Integral Input to State Stable systems in cascade

Systems & Control Letters 57 (2008) 519–527

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Integral Input to State Stable systems in cascade Antoine Chaillet a,∗ , David Angeli b a LSS - SUPELEC - Univ. Paris Sud - EECI. 3, rue Joliot Curie, 91192 - Gif s/Yvette, France b Department of Electrical and Electronic Engineering, Imperial College, SW11 2AZ - London, UK

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Article history: Received 7 October 2005 Received in revised form 21 November 2007 Accepted 17 December 2007

Keywords: Stability analysis Robustness analysis Nonlinear systems Cascades

a b s t r a c t The Integral Input to State Stability (iISS) property is studied in the context of nonlinear time-invariant systems in cascade. Some sufficient conditions for the preservation of the iISS property under a cascade interconnection are presented. These are first given as growth restrictions on the supply functions of the storage function associated with each subsystem and are then expressed as solutions-based requirements. A Lyapunov-based condition guaranteeing that the cascade composed of an iISS system driven by a Globally Asymptotically Stable (GAS) one remains GAS is also provided. We also show that some of these results extend to cascades composed of more than two subsystems. © 2008 Elsevier B.V. All rights reserved.

1. Introduction The stability of nonlinear cascaded systems has been extensively studied in recent years leading to important progress in both analysis and control of nonlinear dynamical systems. For instance, some powerful methods, such as backstepping or forwarding, provide explicit stabilizing feedback based on a cascaded decomposition of the system under consideration. A lot of work also aims at studying the invariance of some stability properties under the cascade interconnection. Even though the Global Asymptotic Stability (GAS) property is not preserved in general by cascade structures [16,14], it is now a well known fact that the cascade composed of two GAS systems is itself GAS, provided that its solutions are bounded. This fact was originally established in [11,10] and later extended to time-varying systems in [9] where some sufficient conditions for actually checking (uniform) boundedness of solutions are also provided. A related approach involves the notion of Input to State Stability (ISS). This property, first introduced in [12], links the norm of the current state to the supremum of the applied input, through a nonlinear inequality which also takes into account a fading term due to initial conditions. This way of formulating external and internal stability notions appears to be a particularly well adapted tool for analyzing cascades, and it is a natural consequence of the ISS definition that the property is preserved under cascade

∗ Corresponding author. E-mail addresses: [email protected] (A. Chaillet), [email protected], [email protected] (D. Angeli). 0167-6911/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.12.010

interconnections (see for instance [15]). Since ISS implies GAS when the input is identically zero (we refer to this property as 0GAS), it follows that the cascade composed of an ISS subsystem driven by a GAS one is itself GAS. Even though this fact has been widely used both in analysis and design, ISS happens to be too strong a requirement in several cases. This has motivated the introduction of Integral Input to State Stability (iISS) [13], which turns out to be a much weaker property. Instead of linking the state to the supremum of the input, it involves a measure of the energy that the latter feeds into the system. Similarly to ISS, it ensures GAS for the zeroinput system and guarantees some robustness to the system. In addition, though more conservative than mere 0-GAS plus forward completeness (see [2]), it happens to hold very often in practice for the subsystems involved in cascades. In this respect, it is of interest to know whether properties that are similar to those that hold in the ISS case are actually true for iISS systems. For instance: (1) Is the cascade of two iISS systems iISS ? (2) Is the cascade of an iISS system driven by a GAS system GAS ? A counter-example shows that the answer to the second question is negative in general. In [3], an additional condition is indeed required to restrict the iISS gain of the driven subsystem in relation to the convergence rate of the state trajectories generated by the driving subsystem. Roughly speaking, the decay rate of the perturbing input for the iISS driven subsystem has to be strong enough with respect to the iISS gain. Based on a similar rationale, sufficient conditions for Uniform Global Asymptotic Stability (UGAS) of time-varying systems were established originally in [8]; more precisely, in that paper the authors proved UGAS of a cascade of time-varying systems under

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520

some growth-order restrictions on the interconnection term and, most importantly, the condition that the state trajectories of the driving system are uniformly globally integrable. The latter is a decay-rate restriction that holds, in particular, for ULES systems (uniformly locally exponentially stable). In the framework of [3], LES plus local Lipschitz continuity of the iISS gain is sufficient to satisfy the required integrability assumption. Thus, while the second question posed above has been studied and partially answered, the first question, to the best of our knowledge, remained open with the notable exception of [4–6] who provide a result of this nature but implicitly require that the dissipation rate of the driving subsystem be nondecreasing. The previously cited counter-example gives a negative answer in general. In this paper, we provide additional conditions which are sufficient for the preservation of iISS under cascade interconnection. These conditions are first given in the case when an explicit iISS Lyapunov-like function is known for each of the two subsystems. Roughly, it suffices that the dissipation term of the driving subsystem dominates the supply function of the driven one in the neighbourhood of the origin. The second step consists in stating this condition in terms of the estimates of the trajectories of the two subsystems when disconnected. More precisely, in the case of a continuously differentiable zero-input driving subsystem, we impose, similarly to [3], the condition that the driven subsystem presents a locally Lipschitz iISS gain, and that the driving one be 0-LES. In addition, we complete the main result in [3] by giving a sufficient condition for the cascade composed of an iISS driven by a GAS one to remain GAS in the case when explicit Lyapunov functions are known. Roughly, it is again required that the dissipation term of the GAS subsystem dominates the supply function of the iISS one around zero. This result may be useful in practice since the iISS and GAS properties are commonly established through Lyapunov arguments. Furthermore, this result naturally extends to multiple cascaded systems, i.e. series of cascaded iISS systems driven by a GAS one. The rest of the paper is organized as follows. We start by recalling some notions related to the concept of iISS. Then we present our main results in Section 3 together with illustrative examples. Sections 4 and 5 are respectively dedicated to the proofs of the main results and the technical lemmas. 2. Preliminary definitions Notation: P D denotes the class of all continuous positive definite functions R≥0 → R≥0 . K designates the set of all continuous increasing functions R≥0 → R≥0 that vanish at 0. A function is said to belong to class K∞ if it is of class K and tends to infinity with its argument. L is the class of all continuous decreasing functions R≥0 → R≥0 that tend to zero when their argument tends to infinity. A function is said to be in class KL if it is of class K in the first argument and of class L in the second argument. Given a positive ε, Bε denotes the open ball of radius ε in the Euclidean space of appropriate dimension. Let a ∈ {0, +∞} and q1 and q2 be class K functions. We say that q1 dominates q2 in a neighbourhood of a (and we write q2 (s) = O (q1 (s))) if there exists a nonnegative constant k such that lim sups→a q2 (s)/q1 (s) ≤ k. We say that q1 strictly dominates q2 (notation: q2 (s) = o(q1 (s))) if k can be taken as 0, and that q1 is equivalent to q2 (i.e., q1 (s) ∼ q2 (s)) if lims→a q2 (s)/q1 (s) = 1. We start by recalling the definition of the Integral Input to State Stability (iISS) as well as some related notions. Consider the system x˙ = f (x, u),

