Adaptive Control of Nonlinear Continuous-Time Systems Using Neural Networks - General Relative Degree and MIMO Cases

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

ADAPTIVE CONTROL OF NONLINEAR CONTINUOUS-TIME SYSTEMS USING NEURAL NETWORKS - GENERAL RELATIVE DEGREE AND MIMO CASES Fu-Chuang Chen and Chen-Chung Liu Department o/Contro/ Enginuring, NatioNJ/ Chiao Tung University,lIsinchu , Taiwan , ROC

Keywords -

Neural Network; Nonlinear Control; Adaptive Control.

Abstract - Multilayer neural networks are used in a non linear adaptive control problem . The plant is an unknown feedback linearizable continuous-time system with relative degree ~ 1. The single-input/single-output system is studied first and then the methodology is extended to control square multi-input/multi-output systems. The control objective is for the pIa nt to track a reference trajectory, and the control law is defined in terms of the outputs of the neural networks . The parameters of the networks are updated on-line according to an augmented tracking error and the network derivatives. A local convergence theorem is given on the convergence of the tracking error and its derivatives , and the norm of the parameter error vectors . This control algorithm is applied to control a two-input/two-output relative-degree-two system.

1

local convergence results are arrived. The detailed control algorithm and convergence analysis for SISO systems are provided in section 2 and section 3. The results are generalized to square MIMO systems in sections 4 and 5. The simulation results are reported in section 6.

2

Linearizing Feedback Control for SISO Systems

The plant is a single-input/single-output "Yth order system with "Y ~ 1.

Introduction

=

X"I

f(x)

+ g(x)u (1)

Multilayer neural networks (1] are promising tools for identification and control applications because they are capable of approximating any "well-behaved" nonlinear function to any desired accuracy (4, 5]. Although many related applications and algorithms have been reported , e.g. (2, 3], it is difficult to study the stability issue of multilayer network based control systems because of the inherent nonlinearities in the multilayer neural networks. Recently, convergence analysis of multilayer network based control system is provided in (6] for discrete-time systems, and in (7] for continuous-time systems . The systems considered in (7] are restricted to be SISO and relative-degree-one . In this paper , the results of (7] will be extended to SISO and (square) MIMO systems with general relative degrees. The multilayer networks will be used to model nonlinear functions in the system to be controlled. The control law is generated from the neural network models to control the system to track a reference command. The network parameters are updated on-line according to an updating law which is defined in terms of an augmented tracking error and the network derivatives. A deadzone function which is specified around the origin of the error states is used in the updating law, in order to tolerate parameter errors and modeling errors. Due to multilayer network nonlinearities , only

where x E R"I, and f, 9 are smooth functions. The states x are assumed available. For example, a single link robot can be represented in the form of (1), with "Y = 2 and x (XI = e, X2 = 9) measurable . Differentiating the output 1/ with respect to time for "Y times (till the input u appears), one obtains the input/output form of (1) as y"l

= f(x) + g(x)u

(2)

The system is said to have relative degree "Y if g(x) is bounded away from zero.

Assumption 1

g(x) is bounded away from zero over a compact set 5 C R"1, that is, Ig(x)1

~ b

> 0, V x

E5

o

(3)

For all x E 5 , with Assumption 1, the control law can be defined as u

-f(x) + r = """' :g(x) '-'-::--'-:--

(4)

and the system (2) becomes y"l = r . If the control goal is for the plant output y to track a reference trajectory 1/m, the control input r can be defined as r

901

= y~ + a"l(y~-I

-

Y"I-I) + ... + aJ(1/m - 1/)

(5)

where Q'1" .. , QI are chosen such that the polynomial s'1 + Q'1S'1-I + ... + QI is Hurwitz . Then the control input (5) results in the error equation e'1 + Q'1e'1-1 + ... + Q1e = 0,

(11) where e

=Y-

Ym . Define the augmented error

=

with e Y - Ym being the tracking error . It is clear that e will approach zero.

