input disturbance suppression for a class of feedforward uncertain nonlinear systems

Systems & Control Letters 45 (2002) 227–236www.elsevier.com/locate/sysconle

Input disturbance suppression for a class of feedforwarduncertain nonlinear systems�

Lorenzo Marconia ; ∗, Alberto Isidorib;c, Andrea Serranid

aDipartimento di Elettronica, Informatica e Sistemistica, University of Bologna, V. Risorgimento 2, 40136 Bologna, ItalybDipartimento di Informatica e Sistemistica, University of Rome “La Sapienza”, Rome, ItalycDepartment of Systems Science and Mathematics, Washington University, St.Louis, USA

dDipartimento di Elettronica e Automatica, University of Ancona, Ancona, Italy

Received 9 January 2001; received in revised form 14 September 2001; accepted 17 September 2001

Abstract

This paper deals with the problem of asymptotically rejecting bounded unknown disturbances a3ecting the input channelof a feedforward uncertain nonlinear system. The problem is solved assuming that the matched disturbance belongs to theclass of signals generated by an autonomous neutrally stable exosystem whose state is not accessible. We design an internalmodel-based regulator capable on one hand to reject the matched disturbance for any initial state of the exosystem and, onthe other hand, to robustly globally asymptotically stabilize the system using state feedback. c© 2002 Elsevier Science B.V.All rights reserved.

Keywords: Nonlinear feedforward systems; Disturbance suppression; Saturated control; Output regulation; Internal model

1. Introduction

This paper presents the design of a global statefeedback regulator able to globally asymptoticallystabilize a certain class of nonlinear systems af-fected by parametric uncertainties, in spite ofthe presence of exogenous disturbances matchedin the input channel. More speci=cally, we con-sider the class of nonlinear feedforward system

� This work was partially supported by AFORS under grantF49620-95-1-0232, by ONR under grant N00014-99-1-0697 andby MURST.

∗ Corresponding author. Tel.: +39-051-209-3788; fax:+39-051-209-3073.

E-mail address: [email protected] (L. Marconi).

described by

x1 = �1x2 + g1(x2; : : : ; xn; �)

x2 = �2x3 + g2(x3; : : : ; xn; �)

· · ·xn−1 = �n−1xn + gn−1(xn; �)

xn = u− d(t);

(1)

where �i; i=1; : : : ; n− 1 are uncertain (possibly timevarying) real parameters, � = col(�1; : : : ; �n−1), andd(t) is a disturbance. We assume that d(t) belongsto the class of signals generated as the output ofan autonomous neutrally stable system (the exosys-tem), whose state is not accessible and whose initialconditions take values on the entire state space. Inthis respect, the problem at issue can be seen as an

0167-6911/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0167 -6911(01)00180 -3

228 L. Marconi et al. / Systems & Control Letters 45 (2002) 227–236

example of global regulation problem (see [2,8]). Itis well known that any solution of the above problemnecessarily relies on the design of an internal model ofthe exosystem (capable of reproducing all the controlactions needed to o3set all the possible disturbancesd(t)), and of a stabilizer able to globally asymptoti-cally stabilize the “augmented” system composed bythe original system and the internal model. As faras the global stabilizer is concerned, we show howthe design procedure proposed in [6] (which extendsprevious results on the stabilization of feedforwardsystems presented in [3,12]) can in turn be general-ized in order to take into account the internal modeldynamics. In particular, we show that the problemin question can be recast into a problem of globallyasymptotically stabilizing the original feedforwardsystem in the presence of unmodeled (stable) actua-tor dynamics (see, besides others, [1]). In the light ofthis, our approach can also be seen as a design proce-dure alternative to that proposed in the literature, withthe key feature of being able to deal with feedforwardnonlinear systems with uncertain linear approxima-tions. The control law which solves the problem isobtained by adding a linear function of the state ofthe system and of the internal model, and a nonlinearfunction obtained, in accordance with classical resultson the stabilization of feedforward systems, by nest-ing saturated functions of the state of the system (see[4,12,5,11]). A key notion used throughout the paperis that of input to state stability with restriction andthe associated small gain theorem (see [12]).

