Abstract-- This paper deals with both state feedback and

output feedback tracking control of discrete-time nonlinear system using CNN. Firstly, state feedback control is presented via backstepping, applied to a strict feedback form. In this CNN is used to approximate unknown functions to design control law by the backstepping technique and solves the non-causal problem in discrete-time system. After this output feedback control is presented by converting strict feedback form into cascade form (Brunovsky form). This paper also presents the respective stability analysis, on the basis of the Lyapunov approach, for the whole controlled system. A single layer functional link CNN is used where the need of hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials and approximation of complex nonlinear systems becomes easier. A simulation example is given to show the effectiveness of control schemes.

Index Terms—Backstepping, Chebyshev Neural Network, Discrete-time nonlinear system, Lyapunov stability, Non-causal system.

I. INTRODUCTION HE important property of Neural Networks (NNs) are its universal approximation property. In general, NNs are

used as function approximators to approximate complex nonlinear function or systems. Artificial neural networks have been applied successfully to speech recognition, image analysis and adaptive control. In adaptive control NN is used to approximate unknown functions when the complete knowledge of system is not available.

NN control approaches have been proposed for nonlinear system with strict-feedback form [1]-[5] and output feedback form [6]-[8]. In case of continuous-time systems, the control laws and the adaptive laws of NN weights updation are generally derived based on Lyapunov approach and therefore guarantee the stability of closed-loop continuous-time systems. In case of discrete-time systems the derivation of control laws for adaptive control is more complex due to non-causal problem.

In this paper both state feedback and output feedback tracking control of discrete-time nonlinear system using Chebyshev neural networks are presented. In this CNN is used to approximate unknown functions or to predict future states

Animesh Kumar Shrivastava and Shubhi Purwar are with Deptt. of Electrical Engg. MNNIT, Allahabad-211004, India (e-mail: animesh_258@yahoo.co.in, spurwar@rediffmail.com).

which are complex non-linear functions of system state variables to design control law by the backstepping technique and solves the noncausal problem in discrete-time system. Output feedback control is presented by converting strict feedback form into cascade form (Brunovsky form) [16] through a diffeomorphism transformation. A single layer functional link CNN is used where the need of hidden layer is eliminated by expanding the input pattern by Chebyshev polynomials

This paper is organized as follows. In section II the nonlinear system dynamics is described. Function approximation using neural network is discussed in section III. Controller design schemes are provided in section IV. A simulation example is given to show the effectiveness of control schemes in section V. Performance and the conclusions are provided in Section VI. References are given in section VII.

II. SYSTEM DYNAMICS System dynamics is given for Single-Input and Single-

Output (SISO) discrete-time nonlinear system in strict-feedback form

_ _

1

_ _

1

( 1) ( ( )) ( ( )) ( ),1,2,3,................, 1,

( 1) ( ( )) ( ( )) ( ) ( ),( ) ( ),

i i i i i i

n n n n n

x k f x k g x k x ki n

x k f x k g x k u k d ky k x k

++ = += −

+ = + +=

(1)

where _

1 2( ) [ ( ), ( ),............, ( )] , 1,2,3,......., ,T ii ix k x k x k x k R i n= ∈ = ( ) ,y k R∈ ( )u k R∈ are the system state-variables, system

output and input respectively; ( )d k denotes the external

disturbance satisfying ( ) , 0;d k d d< >_

( ( ))i if x k and, _

( ( ))i ig x k , 1, 2,3,................,i n= are unknown smooth functions. The objective is to design control law ( )u k such that the system output ( )y k follow the desired trajectory ( )dy k .

State Feedback and Output Feedback Tracking Control of Discrete-time Nonlinear System

using Chebyshev Neural Networks Animesh Kumar Shrivastava, and Shubhi Purwar, Member, IEEE

T

978-1-4244-8542-0/10/$26.00 ©2010 IEEE

2

III. FUNCTION APPROXIMATION USING NN Here single layer Chebyshev neural network is used to

approximate nonlinear function. The output of the single layer neural network is given by

Ty w φ∧ ∧

=

Where 1 2[ ............ ]T

nw w w w∧

= are the weights of the NN. And φ is the basis function which is formed using chebyshev

polynomials. A smooth nonlinear function )(xf can be approximated by CNN on a compact set qRΩ ⊂ i.e.

( ) Tf x w φ= + ∈ Where ∈ is the inherent NN approximation error vector.

