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Proceedings of the 38Ih

Conference on Decision & Control

Phoenix, Arizona USA December 1999

F r M l l 14:OO

Global Output-Feedback Tracking

for a Benchmark Nonlinear System

*

Zhong-Ping Jiang Ioannis Kanellakopoulos

Depar tment

of

Electrical Engineering

Polytechnic University University of California

Brooklyn, NY 11201.

zjiang@control.poly.edu ioannis@ee.ucla.edu

Dep artm ent of Electrical Engineering

Los

Angeles, CA 90095-1594

Abstract: In this paper, the output-feedback global

tracking problem is solved for the well-known nonlinear

benchmark RTAC system, where one of the unmea-

sured states appears quadratically in the state equa-

tions. Our novel observer-controller backstepping de-

sign yields

a

nonlinear output-feedback controller that

forces the translational displacement t o globally asymp-

totically track an appropriate time-varying signal. The

proposed solution is new even for the case of global

output-feedback stabilization, namely when the refer-

ence signal is zero.

1

Introduction

The problem of controlling the nonlinear benchmark

mechanical system usually called RTAC (for Rotational-

Translational Actuator) or TORA (Translational

Os-

cillator with

a

Rotational Actuator)

wa s

introduced

in [lo]. This problem has recently received a consid-

erable amount of attention from several researchers [l,

2,

3, 4,

10,

111.

Due to the weak, sinusoid-type non-

linear interaction between the translational oscillations

and the rotational motion, the dynamic model of this

RTAC is not globally feedback linearizable. There-

fore, the direct application of well-known nonlinear

control schemes such as feedback linearization does not

solve the global stabilization problem. In the afore-

mentioned papers, several novel nonlinear approaches

have been developed, based on integrator backstepping

and passivity techniques, for the state- and output-

feedback stabilization and tracking problems. More-

over, some of these approaches have been validated

through experimental results.

The objective of this paper

is

to address the problem

of output-feedback global tracking for this benchmark

system, following our previous results for the state-

feedback [ll] and output-feedback

[4]

problems. We

' The w or k

of

t he f i r st au tho r w as s uppo r ted in pa r t by a

s t a r t - u p g r a n t from Polytechnic Univers i ty . Th e work of t h e

second author

was

s uppo r ted in pa r t by NSF u n d e r G r a n t ECS-

9502945.

consider the translational position and the rotational

angle as the two outputs and assume that the linear

and angular velocities are not available for feedback.

The main difficulty here is that the system equations

depend nonlinearly on an unmeasured state; this ren-

ders existing global output-feedback stabilization and

tracking methods

[6,

7,

81

not

applicable to the RTAC

system. The semiglobal approach proposed in [4]forces

the translational displacement to track a suitably de-

fined reference signal for any given bounded region of

initial conditions. In this paper, we give a global so-

lution to the output-feedback tracking problem. By

exploiting the physical structure of the RTAC system,

the proposed tracking methodology not only accom-

modates all initial conditions with the same controller,

but also guarantees asymptotic tracking for a larger

class of reference signals than th e semiglobal scheme

of

[4].

Notation: For

a vector

2 E Rn,

T denotes its trans-

pose and

1 )

its Euclidean norm.

For a

time-varying

system

= g ( t , J )

+

~ ( t ) ,

+ .,t)

enotes the tra-

jectory starting from at t = 0 .and

W .,t)

eans

@99(.Lt)I i.e., the trajectory of

=

g t , t ) starting

from 5 at t = 0. I n x n is the identity matrix of or-

der

n

while On is the null matrix of order n. GAS

means global asymptotic stability while UGAS stands

for uniform global asymptotic stability.

2 A benchmark problem

The nonlinear benchmark system considered in this

section was introduced by Wan, Bernstein and Cop-

pola in

[lo]

and is known as RTAC or TORA. It is

a

mechanical system in which the translational oscilla-

tions of the platform are controlled via the rotational

motion of an eccentric mass. Assuming tha t the plat-

form moves in the horizontal plane, the dynamics of

the system are described by

0-7803-5250-5/99/O.00

O

I999

EEE

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w1ierc.I; is the disturbance force actirig

on

the cart,

N

is the control torque applied to the proof mass, x is

the translational position and 8 is the rotational angle.

In l ) , 1.1 represents the total mass of the disk and the

cart, m and I represent the mass and the moment of

inertia of th e eccentric mass, respectively, and T is the

radius of rotation.

