00557583

8/2/2019 00557583

1/7

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 3, MARCH 1997 393

[7] L. Hsu, Variable structure model reference adaptive control using onlyI/O measurement: General case, IEEE Trans. Automat. Contr., vol. 35,no. 11, pp. 12381243, 1990.

[8] L. Hsu, A. D. Araujo, and R. R. Costa, Analysis and design of I/O basedvariable structure adaptive control, IEEE Trans. Automat. Contr., vol.39, no. 1, pp. 421, 1994.

[9] L. Hsu, A. D. Araujo, and F. Lizarralde, New results on VS-MRAC:Design and stability analysis, in Proc. Amer. Contr. Conf., San Fran-cisco, 1993, pp. 10911095.

[10] L. Hsu and R. R. Costa, Variable structure model reference adaptive

control using only input and output measurement: Part I, Int. J. Contr.,vol. 49, no. 2, pp. 399416, 1989.

[11] L. Hsu and F. Lizarralde, Redesign and stability analysis of VS-MRACsystems, in Proc. Amer. Contr. Conf., Chicago, 1992, pp. 27252729.

[12] , Experimental results on variable structure adaptive robot controlwithout velocity measurement, in Proc. Amer. Contr. Conf., Seattle,1995, pp. 23172321.

[13] P. Ioannou and K. Tsakalis, A robust direct adaptive controller, IEEETrans. Automat. Contr., vol. 31, no. 11, pp. 10331043, 1986.

[14] S. M. Naik, P. R. Kumar, and B. E. Ydstie, Robust continuous-timeadaptive control by parameter projection, IEEE Trans. Automat. Contr.,vol. 37, no. 2, pp. 182197, 1992.

[15] K. Narendra and A. Annaswamy, Stable Adaptive Systems. EnglewoodCliffs, NJ: Prentice-Hall, 1989.

[16] S. S. Sastry and M. Bodson, Adaptive Control: Stability, Convergenceand Robustness. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[17] V. I. Utkin, Sliding Modes and Their Application in Variable Structure

Systems. MIR, 1978.[18] A. C. Wu, L. C. Fu, and C. F Hsu, Robust MRAC for plant witharbitrary relative degree using variable structure design, in Proc. Amer.Contr. Conf., Chicago, 1992, pp. 27352739.

Robust Control Design for Nonlinear Uncertain Systems

with an Unknown Time-Varying Control Direction

Joseph Kaloust and Z. Qu

AbstractIn this paper, a continuous robust control design approach is

proposed for first-order nonlinear systems whose dynamics contain bothnonlinear uncertainty and an unknown time-varying control direction.

The so-called control direction is the multiplier of the control term in thedynamic equation, and it effectively represents the direction of motionunder any given control. A nonlinear robust control is designed to on-lineand continuously identify sign changes of the unknown control directionand to guarantee stability of uniform ultimate boundedness. The proposedrobust control design requires only three conditions: the nominal system isstable, the control direction is smooth, and the uncertainty in the systemis bounded by a known function. The necessity of these conditions isestablished in this paper. Continuity of the proposed robust control is

achieved by using a so-called shifting law that changes smoothly the signof robust control and tracks the change of the unknown control direction.

Analysis and design is shown by using the Lyapunovs direct method.

Index TermsLyapunov stability, robust control, uncertain systems.

I. INTRODUCTION

For more than one decade, robust control design of nonlinear

uncertain systems has been an active area of research. Several im-

portant classes of stabilizable uncertain systems have been identified,

Manuscript received March 14, 1995; revised September 8, 1995, April 20,1996, and July 1, 1996.

J. Kaloust is with Loral Vought Systems, Dallas, TX 75265-0003 USA.Z. Qu is with the Department of Electrical and Computer Engineering,

University of Central Florida, Orlando, FL 32816 USA.Publisher Item Identifier S 0018-9286(97)02046-1.

and stabilizing robust controllers have been developed, for example,

the results in [1], [3], [10], [16][19], and [21]. The uncertainties

admissible in these results can be of either matched type or unmatched

type, and they are generally bounded by known nonlinear functions

of the state. However, input-related uncertainties analyzed in the

existing results are assumed to be of fixed and known signs; that is,

the direction of motion of the system is fixed and known under any

choice of control. The assumption of priori knowledge of the control

direction being known makes robust control design straightforward.In [5], the problem of designing robust control was studied for

the class of systems with an unknown but constant and bounded-

away-from-zero control direction. The objective of this paper is

to generalize the results in [5] so that systems with an unknown

and time-varying control direction can be treated. This extension is

nontrivial since the time-varying control direction is allowed to cross

zero an infinite number of times and since identification must now be

done fast and continuously to keep tracking the changing control

direction. As before, the Lyapunovs direct method is utilized to

proceed with the design and to ensure the stability of uniform ultimate

boundedness. Similar to the results introduced in [5], continuity of

the proposed robust control is achieved by designing the so-called

shifting law, which continuously and smoothly steers the sign of our

robust control.In the area of adaptive control, many results dealing with systems

