A converse Lyapunov theorem for, non-uniform in time, global exponential robust stability

Systems & Control Letters 44 (2001) 373–384

www.elsevier.com/locate/sysconle

A converse Lyapunov theorem for, non-uniform in time, global exponential robust stability J. Tsinias ∗ Department of Mathematics, National Technical University Athens, Zografou Campus, 15780 Athens, Greece Received 20 June 2000; received in revised form 26 May 2001

Abstract Lyapunov-like characterizations are established for the concepts of, non-uniform in time, global exponential robust c 2001 Elsevier Science B.V. All rights reserved. stability and input-to-state stability for time-varying control systems.
Keywords: Exponential robust stability; ISS; Non-uniform in time; Stabilization

1. Introduction Smooth Lyapunov functions play a central role in the analysis and design of non-linear control theory and several results have recently appeared in the literature o7ering the Lyapunov characterizations of various types of stability and Lyapunov-based approaches for feedback control design (see for instance [1–16] and references therein). The purpose of this paper is to establish a Lyapunov characterization of a concept of, non-uniform in time, global exponential stability of the origin for systems x˙ = f(t; x; u);

(t; x; u) ∈ R × Rn × Rk

(1.1)

and further to characterize in terms of Lyapunov functions an analogous notion of, non-uniform in time, input-to-state stability. 1.1. Motivation The present work is motivated by the backstepping analysis employed in [14,15] for the feedback stabilization problem concerning systems whose dynamics contain uncertainties and the tracking problem of unbounded signals. In both cases the desired feedback is time varying and exhibits asymptotic stability of the equilibrium of the resulting closed-loop system, which is, in general, non-uniform with respect to time. We brie?y illustrate the main idea in [14] by considering the feedback stabilization problem for autonomous ∗

Corresponding author. Tel.: +30-1-77-21-626; fax: +30-1-77-21-775. E-mail address: [email protected] (J. Tsinias).

c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter  PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 1 5 6 - 6

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systems: x˙ = f(x; v; );

(x; v; ) ∈ Rn × R × Rk ;

f(0; 0; ) = 0;

(1.2)

where v plays the role of control and (·) is a time-varying disturbance. We assume that the admissible set

of parameters (·) is described as follows: there exists a strictly increasing function p : R+ → R+ in such a way that for each ∈ it holds, (t) ∈ p(t)[ − 1; 1]k

(1.3)

for all t after some Gnite time T ¿ 0 depending on . For example, (1.3) occurs, if consists of all measurable (·) which are bounded over R+ . By virtue of (1.3), it suIces to exhibit stabilization, uniformly in w ∈ [ − 1; 1]k at the origin 0 ∈ Rn of the time-varying system: w ∈ [ − 1; 1]k

x˙ = f(x; v; p(t)w);

(1.4)

by means of a time-varying feedback v = ’(t; x). Let us, for example, consider the planar case x˙ = y + x;

y˙ = v;

(1.5)

where is any bounded unknown time-varying parameter. Instead of (1.5), we may consider the system x˙ = y + wp(t)x;

y˙ = v;

w(·) ∈ [ − 1; 1];

(1.6)

where p(t) := t 1=4 . We claim that the trajectories r(t) := (x(t; t0 ; x0 ; w); y(t; t0 ; x0 ; w)) of the closed loop (1.6) with v := − Et(y + tx)

(1.7)

for E ¿ 0 appropriately large, satisfy the estimation |r(t)| 6 K(t) exp(−t 2 + t02 )|r(t0 )|

∀t ¿ t0 ¿ 0; x0 ∈ Rn ; |w(·)| 6 1;

(1.8)

where K(·) is a polynomial expression of t. It should be emphasized here that (1.8) guarantees exponential stability, uniformly in w(·), but, in general, non-uniformly in t. To establish (1.8) we consider the Lyapunov candidate 1 V (t; x; y) := x2 + (y + tx)2 1 + t2 and estimate C1 |(x; y)|2 6 V (t; x; y) 6 C2 |(x; y)|2 (1.9) 1 + t2 for some constants C2 ; C1 ¿ 0. For the derivative V˙ (t; x; y; w) of V (·) along the trajectories of (1.6) with (1.7) we Gnd V˙ (t; x; y; w) 6 − C3 tV (t; x; y)

(1.10)

for certain C3 ¿ 0. The desired (1.8) is then a consequence of (1.9) and (1.10). Going back to the original system (1.5) it follows that for any bounded (·) the corresponding trajectory r(t) = (x(t); y(t)) of (1.5) with (1.7) satisGes |r(t)| 6 K(t) exp(−t 2 +t02 )|r(t0 )| for every initial r(t0 ) and for every t ¿ t0 ¿ T := inf {t ¿ 0 : | (s)| ¡ s1=4 ; s ¿ t}. Following a backstepping design approach the result above is extended in [14] for systems (1.2), whose dynamics have triangular structure. Particularly, for the system x˙i = xi+1 + gi ( ; x1 ; : : : ; xi ); i = 1; : : : ; n; (1.11a) x n+1 := u; (·) ∈ Rk ; whose dynamics gi (·) satisfy |gi ( ; ·)| 6 C(1 + t  )|(x1 ; : : : ; xi )|;

t ¿ 0 away from zero

(1.11b)

