**ProCItdlqi of b e **
**Amdcm C o n h l Conlomneo **

**Balllmon, Maryland Juno 1954 **

**FM7 **

= **2 1 0 **

**A Neural Network Approach of **

**Input-Output Linearization of Affine Nonlinear **

**Systems **

Wei-Song Lin, Hong-Yue Shue and Chi-Hsiang Wang Department of Electrical Engineering, National Taiwan University, Taiwan, R.O.C.

* e-mail *:

**weisong@mailbox.ee.ntu.edu.tw****Abstract**

For practical reasons, in the technique of feedback linearization, the requirements of mathematical modeling and access

### of

internal states of complicated nonlinear systems should be removed. This paper demonstrates that, simply using output feedback, the input-output linearization of affine nonlinear systems with zero dynamics being exponentially stable can be accomplished by using multilayer neural network to estimate the instantaneous values of the nonlinear terms appearing in the feedback linearizing control law. Neither mathematical model nor internal state of the nonlinear system is required. The configuration for training the multilayer neural network as a device of the input-output linearizing controller is established. An example of affine nonlinear system is studied by computer simulations for various cases linearizing control.**1. Introduction **

Feedback linearization based on concepts from differential
geometry **[2,6,9] **can be adopted as a design methodology of
nonlinear control systems. Using nonlinear coordinate
transformation and feedback control, the original nonlinear model
can usually be transformed into an equivalent linear one. With the
terminology's, input-state and input-output linearizations refer
respectively to complete **[5,7,12] **and partial linearizations [4,8,10].

For the linearized system, the well-developed design techniques of
linear systems can be applied to solve for the appropriate
controller. This approach has been successfully applied to a
number of practical control plants such as high performance
aircraft's, industrial robots and biomedical devices, Even though,
there still exists major challenges for the feedback linearization
design such as the requirement of precise mathematical model, the
necessity of state feedback, and no guarantee of robustness in the
appearance of parameter uncertainty or unmodeled dynamics [ **1 **I].

Identifying the nonlinearity of control systems by neural networks through learning contributes another concept of nonlinear system design [I]. It is shown in this paper that, simply using output feedback and without the mathematical model of the system, the input-output linearization of an affine nonlinear system can be accomplished by estimating the decoupling matrix appearing in the approximated incermental linearizing control law with multilayer neural networks.

**2. The Problem Of Neural-Network-Based Input-Output **
**Linearization **

Consider a square, affine nonlinear system with exponentially stable zero dynamics described by the following nonlinear differential equations

dynamics of a nonlinear system is the dynamics of the system when the outputs are constrained to be identically equal to zero.) For simplicity the arguments regarding to time, unless noticed, are omitted by the expressions in the following context. In the designs of input-output linearization, each output,

*Y,, *

of equation ### (1)

is differentiated with respect to time repeatedly until at least one of the control inputs appears in the output equation. If*represents the relative degree associated w t h*