2. Problem formulation and standing assumption

Consider the nth dimensional system (1) and sup-pose that gj(0; : : : ; 0; �) = 0 for all �∈P, with P acompact set, so that the origin of the unforced system(i.e. when u=0 and d(t)= 0) is an equilibrium pointfor all �∈P. Moreover, suppose that the matcheddisturbance d(t) is generated as the output of theexosystem

w(t) = Sw(t); d(t) = q(w(t)); w(0)∈Rs; (2)

where S is an Rs × Rs matrix and q(w) is a smoothfunction of w. As customary in regulation theory (see[2]), the matrix S is assumed to have distinct eigen-values on the imaginary axis, that is, the dynamics of

(2) are critically stable. For instance, signals obtainedby a linear combination of a =nite (known) number ofsinusoids of known frequencies and unknown ampli-tudes and phases, belong to the allowed class of dis-turbances. In this framework we address the problemof designing a state feedback dynamic regulator

�= ’(�; x1; : : : ; xn);

u= �(�; x1; : : : ; xn)

such that, for all initial condition w(0)∈Rs and forall �∈P, the state x = (x1; : : : ; xn) is bounded andlimt→∞ ||x(t)|| = 0. In other words we are lookingfor a control law able on one hand to compensate forthe input disturbance d(t) for all possible values ofthe initial state of the exosystem and, on the otherhand, to globally asymptotically stabilize the origin of(1) robustly with respect to the unknown parameters�. The main assumption regarding (1) and (2) underwhich the previous problem will be solved are forconvenience listed in the following:(a1) The parameters �i; i = 1; : : : ; n− 1 are assumed

to be bounded from below and from above byknown positive (or alternatively negative) num-bers, namely

0¡�Li 6 �i6 �Ui ; i = 1; : : : ; n− 1:

(a2) The nonlinear functions gj(xj+1; : : : ; xn; �) are as-sumed to be locally Lipschitz near the origin intheir arguments. In particular, there exist si ¿ 0and Lj; i ¿ 0; j = 1; : : : ; n − 1; i = j + 1; : : : ; n,such that

|xi|6 si; �∈P ⇒ |gj(xj+1; : : : ; xn; �)|

6n∑

i=j+1

Lj; i|xi|

for all j = 1; : : : ; n− 1.(a3) System (2) is assumed to be immersed into a

linear and observable system, namely there existsa mapping �(w) such that

@�(w)@w

Sw = ��(w);

q(w) = ��(w);

where (�;�) is an observable pair.

L. Marconi et al. / Systems & Control Letters 45 (2002) 227–236 229

3. The internal model

It is easy to realize that the previous problem can becast as a global robust regulation problem in whichthe desired control objective is to force the state to zeroin presence of exogenous disturbances. In this partic-ular case the regulator equations (see [2]), whose so-lution represents respectively the desired steady state�(w) and the associated steady state control law u =c(w), yield the trivial global solution

�(w) = 0; c(w) = q(w) ∀w∈Rs:To design the internal model of the disturbance wetake advantage of the following result given in [7],and already used in the context of nonlinear outputregulation in [9].

Proposition 1. Given any Hurwitz matrix F and anyvector G such that (F;G) is an controllable pair; thenthere exists a unique matrix M solution of the equa-tion

M�− FM = G�:

De=ning

� = �M−1

this result allows us to design a canonical internalmodel which assumes the form

�= (F + G�)�+ Nxn;

u=��+ v; (3)

where N is a vector to be designed and v is a newcontrol variable. Note that by construction there existsa mapping M�(w) de=ned as

M�(w) =M�(w)

such that the following holds:

@ M�(w)@(w)

Sw = (F + G�) M�(w);

q(w) =� M�(w):