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for certain
C;  ¿ 0, it is possible to Gnd a feedback u = ’(t; x) and functions K; m : R+ → R+ with t K(t) exp(− 0 m(s) ds) → 0 as t → ∞, in such a way that for any for which (1.11b) holds, there is a time T ¿ 0 depending on such that the trajectory x(t; t0 ; x0 ; ) of the closed-loop system satisGes |x(t; t0 ; x0 ; )| 6 K(t) exp{−m(t) + m(t0 )}|x0 |

∀t ¿ T ;

x0 ∈ Rn :

(1.12)

To sum-up • the feedback stabilization problem for systems with uncertainties may be reduced to feedback stabilization for a time-varying system, thus, it turns out that the feedback stabilizer will be in general time dependent. Moreover, (1.12) shows that • the origin of the resulting closed-loop system x˙ = f(x; ’(t; x); p(t)w) will be asymptotically stable uniformly in w ∈ [ − 1; 1]k , but in general, non-uniformly with respect to time. The purpose of this paper is to establish a Lyapunov description of the, non-uniform in time, robust global exponential stability (RGES) as precisely presented by (1.12), for time-varying systems whose dynamics satisfy the restriction of growth with respect to state (Theorem 2.3). As a consequence of the main result we obtain a Lyapunov characterization of the, non-uniform in time, input-to-state exponential stability (expISS) (Proposition 3.2). The latter enables us to establish suIcient conditions for the expISS property for composite systems (Proposition 3.3). Notations: In the sequel, we denote by  the set of all C 1 functions ’ : R → R+ which satisGes t 1 6 ’(t) 6 t 2 for t ¿ 0 away from zero (namely, for all t ¿ 0 after some Gnite time) for certain constants 1 ; 2 ∈ R. By |x| we denote the usual Euclidean norm of a vector x ∈ Rn and by x
its transpose. The induced norm of a real matrix A ⊂ Rn×m is denoted by |A| := sup{|Ax|=|x|; x ∈ Rm \ {0}}. 2. Robust exponential stability We consider time-varying systems of the form (1.1), where f : R × Rn × Rk → Rn is C 0 and C 1 except possibly on R × {0} × Rk . The set U of admissible inputs u consists of all measurable functions of time taking values on a compact subset D of Rk . We further assume that there exists a function L of class  such that |f(t; x; u)| 6 L(t)|x|

∀(t; x; u) ∈ R × Rn × D:

(2.1)

For simplicity, we adopt the notation Xt;ut0 (x) for the solution of (1:1a) corresponding to the input u, initiated from x at time t0 . Condition (2.1), in conjunction with the regularity assumption for f(·), asserts that (1.1) is (both forward and backward) complete, namely, for every (t0 ; x) ∈ R × Rn and u ∈ U the solution Xt;ut0 (x) of (1.1) is unique and exists for all t ∈ R. Denition 2.1. We say that zero 0 ∈ Rn is robust globally exponentially stable with respect to (1.1) (RGES), if there exist a pair of functions K; m of class , constant  ¿ 0 and time t0∗ ¿ 0 such that t −1 6 m(t); ˙ |Xt;ut0 (x)| 6 K(m(t)) exp{−m(t) + m(t0 )}|x|

(2.2a) ∀t ¿ t0 ¿ t0∗ ; x ∈ Rn ; u ∈ U

(2.2b)

(m(·) ˙ denotes the derivative of m(·)). Remark 2.2. • From (2.2a) it follows that −1 (t  − t0 ) 6 m(t) − m(t0 ) and the latter in conjunction with our hypotheses, that both K and m are of class , assert by (2.2b) that Xt;ut0 (x) converges to zero with exponential rate of convergence.

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• “Robustness” in deGnition above means that estimation (2.2b) is valid uniformly in u ∈ U , however, as it was pointed out in the introduction, this type of asymptotic stability is in general non-uniform in t. When K(·) is bounded, then obviously RGES is equivalent to the robust uniform global exponential stability (RUGES) (see [13]), which is a special case of the familiar notion of robust uniform global asymptotic stability (RUGAS) (see [4]) and which guarantees asymptotic stability uniformly in both u ∈ U and t ∈ R+ . We next establish a Lyapunov characterization of RGES. Theorem 2.3. Suppose that there exists a C 1 function V : R+ × Rn → R+ being C ∞ except possibly on N KN of class ; a constant N ¿ 0 and time tN0 ¿ 0 such that R+ × {0}, a pair of functions m; N 6 m(t); N˙ t −1

(2.3a)

1 |x|2 6 V (t; x) 6 |x|2 ; N m(t)) K( N

(2.3b)

@V @V f(t; x; u) + 6 − m(t)V N˙ (t; x) V˙ (t; x; u) := @x @t

∀u ∈ D; x = 0 and t ¿ tN0 :

