 Open Access
 Total Downloads : 235
 Authors : Kumar Manu, Ekta Agarwal, Mansi Vashisht
 Paper ID : IJERTV4IS080299
 Volume & Issue : Volume 04, Issue 08 (August 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS080299
 Published (First Online): 18082015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
DiscreteTime Chebyshev Neural Observer for Twin Rotor MIMO System
Kumar Manu Department of Electronics and Communication Engineering
Moradabad Institute of Technology Moradabad, India
Ekta Agrawal Department of Electronics and Communication Engineering
Moradabad Institute of Technology Moradabad, India
Mansi Vashisht Department of Electronics and Communication Engineering
Moradabad Institute of Technology Moradabad, India
Abstract– In this paper, we investigate the problem of designing a Neural Network (NN) observer for the Euler discretized model of Twin Rotor MultiInputMultiOutput (MIMO) system which belongs to a class of nonlinear system. The observer is based on a Chebyshev Neural Network (CNN), trained by using Extended Kalman Filter (EKF) learning algorithm. The state estimation error and output error are guaranteed to be semiglobally uniformly ultimately bounded (SGUUB) and neural network weights to be bounded. Simulation results are also included to illustrate the applicability of the proposed observer.
Keywords– Chebyshev Neural Network, DiscreteTime Nonlinear System, Extended Kalman Filtering, Neural Observer, Twin Rotor MIMO System.

INTRODUCTION
During the past four decades, nonlinear state estimation has been a very important topic for nonlinear control. The concept of an observer for a dynamic process was introduced by D. Luenberger [1]. The generic Luenberger Observer, however, appeared several years after the Kalman Filter [2], which infact an important case of a Luenberger Observer an observer optimized for the noise present in the observation and in the input to the process. Furthermore, state estimation has been studied by many authors, who have obtained interesting results in different directions [3][6]. Most of the approaches need the previous knowledge of the plant dynamics. Recently, neural observers [7][8] has emerged for unknown plant dynamics. Now a days neural networks are very important methodology for solving some very difficult
This paper presents the following main contributions: In Section II the TRMS system is introduced and its discrete model is obtained. In Section III, CNN structure is given. The EKF training algorithm is given in Section IV. The proposed CNN observer is introduced in Section V. The observer performance is demonstrated in Section VI by providing simulation results. Finally concluding remarks are made in the last section.

2DOF TRMS MODEL
The mechanical setup of the twin rotor MIMO system is shown in Fig.1.

ContinuousTime Model
The complete dynamics of the TRMS can be approximately represented in the state space form as follows [9]
d
dt
d a1 2 b1 M g sin 0.0326 sin(2)2
dt I 1 I 1 I 2I
1 1 1 1
1 1 1 1
B K K
1 gy a cos()2 gy b cos()
I I I
problems in engineering, as exemplified by their applications in control nonlinear and complex systems.
In this paper, we develop a Luenberger – like observer for the Euler discretized model of Twin Rotor MultiInputMulti
d
dt
1 1 1
(1)
Output (MIMO) system (TRMS) [9] which belongs to a class of nonlinear system. The observer is based on a Chebyshev
d a2 2 b2
B
1
1.75 k a 2 1.75
k b
Neural Network (CNN) [10][12], which estimates the state
dt I
2 I 2 I I
c 1 1
I c 1 1
vector of the unknown plant dynamics. In CNN, for functional d
2 2 2 2 2
T k
expansion of the input pattern, we have chosen the 10 1 u
T T
Chebyshev polynomials and the network is also named as dt 1
ChebyshevFunctional Link Artificial Neural Network d
(CFLANN) [12]. The learning algorithm for the CNN is
1 1
11 11
T k
20 2 u
based on an extended Kalman filter (EKF) [7][8], [13], [18].
dt 2
T 2 T 2
With the EKF based algorithm, the learning convergence is improved as compared to other previously used algorithms [13]. The state estimation error and output error are guaranteed to be semiglobally uniformly ultimately bounded (SGUUB) and neural network weights to be bounded [7].
The outp21ut is giv21en by
y T
where
: Pitch (Elevation) Angle, : Yaw (Azimuth) Angle,
(2)
1 : Momentum of Main Rotor,
0 1 0 0 0 0
2 : Momentum of Tail Rotor,
u1 and u2 : Inputs,
B
I I
0 1 0 0
b
1 0
and, the system parameters of the TRMS are given in Table
I. 1 1
TABLE I. TRMS PARAMETERS
0 0 0 1 0 0
Parameters
Values
I1 Moment of Inertia of Vertical Rotor
6.8×102 kgm2
I2 Moment of Inertia of Horizontal Rotor
2×102 kgm2
a1 Static Characteristic Parameter
0.0135
b1 Static Characteristic Parameter
0.0924
a2 Static Characteristic Parameter
0.02
b2 Static Characteristic Parameter
0.09
Mg Gravity Momentum
0.32 Nm
B1 Friction Momentum Function Parameter
6×103 Nms/rad
B1 Friction Momentum Function Parameter
1×101 Nms/rad
Kgy Gyroscopic Momentum Parameter
0.05 s/rad
k1 Motor 1 Gain
1.1
k2 Motor 2 Gain
0.8
T11 Motor 1 Denominator Parameter
1.1
T10 Motor 1 Denominator Parameter
1
T21 Motor 2 Denominator Parameter
1
T20 Motor 2 Denominator Parameter
1
Tp Cross Section Momentum Parameter
2
T0 Cross Section Momentum Parameter
3.5
kc Cross Reaction Momentum Gain
0.2
B k b b
A 0 0 0
1
1.75
c 1 2 ;
I I I
2 2 2
T
0 0 0 0
10 0
T
11
T
0 0 0 0 0
0 0
0 0
0 0
0 0
20
T21 ( 6 x 6 )
B k ;
1 0
T
11
k
T
0 2
21
( 6 x 2 )
1 0 0 0 0 0
C 0 0 1 0 0 0
and
;
( 2 x 6 )
a M
0
0.0326
{ 1 2 g sin sin(2)2
I 1 I 2I
1 1 1
K K
gy a cos()2 gy b cos() }
I 1 1 I 1 1
f ( x) 1 1
0
2
I
a2 2
1.75
k a 2
Figure 1. The Twin Rotor MIMO System (TRMS).
I2
c 1 1
0
Consider the state space form
x Ax f (x) Bu
0
(3)
( 6 x1)
y Cx
where
where x , u , y , A x, B x , C x and p m. The function f (x) can be constructed as uncertainties or nonlinearities in plant.
For the state space representation of TRMS by (3)
x T
1 2 ( 6 x1)
For existence purpose, one requires that f (x) be continuous in x.