3

Adaptive Control for S1S0 Systems

In section 2, the control algorithm for the system (1) is described under the assumption that the functions f(x) and g(x) are known exactly. If f(x) and g(x) are unknown, it is proposed to use two neural networks j(x, w) and g(x, v) to model f(x) and g(x), respectively, where wand v being vectors containing network parameters. It has been shown that multilayer neural networks can approximate any "well-behaved" nonlinear function to any desired accuracy [4] . Accordingly, we make the following assumption.

e1

The parameters Q'1" " ,Q1 in (5) and should be selected such that

M(s)

=

/3'1S'1-1

s'1

max Ij(x, w) - f(x)1 ~ c, and

max Ig(x, v) - g(x)1

~

c, \Ix E 5

EJ(s)

u(t)

= - j(x(t) , wt) + r(t) g(x(t), vt)

U(s)

(14)

+ 17(t)] (15)

where 0 0

[

bT

=

0 1

0

0 0

0

0

0

1

-QI

-Q2

-Q'1-1

-Q'1

0

1,eT =

0

0

[

1

/31

P'1

1

Since M(s) is SPR, by the Kalman-Yacubovich-Popov lemma [9], there exist symmetric and positive definite matrices P and Q such that

= -Q

=

f(x) + g(x)u r + [U(x, w) - j(x , wt)) + (g(x, v) - g(x, vt))u] + [(f(x) - j(x, w)) + (g(x) - g(x, v))u] 2 (9) r + [_eTJ + 0(leI )] + [O(c)]

(~Iw.) 1 89~x,;V) Iv.) u

A

Aem(t) + b[-eTJ eTem(t)

0, J.lel

(10)

With r dcfiflcd ill (5), then (9) can be rewritten as

902

(16)

if e~Pem ~ d~

J,

if e~Pem > d~

(17)

where J.I is a positive number representing the learning rate, el is the augmented error defined in (12), and do specifies the size of the deadzone. So far we have described the neural network based adaptive control system . The convergence analysis is provided next . Consider the sets

le

(

e

The network weights are updated according to U pda ting Law

and the parameter error vector at time t as e(t) = et -e. Applying the control u(t), defined in (7) , to the input/output form of the system (2), one obtains

J - [

(13)

D(s)

(7)

(8)

where

= N(s)

= N(s)E(s) = N(s) D(s) = M(s)U(s)

Denote the parameter vector at time t as

y '1

in (12)

If em = [e, ... , e '1-1 f are selected as the states of the system (14), then (14) can be realized as

A

In practice, we need to fix the network size first; therefore, the smallest c for the inequalities in (6) to hold is determined . The parameters wand v which satisfy (6) are unknown. Let Wt and Vt denote the estimates of w and v at time t. Then the control u(t) can be defined as Control Law

+ .. . + /3l s

.• , /31

IS a SPR transfer function and that N(s), D(s) are coprime. Let EI(s) = C{el(t)}, E(s) = .c{e(t)} and U(s) = C{-eTJ + 17(t)}. EI(s) and U(s) are related by M(s), since

(6) 0

(12)

/3'1"

+ Q'1S'1-1 + ... + QIS

Assumption 2 There exist coefficients wand v such that j and 9 approximate the continuous functions f and g, with accuracy ( over a compact set 5 C R'1, that is, 3 w, v such that

= /3'1e'1-1 + ... + /3le

= {eml leml < 0,

e

and le

= {el lel < 6}

(18)

and 6 are positive constants, and consider the where function V(e m , e) defined as

First we want to sh()w t.ltilt. V ~ 0 ill 1.11<' and f are small CII()u!-\h. ~IIPP()"t· I.It;lt.

sd /..

x 11-1 if I>

1.1t(·1I (

Let Ym, . .. , y;:'-I be boulld('d by 1o , ... , I~ _ I , V t ~ 0, respectively. Thus,

,y~-If

E JJ)"

(21) Before further development, we need to make it clear that the control u is bounded if 6 and ( are small enough. The control u (7) would go to infinity if g(x, VI) approaches zero . Since

Ig(x, vd - g(xll

< ~

assuming that

falls into M, which, together with the

enough (for example, if p.

~e+Jr5+ ... +/~_1

where "'(I

)

derivation (25), guarantee that M is an invariant set (i.e., (em{t)).. . . E>( t) will stay ill M for all t ~ 0), If IS large

(20)

x=[y, ...

