Strong Lyapunov Functions for Systems Satisfying the Conditions of La Salle

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1 06 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 Strong Lyapunov Functions or Systems Satisying the Conditions o La Salle Frédéric Mazenc and Dragan Ne sić Abstract We present a construction o a (strong) Lyapunov unction whose derivative is negative deinite along the solutions o the system using another (weak) Lyapunov unction whose derivative along the solutions o the system is negative semideinite. The construction can be carried out i a Lie algebraic condition that involves the (weak) Lyapunov unction and the system vector ield is satisied. Our main result extends to general nonlinear systems the strong Lyapunov unction construction presented in a previous paper that was valid only or homogeneous systems. Index Terms La Salle principle, Lyapunov unctions, stability. I. INTRODUCTION Lyapunov unctions are an indispensable tool in analysis and controller design o nonlinear systems as illustrated by [6] [] and numerous reerences cited therein. In particular, positive deinite Lyapunov unctions whose derivative is negative deinite along solutions o the system (strong Lyapunov unctions) are typically more useul than the ones whose derivative is only negative semideinite (weak Lyapunov unctions). Indeed, in the latter case we can only conclude i the system is stable using the La Salle invariance principle [0] while in the ormer case we can also guarantee certain orm o stability robustness. This is probably the main reason why strong Lyapunov unctions have ound a widespread use in controller design methods, such as backstepping [8], orwarding [], [] and universal stabilizing controllers via control Lyapunov unctions [4]. While it is very desirable to obtain a strong Lyapunov unction, this is highly non trivial to achieve in general. On the other hand, or large classes o systems it is much easier to obtain a weak Lyapunov unction although it is not as useul as a strong Lyapunov unction. For instance, the storage (energy) unction o passive nonlinear systems, such as electromechanical systems, typically turns out to be a weak Lyapunov unction or the closed loop system i the system is stabilized using linear output eedback. Another example are systems that are stabilized using the Jurdjevic Quinn Theorem [] []. Hence, there is a strong motivation in providing constructions o strong Lyapunov unctions (that are usually more useul) rom weak Lyapunov unctions (that are oten easier to construct). This problem has attracted attention o several researchers, such as [] and []. The construction in [] requires a weak Lyapunov unction V ( ) and an auxiliary Lyapunov unction V ( ) that veriies appropriate detectability properties o the system with respect to an appropriately deined output. The construction is very general but its main drawback is that the construction o V ( ) may be still very hard in general and there is no general procedure that achieves this. A dierent approach Manuscript received March 6, 00; revised August 5, 00. Recommended by Associate Editor Z.-P. Jiang. This work was supported by the Australian Research Council under the Discovery Grants Scheme while the irst author was visiting The University o Melbourne, Melbourne, Australia. F. Mazenc is with the Projet MERE INRIA-INRA, UMR Analyze des Systèmes et Biométrie, 4060 Montpellier, France. D. Ne sić is with the Department o Electrical and Electronic Engineering, The University o Melbourne, Melbourne, VIC 05, Australia. Digital Object Identiier 0.09/TAC Results in [] and [] are presented or slightly dierent situations rom ours. Indeed, the results in [] are presented or analysis o the more general property o input-to-state stability (ISS) or systems with disturbance inputs. The results in [] are presented or controller design via the control Lyapunov unctions (CLFs) or systems that satisy the Jurdjevic Quinn conditions. was pursued in [] where the construction o a strong Lyapunov unction is based entirely on a weak Lyapunov unction V ( ) and its iterated Lie derivatives along solutions o an auxiliary system with a scaled vector ield. This approach appears to be more direct than the approach in [] but it can be used under slightly dierent (possibly stronger) conditions. However, results in [] are applicable only to homogeneous systems and we are not aware o any similar results or general nonlinear systems. In this note, we present a construction o a strong Lyapunov unction or general