Global robust stabilization of nonlinear systems subject to input constraints

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt. J. Robust Nonlinear Control 2002; 12:1227–1238 (DOI: 10.1002/rnc.693)

Global robust stabilization of nonlinear systemssubject to input constraints

Rodolfo Su!aarez, Julio Sol!ııs-Daun*,y and Jos!ee !AAlvarez-Ram!ıırez

Divisi !oon de Ciencias B !aasicas e Ingenier!ııa, Universidad Aut !oonoma Metropolitana-Iztapalapa, Apdo. Postal 55-534,

C.P. 09340, Mexico, D.F., Mexico

SUMMARY

Our main purpose in this paper is to further address the global stabilization problem for affine systems bymeans of bounded feedback control functions, taking into account a large class of control value sets: p; r-weighted balls Bm

r ðpÞ; with 15p41; defined via p; r-weighted gauge functions. Observe that p ¼ 1 isallowed, so that m-dimensional r-hyperboxes Bm

r ð1Þ :¼ ½�r�1 ; rþ1 � � � � � � ½�r�m ; r

þm �; r�j > 0 are also

considered. Working along the line of Artstein–Sontag’s approach, we construct an explicit formula fora one-parameterized family of continuous feedback controls taking values in Bm

r ðpÞ that globallyasymptotically stabilize an affine system, provided an appropriate control Lyapunov function is known. Thedesigned family of controls is suboptimal with respect to the robust stability margin for uncertain systems.The problem of achieving disturbance attenuation for persistent disturbances is also considered. Copyright# 2002 John Wiley & Sons, Ltd.

KEY WORDS: bounded control; global stabilization; control Lyapunov function; robust control; robuststability margin

1. INTRODUCTION

Consider the multiple input continuous-time affine system

’xx ¼ f ðxÞ þ u1g1ðxÞ þ � � � þ umgmðxÞ ð1Þ

where x 2 Rn; uj 2 R; and f :Rn ! Rn; gj :Rn ! Rn for j ¼ 1; . . . ;m; are CaðRnÞ ða51Þ

functions. Without loss of generality, we shall assume that the origin is an equilibrium pointof the associated free dynamics of (1), i.e. f ð0Þ ¼ 0: Throughout this paper, stabilization willalways be understood as stabilization at the origin.

The use of Lyapunov direct method for stabilization control design has been developed at leastsince the 1960s. For certain system (1) with stable-free dynamics, Jurdjevic and Quinn [1]presented a successful approach based on a controllability-like rank condition: the ad-condition.Lin [2,3] proposed an improved ad-condition generalizing previous results, and obtained the

Received 2 October 2000Published online 15 April 2002 Revised 28 March 2001Copyright # 2002 John Wiley & Sons, Ltd. Accepted 28 November 2001

*Correspondence to: Julio Sol!ııs-Daun, Divisi !oon de Ciencias B!aasicas e Ingenier!ııa, Universidad Aut !oonomaMetropolitana-Iztapalapa, Apdo. Postal 55-534, C.P. 09340, M!eexico, D.F., M!eexico.

yE-mail: [email protected]

Contract/grant Sponsor: CONACyT; contract/grant number: 400200-5-C036E

global asymptotic stabilization (GAS) of general (affine or not) passive systems using arbitrarilysmall stabilizers. Recently, in Reference [4], we addressed the GAS of passive affine systemswhen controls are subject to both magnitude and rate constraints (given a priori).

A general design method of feedback controllers for system (1) has been achieved byintroducing certain Lyapunov functions [5–7]. Artstein’s theorem [5] states that, given a convexcontrol value set U � Rm; the existence of a continuous (except possibly at x ¼ 0) feedbackcontrol taking values in U for the GAS of system (1) is equivalent to the existence of a control

Lyapunov function (clf): A function V :Rn ! R is called a control Lyapunov function [withrespect to (w.r.t.) system (1) with controls taking values in U ] iff (1) it is a CkðRnÞ ðk52Þfunction which is positive definite ðV ð0Þ ¼ 0 and V ðxÞ > 0 iff x=0Þ; proper (for any c50; V �1ðcÞis a compact set), and satisfiesz

infu2U

faðxÞ � B>ðxÞug50 8x=0 ð2Þ

where we denote the Lie derivatives of V ðxÞ w.r.t. the vector fields defining (1) as

aðxÞ :¼ Lf V ðxÞ; B>ðxÞ :¼ ðb1ðxÞ; . . . ; bmðxÞÞ

with bjðxÞ ¼ �LgjV ðxÞ; j ¼ 1; . . . ;m ð3Þ

Moreover, to guarantee continuity of feedbacks at x ¼ 0; Artstein introduced the concept ofsmall control property (see References [5,8]): For each e > 0; there is d > 0 such that if 05jjxjjRn