In other words; system (3) initialized at the value�(0) = M�(w(0)) reproduces the control law needed too3set the disturbance d(t); for all t¿ 0. System (3)quali=es as an internal model of the disturbance. Now;

consider the global change of coordinates

" = �− M�(w)− Gxn: (4)

Simple algebraic computations show that; choosing

N =−FG − G�G (5)

and

v=−�Gxn + v′;

the internal model (3) and the xn equation in (1) as-sume the following expressions in the new coordi-nates:

xn = v′ +�";

" = F" − Gv′:

In summary; in the new coordinates (x1; : : : ; xn; ");choosing N in (3) as in (5); and applying the controllaw

u=��−�Gxn + v′ =�" +� M�(w) + v′

the (augmented) system (1); (3) reads asx1 = �1x2 + g1(x2; : : : ; xn; �)

x2 = �2x3 + g2(x3; : : : ; xn; �)

· · ·xn−1 = �n−1xn + gn−1(xn; �)

xn = v′ +�"

" = F" − Gv′:

(6)

The origin of (6) is an equilibrium point for the un-forced system (v′ = 0) for all possible value of �.Therefore; the problem of disturbance suppression for(1) has been translated into the problem of globallyasymptotically stabilizing the origin of the augmentedsystem (6) with the constraint that the state " cannotbe used for feedback; since it depends on w which isnot known (see (4)). This will be the goal of the nextsection.

Remark 2. It is worth noting that in deriving the form(6) it is necessary that the gain associated with thecontrol u in the last equation of (1) is exactly knownand bounded away from zero (here; without loss ofgenerality; it has been taken equal to 1). The particu-lar structure of the last two equations of (6) is indeedcrucial for the proposed methodology to lead to a suc-cessful design. Removal of this assumption is not atrivial task and requires further research.


4. The global stabilizer

A crucial observation is that the problem of stabi-lizing (6) by means of the control law v′ can be seenas the problem of stabilizing (1), with d(t) ≡ 0, inpresence of unmodeled (asymptotically stable) actua-tor dynamics described by

" = F" − Gv′;

u=�" + v′;

namely the same problem addressed, for instance, in[1]. However, it is worth noting that the stabilizationprocedure proposed in that paper does not apply di-rectly to (6), because the linear approximation of thesystem depends on uncertain parameters and the non-linear function gj(xj+1; : : : ; xn; �) are not necessarilyquadratic. For these reasons we need to generalize thestabilization procedure proposed in [6] to take into ac-count the additional dynamics ".Consider the change of coordinates (inspired by [1])

xi → xi = xi; i = 1; : : : ; n− 1;

xn → xn = xn −�F−1"

which modi=es the overall system (6) as

˙x1 = �1x2 + g1( ˙x2; : : : ; ˙xn; "; �)

˙x2 = �2x3 + g2( ˙x3; : : : ; ˙xn; "; �)

· · ·˙xn−1 = �n−1xn + gn−1( ˙xn; "; �)

˙xn = #v′

" = F" − Gv′;

(7)

where gj( ˙xj+1; : : : ; ˙xn; "; �); j = 1; : : : ; n− 1, are suit-ably de=ned nonlinear (locally Lipschitz) functionsand

#= 1 +�F−1G:

Let, as in [6], a saturation function to be any di3eren-tiable function $ :R→ R which enjoys the followingproperties:(i) |$′(s)|= |d$(s)=ds|6 2 for all s;(ii) s$(s)¿ 0 for all s =0; $(0) = 0;(iii) $(s) = sgn(s) for |s|¿ 1;(iv) |$(s)|¿ |s| for |s|¡ 1.

