Nonlinear Output-Feedback Tracking Using High-gain Observer and Variable Structure Control*,1

PII: SOO05-1098(97)O01 Ii-8

Pergamon

@

Automatic, Vol. 33, No. 10, pp. 1845-1856, 1997 Elsevier Science Ltd. All rights reserved Printed i“ Great Britai” OC05-1098/97 $17.00+ O.OO

01997

Nonlinear Output-Feedback Tracking Using High-gain Observer and Variable Structure Control* SEUNGROHK OH? and HASSAN K. KHALIL~ A globally bounded output-feedback variable structure controller with a highgain observer is designed for a feedback- linearizable minimum-phase nonlinear system in the presence of unknown disturbance Key Wordw-Variable structure control; high-gain observer; feedback linearization; tracking system; globally bounded control; minimum-phasesystems

Abstract-We consider a single-input–single-outputnonlinear

systemwhichcan be representedby an input&outputmodel.The system, which can be transformed into the normal form, is required to be minimum phase. The model contains unknown bounded disturbances.We assume that the referencesignal and its derivatives are bounded. A high-gain observer is used to estimate derivatives of the tracking error while rejecting the effect of the disturbances. We design a globally bounded output-feedbackvariable structure controller that ensures tracking of the referencesignal in the presenceof unknowntime-varying disturbances and modelingerrors. We give regional as well as semiglobalresults.We do not require exponentialstabilityof the zero dynamics nor global growth conditions. (Q 1997Elsevier ScienceLtd. 1. INTRODUCTION

Variable structure control (VSC)has been successfully used in various applications (DeCarlo et al., 1988; Utkin, 1992; Hung et al., 1993),modeling uncertainty and disturbances (Drazenovic, 1969). Most of the work on VSC assumes state feedback. There have been, however,some efforts to develop output-feedback VSC. These can be classifiedinto two classes of output-feedback schemes. One scheme is static output-feedback VSC which does not use an observer to estimate the states of the *Received13 February 1996;revised 2 December 1996;received in final form 24 April 1997.An earlier version of this paper was presented at the 1995 IFAC Nonlinear Control Systems Design Symposium,held in Lake Tahoe, U.S.A. in June 1995.This paper was recommended for publication in revised form by Associate Editor Alberto Isidori under the direction of Editor Tamer Ba$ar, Corresponding author Dr. Seungrohk Oh. Tel + 8227092869; Fax + 8227092590; E-mail [email protected]. tDepartment of Electronics Engineering,Dankook University, Hannam-Dong Youngsan-Ku Seoul,postal code 140-714, South Korea. jDepartment of Electrical Engineering, Michigan State University,East Lansing, MI 48824-1226,U.S.A. 1845

system (E1-Khazaliand Decarlo, 1992;Heck and Ferri, 1989)but is limited to relative degree one systems.The other scheme is observer-based controllers which could be used for relative degree higher than one systems (Bondaref et al., 1985; Diong and Medanic, 1992;Emelyanov et al., 1992; Raghavan and Hedrick, 1990). Bondaref et al. (1985)and Diong and Medanic (1992)treat linear systems assuming perfect knowledge of the state model, leadingto a standard observerdesign problem. In Emelyanov et al. (1992)and Raghavan and Hedrick (1990)high-gain observers are used to reject disturbance due to modeling uncertainty and imperfect feedback cancellation of nonlinearities. However,the use of high-gain observersof relative degree higher than one results in peaking in the state variables as well as shrinking of the region of attraction (Esfandiari and Khalil, 1992).The same peaking phenomenon is present in the high-gain observer-based VSC designs of Emelyanov et al. (1992)and Raghavan and Hedrick (1990),as we demonstrated in Oh and Khalil (1995).In the same paper (Oh and Khalil, 1995),and motivated by Esfandiariand Khalil (1992),we showedthat a globally bounded VSC with a high-gain observer can be designedto stabilizea classof nonlinear systems with no peaking. Since the idea of combining a high-gain observer with a globally bounded controller was introduced by Esfandiari and Khalil (1992),it has been used in a fewdesignsof continuous output feedback control (Jankovic, 1997; Khalil and Esfandiari, 1993;Lin and Saberi, 1995; Mahmoud and Khalil, 1996;Teel and Praly, 1995, 1994;Khalil, 1996b).In particular, it is a very useful tool to achieve semiglobaloutput-feedback stabilization where the controller is designed to include any givencompact set in the region of attraction. In

