Discrete-Time Chebyshev Neural Observer for Twin Rotor MIMO System

Discrete-Time 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 Multi-Input-Multi-Output (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.

1. 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.

2. 2-DOF TRMS MODEL

   The mechanical setup of the twin rotor MIMO system is shown in Fig.1.

   1. Continuous-Time 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 Multi-Input-Multi-

      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

      Chebyshev-Functional 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	6.8×10-2 kg-m2
      I2 Moment of Inertia of Horizontal Rotor	2×10-2 kg-m2
      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 N-m
      B1 Friction Momentum Function Parameter	6×10-3 N-m-s/rad
      B1 Friction Momentum Function Parameter	1×10-1 N-m-s/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.

   2. Discrete-Time Model

   The discrete-time 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)

3. CNN STRUCTURE Consider a discrete-time 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].

1. NEURAL OBSERVER DESIGN USING CNN Consider an observable discrete-time 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 EKF-based 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

2. 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

   F-DC 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

   1. David G. Luenberger, Observing the State of a Linear System, IEEE Transactions on Military Electronics, vol. 8, no. 2, pp. 74-80, April, 1964.

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

   3. S. Benamorl , H. Hammouri and F. Couenne, A Luenberger-Like Observer for Discrete-Time Nonlinear Systems, Proceedings of the 37th IEEE Conference on Decision & Control, Tampa, Florida, USA, December, 1998.

   4. V. Sundarapandian, Observer Design for Discrete-Time Nonlinear Systems, Mathematical and Computer Modelling, vol. 35, no. 1-2, pp. 37-44, January, 2002.

   5. Tarek Ahmed-Ali, Romain Postoyan, Françoise Lamnabhi-Lagarrigue,

      2.5

      2

      Yaw Angle Error (rad)

      1.5

      1

      0.5

      0

      -0.5

      -10

      x 10

      Sampling Step (k)

      4

      x 10

      Continuous-Discrete Adaptive Observers for State Affine Systems, Automatica, vol. 45, no. 12, pp. 2986-2990, December, 2009.

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

   7. Alma Y. Alanis, Edgar N. Sanchez, Alexander G. Loukianov, and Marco A. Perez, Real-Time Recurrent Neural State Estimation, IEEE Transactions on Neural Networks, vol. 22, no. 3, pp. 497-505, March, 2011.

   8. Alma Y. Alanis, Edgar N. Sanchez and Alexander G. Loukianov, Discrete-Time Nonlinear Recurrent High Order Neural Observer, Proceedings of the 2006 IEEE International Symposium on Intelligent Control, Munich, Germany, October 4-6, 2006.

   9. TRMS 33-949S User Manual, Feedback Instruments Ltd., East Sussex, U.K., 2006.

   10. S. Purwar, I.N. Kar, A.N. Jha, "On-line 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. 364-372, 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.

3. CONCLUSIONS

A neural observer for Euler discretized model of 2-DOF Twin Rotor MIMO System (TRMS) is presented. A CNN is used to design a Luenberger-like observer for a class of MIMO discrete-time 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.

1. Ferdose Ahammad Shaik, Shubhi Purwar , Bhanu Pratap, Real-Time Implementation of Chebyshev Neural Network Observer for Twin Rotor Control System, Expert Systems with Applications, vol. 38, issue 10, pp. 13043-13049,15 September, 2011.

2. 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.

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

4. Animesh Kumar Shrivastava, and Shubhi Purwar, State Feedback and Output Feedback Tracking Control of Discrete-time Nonlinear System using Chebyshev Neural Networks, Proc. International Conference on Power, Control and Embedded Systems (ICPCES), pp. 1-6, November, 2010.

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

6. 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.

7. 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. 107-113, July, 2010.

8. E A Hernandez-Vargas, 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. 377-384, February, 2009.