Adaptive Control of a Class of Nonlinear Systems with Unknown Dead-Zones

Copyright @ 1996 IF AC 13th Triennial World Congreu, San Francisco, USA

2b-06 6

ADAPTIVE CONTROL OF A CLASS OF NONLINEAR SYSTEMS WITH UNKNOWN DEAD-ZONES Ming Tian and Gang Tao Department of Electrical Engineering University of Virginia Charlottesville, VA 22903 Abstract: An adaptive contra) design is proposed Cor systems with an unknown dead-zooe at the input oC a smooth nonJinear dynamics which can be transformed into the parame~nc."trid·feeJbad form. The adaptive controller contaios an adaptive dead-zone inverse to cancel the dead-zone and is designed using the backstepping method for adaptive nonlinear control. The adaptive control scheme ensures closed-loop signal boundedness, Keywords: Adaptive control, dead-zone, nonijnear systems, stability. 1. INTRODUCTION

Dead-zone characteristic is common in many components of control systems, particularly in actuators such as hydraulic servo-valves and electric motors (Dorf, 1990; Kuo, 1995), and caD severely limit system performance. Adaptive control of systems with such non~ smootb nonlinearities was initiated by Recker, et d/. (1991). Most existing adaptive control approaches were not applicable to these systems, which motivated the development oC the adaptive inverse approach Cor plants with an unknown input dead-zone at the input of a linear part (Recker, et al'l 1991; Tao and Kokotovic, 1994; Tao and Tian, 1995). In this paper we will consider the situation when dead-zones appear at the input of a oonlinear part. Cor exampJe, dead-zones in a.ctuators such as electric motors whose dynamics are nonlinear. vVe will further develop the adaptive dead-zone inverse approach for such nonlinear systems.

Recently, significant progresses have been made in adaptive nonlinear control for Ceedback linearizable systems (Isidori, 1989; Sastry and Isidori, 1989; Pomet and Praly, 1989; Praly, .t .1., 1991; KaneU.kopoulos, ,t

2090

d ., 1991; Kristic, et d/., 1992) where the nonlinearities are assumed to be smooth. Adaptive controllers can be systematically designed for the class of feedback linearizable systems which can be transformed into the parametric-puTt-feedback form (Kanellakopoulos, et (11., 1991). For the nonlinear systems transformable into the more restrictive parametnc--6trict~feedback form, Kl'i6tic, tt al. (1992) gives an adaptive control design without overparametrization to ensure globally asymptotic stability, These non linear systems cover a wide class oC real life systems, such as electric motors. However, iD many situa.tions, electric motor systems ma.y contain a deadzone cha.racteristic due to nonlinearities in an amplifier gain or in the flux-ta-current relation of an electric molar (Kuo, 1995). Such syslems can be described by Ihe model shown in Figure 1 where DZ(·) denoles a de.dzone non linearity and SN (,) denotes a smooth nonlinear dyna.mics. j

~DZ(.) Figure 1: Plant with an input dead-zone. It is the goal of this paper to present a solution to adaptive control of non linear systems wit.h an unknown deadzone at the input of a smooth Donlinear dynamics which can be transformed into the parametric-!trict-Jeed6(1ck form, Our solution will be based on the backstepping technique developed in (Kristic, et (11-, Ig92) and the adaptive inverse approach developed in (Tao and Koko-

tovic, 1994; Too and Tian, 1995).

2. PROBLEM STATEMENT

3.1 Dead-zone Inverse

Consider the nonlinear plant in the param etric-strictfeedback form with an input dead-zone:

+ gT lPi(Zl,· ·· . Zt), i = 1" = '1'0(%) + OT '1'.(%) + Po(%)o

%, = ZHl

i. U

= DZ(v)