S. Oh and H. K. Khalil

1846

this paper we generalize our previous work (Oh and Khalil, 1995)in two directions. First, we consider a class of nonlinear systemswhich have zero dynamics; our previous work was limited to systems with no zero dynamics. Second,we solve the tracking problem in the presence of disturbances, as opposed to the stabilization problem studied in Oh and Khalil (1995).We consider a nonlinear system represented by an input–output model which contains unknown external disturbances. We extend the dynamics of the system by adding a seriesof integrators at the input sideof the system and represent the augmented system by a statespace model. The extension of the dynamics of the systemmakes the derivativesof the input available for feedback.We construct a robust high-gain observer to estimate the tracking error and its derivatives. The sliding surface is chosen in the observer space and a globally bounded output-feedback VSC is designed to satisfy the reaching condition. The use of the high-gain observer and the globally bounded control enables us to show that the estimation error decays to arbitrarily small values during a short transient period. After showingthat the trajectories reach the sliding surface in finite time, we show that the output tracks the reference signal with arbitrarily small error. We assume that the zero dynamics of the system are minimum phase, but we do not require exponential stability or global growth conditions. 2. PROBLEM STATEMENT

We considera single-input–single-outputnonlinear systemrepresented by the nth order differential equation Y(“)=f( .) + g(. )u(m),

these integrators by ZI = U,Z2= U(l), Up to z~ = U(m-1),and set v = u(m)as the control input of the augmented system.By taking xl = y, X2= y(l), up to Xn= y(n-1) we can represent the augmented systemby the st~te-spacemodel 1< i < n – 1, .-in =f(x, z, d) + g(x,z, d)o, ij = Zj+ 1, 1
(2)

im= v, where x = [xl, ... , x~]T and z = IzI, ... , zn]T. A similar model, called generalized observer canonical form, was studied in Sira-Ramirez (1993) for state feedback VSC but no integrators were augmented at the input side. The augmentation of integrators is not needed in the state feedbackcase since all state variables can be measured. In the output feedback case, high-gain observers can be used to robustly estimate a part of the state vector which constitutes derivatives of the output. This restricts the controller to use partial-state feedback (see,for example,Lin and Saberi, 1995;Mahmoud and Khalil, 1996).By augmentingintegrators at the input side of the system, we obtain the system (2) whosestate (x,z)comprisesz which is readily available for feedback and x which constitutes derivativesof y; so it can be robustly estimated by a highgain observer.We make the followingassumptions on the system (2). Assumption1. For all (x,z,d)e U x r,, Ig(x,.z,d)l> k, >0. Assumption 1 implies that the system (2) has a uniform relative degree n.

(1)

where u is the control input, y is the measured the ~th derivatives of output, u(i) and ~(i) denote u and y, respectively,and w < n. The functions~(”) and g(.) coulddependon Y,y(i),... ,Y(“–1),u,U(l),... , ~(m-1) and an external time-varying disturbance d(t). We assume that j(”) and g(. ) are sufficiently smooth and definedin a region U. x UZx rl where U. c R“, U, c R“, rl is a known compact subset of F!pand d(t) Grl.We also assume that d(t)e rz where rz is a known compact subset of Rp. We set U = U, x U. and r = rl x rz. The objective of this paper is to design an output feedback VSC which guarantees boundness of all variables of the closed-loopsystem,and output tracking of a given reference signal y,(t) in the presence of modeling uncertainty and time-varyingdisturbance. We represent an extended version of (1) by a state-space model. We augment a seriesof m integrators at the input side of the system and denote the states of

Assumption2. (1) There exist m smooth functions ~i,1< i < m, definedon U x rl such that the mapping [:] = T(x, z,d) is a diffeomorphism of U onto its image, for every d~ 171,that transforms the system (2)into the normal form 1< i S n – 1,

(3)

% = 3(x>q, d) + ~(x>q, d)v>

(4)

ii =

Xi+

y = x~,

1,

(6)

where q = [ql, ... , n~]Tand w = [d i]T. (2) There exists a domain $2Xx$2,c 07?’ x R“’ that containsthe originsuch that $?2.x $3,c T(U, d), Vd~rl and equation (5) has a unique equilibrium point at q = Owhen (x,w) = (0,0),i.e.,0(0,0,0) = 0. Remark 1. When g(”) is independent of d, the diffeomorphism T(”) is also independent of d and the

Nonlinear output-feedback tracking right-hand side of (5) reduces to O(X,q, d). In this case, local existence of the diffeomorphism T( ) followsfrom the theory ofinput-output linearizable systems (Isidori, 1995).Moreover, when Assumption 1 is satisfiedglobally,global existencecan be shown using the tools of Byrnesand Isidori (1991).

1847

(x,w)= (O,O);hence,the systemis minimum phase. It can be seen (Khalil, 1996a, Corollary 5.2) that when (x,w) # (O,O)but bounded, the solution q(t) satisfiesthe estimate Ilv(dll~ Pl(lb’l(o)ll,t) + y(h),

Vt20,

where Remark 2. In the special case when g is constant, one can verify that the change of variables Xn– ~ + i qi = Zi – —,

1S i < m

9

transforms the last m equation of (2) into di

=

~i+l,

1<

~m= –f(x, z,d) 9

i
, z, =q, +x..