Consider the change of coordinates

x1 → z1 = x1;

xi → zi = xi + 'i−1$(Ki−1zi−1='i−1); i = 2; : : : ; n

and the control law 1

v′ =−'n$(Knxn + 'n−1$(Kn−1zn−1='n−1)

'n

)

=−'n$(Knzn +�F−1"

'n

): (8)

In the new coordinates, under the control law (8),the overall system reads as (for convenience we haveleft the arguments of the nonlinear function gi in theoriginal coordinates)

z1 =−�1'1$(K1z1'1

)+ �1z2 + g1( ˙x2; : : : ; ˙xn; "; �)

· · ·

zj =−�j'j$(Kjzj'j

)+ �jzj+1

+Kj−1$′(Kj−1zj−1

'j−1

)zj−1

+gj( ˙xj+1; : : : ; ˙xn; "; �)

· · ·

zn =−#'n$(Knzn +�F−1"

'n

)

+Kn−1$′(Kn−1zn−1

'n−1

)zn−1

" = F" + G'n$(Knzn +�F−1"

'n

)

(9)

which can be seen as feedback interconnection of thez-subsystem and of the "-subsystem. In the followingwe shall concentrate on the z-subsystem and we showhow to tune the parameters 'j and Kj; j = 1; : : : ; n torender the z-subsystems input to state stable (ISS)with respect to the input ". To this end, as in [6], we

1 Note that in the expression of v′ the state xn is used in placeof xn, which is not available for feedback as it depends on theunknown state ".


consider =rst the system

z1 =−�1'1$(K1z1'1

)+ �1z2 + v1

· · ·zj =−�j'j$

(Kjzj'j

)+ �jzj+1

+Kj−1$′(Kj−1zj−1

'j−1

)zj−1 + vj

· · ·zn =−#'n$

(Knzn + v"'n

)

+Kn−1$′(Kn−1zn−1

'n−1

)zn−1

(10)

and we show how to render this system ISS withrestrictions with respect to the inputs v1; : : : ; vn−1

and v".The following two lemmas are adapted from anal-

ogous results in [6]. In particular, in the next result itis shown that, if the '′i s and K

′i s are properly chosen

and if the exogenous inputs are suitably bounded, thenthe state of system (10) enters in =nite time a neigh-borhood of the origin. Then Lemma 2, whose proofemploys the previous result, presents extra conditionsfor system (10) to be ISS.

Lemma 3. Consider system (10) and assume that

||vi(·)||∞6 vi;M; i = 1; : : : ; n− 1

and

||v"(·)||∞6 v";M

for some vi;M; v";M. If 'i and Ki can be chosen so thatthe following inequalities are ful=lled:

'j+1

Kj+1¡'j2;

'nKn

+ v";M¡'n−1

2(11)

for j = 1; : : : ; n− 2 and

v1;M¡�L1'12;

vj;M + 4Kj−1�Uj−1'j−1¡�Lj'j2;

4Kn−1�Un−1'n−1¡#'n (12)

for j = 2; : : : ; n− 1; then there exists a time T ∗ suchthat

z(t)∈* for all t¿T ∗;

where

*={zi:|zi|6 'i

Ki; i = 1; : : : ; n− 1;

|zn|6 'nKn

+ v";M

}:

Moreover

|zj(t)|6 2�Uj 'j; j = 1; : : : ; n− 1;

|zn(t)|6 2#'n (13)

for all t¿T ∗.

Proof. The proof of the lemma is similar to that ofLemma 1 in [6]. In particular; consider the nested sets

*j ={z ∈Rn: |zi|6 'i

Ki; i = j; : : : ; n− 1;

|zn|6 'nKn

+ v";M

}(14)

and note that *=*1. Following [3;6]; the lemma is aconsequence of the following two facts:(a) the sets *i’s; i=1; : : : ; n; are positively invariant;(b) every trajectory starting in Rn\*n enters in =nite

time the set *n; and every trajectory starting in*i \*i−1 for i=2; : : : ; n enters in =nite time theset *i−1.