nonlinear systems borrowing some ideas rom []. The Lie bracket conditions under which we can carry out our construction is very similar to the conditions rom [] but our result applies to general nonlinear systems, as opposed to [] that only applies to homogeneous systems. Furthermore, our construction is dierent rom the one used in [] and, hence, our Lyapunov unctions will be in general dierent rom the ones obtained in [] when applied to the class o homogeneous systems. The note is organized as ollows. First, we present preliminaries in Section II. The main results are stated in Section III and proved in Section IV. Conclusions and suggestions or urther work are presented in the last Section. Several technical lemmas are presented in the Appendix. II. PRELIMINARIES The set o natural numbers (not including zero) is denoted as N. Sets o real, nonnegative real and strictly positive real numbers are denoted respectively as R; R0, and R >0. A unction k: R0! R0 is said to be o class K i it is continuous, strictly increasing, zero at zero and unbounded. Given a; b N, with a b we use the notation [a; b] to denote the set z N : a z bg. We use the convention that i a>bthen [a; b] =;. Also, given an arbitrary sequence c i, we let b i=a c i =0i a; b N are such that b<a. I a real-valued unction k: R! R has continuous derivatives o arbitrary order we say that the unction is smooth and we denote its ith derivative by k (i) ( ). Given V : R n! R and : R n! R n, we use the ollowing notation: L (x) :=L L i 8i N: We say that V : R n! R0 is positive deinite i V (0) = 0 and > 0 or all x 6= 0. It is positive semideinite i V (0) = 0 and 0 or all x 6= 0. The unction V ( ) is negative deinite (semideinite) i 0V ( ) is positive deinite (semideinite). The unction V ( ) is radially unbounded i jxj!implies that!. In this note, we consider systems o the orm _x = (x) () with (0) = 0. We will carry out analysis o () via an auxiliary system o the orm _x = (x) :=()(x) () where : R0! R >0 is a strictly positive unction that will be deined later and V : R n! R0 is positive deinite. We will always assume that all unctions are dierentiable suiciently many times. In the sequel, we use the ollowing deinition: Deinition.: A unction V : R n! R0 is a weak Lyapunov unction or the system () i it is positive deinite, radially unbounded and _V = L 0 8x R n : () /04$ IEEE

2 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE A unction V ( ) is a strong Lyapunov unction or the system () i it is positive deinite, radially unbounded and the unction 0L is positive deinite. I there exists a neighborhood o the origin E R n such that V ( ) is positive deinite on E and the condition () holds or all x E, we say that V ( ) is a weak Lyapunov unction or the system () on the set E. We deine strong Lyapunov unctions on the set E in a similar manner. We use the standard deinitions o exponential, global exponential, asymptotic, and global asymptotic stability (GAS) (see [6]). III. MAIN RESULT In this section, we state the main result o the note which is proved in the next section. The result provides a construction o a strong Lyapunov unction or the system () under the ollowing conditions: i) a weak Lyapunov unction is available or the system (); and ii) the weak Lyapunov unction and the system () satisy a Lie algebraic condition (item ii) o Theorem.). Theorem.: Suppose that there exists V : R n! R 0 such that the ollowing conditions hold. i) V ( ) is a weak Lyapunov unction or the system (). ii) There exists ` N such that or all x 6= 0, there exists i [;`], such that L i 6= 0. Then, there exists a nonincreasing strictly positive unction : R 0! R >0 such that the unction `0 U (x) = 0 L i is a strong Lyapunov unction or (), where = (x) := ()(x). In particular, we can take any ( ) such that L j 8j ` [;`] 8x R n : (5) A local version o Theorem. can also be stated. Due to space reasons, the proo o this act is omitted. Corollary.: Suppose that all conditions o Theorem. hold on a given neighborhood o the origin E R n. Then, there exists a neighborhood o the origin E with E E such that the Lyapunov unction (4) is a strict Lyapunov unction or () on the set E. In the case o globally asymptotically stable analytic systems or which condition i) o Theorem. holds with an analytic unction V ( ), it can be proved that condition ii) holds on arbitrarily large compact sets which do not contain the origin. The proo o this result is provided in the next section. Proposition.: Consider an analytic system () and assume that it is globally