SEMI-GLOBAL EXPONENTIAL STABILIZATION OF LINEAR SYSTEMS SUBJECT TO \INPUT SATURATION" VIA LINEAR FEEDBACKS. Zongli Lin & Ali Saberi

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1 Appeared n Systems & Control Letters, vol 1, pp-9, 199 SEMI-GLOBAL EXPONENTIAL STABILIZATION OF LINEAR SYSTEMS SUBJECT TO \INPUT SATURATION" VIA LINEAR FEEDBACKS Zongl Ln & Al Saber School of Electrcal Engneerng and Computer Scence Washngton State Unversty Pullman, WA 991- ABSTRACT It s known that a lnear tme-nvarant system subject to \nput saturaton" can be globally asymptotcally stablzed f t has no egenvalues wth postve real parts It s also shown by Fuller [] and Sussmann and Yang [1] that n general one must use nonlnear control laws and only some specal cases can be handled by lnear control laws In ths paper we show the exstence of lnear state feedback and/or output feedback control laws for sem-global exponental stablzaton rather than global asymptotc stablzaton of such systems We explctly construct lnear statc state feedback laws and/or lnear dynamc output feedback laws that sem-globally exponentally stablze the gven system Our results complement the \negatve result" of [] and [1] Key Words: Sem-global exponental stablzaton, bounded control, nput saturaton

2 1 1 INTRODUCTION In ths paper we focus on the problem of stablzaton of a general lnear system whch s subject to \nput saturaton" Ths problem has a rch hstory (see, for example, []), and much of the lterature of the 19's and 19's on the absolute stablty was motvated by ths problem (see, for example, [1], [], []) A recent result due to Songtag and Sussmann ([1]) shows that only lnear stablzable systems havng no open-loop egenvalues wth postve real parts can be globally asymptotcally stablzed by a bounded control Another nterestng aspect of ths problem was shown by Fuller ([]) and more recently by Sussmann and Yang ([1]) They showed that for a system of a chan of ntegrators of length n where n and whch s subject to \nput saturaton", there does not exst a lnear control law that globally asymptotcally stablzes the gven system The mplcaton of the results of [1], [] and [1] s straghtforward and can be concluded as follows: Gven a lnear system whch s subject to \nput saturaton", the global stablzaton can be solved f and only f all the egenvalues of the gven system are n the closed left half plane, and, even then, n general one must use nonlnear control and only very smple cases can be handled va lnear feedback control laws (An nterestng nonlnear control law of nested saturaton type for global asymptotcal stablzaton of a chan of ntegrators subject to \nput saturaton" was proposed n [1], whch was later extended by [11]) In ths paper we present a result that complements the \negatve results" of [] and [1] We consder sem-global exponental stablzaton of general lnear systems subject to \nput saturaton" and show that, n contrast to the case of global asymptotc stablzaton, one can sem-globally exponentally stablze such systems va lnear feedback laws Here by sem-global exponental stablzaton, as usual, we mean local exponental stablzaton of the system such that the doman of attracton of the closed-loop system contans an a pror gven bounded set (For precse denton of sem-global exponental stablzaton, see Dentons 1 and ) Relaxng the requrement of global stablzaton to that of sem-global stablzaton, from engneerng pont of vew, makes sense, snce n general a plant's model s usually vald n some regon of the state space Such relaxaton gves us smple lnear control laws and stronger stablty property for the closed-loop system, that s, the exponental stablty of the closed-loop system, rather than asymptotcal stablty Ths paper s organzed as follows We formulate our problem n secton Secton deals wth state feedback case, whle the output feedback case s dealt wth n secton Secton draws the conclusons of our current work PROBLEM STATEMENT We consder a lnear system subject to \nput saturaton" descrbed by _x = Ax + B h (u) (1) y = Cx ()