Claim (a) can be easily proved showing that thefollowing implications hold:

z ∈*i and |zi|= 'i=Ki ⇒ zizi ¡ 0; i = 1; : : : ; n− 1;

|zn|= 'n=Kn + v";M ⇒ znzn ¡ 0;

z ∈*i ⇒ |zi|6 2�Ui 'i; i = 1; : : : ; n− 1;

z ∈*n ⇒ |zn|6 2#'n:

(15)

For i = 1; : : : ; n − 2 the proof of the previous claimfollows exactly that given in [6], and need not be re-peated. Consider now i = n− 1: we have that

zn−1 = �n−1'n−1$(Kn−1

zn−1

'n−1

)+ �n−1zn

+Kn−2$′(zn−2)zn−2 + vn−2:


If z ∈*n−1, since the term |Kn−2$′(zn−2)zn−2| isbounded by 4Kn−2�Un−2'n−2 (in fact if z ∈*n−2 then$′(Kn−2zn−2='n−2)=0, otherwise |zn−2|6 2�Un−2'n−2)we have the following estimate:

|zn−1|6 �Un−1'n−1 + �Un−1'nKn

+4Kn−2�Un−2'n−2 + vn−1;M:

Since (11) implies 'n=Kn ¡'n−1=2, application of(12) evaluated in j = n− 1 yields

|zn−1|¡ 2�Un−1'n−1:

The same arguments can be used to prove that zn−1 ='n−1=Kn−1 and z ∈*n−1 implies zn−1zn−1¡ 0.Now consider the case i = n and the zn dynamics

zn =−#'n$(Knzn + v"'n

)

+Kn−1$′(Kn−1zn−1

'n−1

)zn−1:

The fact that z ∈*n implies zn6 2#'n follows easilyfrom the bound∣∣∣∣Kn−1$′

(Kn−1zn−1

'n−1

)zn−1

∣∣∣∣6 4Kn−1�Un−1'n−1 (16)

and from the last of (12). To show that |zn|= 'n=Kn+v";M implies znzn ¡ 0, note that

|zn|= 'n=Kn + v";M ⇒ #'n$(Knzn + v"'n

)

=#'n sgn(zn):

The above implication, together with (16) and the lastinequality in (12), lead to the result. Once (15) hasbeen proved, claim (a) follows by the same argumentsused in [6].To prove claim (b), we only need to show that

the state enters the set *n in =nite time. To this end,it is enough to observe that if |zn|¿'n=Kn + v";Mthen

$(Knzn + v"'n

)= sgn(zn):

This, and the discussion in the proof of Lemma 2 in[6] give the result.

De=ne

�i = 24�Ui ; �ivi = 24�Ui�Li; i = 1; 2; : : : ; n− 1 (17)

and

�ivj = 48�Ui�LiKi−1�i−1

vj ; i = 2; : : : ; n; j = 1; : : : ; i − 1:

(18)

The following lemma states conditions under whichsystem (10) is ISS with respect to the inputsv1; : : : ; vn−1; v", and gives explicit expressions for thelinear gains.

Lemma 4. Assume; in addition to the inequalities(11); (12); that the Ki’s are such that

6Ki−1

Ki�Li�i−1¡ 1; i = 2; : : : ; n− 1;

and

6Kn−1

Kn#�n−1¡ 1: (19)

Then system (10) is ISS with no restriction on theinitial state; restriction vj;M on the inputs vj; j =1; : : : ; n− 1 and restriction v";M on the input v"; withlinear gains. In particular

lim supt→∞

|zi(t)|

6max{�iv1 lim sup

t→∞|v1(t)|; : : : ;

�ivn−1lim supt→∞

|vn−1(t)|; �iv" lim supt→∞

|v"(t)|};

where �1v1 = 2=�L1K1; �1vj = 2�2vj =K1; j = 2; : : : ; n− 1;

�ivj =

6Ki−1

Ki�Li�i−1vj ; j = 1; : : : ; i − 1;

3(j − i + 1)

�Lj∏j

‘=i K‘; j = i; : : : ; n− 1

(20)

for i = 2; : : : ; n and

�nv" = 2; �jv" = 2n−1∏i=j

3Ki; j = 1; : : : ; n− 1:


