Design ofRBF Neural Controller withDifferencial Reconstruction and PID Conpensationfor a Class ofNonlinear System with Linear Input

Xinyu WangInstitute of Science and

Technologyfor Opto-electronicInformation ,Yantai University,

Yantai, China ,264001E-mail: leijunwei(l26.com

Unmodeled DynamicsJunwei Lei

Dept ofAutomatic Control Engineering,Naval Aeronautical Engineering Academy,

Yantai China,264001E-mail: leijunwei@l26.com

Hongyun YuSchool of Mathematics &Information

Yantai Normal University,China,264001

E-mail: leijunweig126.com

Abstract-Considered both the situation with unknowncontrol function matrices and the situation with linearunmodeled input dynamics ,adaptive neural robust controllerwas designed by using adaptive backstepping method for aclass of multi-input to multi-output nonlinear systems whichcould be turned to "standard block control type", Furthermore,It is possible to make the network more stable and make theselection of simulation parameter more easy due to theintroduction of differential reconstruction which increased thedamp of the system. It was proved by constructing Lyapunovfunction step by step that all signals of the system are boundedand exponentially converge to the neighborhood of the originglobally. Finally, simulation study is given to demonstrate thatthe proposed method is effective and the known information ofsystem was made use of as maximally as possible byintroducing the PID control..

I. INTRODUCTION

In recent ten years, adaptive control of nonlinear systemshas been extensively concerned,and many significant resultshave been obtained[I-5] . The researches made in article[13]--31 have formed an important branch of adaptive

control, which make use of the ability of neural networks toapproximate smooth nonlinear functions to design controlsystem. Backstepping design technique is one of method todesign nonlinear system. It deduces the control principlethrough structuring Lyapunov functions step by step. Greatsuccess was obtained in nonlinear control by using ofbackstepping control technique and neural network.If thecontrol function matrices is known,we could use the methodproposed in paper [4].

The situation of a 2-order system with unknown controlmatrices was considered in paper [6]; the situation of a2-order system with known control matrices and unmodeledinput dynamics was considered in paper [7],but the controlmatrix of bn need to be a diagonal matrix. Both thesituation with unknown control function matrices and thesituation with linear unmodeled input dynamics wasconsidered in this paper, and by using adaptive backstepping

method , adaptive neural robust controller was designed foran a class of multi-input to multi-output nonlinear n-ordersystems which could be turned to "standard block controltype". Finally ,It was proved by constructing Lyapunovfunction step by step that all signals of the system arebounded and exponentially converge to the neighborhood ofthe origin globally.

The known information of system was made use of asmaximally as possible by introduing the PID control , and anew kind of tuning law of neural network was adopt toimprove the ability of neural network to approximate theunknown function. Furthermore, It is possible to make thenetwork more stable and make the selection of simulationparameter more easy due to the introduction of differentialreconstruction which increased the damp of the system.

I. SYSTEM DESCIPTIONS

In this paper, consider a class of nonlinear systemswhich can be transformed to "standard block control type"as follows:

X1i = f, (XI,X2,..., )+b (x,X21.Xi)Xi+

-'n = fn(')+bn(1)y =x (1)

where =[x,xT ,...,xT , x ERi, i = 1,2,..., nare the state variables of the system andn, =n2 ... = nn are the dimensions of each

subsystem, xi = [xl2 ,..., xi i<n

u=Lui u TE Rnn yyERnI are the controlinput and output of the system. Assuming that fi (.), bi ()are unknown smooth matrixes with appropriate dimensions.The goal is to design controller to make sure that the output

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d

y of system (1) tracks the desired trajectory xiAssumption 1: A function vector h : Q2 I R,P for

any a > 0, there always exist a Gauss function arrayB: Rm i- R' and an optimal weight matrix W such

that |h(x)-W*TB(x).<cr, VxeQ.Where Q is

a tight set ofRm, anddef

h(x) -W*TB(x)= Ah(x) (2)

is called reconstruction error. Define W = W-Wwhere. W E RIx3 is the estimated values of W

Assumption 2: gi1 (ui) is existent andcontinuous.