asymptotically stable. Suppose that condition i) o Theorem. holds with an analytic unction V ( ). Then, given an arbitrary compact set E which does not contain the origin there exists ` N such that or any x E with x 6= 0we have that LV i (x) 6= 0or some i [;`]. We note that conditions i) and ii) in Theorem. are not necessary or global asymptotic stability o the system. Indeed, there may exist a system () that satisies condition i) but not condition ii) and which can be proved to be globally asymptotically stable using the La Salle invariance principle. The ollowing example that is taken rom [] illustrates this situation. Example.4: Consider the system _x = x ; _x = 0x 0x B(x ), where B(s) =e 0(=s 0) when jsj 6= and B() = B(0) = 0. Note that B( ) is a smooth unction. Note that item i) o Theorem. is satisied with V (x ;x ) =x x since V _ = 0x B(x ). Moreover, by applying the La Salle invariance principle, one can prove that this system is globally asymptotically stable (see []). However, (4) condition ii) o Theorem. does not hold since or x =(0 ) T and all i N we have that LV i (x )=0. Remark.5: We note that when () is locally exponentially stable, the derivative o (4) along the trajectories o () is typically not locally upper bounded by a negative deinite quadratic unction. Moreover, our construction usually does not produce a Lyapunov unction U ( ) using which we can veriy local or global exponential stability. However, one can use U ( ) to obtain another strong Lyapunov unction W ( ) using which exponential stability can be veriied. For example, i the system satisies conditions o Theorem. and, moreover, its linearization is exponentially stable, then one can construct a Lyapunov unction W ( ) such that (jxj) W (x) (jxj); _W 0 (jxj) or all x R n and some ; ; K and, moreover, there exists ; a; b R >0 such that (s) = as and (s) =bs or all s [0;] (see [4, Lemma 0..5]). Remark.6: The construction in [] applies only to homogeneous systems whereas our construction can be used or general nonlinear systems (). We note that the example o the TORA system in [, Sec. 8] illustrates how results o [] can be sometimes used or nonhomogeneous systems. However, [] does not provide a systematic construction o strong Lyapunov unctions or general nonhomogeneous system, which is what we do. Remark.7: It is possible to use our result in a range o controller design situations, similar to the one presented in [] or systems satisying the Jurdjevic Quinn conditions. Due to space reasons we have not pursued these controller design questions in the current note. Remark.8: An important application o strong Lyapunov unctions is robustness analysis o stability o nonlinear systems (see, or example, [6, Ch. 5]). In order to analyze stability robustness we typically need also that the strong Lyapunov unction U ( ) satisies the ollowing condition: j(@u=@x)j (jxj); 8x R n, or some class K unction ( ). We note that i the weak Lyapunov unction V ( ) is continuously dierentiable suiciently many times, then the unction U ( ) constructed using (4) will satisy this extra condition. Remark.9: An alternative construction o strong Lyapunov unctions or systems satisying conditions o La Salle is presented in [], where a weak Lyapunov unction is combined with another auxiliary Lyapunov unction. More precisely, it is assumed that there exist two positive deinite and radially unbounded unctions V ( ) and V ( ) satisying or all x R n _V 0 (jyj) _ V 0 (jxj) (jyj) (6) where ; ; are K unctions and y = h(x). Note that V ( ) in (6) is typically a weak Lyapunov unction since jh(x)j is typically a positive semideinite unction. The unction V ( ) in (6) is an output-to-state Lyapunov unction (see [7]) that characterizes a particular orm o detectability o x rom the output y. A strong Lyapunov unction in [] takes the orm U (x) =(), where ( ) is a K unction. The main dierence between our approach and [] is that our conditions appear to be stronger but easier to check than those in []. Indeed, searching or V ( ) satisying (6) is much harder than checking our Lie bracket condition ii) in Theorem. and then using (4). IV. PROOFS OF MAIN RESULTS Proo o Theorem.: Note irst that since V ( ) is radially unbounded and positive deinite and LV i (x) are continuous unctions o x or all i [;` ], all conditions o Lemma. in the Appendix hold and the lemma guarantees that we can always ind ( ) such that (5) holds. Moreover, one possible construction o ( ) is Actually, results in [] are presented or the more general case o ISS or systems with inputs.