, " n - 1 (2.1) (2 .2) (2.3)

where () E RP is the vector of unknown constant parameters, l.po,/30 and f{)i, 1:5 i :5 n, arc known smooth nonlinear functions in and Po(z} '# 0,"1% ERn, xi,l :5 i :5 n , ale the state variables. and DZ(·) is the dead-zone characteristic described by

nn,

m.(v(l) - b.) if v(l) ~ b. 0(1) = DZ(v(I» = 0 if b, < v(l) < b. { m,(v(t) - b,) if vet) ~ b, (2.4) where b" b", m,. and rn, are constant parameters. As in (Kanellakopoulos, et al., 1991; Kristic, et al., 1992), %l(t) i. considered as the plant output: yet) = %l(t). The control objective is 10 design a feedback control vet) for the plant with an unknown input dead-zone to ensure that all closed-loop .ignals are bounded and the output yet) track. a reference signal r(I). The control problem has the following features: (i) the output of the input nonlinearity, u(t), is not accessible for measurement; (ii) the parameter. of DZ(-) are unknown; (iii) the parameters of the smooth nonline8l' part SN(·), 9, are unknown. To achieve the control objective, the following assumptions are made: (AI) full state measurements of %,(1) , I ~ j ~ n are available; (A2) the dead-zone parameter. have known bounds: m,. 1 $ mr :5 mr2, mn :5 rn, :S m/21 0 $ mrbr :5 m.r , -m., $ m,b, :::; 0, for some known positive and constant md, m/I! m.r , m.,; (A3) 0 < f3o. < IPo(%)1 < 13., for BOme positive constants 13. ,13,; (A4) the reference signal r(i) and its first n derivatives are known and bounded; (AS) an upper bound M, on 11011 is known, where 11 · 11 i8 the Euclidean vector norm. Th. assumption (AI) il needed for the backstepping design (Kristic, et .1., 1992), the assumption (A2) is used to project the estimates of m r , m/,br,b, for implementing an adaptive dead-zone inverse, The assumption (A3) is needed for a. nonsingular control, and the assumption (A4) is needed for tracking, while the assumption (AS) is needed for robust adaptation of the estimate of 9.

As a part of our adaptive controller, an adaptive deadzone inverse DIO is used 10 generate the control v(l) 10 the plant (2.1) - (2.3) :

(3.1) where Ud(t) is a. signa.l from a nonlinear feedbac.k control law to be designed.

The dead-zone inverse i8 described as: if u.(t)

>0

if o.(t) < 0 if Ud(t) : 0

(3.2)

where ,;,:br, m;, .;;;;"br I ffi'j are the estimates of the dead7.one parameters mrbr • m r , m,b,. ID,. This dead-zone inverse has the desired properties: when implemented with the true parameters ffirb r • mr, m,b,. m" it. cancels the dead-zone effect:

v(l)

= DJ(Ud(t»

=> u(l) : DZ(DI(ud(l)))

=ud(l) (3.3)

where

when implemented with parameter estimates, the deadzone inverse results in:

me)

where

"'N(I)

= (-x,(I)v(I), x,(I), -(1- x,(t»v(I), 1- x,(t)f (3.6)

with

-;(t)

X

= { 01

if vet) ?' 0 otberWl5e

is a measured vector signal dependent. only on the deadzone inverse

m( .).

~ ON(t)

T = (m;(I), m.b.(I), m,(t), m,b,(t»

(3.7)

is an estima.te of the dead·zone parameter ON:

and dN(t) is an unparametriza.ble but bounded error:

3. ADAPTIVE CONTROL DESIGN Now the adaptive control scheme is developed for the plant (2.1) - (2.3), which consists of an adaptive deadzone inverse to cancel the dead-zone and an adaptive backstepping nonlinea.r feedback law.

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dN(t): -m,x.(I)(v(t)-b,)-mlX/(t)(v(t)-b,) (3.9) X

,et) =

{I0 if 0~ ~(t) < b. otherWise

(t) _ Xl

-

{I0

6, < v(t) < 0

if

(3.10)

otherwise.

=

=

Moreover, dN(t) 0 when ON(t) ON. These properties are crucial for our adaptive control design.

Now, (3.16) and (3.18) can be viewed as a second-order system to be stabilized by <>1 given in (3.14) and <>, with respect to V2 = VI + iz~, The derivative of V2 is

3.2 Adaptive Dead-zone Inverse Control

The backstepping technique (Kristic, et al., 1992) has heen extensively used in adaptive and non-adaptive nODlinear control. Briefly, such a technique, for the plant

nonlinear part (2.1) - (2.2), is described by a design procedure of n steps. At each i step, an "error" coordinate is defined and a stabilizing function Q'j is designed to stabilize an itb-order subsystem with respect to a LyaPUDOV function Vt. Finally, at step n, a feedback control law for Ud, an adaptive update law for the estlmate 8 of 0, and an adaptive update law for the estimate ON, are designed to stabilize the closed-loop system.