DiscreteTime Model
The discretetime model of TRMS, obtained using Euler forward discretization method, is given by
x(k 1) Fx(k ) g(x(k )) Gu(k )
x1 (k )
y(k ) Cx(k )
where
k : Sampling Step,
() [( )] ,
(4)
x2 (k )
wi
gi ( x(k ))
x (k )
( + 1) [( + 1)] ,
() [()] ,
F= 6 + ,
G= ,
g(x(k)) = () ,
Ts : Sampling Time,
and, I6 is a 6×6 identity matrix.
n
Figure 2. CNN Structure.
A smooth nonlinear function approximated by CNN [14]
i i i
g (x(k)) w*T (x(k))
Function Expansion
g(x(k))
can be
(7)


CNN STRUCTURE Consider a discretetime MIMO nonlinear system
x(k 1) Fx(k ) g(x(k )) Gu(k )
y(k ) Cx(k )
(5)
where gi (x(k )) is the nonlinear function in ith plant state, i
is the bounded approximation error [14].
i
Assume that there exists an ideal weight vector w* such
that i can be minimized on a compact set i . The
where () , () , () , F x , G x, C x and p m. The function g(x(k)) is a
nonlinear function, which is unknown with entries gi (.)
weight estimation error is defined as
i i i
w (k) w* w (k) (8)
where w* is the ideal weight vector and w its estimate. Since
(i=1, , n).
Here, a single layer CNN is used to approximate the unknown nonlinear function. The output of the CNN is given by [14]
g (x(k)) wT (x(k)) (i 1,……., n) (6)
i i
i
w* is constant, one has
wi (k 1) wi (k) wi (k) wi (k 1)
i i IV. EKF TRAINING ALGORITHM
where
gi (x(k )) is the output of ith neuron, n is the output
In this paper we use the following modified EKF based
i
dimension, wi is the respective online adapted weight vector, given by
training algorithm, for designing the CNN observer [7][8], [13], [18]
wi wi1
wi 2 …………
wiL
wi (k 1) wi (k ) Ki (k )e(k )
i 1, …….., n
(9)
where
Li is the respective number of higher order
Ki (k ) Pi (k )Hi (k )M i (k ) (10)
connections, and
(x(k)) is the basis function which is
P (k 1) P (k ) K (k )H T (k )P (k ) Q (k ) (11)
i i i i i i
formed using Chebyshev polynomials.
The Chebyshev polynomials can be generated by the following recursive formula
with
M (k ) [R (k ) H T (k )P (k )H (k )]1
(12)
i i i i i
0
Tr 1 (x) 2xTr (x) Tr 1 (x),
T (x) 1
e(k ) y(k ) y(k ) (13)
where Tr (x) is a Chebyshev polynomial, r is the order of polynomials chosen and x is a scalar quantity.
The following theorem states the function approximation
capability of CNN
Theorem 1: Assume a feed forward MLP neural network with only one hidden layer and linear activation functions of the output layer. If all the activation functions of the hidden layer satisfy the Riemann integrable condition, then the feed forward neural network can always be represented as a Chebyshev neural network. [11]
The CNN structure is shown in Fig. 2. The basis function
where () is the output error and () x is the weight estimation error covariance matrix at step k, () is the weight state vector, () is the plant output,
() is the neural observer output, n is the number of plant states, () x is the Kalman gain matrix, Qi(k) x is the NN weight estimation noise covariance matrix,
() x is the error noise covariance matrix and
() x is a matrix, in which each entry Hij is the derivative of the ith neural output with respect to ijth neural network weight given as follows
y(k ) T
is given as
Hij (k )
(14)
(x(k)) 1
Tr (x1 (k))
Tr (x2 (k)) ………….
Tr (xn (k))
wij (k )
In this paper, the order of r is taken as 2. where j=1,., Li and i=1,., n. The ijth NN weight
is selected according to the output then Hij is given as
y(k )
. If
y(k) xn (k) ,
y(k ) T
x (k 1) w (k )( x(k )) * J x(k ) (19)
Hij (k ) w
(k )
with
nj
i i i i
where wnj is the njth NN weight.
Pi(k), Qi(k) and Ri(k) matrices are initialized as diagonal matrices with entries Pi(0), Qi(0) and Ri(0) respectively. It is to be noted that Pi(k), Qi(k) and Ri(k) for EKF are bounded [15].