(2.3c) 1=2

Then zero is RGES; particularly; (2:2) holds with ; = ; N m := 12 mN and K(s) = KN (2s). Conversely; if zero is RGES with respect to (1:1); there exist functions V; KN and mN as above; satisfying (2:3): Proof. The direct part of proof is obvious. Indeed, by (2.3b) and (2.3c) we obtain 1 |Xt;ut0 (x)|2 6 V (t; Xt;ut0 (x)) 6 exp{−m(t) N + m(t N 0 )}|x|2 ; N K(m(t)) N which implies (2:2) with ; m(·) and K(·) as in statement of our theorem. N Assume now that zero is RGES and build functions V (·); K(·) and m(·) N satisfying (2.3a) – (2.3c). We distinguish two cases: Case 1: Condition (2:2) is fulGlled with m(t) = lt for some constant l ¿ 0, namely, assume |Xt;ut0 (x)| 6 K(t) exp{−l(t − t0 )}|x|

(2.4)

for some K ∈  and t ¿ t0 ¿ 0 away from zero. To simplify the analysis we may distinguish two subcases: Case 1A: The function K(·) satisGes K(t) 6 t & ;

t ¿ 0 away from zero

(2.5a)

for certain constant 0 ¡ & ¡ 1 such that 0 ¡ & ¡ 14 :

(2.5b)

Without any loss of generality, we may assume that (2.4) and (2.5a) hold for all t ¿ 0, both K(·); L(·) are increasing and min{K(t); L(t)} ¿ 1

∀t ¿ 0;

(2.6)

where L(·) is deGned in (2.1). We proceed to the construction of the desired Lyapunov function as follows. First, deGne  t+((t) |Xs;ut (x)|2 ds; (2.7a) '(t; x) := sup u∈U

((t) :=

t

1 log(At 2 + B) 2l

(2.7b)

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for arbitrary constants A ¿ 0 and B ¿ 1. By virtue of (2.4), '(·) is well deGned on R+ × Rn ; particularly, it holds that '(t; x) 6 ’(t)|x|2 ;

(2.8a)

’(t) := ((t)K 2 (t + ((t)):

(2.8b)

where To get an estimation from below, notice Grst that (2.1) is in conjunction with (2.4) and the fact that K; L are increasing implies  s u |Xs; t (x)| ¿ |x| − L(T )|XT;u t (x)| dT t

¿ |x|(1 − (t) L(t + (t)) K(t + (t))) ¿ 12 |x| (t) :=

∀x ∈ Rn ; u ∈ U; s ∈ [t; (t) + t];

1 : 2L(2t) K(2t)

(2.9a) (2.9b)

We remark that, because of (2:6), (t) 6 t and this establishes the last inequality in (2.9a). Using (2:9) and the fact that ((t) ¿ (t) for large t it follows that for any u0 ∈ U and for t away from zero we have  t+((t)  t+ (t) u0 2 2 '(t; x) ¿ |Xs; t (x)| ds ¿ |Xs;ut0 (x)|2 ds ¿ ’(t)|x| N ; (2.10a) t

t

1 1 ’(t) N := (t) = : 4 8L(2t) K(2t)

(2.10b)

We next show that ' is Lipschitz continuous in R+ × Rn . Using (2.1) and Gronwall’s inequality we get  t u |Xt; t0 (x)| 6 |x| exp L(s) ds ∀u ∈ U; x ∈ Rn ; t ¿ t0 (2.11a) t0

1

and, since f(·) is C (R × (Rn \ {0}) × Rk ), (2.11a) in conjunction with (2.1) asserts that for any interval [T1 ; T2 ] ⊂ R+ , compact region M ⊂ Rn \ {0} and T ¿ T2 , a constant C ¿ 0 can be found such that |Xt;ut2 (x2 ) − Xt;ut1 (x1 )| 6 C{|t2 − t1 | + |x2 − x1 |} ∀u ∈ U ; x1 ; x2 ∈ M ; t1 ; t2 ∈ [T1 ; T2 ]; max{t1 ; t2 } 6 t 6 T:

(2.11b)

(Note that version of (2.11b) has been established in [4, Proposition 5:1] for time-invariant systems, under the general hypothesis of completeness of solutions.) Now, let B be any compact subset of R+ × (Rn \ {0}) and let + be a strictly positive constant. Then deGnition (2:7) of ' implies  t2 +((t2 ) '(t2 ; x2 ) 6 |Xs;uˆt2 (x2 )|2 ds + + t2

for some uˆ ∈ U depending on (t2 ; x2 ), thus  t2 +((t2 )  '(t2 ; x2 ) − '(t1 ; x1 ) 6 |Xs;uˆt2 (x2 )|2 ds − t2

t1 +((t1 )

t1

|Xs;uˆt1 (x1 )|2 ds + +

for any (t1 ; x1 ); (t2 ; x2 ) ∈ B. By assuming that t2 ¿ t1 it follows  t2 +((t2 ) '(t2 ; x2 ) − '(t1 ; x1 ) 6 ||Xs;uˆt2 (x2 )|2 − |Xs;uˆt1 (x1 )|2 | ds t2