As in [lo], after some appropriat e normalized transfor-

mations, equations (1) and 1) are simplified to

, I + Xd = &(e2sino ecosq f Fd

2)

8 = U-&XdCOS8,

where 0

lTL1/2

+

l / ( 2 L 2 d m ) is

-

24

a design pa-

rameter. Notice that 2 2 el

=

8 and z3

0,. =

z

are .available for feedback design. Also note tha t

al (~, ,B, ,O,~, )= 0. Byvirtueof 18), 21) andLemma

1,

completing the squares we obtain

(32)

L 2 m e ; Z3z4J 1 2

2 1 2 cos2

2 3

Step 3

:

Consider the function

33)

,

V3

=

V2(%,el1e2)

+

2z4

Differentiate V3 along the solutions of (18),

(21)

and

(26) and choose the following tracking control law to

make V3 negative definite:

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aal

. aal .. aal .

eT eT

eT

aeT ao a x 3

r

e, (J-- - ^.

COS'

(z3 + 0,) 1 ]} 34)

where

c2 >

0

is a

design parameter, and

2 ,2 ,

2

and stand for the partial derivatives of

a1

with

respect to its first to fourth argument, respectively.

Indeed, when 34) is substituted into

V 3

from (32)

and the completion of th e squares, it follows that

35)

Letting

a = min { c1-

+

A

L 2 m ,

c 2 ,

, L z F }

36)

V3

5

- 2 a & ( ~ ~ , ~ * , e l , e ~ ) 37)

35) implies

The statement and the proof of

our

main s tability re-

sults are given in the next section.

4 Main

results

We are now ready t o st ate our main theorem.

Theorem

1

For any reference signal xT( t ) atisfying

Assumption 1, the problem of global output-feedback

tracking is solvable for the benchmark system [2) with

Fd = 0.

Before proving Theorem 1,we give a technical stability

result for a nonautonomous cascade system of the type

[

=

f ( t , S , Y )

38)

=

fd( t ,O

(39)

Y =

HC

(40)

where

[

E EtRn(,

E Etn L. It remains to verify A l ) . When setting

el =

Z3

= 0 in (28), the time derivative of in 26)

becomes

11

arctan

2 2

2

11

arctan

z2

28,

2

os

I= - 2 ~ ~ 2 s i n

which, toge ther with the choice of in (0,

2-4OmaX/r ,

implies that

VI

is nonpositive and is equal to zero if

and only if

22

= 0. Lyapunov stability theory tells

us that the zero-input (-system is globally stable. Fur-

thermore, from Barbalats lemma (see, e.g., [6, p. 491]),

zz(t)

goes to zero as

t

+ +CO. As in [4, Sec.4.11, by

an application of [5, Lemma 21 to the equation

2 2 = -z1 + E(sin(z3 +e,) -sin@,) ,

45)

we conclude that

z l ( t )

also goes to zero as

t

-+ +CO.

Therefore, the property A l ) has been proved. Finally,

the proof of Theorem 1 is completed with the help of

Lemma 3. AAA

For

a subclass

of reference signals identified in

As-

sumption 1, that is, a set of periodic reference signals

x , ( t ) ,

we can even conclude the

uniform

convergence

of

2 d ( t )

,(t) to zero.

Corollary

1

Under the conditions

of

Theorem

1, i f

x, (t ) and the deduced signal q( t) in Assumption

1

are

periodic,

all

the states

of

the

closed-loop

system are

uniformly bounded. In particular, Xd(t) x,.(t) uni-

formly

converges t o zero as t -+

ca.

5

Concluding

remarks

The problem of output-feedback tracking for the well-

known nonlinear benchmark RTAC system has been

solved globally for the first time. The novelty of this

paper is the introduction of a nonlinear s ta te transfor-

mation which eliminates the quadratic dependence of

the system equations on the unmeasured states. This

transformation then allows the design of a novel ob-

server/controller backstepping scheme, which leads to

the desired global output-feedback tracking.

Lyapunov-based backstepping designs are inherently

robust to some types of disturbances, and can be robus-

tified with respect to many other types. Our controller

can also be robustified against several types of dis tur-

bances using techniques very similar to those in

[ l l] A

complete study of the robustness issue is beyond the

scope of this paper, but it undoubtedly is a topic that

deserves further attention.

References

[l]

G. Escobar, R. Ortega, and H. Sira-Ramirez,

Output-feedback global stabi lization of a nonlin-

ear benchmark system using a saturated passivity-

based controller,

IEEE TCST,

7 289-293) 1999.

[2] R. A. Freeman and

P.

V.

KokotoviC, Tracking

controllers for systems linear in the unmeasured

states,

Autornatica,

vol. 32, pp. 735-746, 1996.

[3]

M.

Jankovic,

D.

Fontaine and P. V. KokotoviC,

TORA example: cascade- and passivity-based

control designs,

IEEE TCST,

4 292-297) 1996.

[4] Z. P. Jiang,

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P.

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[8] L. Praly and Z. P. Jiang, Stabilization by out-

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Global stabilization of the oscillating eccentric

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at-

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Int.

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