5d; there is u with jjujjU5e such that aðxÞ � B>ðxÞu50: (Here jj � jjA stands for the norm in A:)Although Artstein’s theorem made a considerable impact on stabilization theory, it cannot be

used as a control design tool since the proof was not constructive. In Reference [7], Sontagobtained a ‘universal’ construction of Artstein’s theorem, proposing a control formula when thecontrol value set U is Rm: For compact control value sets, other ‘universal’ formulae for theGAS of system (1) has been proposed: (1) Lin and Sontag [9] for the Euclidean unit ball; and (2)recently, Malisoff and Sontag [8] extended that result for Minkowski unit balls Bm

1 ðpÞ :¼ fu 2Rm: jjujjp41g; where jjujjp :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiju1j

p þ � � � þ jumjpp

p-a p-norm, with p ¼ 2k=ð2k � 1Þ for some

k ¼ 1; 2; . . . ðso 15p42Þ; and the explicit formula is

uðxÞ :¼ kpðaðxÞ; jjBðxÞjj2k2kÞðb1ðxÞ2k�1; . . . ; bmðxÞ

2k�1Þ> ð4Þ

with kp : R� ½0;1Þ ! R defined by

kpða; bÞ :¼

aþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2k þ b2k2k

qð1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2k�1Þb2k

q if b > 0

0 if b ¼ 0

8>>>><>>>>:

ð5Þ

The feedback (4)–(5) is smooth in Rn\\{0} and everywhere continuous (an almost smooth

function), whenever aðxÞ and BðxÞ are smooth. Moreover, (4)–(5) was also used to partially solvethe stabilization problem for control value sets Bm

1 ð1Þ and ½�1; 1�2; at the expense of smalloverflows of the feedback values. The ‘overflow problem’ for ½�1; 1�2 was overcome in Reference[10].

zWithout loss of generality, in order to simplify the proposed control formula for asymmetric control value sets, wemodified (2)–(3) by changing the sign.

Copyright # 2002 John Wiley & Sons, Ltd. Int. J. Robust Nonlinear Control 2002; 12:1227–1238

R. SUAREZ, J. SOLIS-DAUN AND J. ALVAREZ-RAMIREZ1228

Define the (possibly asymmetric) control value set Bmr ðpÞ :¼ fu 2 Rm :cp;rðuÞ41g; where

cp;rðuÞ :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir�p1 ju1j

p þ � � � þ r�pm jumj

ppp

if 15p51

maxfr�11 ju1j; . . . ; r�1

m jumjg if p ¼ 1

(ð6Þ

each rj is a function defined by

rjðujÞ :¼rþj > 0 if uj50

r�j > 0 if uj50

(ð7Þ

for j ¼ 1; . . . ;m; and r ¼ ðr�1 ; rþ1 ; . . . ; r

�m ; r

þm Þ 2 R2n

þ : Control bound functions rj are introducedto allow us to consider asymmetries in the control value set, so that cp;rð�Þ is generally anasymmetric gauge function (cf. appendix). In particular, if r�j ¼ rþj for all j ¼ 1; . . . ;m; thencp;rð�Þ ¼ jj � jjp;r -a p; r-weighted norm, and thusBm

r ðpÞ is a symmetric p; r-weighted closed ball (ifr ¼ ð1; . . . ; 1Þ then cp;rð�Þ ¼ jj � jjp and Bm

r ðpÞ is Minkowski unit ball). Whereas, if we set p ¼ 1;we obtain the r-hyperbox Bm

r ð1Þ ¼ ½�r�1 ; rþ1 � � � � � � ½�r�m ; r

þm �:

In this paper, we address the problem of GAS of system (1) by means of an e-parameterized(e > 0) family of continuous feedback control functions ueðxÞ taking values in (convex) p; r-weighted closed balls Bm

r ðpÞ; for 15p41: Hence, in accordance with Artstein’s theorem, theset of admissible control inputs is given by

Umr ðpÞ :¼ fu :Rn ! Bm

r ðpÞ : uðxÞ is a continuous functiong ð8Þ

In addition, if 15p42; controls ueðxÞ are almost smooth, whenever aðxÞ and BðxÞ are smooth.However, if p > 2 and bjðxÞ ¼ 0 for some j; the controls, though continuous, could be non-Lipschitz. In the case aðxÞ40; whenever bjðxÞ ¼ 0; we can redesign ueðxÞ to obtain differentiablecontrols applying the ‘regularization’ method developed in Reference [4]. (In that paper, thefeedback control design involved a suitable rescaling obtained through solving nonlinearparametric programs. The differentiable controls satisfy both magnitude and rate constraints(given by Cartesian products of p; r-weighted symmetric balls), while yielding large gains.