(tx~~)

be larger than 4 e~ (0) Pe m (0)) and if 6, f are small enough such that (24) holds. Since M C le X le, it is clear that V ~ 0, V t ~ O. From (25), since V < I-'"d o < 0 when e~Pem > YAmu(P)

d~, the total time during which adaptation takes place is

finite. Let Ti denote the time interval during which em, for the ith time, stays outside the ellipsoid (i.e., e~Pem > Ig(x, vd - g(x, v)1 + Ig(x, v) _ g(x)ld~). If there are only .finit.e times that e~ would leave (and __ _ then reenter) the elhpsOld, then em will eventually stay cI0(t)1 + f ~ c6 + f, (22)in the ellipsoid. If em may leave the ellipsoid for infinitely

6~ 6 and

f :::: (,

if 6 and (are small enough

such that c6 + ( ~ b < b, then the network g(x, vd is bounded away from zero and has the same sign as g(x) (since Ig(x)1 ~ b > 0, sec Assumption 1). With bounded states x (x E B"I') and bounded control u, there exist Cl and C2 (depending on "'(1,6, and such that

n

If 6 and ( are small enough, there exists (T

many times, since

E::l Ti

Ti _

d~,

::::

if e;"Pe m

> d~,

<

<0

e;"i Pe m •

(25)

From (23), (24) and (25), it is clear that the purpose of employing a deadzone of size do in the updating law (17) is to cover the modeling error and the nonlinearities associated with the parameter errors, such that the condition V ~ 0 can be achieved. So far it has been shown that V < 0 when em E le and E> E le, provided 6, ( are small-enough. Now, in order to have the condition (20) hold for the whole control ~rocess, i.e., em E le and E> E le, V t ~ 0 (such that V ~ 0, V t ~ 0), we will show that the learning rate p. and the size of the initial parameter errors have to satisfy certain conditions. Consider the set

If p. and E>(O) are chosen such that 2

p. -

6

2 e~(O)Pem(O)'

-P 2 d6

}

d~ -

0 as i -

00

(29)

The result (29) says that em will converge to the ellipsoid. Since V is bounded below and V ~ 0, we obtain

Voo

as t -

00

(30)

00,

(31)

where C is a constant. The above derivations can be summarized as a theorem. Theorem 1 Suppose that Ym, ... , y~ are bounded by 10 , •• . , 11" that frl, ... , fr1' and f3I, .'" f31' are selected as in (11) and (12), and that P and Q satisfy (16) . Given any constant p > 0 and any small constant do > 0, there exist positive constants "'(I = "'(1(p,/o, .. . ,/1"P), f·= (·(p,do,/o, .. . ,/1" fri,f3i,P,Q), 6· = 6·(p,d o ,/o, .. . ,/1" fri,f3i,P,Q), and p.. = p.·(p,do,/ o, ... , 11" fri,f3i, P, Q) such that if Assumptions 1 and 2 are satisfied on S :::> B1'1' and

Ix(O)1

<

p,

IE>(O)I

=

6· 100 - 01 < -

p.

< <

(26)

{

-

1E>(t)l- Cast -

J>'max(P)

. < mm

(28)

00

then (19), (30) and the fact that em will converge to an ellipsoid together imply that

2E>TE> = 0, V V = -p.e;"Qe m + 2p.e;"c7] p.(Td o

as i -

0

=

V Then, for the function V(em,E» defined in (19),

is still finite, it follows that

It has been shown that em [e, . . . , e1'-lf is bounded via an invariant set argument . Hence, from (11), it is clear that e1' is also bounded. Let e~iPemi denote the largest value of e~Pem during the Ti interval. Then (28) and the boundedness of e, ... , e1' together imply that

> 0 such that (24)

if e;"Pe m

=4

e e?.(o~~em(O)' then ehas to

(

•

p.

Vi'

.

then

'

1. The tracking error and its derivatives, i.e., em, will converge to the ellipsoid e~Pem ~ d~ as t 00.

6 and 10(0)1 ~ Vi

2.

(27)

903

IE>( t) I will converge to a constant.

