Robust adaptive tracking for nonlinear systems using dynamic neural networks

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

14th World Congress of IFAC

Copyright ((.; 1999 IFAC 14th TrielJuiaJ W-orld Congress, Beijing~ P.R. China

1-3b-03-2

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS USING DYNAMIC NEURAL NETWORKS

Dai Qionghai*, Wu Hongwei*, Sun Fuxin*,· Li Yanda* and Wang Wei**, Chai T. y.*.*

* Automation Department. Tsinghua University, P. R. China *"" The Center ofAutomation, Northeast University, P. R. China Corresponding Address: Dr. Dai Qionghai, Automation Department, Tsinghua University, Beijing, Email:

J00084~

P R. China

qhdai@}jerry~au.tsinghua.edu.cn

Abstract: A dynamic-neural-network-based control is proposed for a class of affine systems. The approximation for the unknown affine system is studied. The robust learning Jaws are derived with respect to the modeling errors. The 0 - protection and hysteresis technique are used to guarantee the stability of the resulting controller. The features of this approach lie in that neither off-line learning nor sufficiently small initial

parameter errors are required. It is proven that the proposed algorithm guarantees signals bounded in the adaptive loop under any bounded initial

conditions~

an

Simulation

results are given to verify the effectiveness of the proposed adaptive control algorithm. Copyright© ]999 IFAC

Keyword: dynamic neural network, affine nonlinear system robust adaptive tracking oa

I~

connected

INTRODUCTION

the

static

back

propagation

neural

networks with linear dynamic systems in series or in

Adaptive control of nonlinear dynamic systems by

parallel. Levin and Narendra (1993) indicated the viability of the neuraI-network-based controllers on

using neural networks has recently been an active area~

Narendra et aJ. (1990; 1994)

the basis of the results in nonlinear control theories.

originally formulated the problem. They proposed dynamic back propagation schemes, which

In (Liu and Chen, 1993) and (Jin et aI., 1993) the

and challenging

convergence

analysis

of the

multi-layer-neural-

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14th World Congress of IFAC

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

network-based direct adaptive control was provided.

architecture capable of modeJing continuous time

A weight learning algorithm was presented by using the gradient descent nlethod with a dead-zone

dynamic systems as below: i = f(x) + g(x)u

function, and the errors were proven exponentially

where x ERn is the state variables,

convergent. But their conclusion required the initial

(1) U E

R m is the

input, f(x) ERn and g(~ E R nxm are the function

parameter errors sufficiently small. In (Dai et aL,

vector and matrix respectively.

1995) the robust stable analysis of indirect adaptive

control with modeling errors was presented. A novel neural-network-based controller was explored with

When some mild assumptions are imposed on the

the robust stability to modeling errors. And the

system (1) and the neural network topologies, the

learning law was given by using Lyapunov theory,

following theorem is derived.

which guaranteed the stability of the resulting

Theorem 1 Suppose i(O):= x(O) and

controller and avoided iterative training procedures.

U E

U c Rm

where U is a compact set. The two-layer dynamic neural network shown in Fig. 1 satisfies

This paper is organized as follows. Section 2 develops the problem of approximating the affine nonlinear systems by using two-layer DNNs. In

Section 3, a class of bound functions for modeling

.where

errors is employed to investigate the robustness

network, W

E

R

properties of the DNN-based controller with respect

weight) V

R

Nxm

to modeling errors. Section 3 also proposes a law for

weight~

is the state vector of the neural

XE RN

E

h'xN

is a N X N matrix of synaptic

is a N X m matrix of synaptic

A and Bare N X N diagonal matrices with elements the scalars Qj, b i for i=l, 2~ , N,

robustly tuning the weights, in which a sort of hysteresis is used to guarantee the stability and

S(x) is a N-dimensional vector and S'(x) is a NXN

robustness of the controller. In section 4, some

diagonal matrix with elements 8(XI) and S'(,'Yi)

simulation effectiveness

results

are

given

to

verifY

respective Iy.

the

monotone

of the proposed adaptive control

s(.)

s'()

and

increasing

are

functions

both

This

smooth two.. layer

algorithm. Finally the conclusion of the paper is

recurrent DNN with dynamical elements in its

dro\vn.

neurons and N

~

n, can approximate the nonlinear

system (1) (proof omitted). 2. APPROXIMATION OF AFFINE SYSTEMS BY

USING DYNAMJC NEURAL NETWORKS

In order to approximate the behavior of a dynamic system, it is clear that any proposed neural network configuration must have some feedback connections.

Such networks are named as dynamic networks. The

neural networks is determined by weights. Therefore the representation input-output

response

of

Fig.. 1. The architecture of the two-layer dynamic

capability of a given network depends on whether

neural network

there exists such a set of weight values that the neural nern:ork configuration can approximate the behavior

In this paper the network with N

of the given dynamic system. In this

section the problem

is considered

=n

and without

any restrictions on the weight matrices W and V

of

is considered. Such a neural network is known as the

constructing a two-layer dynamic neural network

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

generalized

Hopfie Id

network.