measurable locally essentially bounded functions. We say that (1) is iISS if there exist functions β ∈ KL and γ, µ ∈ K such that, for all ξ ∈ Rn and all u, its solution x(·, ξ, u) satisfies |x(t, ξ, u)| ≤  R β(|ξ| , t) + γ 0t µ(|u(τ)|)dτ . The function µ is then referred to as an iISS gain for (1). The system (1) is said to be 0-GAS (resp. 0-LES) if

x˙ = f (x, 0) is GAS (resp. LES). It is said to satisfy the Bounded Energy

Frequently Bounded State (BEFBS) property if there exists σ ∈ K∞ R∞ such that, if 0 σ(|u(τ)|)dτ < ∞ then lim inf t→+∞ |x(t, ξ, u)| < ∞. It was established in [2] that a necessary and sufficient condition for a system like (1) to be iISS is that there exist a positive definite radially unbounded continuously differentiable function V , a K function γ and a P D function α satisfying, for all x ∈ Rn and all u ∈ Rm , ∂∂Vx (x)f (x, u) ≤ −α(|x|) + γ(|u|). 3. Main results 3.1. Lyapunov-based condition for cascades “GAS + iISS” Our first result studies the cascade composed of a GAS subsystem driving an iISS one. It establishes a sufficient condition for it to be GAS. It can be seen as a natural counter-part of [3, Theorem 1] for the case when the GAS of the driving subsystem and the iISS of the driven one are not established through an explicit estimate of their solutions, but instead in terms of Lyapunov functions. Given locally Lipschitz functions f and g, the considered system is

x˙ = f (x, z)

(2a)

z˙ = g(z),

(2b)

where x ∈ Rnx , z ∈ Rnz . We assume that f (0, 0) = 0 and g(0) = 0. Theorem 1. Assume that the origin of (2b) is globally attractive1 and that there exist a constant ε > 0 and two continuous positive definite radially unbounded functions V and W , differentiable on Rnx and Bε \ {0} respectively, and satisfying

∂V (x)f (x, z) ≤ −αx (|x|) + γ(|z|), ∀x ∈ Rnx (3) ∂x ∂W (z)g(z) ≤ −αz (|z|), ∀z ∈ Bε \ {0}, (4) ∂z where αx , αz ∈ P D and γ ∈ K . Then, under the condition that γ(s) = O (αz (s)) in a neighbourhood of zero, the cascade (2) is GAS. The proof is given in Section 4.1. Note that the combination of global attractivity and (4) is actually equivalent to GAS. This statement is motivated by an easier applicability in practice and the possibility it offers for generalizing to multiple cascades: see Theorem 4. While authorizing a slight additional flexibility, as shown in the sequel, the fact of not requiring the differentiability of W at zero is motivated by homogeneity concerns with the trajectorybased result presented in Theorem 9. The Lyapunov-like function constructed for the proof of the latter is indeed possibly nondifferentiable at the origin. A similar result was recently given in [4–6]. In those references, a small gain condition for iISS systems is provided and, as a particular case, cascades composed of an iISS system driven by a GAS one are considered. The condition imposed there, however, involves the K∞ bounds on V and is expressed as a dominance condition on the whole R≥0 , whereas ours needs to hold only in the neighbourhood of zero. Also, in [4–6], it is implicitly required that the dissipation rate α1 of the driving subsystem be of class K .

(1) n

n

m

n

where x ∈ R is the state and f : R × R → R is a locally Lipschitz function. Input signals u : R≥0 → Rm , as usual, are

1 Meaning that the solution of (2b) tends to the origin from any initial state.

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

Although the results proposed in those references apply to a more general context (feedback interconnection), these two features make our condition less conservative and easier to apply. Remark 2. It is worth mentioning that, if an upper bound on W of the form W (z) ≤ α(|z|) is explicitly known, where α designates a K∞ function, the condition in Theorem 1 (namely γ(s) = O (αz (s)) as s → 0) can be considerably relaxed. More precisely, it suffices that there exists p ∈ [0, 1) such that   αz (s) γ(s) = O (5) , αz (s) = o(α(s)p ), as s → 0. p

α(s)

Indeed, consider the function W (·) := W (·)(1−p) . Then W is a positive definite function, differentiable on Rn \{0}, and we get from (4) that

∂W (z)g(z) ≤ −(1 − p)αz (|z|)W −p (z) ∂z αz (|z|) ≤ −(1 − p) =: −α˜ z (|z|). α(|z|)p

521

1, are iISS, provided growth conditions on the supply functions involved in their Lyapunov characterizations. The proof requires a particular behaviour of the involved supply functions in the neighbourhood of zero. Namely, we will assume that, for some ε > 0, the involved supply rates satisfy λ α(a + b) ≤ α(a) + α(b) for all a, b ∈ (0, ε), where λ > 0 denotes a constant. The set of all such functions α is referred to as Iε . Theorem 4. Assume that the origin of (6.N) is globally attractive and that there exists a constant ε > 0 and, for each i = 1, . . . , N, a continuous positive definite radially unbounded function Vi , differentiable on Rni , satisfying

∂Vi fi (xi , xi+1→N ) ≤ −αi (|xi |) + γi (|xi+1→N |), ∀xi ∈ Rni ∂x i ∂VN (xN )fN (xN ) ≤ −αN (|xN |), ∀xN ∈ Bε \ {0}, ∂x N where, for each i = 1, . . . , N, αi ∈ P D ∩ Iε and, for each i = 1, . . . , N − 1, γi ∈ K ∩ Iε. Then, under the condition that

˜ z ∈ P D . Hence Theorem 1 applies with W , and In view of (5), α establishes that (2) is GAS. In this respect, notice that allowing W to be non-differentiable at the origin is useful, as further illustrated by the following example.

