J. TSINIAS In the present paper we extend previous results on the same problem for interconnected nonlinear systems èsee ë1-18ë and references therein

Transcription

1 Journal of Mathematical Systems, Estimation, and Control Vol. 6, No. 1, 1996, pp. 1í17 cæ 1996 Birkhíauser-Boston Versions of Sontag's Input to State Stability Condition and Output Feedback Global Stabilization æ J. Tsinias Abstract A new approach on the output feedback global stabilization problem for triangular nonlinear systems is presented. Our methodology extends some ideas from our recent works è1993è and is quite different and less technical than these proposed by earlier works where similar results are obtained in the presence of the ëinput to state stability condition". The main suæcient conditions we propose are versions of this condition. Key words: input to state stability, feedback, global, stabilization AMS Subject Classiæcations: 93D15 1 Introduction We deal with the output feedback global stabilizability problem for singleinput systems of the form _x = fèx; y 1 è _y i = g i èx; y 1 ; :::; y i è+y i+1 ; 1 ç i ç m è1.1è y m+1 + u x 2é n ; y+èy 1 ; :::; y m è02é m where u and y are the input and the output of è1.1è, respectively, 0 stands for transpose, and we assume that the mappings f and g i are continuous èc æ èvanishing at zero. æ Received November 24, 1993; received in ænal form March 7, Summary appeared in Volume 6, Number 1,

2 J. TSINIAS In the present paper we extend previous results on the same problem for interconnected nonlinear systems èsee ë1-18ë and references thereinè. Our approach is diæerent from this employed in relative existing works èsee for instance ë2,3,9,12ëè and is based on some ideas from ë17,18ë as well as on versions of the ëinput to state stability condition" èi.s.s.c.è concerning the stability behavior of the subsystem _x = fèx; yè with y as input è1.2è In particular, the ærst version ècondition 2.1è is of Lyapunovtype and as it has been recently proved by Lin, Sontag and Wang is equivalent to I.S.S.C. This condition implies that for the case m = 1 the corresponding èn + 1èdimensional system è1.1è is globally asymptotically stabilizable at the origin èg.a.s.è by output feedback which iscontinuous on é n+1 ètheorem 3.1è. The second version ècondition 2.2è is weaker than the I.S.S.C. and guarantees that è1.1è is G.A.S. by means of an output feedback which is smooth on é n ; provided that f and g i are C 1 and the matrix è@f=@xèè0; 0è is Hurwitz ètheorem 3.2 and Corollary 3.4è. Finally, a pair of stability conditions much more weaker than I.S.S.C. as well as Conditions 2.2 are presented in section 4 èconditions 4.1 and 4.2è. These conditions are applicable to global stabilization of è1.1è provided that the mappings g i are bounded ècorollary 4.4è. It should be noted that versions of the previous results have been originally presented in the recent works ë3ë of Jiang, Praly and Teel and ë9ë of Praly and Jiang. In particular, in ë3ë among other things it is shown that è1.1è is G.A.S. by smooth output feedback provided that è@f=@xèè0; 0è is Hurwitz and the subsystem è1.2è satisæes the I.S.S.C. Comparing with ë3,9ë our approach is based to weaker assumptions and is less technical. We also remark that our results can directly be extended for systems of the form _x = fèx; y 1 è; _y i = g i èx; y 1 ; :::; y i è+y i+1 h i èx; y 1 ; :::; y i è; 1 ç i ç m; y m+1 := u; where h i are everywhere strictly positive C æ mappings. 2 Versions of the ëinput to State Stability Condition" It will be useful to recall ærst the deænition of the I.S.S.C. We say that è1.2è satisæes the I.S.S.C. if it is complete, ènamely, for every initial state x 0 and for any essentially bounded measurable control y there exists a solution xèt; x 0 ; yè of è1.2è starting from x 0 at time t =0;which is deæned for every t ç 0è and there exist a function æ : é +! é + of class K, ènamely æ is continuous, strictly increasing vanishing at zeroè and a C æ function æ : é + æé +!é + such that for each æxed t the function æèæ;tèis of class K; for each æxed s the function æès; æè decreases to zero at inænity and further for any x 0, èessentiallyè bounded input y and for èalmostè all t the following holds: jxèt; x 0 ; yèj çæèjx 0 j;tè+æèky t kè; 2 è2.1è