3 08 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 presented in Lemma.. Consider the auxiliary system (). Item o Lemma. states that i we construct a strong Lyapunov unction or the system (), then the same unction is a strong Lyapunov unction or (). Hence, in the rest o the proo we concentrate only on (). Moreover, items and o Lemma. imply that since conditions i) and ii) o Theorem. hold or the system (), the same conditions hold or (). We now show that (4) is a strong Lyapunov unction or () that satisies conditions i) and ii) o Theorem. and or which (5) holds. To simpliy the notation, we introduce the unctions and M i (x) :=0L i i [;`0 ] (7) `0 S(x) := M i(x): (8) Direct calculations show that, or all i [;` 0 ], the derivative o M i ( ) along solutions o () is with _M i j () = 0!(x) = i L i 0!(x) (9) 0 L i : (0) Note that or any a; b R0 and any i N we can write (consider the cases a (=)b and a>(=)b ) ab 0 b (a) () and using (9) and () with b = j j and a = i jl i j jl i j, we can write j!(x)j Li i L i L i : () Using (9) and (), we can write or all i [;`0 ]: _ M i j () 0 Li Then, using (8), (), and (5) with i L i L i : () _Sj () L 0 P (x) = `0 `0 P (x) (4) i ` Li : (5) However This implies P (x) = ` jl j `0 i ` Li i= `0 `0 (L ) i ` (L ) _Sj () L 0 `0 `0 Li (L ) : (6) : (7) Condition i) o Theorem. and the item o Lemma. guarantee that L is negative semideinite and since (( i )=) i and jl i j or all i =; ;...;`, we get _Sj () L 0 `0 5 L 0 4 `0 `0 : (8) Observe that the unction U (x) deined in (4) can be written as U (x) = ( S(x)) (9) and (8) ensures that its derivative along the trajectories o () satisies with _U j () L ( S(x)) (x) (0) (x) = L 0 4 `0 From (5), it ollows that or all x R n we have `0 S(x) M i (x) 0 `0 0 ` 0 ` ` : () 9` : ()

4 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE Hence, since L is negative semideinite and using () we can obtain, using (0), the ollowing: _Uj () L (x) () or all x R n. It is easy to see that the right-hand side o () is negative deinite. Indeed, note irst that both terms on the right-hand side o () are nonpositive unctions. Moreover, i or some x 6= 0we have L < 0 then the irst term is negative and, hence, Uj _ () < 0.On the other hand, i L =0then rom condition ii) o the theorem `0 and Lemma. we have that 0(=4) (Li ) < 0 and, hence, _Uj () < 0. Finally, note that (9) and () imply that U (x) 8x Rn (4) which implies that U (x) is positive deinite and radially unbounded (since V ( ) is positive deinite and radially unbounded). Hence, U ( ) is a strong Lyapunov unction or (). According to item o Lemma. in the Appendix, U ( ) is also a strong Lyapunov unction or the system (), which completes the proo. Proo o Proposition.: Let all conditions o Proposition. be satisied. First, we show that or any x R n, with x 6= 0there exists an integer n = n(x) > 0 such that L n 6= 0. Consider an arbitrary x R n with x 6= 0. Since the system is assumed to be GAS, the solution (t; x) is deined or all t [0; ). For the purpose o showing contradiction suppose that L V ((t; x)) = 0 or all t [0; ). This implies that _ V ((t; x)) 0 and, hence, V ((t; x)) = > 0 or all t [0; ). Since V ( ) is positive deinite and radially unbounded, this contradicts the assumptions that the system is GAS. Hence, there exists t c (0; ) such that L V ((t c ;x)) 6= 0: (5) Since V ( ) and ( ) are analytic unctions, we have that L V ((t; x)) is an analytic unction o t and we can write L V ((t; x)) = i=0 tj j! 8t [0; ): (6) Now, i we assume that LV i (x) =0or all i N we have rom (6) with t = t c that L V ((t c;x)) = i=0 Li (tc=j!) j = 0, but this contradicts (5). Consequently, there must exist an