containing an unknown constant control direction, referred to therein

as the unknown high-frequency gain, have been obtained. For ex-

ample, Nussbaum [15] proposed for linear time-invariant systems an

adaptive control algorithm whose sign changes an infinite number

of times as its argument tends to infinity. In [12], Mudgett and his

coworkers developed an adaptive control design scheme for general

linear systems without requiring the knowledge of high-frequency

gain by following the same principle as that in [15]. In the effort

of extending these results to nonlinear systems having unknown

control directions, two adaptive control schemes have been proposed

for simple first-order nonlinear systems [2], [6]. In [2] and [6], a

hysteresis method was used to construct an adaptive control that

jumps a finite number of times over the infinite time horizon. Alongthe line of nonlinear robust control, a continuous robust control design

scheme was developed in [5] for uncertain nonlinear systems with an

unknown but constant control direction. Compared with the existing

result, the proposed result is the first attempt in addressing the control

design problem in the presence of an unknown but smooth control

direction, whose sign change is of unlimited number and may occur

anytime.

The paper is broken into the following sections. System description,

necessary assumptions, and the problem of designing robust control

are described in Section II. Robust control design is proceeded in

Section III by the Lyapunovs direct method. A simulation example

is presented in Section IV.

II. PROBLEM FORMULATION

We shall consider the nonlinear uncertain system described by

_x = f ( x ; t ) + 1 f

0

( x ; t ) + a ( t ) u ( x ; t )

(1)

wherex 2

8/2/2019 00557583

2/7

394 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 42, NO. 3, MARCH 1997

in the presence of an unknown, time-varying control direction. It is

obvious that (1) can be made of a vector and that, based on the results

in [5], it can also be extended to be of higher order.

The control direction,s i g n [ a ( t ) ]

of functiona ( t ) ;

is unknown,

and it may switch over time among0

1, 0, and 1 arbitrarily. For

identifiability, an upper bound will be imposed on the rate of change

of functiona ( t ) ;

and the necessity of such an assumption can easily

be shown by contradiction. Sincea ( t )

is allowed to change its sign,

the case thata ( t )

may be zero all the time must be considered. Notethata ( t ) 0

implies that there is no control action, no matter whatu

is. In this case, motion and stability of the system are governed solely

by known and unknown dynamics, but these dynamics, if unstable,

must be compensated. For this reason, two necessary assumptions

must be made. First, known dynamicsf ( x ; t )

have to be stable (for

the case thata ( t ) 0

and1 f ( x ; t ) 0

). Second, the uncertainty

in the system has to have the property that for some well defined

function1 f ( x ; t ) ; 1 f

0

( x ; t ) = a ( t ) 1 f ( x ; t ) :

Withouta ( t )

being

the common factor between the terms of control and uncertainty,

stabilizability cannot be guaranteed if the bounding function of the

uncertainty is arbitrary. That is, if functiona ( t )

is multiplied only at

the front of controlu ;

the system is stable only if the uncertainty is

small in the sense that it is dominated by the robust stability margin.

This latter case is not considered in the paper, since stability is athand simply by lettingu = 0

; therefore, the control design is trivial.

With these facts in mind, we can define the following assumptions

for systems in the form of (1).

Assumption 1: Under a continuous controlu

to be devised, the

dynamics of (1) are Caratheodory. That is, (1) has a classical solution

under any continuous control.

Assumption 2: Functionf ( x ; t )

characterizing the dynamics

of the nominal system is exponentially stable. That is, by

the Lyapunov converse theorem, there exists a differentiable

functionV

1

( x ; t )

such that, for some positive constants

1

;

2

;

3

;

and

4

;

1

j x j

2

V

1

( x ; t )

2

j x j

2

; j @ V

1

= @ t j

3

j x j ;

and

@ V

1

= @ t + ( @ V

1

= @ x ) f ( x ; t ) 0

4

j x j

2

:

Assumption 3: Lumped uncertainty1 f

0

( x ; t )

can be rewritten as

1 f

0

( x ; t ) = a ( t ) 1 f ( x ; t )

, and1 f ( x ; t )

is bounded as, for all( x ; t ) ; j 1 f ( x ; t ) j ( x ) ;

where ( 1 )

is known, continuous, and

locally uniformly bounded with respect tox :

Assumption 4: Functiona ( t )

is bounded and normalized in the

sense thatj a ( t ) j :

In addition,a ( t )

is smooth in the sense that

its first-order derivative is bounded from above asj _a ( t ) j

for a

known constant > 0 :

Assumption 5: For all possible choices of the uncertainty

1 f

0

( x ; t )

bounded by( 2 + ) ( x )

for some constant > 0 ;

(1)

withu 0

does not have a finite escape time that is infinitesimal.