We work along the line of Artstein–Sontag’s approach [5,7–9]. For fixed p > 1 and r 2 R2nþ ; we

will suppose that there exists a clf V ðxÞ w.r.t. system (1) with controls taking values inBmr ðpÞ:We

first extend the design of feedback control functions, proposing an explicit formula for a one-parameterized family of continuous controls ueðxÞ taking values in a control value set Bm

r ðpÞ; for15p51; that globally asymptotically stabilizes system (1). One motivation of this research isthat in many applications the control value set is an r-hyperbox. Then, in order to derivecontrols taking values in Bm

r ð1Þ; we will take the control functions structure ueðxÞ as ascheme for an adequate substitution that automatically implements the fact that limp!1Bm

r ðpÞ ¼Bm

r ð1Þ; thus enabling GAS of system (1).We extend the control design w.r.t. existing results with the following desired features:

1. Given a clf and a control value set Bmr ðpÞ; the proposed one-parameterized

family of continuous feedbacks ueðxÞ approximates a feedback control %ooðxÞ that isoptimal w.r.t. the robustness stability margin against bounded uncertainties (see e.g.Reference [11]).

2. The designed family of controls ueðxÞ does not cancel the free dynamics when it is beneficialin achieving the stabilization of the system. If the free dynamics is stable it helps the designtask.


STABILIZATION OF NONLINEAR SYSTEMS 1229

Feature 1 is important because in most physical applications actuators usually displaycontinuous dynamics. In general, control %ooðxÞ is discontinuous and takes values on theboundary of the p; r-weighted ball, @Bm

r ðpÞ: Hence, we propose a one-parameterized family ofsuboptimal continuous controls ueðxÞ; where parameter e > 0 is introduced for approximationpurposes, allowing controls ueðxÞ to take advantage of as much of the available control resourceas desired. Concerning item 2, we have that (1) the use of free dynamics is an important controldesign approach in e.g. electric and mechanical systems (see Reference [12]), and (2) cancellationof nonlinearities may lead to extra control effort producing unnecessary degradation ofperformance (see Reference [13]). Therefore, the suboptimal continuous feedback controls ueðxÞallow us to address the robustness problem for systems with bounded disturbances and thedisturbance attenuation problem in the presence of control constraints, avoiding unnecessaryextra control efforts.

2. DESIGN OF A FAMILY OF ADMISSIBLE CONTROLS FOR Bmr ðpÞ; WITH 15p51

For fixed 15p51; let q be its corresponding dual value, obtained from the duality formula

1

pþ

1

q¼ 1 ð9Þ

Relative to the gauge function cp;rðuÞ defined in (6), consider its polar function cq;1=rðvÞ given inEquation (A2) (cf. appendix). Then, for v ¼ BðxÞ given in (3), define

cq;1=rðBðxÞÞ :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmj¼1

rjðxÞqjbjðxÞj

qq

vuut ð10Þ

where rjðxÞ ¼ rjðbjðxÞÞ with rjð�Þ given in (7), j ¼ 1; . . . ;m: Note that, although rjðxÞ arediscontinuous functions, expressions rjðxÞbjðxÞ are Lipschitz-continuous.

Given the control value set Bmr ðpÞ; define the control function

%ooðxÞ ¼ ð #r1r1ðxÞx1ðxÞq�1; . . . ; #rrmðxÞxmðxÞ

q�1Þ>; where xjðxÞ :¼#rrjðxÞbjðxÞbqðxÞ

ð11Þ

#rrjðxÞ ¼ rjðxÞ signðbjðxÞÞ for j ¼ 1; . . . ;m; and bqðxÞ :¼ cq;1=rðBðxÞÞThe function %ooðxÞ takes values on @Bm