γi (s) = O (αi+1 (s)) , ∀i = 1, . . . , N − 1 γi (s) = O (γi+1 (s)) , ∀i = 1, . . . , N − 2,

Example 3. Consider the following two-dimensional cascaded system, consisting of a particular case of [3, Example 4]:

Remark 5. Although not stated explicitly for the sake of compactness, VN is only required to be differentiable over Bε \ {0}.

x˙ = −sat(x) + xz z z˙ = − , 1 + z2

The proof of the above result is presented in Section 4.2. Note that the requirement that all supply functions be in the class Iε is a little conservative as it needs to hold only locally. For instance, it is satisfied by any function with polynomial behaviour around zero.

where sat(r) := sign(r) min{1, |r|} for all r ∈ R. Direct computations show that the functions V (x) = 21 ln(1 + x2 ) and W (z) = 12 z2 satisfy (3) and (4) with αx (s) = (s sat s)/(1 + s2 ), γ(s) = s and αz (s) = s2 /(1 + s2 ). Since the requirement γ(s) = O (αz (s)) as s tends to zero does not hold, it is not possible to apply Theorem 1 directly. Nevertheless, it is possible to conclude using the previous remark with p = 1/2. Indeed, take α(|z|) = 21 |z|2 ≥

√

p W (z), then α(s)√ = s/ 2 strictly dominates αz (s) around zero, and p αz (s)/α(s) = s 2/(1 + s2 ) dominates γ(s).

We stress that, although the present use of Lyapunov-based conditions simplifies the argument, the above example can also be addressed by existing methods such as [3,9]. 3.2. Lyapunov-based conditions for GAS of multiple cascades As already evoked, the previous result extends to multiple cascades, i.e. x˙ 1 = f1 (x1 , x2→N ) x˙ 2 = f2 (x2 , x3→N )

.. .

(6)

x˙ N−1 = fN−1 (xN−1 , xN ) x˙ N = fN (xN ),

where we used the notation xi→j to denote (xTi , . . . , xTj )T ∈ Rni +···+nj for all integers 0 ≤ i ≤ j. The functions fi : Rni × Rni +···N → Rni are assumed to be locally Lipschitz and to vanish at the origin. The following result establishes GAS of (6) by assuming that2 (6.N) is itself GAS and that the subsystems (6.i), i = 1, . . . , N −

2 We use the notation (6.i) to denote the system x˙ = f (x , x i i i i+1→N ).

(7a) (7b)

as s tends to zero, the cascade (6) is GAS.

3.3. Lyapunov-based condition for cascades of iISS systems The next result studies the cascade connection of two iISS systems, in the case when an iISS-Lyapunov function is explicitly known for each of them. For the sake of generality, it is allowed that the driven subsystem depends also on the input of the driving one. We therefore deal with systems of the form: x˙ 1 = f1 (x1 , x2 , u)

(8a)

x˙ 2 = f2 (x2 , u),

(8b)

where x1 ∈ Rn1 , x2 ∈ Rn2 , u : R≥0 → Rm is a measurable locally essentially bounded function, f1 and f2 are locally Lipschitz and satisfy f1 (0, 0, 0) = 0 and f2 (0, 0) = 0. Theorem 6. Let V1 and V2 be continuous positive definite radially unbounded functions, differentiable on Rn1 and Rn2 \ {0} respectively. Suppose that there exist a K function ν1 and, for all i ∈ {1, 2}, a P D function αi and a K function γi such that, for all (x1 , x2 ) ∈ Rn1 × Rn2 and all u ∈ Rm ,

∂V1 (x1 )f1 (x1 , x2 , u) ≤ −α1 (|x1 |) + γ1 (|x2 |) + ν1 (|u|) ∂x 1 ∂V2 x2 6= 0 ⇒ (x2 )f2 (x2 , u) ≤ −α2 (|x2 |) + γ2 (|u|). ∂x2

(9) (10)

Then, if γ1 (s) = O (α2 (s)) as s tends to zero, the cascade (8) is iISS. See Section 4.3 for the proof. We stress that, unlike Theorem 1, the above result is not easily extendable to cascades involving more than two subsystems as condition (10) is required to hold on the whole Rn2 \ {0} and not just locally as (4), which makes the dissipation rate resulting from the application of Theorem 6 difficult to estimate.

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

522

A direct consequence of Theorem 6, which is of notable interest in practice, concerns the case when the driven subsystem does not depend on the input u. The system then takes the more classical cascade form x˙ 1 = f1 (x1 , x2 )

(11a)

x˙ 2 = f2 (x2 , u).

(11b)

Corollary 7. Let V1 and V2 be as in Theorem 6 and suppose that, for all i ∈ {1, 2}, there exist: a class P D function αi and a class K function γi such that, for all (x1 , x2 ) ∈ Rn1 × Rn2 , (10) holds and

∂V1 (x1 )f1 (x1 , x2 ) ≤ −α1 (|x1 |) + γ1 (|x2 |). ∂x 1 Then, if γ1 (s) = O (α2 (s)) as s tends to 0, the cascade (11) is iISS. Intuitively, one could expect that the cascade inherits the iISS gain of its driving subsystem. This is not true in general, as shown by the following counter-example which is an adaptation of the one initially detailed in [3,9]. Example 8. Consider the two-dimensional cascaded system: x˙ 1 = −sat(x1 ) + x1 x2

3.4. Trajectory-based condition for cascades of iISS systems Our last two results give a sufficient condition for a cascade of iISS systems to remain iISS, without requiring the knowledge of any Lyapunov function. Instead, greater stability properties are required for the subsystems. It is indeed assumed that the driving subsystem be 0-LES, and that the iISS gain of the driven subsystem be locally Lipschitz. See Section 4.4 for the proof. Theorem 9. Assume that the system (8a) is iISS with respect to (xT2 , uT )T with an iISS gain µ1 , and that the system (8b) is iISS and 0-LES. Assume also that f2 (·, 0) is continuously differentiable. Then, under the condition that µ1 is locally Lipschitz, the cascade (8) is iISS. It is interesting to see that we recover a similar condition to the one derived from [3]. Also, similarly to Theorem 6, note that this result applies to cascaded systems like (11), i.e. when the driven subsystem does not depend on u. Under additional regularity conditions on the dynamics of the driven subsystems, the above result may extend to multiple cascaded systems, i.e.