”,+,/g

thus transforming (2)into the normal form (3~6). The mapping T is required to be a diffeomorphism of a domain U onto its image.We recognize that there are no results in the literature that will guarantee the existence of the mapping T of Assumption 2. However, under Assumption 1, one can always find a local diffeomorphism.As one solvesfor the local diffeomorphism,it is possibleto determine the region of validity of the mapping. The essenceof feedbacklinearizationstrategiesis to choose the input v to control the behavior of x. Sinceequation (5)is driven by (x,w),it should have a bounded-input–bounded-state stability property, with (x,w) as input and q as state. Moreover, q should belong to the domain of @(.). We make the followingassumption for such a system.Define the ball, Dv = {qe ElmlIlqll< dq}where dqis chosen such that Dqc @q. Assumption 3. There exists a Cl function W :Dq+ 5!+and class% functions,~i:IO,dJ + R+, i = 1,2,3, and yl: [0, d,) + R+ such that Ul(llql[)<

w(q)< Ctz(llqll),

(7)

(8)

d, =

x

SUP (x, w)a,

xr

[1 .

w

This assumption is equivalent to (local) inputto-state stability (Sontag, 1989; Khalil 1996a).It implies that q = O is asymptotically stable when

[:;1 -0}7 ‘=SUP{ /11is a class %S? function,and y is a class % function. The definitionof classXY and classX functions can be found in Khalil (1996a).In the semiglobal tracking case of Section 4, Assumption 3 must hold globally.However, it is less restrictive than some of the global assumptions used in the literature (see,for example,Emelyanovet al., 1992). In fact, Emelyanov et al. (1992) assumes that i = 0(0,11,0)is globally exponentially stable and @isglobally Lipschitzin (x,w).Under these conditions, it can be shown that there existsa Lyapunov function W(. ) such that Assumption 3 is satisfied. We will see in Section 3 that the controller is designedin the (x,z) coordinates whilethe stability analysisis performedin the (x,~)coordinates. Since the mapping T(. ) could depend on unknown disturbance, we make the following assumption to ensure that for every domain of interest in the (x,q) coordinates, there is a corresponding nonempty domain in the (x,z) coordinates. Assumption 4. For every domain fi. x ~Vc ~. x Dq,there are nonempty domains %Xx42=c Uxx U, and ~1 c rl such that T(4YXx %Z,d) c 5.x Dq,YdGr~. Assumption 4 has the effect of restricting the class of disturbances.In the case of no disturbance, for any ~. x ~V,we can always find 42Xx42=such that (X,Z)G%Xx 4!4= c Uxx u=(x,q)~~X x & c f7Xx D,. When the mapping T depends on unknown disturbance, existenceof the set 4?lX x 4?/=depends on the class of disturbances. Note that this condition implies that the set ~1 depends on the choice of ~. x D,. Define the set Cl,= {qe Rml W(q) s c,} where Cvis a positive constant such that ~z. Yl(d,)< c~< %(dv). (9) Using Assumption 3, it can be seen that Clqc D,. From Assumption 4, we can find sets %?X x ~, c U. x U=and ~1 c rl such that T(%, x 42z,d) c 9Xx t2q,Vdc ~l. We assume that the referencesignal y,(t) is bounded, has bounded derivativesup to the nth order, and yjn)(t)is piecewisecontinuous.

1848

S. Oh and H. K. Khalil

Let

are in the open left-halfplane. We rewrite the observerequations (10)and (11)in the compact form

Wr(t)= [y,(t),y;l)(t), ..., y$n-l)(t)]=,

i?= At + ~{fo(~ + ?/,, Z)

@YR(t) = [y,(t),y:l)(t),..., y;”- ‘)(t),y:n)(t)]=. We also assume that !Y,E Y, and 3Y~~Y~ where Y, and Y~ are compact subsets of R“ and R“+l, respectively.Define et = y —y,, ez = . ,=. . e. =

j —j,, . . Y(n–1)– Y:n-1)

We design the augmented control input u for the system(2)to achievetracking of the givenreference signal. We use a globally bounded controller to eliminate the peaking phenomenon in the state variables(Oh and Khalil, 1995).The actual control input u is the output of the series of m integrators driven by the augmented control input v. Moreover, measurements of z are always available since its components are obtained by integrating o. Let fo(x, z) and go(x, z) be known nominal models of ~(x, z,d) and g(x,z, d), respectively.Suppose~O(x,z) and go(x,z) are sufficientlysmooth and go(x,z) # O for all (x,z)e U. We assume that~o(x,z) and go(x,z) are globallybounded. This can be always achieved by saturating the given nominal functions outside a bounded domain of interest, as it will be illustrated later on. To estimate the derivatives of the tracking error (el = y – y,), we construct the following observer: (lo)

#n =j(el –21)+fo(2+ 4Yr,z)

(13)

1, ... ,n.

