 4 2 5 5 1

1 1 1 1 1

( )0.0326

2 cos 2 sin2

gMB a b

x x x x x xI I I I I

ψ + − + −

( ) ( )12 2

1 4 2 1 4 1 5 1 5

1 1 1

( )0.0326

sin 2 cos2

gykB

x x x x x a x b xI I I

ψ + − − +

( )( )2 221 4 1 1 5 1 5 6

1 1 2

2 ( )0.0326

sin 2 cos2

gyk a

x x x a x b x xI I I

+ − +

( )1 22 1 16 4 1 5 1 5 5

2 2 2 1 1

21.75

c

Bb a bx x k a x b x x

I I I I I

ϕ + − − + − +

( ) ( ) 10

1 4 1 5 1 5

1 11

2cosgy

k Tx x a x b x

I T

− +

(22)

( )1 13 2 22 22 6 6 4 1 5 1 5

2 2 2 2 2

1.75 c

f

B B ka bL h x x x a x b x

I I I I I

ϕ ϕ = − + − − +

( ) 10 202 21 5 1 5 6 6

2 11 2 2 21

2 21.75 ck T Ta b

a x b x x xI T I I T

+ + − +

(23)

The control signal u is composed of the tracking

controller with the saturation compensator given by

ˆeq

u u δ= − (24)

where δ is estimates of unknown function δ .

C. Compensator Design:

Using CNN approximation property, the nonlinear

functions δ and δ can be represented as

( ),T

dW y yδ φ ε= +ɶ (25)

where ε is the CNN approximation error bounded by C

ε ,

W is the optimal weight and φ is the basis function.

( )ˆ ˆ ,T

dW y yδ φ= ɶ (26)

where W is the estimates of the W .

Using (13), (14) and (24), the error dynamics of the closed

loop system is

1076

( ) 3ˆ

re u e K eδ δ= − + −ɺ (27)

Again using (25) and (26), (27) becomes

Te W Keφ ε= − +ɶɺ (28)

where 3 r

K K u= − and ˆW W W= −ɶ is the CNN weights

approximation error.

D. Stability Analysis:

Assumption 1: The desired trajectory d

y and its

derivatives up to third order are bounded.

Assumption 2: The optimal weight W is bounded by

known positive values C

W , so that CF

W W≤ .

Theorem 1: Consider the TRMS plant (6), controller (24) satisfying Assumptions 1 and 2. If the weights of the CNN are updated according to adaptation law

ˆ ˆTW e e Wηφ ρη= −ɺ

(29)

then the error e and CNN weight estimation error Wɶ are uniformly ultimately bounded.

Proof: Consider the Lyapunov function

{ }11 1

2 2

T TV e e tr W Wη−= + ɶ ɶ (30)

Using (25)-(30), Vɺ is negative as long as

( )2

min/ 2 /

C CW eρ ε λ + <

(31)

or, ( )2

/ 2 / 2 /C C C

W W Wε ρ β+ + < ɶ (32)

Thus, Vɺ is negative outside a compact set. The detailed stability analysis is not presented here as it is beyond the scope of this paper. To take into account the limitation of bounded control effort a compensator using CNN is designed. The local state observer design has been presented in the following section.

V. LOCAL STATE OBSERVER FOR TRMS

The TRMS given in (6) has a robust relative degree

{ }3, 3 about 0x = , if there exist smooth function

( ), 1, 2, 3i

x iσ = , such that [21],

( ) ( ) ( )1 0,h x x x uφ σ= + (33)

( ) ( ) ( )1, 1, 2

f gu i i iL x x x u iφ φ σ+ += + = (34)

( ) ( ) ( ) ( )3 3,

f guL x b x a x u x uφ σ+ = + + (35)

where the functions ( ), , 1, 2, 3i

x u iσ = are ( )2

,O x u and

( )a x is ( )0

O x and define

( ) ( ) ( )1 6, ,

T

x x xφ φΦ = ⋯

Theorem 1. The TRMS given in (6) has a relative degree

{ }3, 3 , the local state observer is given by [21]

( )

( )[ ]

1ˆ

ˆ ˆ ˆˆ

ˆ ˆ

xx f x gu L y y

x

y Cx

−∂Φ

= + + − ∂

=

ɺ

(36)

where the matrix 6 2L

×∈ℜ is selected so that the solution of (36) satisfies the following condition

( ) ( ) ( ) ( )ˆ ˆ 0 0 0t

x t x t x x e tαβ −− ≤ − ∀ ≥ (37)

Given 0α > there exist 1

0δ > , 2

0δ > , 3

0δ > , 0β > , and

provided that ( ) 1x t δ< , ( ) 2

u t δ< 0t∀ ≥ and

( ) ( ) 3ˆ 0 0x x δ− < .

It follows from [21] that there exists a neighborhood U

of the origin such that ( )xΦ is a diffeomorphism on U ,

can be chosen as

( ) ( )

1

3

2

4

11

5 1 2 1 1 4 5

1 1 1 1

126 4 1 5

2 2 2

( )

sin cos

1.75

g gy

c

x

x

x

xx

M kBbx x x b x x x

I I I I

Bbx x k b x

I I I

ψ

ϕ

Φ = − − − − −

(38) As a result, we have

( ) ( )2 2 21

5 1 4 1 1 4 5

1 1 1

2 2126 5

2 2

0

0

0

0( , )

0.0326sin 2 cos

2

1.75

gy

c

x uka

x x x a x x xI I I

k aax x

I I

σ

= + − −

(39) This process finally yields a TRMS observer in the form of (23)

where ( )

