Neural network approach to variable structure based adaptive tracking of SISO systems

Neural Network Approach to Variable Structure Based

Adaptive Tracking of

SISO Systems

Li-Chen Fu

Department of Electrical Engineering

Department of Computer Science and Information Engineering

National Taiwan University, Taipei, Taiwan,

R.O.C.

October 16, 1996

Abstract

This paper presents a novel approach to adaptive tracking control of linear SISO systems, which can solve the traditional model reference adaptive control (MRAC) problems. In this approach, a neural net- work universal approximator is included to furnish an on-line estimate of a function of the state and some signals relevant to the desired trajectory. The salient feature of the present work is that a rigorous proof via Lyapunov stability theory is provided. It is shown that the output error will fall into a residual set which can be made arbitrarily small.

1

Introduction

In the past two decades, model reference adap- tive control (MRAC) has evolved as one of the most soundly developed adaptive control techniques, though several constraints remain to be resolved. Not only the stability property has been rigorously established [11]

[lo]

but also the performance has been made continu-

ously improving [3] [14] [4] [7].

Recently, neural networks have shown great promise in the realm of nonlinear control problems because of their universal approximation capability. This power- ful property inspires a great number of neural-network based controllers without significant prior knowledge of the system dynamics. Till now, some developed feedforward neural-network based control schemes, whose network parameters are adapted according to Lyapunov theory, have provided rigorous theoretical analysis. In particular, Sanner and Slotine [13] uti- lized the spatial sampling theory to design a Gaussian radial basis function network. Based on their work, adaptive control theory can be applied to deal with the plant nonlinearities. Moreover, Chen and Liu

[l],

Chen and Khalil [2] used multilayer feedforward net- works along with the backpropagation (BP) rule and on-line update rule, respectively, with a deadzone to overcome the modeling errors. Following the similar line of thoughts, Lewis et al. [9] also utilized multilayer feedforward networks, but applied the el-modification to the BP rule and added an extra robustifying control signal.

Despite that there exists quite a few neural-network based controllers for nonlinear systems as mentioned earlier, there are relatively fewer results about apply- ing the similar concept to solve the traditional MRAC problems. By doing so, it in fact can be shown that the need to have some reference model to be followed by the unknown plant, as has been quite often seen in the literature, can be completely relaxed i.e., the class of MRAC problems can simply be reformulated as adap- tive tracking problems, and, hence, the so-developed approach apparently will be more general.

In this paper, we propose a novel neural-network ap- proach to solve the traditional MRAC problems. First, a feedforward neural network model [9] (cf. [8]) is es- tablished and detailedly analyzed for subsequent con- troller design. It is shown that a bound on the model- ing error can, in fact, be expressed as a sum of a lin- early parametrized form and a small residual error. By adopting a neural-network based sliding mode control, it can be shown that the present developed approach does not require the prior knowledge of the so-called "optimal" neural networks, or the "optimal" neural weights, in contrast with all the other approaches men- tioned previously. Along with the control law, a stable adaptive law is devised by Lyapunov theory, whereby the boundedness of all signals as well as the conver- gence of the tracking errors of the closed-loop system are clearly guaranteed.

2

Problem Formulation

Consider an SISO linear system described by

where m

5

n , y p amd u p denote the plant output and the plant input, respectively, and n and m are some known positive integers.

Such linear plant does not have to be stable, but the sign of the high frequency gain k, has to be known a priori and the plant has to be minimum-phase, which are stated in the following assumptions:

Assumption

(Al).

The value of k , may not be known, but its sign should be known. Without loss of generality, throughout the paper we wall assume that it is positive here.

Assumption

(AZ).

The plant transfer function de- scribed b y (1) is minimum-phase, i.e., all its zeros lie in the open left half complex plane.

Assumption (A3). All the coefficients of 5ip(s) and

d p ( s ) are unknown a priori, but they are co-prime.

Let Y d denote the desired trajectory to be followed

by the plant output y p . It is assumed to be at least ( n

-

m ) times differentiable and satisfies the following assumption :

Assumption (A4). The signals Y d , Y d

,

...,

y f - m ) exist and are uniformly bounded.