Most systems in the form of (1) satisfy Assumption 5. To see

this conclusion, consider the choice of bounding function ( x )

that

( x ) = j x j

p for somep > 0 :

Then, it follows from Assumption 2 and

fromu = 0

that, along every trajectory of (1), the time derivative of

Lyapunov functionV

1

( x )

satisfies

_

V

1

0

4

j x j

2

+ ( 2 + ) j x j ( x )

0

4

2

V

1

+

2 +

p + 1

2

V

p + 1 = 2

1

:

It is obvious from the upper bound on the growth rate(

_

V

1

)

that

Lyapunov functionV

1

, and therefore the statex

, will not become

infinite instantaneously. This result holds for any choice of positive

real numberp :

Note that by the Taylor series expansion, polynomial

functions form a local basis for all differentiable functions. Never-

theless, to avoid the complication of proving Assumption 5 for all

possible expressions of bounding function ( 1 ) ; we choose to make

the assumption. Once ( x )

is given, one can simply proceed with the

proof of Assumption 5 using the above argument.

Assumption 5 is needed for on-line identification of the unknown

and time-varying direction. If Assumption 5 does not hold, no iden-

tification algorithm can identify instantaneously and keep perfectly

tracking an unknown changing control direction so that a robust

control of the correct sign can be implemented. The reason that the

bounding function used in the assumption is( 2 + ) ( x )

rather than

( x )

is that robust control must have its magnitude no smaller than ( x )

for compensating for uncertainty of size ( x )

and that the worst

case of robust control initially having a wrong sign and consequently

being destabilizing must be considered. In the case that the finite

escape time is infinitesimal, there is no control of finite magnitude

that can pull the system back, even if the control direction can be

identified instantaneously.

Based on these assumptions, we can proceed with robust control

design in the next section.

III. ROBUST CONTROL DESIGN

The main approach for designing robust control for nonlinear

uncertain systems has been the Lyapunovs direct method. In its

application, deterministic functions are used to bound uncertainties,

and several classes of uncertain systems have been shown to beglobally stabilizable [1], [3], [10], [16], [19], [21]. More recent

development along this line of work is the recursive-interlacing

design procedure [17], [18] which is generic enough to overcome

the common structural conditions imposed on uncertain systems.

There are two common features among these results. First, global

stabilizing robust controllers are devised if bounding functions of

uncertainties are available. The selection of a robust controller is done

through a Lyapunov argument such that robust control dominates

uncertainties in both magnitude and sign. Second, in these results

the control directions are known in the sense that after choosing a

robust controller, the sign of the robust control determines the signs

of the time derivatives of both the state and the Lyapunov function.

It was shown in [5] that the dominating effect of robust control can

be used to identify a constant unknown control direction and thata continuous robust control can be devised whose sign is changed

smoothly.

In this paper, robust control design is studied for systems with an

unknown and time-varying control direction, which can be viewed

as an extension of [5]. Unlike the problem of dealing with a

fixed unknown control direction, the time-varying control direction

considered here is not bounded away from zero, the number of its

sign changes is unlimited over time, and a successful identification

algorithm must track closely its sign changes and be capable of

handling the potential problem of singularity at zero crossings.

Thus, the proposed extension is not trivial. Like the existing results,

the design of a stabilizing robust control in the presence of a

time-varying and unknown control direction involves a Lyapunov

argument and the concept of having the control dominate the un-known dynamics. As in [5], an initial guess is made on the sign

of the unknown control direction, identification of the unknown

control direction is then carried out on-line continuously, and a

robust control is designed whose sign is modified according to the

identified control direction. The key results in this study are that

despite the fact that the unknown control direction may change

its sign an infinite number of times, identification and tracking

can be done using the concept of size domination, and since a

continuous control usually produces a better transient response than

discontinuous controls, a smooth transition of the sign of the robust

control can be achieved. Intuitively, the basic idea emanates as

follows: begin with some guessa ( t )

ofa ( t ) ;

design a robust control

8/2/2019 00557583

3/7


in terms ofa ( t ) ;

identifya ( t )

on line; formulate shifting law _a ( t )

so that the robust control is continuous; and let all three designs

be coordinated and integrated in such a way that robust stability is

ensured.

Robust control design in the presence of a time-varying con-

trol direction consists of three integrated parts: 1) selection of

robust control using the Lyapunovs direct method in a similar

fashion as that with a known control direction; 2) a scheme that

on-line identifies the unknown time-varying control direction; 3)formulate a continuous shifting law that changes the sign of ro-

bust control smoothly in tracking the identified sign of the time-

varying control direction. These three parts are detailed as fol-

lows.

Part I: In this part, a candidate of robust control is selected. As

shown in [1] and [19], robust control in the absence of an unknown

control direction can be selected to be

u

r

1

= 0

j ( x ) j +

2

j ( x ) j

2

+

3

( x ) ( x )

(2)

where > 0

is a design parameter, and ( x ) = ( @ V

= @ x ) ( x ) :

Note

that

j u

r

j =

j ( x ) j

2

+

2

j ( x ) j

j ( x ) j

2

+

3

( x ) +

4

( x ) :

(3)

The above inequality holds since

( +

4

) ( j ( x ) j

2

+

3

)

j ( x ) j

2

+

3

+

4

j ( x ) j

2

j ( x ) j

2

+

2

j ( x ) j

(4)

in which the CauchyScharwz inequalitya

2

+ b

2

2 a b

is used.