3 where x < n s the state, u < m s the control nput to the saturator, y < p s the measurement output, and h (s) s a bounded vector functon dened as wth h (s) = [ h1 (s 1 ); h (s ); ; hm (s m )] h (s ) 8 < : = s f js j h <?h f s <?h (:) > h f s > h where s = [s 1 ; s ; ; s m ] and h = [h 1 ; h ; ; h m ]; h > We make the followng assumptons on the system (1)-() A1 All the egenvalues of A are located on the closed left half s-plane; A The par (A; B) s stablzable; A The par (A; C) s detectable Remark 1 We should pont out that assumptons A1 and A are equvalent to the noton of asymptotc null-controllablty that was ntroduced n [8] and [9] Before statng the problem at hand, we have the followng dentons Denton 1 (Sem-global exponental stablzaton va lnear statc state feedback) The system (1)-() s sem-globally exponentally stablzable by lnear statc state feedback f for any a pror gven bounded set of ntal condtons W < n, there exsts a state feedback law u = Kx such that the equlbrum x = of the closed-loop system s locally exponentally stable and W s contaned n the doman of attracton of the equlbrum x = Denton (Sem-global exponental stablzaton va lnear dynamc output feedback of dynamcal order l) The system (1)-() s sem-globally exponentally stablzable by lnear dynamc output feedback of dynamcal order l, f for any a pror gven bounded set W < n+l, there exsts a lnear dynamc output feedback control law _z = A con z + B con y; z < l () u = C con z + D con y () such that the equlbrum (; ) of the closed-loop system consstng of the system (1)-() and the controller ()-() s locally exponentally stable and W s contaned n the doman of attracton of the equlbrum (; ) The problem posed n ths paper s to establsh the sem-global exponental stablzaton va lnear statc state feedback laws and/or lnear dynamc output feedback laws for the system (1)- ()

4 STATE FEEDBACK DESIGN In ths secton we examne the problem of sem-global exponental stablzaton va lnear state feedback control of the asymptotc null-controllable lnear systems Our man result s gven n the followng theorem Theorem 1 The lnear system (1)-() satsfyng assumptons A1 and A s sem-globally exponentally stablzable va lnear state feedback Namely, for any a pror gven (arbtrarly large) bounded set W and any (arbtrarly small) numbers h > ; = 1 to m, there s a lnear control law u =?Kx such that (a) The equlbrum x = of the closed-loop system s locally exponentally stable; (b) W s contaned n the doman of attracton of the equlbrum x = Before provng ths theorem, let us dgress to prove a lemma, whch wll play a fundamental role n our proof of Theorem 1 Lemma 1 Consder a lnear sngle nput system n the controllable canoncal form where A = _x = Ax + Bu ; B =?a n?a n?1?a n??a 1 Assume that all the egenvalues are n the closed left half s-plane Let K() < 1n be the state feedback gan such that (A? BK()) =? + (A) Then, there exsts an > such that for all <, kk()e (A?BK())t k e?t (:1) ke (A?BK())t k r?1 e?t (:) for some > and > ndependent of, where r s the largest algebrac multplcty of the egenvalues of A Proof of Lemma 1 : Let p 1 = det(si? A) = py =1 (s? ) n : where = j, = j Then, for each = 1 to p, the n generalzed egenvectors of A are gven by ([]): 1 n? n?1 ; p = 1 (n? 1) n? ; ; p n = 1 C n?1 n? n?n?1 C n?1 n?1 n?n :

5 Smlarly, for each = 1 to p, the n generalzed egenvectors of A? BK() are gven by q 1 = 1 n? n?1 ; q = where =? + and C n s dened as 1 (n? 1) n? ; ; q n = n! Cn = n (n?)!! otherwse : We next form the followng two nonsngular transformaton matrces whch s ndependent of, and It then follows that P?1 AP = P = [p 1 1 ; p1 ; ; p1 n 1 ; ; p p 1 ; pp ; ; pp n p ] Q() = [q 1 1; q 1 ; ; q 1 n 1 ; ; q p 1 ; qp ; ; qp n p ]: J 1 J ; J = J p C n?1 n? n?n?1 C n?1 n?1 n?n 1 1 ; (:) : and Q?1 ()(A? BK())Q() = J 1 J ; J = J p 1 1 : (:) By the denton of generalzed egenvectors, we have the followng relatonshps, and Ap 1 = p 1; Ap j = p j?1 + p j ; j = ; ; ; n ; = 1; ; ; p () (A? BK())q 1 = q 1; (A? BK())q j = q j?1 + q j ; j = ; ; ; n ; = 1; ; ; p () Denotng A n = [?a n?a n?1?a 1 ], t then follows form () and () that, for each = 1 to p, we have,