3 INPUT TO STATE STABILITY where y T ètè equals yètè for 0 ç t ç T and is zero otherwise, and kæk;jæj are the L 1 and the usual Euclidean norm, respectively. ènote: Completeness of è1.2è imposed by the previous deænition does not consist an extra assumption; in particular, it is very easy to see that è2.1è implies completeness of è1.2èè. The previous condition has been used by several authors in order to face the global stabilizability problem èsee for instance ë2,3,9,11ëè. We now propose two versions of the I.S.S.C. Condition 2.1 There exist a positive deænite, uniformly unbounded èp.d.u.u.è C 1 function V : é n! é + and a function æ : é +! é + of class K 1 ènamely, æ 2 K and æèsè! +1 as s! +1è such that èdv denotes the derivative of V è. DV èxèfèx; yè é 0; 8jyj çæèjxjè; x6=0 è2.2è It is important to note that Condition 2.1 is equivalent totheëlyapunov description" of the I.S.S.C. èsee equation è36è in ë11ë or condition è3è and Lemma 1 of the recent work ë2ë of Freeman; Kokotovicè. To be more speciæc, if è2.2è holds then following the same arguments with those given in ë10, Section VIë we can determine a p.d.u.u. C 1 map ^V : é n!é + and a function ç : é +!é + of class K 1 such that D ^V èxèfèx; yè ç,çèjxjè; 8jyj çæèjxjè: The latter, according to the proof of Theorem 1 in ë10ë, implies I.S.S.C. Conversely, as we have mentioned in the introduction, it has been recently proved by Sontag and Wang in ë14ë that I.S.S.C. implies Condition 2.1 provided that f is at least C 1. This is a consequence of a general converse Lyapunov theorem proved by Lin, Sontag and Wang in ë7ë concerning the set stability of control systems with inputs taking values on a compact set. Since I.S.S.C. is stronger than ëbounded input bounded state" èb.i.b.s.è and also ëconverging input converging state" èc.i.c.s.è properties èsee ë12ë for deænitionsè it follows that Condition 2.1 implies B.I.B.S., C.I.C.S. and completeness of è1.2è. The following condition is weaker than I.S.S.C. Condition 2.2 that There exists a function æ : é +!é + of class K 1 such èaè for every initial state x 0 and input y such that xèt; x 0 ; yè exists for all t ç 0 and satisæes jyètèj çæèjxèt; x 0 ; yèjè è2.3è it follows that 3

4 J. TSINIAS xèt; x 0 ; yè! 0 as t! +1 and since æ is C æ vanishing at zero è2.4aè yètè! 0; as t! +1 ; è2.4bè èbè for every initial state x 0, input y and time T such that è2.3è is satisæed on ë0;tè it follows that lim t!t jxèt; x 0 ; yèj é +1 and, since æ is of class K 1, lim t!t jyètèj é +1. Condition 2.2 says essentially the following: for each vector èx 0 ;y 0 è belonging to the region M := fèx; yè 2é n+1 : jyj çæèjxjègand for every input y the trajectory èxèt; x 0 ; yè;yètèè either leaves Mnf0g after some ænite time, or stays in M for all t; approaching zero as t! +1. Example Consider the system _x 1 = gèx 1 è; _x 2 =,x 2 +y; èx 1 ;x 2 è2é k æé and assume that 0 2é k is a global attractor with respect to _x 1 = gèx 1 è. Then we can easily justify that this system satisæes Condition 2.2 with æèsè = 1 2 s1=2, although it may fails to be input to state stable or even B.I.B.S. Note that I.S.S.C. is satisæed under the extra assumption that 0 2é k is globally asymptotically stable with respect to _x 1 = gèx 1 è. The following Propositions summarize some interesting properties of Condition 2.2. In particular, in Propositions 2.3 and 2.4 some rather strict but useful Lyapunovlike descriptions of Condition 2.2 are provided and in Proposition 2.5 it is shown that I.S.S.C. implies Condition 2.2. This in conjunction with the previous example guarantees that Condition 2.2 is weaker than I.S.S.C. Proposition 2.3 èiè Suppose that there exist a pair of positive deænite C 1 functions V 1 ; V 2 : é n! é + ;V 2 being uniformly unbounded, functions ç 2 K and æ 2 K 1, and ç : é +! é + being everywhere strictly positive and continuous with Z +1 and a positive constant R such that 0 dr çèrè =+1 DV 1 èxèfèx; yè é,çèjxjè; 8jyjçæèjxjè; x6=0 è2.5è è2.6aè DV 2 èxèfèx; yè ç çèv 2 èxèè; 8jyjçæèjxjè; jxjçr è2.6bè Then Condition 2.2 is satisæed with the same characteristic function æ; 4

5 INPUT TO STATE STABILITY èiiè it turns out that Condition 2.1 implies Condition 2.2 èwith the same characteristic function æè. Proof èoutlineè: We show that è2.6bè plus è2.5è implies property èbè of Condition 2.2. èits proof consists a slight generalization of a well known result due to Wintner ë19ëè. Suppose on the contrary that there exist an x 0, input y and time T such that the solution xèt; x 0 ; yè of è1.2è exists and satisæes è2.3è on ë0;tè; whereas lim jxèt; x 0 ; yèj!1as t! T. Then there would exist a pair of sequences ft v g and fç v g; èv = 1; 2;:::è with T é t v é ç v, t v! T such that jxèt; x 0 ; yèj ç jxèç v ;x 0 ;yèj = R for t 2 ët v ;ç v ë and jxèt v ;x 0 ;yèj!+1as t v! T. Then by è2.5è,è2.6è and the fact that V 2 is p.d.u.u. on é n we get +1 = lim t v!t Z V 2èxèt v;x 0;yèè V 2èxèç v;x 0;yèè dr çèrè ét a contradiction, hence property èbè of Condition 2.2 is satisæed. The latter in conjunction with è2.6aè implies property èaè of Condition 2.2 èthe proof of this statement follows by using standard Lyapunov based arguments based on property èbè of Condition 2.2 and the fact that ç 2 Kè. Finally, note that if Condition 2.1 is satisæed then obviously è2.6aè and è2.6bè hold with V 1 = V 2 = V being uniformly unbounded, ç =1; R= 0 and the same æ. Amuch more strong Lyapunov like description of Condition 2.2 is given by the following proposition. Proposition 2.4 Assume that there exist a pair of positive deænite C 1 functions V 1 ;V 2 :é n!é + ;V 2 being uniformly unbounded oné n ;a function æ 2 K 1 and a constant R ç 0 such that DV 1 èxèfèx; yè é 0; 8jyjçæèjxjè; x6=0 è2.7aè DV 2 èxèfèx; yè ç 0; 8jyj çæèjxjè; jxjçr: è2.7bè Then the system è1.2è satisæes Condition 2.2 as well as B.I.B.S. and C.I.C.S. properties. Obviously, Condition 2.1 implies both è2.7aè and è2.7bè with V 1 = V 2 = V; R =0and the same æ. Proof: We only show that è2.7bè implies B.I.B.S. The rest part of the proof follows easily and it is left to the reader. First, note that è2.7bè is a special case of è2.6bè hence implies property èbè of Condition 2.2. Consider now any initial x 0 and bounded input y. We show that the solution xèæ;x 0 ;yè is bounded. We distinguish three cases. The ærst is jyètèj çæèjxèt; x 0 ; yèjè for all t after some ænite time. Then the desired conclusion follows by taking into account è2.7bè and the fact that V 2 is p.d.u.u. 5