The

differential

equation of this network is described by

14th World Congress of IFAC

In fact a very reasonable assumption (Craig, 1988) is that

i = Ax+ BWS(x) + BS'(x)Vu (3)

{m (x) =

ao (:td) Xd' xd) + a (e)

(8)

The computer simulation results have shown that the

with ad (Xd' Xd' Xd) and a(e) both continuous polynomial functions. The coefficients of aCe)

model represented by equation (3) is more effective

depend on x,,-, id ~ Xd •

and the learning is also faster, which means that the network

has

strong

approximation

ability

for

To solve this problem it is required to add a robust

nonlinear system (1). In this paper the robust tracking

term

control of nonJinear systems is carried out using the

yER.

UR

(t) , based on the function defined as (9) for

DNN model as (3).

l-exp(-Y/) sat(y) - {

-

with

3. ROBUST ADAPTIVE TRACKING CONTROL BASED ON DNN

r

/y

-(l-exp( ~)

ify~

(9)

ify
a small positive (design) parameter. When

Y ERn, sat(y) is a vector with entries sat(Yi)'

In this section a robust adaptive tracking control of affine nonlinear dynamic systems is studied. When

Take the control input as

DNN is considered as a model of the plant (I),

(10)

modeling errors are defined as following: with auxiliary signal

t(x):::= Ax+ BW*S(~+BS'V"'u -f(x}-g(x)u (4)

u R =- ~m(x)sat(e) (11)

with W . . and V'" denoting the unknown optimal weight values. The modeling errors arise from the

The closed-loop error dynamics become

e = Ae+BWS+BS'Vu+ t(x)+u R

inadequacy of the approximation. Thus the dynamic

equation is expressed as

x== Ax+BW"'S(x)+BS'(x)VAu- ~(~

(5)

with

W=W-W·

and

The

architecture

of the

(12)

V=V-V·. proposed

DNN-based

adaptive tra~king controJ is shovm in Fig .2. In the tracking

~roblem

the control objective is to

force the pJant's state vector x to follow a specified desired traj ectory x d. The tracking error vector between the plant's states and the desired states is defined as fo lIowing: e=xd-x

(6)

Differentiating (6) and considering (5) the following Fig. 2 the architecture of the DNN-based adaptive

expression is obtained:

e = Ae -

BW·S(x) - BS'(x)V· u + t(x) + r

with r ==- Xd -

AXd.

tracking control

(7)

The weight learning Jaws are given in (Dai et al.,

DNN is here used to make the

1995) as (16) when not considering modeling errors:

nonlinear system track the desired trajectory, thus the

problem is to design a control law.

W= -BPS"(x)E; V== -BPS'(x)eu T

(13)

It is developed that the approximation error bound ~m (x). is not a constant, but depends on x d and x.

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

14th World Congress of IFAC

hysteresis plus 0 -protection is considered, shown SI

with

E=diag{el,p~p,en} andS"=

as below:

: [ SI

Due to the presence of modeling errors, the learning laws about W and V given by (13) should be modified. Ioannou and Aniruddha (1991) proposed an alternative leaning scheme which required neither

IIV"II. and replaced

priori knowledge of ~W·II nor the constant tenn equations

for

G'.

laws

are

modified

6~v

L=

-I 6v~V

(15)

0"']

0

0

0"2

>(0',-,[

0

dJ!:. < f. The

L£

0

0

::=

crm _]

o

drifting to infinity.

Gm

0

o

+(m-N-l)om

The hysteresis idea (.MatheIin and Badson, 1995) is

singular value, and

a

b are

a

O"min (.)

+Om

0

Em

:5: (Cm +(m-N~l)om)I

and {tit.} are defined as following

expressions, in which

o Bm

o

applied to guarantee the existence of u. The

Vc

+!3N)"2(CN +!3N)I

UN

O"N+I

modified laws will prevent the weight values from

variables of

diag{o-,}, ,

0

0

and 6 v diagonal matrices, do and d 1

positive constants chosen to satisfy

:=

(14)

6v =-d, 5v -doE with

~l= 1:",

J

a;::; aj "'ii <

:-d1 5w -doE

-BPS'(x)eu T

V is considered

to make E and I:s satisfy the following inequities:

W= -BPS"(x)E -18w I'"

v=

Specifically, the partitioned singular value

[M MslV[N Ns ]] =- [ :

as

followings;

6w

G~.

decomposition of

with an adjusting term. The

learning

where M T 1:""" 1 N is a lower rank approximation of

The constant

denotes the smallest

Bm

and Om are chosen to satisfy the

foIIowing conditions:

is a designed positive constant, parameters

chosen

to

satisfy

l
wIth am > 1 and 0 < 8 m

G; The

am

can be guaranteed.

following

lemma

is

needed

for

further

proceeding.