6 for j = 1; ; ; n, from whch we have K()q j = A n q j? Cj?1 = A n = A n = A n n?j = X n?j k= A n p j = C j?1 n?1 n?j+1 + C j? n?1 n?j+1 = Cn j?1 (A n? K())qj = C j?1 n?1 n?j+1 + C j? n?1 n?j+1 = Cn j?1 n n?j+1 C j?1 j?1 C j?1 j (? + ) C j?1 (? + n?1 ) n?j C j?1 j P C j?1 n?j n?1 X k= C j?1 j?1 P 1k= (?) k 1?k? Cn j?1 (? + ) n?j+1 k= Ck n?j (?)k n?j?k n?j (?) k C k k+j?1 p j+k? Cj?1 n X k=? Cn j?1 (? + ) n?j+1 n?j+1 n?j+1 (?) k C k n?j+1 n?j?k+1 + ( n?j+1 ) X n?j (?) k Ck+j?1C k n j+k?1 n?j?k+1? Cn j?1 (?) k Cn?j+1 k n?j?k+1 + ( n?j+1 ) = ( n?j+1 ) where ( n?j+1 ) s a polynomal n whose coecents of terms of order lower than n? j + 1 are all zero Hence, we have, for < 1, k= jk()q j j j n?j+1 (:) where j s some postve constant ndependent of We next note that both Q() and Q?1 () are contnuous functons of satsfyng Q() = P and Q?1 () = P?1 Now from the contnuty of norm functons t follows that there exsts an, < 1, such that for all <, kq()k kp k + 1; kq?1 ()k kp?1 k + 1: (:8) Fnally () and (8) show that, for all <, k K ()e (A?BK())t k = kk()q()e Q?1 ()(A?BK())Q()t Q?1 ()k

7 (kp?1 k + 1) (kp?1 k + 1) (kp?1 k + 1) (kp?1 k + 1) (kp?1 k + 1) = e?t px =1 px =1 kk()[q 1; q ; ; q n ]e J t k kk()[q 1; q ; ; q n ] px Xn jx =1 j=1 l=1 px Xn jx =1 j=1 l=1 px Xn jx =1 j=1 l=1 l n?l+1 t j?l e?t =(j? l)! l (j? l)j?l n?j+1 l (j? l)! (j? l)j?l (j? l)! e?t where = (kp?1 k + 1) P p P n P j l (j?l)j?l =1 j=1 l=1 (j?l)! e t te t t e t =! t n?1 e t =(n? 1)! e t te t n? e t =(n? )! k e t e?t s ndependent of Ths shows (1) Usng (8), () can be shown as follows For all <, k e (A?BK())t k = kq()e Q?1 ()(A?BK())Q()t Q?1 ()k (kp k + 1)(kP?1 k + 1) (kp k + 1)(kP?1 k + 1) (kp k + 1)(kP?1 k + 1) (kp k + 1)(kP?1 k + 1) (kp k + 1)(kP?1 k + 1) = e?t r?1 px =1 px =1 ke J t k k px Xn =1 j=1 l=1 px Xn e t te t t e t =! t n?1 e t =(n? 1)! e t te t n? e t =(n? )! k e t jx jx =1 j=1 l=1 px Xn jx =1 j=1 l=1 t j?l e?t =(j? l)! (j? l) j?l e?t (j? l)!j?l (j? l) j?l e?t r?1 (j? l)! s ndepen- where r = maxfn 1 ; n ; ; n p g and = (kp k + 1)(kP?1 k + 1) P p =1 dent of P n P j (j?l) j?l j=1 l=1 (j?l)! Proof of Theorem 1 : The dea of the proof s as follows We rst consder the gven system (1)-() wthout the saturaton element h (e, h (u) = u) Namely, we consder the lnear system _x = Ax + Bu: (:9)

8 Then we show that there exst a famly of state feedback gan matrces K() < mn ; >, such that the closed-loop system consstng of (9) and the state feedback law u =?K()x (:1) s exponentally stable for all >, e, all the egenvalues of A? BK() are n the open left half s-plane Moreover, for any a pror gven bounded set W and any h > ; = 1 to m, there exsts an > such that for all = 1 to m, ju (t)j h ; 8 (; ] and 8x() W; where u (t) s the th element of u =?K()x Havng done ths, now t s straghtforward to verfy that the applcaton of the state feedback law (1) wth (; ] to the orgnal system (1)-() results n a closed-loop system that remans lnear for all x W snce ju j h and h s lnear n ths regon The results of the theorem thus follow We now begn the proof by ntroducng the followng state transformaton x =?~x; ~x = [~x 1; ~x ; ; ~x p; ~x c] ; ~x < n ; ~x c < nc where? s such that (??1 A?;??1 B) s n the followng trangular canoncal form ([])??1 A? = A 1 A 1 A 1p A 1c A A p A c A p A pc A c ;??1 B = Here 's represent submatrces of less nterest and for = 1; ; ; p, A = ; B =?a n?a n?1?a n??a 1 B 1 B B p Clearly, (A ; B ) s controllable, all the egenvalues of A are on the closed left half s-plane, and all the egenvalues of A c have strctly negatve real parts For = 1; ; ; p, let K () = [K 1(); K (); ; Kn ()] be chosen such that (A? B K ()) =? + (A ) For the later use, we make the followng observaton Observaton 1 For all, < 1, kk ()k < k, for some postve constant k Proof : Let f l ; l = 1; ; ; n g be the egenvalues of A, then Yn l=1 (s? l) = s n + a 1 s n?1 + + a n 1