6 J. TSINIAS on é n : Suppose next that there exist sequences ft v g; fç v g; fs v g with 0 é t v ç ç v ç s v ét v+1 and t v! +1 such that jyètèj ç æèjxèt; x 0 ; yèjè for t 2 ët v ;s v ë; jyètèj ç æèjxèt; x 0 ; yèjè for t 2 ës v ;t v+1 ë and jxèç v ;x 0 ;yèj = maxfjxèt; x 0 ; yèj; t 2 ët v ;s v ëg and assume on the contrary that limjxèç v ;x 0 ;yèj=+1: By è2.7bè and the fact that V 2 is p.d.u.u. on é n we get limv 2 èxèt v ;x 0 ;yèè ç limv 2 èxèç v ;x 0 ;yèè=+1 and so limjxèt v ;x 0 ;yèj!+1:on the other hand jyèt v èj = æèjxèt v ;x 0 ;yèjè and since æ 2 K 1 it follows that limjyèt v èj =+1;a contradiction. The third case is jyètèj çæèjxèt; x 0 ; yèjè for all t after some ænite time. Then the desired conclusion follows directly from the boundedness of the input y. Hence B.I.B.S. property is satisæed. Similarly we can establish that è2.7aè implies C.I.C.S. Proposition 2.5 If the system è1.2è satisæes the I.S.S.C. then it also satisæes Condition 2.2 with æ being any real function of class K 1 such that èæ æ æèèsè é çs; 8 sé0 è2.8è where æ is the function deæned in è2.1è and ç being a positive constant with çé1. Proof: The proof can be followed by using Proposition 2.4 and the equivalence of Condition 2.1 with I.S.S.C., which aswehave already mentioned was established in ë14ë. For reasons of completeness a simple proof, which follows directly from the deænition of the I.S.S.C., is presented here. Consider any function æ of class K 1 which satisæes è2.1è and let an initial state x 0, input y and time T ç +1 such that xèt; x 0 ; yè exists and è2.3è holds on ë0;tè. We claim that limjxèt; x 0 ;yèj é +1 as t! T. Indeed, by è2.1è it follows jxèt; x 0 ; yèj çæèjx 0 j;tè+èæææèèkxèæ;x 0 ;yè t kè è2.9è for all t 2 ë0;tè. Suppose on the contrary that limjxèt; x 0 ; yèj =+1as t! T. Then there would exist a sequence ft v g with t 2 ë0;tè and t v! T such that jxèt; x 0 ; yèj çjxèt v ;x 0 ;yè tv jfor all t ç t v : This implies kxèæ;x 0 ;yè tv k= jxèt v ;x 0 ;yèj and so by è2.8è and è2.9è we get è1, çèlimjxèt v ;x 0 ;yèj ç lim æèjx 0 j;t v è é +1 as t v! T; a contradiction. Therefore property èbè of Condition 2.2 is satisæed. We now establish property èaè of Condition 2.2, namely we show that è2.4è is fulælled for any x 0 and input y satisfying è2.3è. Suppose on the contrary that limjxèt; x 0 ; yèj = çé0ast!+1. The analysis given above implies that çé+1. It turns out that for every æé0 there exists a T = T èæè é 0 such that jxèt; x 0 ; yèj éç+æfor all t ç T: 6 è2.10è