Proof. This result is an easy extension of Lemma 2in [6]: the only di3erence is that the zn dynamic is fedby the exogenous input v":

zn =−#'n$(Knzn + v"'n

)+ wn−1;

where

wn−1 = Kn−1$′(Kn−1zn−1

'n−1

)zn−1:

Easy computations show that the zn dynamic is ISSwith respect to the inputs v" and wn−1 without restric-tion on the input v"; restriction #'n on the input wn−1

and linear gain given by

||zn||a6max{2||v"||a; 2

Kn#||wn−1||a

}:

From this result above; the proof of the lemma followsthe same arguments in [6].

As shown in [6], inequalities (11), (12) and (19)can be ful=lled choosing

'i = Kici; Ki = K‘i; i = 1; : : : ; n; K ¿ 0 (21)

with

ci+1 =g‘i+1

2ci; i = 1; : : : ; n− 1;

0¡g¡ 1; c1¿ 0: (22)

As a matter of fact, this choice renders inequalities(11) ful=lled for any ‘¿ 0 for a suQciently smallvalue of v";M. Simple computations show that inequal-ities (12) are all satis=ed for a suQciently small vj;Mif ‘ is taken suQciently large. Finally, a large valueof ‘ renders the small gain conditions (19) ful=lled.The following proposition shows how, given a set

of parameters ful=lling the conditions in question, itis possible to construct an in=nite family of similarparameters.

Proposition 5. Suppose the sets {('?i ; K?i ): i =

1; : : : ; n}; {(v?i;M; v?";M): i=1; : : : ; n− 1} are such that(11)–(12) and (19) hold. Then; for any /¿ 0; thechoice

('i; Ki) = (/i'?i ; /K?i ); i = 1; : : : ; n (23)

ful=lls (11)–(12) and (19); with vi;M and v";M givenby

vi;M = /iv?i;M; i = 1; : : : ; n− 1; v";M = /n−1v?";M:

(24)

The last part of this paper is devoted to show theglobal asymptotic stability of system (9). To thisend, we use the result presented in Lemma 2 and thesmall gain theorem to show that the system obtainedsetting

vj = gj( ˙xj+1; : : : ; ˙xn; "; �); v" =�F−1"

is indeed globally asymptotically stable. Since sys-tem (10) is ISS with restrictions, we need to takecare =rst of all that the restrictions are ful=lled andthen that a small gain condition can be enforced. Forconvenience, we have split the last part of the paperin three parts: in the =rst step it is shown that, forthe functions gj( ˙xj+1; : : : ; ˙xn; "; �), the restrictions areful=lled in =nite time, namely there exists a T?¿ 0such that

|gj( ˙xj+1(t); : : : ; ˙xn(t); "; �)|¡vj;M;

j = 1; : : : ; n− 1; (25)

for all t¿T?.The second step focuses on how it is possible to

enforce a small gain condition between the inputs vjand the outputs gj( ˙xj+1; : : : ; ˙xn; "; �).

The last step considers the interconnection v" =�F−1" and shows how the condition on the restric-tion and the small gain can be satis=ed at once. Ineach step, the parameter / introduced in the previousproposition is used to achieve the result. In what fol-lows ('?i ; K

?i ); i=1; : : : ; n, represents any admissible

choice of the set of parameters which ful=ll (11), (19)and (12) for some set of v?i;M’s and v

?";M.