9 8 Hence the characterstc equaton of A? B K () s gven by Yn l=1 (s +? l) = s n + (a 1 + K n ())s n?1 + + (a n + K 1 ()) = from whch t s straghtforward to see that for any j; j = 1; ; ; n, K j () s a polynomal n and K j () = It then follows trvally that for < 1, there exsts some k > such that kk ()k k We then form a state feedback gan K() as follows K() = K 1 ( (r +1)(r +1)(r p+1) ) K ( (r +1)(r +1)(r p+1) ) K p?1 ( rp+1 ) K p () where r s the largest algebrac multplcty of the egenvalues of A It s trval to see that all the egenvalues of A? BK() have strctly negatve real parts and hence the system (9) under the control u =?K()x s stable for all > We next show that for the gven bounded set W and the gven vector h, there exsts an >, such that for all = 1 to m, ju (t)j h ; 8 (; ] and 8x() W To ths end, let us wrte out the closed-loop system dynamcs (9)-(1) n the new state ~x as follows, _~x = (A? B K ())~x + _~x c = A c ~x c px l=+1 A l ~x l + A c ~x c ; = 1; ; ; p Snce all the egenvalues of A c are n the open left half s-plane, there exsts a > and an c >, both ndependent of, such that k~x c k c e?t k~x c ()k We then vew ~x c as an nput sgnal to the dynamcs of ~x p By Lemma 1, there exsts an p, < p mnf1=; g, such that for all < p, kk p ()e (Ap?BpKp())t k p e?t for some p > We then have kk p ()~x p k kk p ()e (Ap?BpKp())t ~x p ()k + kk p () p e?t k~x p ()k + c p ka pc k e (Ap?BpKp())(t? ) A pc ~x c ()d k e?(t? ) e? d k~x c ()k p e?t k~x p ()k + c p ka pc ke?t e?(?) d k~x c ()k p e?t k~x p ()k + c p ka pc k e?t k~x? c ()k p p e?t kx()k??1

10 9 for some postve constant p ndependent of Also we have, for all < p, k~x p k ke (Ap?BpKp())t ~x p ()k + k e (Ap?BpKp())(t? ) A pc ~x c ()d k p rp?1 e?t k~x p ()k + c p ka pc k (? p) rp?1 e?t k~x c ()k p rp?1 e?t kx()k for some postve constant p ndependent of Vewng ~x c and ~x p as nputs to the dynamcs of ~x p?1 and usng the nequaltes (1) and (), we have, for some < p?1 p, and for all <, p?1 kk p?1 ( rp+1 )~x p?1 k kk p?1 ( rp+1 )e (A p?1?b p?1 K p?1 ( rp+1 ))t ~x p?1 ()k +kk p?1 ( rp+1 ) +kk p?1 ( rp+1 ) e (A p?1?b p?1 K p?1 ( rp +1 ))(t? ) A p?1p ~x p ()d k e (A p?1?b p?1 K p?1 ( rp +1 ))(t? ) A p?1c ~x c ()d k p?1 rp+1 e?rp +1 t kx()k + p p?1 ka p?1p k rp?1 + c p?1 ka p?1c k rp+1 e?rp +1 (t? ) e? d kx()k p?1 rp+1 e?rp +1 t kx()k + p p?1 ka p?1p k e?rp +1 t + c p?1 ka p?1c k rp+1 e?rp +1 t rp+1 e?rp +1 (t? ) e? d kx()k e?(?rp +1 ) d kx()k p?1 e?rp +1 t kx()k + p p?1 ka p?1p ke?rp +1 t kx()k + c p?1 ka p?1c k rp +1 t? ( e? kx()k p?1 )rp+1 p?1 e?rp +1 t kx()k for some postve constant p?1 ndependent of, and smlarly, k~x p?1 k ke (A p?1?b p?1 K p?1 ( rp+1 )t ~x p?1 ()k + k e?(?rp +1 ) d kx()k e (A p?1?b p?1 K p?1 ( rp+1 )(t? ) A p?1p ~x p ()d k +k e (A p?1?b p?1 K p?1 ( rp +1 )(t? ) A p?1c ~x c ()d k p?1 (rp+1)(r p?1?1) e? Z rp +1 t k~xp?1 ()k + p p?1 ka p?1p k t e?rp +1 (t? ) e? d (rp+1)(r kx()k p?1?1) rp?1 + c p?1 ka p?1c k e?rp +1 (t? ) e? d kx()k (rp+1)(r p?1?1) p?1 +1 (rp+1)r p?1?1 e?rp t p p?1ka p?1pk kx()k + (rp+1)r p?1?1 c p?1 ka p?1c k + +1 (? ( p?1 )rp+1 ) (rp+1)r p?1?1 e?rp t kx()k p?1 e? rp +1 t kx()k r p?1(r p+1)?1 kx()k