Now we are ready to sate the problem to be solved in this paper. Given a linear SISO plant described by eq. (1) satisfying assumptions ( A l ) - (A3) and given a desired trajectory Y d satisfying assumption (A4),

devise control input u p using only input and output measurements such that the plant output y p will follow

yd as closely as possible.

h

(1)

3

Feedforward Neural Networks

In a neural network, the simple nonlinear elements, called neurons, are interconnected, and the strengths of the interconnections are denoted by adjustable pa- rameters called neuron weights. A three-layer feedfor- ward neural network is used to design a robust adap- tive neural controller. The structure of the neural net- work is shown in Fig. 1, which performs as an approx- imator described in a matrix form as follows:

h

h n ( f , @ h , p h )

=

@ h U ( c h ? a )

,

( 2 ) where @h E R m x p and

i&

E Rpx("+l) are output- hidden weight matrix and hidden-input weight ma- trix, respectively, ?

c

R n x l is the input vector,

C, E ( Z T , -l)T E R("+l)xl is the augmented neural input vector (the -1 term denotes the input bias),

E R , i = l ; . . , p

h 1

uj (V*, 5,)

1

+

exp(-i4,f,) (1) \ ,

is a sigmoid function, and

u l ( v h 1 2 a )

a(v&)

=

(

where v h includes the threshold, a ; ( v h , % , )

zi

for

each i = 1,

.,

p , and U ( V h ? , ) ? g.

Theorem 1. Let 2 E

V

( a compact subset of R " ) ,

h(?) :

D

+-

Rm is a continuous function vector. For an arbitrary constant E

>

0 , there exists an integer p

(the number of hidden neurons) and real constant op- timal weight matrices w h E R m x p and v h E RPx("+l)

such that

)

,

with p h E

(

lh'

)

,

up ( v h , f a ) VhP

h h

h

where E h ( ? ) as the approximation error vector satisfy-

ing llEh(?)ll

<

E ,

v%

€ 2). The optimal approximator

can be described as

hn(31 w h , v h ) E W h U ( V h 3 a )

where

11 ' 1 1

denotes the 2-norm, and 5 , (?*, -l)*.

Proof: Please see Funahashi [5].

In general, studying the stability of a multilayer neural-network based control system is difficult be- cause the corresponding dynamics are nonlinear in adjustable neural network weights. Such a structure

is generally not suitable for a d y t i v e control. Fortu- nately, the approximation error h , can be expressed in linearly parameterized form modulo a residual term. This is stated in the following lemma.

Lemma 1. [/?]hang] Define the estimation errors of the weight matrix as:

-

h

-w h w h

-

w h , v h v h - v h

and the estimation_ error of the hidden layer vector as

U E U ( V h 5 , ) - a ( V h % c , ) , then the function approxima- tion error h , is

-

-

A A h h, ? h(?) - h,(%, W h , v h )

-

= w h ' ($ - Z ' p h 2 , )

+

F h s ' V h f ,

+

d h (3)

where the residual term d h can be expressed as d h = w h z ' v h ? ,

+

w h o ( c h ? , )

+

Eh(?)

,

wzth o ( . ) representang a s u m of hzgh order terms of the argument an a Taylor series expansaon

Lemma 2. [/?]hang] T h e resadual term d h can be bounded by a lanear-an-parameter functzon, z.e.,

lldhll

<

QTYLY (4)

where the unknown parameter vector CUh E 'R4'l as composed of optamal weaght matrzces and some bounded constants, and the k o w n functzon-vector

Y,

2s equal t o (1, 115alll ll2all IlwhllF, I l E a l l I I V h l l F ) T .

4

Variable Structure Based Adaptive

Tracking

Controller

Referring to the SISO plant described by ( l ) , we let

( A P , b,, c p ) be its minimal realization given below:

-U, 1

0

" '

0

--a1 0

0

. . .

= A p x p

+

b,u, (5) y, =

(

1 0 0 " ' 0 ) X P =cpTxp - where bi = 0, ...., bL-m-l - 0 , b& = k,, b;-,+, = k p b l , ...., b;

=

k,b,. Let A0 be a Hurwitz matrix chosen as follows:

-4; 1 0 " '

AO =

[

0 . 1 .

1::

f

1

- U ; 0 0

. . .

which will have the desired property to be clear later on, Then, it is straightforward to derive the following from eq. ( 5 ) :

x p = Aozp

+

(A,

-

Ao)xp

+

bpup

= AOxp +Gyp

+

bpUp

Y p = + P

where we used the fact that

(AP - A 0 ) z p =

(

U; - U, ' . . U; - )T cTxp

I

-

- ;rc;xp = ayp

Now, we again adopt the mixed use of a timed signal and a Laplace transfer function to obtain the explicit

expression of 2, in terms of yp and up as follows:

x , = ( S I - A o ) - ' Z ( y p )

+

(SI

- Ao)-l b p ( u p ) + E ( 6 )

- -

-

where i ( s ) = d e t ( s 1 - Ao) is the characteristic poly- nomial of A0 with degree n , d,3(s) are polynomials of

s, i = 1 , 2 , j = 1 , 2 ,

...,

n , of which each has degree no greater than ( n - l), and E stands for the exponentially decaying terms to capture the initial condition effect.

On the other hand, define an n x n Hurwitz matrix A as:

h

0 1 . . .

