Discrete-Time Chebyshev Neural Observer for Twin Rotor MIMO System

DOI : 10.17577/IJERTV4IS080299

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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.


    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.


    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 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-



      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



      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


      : Pitch (Elevation) Angle, : Yaw (Azimuth) Angle,


      1 : Momentum of Main Rotor,

      0 1 0 0 0 0

      2 : Momentum of Tail Rotor,

      u1 and u2 : Inputs,


      I I

      0 1 0 0


      1 0

      and, the system parameters of the TRMS are given in Table

      I. 1 1


      0 0 0 1 0 0



      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


      b1 Static Characteristic Parameter


      a2 Static Characteristic Parameter


      b2 Static Characteristic Parameter


      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


      k2 Motor 2 Gain


      T11 Motor 1 Denominator Parameter


      T10 Motor 1 Denominator Parameter


      T21 Motor 2 Denominator Parameter


      T20 Motor 2 Denominator Parameter


      Tp Cross Section Momentum Parameter


      T0 Cross Section Momentum Parameter


      kc Cross Reaction Momentum Gain


      B k b b

      A 0 0 0



      c 1 2 ;

      I I I

      2 2 2


      0 0 0 0

      10 0




      0 0 0 0 0

      0 0

      0 0

      0 0

      0 0


      T21 ( 6 x 6 )

      B k ;

      1 0





      0 2


      ( 6 x 2 )

      1 0 0 0 0 0

      C 0 0 1 0 0 0



      ( 2 x 6 )

      a M



      { 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




      a2 2


      k a 2

      Figure 1. The Twin Rotor MIMO System (TRMS).


      c 1 1


      Consider the state space form

      x Ax f (x) Bu



      ( 6 x1)

      y Cx


      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 )


    k : Sampling Step,

    () [( )] ,


    x2 (k )


    gi ( x(k ))

    x (k )

    ( + 1) [( + 1)] ,

    () [()] ,

    F= 6 + ,

    G= ,

    g(x(k)) = () ,

    Ts : Sampling Time,

    and, I6 is a 6×6 identity matrix.


    Figure 2. CNN Structure.

    A smooth nonlinear function approximated by CNN [14]

    i i i

    g (x(k)) w*T (x(k))

    Function Expansion


    can be


  3. CNN STRUCTURE Consider a discrete-time MIMO nonlinear system

x(k 1) Fx(k ) g(x(k )) Gu(k )

y(k ) Cx(k )


where gi (x(k )) is the nonlinear function in ith plant state, i

is the bounded approximation error [14].


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


w* is constant, one has

wi (k 1) wi (k) wi (k) wi (k 1)



gi (x(k )) is the output of ith neuron, n is the output

In this paper we use the following modified EKF based


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 …………


wi (k 1) wi (k ) Ki (k )e(k )

i 1, …….., n



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


M (k ) [R (k ) H T (k )P (k )H (k )]1


i i i i i


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 )


(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 )



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 )]


    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 )


    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


    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).






    with Unknown Nonlinearities

    wT (x(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.


    Actual Pitch Angle

    CNN +

    x(k 1)





    Pitch Angle (rad)


    Observed Pitch Angle






    CNN Observer

    Figure 3. CNN Observer Scheme.




    0 0.5 1 1.5 2 2.5

    The observer gain matrix D

    is chosen such that

    Sampling Step (k)


    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




    Yaw Angle (rad)









    Actual Yaw Angle

    Observed Yaw Angle

    0 0.5 1 1.5 2 2.5

    Sampling Step (k)


    x 10

    Figure 4. Actual and Observed States of TRMS with CNN Observer.


    x 10


    Pitch Angle Error (rad)






    0 0.5 1 1.5 2 2.5


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      Yaw Angle Error (rad)







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      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.


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.

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