7 INPUT TO STATE STABILITY Let ^x 0 = xèt;x 0 ;yè and ^yètè = yèt + T è. By è2.1è, è2.8è and è2.10è it follows that jxèt; ^x 0 ;^yèj ç æèj^x 0 j;tè+æèk^y t kè ç æèj^x 0 j;tè+èæææèèkxèæ; ^x 0 ;^y t èkè ç æèj^x 0 j;tè+èæææèèç + æè; 8 t ç 0: therefore ç = limjxèt; x 0 ; yèj é lim æèj^x 0 j;tè+èæææèèç + æè =èæææèèç + æè. Since the latter holds for every æé0we conclude that ç ç èæ æ æèèçè; ç6=0 which contradicts è2.8è, hence property èaè of Condition 2.2 is fulælled. We conclude this section by the following proposition that we shall need in the next section. èanalogous result is provided in ë3ë in the presence of the I.S.S.C.è. Proposition 2.6 Suppose that f is C 1 near zero and there exists a C 1 Lyapunov function æ of the origin with respect to _x = fèx; 0è such that inffdæèxèf èx; 0è=jxjjDæèxèj; x 6= 0 near zerog é 0: è2.11è Then, if è1.2è satisæes Condition 2.2, there exist a smooth function ^æ : é +!é + of class K 1 and positive constants c and ç such that ^æèsè =cs for s 2 ë0;çë; ^æèsè ç æèsè otherwise and further Condition 2.2 is satisæed with ^æ instead of æ. In particular, è2.11è holds if we assume that è@f =@xèè0; 0è is Hurwitz. Proof: Since f is C 1 is written fèx; yè = fèx; 0è + è@f=@yèèx; ^yèy for x; y and appropriate ^y near zero, hence by è2.11è we can ænd a constant c é 0 such that Dæèxèfèx; yè é 0 for all jyj ç cjxj; x near zero. This implies the existence of a neighborhood N of 0 2é n such that the solution xèt; x 0 ; yè is deæned for all t ç 0 and is contained in N approaching zero as t! +1 for every x 0 2 N and input y with jyètèj çcjxèt; x 0 ; yèj. The desired conclusion follows then by combining the previous discussion and properties èaè and èbè of Condition Output Feedback Stabilizability Our ærst result is the following theorem. Theorem 3.1 If Condition 2.1 is satisæed, then the system è1.1è with m=1 is G.A.S. by means of a continuous output feedback u = çèyè with çè0è = 0 which is smooth on é n nf0g. Proof: In order to simplify the notation we denote y = y 1 ; g = g 1 : Consider a pair of C æ functions a; b : é +! é of class K 1 such that according to the previous notations the following holds: jgèx; yèj çaèjxjè+bèjyjè; 8èx; yè 2é n+1 è3.1è 7

8 J. TSINIAS èsee ë9ë for a proof of the previous inequalityè. We deæne ç 1 èsè = æèsè and ç 2 èsè =è1=2èæèsè; æbeing the function introduced in Condition 2.1. Without any loss of generality wemay assume that a; b are C 1 on énf0g and æ is C 1 on é. The latter is a direct consequence of è2.2è and Lemma 3.1 in ë13ë. We establish that there exists a C æ map y! çèyè which isodd, C 1 on énf0g; its restriction ç : é +!é + is of class K 1 and such that the function çèyè +ç,è1 + Eèy, bèyè, çèyè ; y ç 0,è1 + Eèy + bè,yè, çèyè ; yé0; è3.2è E being an arbitrary positive constant and b as deæned in è3.1è, is an output feedback stabilizer. Particularly, ç can be chosen in such away that ç ç 2aèjxjè + max jdç 1 èjxjèfèx; yèj; ç 2 èjxjè ç y ç 3 2 ç 1èjxjè é 1 4 çèç 2èjxjèè; 8x 6= 0: è3.3è The existence of ç satisfying the previous properties follows from the fact that ç 2 is of class K 1 and the mappings aèjxjè; ç 1 èjxjè;ç 2 èjxjè;dç 1 èjxjè and fèx; yè are continuous vanishing at zero. In order to establish that the map ç as deæned by è3.2è is the desirable output feedback we ærst prove that the following properties are satisæed: èiè For every initial state èx 0 ;y 0 è2é n+1 wehave d dt èy2 ètèè = yètèègèxètè;yètèè + çèyètèèè ç,jyètèj 2 è3.4è for each t ç 0 such that jyètèj éç 2 èjxètèjèand for which the corresponding ènot necessarily uniqueè solution of the closed - loop system exists; èxètè;yètèè = èxèt; x 0 ;y 0 è;yèt; x 0 ;y 0 èè _x = fèx; yè; _y = gèx; yè+çèyè è3.5è èiiè the region M + fèx; yè 2é n+1 : jyj çç 1 èjxjèg is positively invariant with respect to è3.5è. è3.6è 8

9 INPUT TO STATE STABILITY We ærst prove property èiè. We substitute the term çèyèy = jyjçèjyjè =,è1 + Eèy 2,jyjbèjyjè,jyjçèjyjè è3.7è in è3.4è and take into account è3.1è. It suæces then to show that jyjèaèjxjè+bèjyjèè, è1 + Eèy 2,jyjbèjyjè,jyjçèjyjèé,jyj 2 ; or jyjèaèjxjè,çèjyjèè é 0 for jyj éç 2 èjxjè which follows from è3.3è. Next we establish property èiiè. Let èx 0 ;y 0 è2m. Since 0 2é n+1 is an equilibrium for è3.5è and belongs to M; we may assume that èx 0 ;y 0 è 6=0. Weshow that the corresponding solution of è3.5è lies in M for every t ç 0 for which it exists. Equivalently y 2 ètè ç ç 2 1èjxètèjè ory0+2z 2 t yèsèègèxèsè;yèsèè + çèyèsèèè ds ç ç 2 1èjx 0 jè+2z t 0 0 ç 1 èjxèsèjèdç 1 èjxèsèjè x0 èsè jxèsèj fèxèsè;yèsèèds for every t ç 0. Since jy 0 jçç 1 èjx 0 jè it suæces to prove that yègèx; yè+çèyèè éç 1 èjxjèdç 1 èjxjè x0 fèx; yè jxj for ç 2 èjxjè ç y ç 3 2 ç 1èjxjè orby è3.1è and è3.7è that jyj èaèjxjè +bèjyjèè, è1 + Eèy 2,jyjbèjyjè,jyjçèjyjè çç 1 èjxjèdç 1 èjxjè x0 fèx; yè jxj which follows from è3.3è and the fact that 1 2 ç 1èjxjè ç y ç 3 2 ç 1èjxjè: From property èiè it follows that for any initial state èx 0 ;y 0 è the y- component of the solution èxètè;yètèè of è3.5è satisæes the inequality jyètèj çjy 0 je,1 2 t è3.8aè as long as the solution exists and ç 1 èjxètèjè é jyètèj è3.8bè By è3.8è and the positively invariance of M we distinguish two cases. The ærst is jyètèj çç 1 èjxètèjè after some ænite time. The second is jyètèj é ç 1 èjxètèjè for all t ç 0 èfor which the solution existsè. This in conjunction with è3.8è implies that yètè! 0 and so ç 1 èjxètèjè! 0 provided that the solution exists for all t ç 0. Since ç 1 is C æ vanishing at zero it follows 9