1

whose characteristic polynomial apparently is the same as that of Ao. Then, construct two n-dim signal vector w1 and w2 as follows:

&I = A w l + b A u p

W z = A w a + b ~ y p

where bA = [0,

...,

1]*, which by some manipulations will lead to explicit expressions of w1 and w2, i.e.,

/ l \

By comparing (6) and ( 7 ) , ( 8 ) , we can easily con-

clude that

x P = T i w i

+

TZWZ

+

E

for some proper n x n constant matrices TI and T2. yp - ?&, and define a sliding surface variable

s

as :

s

= e t - ' )

+

w h e r e r = n - m a n d X i , i = 1 , 2 ,...., r-1,arechosen such that the polynomial

Let e , denote the output tracking error, i.e., e , =

+

.

.

+

~ , . - 2 e p )

+

Xr-leo

F ( s ) = sr-l

+

X I S r - - : !

+

' ' .

+

X r - 2 S

+

x,-j

r-1 . - 7 - 1 .

-= I I ( S + Q i ) = r r i ; ( S )

is Hurwitz, for some cri

>

0, i = 1 , 2 ,

...,

P - l. Obvi- ously, S = @(.s)(e,,). Using the fact that

By some observation of the above set of error equa- tions ( l l ) , if we define the signals u i , i = 0 , 1 ,

...,

r - 2, where uo = up, as follows:

yf) = cFA;x,

+

c;fA;-'bpup = c;A;xp

+

kpup we can obtain the following equality:

1 - 2

e , = --Me,a + I C p

for some

>

0, where 5 =' [ ~ : ~ , w : ~ , Y f d ] ~ with W f l = h ( u l ) , W f 2 = i;iSi("Z)i 1

and e,A will be defined later in the sequel.

and Yfd = A ( y d ) ,

P ( S )

ur-2 = - s g n ( e r - l ) ( ~ ~ ~ ( ~ r - i ) t ~ ~ m

+

?>

(12) To proceed with the controller design, we define a

set of auxiliary error signals: _{for some}

1)

>

0, then we can readily conclude that e,,

i = 1 , 2 ,

...,

P - 1, will reach the zero value in finite time using the following argument:

1 1 1 1 el = r,(s)("P) - - ( U d F ( r s ) d

-

e;/

<

--cr,/eiI

-

17 d t ' - e2 = -(u1)

- -(m)

12 (5-1 F ( r s )

1 1 when le,/

#

0, i = 1 , 2 , ..., r - 1. Thus, we can view

L - l ( S ) ( U Z )

-

- ( U r - l ) F ( r s ) (10) (12) equivalently as e r - l =

h h

~ I ( s ) ' . . l r - l ( s )

m

F ( S ) - l F ( 5 )

where F ( r s ) is any Hurwitz polynomial in T S for some

r

2

0, with degree at least two and F ( 0 ) = 1. In the literature, l / F ( r s ) is referred to as an averagzngfilter, which is obviously a low-pass filter whose bandwidth can be arbitrarily enlarged as r

-+

0. In other words, if r is smaller and smaller, the filter l / F ( r s ) is flatter and flatter.

up = (ur-1) = n ( u r - 1 )

after

( l o ) , we can rewrite it as

fininte period of time, Referring back to

1

Rewrite eq. (11) as: e,, =

where 6, = G ( r s ) ( u r - l ) = -(1 - 7 1

( T J ) - 1 ) ( u r - 1 ) sat- isfies

h

F ( r s )

IbT(t)I

I

IG(TS)(ur-l (t))I

L-

r f c I

I

(Ur-l)tIIco e r - l = - a r - I e r - l + ur-2

-

-(ur-1)(11) _{for Some} f c

2

0.

By Theorem 1 and Lemma 1, we assume that and note that there exists an optimal neural approximator q5n =

W+a(V+%,) of the function q5(%) and then design a l-(uj(t))l

I

211(uj)tlIm

,

j = 1 , 2 ,

....,

P - 1 , V t

2

0 neural approximator & ( W + ,

v+,

5,) to model such

function $(?), and the approximation error $n is where

I l ( . ) t l l m

denotes the truncated sup-norm. The

-

-

-

pensatzon szgnas to be defined later.

h F(7-s) 1

-

b

(SI $n(W+, V+, % a )

=

d J - &

signals U , , i = 1 , 2 , ..., P - 1, are called auxzlzary com-

-

where 5, E ( Z T , - l ) T E R ( 2 n + 2 ) is the augmented neural input vector, W , W , - W,, V, G V, - V,

are the estimation errors of W , and V,, respectively. For clarity, we rewrite all the variables explicitly as

follows:

-

A - h

P,

= ( v ; , . . . , v , p ) -T T E ~ p x ( 2 n + 2 )

,

h

w,

=

(E,l,

" ' , E , p )

E R I X P , U = (U,(P,12a)l..