III. DIFFERENCIAL RECONSTRUCTION

According assumption 1, there always exist an optimalweight matrix W and a fimction array which is formed bytrigonometric functions such that:

n nh(x) * W sin(nwt) + E W* cos(nwt)

i=1 i=l

Where W is the i+n item of WV, so is Wbi+n.Choose the difference of network to reconstruct it, then

suppose that there exist an optimal weight matrix

Wb which can approximate the unknown function

h(x) such that:n n

h(x) = JY sin(ncot) + Z W+i cos(ncot)i= i=1

i=l ~dt i=1 i,+ dtIt is easy to predigest the formula as follows:

n

h(x) = (bi- k no,i+n) sin(noit)i=l

n+ (W,, + kdWb +ncf) cos(nwt)

i=1Then two equations was get as following:

W*.ikW*k nw=W*Wb,i- dWb,i+n ai

W + k Wb,i+n WWbi db,Wani+nIt is easy to get the solutions such as:

wbi Wa! + Wa,i+nbi 2

* *

wb,i+ = ai+n2 (i = 1,2, ..., n)b,i+n 2nwo

Then it is proved that the aptimal weight matrix Wb isexist contemporary.

IV. DESIGNS AND STABILITY ANALYSIS OF CONTROLLER

Firstly, consider the following subsystem:

X1i = f, (x,) + bj (XI )X2Define a new error variable z1 = x -x , th

en we get:

(3)

=f (xl) + bl(xI)x2 -*d (4)Adopt the following expected feedback control law,we

have:

xa= b, (xl ) [-fi (xl ) + _id k,z, + u,I + upj (5)

Where E is an unit matrix ,and uv, is a robust item, weassume that:

UP1 1(1)kll+kl|lt dlzl ) + Uht (6Where:Uht=bi (x1)W1TB,(w1) - W1T (B1(wj) + K JB,(w,)dt)

Because bl (x1), f1 (xl) is unknown ,so we consider to use

neural network to approximate dd so

X2 =W1 Bi(w) + Ahi(wi)

=WI B1(w1) + W, B1 (w1) + Ah, (w1)Where w1 = lx dxT xruT J, we choose the

expect value of x2 as (8):d AT~ '-

X2 =WI B.l(wl)+kp,lZl - ki1 zidt+kdl l (8)

define a new error variable z2 = - x2d andchoose Lyapunov fumction as:

VIl=2l Z1 +M Il wliWli (9)

Where WI IWI I In

zj =[z11 ... Zjnj , according (3-8), we get:T.=Z (xf dd _-Cz1 z1 (fXI) + b, (x1 )xdZ1Z1Z1 (f(Xl)+l(xl)2 -1 d(10)+b X)(2 -2 ) l(Xl )(X2 -2 )

1821

V1=z (-klz +bl(X1)Z2 +

uvT

1(BI (wl ) + Ki J (wl )dt)n,

-b (x)Ah, (w,)) + ZWli Fwi Wlii=l

Choose the tuning law of neural network as:A

= -Fw1j (B1 (w1 ) + K, JR1 (w1 )dt)z1, (Substitute (12) to (11):

I=-|lZ111 + Z[[b(xj)Z2 + UV (A-(x_)M( )] (

Assume:

b1(X1)Ah1(W1) =[11 (W1) ... ln (WI)Uvl= vll *.. Uvlnl according assumption 1

we set:

_Pii (X1 )ZIi 1zliu p~(x1)lz1I2vli-+l-()t

Pli (xI) > loli (wI )IThen we get:

zli(uvi- oli (w)) . ~Ie-atSubstitute (15) to (13) ,we get:

V1 ~~<Tb||lllb(XI)Z2 + njlle-'tV1I .! -ki liz1 112 + ZTbx)Consider the second subsystem:

i2 -x22)bZ2 f (X2) + 2 (12 )X3 - 2

Adopt the expected feedback control law as:

xd4 = b2(2-1 [-2(2)+ d k2z2

-bl[GX)zi +UV2 + (b2(2)-E)W B2 (W2)]Using the same method as before, we consider t

neural network to approximate xd ,at the same time s

output of neural network as the expected value of x.get:

4d ArTB2(W)+WTB2(W)A ()X3d W2 B2(2 W2 B(2) + Ah2(W2)

=X + W2 B2 (W2)+ Ah2 (W2)Choose Lyapunov function as:

17' i-, 1 T 1W,V2 =V + 2Z2 Z2 + 2i2 =

Choose the tuning law ofweight as form as (21):

W2i = -1w2jB2 (W2 )Z2iit is easy to get that:

V2 < -kII|ziII2-k211z2112 +Z2jb2(-2)z3 + n22e-al

Following the same procedure,we get:n-i n-i

Vn-.<-ZklI|zI2 +Zn_lbn-1(in-_)zn + n 4eti (23)i11) i=l i=l

fnn(n ) +bn (Xn id8- (24)Choose the expected feedback control law as:

ud =nG {bn (Xn)[-fn(2n)+4 (25)