4.1. Ful=llment of the restrictions

To prove (25) we =rst show that there exists a timeMT such that

z(t)∈*j ⇒ |gj( ˙xj+1(t); : : : ; ˙xn(t); "(t); �)|¡vj;M(26)

for all t¿ MT , where *j is de=ned in (14). Since Fis Hurwitz and |v′|6 'n, for any /¿ 0 there exists


MT ¿ 0 such that

||"(t)||6 (/n +M'n) = (1 +M'?)/n

for all t¿ MT ; (27)

where M is some positive number. Moreover, since˙xn = #v′ and |v′|6 'n, we have that

| ˙xn|6 #'n = #'?n /n: (28)

As a result, it is readily seen that claim (26) is truefor j= n− 1. Since gn−1( ˙xn; "; �) is locally Lipschitz,there exist positive constants Gn−1; n, Gn−1; " such that

|gn−1( ˙xn(t); "(t); �)|6Gn−1; n| ˙xn(t)|+ Gn−1; "|"(t)|6 (1 + Gn−1; n#'?n +M'?n )/

n

for all t¿ MT . Hence, (26) is true for j = n− 1 if

(1 + Gn−1; n#'?n +M'?n )/n6 v?n−1;M/

n−1

which can be always satis=ed taking / suQcientlysmall. Now, suppose (26) is true for j=‘+1; : : : ; n−1,that is

|gj( ˙xj+1(t); : : : ; ˙xn(t); "(t); �)|6 vj;M;

j = ‘ + 1; : : : ; n (29)

for all t¿ MT . To show that (26) holds for j=‘, recallthat

˙xj = �j(zj+1 − 'j$(Kzjzj='j))

+gj( ˙xj+1; : : : ; ˙xn; "; �):

Keeping in mind (11) and (29), we have that if z ∈*‘,then

| ˙xj|6 �Uj

('jKj

+ 'j

)+ vj;M6 3

2�Uj 'j + vj;M

=(32�

Uj '

?j + v?j;M

)/j = #j/j

for all j = ‘ + 1; : : : ; n − 1. As a result, byvirtue of the bounds (28)–(27) and the fact thatg‘( ˙x‘+1(t); : : : ; ˙xn(t); �) is locally Lipschitz, we havethat z ∈*‘ implies

|g‘( ˙x‘+1(t); : : : ; ˙xn(t); �)|

6n∑

i=‘+1

G‘;i| ˙xi(t)|+ G‘;"|"(t)|

6n∑

i=‘+1

G‘;i#i/i + G‘;"(1 +M'?n )/n

for some positive constants Gj‘; i and G‘;", and for allt¿ MT . Hence (26) is true for j = ‘ ifn∑

i=‘+1

G‘;i#i/i + G‘;"(1 +M'?n )/n ¡v?‘;M/

‘

which indeed holds taking / suQciently small. By in-duction, we prove that (26) is true for all j=1; : : : ; n−1. To show that (25) also holds, we repeat the rea-soning in [6] observing that in the proof of claim (a)in Lemma 1, the assumption |v‘|6 v‘;M was used inconjunction with the assumption that z ∈*‘. Thus, inview of (26), the proof of claim (a) can be repeatedverbatim and, as in claim (b) of Lemma 1, it can beconcluded that the state of the system enters in =nitetime the set *1. At this point, since *1 ⊂ *j for allj¿ 1, implication (26) again gives the result.

4.2. Small gain conditions

We consider now the interconnection of the outputsgj( ˙xj+1; : : : ; ˙xn; "; �) on the inputs vj for j=1; : : : ; n−1,and we show that the overall system is ISS with respectto the input v". Clearly, since

|gj( ˙xj+1(t); : : : ; ˙xn(t); "; �)|

6n∑

i=j+1

Gj; i| ˙xi(t)|+ Gj;"|"(t)|

for some positive constants Gj; i, Gj;", this goal can beachieved if the gain between vj, j=1; : : : ; n−1 and ˙xi,i= j+1; : : : ; n and " can be rendered arbitrarily small.As far as the gains between vj and ˙xi are concerned,note that

˙xi = �izi+1 − �i'i$(Kizi'i

)+ vi;

and thus, by the de=nition of $(·), the following esti-mate holds:

| ˙xi|6 �Ui |zi+1|+ 2�Ui Ki|zi|+ |vi|:


Since by Lemma 2 (see, in particular, the upper partof (20)) we have that

||zi+1(·)||a6 �i+1vj ||vj(·)||a6 /i−jhi; j||vj(·)||a

and

||zi(·)||a6 �ivj ||vj(·)||a6 /i−j−1h′i; j||vj(·)||afor some positive constants hi; j ; h′i; j ; for j=1; : : : ; n−1;i=j+1; : : : ; n; it follows that there exist constants �i;jsuch that

|| ˙xi(·)||a6�i;j/i−j||vj(·)||a:As far as the gain between vj and " is concerned, notethat, since F is Hurwitz,

||"(·)||a6 �||v′(·)||afor some positive �. By de=nition of saturation func-tion we have that

|v′| = 'n$(Kn

|zn + v"|'n

)6 2Kn(|zn|+ v"|)

6max{4Kn|zn|; 4Kn|v"|}:Since, by Lemma 2

||z(·)||a6max{�nv1 ||v1(·)||a; : : : ; �nvn−1||vn−1(·)||a;

�nv" ||v"(·)||a}it follows that

||v′(·)||a6max{4Kn�nv1 ||v1(·)||a; : : : ;4Kn�

nvn−1

||vn−1(·)||a; 4Kn�nv" ||v"(·)||a}and

||"(·)||a6 �max{4Kn�nv1 ||v1(·)||a; : : : ;4Kn�

nvn−1

||vn−1(·)||a; 4Kn�nv" ||v"(·)||a}:By the expression of �nvi and �nv" in Lemma 2, wededuce the existence of positive constants hj; j =1; : : : ; n− 1 and h" such that the bound

||"(·)||a6max{/n−jhj||vj(·)||a; /h"||v"(·)||a} (30)

holds for all j=1; : : : ; n−1. Hence, the overall systemis ISS with respect to the input v" with restriction v";M,if the small gain conditionn∑

i=j+1

Gi;j�i; j /i−j + Gj;"hj/n−j ¡ 1 (31)

is satis=ed for all j = 1; : : : ; n − 1. Clearly, (31) canbe satis=ed choosing / suQciently small.

4.3. Stability of the interconnection

To prove global asymptotic stability it suQces toshow that / can be chosen small enough so that thefeedback interconnection v" =�F−1" is a small gaininterconnection. As before, we need to check =rst ifthe restriction |�F−1"|6 v";M holds for a suQcientlysmall /. Keeping in mind (27), there exists a positivenumber L such that

|�F−1"(t)|6L/n; for all t¿ MT :

Hence the restriction is ultimately ful=lled if

L/n6 v";M = v∗";M/n−1;

namely if / is small enough. Finally, by virtue of (30),it follows that global asymptotic stability is establishedif the following small gain condition is satis=ed:

||�F−1||h"/¡ 1

which again occurs for / suQciently small.

Remark 6. Due to the particular stabilization methodemployed; which is based on low-amplitude control;the closed loop system cannot be expected to be ro-bust with respect to arbitrary additive disturbances atthe plant input (for instance; in the sense of [10]).Nonetheless; the closed loop system is input tostate stable with restriction with respect to the inputchannel; and this provides robustness with respect tosmall bounded measurable disturbances; those whosemaximum norm satis=es the restriction on the inputchannel imposed by the control law.

5. Conclusions

In this paper we have shown how to design a globalstate-feedback regulator for a class of uncertain feed-forward systems able to asymptotically reject an en-tire family of matched disturbances and to globallyasymptotically stabilize the origin of the system. Thedisturbance to be rejected has been modeled as theoutput of an autonomous exosystem, whose state isnot accessible and whose initial state can take valueson the whole state space. This has allowed to cast the


problem as a global regulation problem. The controllaw consists of a dynamic part (the internal model,needed to o3set asymptotically the e3ect of the dis-turbance), and of a nonlinear static part, derived usingnested saturation functions.

References

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input disturbance suppression for a class of feedforward uncertain nonlinear systems

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