.

,

up(P,pza))

E R P X l

,

2

= diag{a:(V,,z,), . . .

,

cT;(P,pZa)} E R P X P h

Furthermore, from Lemma 2 the upper bound of the residual term d, can be found as follows:

Ild,ll

F

@$Ye

where 0 4 E

R4x1

is an unknown parameter vector, and Y, E

R 4 x 1

is a known vector defined to be

Consequently, the control law U,.-, is designed as

h

U,-1 =

-4n

+ v p

v p = - s a t ( e , / c ) ( B T Y ,

+ P O )

,

(13)

for some P o

2

0 , where a deadzone size c is used to avoid discontinuous control transitions, and s a t ( . ) is the saturation function and the adaptive law is de- scribed as

where the adaptive gain matrices

r w ,

r,,,

and

ra

are all symmetric positive definite with

eo& = e, - E s a t ( e , / c )

being used to carry out a smooth adaptive law with a deadzone size E

.

Such a deadzone technique has often been incorporated to deal with the phenomenon of parameter drift in robust adaptive control theory [ l l ] . Note that the function e,A has several useful properties as listed below:

The block diagram of the proposed controller is shown in Fig. 2, and the effect of this controller is stated via the following theorem.

Theorem 2. Consider the 5'15'0 system ( I ) satisfying Assumptions ( A l ) - ( A 4 ) . Define the control law as

given in (14) and the adaptive law as i n (lS).If Y d , . '

',

y$") are all bounded, then f o r suficient number of hid- den neurons the tracking error e

=

y - Y d wall asymp- totically converge to a neighborhood of zero whose size is of an order of E , and all adjustable parameters as

well as the internal signals will remain bounded. Before we prove Theorem 2, we first present the following useful lemma.

Lemma 3. Let u p be the variable defined in eq. (12), then it follows that

f o r some

P I ,

,&

2

0 , provided that

P+,

E,+,

and a ,

are uniformly bounded.

Proof of Theorem 2: From the definition and prop- erty of the function, +(.), in equation (9), Theorem 1 guarantees the existence of a sufficiently large com- pact set V (enclosing the origin) and an optimal neural network with approximation accuracy E over the set

D

and satisfying the Remark 1. Now, suppose the initial involved state is well within the compact set 'D, and then choose a Lyapunov function V = VI

+

V2, where

1 2

v,

= -e,A 2

According to the properties of eoA and the adaptive law ( 1 4 ) , it follows that V = V2 and V = 0 for leo[

5

E . In the case where

leal

>

E , the time derivative of VI

can be found referring to the result of Lemma 2 as

v,

= eoaeo

h

I

5

W+ . (Z - Z'V,z,)e,a

+

tr{F+t,e,AW~Z'~

Now, evaluating the time derivative

v,

=

-

0 If le,l

2

E

,

then e,A = 0 = e , ~

which, along with the adaptive law following: 1- 2 V

5

--Me:, of VZ, we obtain -

z;r;Iii,

(14), leads to the

provided that

& ( t )

5

Po for

t

E [O,T] for some posi- tive T. From the above and Lemma 3, we know that

e, and, hence, up are uniformly bounded over [0, TI,

which in turn implies that the whole state X defined in Lemma 3 is also uniformly bounded. As a result, 6, is uniformly bounded as well. By letting T to be cho-

sen small enough so that 16,

I

can be arbitrarily made smaller, it can then be verified that eq. (16) will hold for all time t

2

0, and therefore

-la

~ c i t

<

oo implies e,A E L2

Note that all the derivations so far are independent

of the set

V

and, hence, the foregoing analysis and boundedness conclusion can be easily justified by tak- ing the set 2) large enough. Finally, because of the boundedness of x p and x,, it is easy to obtain the fact that Ya, v p , u p , and e , , ~ are all bounded. As a result of

Barbalat’s Lemma, we can conclude that e,A

-+

0 as

t

-+

m , which thus implies the ultimate boundedness of the tracking errors. Q.E.D.

5

Conclusions

In this paper, a novel approach using neural net- works to solve the adaptive tracking problem for lin- ear SISO systems was proposed. First, a feedfor- ward neural network with sigmoid hidden units were detailedly analyzed for subsequent controller design. Based on the above-mentioned analytic results, an on- line tuning multilayer neural-network based controller has been developed. This scheme combines neural net- works and sliding mode control technique, where the former is mainly applied to model some unknown func- tions of inaccessible states whereas the latter is used to overcome some modeling residual term. It has been shown that this approach can solve traditional MRAC problems but do not require the a przori setting of

the reference model. Rigorous proof using Lyapunov stability theory was provided. Another salient feature is the the controller design never requires the unreal- istic information on the optimal neural network (e.g. the upper bound on the norm of the optimal weight matrices)

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