**r,***Y,.*Then the ("-order derivative of

*with respect to time can be written as follows :*

**y ,**where * h, *denotes the

**j I h****component of h, L,h,(x): R"**

### -+

**R**and

**L,,h,(x): R "**

### -+

**R stand for the Lie derivatives of h,(x) with **

**respect to f(x) and gJ(x), respectively. Equation (2)**can be rewritten in vector-matrix form as follows

**y'" **= **c(x)+ D(x)u ** **(3) **

where **Y(') ** = **[y!q), **#),

### . .

### .,

*J A ; ) ] ~ with r,*

**,r,**### ,. .

*being the relative*

**.,r,,,**degrees of the system, the decoupling matrix is

### I

** * .*

**L&-'h,(x)**

**L&'h,(X)**### [ -

**L,,****L;1.-'hlIl(x)**

### .. .

**L,," Lf"h,,,(****X )**

**D(x)**= and

**c( x) =**

*[L;*

*h, *

**(x),**

### .

### .

### .)

**Lth,,(X)]'.**

If the mathematical model of the nonlinear system is given, the
design of input-output linearization is to solve for the control law,
**U, **so that the closed-loop characteristics of the system
characterized by equation (3) is linearized. Unfortunately the
availability of mathematical model is not the practical case of
most complicated nonlinear systems. For many real world
applications, the mathematical models are usually severe difficult
to be identified. This reality keeps the feedback linearization
approach of many complicated nonlinear systems remaining in the
theoretical phase. Instead of using the mathematical model, our
consideration is to accommodate the nonlinearity required in the
linearizing control law by the learning ability of artificial neural
networks. Fig. 1 shows a block diagram of this substitutive
**configuration. Comparing with equation (3), the neural network **
**may provide the estimations of nonlinear functions, D(x) and c(x). **
through an appropriate learning process. This configuration
releases the designer from the necessity of mathematically
modeling the nonlinear plants. The goal is to establish the design
procedures and show the feasibility of neural-network-based
input-output linearization of nonlinear systems. Only square,
affine nonlinear systems with zero dynamics being exponentially
stable are considered.

### [L

### L

### F

### -

### H

### F

### T

**Controller ** **Controller **
**feedback linearization loop **

### y-r-l

**pleplaccment loop **

**Fig. 1 A block diagram of **

**neural network based feedback linearixzation **

**3. The Incremental Linearizing Control Law **

If **the decoupling matrix, D(x), ** of the nonlinear system
characterized by **(1) **is nonsingular, the exact linearizing control
law is

**where v ER"' is the auxiliary input vector. Applying **(4) to (3)
yields the following **m ****decoupled linear, SISO, systems **

To stabilize the linearized system described by * ( 5 ) , *many well-
developed linear design methods can be applied. Pole-placement
by output feedback may give control laws as

for * i=1,2, ..,m to *achieve closed-loop characteristics governed by

**U **= **D-'(x)[-c(x)+v] ** (4)
**V. ** **( 5 ) ****y(r) ** =
* v, *=

**w,**- a , l y ~ ~ - ~ ) - . . . - u l ~ ~ l ~ , - a , $ y I

**y,oi)+a,,yjr.-l)+**### ...+

*u,,_,j,+a,,y,*= w, (6)

*(7)*

**where w,qs are reference inputs, ** *aI/Is, j=1,2 *,..., * r * are the
parameters to be chosen so that

**SG***+a,, *

*sG-'+...+a,$*are Hurwitz polynomials.

**For the nonlinear system under consideration is with c(x) and **
**D(x) being uniformly continuous functions of time over the **
interval

**o **

I **o**

*I*

**t***L .*Then

**c(x),,,**

### +

**c ( x ) ( ~ - ~ ) and**

**D(x)(,)**

### -+

**D(x)(,~-~)**are true for sufficiently small sampling intervai of

*-*

**T =****t,***and the subscripts denote the time instant at*

**tk-l,**where 0 5**T**5 L*This knowledge motivates the time-delay control law of input-output linearization as follows E1 31,*

**t,.**It has been shown that if one chooses an appropriate

**fi **

such that
*=*

**U ( k )****D-'(x>(,)**

**K-Y;L) **

+ **D(X)(k-l) U(*-I))**+

**V(k)I**(8)

(9)
**111 - D( x)D-' **

### 11

< 1**and further, keeps the auxiliary inputs, v, as uniformly continuous **
functions of time, where **I **represents **an **identity matrix with
appropriate dimension. Then the system defined by (1) can be
input-output linearized by applying the following approximated
time-delay control law,

**U(,) **= **D-WY;;!,) ** + **DU(kbI))+ V(k)I ** (10)
**Here, using D(x)(,, = D(x)(~+ **and after rearrangement of (8), we
obtain the incremental linearizing control law as follows,