The Chebyshev polynomials can be formed using following recursive formula [11]

1 1 0( ) 2 ( ) ( ), ( ) 1i i iT x xT x T x T x+ −= − = (2) The NN weight updation algorithm for the discrete time model is given by:

( ) ( 1) ( ). ( )

( ) ( ) ( ) ( ) ( ) ( 1)

( 1) ( )( )

1 ( ) ( 1) ( )

( ) ( 1) ( ) ( ) ( 1)

w n w n k n e n

Te n y n y n y n n w n

P n nk n T n P n n

TP n P n k n n P n

φ

φ

φ φ

φ

∧ ∧= − +

∧ ∧= − = − −

−=

+ −

= − − −

(3)

where ‘e’ is the error between actual plant output and estimated plant output and φ is the basis function formed by

the chebyshev polynomials and 1( ) ( ( ) ( ))TP n n n −= Φ Φ .

IV. CONTROLLER DESIGN

A. State-feedback NN Controller Design

1) Case 1: For system described in (1) when _

( ( ))f x ki i

, 1, 2, 3, ................,i n= are unknown. We will approximate these unknown functions by NN.

(.)T

f wii iφ∧ ∧

= , where (.)fi∧

, wi∧

are the estimates of (.)fi ,

and weights and iφ , , 1, 2, 3, ................,i n= are the basis

functions. All weights are continuously updated using the updation law given by (3).

Let ( )1z k be the deviation of the output ( )y k from the

target i.e. ( ) ( ) ( ) ( ) ( )1 1z k y k y k x k y kd d= − = −

Consider a positive definite lyapunov function ( ) ( )1 1V k z k=

( ) ( 1) ( ) ( 1) ( )1 1 1 1 1V k V k V k z k z kΔ = + − = + −

( ) ( 1) ( 1) ( )1 1 1V k x k y k z kdΔ = + − + −

( ) ( ) ( ). ( ) ( 1) ( )11 1 2 1V k f k g k x k y k z kd∧

Δ = + − + −

Then we use ( )2x k as virtual control the ideal fictitious

control is chosen as 1( ) ( ) ( 1) ( )1 1 12 ( )1

x k f k y k c z kd dg k

∧⎛ ⎞−= − + +⎜ ⎟⎝ ⎠

(4)

Now

( )( ) ( ) ( )1 1 1 1

( ) 1 ( )1 1 1

V k c z k z k

V k c z k

Δ = −

Δ = −

Where 1c is a design constant, for 11c < then ( )1V kΔ

will become to negative definite. Now let ( )2z k be the

deviation of the ( )2x k from the desired fictitious control

( )2x kd i.e. ( )2z k = ( ) ( )2 2x k x kd− . Now consider a

positive definite lyapunov function

2 1 2( ) ( ) ( )V k V k z k= +

( ) ( 1) ( ) ( ) ( 1) ( )2 2 2 1 2 2

( ) ( ) ( ) ( ). ( ) ( 1) ( )22 1 2 3 22

V k V k V k V k z k z k

V k V k f k g k x k x k z kd

Δ = + − = Δ + + −

∧Δ = Δ + + − + −

Then we use )(3 kx as virtual control the ideal fictitious control is chosen as

⎟⎠⎞

⎜⎝⎛ ++−−=

∧)()1()(

)(1)( 2222

23 kzckxkf

kgkx dd (5)

Now

( ) )(1)()(

)()()()(

2212

22212

kzckVkV

kzkzckVkV

−+Δ=Δ

−+Δ=Δ

Where 2c is a design constant to be chosen later. For 12 <c

then )(2 kVΔ will become to negative definite. We will continue this process until the final control input )(ku is designed. The ideal fictitious control is given by

⎟⎠⎞

⎜⎝⎛ +++−−=

∧)()()1()(

)(1)( kzckdkxkfkg

ku nnndnn

d

(6)

2) Case 2: For system described in (1) when ))((_

kxg ii

ni .......,,.........3,2,1, = are unknown. We will approximate these unknown functions by NN.

iT

ii wg φ∧∧

=(.) , where∧(.)ig , iw

∧ are the estimates

of (.)ig and weights and iφ ni .......,,.........3,2,1, = are the basis functions. All weights continuously updated using the updation law given by equation (3). The derivation for control input )(kud is same as discussed for case 1. The ideal fictitious control is given by

3

( ))()()1()()(

1)( kzckdkxkfkg

ku nnndn

n

d +++−−= ∧

(7) 3) Case 3: For system described in (1) when

both ))((_

kxf ii , ))((_

kxg ii ni .......,,.........3,2,1, = are unknown. We will approximate these unknown functions, by

NN. if

Tifi wf φ

∧∧=(.) ,

igT

igi wg φ∧∧

=(.)

Where∧(.)if ,

∧(.)ig are the estimates of (.)if , (.)ig

andifw

∧,

igw∧

are the estimates of the weights andifφ ,

igφ

are the basis functions ni .......,,.........3,2,1, = . All weights continuously updated using the updation law given by (3).