 +

t2

t1

|Xs;uˆt1 (x1 )|2 ds +



t2 +((t2 )

t1 +((t1 )

|Xs;uˆt1 (x1 )|2 ds + +

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and a similar estimation is obtained for the case t1 ¿ t2 . By virtue of (2.11a) and (2.11b) and the fact that ((·) is C 1 , the previous inequality guarantees: '(t2 ; x2 ) − '(t1 ; x1 ) 6 C{|t2 − t1 | + |x2 − x1 |} + + for certain constant C ¿ 0 and, since + is arbitrary, '(t2 ; x2 )−'(t1 ; x1 ) 6 C{|t2 −t1 |+|x2 −x1 |} for all (t1 ; x1 ); (t2 ; x2 ) ∈ B. Likewise, we obtain '(t1 ; x1 ) − '(t2 ; x2 ) 6 C{|t2 − t1 | + |x2 − x1 |}. This establishes that ' is Lipschitz continuous, thus, di7erentiable almost everywhere in R+ × Rn . ˙ x; u) of ' along a trajectory X u (x) of (1.1) corresponding to We next evaluate the time derivative '(t; t; t0 some input u ∈ U : 1 u ˙ x; u) a:e: '(t; = lim+ {'(t + h; Xt+h; t (x)) − '(t; x)}: h→0 h From deGnition (2:7) of '(·) it follows that for any h there corresponds an input uh (·) ∈ U in such a way that  t+h+((t+h) u u h '(t + h; Xt+h; (x)) 6 |Xs;ut+h (Xt+h (x))|2 ds + h2 ; (2.12a) t t+h

 '(t; x) ¿

t

 u h (s) :=

t+((t)

|Xs;uth (x)|2 ds;

u(s);

t 6 s 6 h + t;

uh (s);

h + t ¡ s:

(2.12b)

(2.12c)

uN h u h By taking into account (2.1) and (2.4) and the fact that Xs;ut+h (Xt+h; t (x)) = Xs; t (x) we get  t+h+((t+h)  t+h a:e: 1 h ˙ x; u) 6 lim 1 |Xs;ut+h (x)|2 ds − lim+ |Xs;uth (x)|2 ds '(t; h→0+ h t+((t) h→0 h t 2 ˙ 6 {(1 + ((t))K (t + ((t)) exp(−2l((t)) − 1}|x|2 :

The latter by virtue of (2:5) and (2.7b) implies that for almost all x = 0; u ∈ D and t ¿ 0 away from zero we have ˙ x; u) 6 1 |x|2 : '(t; (2.13) 2

In order to build a function, '(·) being C ∞ (R+ × (Rn \{0})) and satisfying (2:8), (2:10) and (2.13) we can apply Theorem B:1 in [4] by considering the time-extended system x˙ = f(t; x; u); t˙ = 1. Hence, we may assume in the sequel that '(·) is C ∞ (R+ × (Rn \{0})) and satisGes (2:8), (2:10), (2.13) with same ’ and ’. N Moreover, because of (2:8) and (2:10), ' is everywhere C 1 . We next establish that the function 1 V (t; x) := '(t; x) (2.14) ’(t) satisGes requirements (2.3b) and (2.3c). Indeed, from (2:8), (2:10) and (2.14) it follows that |x|2 6 V (t; x) 6 |x|2 (2.15a) ˆ K(t) with ˆ := 8K(2t) L(2t)((t) K 2 (t + ((t)): K(t) (2.15b) Finally, we estimate ’(t) ˙ 1 ˙ '(t; x; u) − 2 '(t; x): V˙ (t; x; u) = ’(t) ’ (t) Since ’˙ is non-negative, it follows by taking into account (2:5), (2.7b) and (2.16): 1 1 ˙ '(t; x; u) 6 − |x|2 V˙ (t; x; u) 6 ’(t) 2((t)K 2 (t + ((t)) 6−

1 2 |x| ; t-

t ¿ 0 away from zero

(2.16)

(2.17)

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for certain constant -: 0 ¡ - ¡ 1. DeGne m(t) N := 1=()t N N; N := 1 − - and pick KN ∈  such that N m(t)) ˆ K( N = K(t):

(2.18)

N It turns out from (2:15), (2.17) and (2.18) that (2.3a) – (2.3c) hold with m(·) N and K(·) as above. Case 1B: The constant & in (2.5a) satisGes & ¿ 14 :

(2.19)

This case is reduced to the previous one as follows. Let - be a constant with 0 ¡ - ¡ 14 &

(2.20)

and consider the map y = T (x) := |x|-=&−1 x:

(2.21)