In the presence of the time-varying, unknown control direction, we

propose the following robust control:

u ( x ; a ( t ) ) =

a ( t )

( t )

1 u

r

(5)

whereu

r

is defined in (2),a ( t )

represents on-line estimate of function

a ( t ) ;

and ( t ) > > 0

is a design time function such that for all

t t

0

j a ( t ) j ( t ) :

(6)

Function ( t ) ;

to be defined shortly, is to normalize robust control

(5) so that its magnitude is compatible with control (2) and therefore

with the class of uncertainties under consideration.

It follows from (3) and (6) that

j a ( t ) [ 1 f ( x ; t ) + u ] j ( x ) + j u

r

j ( 2 +

4

) ( x ) :

(7)

Therefore, by choosing

small such that 4 ;

we can always

ensure that the condition in Assumption 5 holds, which is why (6)

was imposed on control (5). Note that besides the requirement of

4 ;

in (5) or (2) can be chosen freely by the designer, and the

smaller

is, the smaller the ultimate bound on the state.

As introduced in [5], the concept of size domination means that in aLyapunov argument, the term associated with the control dominates in

magnitude, and thus in sign, the term(s) associated with uncertainties.

If size domination is possible, the sign of the time derivative of the

Lyapunov function is dictated by that of the term associated with the

robust control, and therefore stability of the controlled system can be

ensured. Sincea ( t )

may become zero and since ( t )

in robust control

(5) must be bounded from zero, we choose function ( t )

to be

( t ) =

j a ( t ) j

ifj a ( t ) j

ifj a ( t ) j < :

(8)

The exact role of function ;

a design constant, will become clear

shortly. Obviously, choice (8) of ( t )

makes (6) hold. In the case

that the control direction is time invariant, setting ( t ) = = =

j a ( t

0

) j

reduces control (5) to that in [5] (in [5],a ( t

0

)

is set to be

either 1 or0

1 so that ( t ) = ) :

As will be shown in the upcoming stability analysis, size domi-

nation of robust control over uncertainty is achieved in a Lyapunov

argument if

@ V

@ x

u >

@ V

@ x

1 f ( x ; t )

(9)

or equivalently

j a ( t ) j

( t )

1

j ( x ) j

2

+

2

j ( x ) j

j ( x ) j

2

+

3

j ( x ) j j ( x ) j :

Therefore, we know that the size domination is accomplished under

the following conditions:

j ( x ) j

(10)

and

j a ( t ) j :

(11)

Note that estimatea ( t )

has to be able to cross zero in order to track

the identified version ofa ( t )

and that wheneverj a ( t ) j < ;

function

a ( t )

may soon have zero crossing. The size domination in terms of

(9) is no longer possible, and the sign ofa ( t )

may not be identified

correctly. Since instability will occur if the robust control provides

a wrong control sign for a period of time,a ( t )

must tracka ( t )

closely. However, it is impossible fora ( t )

to correctly tracka ( t )

at all time. To eliminate the possibility of instability, we shall devise

a sign identification scheme which successfully identifies the sign of

a ( t )

unless eithera ( t )

is about to change its sign orx

is very small. It

is by limiting the occurence of identification failure to be only within

the described regions in the spaces ofx

anda ( t )

that the stability of

the overall system can be guaranteed.

Part II: In this part, a scheme to on-line identify the control

direction is synthesized using the concept of size domination. Since

the control direction is unknown, any control will fail to ensure

stability without a correct identification of the control direction. If

a ( t )

is constant, identification of the control direction needs to be

performed only for a very small interval of time as was proposed

in [5], and robust control can then be operated forever based on the

one-time identification. Since the control direction considered in this

paper is a smooth time function, its sign changes, possibly an infinite

number of times, must be accounted for continuously.

In order to identifys i g n [ a ( t ) ] ;

multiplying@ V

= @ x

on both sides

of (1) and then integrating the both sides yield, for all[ t

; t

2

]

[ t

0

; )

t

t

@ V

@ x

d x 0

t

t

@ V

@ x

f ( x ( ) ; ) d

=

t

t

a ( )

@ V

@ x

1 f ( x ( ) ; ) +

@ V

@ x

u ( x ( ) ) d :

(12)

Note that the integrand functions are all continuous and that the

sign of term( @ V

= @ x ) u

is the negative of the sign ofa ( t ) :

Recallthat under conditions defined in (10) and (11),

u ( x )

will dominate

uncertainty 1 f ( x ; t ) in magnitude as specified by (9). Therefore, it

follows from the mean value theorem that under conditions (10) and

(11), the sign ofa ( t )

can be identified by the following equation if

botha ( t )

anda ( t )

do not change their signs in the intervalt 2 [ t

; t

2

] :

s i g n [ a ( t ) ] = 0 s i g n [ ^ a ( t ) ] s i g n

t

t

@ V

@ x

d x

0

t

t

@ V

@ x

f ( x ( ) ; ) d :