10 J. TSINIAS that èxètè;yètèè! 0 as t! +1. We can use the previous arguments in order to show that è3.5è is complete. Indeed, by è3.8è the solution èxètè;yètèè exists for all t ç 0 such that èxètè;yètèè 62 M. Finally, weinvoke Proposition 2.3 which asserts that Condition 2.1 implies Condition 2.2 with the same characteristic function æè= ç 1 è: It follows that the solution of è3.5è also exists for all t ç 0 with èxètè;yètèè 2 M and thus by the positively invariance of M for every t ç 0. We are now in a position to prove that 0 2é m+1 is globally asymptotically stable with respect to è3.5è. Using the positively invariance of M and Condition 2.2 it follows that xèt; x 0 ;y 0 è!0ast!+1and consequently yètè! 0 as t! +1 for every initial èx 0 ;y 0 è 2 M. The previous discussion, in conjunction with the fact that because of è3.8è each trajectory with initial state outside M is tending to zero as t! +1 or enters M after some ænite time, guarantees that èxètè;yètèè! 0ast! +1for all èx 0 ;y 0 è 2é n+1. Next, we show that, 0 2é n+1 is stable with respect to è3.5è. By è2.2è and the fact that ç 1 is C æ vanishing at zero, we can ænd for each æé0aæé0 with æ + ç 1 èæè ç æ=2 such that jèxèt; x 0 ;y 0 è;yèt; x 0 ;y 0 èèj çjxèt; x 0 ;y 0 èj+ç 1 èjxèt; x 0 ;y 0 èjè é æ 2 è3.9è for all t ç 0 and èx 0 ;y 0 è 2 M with jx 0 j é æ. Finally, from è3.8è, è3.9è, the positively invariance of M and since ç 1 is strictly increasing it follows that for every èx 0 ;y 0 è 62 M with jx 0 j éæand jy 0 j éç 1 èæèwehave ç 1 èjxèt; x 0 ;y 0 èjè é jyèt; x 0 ;y 0 èj é ç 1 èæè as long as the solution remains outside M; which in conjunction with è3.9è implies that jèxèt; x 0 ;y 0 è;yèt; x 0 ;y 0 èèj éç 1 èæè+æ+ æ 2 éæ è3.10è for all t ç 0. By è3.9è and è3.10è it follows that for each æé0wehave jèxèt; x 0 ;y 0 è;yèt; x 0 ;y 0 èèj é æ for all t ç 0 and èx 0 ;y 0 è belonging to the region fèx; yè 2é n+1 : jxj éæ; jyjéç 1 èæèg. We conclude that 0 2é n+1 is globally asymptotically stable with respect to è3.5è. Next we generalize Corollary 2.2 and Proposition 4.3 in ë3ë by proving that if the dynamics f and g i are C 1,è@f=@xèè0; 0è is Hurwitz and Condition 2.2 is satisæed, then the system è1.1è is G.A.S. by a smooth output feedback. First, we examine the case m =1. As in the proof of Theorem 3.1 we use the notations y = y 1 and g = g 1. Theorem 3.2 Consider the system è1.1è with m =1and suppose that the mappings f and g = g 1 are C 1, the matrix è@f=@xèè0; 0è is Hurwitz and either Condition 2.2 or one of its stronger versions èè2.5è plus è2.6è, è2.7è, Condition 2.1, I.S.S.C.è is satisæed. Then è1.1è with m =1is G.A.S. by means of a C 1 output feedback u = çèyè; çè0è=0;which is linear near 10