r ðpÞ: In fact, from (11) and (9), we obtain

cpp;rð %ooÞ ¼

Xm

j¼1

%oopj

rpj¼Xm

j¼1

rjbjbq

!ðq�1Þp

¼1

bqq

Xm

j¼1ðrjbjÞ

q � 1 ð12Þ

Hereafter, unless otherwise specified, we will denote bqðxÞ :¼ cq;1=rðBðxÞÞ:For U ¼ Bm

r ðpÞ; the minimum-type problem (2) is satisfied iff there exists a uðxÞ taking valuesin @Bm

r ðpÞ such that aðxÞ5B>ðxÞuðxÞ; 8x=0: The equivalence between (2) and the followinginequality:

aðxÞ5bqðxÞ 8x=0 ð13Þ

is accomplished by means of the control %ooðxÞ since B>ðxÞ %ooðxÞ4bqðxÞcp;rð %ooðxÞÞ (cf. Remark 3 inthe appendix) and cp;rð %ooðxÞÞ � 1 (from (12)). Then, control %ooðxÞ is the unique optimal feedback



control in the sense that it is a solution to the equation infu2Bmr ðpÞfaðxÞ � B>ðxÞug ¼ aðxÞ � B>

ðxÞ %ooðxÞ: Uniqueness is understood modulo the set N ¼ fx 2 Rn : BðxÞ ¼ 0g because %ooðxÞpN isarbitrary. Hence, %ooðxÞ is a singular control (so %oo =2 Um

r ðpÞ given in (8)).Define the one-parameterized family of control functions by means of the formula

ueðxÞ :¼reðaðxÞ; bqðxÞÞ %ooðxÞ if bqðxÞ > 0

0 if bqðxÞ ¼ 0

(ð14Þ

where %ooðxÞ is given in (11), re :R� ½0;1Þ ! R is defined by

reða;bÞ :¼eðaþ jajÞ þ 2bq

2ðebþ bqÞð15Þ

and e > 0 is a tuning parameter. Hereafter, we will denote reðxÞ :¼ reðaðxÞ; bqðxÞÞ:The continuity of ueðxÞ at x ¼ 0 is proved as follows: From Remark 3, we have a5B>ue4

bqcp;rðueÞ: Then, due to the small control property, given E > 0 there exists d > 0 such thata5Ebq; for all jjxjj5d: Therefore, limx!0 a=bq ¼ 0: Thus,

04 limx!0eðaþ jajÞ2ðebq þ bqqÞ

#rrjbjbq

��q�1

4 limx!0eðaþ jajÞ2ðebq þ bqqÞ

¼ 0

limx!0

bqqðebq þ bqqÞ

" ##rrjbjbq

��q�1

¼ limx!01

ðeþ bq�1q Þ

" #j#rrjbjj

q�1 ¼ 0

from where, the continuity at x ¼ 0 follows. On the other hand, the continuity of uej atbj ¼ 0 is a consequence of the continuity of #rrjð#rrjbjÞ

q�1 and the continuity of reb1�qq at bq ¼ 0

which follows if we observe that reb1�qq � 1=ðeþ bq�1

q Þ when bq is sufficiently small,since from (13), bq ¼ 0 implies a50: Then, ueðxÞ are continuous control functions, forany e > 0:

From (9), we have that 15p42 iff q52: Then, for q52; it is not difficult to prove that ueðxÞ isan almost smooth function, whenever aðxÞ and BðxÞ are smooth. On the other hand, if p > 2; then05q� 151: Therefore, the formula given by Equation (14) could be non-smooth wheneverbjðxÞ ¼ 0 for some j; because all terms in Equations (11) and (14) involve potential functionswith exponent q� 1: For many purposes, it is desirable to guarantee some kind of regularityof the control law. If there exists a clf w.r.t. system (1) such that aðxÞ40; when-ever bjðxÞ ¼ 0; j ¼ 1; . . . ;m; we can apply the smooth approximation to odd root-likefunctions developed in Reference [4], to redesign (14) in order to obtain differentiable controlfunctions.

3. CLF STABILIZATION FOR CONTROL VALUE SET Bmr ðpÞ; WITH 15p51

Considering a fixed control value set Bmr ðpÞ; 15p51; we claim that the one-para-

meterized family of feedback control functions (14) fulfils the two items mentioned at theIntroduction.