**U@) **= **0. **
**Au(,) **= **~ ( k ) **

### -

**U(,-,)**=

**D-'(x)(k)[v(k)**

### -

**Y&)],**(11) Using multilayer neural nenvork

**10**give the estimation of

**D(x)(,,**as

**6(k) **

at every sampling instant, the approximation of the
incremental linearizing control law is obtained as follows,
In the digital realization of the linearizing controller, the uniformly
continuous property of the auxiliary input is usually not
guaranteed. Under this consideration, the following theorem
shows the situations of the tracking error in applying the
approximated incremental linearizing control law.
**6(k)**

**Theorem ****1 Consider the nonlinear system and its controller ****being well defined by (1) and (12), respectively and ****c(x) ****and ****D(x) **

**AU(k) **= **U(k) -U(,-,) ** = **D - l ( k ) [ V ( , ) **

**-Y;;!l)l, **

**U@)**=

**0.**(12)

(16)

**W ( k ) **

**- Y g I **

= **P **

**-D(X)(k)D;kl)ltv(k-l)**

**- Y ; L ) l + **

**[I**

### -

**D(x)(k)D;&k)**

### -

**V(k-I)l**

Taking norm on vectors and induced norm on matrices of equation (1 6), the following inequality is obvious

**Since 2p 1, (20) depicts that the tracking error converges **
exponentially to zero for sufficiently large value of **k. **

case (iii)- When

**IIv(~) **

### -

### ~ ( ~ - ~ ) l l >

**I(v(~_~) **

### -

**yi;$11**

### ,

the substitution of**IIv(,) -vg-,)1I**for

**]Iv(~-~) **

**--y{;!,)]] on the right side of (17), we**obtain

**IIv(k) **-Y$ll'

### 2111-

D(x)(k)D~~)ll,**IIv(k)**-v(k-l)l]

**(21)**

This proves the last case.+

For nonlinear systems being exactly linearized, the tracking error
should always be kept at zero. However this is only possible when
perfect model of the system is available. Alternatively, by using
the approximated incremental linearizing control law, theorem 1
has revealed that if the decoupling matrix can be estimated
accurately to some extent and the auxiliary input is manipulated
finely. The tracking error of the linearizing control can either be
constrained within a bound as in case (iii), or even converge
exponentially to zero as in the cases of (i) and (ii). As a result,
instead of * (S), * the imperfect input-output linearizing control
results in a perturbed linear system described by

where

*5 *

represents the vector of perturbation. For case (iii), the
perturbation may be so large that the linearizing control being
divergent, if the decoupling matrix and the step size of the
auxiliary inputs are not chosen appropriately.
**4. Neural Network Based Linearization **
**4.1. Issues of The Multilayer Neural Networks **

A neural network is a massively parallel, interconnected
network of elementary units called neurons. A neural network
with multiple neurons is usually organized into a sequence of
layers, called multilayer neural network. The general structure of
**an (q+1) multilayer neural network with ****no ****inputs and n4 outputs **
can be illustrated as shown in Fig. 2 and Fig. 3. The input layer
usually acts as

### an

input data holder and signals flow from the input layer through the hidden layers to the output layer. The output of each neuron in the*layer can be expressed as*

**p f h****y'" **= **v **

### + 5

_{(22) }

**1 **

**Fig. 2 A hidden neuron i of layer p. ** **Fig.3 A multilayer neural network **

(23)
In applications of system modeling, it is common for the dynamic
range of output data to be greater than 1, the activating function of
the output node is therefore chosen to be linear. Thus the **i* **

**output node performs a weighted s u m of its inputs as follows **

**4.2. ** Training of The Multilayer Neural Network for **Input- **
Training of the neural network is to determine * w's *and

**b's **

such
that **b's**

*of (24) is as close to the desired output as possible. Using Stone-Weierstrass theorem [3] it can be shown that a given nonlinear function under certain conditions can be represented by*

**x ; ( t )**Output Linearization

a corresponding series such **as **Voiterra series or Wiener series.
The practical consequence of Stone-Weierstrass theorem is that an

infinitely large neural network can model arbitrary piecewise continuous function. A finite network, however, may only accurately model such functions over a subset of the domain. Our interest is mainly in networks which permit on-line identification and control of dynamic systems in terms of finite dimensional nonlinear differential equations.