11 1 for some postve constant p?1 ndependent of Contnung n the same manner, we can show that for each = p? ; p? ; 1, there exsts such that for all < +1, kk ( (r +1+1)(r p+1) )~x k e?(r +1 +1)(r + +1)(rp+1) t kx()k and k~x k e? (r +1 +1)(r + +1)(rp +1) t kx()k r (r +1 +1)(r + +1)(r p+1)?1 for some postve constants and ndependent of We then have, for all < 1, ku()k = kk()xk px =1 kk ( (r +1+1)(r p+1) ))~x k px =1 e?(r +1 +1)(r + +1)(rp+1) t kx()k (:11) Let R > be such that W fx : kxk Rg, h = mnfh 1 ; h ; ; h m g, and let = mnf 1 ; h =(R P p =1 )g Then t mmedately follows from (11) that, for all = 1 to m, ju (t)j h ; 8 (; ] and 8x() W whch concludes our proof We next demonstrate our desgn strategy by the followng example Example 1: Consder the lnear system (1)-() wth A = ?1?1???1?1? ; B = where h s any bounded functon satsfyng () It s straghtforward to verfy that ths system s stablzable wth ve controllable modes at f?1;?j;?j; j; jg, where j = p?1, and an uncontrollable mode at? Followng the algorthm gven n the proof of Theorem 1, we construct the followng famly of lnear bounded state feedback control laws whch sem-globally stablzes the gven system, u =?[ ]x where > s a tunable parameter to be chosen accordng to the a pror gven bounded set W and the value of h Let the set W be gven by W = fx : kxk 1g and h = : Then an estmate of s gven by = :1 The smulaton results are shown n Fgure 1 1 ;

12 11 OUTPUT FEEDBACK DESIGN Ths secton deals wth the problem of sem-global exponental stablzaton of detectable and asymptotc null-controllable systems Our man result s gven n the followng theorem Theorem 1 Let assumptons A1 to A be satsed Then the system (1)-() s sem-globally exponentally stablzable va lnear output feedback of dynamcal order n Namely, for any a pror gven (arbtrarly large) bounded set W < n and any (arbtrarly small) numbers h > ; = 1 to m, there exsts a lnear output feedback law of dynamcal order n such that (a) The equlbrum (; ) of the closed loop system s locally exponentally stable; (b) W s contaned n the doman of attracton of the equlbrum (; ) Proof : We rst construct a famly of dynamc output feedback controllers parameterzed n as follows _^x = A^x + B h (?K()^x) + L(y? C ^x) (1) u =?K()^x () where K() s as gven n the proof of Theorem 1, and L s any matrx such that all the egenvalues of A? LC are n the open left half s-plane Followng the same dea of the proof of Theorem 1, we rst consder the closed-loop system wthout saturaton element Namely, we set h (u) = u n both the system (1)-() and the proposed controller (1)-() Then we show that for the gven set W and h > ; = 1 to m, there exsts an > such that for all = 1 to m, ju (t)j h ; 8 (; ] and 8[x() ; ^x() ] W: (:) where u (t) s the th element of u =?K()^x Havng done ths, the result of the theorem then follows readly snce the closed-loop system wth saturaton element, namely, (1)-() and (1)-(), remans lnear for all [x() ; ^x() ] W To show ths, we observe that the closed-loop system wth h (u) = u can be rewrtten as _x = (A? BK())x + BK()e () _e = (A? LC)e () u =?K()x + K()e () where e = x? ^x The stablty of ths above system for all > follows from the separaton prncple Next we need to show that for the gven W and the gven h >, = 1; ; ; m, there exsts an > such that () holds To ths end, as n the proof of Theorem 1, we wrte the system ()-() as _~x = (A? B K ())~x + _~x c = A c ~x c px l=+1 A l ~x l + A c ~x c +? BK()e; = 1; ; ; p () (8) _e = (A? LC)e: (9)