1

1 2 3 4

5 6 7 8

1 0 0 0 0 0

0 0 1 0 0 0

0 1 0 0 0 0ˆ

0 0 0 1 0 0ˆ

0 0

20

9

x

xd d d d

d d d d

−

∂Φ = ∂

(40)

and , , , , , , a b c d e f g and h are the nonlinear functions

given by

( ) ( )

( )1 1 4 5

1

1 4

ˆ ˆ ˆ ˆ

ˆ ˆ

1600cos 231sin1

231 20 cos

x x x xd

x x

− =

− ;

( )2

1 4ˆ ˆ

100 1

77 20 cosd

x x

=

− ;

( )

( )1 5

3

1 4

ˆ ˆ

ˆ ˆ

cos

20 cos

x xd

x x=

−;

( )4

1 4ˆ ˆ

3400 1

231 20 cosd

x x

=

− ;

1077

( ) ( )

( )1 1 4 5

5

1 4

ˆ ˆ ˆ ˆ

ˆ ˆ

16000cos 231sin14

9000 20 cos

x x x xd

x x

−− =

− ;

( )6

1 4ˆ ˆ

14 1

30 20 cosd

x x

=

−

−;

( ) ( )

( )1 5 1 4

7

1 4

ˆ ˆ ˆ ˆ

ˆ ˆ

0.258cos 1.11cos 22.2

20 cos

x x x xd

x x

− + −=

−;

( )8

1 4ˆ ˆ

23817 1

45 20 cosd

x x

=

− .

The local state observer for TRMS is very simple to design. The proposed observer based controller is validated through simulation results in next section.

VI. SIMULATION RESULTS

The final actuator output ( )u t is obtained by applying

(7) to (24) as shown in Fig. 3.

( )6neq

u u y

−+

equ

dy

δ

y

dy

x

, , ,d d d dy y y yɺ ɺɺ ɺɺɺ

( )14n

eq

( )26neq ( )36n

eq

yɶ

+

+−

−

yɶ

Fig. 3. Block diagram of observer based controller with compensator

A detailed simulation study of state observer based robust feedback linearization controller is carried out. Simulation results show reliable performance.

The desired trajectories for main rotor and tail rotor are chosen as

( ) ( ){ }1 20.13 sin 0.0225 0.5 sin 0.0675

d dy y t tπ π π= = − +

The initial conditions of the plant and observer are

[ ]0.02 0 0.02 0 0 0 and [ ]0 0 0 0 0 0

respectively. The tuning of the CNN weights is done online.

Also we consider presence of some disturbances d in the system (6) as

[ ]0.0043 0.0167 0.0013 0.0029 0.0115 0.0119T

disτ = − − −

For faster convergence of error to zero, observer gain L is chosen as

100 10 10 10 10 10

10 100 10 10 10 10

T

L

=

The design parameters of observer based controller are chosen as

{ }130K diag= , { }2

5K diag= , { }350K diag= ,

0.5µ = , 5γ = , 0.1η = and 0.5ρ = .

Fig. 4. Pitch angle tracking (

1x and

1dy )

Fig. 5. Yaw angle tracking (

3x and

2dy )

Fig. 6. Pitch angle tracking error (

1 1dx y− )

Fig. 7. Yaw angle tracking error (

3 2dx y− )

Fig. 8. Control effort without compensator ( 1equ )

Fig. 9. Control effort without compensator ( 2equ )

Fig. 10. Control effort with compensator (

1u )

Fig. 11. Control effort without compensator (

2u )

0 20 40 60 80 100-0.5

0

0.5

Time (sec)

Pitch T

rackin

g (

rad)

y

1d

x1

0 20 40 60 80 100-0.5

0

0.5

Time (sec)

Yaw

Tra

ckin

g (

rad)

y

2d

x3

0 20 40 60 80 100

0

0.05

0.1

0.15

Time (sec)

Pitch E

rror

(rad)

0 20 40 60 80 100-0.05

0

0.05

0.1

0.15

Time (sec)

Yaw

Err

or

(rad)

0 20 40 60 80 100

-40

-20

0

Time (sec)

Contr

ol E

ffort

u1

eq (

volt)

0 20 40 60 80 100-15

-10

-5

0

Time (sec)

Contr

ol E

ffort

u2e

q (

volt)

0 20 40 60 80 100

-2

0

2

Time (sec)

Actu

ato

r O

utp

ut

u1

0 20 40 60 80 100

-2

0

2

Time (sec)

Actu

ato

r O

utp

ut

u2

1078

The initial weights of the neural network are selected as

zeroes. The inputs to CNN are d

y and yɶ . Fig. 4 and 5

shows that the pitch and yaw angles closely follow the

desired trajectories 1d

y and 2d

y . The tracking errors for

pitch and yaw angles are shown in Fig. 6 and 7 respectively. In addition, Fig. 8 and 9, indicate that the control efforts

without compensator 1eq

u and 2eq

u . Fig. 10 and 11 show

that control efforts with compensator 1

u and 2

u are within

the limits 2.5± as stated in section III. These simulation results show that the tracking performances of the proposed observer based controller is quite satisfactory.

VII. CONCLUSION

A state observer based feedback linearization controller for the TRMS is presented in this paper. Here, the system nonlinearities need to be known. The controller does not require a saturation model to be known. After initial time period, the observer based controller learns saturation nonlinearity and adjusts its weights to prevent the control signal from being saturated. Finally, simulation studies are presented to illustrate the effectiveness of the proposed observer based control. To test the applicability of the local State observer based feedback linearization controller in real time is the proposed future scope of work. For real time implementation, the experiments have to be carried out on the real-time TRMS system using MATLAB real-time tool box and Advantech 1711PCI card.

VIII. ACKNOWLEDGEMENT

The authors acknowledge the contribution of Department of Science and Technology, Government of India, New

Delhi, India through Project / 3 / / 004 / 2008SR S EECE .

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