Linearizing Feedback Control for Square MIMO Systems

4

The system studied for the MIMO case in section 4 and 5 is of the general form

[ :~: 1 [::::: 1 where G(x)

=

[

: 9pl

(x)

K. Funaha.hi, liOn the Approximatbe Realia&&ion of Condnuou. Mapping. by Neural Network.", Neural Networkt, Vol. 2, pp. 183192, 19811.

[5] K . Hornik. M . Slinchcomb. &< H . Whil., .. Mullil .. y.r F••dfor .... rd Network. are UniveraaJ Approximaton", Neural Network., Vol. 2, pp. 359-366. 1989 . [6] F.-C. Ch.n &< H . K. Kh"lil. "Adaplive Conlrol of Nonlinear Sy •. tern. u.in, Neural Network. - A Dead-Zone ApprO&f;h," Proceedin,. 1991 American Control Conference, 661- 672. [7] F .·C . Ch.n &< C .. C. Liu."Adaplively Conlrollin, Nonlinear Continuou.-Time Sy.tem. u.inl Neural Network.," Proceedinl' 1992 American Control Conference, 46-50.

+G(x) [ ::

911(X)

[4]

1

91p(X)

(32)

[8] S . Sa.lry &< M . Bod.on. Adapliv. Conlrol. Pr.nlice-Hall. 1989. [9] B . D . O . Anderaon &< S. Vonsp .. nillerd, Nelwork An ..ly.i ... nd Synthe.i., Prentice-Hall, 1973 .

1

[10]

R. A. Horn &< C . R. Johnaon. MalriK An .. ly.i •• C .. mbrid,e Univeuity Pra .. , 1985.

[11]

J . . J . E . Sloline &< Weipins Li. Pr.nlice-Hall. 1991.

Applied Nonline .. r Conlrol.

9pp(X)

Detailed discussions will be available in a forthcoming paper in IJC.

5

Adaptive Control for Square MIMO Systems

6

Simulations

IOr-------~-------~------~------~--_;--,

_

8

: reference corn.m&nd

=0 .7 =0.6 : 6 =o . ~

----- : 6

6

: 6

........

o

;.

~

~

\

The plant used in the simulation is a two link rigid robot which moves in a horizontal plane [11], which is of the form (32), with p = 2, "'(1 = 2 and "'(2 = 2. The neural networks 911 //22 are used to model/I, h, 911 and 922 . The functions 921 and 912 are assumed known a priori. The networks are pretrained until the output error level can hardly reduce any further. The network weights obtained after the pretraining are treated as the correct ones.

il J2,

.' .... . }

·2 ·4

-8

_10L-_ _ _

o

~

___

~

____

w

~....:.....

__

~

_ _ _...J

~

I(sec)

Figure 1: The local property of the parameter errors

Example 1 To show the convergence result is local in parameter errors. The pretrained network parameters are perturbed by some numbers randomly selected from [-6,6] . Fig. 1 shows the first output of the system (i.e., ql, the angle of the first link of the robot,) for 6 = 0.5,0.6 and 0.7 and the reference command. It is clear that the system output diverges when the initial parameter errors are too large. 3 .

Example 2 To show that the convergence result is not local in the states of the system. The pretrained network parameters are perturbed by some numbers randomly selected from [-0 .15,0.15]. Fig. 2 shows the reference trajectory and the ql 's for the three initial conditions of 4.0, 2.0 and -2.0.

- - : ~(erence cornrnand

2

References [11 D . Rum.lh"rl, G . E . Hinlon

&< R . J. Willi"m" " Learnins Inl.rn,,1 RepresentAtions by Error PropaKa.tion ," In Rumelhart and Mc· CI.II"nd (Ed) , Par"II.1 Dislribul.d Proc ••• ing , Vo!. 1. MIT Pr .... 1986 .

[2] IEEE Control Sydems Magazine, !Special iuue&1 on neural network conlrol 'yH.ma, April 1988. April 1989. April 1990 .

[31 K. S. Nu.ndr"

&< K . Parlh ... aralhy. "Id.nlific .. lion .. nd Conlrol of Dynamical System" UsinS Neura.l Networks ," IEEE Tra.n . on N.ural N.lwork •• pp . 4- 27. March 1990.

904

w

12

14

I(SCC)

Figure 2: The nonlocal property of the system states