z,

Introduce tuning function

T2

as

(3.20) and choose stabilizing function

0'2

as

The backstepping design steps in (Kristit, et al., 1992) for

OUt

tracking problem are as follows:

Step 1: Letting

= %1 -

%1

rand

%2

= X2 -

al,

we have

(3.11)

(3.21 ) where

At this step, (3.11) is viewed as a first-order system to be stabilized by 01 with respect to the Lyapunov function

V,

12 1~ = 2z, + 2'(0 -

0)

T

r- l(0- -

- = {o

(3.12)

0),

10> 0,0'0 where .

V,

r = rT > O.

The derivative of VI is

= z,(z, + a, + 0

~T

'1" -

-

T

l-

r) + (0 - 0) r- (0 -

rZ 11"1).

(3.13) Choosing the first stabilizing function as 01

=

Then we rewrite

V2

2:2

V, = -c,z, - c'z,

fiT /PI + r

-CIZl -

> 0, nm

~

/m (,)

This modification term robust adaptation of O.

,

if 11811:; M, if 11811 > M, (3.22) (3.23) n-1, M, > 11911.

10(1- e-Q.(l1811-M.)·~)

Im(O)

is introduced to ensure

as actl :.

+ z,z3 + -z,-_ (0 80

(3.14)

+ (0 - Wr-1(0 - 1'2)

and defining the first tuning function as

~ ~

7"2+ r/m(O)O) (3.24)

(3.15) we obtain

(3.16) •

V,

= -c,zl2 + z'z, + (0- -

The ~cond term and

%IZ2

0)

T

r- I!.. (0 -

(3.25) Td·

(3.17)

will be cancelled at the next step,

eis to be given later on.

Step 2: Introducing .

Z2

..

= x2 -

0'1

Z3

= %3 aT

= Xa + 17

_ 80'_10_ {)O

0'2,

/P2 -

Xl

80 1 ;._ 80..1 ;:, {)r

{)r

= XHl -

Cti

and lti

= Vi-I +

i-I ()

we have

aa1 ( - ( ) x2

Step i: Introducing Zi+I 2'zi, we h ave " .' Z,. = Xj-ai_l

+ eT) !.pI (3.18)

2092

=

aT

Xi+l+17

'"

Dt;_, (

aT)

I.p,- ~ - ( ) - - ZI:+l+U .1:=1 XI;

!.pI:

(3.26)

Now, the i-th order system (ZI,' .. , Zi) is to be stabilized = Vi-I + ~zl. We have

by ai with respect to Vi V;.

i-I

i-2

_=1

i;:1

,- " 8".' =- " L.J C.Z, (L.J %1+'-)(9 -

Step n: Introducing Vn

. = - "2,, a". )(Tn_. - 0' + ffm(O)O) -L.J e." + (L.J Z,+.-_

Vn Ti_.

- + fJm(O)9)

= Vn_l + ~z~, we have

n-l

n-2

'=1

'=1

80

l)()

_..!:-a"i_. (') + iJT(. _ ~ 8",_. )] L- 8r(l:-I) r CP. L.J 8xJ:
,1:;:1

~

+ (0 -

:..

1-1 IJCXI_1

i

off-'[O - f

L: Z'('I" - L: -8-'1")]' (3.27) 1=1

Introducing tuning function

i;:1 Ti

xI;;

At thi, ,tep, if the dead-zone DZ(.) were absent and u(t) were the control, then we could design the adaptive update law for i; as

as

i

1-1

/=1

1:=1

a

= f L: Z'('I" - L: ;'-. '1',) and choosing stabilizing function

(3.28)

XJ: ai

as with

0'0

= I,

and the control u as

n-l

e '"' n

"'" ,h~n-l

80 n _ 1

8CXn _l

+ L.J -;;;-x.+ 1 + ~ + L.J" 1;;:1

(3.29)

we rewrite

Vi as

u

u(J

le

n- 2

(8-T" - L.J

J:=1 uT

n-l

!to

v'"

"

(._I)r

(ok)

a

"n-' )] . (3.34 )

%'+1-- f)('I'n - L.J -a-'I"

1;=1

IJD

1;=1

xI:

For the resulting dosed-loop system without the deadzone, we would have n

Vn (3.30)