Lemma 1For all x 2 Rn; 04reðxÞ51: Furthermore, if e ! 0 then reðxÞ ! 1:



ProofWe should consider two cases:

If a40; then from (14), we have that for all e > 0;

04reðxÞ ¼bqq

ebq þ bqq51 ð16Þ

If a > 0; then from (14) and (13), we have that for all e > 0;

04reðxÞ ¼eaþ bqqebq þ bqq

5eþ bq�1

q

eþ bq�1q

� 1 ð17Þ

In both cases, it follows from the above inequalities that reðxÞ ! 1; as e ! 0: &

Remark 1At this point, we have reached the first item, i.e. the family of controls ueðxÞ given in Equation(14) is suboptimal w.r.t. the gauge function cp;r: This is a simple consequence of Lemma 1: Ife ! 0; we have that ueðxÞ ! %ooðxÞ }the optimal control (11).

Remark 2If a40; then from (16) and (14), we have that

ueðxÞ ¼bq�1q ðxÞ

eþ bq�1q ðxÞ

%ooðxÞ ð18Þ

Thus, the designed control prevents the cancellation of the free dynamics which in this case isbeneficial in achieving the stabilization of the system.

On the other hand, the worst control relative to %ooðxÞ that stabilizes system (1) is

%oðxÞ ¼

aðxÞbqðxÞ

%ooðxÞ if aðxÞ > 0

0 if aðxÞ40

8><>: ð19Þ

This feedback control guarantees the stability of system (1), but asymptotic stability. In fact,recalling that V ðxÞ is a clf, we have two cases:(1) if a40; then dV =dt ¼ a� B>

%o ¼ a40; whereas,

(2) if a > 0; then

_VV ¼ a� B>

%o ¼ a�

abq

Xm

j¼1bj %ooj ¼ a�

abq

Xm

j¼1

rqj bqj

bq�1q

� 0

The clf stabilization result based on the family of feedback control functions (14) satisfyingitems 1 and 2 is given in the next theorem.

Theorem 2Assume that V ðxÞ is a clf [w.r.t. system (1) with controls taking values in Bm

r ðpÞ; for 15p51�satisfying the small control property. Then, for any e > 0; the formula ueðxÞ given in (14) definesa parameterized family of suboptimal admissible controls (i.e. ue 2 Um

r ðpÞ) that globally



asymptotically stabilizes system (1). Moreover, if e ! 1; then uðxÞ !%oðxÞ; so that the resulting

closed-loop system is globally stable.

ProofFirst of all, we prove that ueðxÞ take values in Bm

r ðpÞ: In fact, cp;rðueðxÞÞ51 is implied by:

(1) 04reðxÞ51 (Lemma 1), (2) for any e > 0;cp;rðueðxÞÞ5reðxÞcp;rð %ooðxÞÞ and (3) from Equation(12), cp;rð %ooðxÞÞ � 1}recall that %ooðxÞ takes values precisely in @Bm

r ðpÞ:We now proceed with the solution to the global stabilization problem. If bq ¼ 0;

then Equation (13) implies that a50; and thus, the GAS condition dV =dt50 is clearlyattained.

When bq=0; we face two cases:If a40; then from Equation (18), it easily follows that dV =dt ¼ a� bqq=ðeþ bq�1

q Þ50:If instead, a > 0; then from (14) and (12) and recalling again Equation (13), we obtain

_VV ¼ a� B>ue ¼ a�rebqq

Xm

j¼1jrjbjjq ¼

bq�1q

eþ bq�1q

ða� bqÞ50 ð20Þ

Therefore, we achieve the GAS, of the system.Finally, if a40; then lime!1 ueðxÞ ¼ 0; whereas, if a > 0; for any fixed x 2 Rn; we obtain

lime!1 ueðxÞ ¼ lime!1aþ e�1bqq

ð1þ e�1bq�1q Þbq

%ooðxÞ ¼abq

%ooðxÞ ¼%oðxÞ ð21Þ

In both cases, we have that lime!1 ueðxÞ ¼%oðxÞ }the worst stabilizing control given in