11 zero and such that the matrix INPUT TO STATE è0; @y! è0; è0; 0è è0è è3.11è is Hurwitz and furthermore the system _x = fèx; yè; _y = gèx; yè+çèyè+w è3.12è with w as input satisæes Condition 2.2. Proof: Notice ærst that, since g is C 1 near zero, there exists a pair of smooth functions a and b of class K 1 such that è3.1è holds and further both a and b are linear near zero. è3.13è Next we recall Proposition 2.6 which asserts that since è@f =@xèè0; 0è is Hurwitz, we can determine a C 1 function ^æ : é +!é + of class K 1 which is linear near zero such that Condition 2.2 is satisæed with respect to è1.2è with ^æ instead of æ. We deæne ç 1 èsè = ^æèsè; ç 2 èsè = è1=2èç 1 èsè. Since f; ç 1 and ç 2 are C 1 vanishing at zero and because of è3.13è, there exists a constant Ré0 such that maxfjdç 1 èjxjèfèx; yèj; ç 2 èjxjè ç y ç 3 2 ç 1èjxjèg çrjxj; è3.14è for x near zero. From è3.13è and è3.14è it follows that there exists an odd C 1 function ç which is linear near zero such that the map ç : é +!é + is of class K 1 and è3.3è is satisæed. This in conjunction with è3.13è implies that the output feedback ç : é! é as deæned in è3.2è is smooth and linear near zero. Moreover, since è@f =@xèè0; 0è is Hurwitz, we can select the constant E in è3.2è suæciently large such that the matrix è3.11è is also Hurwitz. Using è3.3è and following exactly the same arguments with those given in the proof of Theorem 3.1 we can establish that è3.8aè is fulælled as long as è3.8bè and jwètèj ç 1 2 aèjxètèjè+1 4 çèjyètèjè è3.15è hold and for each t ç 0 for which the solution èxètè;yètèè = èxèt; x 0 ;y 0 ;wè;yèt; x 0 ;y 0 ;wèè of è3.12è exists. Furthermore è3.3è implies that the set M as deæned by è3.6è is positively invariant with respect to è3.12è, with inputs w satisfying è3.15è; namely, for each èx 0 ;y 0 è2m it holds jyèt; x 0 ;y 0 ;wèjçç 1 èjxèt; x 0 ;y 0 ;wèjè 11

12 J. TSINIAS as long as è3.15è is satisæed. Let, : é +!é + be a function of class K 1 such that,èjèx; yèjè ç 1 2 aèjxjè+1 çèjyjè; 8èx; yè 2én+1 è3.16è 4 èthe existence of, is guaranteed by the fact that both a and ç : é +!é + are of class K 1 è. Using the previous properties, as well as property èaè of Condition 2.2 imposed for the subsystem è1.2è, we can establish as in the proof of Theorem 3.1 that for each initial state èx 0 ;y 0 è 2 é n+1 the trajectory èxètè;yètèè of è3.12è is tending to zero as t! +1 for every input w such that jwètèj é,èjèxètè;yètèèjè è3.17è and providing that the solution exists for all t ç 0. It follows that the system è3.12è satisæes property èaè of Condition 2.2 with, instead of æ. Moreover, as in the proof of Theorem 3.1 we can show that è3.5è ènamely, the system è3.12è with zero inputè is complete and so zero 0 2 é n+1 is a global attractor with respect to è3.5è. This in conjunction with the fact that the matrix è3.11è is Hurwitz implies that 0 2 é n+1 is globally asymptotically stable with respect to è3.6è. Finally, we establish that the system è3.12è also satisæes property èbè of Condition 2.2 with, instead of æ. Consider any vector èx 0 ;y 0 è2é n+1 ; time T and input w satisfying è3.17è for all t 2 ë0;tè and assume that the corresponding solution of è3.12è exists on ë0;tè. We show that lim jèxètè;yètè; wètèèj é +1: è3.18è t!t We distinguish two cases. The ærst is ç 1 èjxètèjè é jyètèj for t near T which in conjunction with è3.8è and è3.15è-è3.17è implies that yètè and therefore xètè and wètè are bounded for t near T. The other case is èx 0 ;y 0 è 2 M and jyètèj ç ç 1 èjxètèjè for all t 2 ë0;tè. Then by using property èbè of Condition 2.2, which has assumed for the system è1.2è, it follows that the solution of è3.12è as well as the corresponding input w satisfying è3.17è are both bounded on ë0;të. The previous discussion asserts that è3.18è is satisæed and so the the proof is completed. Remark 3.3 If the assumption ëè@f =@xèè0; 0è is Hurwitz" in the statement of Theorem 3.2 is dropped and if we further assume that all the Dini derivatives of æ exist at zero, then similar to the proof of this theorem it can be shown that there exists a continuous output feedback u = çèyè such that è3.12è also satisæes Condition 2.2 and the origin will be a global attractor with respect to è3.5è. However in that case ç will be in general C 1 on é n nf0g and zero may fails to be stable with respect to è3.5è. We are now in a position to prove our result concerning the output stabilizability problem for the general case è1.1è with output y =èy 1 ;y 2 ; :::; y m è 0. 12