NEURAL OBSERVER DESIGN USING CNN Consider an observable discretetime system given by (5).
Equation (5) can be be rewritten as
x(k ) [x (k )………..x (k )……….x (k )]T
1 i n
Ji (Fi DiC)
i i i
w w* w (k )
i i i
* w (k )[x(k ) x(k )]
i
where * is a bounded error term [16].
The dynamics of (8) is given by
wi (k 1) wi (k) Ki (k)e(k) (20)
The main result is establish as the following theorem
Theorem 2: For system (15), the nonlinear observer (16) trained with the EKFbased algorithm (9)(14), ensures that
xi (k 1) Fi x(k ) gi (x(k )) Giu(k )
(15)
the output error (17) and the estimation error (18) are semi
y(k ) Cx(k )
globally uniformly ultimately bounded (SGUUB). [7]
Proof: Refer to [7].
For the system (15) the proposed CNN observer, shown in Fig. 3, is given as
x(k ) [x (k )………..x (k)……….x (k)]T

SIMULATION RESULTS
A detailed simulation study of the proposed observer is carried out with the inputs as
1 i n
u (k) u (k) 0.2 sin(0.5kT ) 0.3sin(0.6kT ) . The
i i i i i
x (k 1) F x(k ) wT (x(k )) G u(k ) D e(k )
(16) 1 2 s s
y(k ) Cx(k )
i 1,………, n
sampling time Ts is .001s. All initial values of states are set to zero. All the NN weights are initialized as zero. The
where Di 1x, Fi 1x and Gi 1x.
covariance matrices are initialized as diagonals and the non
zero elements are Pi(0)=100, Qi(0)=.000001 and Ri(0)=100 respectively (i=1,,n).
u(k)
w(k)
x(k)
G
TRMS MODEL
with Unknown Nonlinearities
wT (x(k))
C
1/Z
D
y(k)

e(k)
The simulation results are shown in Fig. 4 and Fig. 5. Fig. 4 shows the actual states and observed states respectively. Fig. 5 shows the state estimation errors.
0.2
Actual Pitch Angle
CNN +
x(k 1)
x(k)
C
y(k)
0.15
Pitch Angle (rad)
0.1
Observed Pitch Angle
F
0.05
0
EKF
0.05
CNN Observer
Figure 3. CNN Observer Scheme.
0.1
x
0.15
0 0.5 1 1.5 2 2.5
The observer gain matrix D
is chosen such that
Sampling Step (k)
4
x 10
FDC is convergent. (Note that a matrix P is called convergent if all the eigen values of P lie inside the open unit circle in complex plane.). The weight vectors are updated online with a modified EKF algorithm (9)(14). The output error is defined by
e(k) y(k) y(k) (17)
and the state estimation error as
x(k) x(k) x(k) (18)
The dynamcs of (18) can be given as
2
1.8
1.6
Yaw Angle (rad)
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Actual Yaw Angle
Observed Yaw Angle
0 0.5 1 1.5 2 2.5
Sampling Step (k)
4
x 10
Figure 4. Actual and Observed States of TRMS with CNN Observer.
9
x 10
1
Pitch Angle Error (rad)
0.5
0
0.5
1
1.5
0 0.5 1 1.5 2 2.5
REFERENCES

David G. Luenberger, Observing the State of a Linear System, IEEE Transactions on Military Electronics, vol. 8, no. 2, pp. 7480, April, 1964.