(13)

With size domination of robust control over uncertainty, (13) can

be used to identify the correct control direction, so long asa ( t )

and

8/2/2019 00557583

4/7


a ( t )

do not change their signs in the intervalt

2[ t

1

; t

2

] :

If there

is a sign change in eithera ( t )

ora ( t ) ;

the result from (13) must be

considered to be wrong. Sincea ( t )

is known, its sign change will

be known as well so that proper precaution can be built into the

identification and robust control schemes to overcome any possible

wrong identification. In fact, as will be shown in the upcoming

stability analysis, the solution is simply to make the convergence of

a ( t )

to an estimate ofs i g n [ a ( t ) ]

fast enough, regardless of whether

the estimate ofs i g n [ a ( t ) ]

is right or wrong.On the other hand, sincea ( t )

is unknown, its sign change cannot

be predicted. Therefore, (13) may potentially produce a wrong sign

at any time, which is a major stability problem. Our solution to

this stability problem is to select judiciously the integration interval

[ t

1

; t

2

]

and to use the fact that the nominal system is exponentially

stable. Specifically, we choose the integration interval as follows: first

lett

2

= t

andt

1

= t

0T

so that the integration interval is a sliding

window of the current time instant. Second, select the interval such

that its lengthT

satisfies the following inequality:

T :

(15)

It is when ja ( t

2

)

j<

that there may be a sign change occurring

in the interval. Thus, assuming that a correct identification of control

direction implies stability (which will be guaranteed by designs of the

robust control and the shifting law), robust stability can be established

by studying whether the controlled system is stable if ja ( t )

j< :

Recalling that the nominal system obtained by settinga ( t ) = 0

is

exponential, we know that robust stability can be established for thesystem (with the robust control of a wrong sign) by simply letting

be sufficiently small. Details on selecting

will be given in the

stability proof. In fact, such a treatment of zoning is intuitive since,

if ja ( t )

j<

1 ;

sign identification is not possible anyway asa ( t )

is time varying.

Summarizing the above discussions, we know that the control

direction found from (13) is correct if (10), (11), and (15) hold.

Part III: As soon as an estimate ofs i g n [ a ( t ) ]

becomes available,

a shifting law can be constructed so that the sign of robust control is

changed smoothly from a possibly wrongly guessed control direction

to the identified control direction and is then updated all the time.

The shifting law given in (16), shown at the bottom of the page,

is chosen such that the overall system is robustly stable during the

time period in which smooth transitions of the sign of robust controlare accomplished, where the initial guess of the control direction

a ( t ) ;

denoted bya ( t

0

) ;

is chosen to be either 1 or 0 or 0 1.k

s

is

a constant gain satisfying

k

s

4

p

(17)

and functionw ( t )

is the identification result ofs i g n [ a ( t ) ]

defined by

w ( t ) =

0s i g n [ ^ a ( t ) ] s i g n

t

t 0 T

@ V

1

@ x

d x

0

t

t 0 T

@ V

1

@ x

f ( x ( ) ; ) d :

Implementation of shifting law (16) requiresw ( t ) :

No matter

whether the identification resultw ( t )

is the correct value ofs i g n [ a ( t ) ]

or not, w ( t ) has only three discrete values: 0 1, 1, and 0. As soon asw ( t )

picks up its new value,w ( t )

will remain to be a constant until

one of inequalities (10), (11), and (15) fails. During the period that

w ( t )

is constant, we have according to (16) that

d [ a ( t )

0w ( t ) ]

2

d t

= [ a ( t )

0w ( t ) ]

d a

d t

0

k

s

p

ja ( t )

0w ( t )

j:

Solving the above differential inequality, we know that the estimate

a ( t )

will converge tow ( t )

in a continuous fashion, in a finite time,

and without any overshoot, that is

ja ( t )

j 1 :

(18)

Ifw ( t ) = s i g n [ a ( t ) ] ;

the estimatea ( t )

will converge quickly to

the correct control direction. In fact, the gaink

s

defined in (17) for

shifting law (16) is chosen such that the time fora ( t )

to converge

tos i g n [ a ( t ) ]

is less thanT

defined in (14). This choice ensures that

a ( t )

is capable of trackings i g n [ a ( t ) ]

except for the case thata ( t )

is

in the zone of ja ( t )

j<

and is on the verge of changing its sign.

In the case that the identification result is wrong, the estimate may

converge to a wrong direction. Stability analysis must be done to

show that any wrong identification is temporary and that the system

state will not escape from the stability region anytime. Since (10) has

been incorporated into the shifting law, we havew ( t ) = s i g n [ a ( t ) ]

by implementing the shifting law if (11) and (15) hold. That is,

stability must be concluded for the cases that these two inequalities

fail.