13 1 where??1 = [? 1 ;? ; ;? p] Vewng [~x c; e ] as a new ~x c and usng Observaton 1, t can be shown exactly as n the proof of Theorem 1, that there exsts an 1 such that for all = 1 to m, ju 1 (t)j h =; 8 (; 1 ] and 8[x() ; ^x() ] W; where u 1 (t) s the th element of?k()x Also notng that ke(t)k k e ke()k for some k e > and ke()k kx()k + k^x()k, t then follows from Observaton 1 that there exsts an, such that for all = 1 to m, ju (t)j h =; 8 (; ] and 8[x() ; ^x() ] W; where u (t) s the th element of K()e Takng = mnf 1; g, we complete our proof We next demonstrate our output feedback desgn by the followng numercal example Example 1 Consder the system (1)-(), where A and B are as gven n Example 1, C = [ 1 ], and h s any bounded functon satsfyng () It can be easly vered that all the assumptons A1 to A are satsed Choosng the observer gan L=[?9? ], we place the egenvalues of A?LC at f?1j;?1j;?1;?g Together wth the state feedback gan obtaned n Example 1, K()=[ + + e ] ; we obtaned the followng observer-based controllers whch sem-globally exponentally stablze the gven system _^x = A^x + B h (?K()^x) + L(y? C ^x) u =?K()^x where s a tunable parameter to be chosen accordng to the a pror gven bounded set W and the value of h Let the set W be gven by W = fx : kxk 1g and h = : Then an estmate of s gven by = :1 The smulaton results are shown n Fgures and CONCLUSIONS In ths paper, we establshed the exstence of the lnear control laws, both state feedback and output feedback type, for sem-global exponental stablzaton of lnear tme-nvarant systems subject to \nput saturaton" The explct constructon of such control laws was also gven References [1] MA Azerman and FR Gantmacher, Absolute Stablty of Regulator Systems, Holden-Day, San Francsco, CA, 19 [] BDO Anderson and JB Moore, Lnear Optmal Control, Prentce-Hall, Englewood Cls, 191

14 1 [] CT Chen, Lnear System Theory and Desgn, Holt, Rnehart and Wnston, New York, 198 [] AT Fuller, \In the large stablty of relay and saturated control systems wth lnear controllers," Internatonal Journal of Control, vol 1, pp 8-, 19 [] T Kalath, Lnear Systems, Prentce-Hall, Englewood Cls, NJ, 198 [] KS Narendra and JH Taylor, Frequency Doman Crtera for Absolute Stablty, Academc Press, 19 [] VM Popov, Hyperstablty of Control Systems, Sprnger-Verlag, Berln, 19 [8] WE Schmtendorf and BR Barmsh, \Null controllablty of lnear systems wth constraned controls," SIAM J Control and Optmzaton, vol 18, pp -, 198 [9] ED Sontag, \An algebrac approach to bounded controllablty of lnear systems," Internatonal Journal of Control, vol 9, pp , 198 [1] ED Sontag ans HJ Sussmann, \Nonlnear output feedback desgn for lnear systems wth saturatng controls," Proc 9th IEEE Conf Decson and Control, pp 1-1, 199 [11] ED Sontag and Y Yang, \Global stablzaton of lnear systems wth bounded feedback," Report SYCON-91-9, Rutgers Center for Systems and Control, 1991 [1] HJ Sussmann and Y Yang, \On the stablzablty of multple ntegrators by means of bounded feedback controls," Report SYCON-91-1, Rutgers Center for Systems and Control, 1991 [1] AR Teel, Feedback Stablzaton: Nonlnear Solutons to Inherently Nonlnear Problems, PhD dssertaton, College of Engneerng, Unversty of Calforna, Berkeley, CA, 199

15 1-1 1 a b c d e f g Fgure 1: Example 1 = :1 a) x 1 ; b) x ; c) x ; d) x ; e) x ; f) x ; g) u

16 a c e g b d f h Fgure : Example 1 = :1 a) x 1 ; b) x ; c) x ; d) x ; e) x ; f) x ; g) ^x 1 ; h) ^x

17 k j l m Fgure : Fgure contnued ) ^x ; j) ^x ; k) ^x ; l) ^x ; m) u