Let )(1 kz be the deviation of the output )(ky from the target i.e.

1 1( ) ( ) ( ) ( ) ( )d dz k y k y k x k y k= − = − Consider a positive definite lyapunov function

)()( 11 kzkV =

1 2 11 1( ) ( ) ( ). ( ) ( 1) ( )dV k f k g k x k y k z k∧ ∧

Δ = + − + −

Then we use )(2 kx as virtual control the ideal fictitious control is chosen as

⎟⎠⎞

⎜⎝⎛ ++−−=

∧

∧ )()1()()(

1)( 111

1

2 kzckykfkg

kx dd (8)

Now

( ) )(1)(

)()()(

111

1111

kzckV

kzkzckV

−=Δ

−=Δ

Where 1c is a design constant to be chosen later. For 11 <c

then )(1 kVΔ will become to negative definite. Let )(2 kz be

the deviation of the )(2 kx from the desired fictitious control

)(2 kx d i.e. )()()( 222 kxkxkz d−= . Now consider a positive definite lyapunov function

2 1 2( ) ( ) ( )V k V k z k= +

2 2 2 1 2 2

2 1 3 2 22 2

( ) ( 1) ( ) ( ) ( 1) ( )

( ) ( ) ( ) ( ). ( ) ( 1) ( )d

V k V k V k V k z k z k

V k V k f k g k x k x k z k∧ ∧

Δ = + − = Δ + + −

Δ = Δ + + − + −

Then we use )(3 kx as virtual control the ideal fictitious

control is chosen as

⎟⎠⎞

⎜⎝⎛ ++−−=

∧

∧ )()1()()(

1)( 2222

2

3 kzckxkfkg

kx dd (9)

Now

( ) )(1)()(

)()()()(

2212

22212

kzckVkV

kzkzckVkV

−+Δ=Δ

−+Δ=Δ

Where 2c is a design constant, for 12 <c then )(2 kVΔ

will become to negative definite. We will continue this process until the final control input )(ku is designed. The ideal fictitious control is given by

⎟⎠⎞

⎜⎝⎛ +++−−=

∧

∧)()()1()(

)(

1)( kzckdkxkfkg

ku nnndn

n

d

(10)

B. Output-feedback NN Controller Design In this subsection we will design controller based on

output feedback. For this we define new system states

1 2( ) [ ( ), ( ),............ ( )]Tny k y k y k y k=

1 1_ _

1 21 1 2 1 2 1

( ) ( )

( 1) ( 1) ( ) ( ( ), ( )) ( ( ))

x k y k

x k y k y k f x k x k F x k

=

+ = + = = =

_ _2 31 2 3 2 3 2( 2) ( 1) ( ) ( ( ), ( )) ( ( ))x k y k y k f x k x k F x k+ = + = = =

_

1

::

( ) ( 1) ( ( ), ( ), ( ))nn nx k n y k f x k u k d k+ = + =

(11)

In (11) for above n equations it can be written as _

( ) ( ( ) )ny k T x k= ,where a non-linear co-ordinate transformation

_ _ _ _2 3 41 1 2 3

_

1

( ( )) [ ( ), ( ( )), ( ( )), ( ( ))

,........., ( ( ))]

n

Tnn

T x k y k F x k F x k F x k

F x k−

= (12)

For equivalent transformation of coordinates, the coordinate transformation mapping T must be diffeomorphism. Lemma 1. If the Jacobian Matrix

1 1

1_ 1 1

1

1 1

1

. .

. .( ( ))( )

. . . .

. .

n

nn

n n

n

x xx xF F

dT x k x xdx k

F Fx x

− −

∂ ∂⎡ ⎤⎢ ⎥∂ ∂⎢ ⎥⎢ ⎥∂ ∂⎢ ⎥∂ ∂= ⎢ ⎥⎢ ⎥⎢ ⎥∂ ∂⎢ ⎥⎢ ⎥∂ ∂⎣ ⎦

is nonsingular at some point np R∈ , then there exist the coordinate transformation mapping T such that it is diffeomorphism. Lemma 2. For the system (1), the nonlinear coordinate

transformation _

( ( ))nT x k in (12), is a smooth map and a diffeomorphism, namely, then there exists a unique

4

transformation function 1( ( ))T y k− is such that _

1( ) ( ( ))nx k T y k−= , and both T and 1T − are one-to-one. Now equation (11) can be written into cascade form (Brunovsky form) which is equivalent to the original system (1),for simplicity we are neglecting disturbance i.e. ( ) 0d k = .