12) -bn-I (xn-I)Zn-I -knZn + Un ]}Unv = UnvO + Unvi +Unv2

13) ~uo =-bn(jn)G(kpz, +ki Jzidt+kz+W B(w))

Unvl = WTB(w) (26)

(14)

whereUnv2 = [unv2l ... Unv2n,r,assumebn (1n)G(Ah(w)) = [onl(w) ... Onn(w)fjIZn = [Znl ... Znn ]T Choose Unv2i as (27):

Pi (W)zni Izninv2i - 2(W)I 12 +n 2 -2at

Pi (W) > Joni (W)l(27)

(15) Then:

Zni(Uvn2i wni(w)) < 5ne (

(16) Because G1 is an unknown function and Ud

couldn't be get, so we consider using neural netwo(17) rk to approximate it. According assumption I,we g

et:

(18)ud d(W*TB(w)) +;h(W)

=FWTB(w)+k d(WTB(w))+WeB(w)+Ah(w);et the Choose the hybrid control as:

we u = W B(w)+k z1 +k, Jz1dt +kThen:

(19) Zn Zn = n n (in )+bn (In,)( ) xn]+ Znbn (In IG(u) - G(u)]

According assumption 2,we get:zzi ZT bTkz- u]

(20) ZnTn ,n [n-b(jn-1 )Zn-I (nZn + Unv ]+Z"nbn (3n G(u-u")

According to (29-30),we get:

(21) u-u" = kpz +k Jzi dt + kd

-WTB(w)-Ah(w)(22) Substitute it to (32),we get:

(28)

(29)

(30)

(31)

(32)

(33)

1822

T. T T2 ZTUZZ -ZO4 X1)1 KZI Zn Zn nbnnl n- )Zn-I ikn n n nv

+Zn bn(n )G(kpz + ki zIdt +

kdZl - WTB(w) - Ah(w))Choose Lyapunov function as:

1~~~~V = Vn-I + 2 zn Zn + 2 Wi FwWXChoose the tuning law of weights as:

Wi = '-rwB(W)Zni

Where WT=[!... T T=~WherW =W1 ** *Wn ] Zn = [Znl ...According to (23-36),we get:

n n

v<. kikzn2 + nje-a'i=l i=1

(34)-4

._

c-Eb

rA

(35)

(36)

znnTFigure 1. The trace of Yd (no PID control )

(37)

V. SIMULATION STUDY

Consider the following 2-order system:*1 =xle 5x1 +(I+X12)X22= x2x, + (3+ cos(x1x2))v

v=2uThe initial state and the initial value of the weight are 0.

We choose the expected trajectory as: Yd = 2 + sin(t),using a single input neural network, we choose input asw1 = z1'w2 = z2 WW and W2 have the form as

following(where n= 25, c = 0):W V=... Wn Wn+l ... W2n W2n+1]

B(x) = [sinx * - * sinnx cosx ... cosnx c]Where FWI1= 1/5 ,rw2l 1 1/100 , Uvnl = 0,

Uvn2 =0 ,kp =kpl ==-2 , ki =kil =0kd =kdl=-1.2,Ki = 0 l, Kdd= 0.01The result were showed by picture 1-5. Figure 1 are the

simulation result without PID control. We can find that theweights are stable finally and the control u is not very big soit is practical and the known information of system wasmade use of as maximally as possible by introduing the PIDcontrol.

Figure 2. The value ofweight W,

0)

3o

02

0)

~~-0.04

-0.06

yx_.. .... .........

0 10 20 tf )30 40 50

Figure 3. The value ofweight W2

1823

U.U4,

-n ri-. -t

k

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o-

Figure 5. The trace of Yd (with PID control)

VI. CONCLUTIONS

Both the situation with unknown control functionmatrices and the situation with linear unmodeled inputdynamics was considered in this paper ,and by usingadaptive backstepping method adaptive neural robustcontroller was designed for an a class of multi-input tomulti-output nonlinear n-order systems which could beturned to "standard block control type". Finally, It wasproved by constructing Lyapunov function step by step thatall signals of the system are bounded and exponentiallyconverge to the neighborhood of the origin globally.Simulation showed that the known information of systemwas made use of as maximally as possible by introduing thePID control , and a new kind of tuning law of neuralnetwork was chose to improve the ability of neural networkto approximate the unknown function. Furthermore, It ispossible to make the network more stable and make theselection of simulation parameter more easy due to theintroduction of differential reconstruction which increasedthe damp of the system.

REFERENCES

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