Global stability of (1) under robust control (5) and shifting law (16)

is analyzed by the Lyapunovs direct method using the Lyapunov

function candidate

V ( x ; a ( t ) ) = V

1

( x ; t ) + V

2

( a ( t ) ; t )

(19)

whereV

1

(

1)

is that given in Assumption 2, and

V

2

( a ( t ) ; t ) =

1

2

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

2

=

1

2

[

ja ( t )

j

3 = 2

a

2

( t ) +

ja ( t )

j

3 = 2

02 s i g n [ a ( t ) ]

ja ( t )

j

3 = 2

a ( t ) ] :

Note that for any differential functionq ( t ) ;

jq ( t )

j

p ands i g n [ q

( t ) ]

jq ( t )

j

p

with p > 1 are also differentiable.The following theorem shows the stability of uniform and ultimate

boundedness for (1).

Theorem: Assume that (1) satisfies Assumptions 15. Then, under

robust control (5) and shifting law (16), the system state can be made

uniformly and ultimately bounded by letting be small enough.

_

a ( t ) =

0s i g n [ ^ a ( t )

0w ( t ) ]

1

1

p

k

s

+ 1 +

1

4

j ( x )

j

( t )

if j ( x )

j

0

if j ( x )

j<

(16)

8/2/2019 00557583

5/7


Proof: Differentiating (19) along the trajectory of (1) yields

_

V =

@ V

1

@ t

+

@ V

1

@ x

f ( x ; t )

+ a ( t )

@ V

1

@ x

1 f ( x ; t ) + a ( t )

@ V

1

@ x

u

+

3

4

[ + ^ a

2

( t ) ] j a ( t ) j s i g n [ a ( t ) ]

0

3

2

a ( t )

ja ( t )

j_a ( t )

+

a ( t )

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

_

a

0

4

jx

j

2

+

ja ( t )

j 1 j ( x )

j

0a ( t )

a ( t )

( t )

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j+

3

4

[ + ^ a

2

( t ) ]

2 ja ( t )

js i g n [ a ( t ) ]

0

3

2

a ( t )

ja ( t )

j_a ( t )

+

a ( t )

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

_

a :

(20)

Note thata ( t )

is bounded by one as shown by (18). Thus, stability

may also be studied by Lyapunov function candidate V 1 ( x ; t ) : Itfollows that

_

V

1

=

@ V

1

@ t

+

@ V

1

@ x

f ( x ; t ) + a ( t )

@ V

1

@ x

1 f ( x ; t )

+ a ( t )

@ V

1

@ x

u

0

4

jx

j

2

+

ja ( t )

j 1 j ( x )

j

0a ( t )

a ( t )

( t )

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j:

(21)

We shall study the stability result by investigating the following four

cases.

Case I j ( x )

j<

: In this case, (10) does not hold, and therefore

robust control does not dominate the uncertainty in size. In this case,w ( t )

may not bes i g n [ a ( t ) ] :

It follows from Assumption 4, shifting

law (16), (6), and (18) that the time derivative (20) of Lyapunov

functionV

is bounded from above as

_

V

0

4

jx

j

2

+

ja ( t )

j 1 j ( x )

j+

ja ( t )

j

1

ja ( t )

j

( t )

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j

+

3

4

[ + ^ a

2

( t ) ]

ja ( t )

j+

3

2

ja ( t )

j ja ( t )

j j_a ( t )

j

+

a ( t )

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

_

a

0

4

jx

j

2

+ 2 + 3 :

Noting thata ( t )

is frozen because of _a ( t ) = 0 ;

there is no need

to have a negative definite term associated with the variables in

V

2

( a ( t ) ; t ) :

In fact, it follows from the same inequalities that the time

derivative (21) of Lyapunov functionV

1

is bounded from above as

_

V

1

0

4

jx

j

2

+

ja ( t )

j 1 j ( x )

j+

ja ( t )

j

1

ja ( t )

j

( t )

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j

0

4

jx

j

2

+ 2 :

From either of the above results, one can conclude using the stability

theorem in [1] that the system is uniformly and ultimately bounded.

Case II ja ( t )

j

: Note that although the estimatea ( t )

of the

control direction may converge to a wrong value,a ( t )

is bounded

by one as indicated by (18). It follows from Assumption 2, (6), (4),

(8), and (18) that the time derivative (21) of Lyapunov function is

bounded from above as

_

V

1

0

4

jx

j

2

+

1 j ( x )

j+ +

4

j ( x )

j

0

4

jx

j

2

+

12 +

4

3

jx

j ( x ) :

It follows from the results in [20] that the system is semiglobal stable

in the sense that the system is uniformly ultimately bounded and that

the stability region can be made arbitrarily large by letting

be

small enough. This is true whether the on-line identification is done

correctly or not.

Case III ja ( t )

j

: In this case, the estimatea ( t )

is confined to

be in a smaller set than its ultimate bound defined by (18). Since

a ( t )

is small, the contribution of the robust control may not be large

enough to compensate for the uncertainty. Thus, this is the case where

instability can potentially occur. However, by the choice of gaink

s

in (17), this case will last a very small period of time during which

Assumption 5 is satisfied [as ensured by (7)]. Therefore, by choosing

small enough, not only can the time interval during which this caseoccurs be small enough, but also the state of the system will stay in

a finite region, specifically, the semiglobal stability region defined in

Case II.