+ g(e + W,, z, d)u– y$’)},

(14)

i = Alz + Blu,

(15)

c~=(A – LC)~ + e13~(e + ~,, z, d) –fo(i? + Y,, z) (16) + {g(e+ Wr,z,d) – 90(2+ @r, z)}ul, where (Al, Ill) is a controllable canonical pair and ~ = ILl, Cz,... , ~n]=.It can be verified that (A – LC) is a Hurwitz matrix. We choose the sliding surface (17) o(8)= Mi?, where M = [ml, ... , m“.-1,1] is chosen such that

2=

I

o

... 1

... ...

0 0

...

...

0

1

0

I

;

o l-ml

1 0

‘m2

..

-7rln-2

-q-l]

is Hurwitz. Define 4 = Pl, ... , e“r=

1]=, 4 = [t?l, ... , i?.- JT, [cl,

...

,L-ll=,

and rewrite equation (10)as (18) where

‘=[],..l).: ‘=[l::],.l) @&)= diag[&”-2,&”-3,... ,s, 1],

(11)

where ~iis the estimate of ei,e is a positiveconstant to be specified,and 2 = [21, ... ,.%]=.The positive constants li are chosen such that the roots of Sn+ 11s”–1+ ... + 1.–1s+ in= o

=

d = Ae + B{f(e + 9,, z, d)

3. CONTROLLER DESIGN

+ go(f?+ w,, Z)u— y:”),

j

&tj,

The closed-loopequation can be rewritten as

Assumption 5. There is a domain D. = {ee WI Ilell< d.} such that VeED. and VYre Y,, e + Y, e 42Xwhere d. is a positive constant.

i=l , ...,/? -1,

where (A, B) is a controllable canonical pair, L = [11,... ,l.]=, C = [1,0, ... ,0], and D(e)= ~] . Let ~j=ej–2j be the estimation diag[~,~, ... ... error, and define the scaled variables [j=

and e = [el, e2, ... , e.]=. Sinceour goal is to design a controller such that e ~ O,we make the following assumption.

li ti = di+1 + #el —21),

+ go(t + &,, z)v – yfn)}+ D(8)LC(e– .2), (12)

e=[1,0, ... ,0,0]1X(n1). We consider a control input of the form v = q(i?+ !Y,,z) + v(4+ 47,,z)sgn(a(~))where P(”) and V(”)are continuous and globallybounded functions. We will specify u later on. Since X is a Hurwitz matrix, for any positive-definitematrix Q,

Nonlinear output-feedback tracking there is a symmetric positive-definite matrix P such that Pi? + ~TP = – Q (Khalil, 1996a).Let V(q)= qTPqand choosec., >0 and r >1 such that E,= {e=R*I IMelS c,,, ~

< c,,}

when11~11 < )&and (e,2)GE, x E,. Thiswillbedone by showingthat ad < – p21crl as long as a # O(Filipov, 1988)where p2 is a positive constant. We have 06 = agi = OM[A2+ B{Jo(i?+ %IV,, z)

is in the interior of De, where c~r= r(a2/al)cer, al = (A~in(Q)/l~,X(P)), and a2 = (211P@l/~m). By inequality (9),

+ go(d+ ~,, z)v– y;} +D(s)LC(e–2)] = O&l Ae + B{fo(e + 4Y,,Z)+ go(d [

Iblll< 71(4)= ~2(llm < c,= w(q) < c,. Hence, {qe Rmlllqll< yl(d,)}c Ctn.Therefore, by Assumptions 3–5, e(t)e E,,

where

Vt >0

and Z(0)C’42Z * ~(t)Gf2q,

Vt>0.

(19)

q Ef2q and x c $2X* z e f2z,c U=.

(20)

The set E, x Clz,is taken as the region of interest in our analysis.Achievinggloballybounded functions ~O(e+ ~,,z) and go(e+ ~., z) can be done by saturating these functionsoutside the set E, x !2=,x Y, where s Cq,}c D.

and c,, < c,,, c~.< c~,.Note that E, can be always defined since E. is in the interior of D.. Let E. = {ee WI Ilhlell< CGO, ~

+ 9,, z)v– y;} + ~@)LC~ , 1

(22)

~(&)= diag[sn- l,en-2, ...,8, 1].