R. E. Kalman, "On the General Theory of Control Systems," Proc. of the First IFAC Moscow Congress, 1960.

S. Benamorl , H. Hammouri and F. Couenne, A LuenbergerLike Observer for DiscreteTime Nonlinear Systems, Proceedings of the 37th IEEE Conference on Decision & Control, Tampa, Florida, USA, December, 1998.

V. Sundarapandian, Observer Design for DiscreteTime Nonlinear Systems, Mathematical and Computer Modelling, vol. 35, no. 12, pp. 3744, January, 2002.

Tarek AhmedAli, Romain Postoyan, FranÃ§oise LamnabhiLagarrigue,
2.5
2
Yaw Angle Error (rad)
1.5
1
0.5
0
0.5
10
x 10
Sampling Step (k)
4
x 10
ContinuousDiscrete Adaptive Observers for State Affine Systems, Automatica, vol. 45, no. 12, pp. 29862990, December, 2009.

Rui Huanga, Sachin C. Patwardhan, Lorenz T. Bieglera, Stability of a Class of DiscreteTime Nonlinear Recursive Observers, Journal of Process Control, vol. 20, no. 10, pp. 11501160, December, 2010.

Alma Y. Alanis, Edgar N. Sanchez, Alexander G. Loukianov, and Marco A. Perez, RealTime Recurrent Neural State Estimation, IEEE Transactions on Neural Networks, vol. 22, no. 3, pp. 497505, March, 2011.

Alma Y. Alanis, Edgar N. Sanchez and Alexander G. Loukianov, DiscreteTime Nonlinear Recurrent High Order Neural Observer, Proceedings of the 2006 IEEE International Symposium on Intelligent Control, Munich, Germany, October 46, 2006.

TRMS 33949S User Manual, Feedback Instruments Ltd., East Sussex, U.K., 2006.

S. Purwar, I.N. Kar, A.N. Jha, "Online System Identification of
1
0 0.5 1 1.5 2 2.5
Complex Systems using Chebyshev Neural Network," Applied Soft
Sampling Step (k)
4
x 10
Computing, vol. 7, no.l, pp. 364372, 2007.
Figure 5. State Estimation Errors in TRMS with CNN Observer.
In the neural observer the unknown nonlinearities are estimated using CNN. It can be seen from Fig. 4 that the response of the observer is good using neural network. Fig. 5 shows that the observer error of TRMS with CNN observer is very small and bounded.


CONCLUSIONS
A neural observer for Euler discretized model of 2DOF Twin Rotor MIMO System (TRMS) is presented. A CNN is used to design a Luenbergerlike observer for a class of MIMO discretetime nonlinear system. The CNN Observer proposed is trained with an EKF based algorithm. With the EKF based algorithm, the learning convergence is improved as compared to other previously used algorithms. Simulation results show the effectiveness of the proposed CNN Observer.

Ferdose Ahammad Shaik, Shubhi Purwar , Bhanu Pratap, RealTime Implementation of Chebyshev Neural Network Observer for Twin Rotor Control System, Expert Systems with Applications, vol. 38, issue 10, pp. 1304313049,15 September, 2011.

Jagdish C. Patra and Alex C. Kot, Nonlinear Dynamic System Identification using Chebyshev Functional Link Artificial Neural Networks, IEEE Transactions on Systems, Man, and Cybernetics Part B: Cybernetics, vol. 32, no. 4, August, 2002.

S. Haykin, Kalman Filtering and Neural Networks, New York: Wiley, 2001.

Animesh Kumar Shrivastava, and Shubhi Purwar, State Feedback and Output Feedback Tracking Control of Discretetime Nonlinear System using Chebyshev Neural Networks, Proc. International Conference on Power, Control and Embedded Systems (ICPCES), pp. 16, November, 2010.

Y. Song and J. W. Grizzle, The Extended Kalman Filter as Local Asymptotic Observer for DiscreteTime Nonlinear Systems, Journal of Mathematical Systems, Estimation and Control, vol. 5, no. 1, pp. 59 78, BirkhauserBoston, 1995.

A. N. Lakhal, A. S. Tlili, and N. Benhadj Braiek, Neural Network Observer for Nonlinear Systems Application to Induction Motors, International Journal of Control and Automation, vol. 3, No. 1, pp. 1 16, March, 2010.

F.Alonge and F. DIppolito, Extended Kalman Filter for Sensorless Control of Induction Motors, First Symposium on Sensorless Control for Electrical Drives (SLED), Padova, pp. 107113, July, 2010.

E A HernandezVargas, E N Sanchez, J F Beteau, C Cadet, Neural Observer Based Hybrid Intelligent Scheme for Activated Sludge Wastewater Treatment, Chemical and Biochemical Engineering Quarterly, vol. 23, no. 3, pp. 377384, February, 2009.