x˙ 2 = −x2 + u.

x˙ 1 = f1 (x1 , ζ2→N )

First, we show that this cascade is iISS. To this end, let V1 (x) = ln(1 + x21 )/2 and V2 (x2 ) = |x2 |. Using the same notations as in Corollary 7, we see that their derivatives satisfy the following upper bounds:

.. .

x˙ 2 = f2 (x2 , ζ3→N )

(12)

x˙ N−1 = fN−1 (xN−1 , ζN→N )

∂V1 x1 satx1 (x1 )f1 (x1 , x2 ) ≤ − + |x2 | ∂x 1 1 + x21 ∂V2 (x2 )f2 (x2 , u) ≤ −|x2 | + |u|. x2 6= 0 ⇒ ∂x 2

where ζi→N := (xTi , . . . , xTN , uT )T ∈ Rni × · · · × RnN × Rm , for all 2 ≤ i ≤ N. By convention, ζN+1→N := u. The following result is proved in Section 4.5.

The previous result easily applies and establishes iISS. Next, we exhibit an iISS gain for the driving subsystem. Its solution satisfies Z t −(t−τ) |x2 (t, ξ2 , u)| ≤ |ξ2 | e−t + u(τ) dτ e 0 Z t    −(t−τ)/2 = |ξ2 | e−t + e−(t−τ)/2 u(τ) dτ. e

Theorem 10. Assume that, for all i = 1, . . . , N each subsystem (12.i) is iISS with respect to ζi+1→N and 0-LES. Assume also that, for all i = 1, . . . , N − 1, the function fi is locally Lipschitz, the function fi (·, 0) is continuously differentiable, ∂fi (·, 0)/∂xi is bounded in a neighbourhood of the origin and the iISS gain of (12.i) is locally Lipschitz. Then the cascade (12) is iISS.

x˙ N = fN (xN , u),

0

The two functions in brackets in the latter integral are in L4 (and actually in Lp for any positive p). Hence, we can apply Holder’s inequality to get that 1/4 3/4 Z t Z t |x2 (t)| ≤ |ξ2 | e−t + e−2(t−τ)/3 dτ e−2(t−τ) |u(τ)|4 dτ 0

0

≤ |ξ2 | e−t +

3 2

Z 0

t

u(τ)4 dτ

1/4

.

This shows that an admissible iISS gain for the driving subsystem is the function µ(s) = s4 . However, if it were an iISS gain for the whole cascade as well, then R[13, Proposition 6] would ∞ notably ensure that, if the integral 0 u(τ)4 dτ is finite, then lim supt→∞ |x(t, ξ, u)| = 0, where x := (x1 , x2 )T and ξ := (ξ1 , ξ2 )T . We show that this is not the case. Consider indeed the feedback input u = x2 − x32 and pick ξ2 = 1 as the initial state. The driving subsystem then becomes x˙ 2 = −x32 √ and its solution is x2 (t, 1, u) = √ 1 + 2t. Consequently u(t) = 1/ 1 + 2t − 1/(1 + 2t)3/2 , and so R1/ ∞ 4 0 u(τ) dτ is finite. √ Considering the driven subsystem, we obtain x˙ 1 = −sat x1 + x1 / 1 + 2t. It was shown in [3,9] that, for any initial condition ξ1 ≥ 3, the trajectory of this system diverges.√Indeed, as long as x1 (t, ξ1 ) ≥ 1,  we Rhave √that x˙ 1 = −1 + x1 / 1 + 2t, √ t hence x1 (t) = e 1+2t−1 ξ1 − 0 e1− 1+2τ dτ . But, considering the √ √ Rt change√of variable s = −1 + 1 + 2τ , we get that 0 e1− 1+2τ dτ ≤ R ∞ 1− 1+2τ R ∞ −s dτ =√ 0 e (s + 1)ds = 2. Thus, if ξ1 ≥ 3, then it holds 0 e that x1 (t, ξ1 ) ≥ e 1+2t−1 ≥ 1 at all time, and so limt→∞ x1 (t, ξ1 ) = ∞. Thus, µ : s 7→ s4 is not an iISS gain for the cascade.

4. Proof of the main results 4.1. Proof of Theorem 1 The conditions in Theorem 1 give local asymptotic stability (LAS) of (2), as a cascade of two LAS subsystems [17]. Hence we are only left to show global attractivity. Let us first consider the particular case when the initial condition of the driving subsystem is ζ = 0. Since 0 is an equilibrium of this subsystem, we then have that z(t, 0) = 0 for all t ≥ 0, and consequently convergence of solutions to zero follows directly from (3) (see e.g. [7, Corollary 3.3]). In the sequel, we therefore consider that ζ ∈ Rnz \ {0}. We underline the fact that, due to the regularity condition imposed on g, it then holds that |z(t, ζ)| 6= 0 for all t ≥ 0. To see this more clearly, notice that since g is assumed to be locally Lipschitz and g(0) = 0, we have that |g(z)| ≤ L(|z|) |z| for some continuous nondecreasing function L. In addition, (4) ensures that the trajectories of the z-subsystem are bounded, so zm := supt≥0 |z(t, ζ)| is finite and positive for all ζ 6= 0. From these observations, it holds that

∂ (|z(t, ζ)|2 ) = 2g(z(t, ζ))T z(t, ζ) ≥ −2L(|z(t, ζ)|) |z(t, ζ)|2 ∂t ≥ −a |z(t, ζ)|2 , where a := 2L(zm ) is a positive constant since ζ 6= 0. In other words, |z(·, ζ)|2 satisfies the differential inequality y˙ ≥ −ay. From the

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

comparison lemma, we conclude that |z(t, ζ)|2 ≥ |ζ|2 e−at , which is, as claimed, positive at all time. Before going further, let us introduce the following “changing dissipation rate” result, in the spirit of [15]. The novelty here stands in the fact of allowing dissipation inequalities that involve continuous positive definite functions and that may hold only on a restricted domain of the state-space. The proof is omitted due to space constraints but follows along the same lines as [15]. Proposition 11. Let D be an open subset of Rn , f : Rn → Rn be a locally Lipschitz function such that f (0) = 0 and V : Rn → R≥0 be a continuous positive definite radially unbounded function, differentiable on D, satisfying

∂V (x)f (x) ≤ −α(|x|), ∀x ∈ D, ∂x ˜ ∈ P D satisfies α( ˜ s) = O (α(s)) as s where α ∈ P D . If α tends to zero, then there exists a continuous positive definite radially unbounded function V˜ , differentiable on D and such that

∂V˜ ˜ |x|), (x)f (x) ≤ −α( ∂x

∀x ∈ D.