Let Yt;ut0 (y) := T (Xt;ut0 (x)); y = T (x) with Xt;ut0 (x) being any trajectory of (1:1a). Then by (2.4), (2.20) and (2.21) it follows   lu N (2.22a) |Yt; t0 (y)| 6 K(t) exp − (t − t0 ) |y|; & where ˆ := K -=& (t) 6 t &ˆ; K(t)

t ¿ 0 is away from zero

(2.22b)

for some positive constant &: ˆ 0 ¡ &ˆ ¡ 14

(2.22c)

and further Yt;ut0 (·) is a solution of ˆ y; u) := |y|1−&=- f(t; |y|&=-−1 y) + y˙ = f(t;



 − 1 |y|−&=-−1 (y
f(t; |y|&=-−1 y; u))y: &

(2.23)

Obviously, the dynamics of (2.23) satisfy (2.1) and from (2.20) and (2.22a) we deduce that zero is RGES ˆ for (2.23). Moreover, the function K(·) satisGes (2.22b) and (2.22c), hence, the present case can be reduced N to the previous one. Particularly, a triple of functions V (·); K(·) and m(·) N can be determined in such a way that (2.3a) – (2.3c) hold for system (2.23). Then, it can be easily veriGed that the map Vˆ (t; x) := V &=- (t; T (x)) N := KN &=- (·) and m(·) N := (&=-)m(·). N satisGes all requirements (2.3a) – (2.3c) for the original system (1.1) with K(·) Case 2: The function m(·) in (2:2) is an arbitrary function of class . Without any loss of generality, assume that the derivative m˙ of m is everywhere strictly positive. The case is also reduced to Case 1 by making scaling of time t → m−1 (t). Particularly, let Yt;ut0 (y) := u Xm−1 (t); m−1 (t0 ) , (x); y = x. Obviously, Yt;ut0 (·) satisGes (2.4) and is a solution of ˆ y; u) := y˙ = f(t;

1

m(m ˙ −1 (t))

f(m−1 (t); y; u)

(2.24)

whose dynamics satisfy the restriction of growth (2.1). Thus, we may invoke the result of Case 1 to determine N a triple of functions V (·); K(·) and m(·) N and a constant N ¿ 0 such that (2.3a) – (2.3c) hold; speciGcally: |x|2 6 V (t; x) 6 |x|2 ; N m(t)) K( N

(2.25a)

@V ˆ @V 2 : V˙ (t; x; u)|(2:24) = f(t; x; u) + 6 − m(t)|x| N˙ @x @t

(2.25b)

Then, if we deGne Vˆ (t; x) := V (m(t); x), it follows from (2:25) that (2.3a) – (2.3c) are fulGlled for the original N system with Vˆ (·) and mN ◦ m, instead of V (·) and m, N respectively, and same K(·).

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Remark 2.3. For the case where RGES holds, with K(·) in (2.2b) being bounded, it is possible to construct N V (·) and K(·) in such a way that, in addition to (2.3a) – (2.3c), it holds that N m(t)) K( N = CL(t);

t away from zero

(2.26)

for some constant C ¿ 0. Indeed, pick A = 0 in (2.7b) and thus ( is appropriate large constant. Then, (2.26) is a consequence of (2:15) and (2.18). Remark 2.4. It should be noticed that inequality (2.3b) in the Lyapunov description (2:3) of RGES can be substituted by the following: 2 ˆ m(t))|x| N |x|2 6 V (t; x) 6 K(

(2.27)

for certain Kˆ ∈ . To be more precise, the Lyapunov characterization (2.3a) – (2.3c) of RGES is equivalent to N Kˆ ∈  such that (2.3a) and (2.3c) hold, as well as (2.27), the existence of functions V : R+ × Rn → R+ ; m; instead of (2.3b). To justify the claim, suppose that (2.3b) and (2.3c) hold and let Kˆ : R → R+ be of class  ˆ ¿ K(·) N ˆ 6 1 K(t) ˆ ˆ m(t))V such that K(·) and (d=dt)K(t) for t ¿ 0 away from zero. DeGne W := K( N (t; x). Then 2 ˆ ˆ m(t)) by virtue of (2.3a) – (2.3c) inequality (2.27) holds with V := W and K as above and further W˙ 6 K( N V˙ + 1 ˙ 1 ˙ ˆ ˙ ˙ (d=dt)K(m(t)) N m(t)V N 6− 2 m(t)W N , thus (2.3b) is fulGlled as well with V := W and 2 m, N instead of m. N Analogous discussion can be applied for proof of the converse statement above. Note Gnally that the presence of the time varying function in Grst (second) inequality in (2.3b) ((2.27), respectively) is related to the lack of uniformity in time in convergence of solution. The only reason we adopt here (2.3b), instead of (2.27), is just to keep the same type of Lyapunov characterizations of RGES and the notion of input-to-state stability with those used in [15,14]. 3. The notion of non-uniform expISS This section is devoted to a concept of, non-uniform in time, exponential input-to-state stability and its Lyapunov characterization. Let us Grst begin with case (1.1), where, instead of (2.1), the following holds: |f(t; x; 0) − f(t; x; u)| 6 |u|L(t)

∀(t; x) ∈ R × Rn ; u ∈ Rk :

(3.1)

Let us further assume that zero 0 ∈ Rn is globally exponentially stable with respect to x˙ = f(t; x; 0);