The method commonly used to evaluate the gradient of a performance h c t i o n with respect to

### a

weight vector of multilayer neural networks is called back propagation. If**J(0)**is the performance index which

**has **

to be optimized with respect to the
parameter vector 6. Then **6 **

can be adjusted by according to the
steepest decent method **as**the following equation

where 0 < q < 1, the step size, is a chosen parameter. In a multilayer neural network, the performance index

**J **

is usually
chosen **J**

**as**

**c(x, **

**c(x,**

**-xP)'**where are the desired values of

**~ Y l s .****Thus **

the weights and the thresholds are respectively updated
according to
**4 **

**,=I **

**wPJ **

*= wP, (t - 1)*

**( t )**### +

**AwP,**(t)

* b,!( t ) *=

**b,"( t**### -

1)### +

*(t)*

**Ab:****with the increments AwP, (t) and **Ab,! (t ) being given by
* Aw;(t) = *q ,6p

**(t)X,P-'****( t )**### +

**IX~AW;**(t

### -

1)* Ab: *(t ) =

**qb6**### p

*)*

**( t**### +

**a,Ab,"****( t**### -

1) (27)* where the subscripts w and b *represent the weight and threshold
respectively,

*and*

**a,***are momentum constants, q, and*

**ah**

**qb**represent the learning rates and * s p ( t ) *is the error signal of the

**i'h**neuron of the * p ' h *layer. When the activating function of the output

neuron is linear, the error signal at

### an

output node is and for the neurons in the hidden layer**s y ( 0 **

= **s y ( 0**

**X,(t)--XP(t)**(28)

6; * ( t ) *=

**y (**### Y

### ;

*( t ) ) ! f 6 Y 1*

**(t)w;'(****t**### -

**l),**

*=*

**p**

**q**### -

1,...,2,1 (29)**,=I**

**where y(K) denotes the first derivative of y(K) with respect to **y.

To realize the estimations of nonlinear functions D(x) and c(x)
in equation (3) for input-output linearizing control, Fig. 4 shows a
configuration for training the multilayer neural network. In order
to calculate the derivatives of plant outputs correctly, the training
input * u(t) *are generated by sending sequences of random signals
through digital lowpass filters with bandwidth

**o **

and sampling
period T subject to the constraint **o**

*= 0.1 rad. The lowpass filter is of 2nd-order Butterworth type and can be expressed as*

**o~****~ ( k ) **

= 1.8588~(k **~ ( k )**

### -

1)### -

**0.8681~(k**

### -

**2)**

**(30) **

### +

0.00233(r(k)+ 2r(k**-l)+**r(k

### -

2))where r(k) are random signals uniformly distributed in the
operating range. The weights of the neural network are updated by
using the steepest decent method represented by (27) with
appropriate selections of the step size q and the momentum * a *to

minimize the performance index
**m **

* J *=

**C(rjli'**

### -

*j y ) * *

**,=I**

**where i?, and **

**Gv **

are the elements of 2 and **Gv**

**D,**

*respectively, j f c ) is*

**the ith component o f f ( " ) (the approximation of y(')) and r, is the***i s *relative degree of the plant. Because the outputs of the neural
network are **6 ****and D rather than **

### 8'".

The error term described by**(28)**for the back propagation learning algorithm are modified to