13 INPUT TO STATE STABILITY Corollary 3.4 Consider the general èn + mè-dimensional case è1.1è and assume that ç and g i are C 1. Then under the same hypothesis of Theorem 3.2 the system è1.1è is G.A.S. by means of a smooth output feedback which is linear near zero. Proof: The proof of the general case is based on Theorem 3.2 and follows by using standard induction arguments like those given in ë3, Proposition 4.3ë and in other relative works èsee for instance ë6,8,13ëè. For reasons of simplicity we consider the case m = 2. The general case follows similarly by induction. First, we invoke Theorem 3.2 in order to establish that there exists a C 1 map y 2 = ç 1 èy 1 è with ç 1 è0è = 0 which is linear near zero and such that if we deæne X + èx 0 ;y 1 è 0 ; FèX; Y è + èf 0 èx 1 ;y 1 è;g 1 èx; y 1 è+ç 1 èy 1 è+yè 0 ; the origin 0 2é n+1 is globally asymptotically stable with respect to _X = F èx; Y è è3.19è with Y = 0; è@f=@xèè0; 0è is Hurwitz and the system è3.19è with Y as input satisæes Condition 2.2. Then we apply the transformation x = x; y 1 = y 1 ; Y = y 2,ç 1 èy 1 è.in the new coordinates the original system è3.19è takes the form _X = F èx; Y è; Y _ = GèX; Y è+u è3.20aè where GèX; Y è +,Dç 1 èy 1 èèg 1 èx; y 1 è+ç 1 èy 1 è+yè + g 2 èx; y 1 ;ç 1 èy 1 è+yè è3.20bè The desired conclusion follows then by using the properties of è3.19è and applying once again Theorem 3.2 for è3.20è. 4 Weaker Stability Conditions It is worth remarking that the assumption æ 2 K 1 that has been imposed in both Conditions 2.1 and 2.2 can be relaxed in some cases. For instance, let us assume that there exists a pair of increasing bounded functions a; b : é +! é + satisfying è3.1è. Then the results of Theorems 3.1 and 3.2 concerning the case è1.1è with m = 1 are also valid if instead of Conditions 2.1 or 2.2 the system è1.2è satisæes one of the following weaker versions. Condition 4.1 The system è1.2è is complete and there exist a p.d.u.u. C 1 function V : é n!é + and a bounded function æ 2 K such that è2.2è holds. 13

14 J. TSINIAS Condition 4.2 The system è1.2è is complete and there exists a bounded function æ 2 K such that properties èaè and èbè of Condition 2.2 are satisæed. Indeed, in that case we can determine an increasing function ç 1 : é +! é + and positive constants ç and ç such that ç 1 èsè ç æèsè for all s ç 0 and ç 1 èsè =çfor every ç ç ç. Furthermore we can assume that without any loss of generality both a and b satisfying è3.1è are constants on ëç;+1è. The previous assumption guarantees the existence a bounded C æ map ç whichis constant for jyj éç;its restriction ç : é +!é + is increasing and satisæes all the required properties including è3.3è. The desired feedback will given again by è3.2è whereas its derivative will be constant for jyj éç. Moreover in this case the resulting system è3.12è will be complete. This follows from completeness assumption for è1.2è and the fact that ç is bounded. Indeed, boundedness of ç together with è3.2è implies that for every pair of mappings xètè and wètè; wbeing essentially bounded, the one-dimensional system _y = gèx; yè + çèyè + wis complete. The latter in conjunction with the completeness of è1.2è implies completeness of è3.12è. Note at this point that both Conditions 4.1 and 4.2 are weaker than I.S.S.C. as well as Condition 2.2 and as in Section 2 we can easily establish that Condition 4.1 implies Condition 4.2. We can also derive an extension of Corollary 3.4 in the case æ 2 K and under the extra assumption that each g i is bounded. However, a more careful analysis, than this given in Theorem 3.2 and Corollary 3.4, is required. We ærst need the following lemma whose proof consists of a slight modiæcation of the procedure used in the proof of Theorem 3.2. Lemma 4.3 Consider the system è1.1è with m =1and suppose that f and g 1 are smooth and g 1 is decomposed asg 1 èx; yè =G 1 èx 1 ;x 2 ;yè+g 2 èx 2 ;yè with x = èx 1 ;x 2 è 2 é n1 æé n2, n 1 +n 2 = n for the case n 2 ç 1 and g 1 èx; yè =G 1 èx; yè+g 2 èyè otherwise, where G 1 and G 2 are smooth vanishing at zero and G 1 is bounded. Furthermore assume that è@f=@xèè0; 0è is Hurwitz and the system è1.2è satisæes Condition 4.2 èor its stronger version Condition 4.1è. Then there exists a linear map çèyè such that the system è1.1è with m = 1 is G.A.S. by means of the smooth feedback u = çèyè, G 2 èx 2 ;yè: Furthermore the derivative of the closed-loop dynamics èf0;g 1 + çè0 at the origin will be Hurwitz and the system _x = fèx; yè; _y = G 1 èx; yè+çèyè+w with w as input satisæes Condition 4.2. Proof èoutlineè: Let P be the positive deænite solution of the matrix è0; 0è è0; 0è ç 0 P =,I and deæne æèx; yè := x 0 Px y 2 ; uèyè:=çèyè,g 2 èx 2 ;yè;çèyè:=,ey;e é 0: Then it can be easily justiæed that there exists a positive E 1 such that for all E ç E 1 the derivative ofæ 14