Even though the mapping T depends on the unknown disturbance d, with known bounds on 9, and d we can find a compact set !2=,such that

E, = {e=R“llA4els c,,, @

1849

< C,o},

C.. < c., and c~o< c~,, @[= (e R“l11~11 < & , ~ = EOx %Lx ~,, 1 { where c is an arbitrary positive constant

(21)

Lemma 1. Consider the singularly perturbed system (14)--(16)with any globally bounded control v and suppose that Assumption 1–5 are satisfied. Suppose (e(0),((O),z(O))ef2and Vd(t)~~l, Vt >0. Then there exist El and T1 = Tl(e) < T4, where T4 is the first time e(t) exits from the set E,, such that for each &e (O,El), we have 11~11 < Fe Vt ● ITl, TJ and z(t)~Qz, for all te [0, T4) where k is a positive constant. Proof The proof of this lemma is a slightmodification of a similar lemma in Oh and Khalil (1995).It ❑ is given in Oh (1994). Lemma 1 implies that the fast variable ( decays very rapidly during a short time period [0, Tl]. Lemma 1 also impliesthat Zeflz, as long as eeE,. Next, we design a globally bounded control input u such that a sliding-mode condition is satisfied

We need an estimate of ~M~(e)LC~ to design the control input u such that the sliding mode condition is satisfied.We make the followingassumption on the uncertainty. Assumption 6. For all (x, z,d)e U x rl there are a scalar nonnegative locally Lipschitz function p(x,z) and a nonnegativeconstant k such that l~(x,z,d) ‘~o(x>d

(23)

< P(X,Z),

19(x>z, d)g; ‘(x, z) – 11s k <;,

t

(24)

where kt = 1$ ICe(~-~c)~Bldt. Remark 3. To be able to designa controller, uncertainty on the input coefficientshould be less than 1over the domain of interest,e.g.,k <1 in equation (24).This is implied by (24)since k,> 1. It can be seen that when we choose all eigenvaluesof (A-LC) to be real and negative,k, = 1. Using smoothness of ~(”), ~o(”), and g(”), and boundedness of j,, d, and v, it can be shown* that, for (e,2)~ E, x E,, $@jLC~l

< p(? + g,,

Z) +

kk.k, + k6&,

Vte

T1 + eln~, T4 , (25) [ ) where k. is an essential upper bound on the absolute value of go(i?+ &,, z)u(2+ W,,z) to be specified and k6 is a some positive constant. Consider the function l/@?,@Yr,z) = 9; 1(~+ ~,>4[ – M(A8 + Bfo(t + 4Y,,z)) + y:.) – (p(~+ w,, z) + PI) sgn(~)l> (26)

*The details of showing(25) that can be found in Oh and Khalil (1995).

1850

S. Oh and H. K. Khalil

where PI is any positive constant that satisfies ~~ ~

k k,k, + 2p2 1 – kk,

(27)

for some Pz >0 and k, is definedby the inequality I – M{A2 + B’.(2 + ?/,, Z)}+ y!”) (28)

– P(2+ ~,, z)sgn(a)l
for almost every (d,@Y~, z)c E, x Y~x Q,,. Notice that k, can be calculated since~0(”) and P(”) are known. We take the control input v(&!V~,z) as ~(e,%~,z), saturated outside the set E, x Y~x flz,. In particular, let vl(4~R,4

Wr,z) x [lI’’f{A@ + lilf~(t+ @Zr, z)}+ y:’)],

=

v2(4~r,4

=

–9;1(2

–9;’(2

+

+

SI=

max

PI}>

1~1(~,~R,

Theorem 1. Suppose that Assumptions 1–6 are satisfied, (e(0),C(O), z(0))e fl, and d(t)e ~1 Vt >0. Then, there is es >0 such that for each EC(O,es),all state variables of the closed-loopsystemformed of the plant (1) and the e-dependent controller (10), (11),and (29),are bounded and there existsa finite time tl >0 such that Ile(t)ll< kp& ‘v’t> tl where kp is a positive constant. Proof The proof has a lot of similarity with our previous work (Oh and Khalil, 1995);so we do it briefly. To show z(t)eflz, ‘v’t>0 and 11~11 TI(8), it is sufficient to show e(t)e E, Vt >0. Define

4Y,,Z)

x {p(2+ ~,, z) +

the sliding-mode condition when 11~11 < I% and (e,6,z)~E, x E, x !2Z~for teITl,T4).It is well known that Lemma 2 implies that sliding mode exists after a finite time. Using the existence of sliding mode in the observer space, it is shown in the following theorem that the output tracks the referencesignal with arbitrarily small error.

Z)l>

(2, !V.R,z)sE, x Y, x %

s~=

max (2, W,, z)=

IV2(4

y,,

!& = {de R“lIll!fi?ll< ce~, m

~

%2},

<

Cqz <

(w+s’sat(wsgn(o)

s~@

where ceo< ~ez< C~Z< c=,, Cqo <

(29)

where sat(”) is the saturation function. Inside the set E, x YR x Q=,,we have V(4,%R,Z)

= ~(i?, @R, Z).