(9 -

Of fm(9)9

(3.35)

i'=l

the closed-loop signal boundedness and the asymptotic tracking: lim'_oo[y(t) - r(t)] O. However, in our control problem, the dead-zone DZ(·) is present and the control is v(t) = Dr(Ud(t», generated from the adaptive dead-zone inverse whose parameter 9N is also to be updated. Thus, we need to design Ud(t) and choose an adaptive law for ON to stabilize the closedloop sysiem with respect to

and i, as

z, = -z,_. - CiZ, + zH' -

= - L: c,z~ -

=

80'1:"

-.!;-(9 - T,

88

--

+ f Jm(O)O))

(3.31)

2093

(3.36)

As the feedback control law, we choose

Uti

1 = -(J [-Zn_l o

n-1

- enzn -!.po

80

"T

8z.l:

.1:=1

+ 8an,,-16 + ~ -(.

+ "~ -8-"n-l X.l:+l

L., k;:;1

80'n_l r(.I: )

8 r C·- t )

~ Zk+l-.oa,)( r 'l'n

- L.,

~ {Ja n _l - L., -8--'1")

8()

"'=1

A

•

k=l

XI>

DO'l

n-2

-rlm(O)OLZ'+l. I.

(3.37) n

80

.=1

o

In view of (3.5), (3.32) and (3.37), we have n

V =- L

c.zi - Im(O)(O -

(=1

ON

O)TO

+ (ON - ON )TrNl(ON + rH Zn{JOWN) + zn13od N . (3.38)

(3.39)

with ON'(O) E [0;",0),.,], where rN

-rNZn{JOWN + IN

~ "";(ZI' _. - , Xi, 8)

= 'Pi -

(3.44)

"'" L-

0';-1 --;:>,.

(3.45)

l}xk

Theorem 3.1 The adaptive dead-zone inverse control sch,m" cons;sl;ng of Ih, f«dbac~ law (3.37) updat,d from (3.33) and Ihe dead·zon, inverse (3.£) updated from (3.99) - (3.49), ensures thal all closed· loop signals art bounded.

= diag(-YNl.'YN2,'YN3,"YN4),

and Ihe jlh component of IN(I), ;

i-I {j

4"=1

= (iJNl
= -rNZn{JOWN + IN

ON

=

where

k=l

Then, the a.daptive law for ON is chosen as

= rLZ,W,- rIm (0)0

"YNi

>0

(3.40)

Proof: From (3.38) and (3 .39), we have

= 1,2, 3, 4, is

n-l

V= -

L c.z~ -

Im(O)(O - efo - Cnz~

+ znf30dH

k=l

(3.46) with 8Ni , 8};, being the ith component.s of

o~

= (mrl,O,mIl

1

(3.41)

- Im(O)(S - efo 50 0, (ON - oNfr,/fN 50 O. (3.41)

-m.,)T, 8~ = (mt"2,ml r , mI2,O)T (3.42)

and 9N,(t) being the jth component of (3.43) The term fN(t) is for robustness of the parameter update law with respect to the bounded dN(I) as well as for parameter projection to ensure tbat the estimate 8N 8atisfies the same conditions of (A2) for implementing the dead-zone inverse (3.2).

With the .bove adaptive laws (3.33), (3.39), and the feedback control law (3.37), the overa.1l dosed· loop system is described by %1

=

%2 -

Ct':) -

The modification (3.22) - (3.23) and the parameter projection (3.40) - (3.43) en.ur. that

Sin;. dN and {Jo are bounded, and Im(O)(o-ef 0 and/or -(ON - eN )Tr;/ IN grows unboundedly if 0 and/or ON grows unboundedly, it follows from (3.36), (3.46) and the assumption (A3) that 9,iJN E LC;tI:J, and ZIc E Lr.If\ k = 1, ,- -,no Since ZI = %1 - r is bounded, we have Xl E L OO and hence ;:>1(~d E L~ . It follows from (3 .14) that "1 E By

=

Ot, we have X2 E LOO and hence It follows from (3.21) that <>, E Loo . Cont.inuing in the same way, we prove that a. e Loo, i 1" , ',n-l,%j E LOO, i 1, - - ·,n. Then from (3.37), we have Ud E Loo_ ~ince t.he parameter projection (3.40) - (3.43) ensures BNi E [8~i , 9~i]' i 1,2,3, 4, t.hat. is, m;: > mo, m, >

LOO _

Z2

1"("1, ",) E

1:2 -

L~.