Notice that, by assumption, it holds that 2γ(s) = O (αz (s)) in a neighbourhood of 0. Apply Proposition 11 to the dissipation ˜ ·) = 2γ(·) on the set D = inequality (4) with the function α( Bε \ {0}. Then there exists a continuous positive definite radially ˜ , differentiable on Bε \ {0}, and satisfying, unbounded function W ˜ for all z ∈ Bε \ {0}, ∂∂Wz (z)g(z) ≤ −2γ(|z|). By summing this inequality with (3), we get that, for all (x, z) ∈ Rnx × (Bε \ {0}), ∂V ∂V ∂x f (x, z) + ∂z g (z) ≤ −αx (|x|) − γ(|z|) =: −φ(x, z), where V (x, z) := ˜ ˜ : it V (x) + W (z). Notice that V inherits the properties of V and W is continuous positive definite and radially unbounded. Therefore, (see e.g. [7, Lemma 3.5]) there exist two class K∞ functions αV and αV such that

αV (|(x, z)|) ≤ V (x, z) ≤ αV (|(x, z)|).

(13)

Moreover, let ϕ(s) := inf |(x,z)|≥s φ(x, z) for all s ≥ 0. Then ϕ ∈ P D and we have that, for all (x, z) ∈ Rnx × (Rnz \ {0}),

∂V ∂V (x, z)f (x, z) + (x, z)g(z) ≤ −ϕ(|(x, z)|) ∂x ∂z 1 ≤ −ϕ ◦ α− V (V (x, z)). Now, notice that, due to the GAS of (2b), given any z0 ∈ Rnz , there exists a time T (z0 ) ≥ 0 such that |z(t, z0 )| ∈ Bε for all t ≥ T . Since, as previously shown, z(t, ζ) 6= 0 for all t ≥ 0, we therefore get 1 that, for all t ≥ T , v˙ (t) ≤ −ϕ ◦ α− V (v(t)), where v(·) should be 1 understood as V (x(·, ξ, z), z(·, ζ)). Noticing that ϕ ◦ α− V is of class P D , we can apply [2, Corollary IV.3] to establish the existence of a class KL function βV such that v(t) ≤ βV (v(T ), t − T ) for all t ≥ T . Global attractivity then follows from (13) and the fact that iISS of (2a) guarantees the existence of the solutions of (2) over in the time interval [0, T ]. 4.2. Proof of Theorem 4 From Theorem 1, we know that (6.N − 1)-(6.N) is GAS. In order to be able to propagate this property to the upper levels, we need to provide a local estimate of the resulting dissipation rate for (6.N − 1)-(6.N). To this end, invoking (7), we apply Proposition 11 to (6.N) on the set D = Bε \ {0}with 2γN−1 as a new dissipation rate. This ensures the existence of a positive definite function V˜ N , differentiable on Bε \ {0}, such that, for all xN ∈ Bε \ {0},

∂V˜ N (xN )fN (xN ) ≤ −2γN−1 (|xN |). ∂x N

523

Summing this inequality with (6.N − 1) then yields

∂V˜ N ∂VN−1 fN (xN ) + fN−1 (xN−1 , xN ) ≤ −αN−1 (|xN−1 |) − γN−1 (|xN |) ∂x N ∂xN−1 ≤ −α˜ N−1 (|xN−1 |) − α˜ N−1 (|xN |) ˜ N−1 (s) := min{αN−1 (s), γN−1 (s)} for all xN−1→N ∈ Bε \ {0}, where α for all s ∈ [0, ε). Since both αN−1 and γN−1 belong to class Iε , there ˜ N−1 (a + b) ≤ α˜ N−1 (a) + α˜ N−1 (b) for exists λN−1 > 0 such that λN−1 α all a, b ∈ (0, ε), therefore

∂VN−1 1 ∂V˜ N α˜ N−1 (|xN−1→N |). fN (xN ) + fN−1 (xN−1 , xN ) ≤ − ∂x N ∂xN−1 λN−1 Thus, the subsystem (6.N − 1)-(6.N) satisfies (4) with a local ˜ N−1 . In addition, in view of (7), dissipation rate equivalent to α α˜ N−1 (s) = O (γN−2 (s)) as s tends to zero. This completes by fulfilling the requirements of Theorem 1, and we conclude that (6.N −2)-(6.N) is GAS. Repeating these operations, thanks to (7) and Theorem 1, we show that each subsystem (6.i)-(6.N) is GAS with a local dissipation rate equivalent to min{αi , γi } and the conclusion follows. 4.3. Proof of Theorem 6 The proof consists in showing separately that (8) is 0-GAS and satisfies the BEFBS property and then applying [1, Theorem 3] which shows equivalence between iISS and the combination of these two properties. The first step is direct in view of Theorem 1, by picking f (·, ·) = f1 (·, ·, 0) and g(·) = f2 (·, 0). To establish the second one, we introduce the following result. Proposition 12. Let ω : R≥0 → R be a continuous function. Suppose that y : R → R is a locally Lipschitz function satisfying y(t)

>0

⇒

y(t) = 0

⇒

y˙ (t) ≤

ω(t) ω(t) ≥ 0.