Equation (19) relative to %ooðxÞ: &

4. DESIGN OF A FAMILY OF ADMISSIBLE CONTROLS FOR Bmr ð1Þ

In this section, we construct a parameterized family of continuous feedback control lawstaking values in Bm

r ð1Þ ¼ ½�r�1 ; rþ1 � � � � � � ½�r�m ; r

þm �: To achieve this aim, the control

functions structure ueðxÞ given in Equation (14) is taken as a scheme for anadequate substitution that automatically implements the fact that limp!1 Bm

r ðpÞ ¼ Bmr ð1Þ

(cf. appendix).Assume that V ðxÞ is a clf [w.r.t. system (1) with controls taking values in Bm

r ð1Þ� such that

infu2Bmr ð1Þ faðxÞ � B>ðxÞug50 8x=0 ð22Þ

Denote by Vd ¼ fx 2 Rn: V ðxÞ4d; d > 0g a level (compact) set of V ðxÞ: From the continuity ofthe function Lfþ�gjujV ðxÞ ¼ aðxÞ � B>ðxÞu; there exists a real number 04an51 such thatEquation (22) is satisfied for all x 2 Vd\{0} with controls u taking values in Bm

anrð1Þ Bmr ð1Þ:

Observe that Equation (22) is also satisfied with u 2 Bmarð1Þ if an4a41: For each d > 0; let aðdÞ

be the infimum of such numbers an: The function aðdÞ is continuous and non-decreasing. Let#aaðdÞ be a continuous non-decreasing function that satisfies aðdÞ5#aaðdÞ51 (this function dependson the design objectives and for instance it can be chosen as #aaðdÞ ¼ ð1þ aðdÞÞ=2Þ: Consider theminimum positive number #ppðdÞ such that

Bm#aaðdÞrð1Þ Bm

r ð #ppðdÞÞ ð23Þ



A short calculation shows that #ppðdÞ ¼ �lnm=ln #aaðdÞ: For some dn > 0 and pn > 1; define thecontinuous non-decreasing function

pðdÞ :¼#ppðdÞ if d > dn

pn if d4dn

(ð24Þ

Then, Equation (22) is satisfied for all x 2 Vd using controls u taking values in Bmr ðpðdÞÞ:

Therefore, we define the one-parameterized family of control functions

ueðxÞ :¼reðaðxÞ;b #qqðxÞÞ %oopðxÞ if b #qqðxÞ > 0

0 if b #qqðxÞ ¼ 0

(ð25Þ

where re :R� ½0;1Þ ! R is given in Equation (15), b #qqðxÞ ¼ c #qq;1=rðxÞ; #qqðdÞ ¼ pðdÞ=ðpðdÞ � 1Þ(from the duality formula (9) w.r.t. pðdÞÞ; #qqðxÞ ¼ #qqðV ðxÞÞ; and

%oopðxÞ ¼ ð#rr1ðxÞx1ðxÞ#qqðxÞ�1; . . . ; #rrmðxÞxmðxÞ

#qqðxÞ�1Þ> ð26Þ

where xjðxÞ :¼ #rrjðxÞbjðxÞ=b #qqðxÞ for j ¼ 1; . . . ;m: Then, ueðxÞ takes values inBmr ð1Þ (cf. Proposition

A4 in the appendix), and the derivative of the Lyapunov function V along the trajectoriesof the closed-loop system (1) and (25) is negative, i.e. the resulting system is GAS. In general,control (25) is an everywhere continuous function, if the clf V ðxÞ satisfies the small controlproperty.

Example 1Consider the following control system with bounded input:

’xx1 ¼ �x1 þ x1jx1jðx22 þ x23Þ

’xx2 ¼ �x2 þ x21jx1jx2u1

’xx3 ¼ �x3 þ x21jx1jx3u2 ð27Þ

with u ¼ ðu1; u2Þ 2 ½�1; 1�2: Denote x ¼ ðx1; x2; x3Þ: Then, function V ðxÞ ¼ 12jjxjj22 satisfies

’VV ¼ aðxÞ � b1ðxÞu1 � b2u250 ð28Þ

for all x 2 R3=f0g iff u 2 @½�1; 1�2; where aðxÞ ¼ �x21 � x22 � x23 þ x21jx1jðx22 þ x23Þ; b1ðxÞ ¼ �x21jx1jx

22;

b2 ¼ �x21jx1jx23 and bq ¼ jx1j

3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 #qq2 þ x2 #qq3 :

qq

This implies that no control with values in B2ð1;1ÞðpÞ; for all 14p51; can stabilize (27). After

some calculations, we obtain aðdÞ ¼ ð2ð35dÞ1:5 � 5Þ=2ð3

5dÞ1:5 if d 2 ð3:07;1Þ; and 0 if d 2 ð0; 3:07�:

If we propose #aaðdÞ ¼ ðd1:5 � 1Þ=d1:5; then

#qqðdÞ ¼ln 2

ln 2þ lnðd1:5 � 1Þ � ln d1:5if d > 3:07 ð29Þ

and the corresponding control defined by Equation (25) renders system (27) GAS.