(30)

Hence, k. in inequality (25)can be taken by (31)

k. = k, + P1. Using (22),(25),(27),(30)and (31), ad s [ – {P(2+ ~r, Z) +

~ ~q,}

and

E, x Y, x Q.

and take U(i?,@yR, .%’) =

Clz= {ec R“lIIMellS Z.2, m Z)l>

K@

+

@r,

Z) +

+ A}

+

kk~(k,+ PI)

k~c]lals – PJJI

(32)

for sufficientlysmall 8.We summarize our findings in the followinglemma. Lemma 2. Consider the singularly perturbed system (14X16) with the control input u(2,9,,z) defined by (29). Suppose that Assumption 6 is satisfied, and suPPose IILII< I?: and (e,2,z)e E, x E, x LIZ.for t~ ITl, T.). Then as long as o # Othe sliding mode condition 06< – pz]ol is satisfiedfor all t● ITl + gln~, T4). So far we have shown that llrll
13q2 =

~?e2~e2(?e2

>

~gz

Cqr,

1), and cq2= ~rc2c.2(r,z > 1).

Since {e(0),C(O)} efl, there exists a finite time T2 < Td such that e~fiz for all O5 ts T2. For sufficientlysmall e, Tl(e) + &ln(~)s T2. Note that 2 might be outside f22 for t< Tl(c) + eln(~),but since tj = ej – g“-j~j, by ll~(t)ll < &~Vt~[T1, T4) with sufficientlysm~lle,~eflz at t= Tl(s) + &ln(~). Using ad <0 and V <0 on the boundary of Qz, it can be verifiedthat 2ef22 for tG ITl(e) + &ln~, TJ. Using a contradiction argument, it can be shown that eefl, for Vt > Tl(e) + sln~. Hence, ee~, for Vt >0. Therefore, the sliding mode condition is satisfiedVt > T where T = Tl(e) + ein ~. This implies that a = Oholds for some finite time. On the manifold a = O,we have (33)

tin_ ~= e.

= – (mltl + m2i22+ ... + m.- 12.-1) + L = –{ml(el–8“-1(1)

+ ...

+ m.- l(e.- 1 – K-1)} + L.

(34)

Substitution of equation (34) into equation (14) yields ~ = ~q + AC = & + A@)C,

(35)

Nonlinear output-feedback tracking where

Choose c=.and r such that

o 0 A=

1851

0 0

1“ o —ml

...

... 0

... ...

...

00

—m2 ...

0 0

—mn. ~ 1

E,~ {ee WI Ikfel< c,,, fi%}, c., > Co, c~,> c~o,Cq,= r~cc,, and r > 1. Sincethe mapping T is a global diffeomorphism and d(t) belongs to the compact set rl, we can find a compact set fi. x fin such that T—

1) xfr

Since ~ and (A – LC) are Hurwitz, for given positive-definite matrices Q1 and Qz, there are positive-definite matrices PI and P2 such that P1~ + ~TP1 = – Q1 and PZ(A – IX) + (A – JX)TP2 = – Q2. Let W_(q,() = qTP1q+ ~TP,C.The derivativeof ?f(q, () along the trajectories of equations (16)and (35)satisfies *(q, ~)S – ll~(q,c),

for W(q, L)> rse

for some positive constants 11 and r3. Let rd > rs, and define Cl.= {(q,~)lW(q, L)S r48}. It can be shown that there is a finite time tl such that (q(t), c(t)) GQ, Vt > tl. Using (33)and (34),and ll~(t)ll< ~c for all t2 T, itcan be shown that hell< k,~ for all t > tl where k, is a positive •1 constant.

(e,4Y,,z)c E, x Y, x 42===(x, q)efiX x fi,. Choose c~such that 6,= Qn~ {qe RmIw(q) < c,} and Cq> U20yl(~a), where pa = max(X,~)=ax ~r Il[illl.Notethat we can always find such c~ since W(q) is radially unbounded. Since E. is a compact set, we can take d, such that E, c D. = {e=Rn\hell< d.}. We can also take d, such that the set Dv = {qGRmlllqll< d~}contains L2qin its interior, i.e., dv> ct~l(c~).Observe that we can choose the constants c~, d., and d~ with no restriction on ~, and d since the system (2) is defined globally. Hence, we do not require Assumptions4 and 5 in the semiglobalcase. Using Assumption 3 and the fact that the mapping T is a globaldiffeomorphism, we can tind a compact set & such that Z(0)=42Z and e(t)e E, Vt >0 =-q(t)c Qq