Case IV: j ( x )

j ;

ja ( t )

j> ;

and ja ( t )

j> :

This case is the

complement of the union of the previous three cases. Since conditions

(10), (15), and (11) hold, identification is done correctly. That is

_

a ( t ) =

0s i g n [ ^ a ( t )

0s i g n [ a ( t ) ] ]

1

p

k

s

+ +

4

j ( x )

j

( t )

:

(22)

It follows from Assumption 4, shifting law (22), (4), and (18) that

the time derivative (20) of Lyapunov functionV

is bounded from

above as

_

V

0

4

jx

j

2

+

ja ( t )

j 1 j ( x )

j

0a ( t ) a ( t )

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j

( t )

+ 3 +

a ( t )

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

_

a :

Note that < ( t )

and that

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

since j ( x )

j> :

Hence

_

V

0

4

jx

j

2

+ [

ja ( t )

j 0a ( t ) a ( t ) ]

1

j ( x )

j

2

+

2

j ( x )

j

j ( x )

j

2

+

3

j ( x )

j

( t )

+ 3

+

a ( t )

ja ( t )

j

[ a ( t ) a ( t )

0 ja ( t )

j]

_

a

0

4

jx

j

2

0k

s

ja ( t ) a ( t )

0 ja ( t )

j j+ 3

from which stability of uniform and ultimate boundedness can be

concluded.

In summary, one can conclude the stability of uniform ultimate

boundedness for (1) by simply combining the above four cases.

In the next section, a simulation example is presented to illustrate

our proposed robust control design.

8/2/2019 00557583

6/7


Fig. 1. System performance under a ( t ) = s i n ( 2 t ) and x ( 0 ) = 2 :

Fig. 2. Shifting law a (solids i g n [ a ( t ) ] ; dasheda ( t ) ; dottedu ) :

Fig. 3. a ( t ) = 0 s i n ( 2 t ) and x ( 0 ) = 2 (dottedx ; dasheda ; solids i g n [ a ( t ) ] ) :

IV. SIMULATION

To illustrate the proposed robust control scheme under an unknown

time-varying control direction, simulations were performed for (1)

with the following choices:

known dynamicsf ( x ; t ) = 0 k x

withk = 0

:

5

andV

1

=

1

2

x

2

;

uncertainty1 f = 0 s i n ( 2 t ) + [ 5 + 5 c o s ( t ) ] x

2

+ 0 x

3 which

is chosen to be unstable;

bounding function ( x )

is chosen to be ( x ) = 0 ( + x

2

+ j x j

3

) ;

unknown control direction is given by time-varying function

a ( t ) = 6 s i n ( 2 t )

(both cases were simulated). The rate of

change is bounded by = 2

;

design parameters in shifting law (16) and in robust control (5)

are = 0 : 0 ; k

s

= 2 0 ;

and = 0 : ;

initial guess of the control direction is set to bea ( 0 ) = 0

(for

all the cases simulated);

initial conditions of the state were chosen to bex ( 0 ) = 6

(both cases were simulated).

The simulations were carried out using SIMNON. System perfor-

mance under all possible combinations of initial conditions of state

x

and unknown control directiona ( t )

are shown in Figs. 1, 3, 4,

and 5. In Fig. 1, trajectories of the state, the estimate of the control

direction, and the robust control are shown separately. In Figs. 35,

trajectories of x ; a ; and s i g n [ a ( t ) ] are combined into one plot foreach case. Control

u

is omitted from these figures for briefness

since control trajectories are similar to that in Fig. 1 and since

the magnitude of the control, ifu

is incorporated into the figures,

will make the other trajectories obscure. It should be noted from

Fig. 4 that the speed ofa

trackings i g n [ a ( t ) ]

becomes slower as the

statex

converges to the origin. This phenomenon is expected from

(10).

To better illustrate the behavior of the shifting law and its corre-

sponding stability property, a portion of Fig. 1 (aroundt = 2 )

is

enlarged to be Fig. 2 in whichs i g n [ a ( t ) ]

first switches its sign,a ( t )

soon follows, and in between, the control temporarily has a wrong

sign and drives the system in a wrong direction (see the first plot

8/2/2019 00557583

7/7


Fig. 4. a ( t ) = 0 s i n ( 2 t ) and x ( 0 ) = 0 2 (dottedx ; dasheda ; solids i g n [ a ( t ) ] ) :

Fig. 5. a ( t ) = s i n ( 2 t ) and x ( 0 ) = 0 2 (dottedx ; dasheda ; solids i g n [ a ( t ) ] ) :

in Fig. 1). Obviously, the simulation shows that this increase inj x j

does not yield any instability. In fact, the temporary increase of the

magnitude of the state makes it possible to identify the current sign of

a ( t )

, identification in turn will provide the correct control direction,

and the state is then forced to converge back to the origin. Similar

results can be seen around all points at which the sign ofa ( t )

is

switched.