1 1

1 2

2 3

( ) ( )( 1) ( )( 1) ( )

x k y ky k y ky k y k

=+ =+ =

1_

1

::

( 1) ( )

( 1) ( ( ), ( )) ( ( ( )), ( ))( ( )) ( ( )) ( )

n n

nn n n

y k y k

y k f x k u k f T y k u kf y k g y k u k

−

−

+ =

+ = == +

(13)

From the above set of equations written in cascade form we can observe that the control input ( )u k has an effect on

( 1)ny k + i.e. 1( )y k n+ . Now '( ) ( ( ))u k f y k= but practically we don’t have the all values of ( )y k i.e.

2 3( ), ( ),......., ( )ny k y k y k . We can predict them using NN.

[ ]

,T T

w wf wg

f wf g wgφ φ

∧ ∧ ∧

∧ ∧ ∧ ∧

=

= =

1( ) . ( )y k f g u k n∧ ∧ ∧

= + −

where 1( ), ,y k f g∧ ∧ ∧

and ,f gw w∧ ∧

are the estimates of

1( ), ,y k f g and weights and φ is the basis function. All weights continuously updated using the updation law given by equation (3). Now the estimation of future values of the outputs i.e. 2 3( ), ( ),......., ( )ny k y k y k

12

13

( ) ( 1) . ( ( 1))

"( ( ), ( 1), ( 2),............, ( ( 1))

( ) ( 2) . ( ( 2))

"( ( 1), ( ), ( 1),............, ( ( 2))

T T

T T

y k y k wf wg u k n

where f y k y k y k y k n

y k y k wf wg u k n

where f y k y k y k y k n

φ φ

φ

φ φ

φ

∧ ∧ ∧ ∧

∧ ∧ ∧ ∧

∧

= + = + − −

= − − − −

= + = + − −

= + − − −

1

::

( ) ( ( 1)) . ( 1)T T

ny k y k n wf wg u kφ φ∧ ∧ ∧ ∧

= + − = + −

"( ( ( 2)), ( ( 3)),............, ( 1))where f y k n y k n y kφ∧ ∧

= + − + − −

Let )(1 kz be the deviation of the output 1( )y k from the target i.e.

1 1

1 1

( ) ( ) ( )( ) ( ) ( )

d

d

z k y k y kz k n y k n y k n

= −+ = + − +

Consider a positive definite lyapunov function

1 1

1 1 1 1 1

1 1

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) . ( ) ( ) ( )d

V k z k

V k V k n V k z k n z k

V k f g u k y k n z k∧ ∧

=

Δ = + − = + −

Δ = + − + −

let choose ( )u k as

1 11( ) ( ) ( )du k f y k n c z k

g

∧

∧− ⎛ ⎞= − + +⎜ ⎟

⎝ ⎠ (14)

,T T

f wf g wgφ φ∧ ∧ ∧ ∧

= =

"( ( ( 1)), ( ( 2)),................., ( ))where f y k n y k n y kφ∧ ∧

= + − + − Now

( ) )(1)(

)()()(

111

1111

kzckV

kzkzckV

−=Δ

−=Δ

Where 1c is a design constant to be chosen later. For 11 <c

then )(1 kVΔ will become to negative definite. So the equation (14) shows the final control input.

V. SIMULATION EXAMPLE Consider a non-linear discrete-time SISO second-order

system 2 ( )1( 1) 1.5 ( )1 221 ( )1

x kx k x k

x k+ = +

+

( )1( 1) 0.8 ( ) ( )2 2 21 ( ) ( )1 2

x kx k u k d k

x k x k+ = + +

+ +

( ) ( )1( ) 0.1cos(0.05 ) cos( ( ))1

y k x k

d k k x k

=

=

The control objective is to make the output ( )y k follow a desired trajectory

( ) 0.5 cos( / 20) 0.5sin( /10)y k k kd π π= +

Fig.1 Simulation results for State-feedback NN Controller case 1

Fig. 2 Simulation results for State-feedback NN Controller case 2

Fig.3 Simulation results for State-feedback NN Controller case 3

Fig.4 Simulation results for Output feedback NN controller

VI. CONCLUSIONS This paper presents both state feedback and output

feedback tracking control of discrete-time nonlinear system using CNN. First, state feedback control is presented via backstepping, applied to a strict feedback form. After this output feedback control is presented by converting strict feedback form into cascade form (Brunovsky form). The stability analysis is also presented on the basis of the Lyapunov approach, for the whole controlled system, including CNN learning algorithm. This paper has presented the application of CNN to approximate unknown complex nonlinear functions to design control law. The training of the NN is performed online using recursively minimize least-squares loss function. Initially, in the neural network all weights corresponding to function f (.) are assumed to be zero but not for function g(.) , otherwise singularity problem arise at the starting . A single layer functional link CNN is used, as the neural network is single layered, it is computationally fast and simple.

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