(14) (15)

Then y˙ (t) ≤ ω(t) actually holds for almost all t. Proof. Let χ be defined as: χ(y) = 1, if y = 0 and χ(y) = 0 otherwise. Then, by the area formula for Lipschitz functions we have: Z +∞ Z +∞ χ(y(t))|˙y(t)|dt = χ(y) card 0

−∞

{t ≥ 0 : y(t) = y, ∃ y˙ (t)} dy = 0. Let K denote the set {t ≥ 0 : y(t) = 0}. The above argument shows that {t ∈ K : y˙ (t) 6= 0} has zero-measure. Hence, for almost all t in K we have y˙ (t) = 0 and consequently, by virtue of (15), for almost all t ∈ K it holds y˙ (t) ≤ 0 ≤ ω(t). Since, by assumption (14) the inequality holds for all t 6∈ K , then, the claim easily follows.  Let us go back to the proof of Theorem 6. Consider any initial state (ξ1 , ξ2 ) ∈ Rn1 × Rn2 and any input u. Notice that, due to the positive definiteness of V1 and V2 , α2 (|x2 (·, ξ2 , u)|) vanishes whenever V (x2 (·, ξ2 , u)) = 0. By considering ω(·) = −α2 (|x2 (·, ξ2 , u)|) + γ2 (|u(·)|) in the previous proposition, we get that V˙ 2 (x2 (t)) ≤ −α2 (|x2 (t, ξ2 , u)|) + γ2 (|u(t)|),

∀a.a. t ≥ 0.

(16)

We establish BEFBS under the following “Bounded Energy” assumption: Z ∞ max{γ2 (|u(τ)|); ν1 (|u(τ)|)} ≤ c, (17) 0

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

524

where c is a positive constant. Integrating Inequality (16) indeed yields, for all t ≥ 0, t

Z 0

α2 (|x2 (τ)|)dτ ≤ V2 (ξ2 ) − V2 (x2 (t)) +

t

Z 0

γ2 (|u(τ)|)dτ

≤ V2 (ξ2 ) + c.

(18)

Moreover, from the assumptions on V2 , there exist α2 , α2 ∈ K∞ such that

α2 (|x2 |) ≤ V2 (x2 ) ≤ α2 (|x2 |),

∀x2 ∈ Rn2 ,

(19)

1 so Inequality (16) implies that V˙ 2 (x2 (t)) ≤ −α2 ◦ α− 2 (V2 (x2 (t))) + 1 is a class P D γ2 (|u(t)|), for almost all t ≥ 0. Since α2 ◦ α− 2 function, [2, Corollary IV.3] establishes the existence of a class KL function β2 such that V2 (x2 (t, ξ2 , u)) ≤ β2 (V2 (ξ2 ), t) + Rt 2 0 γ2 (|u(τ)|)dτ . Using again the bounds on V2 provided by (19),3 it follows that

 Z

1 −1 |x2 (t)| ≤ α− 4 2 (2β2 (α2 (|ξ2 |), t)) + α2

0

t



γ2 (|u(τ)|)dτ .

(20)

By the way, as we will need it later, notice that a similar reasoning based on (9) leads to the following bound on the trajectories of (8a), where β1 ∈ KL: 1 |x1 (t)| ≤ α− 1 (2β1 (α1 (|ξ1 |), t)) t

 Z

1 + α− 4 1

0

 [γ1 (|x2 (τ)|) + ν1 (|u(τ)|)] dτ .

(21)

In view of (17) and (20), [13, Proposition 6] ensures that limt→∞ |x2 (t, ξ2 , u)| = 0. Notably, there exists a finite time T ≥ 0 such that |x2 (t, ξ2 , u)| ≤ 1 for all t ≥ T . Furthermore, since γ1 (s) = O (α2 (s)) in a neighbourhood of zero, there exists a positive constant k such that γ1 (s) ≤ kα2 (s) for all s ∈ [0; 1]. Using this with (17) and (18), we can achieve the following computation: Z

∞

0

γ1 (|x2 (τ, ξ2 , u)|)dτ Z

T

≤ 0

Z

T

≤ 0

Z

≤ 0

T

γ1 (|x2 (τ, ξ2 , u)|)dτ +

∞

Z

γ1 (|x2 (τ, ξ2 , u)|)dτ + k

T

Z T

γ1 (|x2 (τ, ξ2 , u)|)dτ

∞

α2 (|x2 (τ, ξ2 , u)|) dτ

γ1 (|x2 (τ, ξ2 , u)|)dτ + k(V2 (ξ2 ) + c).

Since T is finite, this shows that, under the bounded energy R∞ condition (17), the integral 0 γ1 (|x2 (τ, ξ2 , u)|)dτ is bounded as well. Finally, notice that, since β1 is a KL function, (17) and (21) imply that

4.4. Proof of Theorem 9 The proof consists in designing a Lyapunov-like function for the driving subsystem, in order to follow a similar reasoning as in the proof of Theorem 6. Namely, we will show 0-GAS and BEFBS. We stress that any locally Lipschitz K function can be upper bounded by a differentiable K function. Based on this, we will consider without loss of generality that µ1 is differentiable. Since the driving subsystem is iISS, it is 0-GAS. Hence, there exists a locally Lipschitz class K function η2 and a positive constant k2 such that the trajectories of the zero input driving subsystem satisfy, for any ξ2 ∈ Rn2 , |x2 (t, ξ2 , 0)| ≤ η2 (|ξ2 |)e−k2 t for all t ≥ 0, which means, using the terminology of [3], that x˙ 2 = f2 (x2 , 0) is GAS(α2 ) with α2 (s) := s. In addition, since µ1 , α2 and η2 are all locally Lipschitz and positive definite, there exists a positive constant Rλ such that max {µ1R(s); α2 (s)} ≤ λs for all s ∈ [0; 1]. Then 1 1 we have 0 µ1 ◦ α2 (s)/s ds ≤ 0 λ2 s/s ds ≤ λ2 and the 0-GAS of the cascade (8) follows from [3, Theorem 1]. The proof of the BEFBS property is based on the following two lemmas. The first one ensures the existence of a converse storage function for GAS and LES systems, with a prescribed dissipation rate. It is proved in Section 5.1. Lemma 13. Let f : Rn → Rn be a continuously differentiable function such that the system x˙ = f (x) is GAS and LES. Let µ be a given differentiable function of class K∞ . Then there exists a continuous function V : Rn → R≥0 differentiable over Rn \{0}, class K∞ functions α and α, and a continuous function c : Rn → R≥0 such that, for all x ∈ Rn \ {0},

α(|x|) ≤ V (x) ≤ α(|x|) ∂V (x)f (x) ≤ −µ(|x|) ∂x ∂V ≤ c(x). ( x ) ∂x

(22) (23) (24)