V. CONCLUSION

Robust control of a class of nonlinear uncertain systems is studied.

A system under consideration contains not only nonlinear uncer-

tainty bounded by a known nonlinear function of the state but

also an unknown time-varying control direction. Without any priori

knowledge of the control direction except for smoothness, a three-

part design is proposed for constructing robust control via the

Lyapunovs direct method. A resulting robust control is continuous

and consists of an on-line algorithm to identify continuously the

control direction, a shifting law that changes the sign of robust

control smoothly, and a robust control modified from the standard

robust controller. Essentially, the design only requires the bounding

function of unknown dynamics and the maximum rate of change of

the control direction. Stability of uniform ultimate boundedness is

guaranteed.

REFERENCES

[1] M. J. Corless and G. Leitmann, Continuous state feedback guaranteeinguniform ultimate boundedness for uncertain dynamical systems, IEEE

Trans. Automat. Contr., vol. 26, pp. 11391144, 1981.[2] F. Giri, P. A. Ioannou, and F. Ahmed-Zaid, A stable adaptive con-

trol scheme for first order plants with no priori knowledge on theparameters, IEEE Trans. Automat. Contr., vol. 38, no. 5, pp. 766771,1993.

[3] S. Gutman, Uncertain dynamical systemsA Lyapunov min maxapproach, IEEE Trans. Automat. Contr., vol. 24, no. 3, 1979.

[4] U. Helmke and D. Pratzel-Wolters, Stability and robustness propertiesof universal adaptive controllers for first order linear systems, Intern.

J. Contr., vol. 48, pp. 11531182, 1988.[5] J. Kaloust and Z. Qu, Continuous robust control design for nonlinear

uncertain systems without a priori knowledge of the control direction,IEEE Trans. Automat. Contr., vol. 40, no. 2, pp. 276282, 1995.

[6] R. Lozano and B. Brogliato, Adaptive control of a simple nonlinearsystem without a priori information on the plant parameter, IEEE Trans.

Automat. Contr., vol. 37, no. 1, pp. 3037, 1992.

[7] R. Lozano, J. Collado, and S. Mondie, Model reference adaptive controlwithout a priori knowledge of the high frequency gain, IEEE Trans.Automat. Contr., vol. 35, no. 1, pp. 7178, 1990.

[8] R. T. MCloskey and R. M. Murray, Exponential convergence ofnonholonomic systems: Some analysis tools, in Proc. 1993 Conf.

Decision Contr., 1993, pp. 943948.[9] , Experiments in exponential stabilization of a mobile robot

towing a trailer, in Proc. 1994 Amer. Contr. Conf., 1994, pp. 988993.[10] R. Marino and P. Tomei, Robust stabilization of feedback linearizable

time-varying uncertain nonlinear systems, Automatica, vol. 29, pp.181189, 1993.

[11] B. Martensson, Remarks on adaptive stabilization of first order nonlin-ear systems, Syst. Contr. Lett., vol. 14, pp. 17, 1990.

[12] D. R. Mudgett and A. S. Morse, Adaptive stabilization of linear systemswith unknown high frequency gain, IEEE Trans. Automat. Contr., vol.30, pp. 549554, 1985.

[13] R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to

Robotic Manipulation. New York: CRC, 1994.[14] R. M. Murray and S. S. Sastry, Nonholonomic motion planning:Steering using sinusoids, IEEE Trans. Automat. Contr., vol. 38, no.5, pp. 700716, 1993.

[15] R. D. Nussbaum, Some remarks on the conjecture in parameter adaptivecontrol, Syst. Contr. Lett., vol. 3, no. 5, pp. 243246, 1983.

[16] Z. Qu, Global stabilization of nonlinear systems with a class ofunmatched uncertainties, Syst. Contr. Lett., vol. 18, no. 3, pp. 301307,1992.

[17] , Robust control of nonlinear uncertain systems without general-ized matching conditions, IEEE Trans. Automat. Contr., vol. 40, no. 8,pp. 14531460, 1995.

[18] , A new generic procedure of designing robust control fornonlinear uncertain systems: Beyond the back-stepping design, inProc. 1994 Wkshp. Robust Control via Variable Structure LyapunovTechniques, Benevento, Italy, Sept. 1994, pp. 174181.

[19] , Robust control of nonlinear uncertain systems under generalized

matching conditions, Automatica, vol. 29, pp. 985998, 1993.[20] Z. Qu and J. Dorsey, Robust control of generalized dynamic systemswithout matching conditions, Trans. ASME, J. Dyn. Syst., Meas., Contr.,vol. 113, pp. 582589, 1991.

[21] J. J. E. Slotine and K. Hedrick, Robust inputoutput feedback lineariza-tion, Intern. J. Contr., vol. 57, pp. 11331139, 1993.

Documents

00557583