(min (~(Tk));=O"./c, 1; is the time that

& = -....!!!-. The existence of

Lemma I For weight matrices W and V ~ the following equalities are established (proof omitted).

O'min (V(t))>% \Ilk
tr{WTW}

Furthermore, to relax the restriction that the pseudoinverse rVTV]~l must be bounded, the selective

tr{VTV}

=~II\\f+~llwlr -~~w·t, =~llvf +tllif -tllvof

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

The Lyapunov synthesis method will be used to

14th World Congress ofIFAC

eTp(

derive the stable adaptive Jaws. Therefore we

~x) +UR) ~';m (x)lef Pvec(exp{Jj,{})

consider the Lyapunov function candidate as below:

with vec(y;) the vector with components Y .And by

I T 1 -... T -.. 1 -- T ~ I T v=-e Pe+-tr{W W}+-tr{V V}+-tr{owow}

employing Lemma 1, the following equality is

2

1

2

2

j

2

obtained:

T

+-tr{ Qv Dv} 2

V ::::

(16)

where

P >0

is chosen to satisfy the Lyapunov

equation PA + ATp == -I. The following theorem provides a bound for the adaptive scheme when

-~(l- 4~o )lIe 11 2 +';m (x)leIT pvec(exp{-';!tn

+8~1 (~w·r +llv·r) From (8) it is yielded that

applied to the true system.

Theorem 2 Let :x d , id' Xd be bounded. Cons ider the plant given by (5), and Jet the modeling error bound be of the form (8). The controller is taken as

of DNN are adjusted according to (14) and (15). For any

WeO)

and

YeO)

satisfying

(V(O))~ Gm'

,O'min

Since the exponential term dominates any polynomial it may be argued that for a suitably small y

one~

(10), the auxiliary signal is as (11), and the weights

-;(1- 4~o )1~112 2

vs

+a{)n ~XP{l} +at n J{XP{l}

+8~1 (1Iw·r +/Iv·r) with

uJ

a

continuous

function

x d , id' Xd

of

Proof Differentiate (16) along the solution of (12),

depending on the coefficients of a(e).

the followings are obtained

Since the last terms are independent of e and the desired trajectory is bounded,

I ., T ......, T ,..... T v=--II~I- +e P(BWS)+e P(BS'Vu)+e P ~(x) 2

+ eT Pu R +tr{W

T

an Y /

bounded,

W} + tr{VTV} + tr{ 8"{;r DW } + tr{ 5~ 0v } _1 8d)

(17)

o

/exp{l}'

(lIw ·r +llv.r)

are

a o and a J are a,n ;{xP{J} and respectively_

bound,

Therefore for

Select (8) to guarantee the last term

to satisfy the following inequality:

e;<\g;(x)\ + U Ri )

~ ej(~m exp{-j{ }-8

j)

(18)

et 20

v< 0, implying that

-eiClC;i(x){ -URi)

e, W, V, 6w , 6v

E

Lcn

•

s-e,(.;'mexp{J:}-8;) e, <0 with 0, == ';m(x) -lq(x)1 ~ 0, 2; (';;(x) + URi)

S

4. SIMULATION RESULTS

so that

In order to evaluate the quality and feasibility of the

4m(:l(~eil exp{- j{}-

proposed DNN controller, it is applied to control the speed of a DC motor. The dynamics of a separately excited DC motor are mode led as follows:

Thus

4480

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ROBUST ADAPTIVE TRACKING FOR NONLINEAR SYSTEMS US!. ..

14th World Congress of IFAC

to be robust stable. The learning law is generated using Lyapunov theory. The cS -protection and the hysteresis technique are used to make the adaptive

f(x) ==

system robust and avoid iterative training procedure. Simulation results indicate that the proposed DNNbased controller have good practical use for the It should be illustrated that the robust DNN adaptive

adaptive tracking control of nonlinear systems with

controller requires no knowledge of the dynamics or

modeling errors.

the structures. In the example x 3 can not be measured, which is considered as an unmodeled XJ~ X2

dynamic, and only

REFERENCES

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Narendra

and

Parthasarathy (1990).

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Identification and control of dynamic systems

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T"

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"

:- ••_

-~

:

:

~ -

'"

~.- • •-

: .. -

-

:

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~t5

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:

:

~

n

:

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1

:

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'

:

1~

14

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13

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X m2

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P.N.~

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5. CONCLUSION

adaptive control: a unified approach. Proc.

The DNN-based robust adaptive tracking control of continuous-time

nonlinear

dynamic

systems

IEEE, Vo!. 79, No. 12, 1736-1767.

is

Michel, de, Mathelin and Marc, Bodson (1995).

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approximate affine dynamic systems. A neural-

Multivariabl model reference adaptive control

network-based

without on the high-frequency gain matrix.

controlIer

is

explored.

And

the

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analysis of the function bound about modeling errors J.1.,

is employed to investigate the robustness properties

Craig

(1988).

Adaptive

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ofDNN controlJer. The closed-loop system is proven

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ISBN: 0 08 043248 4