=

=

(0 - 8)T '"'1

2094

=

>

mo, for Borne constant mo

0, from (3.2), we have

we conclude t.hat all signals in the closed-loop system (3.44) are hounded . '11

Extensions of the adaptive inverse control design of this paper to other nonlinearities such 88 backlash, hysteresis a.nd piecewise-linear cha.racteristic, are important future topics.

4. SIMULATION RESULTS

ACKNOWLEDGEMENTS

To show the system tracking performance, we now present the simulation results for the exam ple:

This work was supported by the National Science Foundation under Grant ECS-9307545.

.(t) E Loo, and from (2.4) that u(t) E Loo. Therefore,

Z2

+ 1.5.r1 (4.1)

u

DZ(v).

= 1.21 b,.

where DZ(-) has the parameters m,. 0.5, rn, 2, b, -0 .8.

=

=

.. •

•

/'

I

s ·

.

,

·

;

.. --..

·

:

• •

"

,

,

•

•

. •

•

•

i

,

....

;•

s " .......

• •

,

•

..·

.............

,

·. .. .

· ..

Figure 2: Ttacking error 11 - r . The tracking error 11 - r is shown in Figure 2 for r = 5 with 9(0) 2 .nd 9N (0) (1,0.8,1.6, -1.2jT, which indicates that asymptotic tracking is obtained, compared with the case of no dead-zone inverse where the tracking error remains large for large t (not shown here).

=

=

5. CONCLUSIONS We have proposed an adaptive contra) design for systems with an unknown dead-zone at the input of a smooth nonlinear dynamics in the parametric-strictfeedback form with unknown parameters. Based on tbe backstepping design for adaptive nonlinear feed back control and the adaptive dead-zone inverse to cancel the unknown dead-zone effect , our controller ensures the closed-loop signal boundedness in the presence of the unknown dead-zone which may result in instability without a dead-zone inverse. Simulation results show that our adaptive control scheme improves the system performance significantly.

2095

REFERENCES Dorf, D.F.(1990). Modern Control Systems, 5th ed ., Addison-We,ley. Isidori, A.(1989). Nonlinear Control Systems, 2nd ed., Springer- Verlag . Kanellakopoulos, L, P. V. Kokotovic, and A. S. Morse(1991) . Systematic design of adapt.ive controller. for feedback lineariz.ble systems. IEEE Trans. on Automatic Control, vol. AC-36, no. 11 , pp. 1241-1253. Kristic, M., I. Kanellakopoul08 and P. V. Kokotovic (1992). Adaptive nonlinear control without overparametrization. Systems and Control Letters, vol. 19, pp. 177-185. Kuo, B.C.(1995). Automatic Control Systems, 7th ed., Prentice Hall , Englewood Cliffs, NJ. Pomet, J.-B. and L. Praly(1989). Adaptive nonlinear control: a.n estimation-based algorithm. In: New Trends in Non/inear Control Thtory (J . Deseusse, M. Flies., A.lsidori and D. Leborgne (Ed.». Springer-Verlag, Berlin. P ra,ly, L., G. Bastin, J.-B. Pomet, and Z. P. J iang( J991). Adaptive stahilization of nonline.r systems. In : Foundations of Adaptive Control (P. V. Kokotovic (Ed.» . Springer-Verlag, Berlin. Recker, D., P. V. Kokotovic, D. Rhode and J . Winkelman(1991). Adaptive nonlinear control of systems containing a dead-zone. Proc. of th, 30th IEEE CDe, pp. 2111-2115, Brighton, England. Sastry, S.S. and A. Isidori(1989). Adaptive control of linearizable systems. IEEE Tran.5. Automatic Control, voL AC-34, no. 11, pp. 1123-1131. Too, G .• nd P. V. Kokotovic(1994). AdaptiVe control of plants with unknown dead-zones. IEEE Trans. on Au'omatic Con irol, val. AC-39, no. 1, pp. 59-68, 1994. T.o, G. and M. Tian(1995) . Design of adaptive deadzone inverse for nonminimum phase plants. Proc. of th, 1995 ACe, pp . 2059-2063, Se.Ule, WA.