15 INPUT TO STATE STABILITY along the trajectories of the linearization at zero of the resulting closed-loop system _x = fèx; y 1 è; _y = G 1 èx; yè+çèyè+w; çèyè =Ey è4.1è is strictly negative and for suæciently small cé0;being independent of E; the compact set N := èx; yè 2é n+1 :æèx; yè ç c è4.2è is positively invariant with respect to è4.1è. As in Theorem 3.2, consider the mappings ç 1 èsè and ç 2 èsè and without any loss of generality assume that ç 2 èsè = 1 2 ç 1èsè= 1 2æ; 8s ç ç for arbitrary small positive constants ç and æ: Pick ç such that the sphere of radius 2ç centered at 0 2 é n+1 is contained in N: Let E be a positive constant satisfying EémaxfE 1 ; 2æ 1 sup G 1 èx; yèg +1 whose existence is guaranteed by the boundedness of G 1 : By a slight modiæcation of the approach in Theorem 3.2 we can ænd a bounded function, 2 K in such away that èiè for every initial y 0 with jy 0 j éæ=4 property è3.8aè holds for all inputs! satisfying è3.17è with respect to è4.1è with E as deæned by è4.3è; èiiè the set N ë M; with M; N as deæned by è3.6è and è4.2è, respectively is positively invariant with respect to è4.1è. The previous properties imply as in the proof of Theorem 3.2 the desired conclusion. Corollary 4.4 Suppose that f is C 1 ; the subsystem è1.2è satisæes the same hypotheses of Lemma 4.3 and the mappings g i, i =1; :::; m are C 1 and bounded. Then the system è1.1è is G.A.S. by means of a linear feedback depending only on y 1 ; :::y m. Proof: For reasons of simplicity we consider the case m =2. For the case m =1we deæne G 1 = g 1 ; G 2 = 0 and apply the result of Lemma 4.3. Let ç 1 èy 1 è be the corresponding linear feedback stabilizer. We proceed as in the proof of Corollary 3.4 following the same notations in order to obtain the system è3.20è. Note that G is decomposed as where GèX; Y è + G 1 èx; Y è+g 2 èy 1 ;Yè G 1 èx; Y è +,Dç 1 èy 1 èèg 1 èx; y 1 è+g 2 èx; y 1 ;ç 1 èyè+yè; G 2 èy 1 ;Yè=,Dç 1 èy 1 èèy + ç 1 èy 1 èè: 15

16 J. TSINIAS The desired conclusion then follows from Lemma 4.3, the properties of è3.19è and the fact that g 1 and g 2 are bounded. Particularly, there exists a linear map çèy è as in the statement of Lemma 4.3 such that the desired feedback stabilizer for è3.20è has the form ç 2 èy 1 ;Yè=çèYè,G 2 èy 1 ;Yè; namely it depends only on Y = y 2, ç 1 èy 1 è and y 1 èthe last variable of Xè. It turns out by the deænition of G 2 that ç 2 is linear and depends only on y 1 and y 2 : Acknowledgements I would like to thank Professor E.D. Sontag for his helpful comments. References ë1ë C.I. Byrnes, A. Isidori, J.C. Willems. Passivity, feedback equivalence and the global stabilization of minimum phase nonlinear systems, IEEE Tr. on AC 36è11è è1991è, ë2ë R.A. Freeman, P.V. Kokotovic. Global Robustness of Nonlinear systems to state measurement disturbances, to appear in Proceedings of the 32nd CDC, ë3ë Z.P. Jiang, L.Praly, A.R. Teel. Small-gain theorem for ISS system and applications, to appear in M.C.S.S. ë4ë I. Kanellakopoulos, P.V. Kokotovic, A.S. Morse. A toolkit for nonlinear feedback design, Systems and Control Let. 18è2è è1992è, ë5ë H.K. Khalil and A. Saberi. Adaptive stabilization of a class of nonlinear systems using high-gain feedback, IEEE Tr. on AC 32è11è è1987è, ë6ë P.V. Kokotovic, I. Kanellakopoulos and A.S. Morse. Adaptive Feedback linearization of nonlinear systems, in Lecture Notes in Control and Information Sciences 160. New York: Springer-Verlag, ë7ë Y. Lin, E.D. Sontag and Y. Wang. A smooth converse Lyapunov theorem for Robust stability, to appear in SIAM J. Control and Opt. ë8ë R. Marino and P. Tomei. Robust stabilization of feedback linearizable time-varying uncertain nonlinear systems, Automatica 29è1è è1993è, ë9ë L. Praly and Z.P. Jiang. Stabilization by output feedback for systems with I.S.S. inverse dynamics, Systems and Control Let. 20è5è è1993è,

17 INPUT TO STATE STABILITY ë10ë E.D. Sontag. Smooth stabilization implies coprime factorization, IEEE Trans. on AC 34 è1989è, ë11ë E.D. Sontag. Further facts about input to state stabilization, IEEE Trans. AC 35 è1990è, ë12ë E.D. Sontag. Remarks on stabilization and input-to-state stability, IEEE Conf. Decision and Control, Tampa FL. 1989, IEEE Publications, ë13ë E.D. Sontag and H. Sussmann. Further comments on the stabilizability of the angular velocity of a rigid body, Systems and Control Lett. 12 è1989è, ë14ë E.D. Sontag and Y. Wang. On characterizations of the input-to- state stability property, Systems and Control Let. 24 è1995è, ë15ë A.R. Teel. Semi-global stabilization by minimum phase nonlinear systems in special normal form, Systems and Control Let. 19 è1992è, ë16ë A.R. Teel and L. Praly. Semi-global stabilization by linear dynamic output feedback for SISO minimum phase nonlinear systems, to appear in Proceedings of IFAC 93. ë17ë J. Tsinias. Sontag's input to state stability condition and global stabilization using state detection, Systems and Control Let. 20 è1993è, ë18ë J. Tsinias. Versions of Sontag's input to state stability condition and the global stabilizability problem, SIAM J. Control and Opt. 31è4è è1993è, ë19ë A. Wintner. The nonlocal existence of ordinary diæerential equations, Amer. J. Math. 67 è1945è, National Technical University, Department of Mathematics, Zografou Campus 15780, Athens, Greece Communicated by Alberto Isidori 17