integer n = n(x) such that L n 6= 0. Now, we show that or the given set E there exists ` N such that or any x E we have LV i (x) 6= 0or some i [;`]. Assume or the purpose o showing contradiction that there exists a sequence x p E and a strictly increasing sequence o positive integers n p such that LV i (x p )=0 8i [;n p 0 ] V (xp) 6= 0: (7) L n Since E is a compact set, there exists a subsequence o x p, that we still denote as x p and x E such that lim p! x p = x. From the irst part o the proo, we know that there exists an integer n = n(x ) such that L n V (x ) 6= 0. Since L n is a continuous unction o x, there exists >0 such that L n 6= 0 8x z : jx 0 zj g: (8) Since x p converges to x, there exists P N such that jx p 0x j or all p P and hence L n V (x p ) 6= 0 8p P : (9) The sequence n p is strictly increasing and such that lim np =: p! It ollows that there exists P N with P P such that L n V (x p )=0 8p P : (0) However, (9) and (0) yield a contradiction and this completes the proo. V. CONCLUSION We have presented a partial construction o a strong Lyapunov unction using a weak Lyapunov unction. The construction can be carried out i a mild Lie algebraic condition that involves the weak Lyapunov unction holds. Our construction would be useul or analysis o stability robustness or important classes o systems, such as passive nonlinear systems since the storage unction o the passive system typically turns out to be a weak Lyapunov unction. Also, using our result in design or redesign o controllers or systems such as the ones satisying Jurdjevic Quinn conditions is an interesting topic or urther research. APPENDIX TECHNICAL LEMMAS Lemma.: Let V : R n! R 0 be a positive deinite unction and : R 0! R >0 a strictly positive unction. Then, the ollowing hold. ) V ( ) is a weak Lyapunov unction or () i and only i it is a weak Lyapunov unction or the system (). ) V ( ) is a strong Lyapunov unction or () i and only i it is a strong Lyapunov unction or (). ) System () satisies condition ii) o Theorem. i and only i () satisies condition ii) o Theorem.. Proo: Note that _V j (x)(x) _V j () () Then the proo o the irst two statements o Lemma. ollows rom the ollowing observation. Since () is strictly positive or all x R n, we have that (@V=@x)(x)(x) is negative deinite (semideinite) i and only i ()(@V=@x)(x)(x) is negative deinite (semideinite). Now, we prove the third claim in Lemma.. By induction, one can show that or any i N there exist polynomials Q i : R i! R and P i : R i! R with P i (0; 0;...; 0) = Q i (0; 0;...; 0) = 0 and or all j [0;i] there exist continuous unctions i j : R n! R and j i : R n! R such that P i (x) :=() P i i (x)l ;...; i i(x)l i () = P i (x) () Q i(x) := () Li Q i (x)l i ;...;i(x)l i i (4) = Q i(x): (5)

5 00 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 49, NO. 6, JUNE 004 According to (), the property is satisied at the step i = 0. I one assumes that it is satisied at the step i, then direct calculations on () and (4) show that it is satisied at the step i. The next part o the proo consists in proceeding by contradiction. Suppose that there exists ` N such that: i) or any x R n, there exists an integer n = n(x) with n [;`] such that L n 6= 0; ii) there exists x c R n such that, or all m [;`];L m V (x c)=0. According to (4), or all i [0;l0 ] V (x c )= (V (x c )) Li V (x c ) Q i (x i c )L V (x c );...;i(x i c )L i V (x c ) : (6) Since or all m [;`];L m V (x c) = 0, it ollows that or all i [0;l0 ] V (x c )=0: (7) This contradicts the assumptions that or any x R n, there exists an integer n = n(x) with n [;`] such that L n 6= 0. This proves that i item ii) o Theorem. holds or (), then