5. ROBUSTNESS

The concept of robustness stability margin is useful for knowing under which uncertainties onthe dynamics of the system, a bounded controller retains the asymptotic stability property,guaranteeing robustness. Consider the following system with uncertain dynamics:

’xx ¼ f ðxÞ þ Df ð30Þ

where jjDf jj1 ¼ supx2Rn jjDf ðxÞjj51: The robust stability margin (rsm) is defined by g ¼inffjjDf jj1: ð30Þ is not GASg; i.e. g is the size of the smallest destabilizing perturbation. Thersm (with parametric uncertainty) is hard to compute even for linear systems (see Reference[14]). An approximation to the rsm is based on the existence of a Lyapunov function V ðxÞ for thenominal system ’xx ¼ f ðxÞ such that aðxÞ ¼ Lf V ðxÞ50; 8x=0: Then, the rsm can be approximatedby the rsm w.r.t. the Lyapunov function V ðxÞ; and it is defined by

gV ¼ infx2Rn �aðxÞ

jjrV ðxÞjj8x=0 ð31Þ

Assume that there exists a clf V ðxÞ w.r.t. system (1) and a control value set Bmr ðpÞ; and let uðxÞ

be a feedback control function satisfying aðxÞ � B>ðxÞuðxÞ50: In a more general setting, we candefine the rsm w.r.t. V ðxÞ as the function (see Reference [11])

gV ðxÞ ¼ infx2Rn �aðxÞ � B>ðxÞuðxÞ

jjrV ðxÞjj8x=0 ð32Þ

Clearly, if gV ðxÞ dominates the uncertainty Df (i.e. jjDf ðxÞjj4gV ðxÞÞ; then system

’xx ¼ f ðxÞ þ Df þ u1g1ðxÞ þ � � � þ umgmðxÞ ð33Þ

is robustly stable. Since the control %ooðxÞ given by Equation (11) satisfiesinfu2Bm

r ðpÞfaðxÞ � B>ðxÞug :¼ aðxÞ � B>ðxÞ %ooðxÞ; it also maximizes the rsm w.r.t. the clf V ðxÞ;whereas, the feedback

%oðxÞ given by Equation (19) is the worst controller in the sense that

minimizes it (e.g. gV � 0; if a > 0Þ; while maintaining stability. Since both %ooðxÞ and%oðxÞ are

singular, a family of suboptimal continuous controls ueðxÞ based on (14) are introduced.

5.1. Systems subject to bounded disturbances

Consider the affine system with disturbances

’xx ¼ f ðx; dÞ þ u1g1ðxÞ þ � � � þ umgmðxÞ ð34Þ

where x 2 Rn; uj; di 2 R and f :Rn � Rk ! Rn; gj :Rn ! Rn; are C1ðRnÞ functions, for j ¼

1; . . . ;m; and i ¼ 1; . . . ; k: Assume that u ¼ ðu1; . . . ; umÞ and d ¼ ðd1; . . . ; dkÞ are subject to theconstraints cp;r1 ðuÞ51 and cp;r2 ðdÞ51; respectively.

For fixed p > 1 and r1; r2 2 R2nþ ; a function V :Rn ! Rþ is called a robust control Lyapunov

function [w.r.t. system (34) with controls taking values in Bmr1ðpÞ� (see Reference [13]) iff it is a

positive-definite and proper function such that

infu2Bmr1ðpÞ supd2Bk

r2ðpÞ faðx; dÞ � B>ðxÞug50 8x=0 ð35Þ



whenever V ðxÞ > cV ; for some cV > 0; where

aðx; dÞ :¼ Lf V ðx; dÞ; B>ðxÞ :¼ ðb1ðxÞ; . . . ; bmðxÞÞ;

bjðxÞ :¼ �LgjV ðxÞ; j ¼ 1; . . . ;m ð36Þ

Since Bkr2ðpÞ is a compact set, the next inequality follows from (35)

infu2Bmr1ðpÞ f #aaðxÞ � B>ðxÞug50 8x=0; where #aaðxÞ :¼ supd2Bk

r2ðpÞ faðx; dÞg ð37Þ

Define the one-parameterized family of feedback control functions by means of theformula

ueðxÞ :¼ reð #aaðxÞ;cq;1=r1ðBðxÞÞÞ %ooðxÞ ð38Þ

where re :R� ½0;1Þ ! Rþ is defined by Equation (15) and %ooðxÞ is defined by Equation (11).Observe that ueðxÞ is not necessarily continuous at the origin.