4. SEMIGLOBALTRACKING - z(t) c cl,,

In this section we consider a globally defined input–output model (l), i.e., U = R“+m.For any given compact set of initial conditions of (x,z), for any ~R e YR, and for any bounded disturbance with w(t)c r, Vt >0, we willdesignan output feedback controller that ensures tracking of the reference signal y,(t), where YR and r are any given compact sets. Since U = R“+m,we need to modify the assumptions in the Section 3. We require Assumptions 1 and 2 to hold for U = R“+m, ai(i= 1,2,3) and yl to be class %~ functions, and inequalities (7) and (8) to hold for V(x,q, w)=R“+mx r. We also require that for every compact set 42c R“+m,there exist a nonnegative locally Lipschitz function P(x,z) and nonnegative constant k such that inequalities(23)and (24)hold V(x, z,d)e4Y x rl. Note that E(x,z) and F could depend on 42.To achieveour goal,we need to show that the region of attraction of the closed-loop system (14~16) can be made arbitrarily large. We assume that the initial conditions e(0)and z(0)belong to the given compact sets & and Cl=,respectively.Since Ilkfeland V(q)are radially unbounded, we can choose C.. and c=.large enough such that

and x(t)~f2x Vt >0

Vt>0.

Define E,= {e=WI IMel < c,,, ~

< c,,},

where c,, < c,. and c,, < cQ~.From the global version of Assumption 6, we can find the nonnegative locally Lipschitz function p(e + !Y,,z) and nonnegative constant k over E. x Y, x Cl,,x rl. With the same designprocedure as in Section 3,it can be verified that fl = E. x%( x 42, is in the region of attraction. We summarize our conclusion in the followingtheorem. Theorem 2. Suppose that Assumptions 1 and 2 hold globally, ai(i = 1,2,3) and yl are class X_~ functions, inequalities(7) and (8) in Assumption 3 holdV(x,~,w)eR“‘m x r, and for everycompact set 42c R“+m,there exista nonnegativelocally Lipschitz function ~(x,z) and a nonnegative constant E such that inequalities (23) and (24) in Assumption 6 hold V(x, z,d)e% x rl. Let !2 = E. x 42Cx 42,be any given compact set. Then, there is 84>0 such that for each CC(0,E4)and (e(0),L(O),z(O))efl,the state variables of the closedloop system formed of the plant (1) and the e-dependent controller (10), (11), and (29), are

S. Oh

1852

and H. K. Khalil

bounded and there is a finite time tl >0 such that IIe(t)lls k,~ for all t > tl where k, is a positive constant.

where r is a desired constant angular velocity,we can rewrite (37)in error coordinates as

@l= e2, Z2= – (61+ 04)e2–

t9164(e1 +

r) + /33652

5. EXAMPLE

As an example we consider speed control of a field-controlledDC-motor. The motor is modeled by Sira-Ramirez (1993).

fil =

Va EW1– ZW2U + z’

—R.

K

BK fi2 = ——W2 + —Wlu, JJ

Y =

–

(01

+

e4)j

–

o164y

+

7

z

The followingchange of variables transforms (38) into the normal form and shifts the equilibrium point of interest to the origin

(36) q=

where WI represents the circuit current, W2 is angular velocity of the rotating axis, V. = 5V is a fixed voltage applied to the armature circuit, and u is the field winding input voltage. The constants R, = 7f2, L. = 120mH, and K = 1.41x 10-2 Nm/A represent the resistance, the inductance of the armature circuit and the torque constant, respectively. The parameters J = 1.06x 10-6 Nms2/rad and B = 6.04x 10-6 Nm s/rad are the load’smoment of inertia and the associated viscous damping coefficient,respectively. Our objectiveis to design a feedback controller, that uses only measurement of the angular velocity W2,so that W2asymptoticallytracks a desiredangular velocity. For simplicity, let 01 = R=/L,, 0, = K1/La, 93 = V,/L, 04= B/J, and 65 = K/J. Then the input–output model, with the angular velocity as the output, can be obtained from the state-space model (36)as

(38)

e2 + t94(e1 + r)v

– 0265(el+ r)z2+

z* e2 + 04(el + r) – 04r’ z

(39)

where z*, the value of z at equilibrium,is given by

‘“=~[”-1 It can be seen that the root with the negativesignis the equilibrium point of minimum-phase zero dynamics.Hence, we use it in (39).In the new coordinates, equation (38)is given by

+ {( &) + 6305 q +

(e2

04(e1+

4

r) q + ~

x 6205(e1 +

()

z*

ti= 6’I V + ~

+ j + 04YU u“

2(e2+ 64(e1+ r))2

t14r t14r~+ z*v’

()

9305u– e2e5yU2

}

4

()

+

r))

Z*

3

+ 0205(el + r) q + — 4

04r

x (e2 + (34(e1+ r))

After taking the new state variables as xl = y, X2= j, and z = u and setting v = U,the augmented system is given by

– e2t15x1z2+

X2 + 64X1

z

To estimateel and e2,a robust observeris designed as 1 21= 22 +; (el —i?l),

(37) v,

j2 = ~ (el – ~1)– (O1+

04)82 – 0104(21 + r)

&

The system (37)has a uniform relative degree 2 in the domain

u = {(X,2)1X2 +

6.X, # O,z #

where x = [xl x,]=. By taking et = x1 — r,

e2 = X2,

o},

+ 6365Z – d265(tI +

r)z2+

22 + f14(i?1 + r) u. z

We choose the sliding surface as o = 21 + 82. Let the region of interest be f2, = {(e,z)e R31I el + e21S 120, . 120< el <120, 0.1< z S 0.5}.