The second lemma, proved in Section 5.2, is a direct extension of [2, Proposition II.5] and creates a bridge between the notions of 0-GAS and iISS in terms of a (not necessarily radially unbounded) Lyapunov-like function. The novelty here is to specify the behaviour of the dissipation rate around the origin. Lemma 14. Let f : Rn × Rm → Rn be a locally Lipschitz function. Suppose that there exists a continuous function V : Rn → R≥0 differentiable out of the origin such that, for all x ∈ Rn \ {0}, (22) and (24) hold together with

∂V (x)f (x, 0) ≤ −µ(|x|), ∂x where α, α designate K∞ functions, µ ∈ K , and c : Rn → R≥0 is a continuous function. Then there exists a continuous positive definite function W : Rn → R≥0 differentiable on Rn \ {0} such that, for all x ∈ Rn \ {0} and all u ∈ Rm ,

 Z ∞  1 lim sup |x1 (t)| ≤ α− 8 γ1 (|x2 (τ)|)dτ 1 t→∞ 0  Z ∞  −1 + α1 8 ν1 (|u(τ)|)dτ .

∂W (x)f (x, u) ≤ −ρ(|x|) + δ(|u|), ∂x where δ ∈ K and ρ ∈ P D satisfy ρ(s) ∼ µ(s) in a neighbourhood of

In a nutshell, under the bounded energy assumption (17), the upper limit of the norm of the trajectories of (8a), as t goes to infinity, is finite. This establishes the BEFBS of (8). The conclusion follows from [1, Theorem 3].

As already seen, the driving zero-input subsystem x˙ 2 = f2 (x2 , 0) is GAS and LES. Apply Lemma 13 to it with the function µ(s) := µ1 (2s), where µ1 ∈ K∞ is the iISS gain of the

0

3 And the “weak triangular inequality”: α(a + b) ≤ α(2a) + α(2b) for any nonnegative a and b, if α is a K function.

zero.

driven subsystem.4 The conditions of Lemma 14 are then also

4 If µ 6∈ K , the whole reasoning can be done with any locally Lipschitz ∞ 1 µ˜ 1 ∈ K∞ such that µ˜ 1 (s) = µ1 (s) for all s ∈ [0; 1] and µ˜ 1 (s) ≥ µ1 (s) for all s ≥ 1.

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

fulfilled, and we conclude the existence of a continuous function W , differentiable out of 0, such that, for all x2 6= 0 and all u, ∂W ∂x2 (x2 )f2 (x2 , u) ≤ −ρ(|x2 |) + σ(|u|), where σ is a class K function and ρ ∈ P D satisfies ρ(s) ∼ µ1 (2s) around zero. By letting ω(t) = −ρ(|x2 (t, ξ2 , u)|) + σ(|u(t)|) in Proposition 12, it follows that, for ˙ (x2 (t, ξ2 , u)) ≤ −ρ(|x2 (t, ξ2 , u)|) + σ(|u(t)|). almost all t ≥ 0, W Integrating this inequality yields Z ∞ Z ∞ ρ(|x2 (τ, ξ2 , u)|)dτ ≤ W (ξ2 ) + σ(|u(τ)|)dτ. (25) 0

0

On the other hand, by the assumption of iISS on the driving subsystem, there exists β2 ∈ KL, and γ2 , µ2 ∈ K such that, for all ξ2 ∈ Rn2 and all u, Z t  |x2 (t, ξ2 , u)| ≤ β2 (|ξ2 | , t) + γ2 µ2 (|u(τ)|)dτ . (26) 0

We show that the state is “frequently bounded” under the following bounded energy assumption: Z ∞ max{µ1 (2|u(τ)|); µ2 (|u(τ)|); σ(|u(τ)|)}dτ < ∞. (27) 0

To this end, first notice that, by virtue of [13, Proposition 6], this assumption together with (26) ensures that lim |x2 (t, ξ2 , u)| = 0.

(28)

t→∞

Moreover, provided (27), the integral of the left-hand side of (25) is finite. Hence, since R ∞Lemma 14 ensures that ρ(s) ∼ µ1 (2s) as s tends to zero, we have 0 µ1 (2 |x2 (τ, ξ2 , u)|)dτ < ∞. Finally, since µ1 is the iISS gain of the driven subsystem, there exists β1 ∈ KL and γ1 ∈ K such that

|x1 (t, ξ1 , x2 )| ≤ β1 (|ξ1 | , t) + γ1

Z

≤ β1 (|ξ1 | , t) + γ1

Z

≤ β1 (|ξ1 | , t) + γ1

Z

t

0 t

0 t

0

µ1 (|(x2 (τ, ξ2 , u), u(τ))|) dτ



µ1 (|x2 (τ, ξ2 , u)| + |u(τ)|) dτ µ1 (2 |x2 (τ)|) dτ +

Z 0

t

5.1.1. Upper bound on V Since the system is assumed to be GAS and LES, [3, p.1892] ensures the existence of a locally Lipschitz K∞ function η such that |x(τ, ξ)| ≤ η(|ξ|)e−τ . Based on this observation, we get that Z ∞
V (ξ) ≤ (29) µ η(|ξ|)e−τ dτ =: α(|ξ|). 0

We claim that α is of class K . Indeed, it is clear that α(0) = 0. For any ξ 6= 0, consider the change of variable s = η(|ξ|)e−τ . Then we can see that Z η(|ξ|) µ(s) ds. (30) α(|ξ|) = s

0

However, since µ is differentiable, it is locally Lipschitz, so there exists a nonnegative Lξ such that µ(s) = µ(s) − µ(0) ≤ Lξ s for all s ∈ [0; η(|ξ|)]. This shows that the previous integral is finite, and therefore that α is finite over R>0 . Moreover, it can be seen from (30) that α is continuous and increasing. 5.1.2. Lower bound on V The lower bound on V is obtained as follows. Notice first that, since f is continuously differentiable, it is locally Lipschitz, so there exists a continuous nondecreasing function L such that |f (x)| ≤ L(|x|) |x|. Hence

∂ (|x(τ, ξ)|2 ) = 2f (x(τ, ξ))T x(τ, ξ) ∂τ ≥ −2L(|x(τ, ξ)|) |x(τ, ξ)|2 ≥ −b |x(τ, ξ)|2 , where b := 2L(supτ≥0 |x(τ, ξ)| + 1) is a positive constant which is finite since the system is assumed to be GAS. Thus, |x(·, ξ)|2 satisfies the differential inequality y˙ ≥ −by. From the comparison lemma, we conclude that |x(τ, ξ)|2 ≥ |ξ|2 e−bτ . Therefore Z ∞   µ |ξ| e−bτ/2 dτ V (ξ) ≥ 0

Z 1/b



≥ 0



µ1 (2 |u(τ)|) dτ .