(3.2) 1

+

n

namely, (2:2) hold with u = 0. Then according to Theorem 2.3 there exist a map V (·) ∈ C (R × R ) and N mN ∈  such that (2.3a) – (2.3c) are fulGlled with respect to (3.2). Particularly, we have functions K; @V @V f(t; x; 0) + 6 − m(t)V: N˙ (3.3) @x @t Let us in addition assume that     @V  (t; x) 6 0(t)|x| ∀(t; x) ∈ R+ × Rn (3.4)   @x for certain 0 ∈ . Claim. Existence of a C 1 function V satisfying (2.3b) and (2.3c) as well as (3.4) is guaranteed, if we strengthen (2.2b) by imposing    @  Xt; t0 (x) 6 K(m(t)) exp{−m(t) + m(t0 )}; x ∈ Rn ; t ¿ t0 (3.5)   @x for the solution Xt; t0 (x) of (3:2). (This for example occurs if (2.2b) holds and f(t; x; 0) is linear with respect to x.) To establish the existence of such a V we just repeat the same arguments with those used in the proof of Theorem 2.3, except those

J. Tsinias / Systems & Control Letters 44 (2001) 373–384

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concerning the ‘smoothing part’ of proof based on [4]. For instance, for Case 1A (see (2.4) and (2:5)) we
t+((t) may deGne V (t; x) := t |Xs; t (x)|2 ds and this, by virtue of (3.5), satisGes all desired properties (2.3b) and (2.3c) and (3.4) as well. Details are left to the reader. From (2.3b), (3.3) and the additional hypothesis (3.4) we Gnd    @V  @V @V @V m(t) N˙ @V f(t; x; u) + 6 f(t; x; 0) + + |u|L(t)   6 − V (t; x) (3.6a) V˙ (t; x; u)|(1:1) = @x @t @x @t @x 2 for t ¿ 0 away from zero and for u(·) satisfying m(t) N˙ |u(t)| 6 |x|; (3.6b) N m(t)) 2L(t)0(t)K( N which by virtue of (2.3b) implies that (2.2b) holds for every input u(·) and time t for which (3.6b) is fulGlled. The previous analysis leads to the following deGnition. Denition 3.1. We say that (1.1) satisGes the (non-uniform) exponential input-to-state stability property (expISS), if there exist functions K(·); m(·) and 1(·) of class  such that (2.2a) holds and (2.2b) is satisGed for those inputs u(·) and t ¿ t0 ¿ 0 for which |u(t)| 6 1(t)|Xt;ut0 (x)|;

t ¿ t0

and further the corresponding trajectory

(3.7) Xt;ut0 (x)

is deGned for all t.

The following proposition is a direct consequence of Theorem 2.3 and provides the Lyapunov characterization for expISS. Its proof is quite analogous to this given in [7,11] for the uniform ISS property. Proposition 3.2. Consider system (1:1) and assume that its dynamics are C 1 (R+ × (Rn \{0}) × Rk ) and in such a way that |f(t; x; u)| 6 L(t)(|x| + |u|) ∀(t; x; u) ∈ R × Rn × Rk

(3.8)

for certain L ∈ . Then the following statements are equivalent: (i) System (1:1) satis:es the expISS. (ii) The origin 0 ∈ Rn is RGES for the system; x˙ = f(t; x; 1(t)|x|u);

u ∈ [ − 1; 1]k :

N and (iii) There exist a Lyapunov function V ∈ C ∞ (R+ × (Rn \{0})) being everywhere C 1 ; functions K(·) m(·) N of class ; a constant aN ¿ 0 and time tN0 ¿ 0 such that (2:3a) and (2:3b) hold for t ¿ tN0 and further N˙ (t; x) V˙ (t; x; u)|(1:1) 6 − m(t)V for those t ¿ tN0 and inputs u(·) for which |u(t)| 6 1(t)|x|

∀t ¿ tN0 :

Proof. Implications (ii) ⇔ (i) and (iii) ⇒ (i) are straightforward and (ii) ⇔ (iii) follows from Theorem 2.3. Details are left to the reader. We conclude this section by establishing the following proposition that constitutes a version of a well-known result (see [3]) concerning uniform ISS for interconnected systems. Proposition 3.3. Consider the pair of systems 31 : x˙ = f(t; x; u); 32 : y˙ = g(t; x; y; u); x ∈ Rn 1 ;

y ∈ R n2 ;

u ∈ Rk :

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Assume that the dynamics f(·) and g(·) satisfy the same hypothesis (3:8) with this imposed in Proposition 3:2; with u and (x; u) as inputs; respectively. Suppose that: (H1) The subsystem 31 satis:es the expISS from the input u; namely |Xt;ut0 (x)| 6 K1 (t)exp{−m1 (t) + m1 (t0 )}|x| ∀|u(t)| 6 11 (t)|Xt;ut0 (x)|;

t ¿ t0

(3.9)

for some K1 ; m1 ; 11 ∈  such that (2:2a) holds for m = m1 ; (H2) The subsystem 32 satis:es the expISS from the input (x; u); i:e: |Yt;x;t0u (y)| 6 K2 (t)exp{−m2 (t) + m2 (t0 )}|y|