The next robust stabilization result follows from the proof of Theorem 2 (see also Reference[13]).

Proposition 1Assume that V ðxÞ is a clf [w.r.t. system (34) with controls taking values in Bm

r1ðpÞ�; satisfying

condition (35). Then, the family of feedback functions ueðxÞ defined in (38) takes values inBmr1ðpÞ

and the closed-loop system (34) and (38) is ultimately bounded, i.e. there exists a globallyasymptotically stable compact set containing the origin.

Corollary 2If f ð0; dÞ ¼ 0 for all d 2 Bk

r2ðpÞ and cV ¼ 0; then the family of feedback functions ueðxÞ defined in

(38) takes values in Bmr1ðpÞ; and renders system (34) GAS.

Under different conditions, Liberzon [15] addressed the GAS of system (34) withf ðx; dÞ ¼ f ðxÞ þ

Pkj¼1 fjðxÞdj and control value set Bm

1 ð2Þ using almost smooth or switchingcontrols.

ACKNOWLEDGEMENTS

This work was partially supported by CONTACyT Grant 400200-5-C036E.

APPENDIX SOME RESULTS ON CONVEXITY

In this section, we introduce some important examples of gauge functions, polar functions andpolar sets. For more details on gauges and polars, cf. [16, p. 124] et seq.

For 15p51 and functions rj given in (7), we define the function cp;r :Rm ! R by

cp;rðuÞ :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmj¼1

r�pj ðujÞjujj

pp

vuut ðA1Þ



The next result can be deduced along the lines of the proof of Proposition 1 in Reference[17].

Proposition A.1The function cp;rðuÞ is a gauge in the sense that (1) it is convex, (2) it is positively homogeneousof order 1 and (3) it is positive semi-definite. Furthermore, cp;rðuÞ is positive definite and finiteeverywhere, and the set Bm

r ðpÞ :¼ fu 2 Rm: cp;rðuÞ41g is a compact convex set with 0 2intBm

r ðpÞ:

In general, cp;r is an asymmetric gauge. If r�j ¼ rþj ; for all j ¼ 1; . . . ;m; then cp;r is an r-weighted p-norm, and thus Bm

r ðpÞ is a symmetric p; r-weighted closed ball.On the other hand, we define

cq;1=rðvÞ :¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmj¼1

rjðvjÞqjvjjq

q

vuut and Bm1=rðqÞ ¼ fv 2 Rm : cq;1=rðvÞ41g ðA2Þ

where 15q51 satisfies the duality formula 1=p þ 1=q ¼ 1:

Proposition A.2The function cq;1=r is also a gauge. Furthermore, cp;r and cq;1=r are polar functions to eachother, and Bm

r ðpÞ and Bm1=rðpÞ are polar sets to each other, i.e.

cn

p;rðvÞ :¼ supu=0

v>ucp;rðuÞ

¼ cq;1=rðvÞ

and Bmr ðpÞ

n :¼ fv 2 Rm: cn

p;rðvÞ41g ¼ Bm1=rðqÞ ðA3Þ

and vice versa.

Remark 3Gauges m and mn which are polar to each other have the following important property:

v>u4mnðvÞmðuÞ 8u 2 dom m and 8v 2 dom mn ðA4Þ

Expression (A4) is the ‘best’ inequality in the sense that it cannot be tightened by replacing m ormn by lesser functions on larger domains. E.g. if m ¼ cp;r; then mn ¼ cq;1=r and (A4) reduces to ageneralized H .oolder’s inequality: v>u4cq;1=rðvÞcp;rðuÞ:

A result for the extreme cases p; q ¼ 1;1 is given next.

Proposition A.3ðBm

r ðpÞÞn ¼ Bm

1=rðqÞ and vice versa, if p ¼ 1; q ¼ 1 or p ¼ 1; q ¼ 1:

Finally, we have the approximating property of the balls Bmr ðpÞ to the r-hyperbox Bm

r ð1Þ:

Proposition A.4The family of sets fBm

r ðpÞg satisfies that Bmr ðpÞ intBm

r ðp0Þ; for p5p0; and converges to

Bmr ð1Þ in the sense that for each 05E51; there exists a pn such that ð1� EÞBm

r ð1Þ Bmr

ðpÞ Bmr ð1Þ; for all p5pn:



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Global robust stabilization of nonlinear systems subject to input constraints