Nonlinear output-feedback tracking 1

I

1

1

1

9 ,..

200 0

2

1

1

4

6 time

,

10

8

12

Fig. 1. Tracking of the desired angular velocitywith e = 0.01.

I 5000 r 1 I 4000 ~~ I I

3000 ~ I I

.,.,..

2000 : I I

1000; I

o 0

0.5

1

-looo~ o

1

0.5

time Fig.2. Plots ofstate variable e(t)and its estimate c?(t)withe=O.Ol(solid finesarethe profilesofstate variablesand dashed finesarethe

profiles of their estimates).

Define

(@maxt(”)> vmax> L@)= *(”) I)min< *() < l)-

I

+ min

– 03052+ 62(35(21 + r)z2

– 22– 50sgn(a(2))} and take the control input u as

W“)

<

*rein,

where~m, = max(e,,W 4(2,z),and ~~in= min(e,,W ~(t, z). Using the optimization toolbox of MATLAB,we determined ~~,, = 3.1and tj~i~= – 241. We simulated the response for e(0)= [80 O]T,

S. Oh and H. K. Khalil

1854

0.26

0.24

0.22 I

7 r

I

I

L 0.2 I

I

E ; O.l I I

-33:E 0.1(

o.1~

0

0.2

46

0.4

0.12

10

8

10

5

time

time

time

Fig. 3. The profiles of control input and the actual control input with e = 0.01

-

6W0

I

)..

g g * a

“1

‘3

a)

la .g

.50

.

a

-1oo0

-1000 0

)

50

I

100

estimateof error(el)

150

)

estimateof error(el)

Fig. 4. The phase portrait of estimate of error 2 with e = 0.01.

z(0)= 1.7,and 4(0)= [0 0]= with &= 0.01.Figure 1 shows that the actual angular velocity tracks the desired angular velocity of 200rad/s. Figure 2 shows the performance of the high-gain observer with globallybounded VSC.After a short period of time, errors between e(t) and its estimate t?(t)become arbitrarily small. Moreover, due to the use of a globally bounded control, tz(t) exhibits peaking

instead of ez(t).One observesthat the control input v(t)is saturated over the set f2, in Fig. 3 and the actual control input u(t), i.e.,the field voltage, converges to a stable equilibrium point. We observe also that chattering of the actual control input u(t) is reduced since the actual control input is the output of an integrator. Figure 4 showsthat attractivity to the sliding surface is achieved after error

Nonlinear output-feedback tracking

o

0.5

1855

1

time Fig.5. Plots of state variable e(t)and its estimate ~t)withe=O.006.

e(t) —i?(t)ha sbecomesmallenough. Toinvestigate the effect of q we simulated the system with 8=0.006. As we decrease E, we observe in Figs. 2 and 5 that peaking ofez(t) increases, while other variables, i.e., cl(t), ez(t), and i?i(t),retain values similar to those obtained with &= 0.01.

6.CONCLUSIONS

We have designeda globallybounded output-feedback VSC that ensures tracking of a referencesignal with arbitrarily small error in the presence of unknown bounded disturbances and modeling uncertainty. We can achieve the desired tracking accuracy by choosing the design parameter e sufficiently small. Decreasing the parameter e does not induce peaking of the system’sstate variables due to the use of a globally bounded control, but our ability to decrease &will be limited by practical factors like measurement noise and sampling rates. In our work we start from the input–output model (1)since it givesa convenientstarting point for our output feedback design. Conditions under which a state model will have the input–output model (1) are given in Mahmoud and Khalil (1997).It is well known that chattering of the control input can be a barrier for practical implementation. It was shown (Utkin, 1992)that the use of an observer eliminateschattering for linear systems.Such property was not shown for the more general case of uncertain nonlinear systems. In our case, when m >1, the actual control input u is the output of

a series of m integrators driven by the discontinuous control input rJ. Integration introduces a smoothing effect that eliminates chattering in u. For the m = Ocase, it is clear from the derivation that replacing a signum nonlinearity by a saturation nonlinearity to eliminate the chattering, we can recover the performance to a boundary layer. Then standard analysis inside the boundary layer will show ultimate boundness (Khalil, 1996a). Acknowledgements—This work is supported in part by the National ScienceFoundation under Grant no. ECS-9121501and ECS-9401187,and by a fellowshipfrom Korea Electric Power Corporation.

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