∀|(u(t); x(t))| 6 12 (t)|Yt;x;t0u (y)|;

t ¿ t0

(3.10)

for certain K2 ; m2 ; 12 ∈  such that (2:2a) holds for m = m2 . Under the previous assumptions the overall system 31 ; 32 satis:es the expISS from the input u. Proof. We proceed among the same lines in [13, Proposition 2:2]. From hypothesis (H1) and (H2) it follows by virtue of Proposition 3.2(iii) that there exist a pair of C 1 maps V (t; x) and W (t; y), functions KN i ; mN i ; i = 1; 2 of class  and constants ai ¿ 0; i = 1; 2 such that each mi satisGes (2.3a) with N = i and the following hold for every non-zero x; y and t ¿ 0 away from zero: 1 |x|2 6 V (t; x) 6 |x|2 ; (3.11a) KN 1 (mN 1 (t)) 1 |y|2 6 W (t; y) 6 |y|2 ; (3.11b) N K 2 (mN 2 (t)) V˙ (t; x; u)|31 6 − mN˙ 1 V (t; x)

for |u(t)| 6 11 (t)V 1=2 (t; x);

W˙ (t; y; x; u)|32 6 − mN˙ 2 W (t; y)

for |(u(t); x(t))| 6 12 (t)W 1=2 (t; y):

From (3.12a) and (3.12b) it follows that for every x0 ; y0 and t0 away from zero:

V 1=2 (t; Xt;ut0 (x0 )) 6 exp − 12 (mN 1 (t) − mN 1 (t0 )) V 1=2 (t0 ; x0 );

(3.12a) (3.12b)

(3.13a)

for |u(t)| 6 11 (t)V 1=2 (t; Xt;ut0 (x0 ));

t ¿ t0 ;

(3.13b)



W 1=2 (t; Yt;x;t0u (y0 )) 6 exp − 12 (mN 2 (t) − mN 2 (t0 )) W 1=2 (t0 ; y0 )

(3.14a)

|(u(t); x(t))| 6 12 (t)W 1=2 (t; Yt;x;t0u (y0 ));

(3.14b)

for t ¿ t0 ;

where for simplicity we have denoted (x; u) := (x(t); u(t)). We distinguish two cases: Case 1: We make the additional hypothesis 4KN 1 (mN 1 (t)) 6 121 (t);

t ¿ 0 away from zero

(3.15)

and let us assume in the sequel that without any loss of generality m := mN 1 = mN 2 and 1 := 11 = 12 . DeGne S1 := {(t; x; y):V (t; x) ¿ W (t; y)};

(3.16a)

S2 := R+ × Rn1 × Rn2 \ S1 ;

(3.16b)

6(t; x; y) := 1 max

1

2V

1=2

; W 1=2 − 12 V 1=2 ;

(3.16c)

J. Tsinias / Systems & Control Letters 44 (2001) 373–384

 '(t; x; y) :=

V 1=2 ;

(t; x; y) ∈ S1 ;

W 1=2 ;

(t; x; y) ∈ S2 :

383

(3.16d)

From (3:11) and deGnitions (3.16a) – (3.16d) it follows that there exist functions 7; K ∈  such that 7(t)'(t; x; y) 6 6(t; x; y);

(3.17a)

|(x; y)| 6 '(t; x; y) 6 |(x; y)|: K(m(t))

(3.17b)

By taking into account (3:16) and (3:17) we obtain |u| 6 7'; (t; x; y) ∈ S1 ⇒ |u| 6 12 1V 1=2 ;

(3.18a)

moreover, using hypothesis (3.15) and (3.11a) we have |u| 6 7'; (t; x; y) ∈ S2 ⇒ |u| 6 1W 1=2 − 12 1V 1=2 ⇒ |(u; x)| 6 1W 1=2 − 12 1V 1=2 + |x| (3:11a); (3:15)

6

1W 1=2 :

(3.18b)

It turns out from (3:18) that each trajectory r(t) := (Xt;ut0 (·); Yt;ut0 (·)) of the composite system 31 ; 32 satisGes |u(t)| 6 7(t)8(t; r(t));

(t; r(t)) ∈ S1 ;

t ¿ t0 ⇒ (3:13a);

(3.19a)

|u(t)| 6 7(t)8(t; r(t));

(t; r(t)) ∈ S2 ;

t ¿ t0 ⇒ (3:14a):

(3.19b)

Combining (3.19a) and (3.19b) and recalling (3.16d) we conclude:

8(t; r(t)) 6 exp − 12 (m(t) − m(t0 )) 8(t0 ; r(t0 )) for |u(t)| 6 7(t)8(t; r(t));

t ¿ t0 away from zero:

The latter in conjunction with (3.17b) implies expISS. Case 2: We complete the proof by establishing expISS for 31 ; 32 from u under the absence of (3.15). Without loss of generality, we again may assume that 11 = 12 . We apply the transformation (t; x; y) → (t; (t)x; -(t)y) for certain decreasing C 1 functions (·); -(·) of class  such that      -(t)   (t) ˙  ˙   ; -(t); (3.20) (t);    ∈ (0; 1); t away from zero  -(t)  (t)  yet to be speciGed. The composite system 31 ; 32 becomes    x ˙ -˙ x y 3N 1 : x˙ = x + f t; ; u 3N 2 : y˙ = y + -g t; ; ; u    and consider Grst the subsystem  ˆ x; u) := ˙ x + f t; x ; 11 |x|u ; x˙ = f(t;   

u ∈ [ − 1; 1]k :

(3.21)

It follows from hypotheses (3.8), (3.20) and (3.21) that for any (·) as above the map fˆ satisGes ˆ x; u)| 6 L(t)|x|; |f(t;

∀x ∈ Rn ; u ∈ [ − 1; 1]k ; t away from zero

for certain L ∈  being independent of 

(3.22)

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Moreover, since (·) is decreasing, zero is RGES for (3.21); equivalently (by virtue of Proposition 3.2), (3.9) holds for 3N 1 with same K1 ; m1 and 11 = instead of 11 . For simplicity, assume that m1 (t) ≡ t, take into account (2.15b), (2.18) in proof of Theorem 2.3 and the fact that zero is RGES for (3.21), in order to construct a C ∞ map V (t; x) and functions KN 1 ; mN 1 ;  ∈  in such a way that (2.3a), (3.11a), (3.12a) and (3.20) hold for 3N 1 with 1N1 := 11 = instead of Y1 , and simultaneously 4KN 1 (mN 1 (t)) 6 1N21 (t);

t ¿ 0 away from zero

(3.23)

Note, that the latter is feasible by virtue of (2.15b), (2.18) and (3.22). It turns out that, for appropriate (·) ∈ , both (H1) and (3.23) are fulGlled for 3N 1 and, because of (3.20), 3N 2 satisGes (H2). Particularly, by picking -(·) = 12 2 (·) and taking into account that 11 = 12 it can be easily veriGed as previously that (3.10) holds for 3N 2 with the same K2 ; m2 and 12 = = 11 =, instead of 12 . We then apply the analysis made in Case 1 for the composite system 3N 1 ; 3N 2 and the proof is completed by noticing that expISS holds for 3N 1 ; 3N 2 with u as input, if and only if this property is fulGlled for the original composite system 3N 1 ; 3N 2 . References [1] A. Bacciotti, L. Rosier, Lyapunov and Lagrange stability: inverse theorems for discontinuous systems, Math. Control Signals Systems 11 (1998) 101–125. [2] A. Bacciotti, L. Rosier, On the converse of Grst Lyapunov theorem: the regularity issue, Systems Control Lett. 41 (2000) 265 –270. [3] Z.P. Jiang, A.R. Teel, L. Praly, Small gain theorem for ISS systems and applications, Math. Control, Signals Systems 7 (2) (1995) 95–120. [4] Y. Lin, E.D. Sontag, Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim. 34 (1996) 124–160. [5] L. Rosier, Smooth Lyapunov functions for discontinuous stable systems, Set-Valued Anal. 7 (1999) 375 – 405. [6] E.D. Sontag, H.J. Sussmann, Nonsmooth control—Lyapunov functions, Proceedings of the IEEE Conference Decision and Control, New Orleans, IEEE Publications, New York, 1995, pp. 2799 –2805. [7] E.D. Sontag, Y. Wang, On characterizations of the input-to-state stability property, Systems Control Lett. 24 (1995) 351–359. [8] E.D. Sontag, Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automat. Control 41 (1996) 1283–1294. [9] E.D. Sontag, Y. Wang, Lyapunov characterizations of input to output stability, SIAM J. Control Optim. 39 (2001) 226 –249. [10] A.R. Teel, L. Praly, A smooth Lyapunov function from a class-KL estimate involving two positive semideGnite functions, COCV-ESAIM, 5 (2000) 313–367. [11] J. Tsinias, Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation, COCV-ESAIM 2 (1997) 57–85. [12] J. Tsinias, Stochastic input-to-state stability and applications to global feedback stabilization, Int. J. Control 71 (5) (1999) 907–931. [13] J. Tsinias, The concept of exponential input to state stability for stochastic systems and applications to feedback stabilization, Systems Control Lett. 36 (3) (1999) 221–230. [14] J. Tsinias, Backstepping design for time-varying nonlinear systems with unknown parameters, Systems Control Lett. 39 (2000) 219–227. [15] J. Tsinias, I. Karafyllis, ISS property for time-varying systems and applications to partial-static feedback stabilization and asymptotic tracking, IEEE Trans. Automat. Control 44 (11) (1999) 2179–2184. [16] J. Tsinias, J. Spiliotis, A converse Lyapunov theorem for robust exponential stochastic stability, Lectures Notes in Control and Information Sciences, Vol. 246, Springer, Berlin, 1999, pp. 354 –374.