the same condition holds or (). The statement in the opposite direction is obtained in the same way by using (). Lemma.: Consider : R n! R n and a unction V : R n! R 0. Suppose that or some LN and all i [; L] there exist i : R 0! R >0 such that L i i () 8i [; L] 8x R n : (8) Then, given any c > 0 there exists a strictly positive nonincreasing unction : R 0! R >0 such (x) :=()(x) satisies L i c 8i [; L] 8x R n : (9) Proo: Let LN come rom the lemma. To simpliy notation we use V = in this proo. Note irst that given any unction and or any i [; L] there exists a polynomial G i : R i! R such that L i V (V ) = Gi L V;...;L i V; (V );...; (i0) (V ) (40) where (j) (V ) := (d j =dv j )(V ). This implies that there exist two polynomials 0 : R N! R and 0 : R L! R with positive coeicients and min0 (0; 0;...; 0); 0 (0; 0;...; 0)g > 0 such that, or all i [; L] and or all x R n L i V (V ) 0 jl V j;...; L L V 0 j(v )j;...; (L0) (V ) : (4) Since (8) holds, there exists a nondecreasing unction : R 0! R >0 such that such that or all i [; L] and or all x R n L i V (V ) (V ) 0 j(v )j;...; (L0) (V ) : (4) We denote := sup z [0;];...;z [0;]g 0 (z ;...;z L). Let c>0 be given and deine (v) = v v s s s s (s L) ds L ds (4) where later (9) holds. Note that the unction ( ) is positive and nondecreasing. It ollows that the unction s s (=(s L)) ds L is a nonnegative nonincreasing unction o s L such that s s (s ds L L) (s : L) It ollows by induction that ( ) is nonincreasing and such that (j) (v) 8j [0; L] 8v 0: (44) (v) So, in particular, since (0), then we have (j) (v) 8j [0; L] 8v 0: (45) It ollows rom (4), (44), (45) and the deinition o and that or all x R n we have L i V (V ) (V ) 0 j(v )j;...; (L0) (V ) which completes the proo. 0 j(v )j;...; (L0) (V ) c c (46) ACKNOWLEDGMENT The authors would like to thank J. B. Pomet or useul discussions. REFERENCES [] D. Angeli, Input-to-state stability o PD-controlled robotic systems, Automatica, vol. 5, pp , 999. [] L. Fauboug and J. B. Pomet, Control Lyapunov unctions or homogeneous Jurdjevic Quinn systems, ESAIM: Control, Optim., Calculus Variations, vol. 5, pp. 9, June 000. [] L. Grüne, E. D. Sontag, and F. Wirth, Asymptotic stability equals exponential stability, and ISS equals inite energy gain I you twist your eyes, Syst. Control Lett., vol. 8, pp. 7 4, 999. [4] A. Isidori, Nonlinear Control Systems II. London, U.K.: Springer-Verlag, 999. [5] V. Jurdjević and J. P. Quinn, Controllability and stability, J. Di. Equat., vol. 8, pp. 8 89, 978. [6] H. K. Khalil, Nonlinear Systems, nd ed. Upper Saddle River, NJ: Prentice-Hall, 996. [7] M. Krichman, E. D. Sontag, and Y. Wang, Input-output-to-state stability, SIAM J. Control Optim., vol. 9, pp , 00. [8] M. Krstić, I. Kanellakopoulos, and P. V. Kokotović, Nonlinear and Adaptive Control Design. New York: Wiley, 995. [9] J. Kurzweil, On the inversion o Liapunov s second theorem on stability o motion, AMS Trans., ser. II, vol. 4, pp. 9 77, 956. [0] J. P. La Salle, Some extensions o Lyapunov s second method, IRE Trans. Circuit Theory, vol. CT-7, no. 4, pp , Dec [] F. Mazenc and L. Praly, Adding an integrator and global asymptotic stabilization o eedorward systems, IEEE Trans. Automat. Contr., vol. 4, pp , Nov [] R. Outbib and J. C. Vivalda, On the stabilization o smooth nonlinear systems, IEEE Trans. Automat. Contr., vol. 44, pp. 00 0, Jan [] R. Sepulchre, M. Janković, and P. V. Kokotović, Constructive Nonlinear Control. New York: Springer-Verlag, 996. [4] E. D. Sontag, A universal construction o Artstein s theorem on nonlinear stabilization, Syst. Control Lett., vol., pp. 7, 989.

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