**s:, **

= *[y?*

### -

*j!"] *

### ,

for*=*

**i***I,Z,.*

### .

**.,m**

_{(32) }* Sq di, *-

*-[y,(';)-j!G)]uJ,fori=1,2*

### ,...,

**m**and j = 1 , 2### ,...,

*m*

* where S z *,

**S:g**are the error signals for and

### (ill,

respectively. The algorithms for training the weights of hidden layers of the neural network are kept unchanged. The most important thing of this training algorithm is that### -

**L**measurements of

**Backward**Approximation

**yq **

outputs. This
feature is
practically
necessary.
**4.3. Estimation of Relative Degree **

*To obtain the form described by ( 3 ) for the design of input- *
output linearization, the knowledge of relative degrees is
indispensable. This paragraph presents a numerically feasible
method to determine the relative degrees of a nonlinear system.

*Theorem 2 * *Assume that a nonlinear system characterized by *
*equation (1) has continuous states and with relative degrees *
*r,,r,,. .-,r,,,. Let the control input u,(t) associated with the relative *
*degree of the irh output Y , be a piecewise continuous function *
*with a discontinuity at t = t,, and u J ( t ) *= 0

*for all j + p of u(t).*

*Then y , and its derivatives to the order of r,*

### -

*1 are continuous*

*function of time, except that the * *order derivative, y(c), has a *

*discontinuity at t = t,. *

*proof *: Differentiating the *i l h *output equation of system (1) up to

*Fig. ***4 ***The configuration for training the neural network *

*rl *order, we have

*yp' *= *c: ( x ) * *for k *= O,l;..,r, -1; and *(33) *

*(34) *
*y? = c,? ( x ) *

### +

*d , ( x ) u*

*where c : ( x ) *= *L:ht(x) *for

**k **

= O,l,. **k**

### .

*.,rl,*and

*d , ( x ) = [L,,q-'h,(X) ***e * + ** *LCmL:-'h,(x)] with L,,L:-'h,(X);t 0 *

*are smooth functions of x. For continuous c,k(x) and from (33), *
we obtain

*Similarly, from (34) and the continuity property of c,"(x) and *

*d , ( x ) , *we achieve

* y y ( t; *)

### -

*y,'"(*

**t; )**= *L,, *

### q-'

*h(x(ti ))[U, (t;*) -

= *c,'(x(t; *))- * C: ( ~ ( t ; ) ) *+

**d,(x(j;))u(j:)-d,(x(t;))u(l;)**

*(36)*

*( t i )] Z 0 *

Practically, to discover the relative degrees of a nonlinear system,
one can drive the system with step or square wave inputs, then
investigate the outputs and their derivatives for the appearances of
*discontinuities. The lowest order of the derivative of the ilh output *
which appears discontinuity corresponding to the jump or jumps

### *

**in any one input is the i'h relative degree of the system. A specid ***case of SISO systems is that the nonlinear function D(x) at the *
*occurrence of discontinuity can be calculated from (36) as *

*(37) *
*D(x(t, *

### 1)

=*[ ~ ' " ( t ;*)

### -

*~ ( " ( t ; )I/[uCt;')*

### -

*11 *

**5. Simulation Results **
Given a third order system described by

*Y *= XI

where

exp(-x$)-l

*c ( x ) *= a x ,

### +

**8 exp(-x,/2)+1**

*D ( X ) *= *4 - *exp(-O.15x:

### +

1.1)**A ***three-layer neural network (q=2) with 10 neurons in the hidden *
Iayer is adopted as the learning device of all the following
simulations. The training algorithm updates the weights of the
neural network by the steepest decent method and using step size
q=O.Ol and momentum ~ ~ 0 . 0 1 .

**case (a)- using fixed value for all **

**fi(k). **

**fi(k).**

*With a unit step input u ( t ) = u,(t), the value of D(x) at f *= 0 can

*be estimated by using (37). The result is*

**I) **

### -

..*(39) *

*jj,+4yd+6y,=6w * *(40) *

**( 0 ) **

### -

*(Y(0'*)

### -

### D-

*) ) / ( U @ +*) -

*4 0 -*

### 1)

=*(0.9957*

### -

*0)/(1- 0 )*1

**w**For the desired closed-loop dynamics being described by,
the controller with the approximated incremental linearizing
control is
* AU(k) *=

**D ) ; i ) [ V ( k )**### -

*&k-i)l*

*vck)*=

*-4Y(k-i1 - ~ Y W*+

*6 w ~*(41) (42) where

Using

**D,,, **

= 1 for all time in (41), the simulation result is shown
*in Fig. 5 with the sampling period T*= 0.01 sec., initial state

*xo =[0.5 * -1 *O.2]', and w(*, *= 1. Fig. *5 *shows that the tracking
*of desired value, Y,, is in good quality. However, if w ( ~ ) *= *4 , *the
feedback linearizing control system becomes unstable as shown in
**Fig. 6. This demonstrates the case (iii) of theorem 1. **

**case (b)- estimating D(x) on-line by neural network **

Consider that the sequence of training input for the neural
*network is generated by sending the random signal r(k) uniformly *
distributed in the interval **[-26, 341 ****through the lowpass filter (30) **
*with sampling period T *= 0.01 sec. ~ ( x ) is estimated by training
the neural network for 30000 steps. Fig. 7(a) shows the
*satisfactory response of the system subject to the controller (40) *
*even for w ( ~ , *= *4 . ***Fig. 7(b) shows that D(k) is not precisely equal **
**case (c)- **effect of noise

To investigate the effect of noise on the approximated
*incremental linearizing control, a noise n ( t ) *= 0.3sin(20t) is
superposed on the output of the plant. Fig. 8 shows the output
response of the system by using the control law

to *D ( x ) ( k ) ' *

* AU(k) *=

**D&Y(k-I)**+

*40W(k) - lo&-I) - 40Y(k-,)l*

*(43)*

**5 **

where

**yk, **

= **yk,**

**~ u , ( k T**### -

**i)****and Dtk)**is the estimation of

**D(x),,,**being obtained by the neural network. The desired closed-loop characteristics is described by

The result depicts that the approximated incremental linearizing control can achieve good performance even in the presence of severe output noise.

**1x0 **

**ji, **

### +

1 Oy,### +

40y, = 40w (44)**6. Conclusion **

Using output feedback and the learning ability of multilayer
networks, it has been shown that the input-output linearization of
affine nonlinear systems can be succeeded without mathematically
modeling the nonlinear behavior. The configurations for the
training and control of approximated exact and incremental
linearizing control laws have been established. The incremental
linearizing control is able to relax the tolerance of estimating error
introduced by the neural network due to absence of internal states,
disturbance of noise or inadequate capacity of the network itself.
All the establishments are based upon that the affine nonlinear
system being with exponentially stable zero dynamics. Given a
nonlinear problem, there are still questions about what capacity of
a neural network is necessary by the neural-network-based design
**left to be answered. As a conclusion, the approach with output **
feedback and neural learning has made feedback linearizing
control more practical.

**8 5 **

* Acknowledgments *-

**Financial support for this research .from***National Science Council of Taiwan R. O.C. under MSC 82-0404-*

*EOO2-259 is grate&& acknowledged*

I

**a**

### <

**-**

**ya**### -

**0 s ** **1 ** **1 5 ** **2 ** **2 5 **

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**1 . 5 **

### I

Fig. **5 Response when D,,, **= **1, **wl

Fig **6 ****Unstable results when **fi,,, = 1, **W 4 **

**5 ****1 **

### -

### - -

- -### -

- - - -### _-

- - -### - -

## :>

## I

_ / - - - -**1**

**B**

**E**

**B S**

**3 . 5**

**I 5****3**

**(b**)

**Time(sec)**

Fig **7 Stable results when estimating D(x) as **fi,,,

**fg 8 Response when disturbed by noise **