Thus, under the Bounded Energy condition (27), we get that lim supt→∞ |x1 (t, ξ1 , x2 )| is finite, and consequently, with (28), that the cascade (8) satisfies the BEFBS property. We may therefore conclude iISS by virtue of [1, Theorem 3]. 4.5. Proof of Theorem 10 Consider first the cascade (12.N −1)-(12.N). From Theorem 9, we know that it is iISS with respect to u. In addition, it is not difficult to see that, in view of the regularity assumptions made on fN−1 and fN , (12.N − 1)-(12.N) is also 0-LES, as a cascade of 0-LES systems. Recalling that fN−1 (·, 0) and fN (·, 0) are continuously differentiable, we can again apply Theorem 9 with (12.N − 1)-(12.N) as driving subsystem and we get that (12.N − 2)-(12.N) is iISS. The iteration of this procedure at each level of the cascade establishes the result. 5. Proof of the lemmas 5.1. Proof of Lemma 13 The result is inspired by the converse theorems based on Massera’s lemma (see e.g. [7, Theorem 3.14] and its proof). The novelty consists in allowing to assign a prescribed dissipation term to the constructed function. We define Z ∞ V (ξ) := µ(|x(τ, ξ)|)dτ. 0

525

µ |ξ| e−bτ/2 dτ ≥ µ |ξ|2 e−1 . 








Hence, α(s) := µ s2 e−1 is an appropriate K∞ lower bound5 for V .

5.1.3. Gradient of V The next point consists in showing that V is differentiable. To this R t end, notice that the solution of x˙ = f (x) satisfies x(t, ξ) = ξ + 0 f (x(τ, ξ))dτ . We introduce the notation xξ (t, ξ) := ∂(x(t, ξ))/∂ξ, and differentiate previous equality with respect to ξ to get that R t the ∂f xξ (t, ξ) = 1 + 0 ∂x (x(τ, ξ))xξ (τ, ξ)dτ . Differentiating with respect to t, we obtain that xξ (·, ξ) is a solution of

∂ (xξ (t, ξ)) = A(t, ξ)xξ (t, ξ), xξ (0, ξ) = 1, (31) ∂t where A(t, ξ) := ∂∂xf (x(t, ξ)). We define A∞ := limt→∞ A(t, ξ). Since x(t, ξ) tends to zero and f is continuously differentiable, we can see that A∞ = ∂∂xf (0), which shows that A∞ is independent of ξ. Also,

since the system is assumed to be LES, it follows from [7, Theorem 3.13] that A∞ is a Hurwitz matrix. Hence, for any positive definite symmetric matrix Q , there exists a positive definite symmetric matrix P such that AT∞ P + PA∞ = −Q . Consider the Lyapunov function V (xξ ) := xTξ Pxξ . Then its derivative along the solution of (31) yields  

V˙ (xξ ) = xTξ A(t, ξ)T P + PA(t, ξ) xξ h

i

= −xTξ Qxξ + xTξ (A(t, ξ) − A∞ )T P + P(A(t, ξ) − A∞ ) xξ 2

2

≤ −qm xξ + |A(t, ξ)| xξ ,

(32)

5 Note that, in view of (29), this establishes in turn that α is a class K

∞ as well.

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

526

where qm > 0 is the minimum eigenvalue of Q and A(t, ξ) := (A(t, ξ)−A∞ )T P+P(A(t, ξ)−A∞ ). We can see that limt→∞ |A(t, ξ)| = 0. Hence, for all ξ ∈ Rn , there exists a finite time T (ξ) such that |A(t, ξ)| ≤ qm /2 for all t ≥ T (ξ), and consequently V˙ (xξ (t, ξ)) ≤ 2 −qm xξ (t, ξ) /2 for all t ≥ T (ξ). From this, we conclude that there exist two constants k1 , k2 > 0 such that xξ (t, ξ) ≤ k1 xξ (T (ξ), ξ) e−k2 (t−T (ξ)) ,

∀t ≥ T (ξ).

(33)

It is worth mentioning that the forward completeness is ensured by (32) and the fact that |A(·, ξ)| is bounded (since it is continuous and has a finite limit). Therefore c1 (ξ) := supt∈[0,T (ξ)] xξ (t, ξ) is well defined over Rn . Recalling that supt≥0 |x(t, ξ)| ≤ η(|ξ|) and using also (33), it holds that, for all ξ ∈ Rn , Z 0

∞

xξ (τ) dτ =

Z Z T (ξ) xξ (τ) dτ +

∞

T (ξ)

0

xξ (τ) dτ

k1 xξ (T (ξ)) . k2

≤ c1 (ξ)T (ξ) +

Since x(t, ξ) 6= 0 for all ξ 6= 0 and all t ≥ 0 (see Section 5.1.2), it follows that Z ∞ Z ∞ 0 x(τ) 0 ≤ µ (|x(τ)|) xξ (τ) dτ | | µ ( x (τ) ) x (τ) d τ ξ |x(τ)| 0 0 Z ∞ xξ (τ) dτ ≤ sup |µ0 (s)| s∈[0;η(|ξ|)]

0

  k xξ (T (ξ)) . ≤ sup |µ0 (s)| c1 (ξ)T (ξ) + qm s∈[0;η(|ξ|)]

Thus, the left-hand side of the previous inequality exists and is finite for all ξ ∈ Rn \ {0}. However, the norm of this integral also satisfies Z ∞ x(τ, ξ) xξ (τ, ξ)dτ µ0 (|x(τ, ξ)|) |x(τ, ξ)| 0 Z ∞ ∂V ∂ = (µ(|x(τ, ξ)|)) dτ = (ξ) .

∂ξ

0

∂ξ

This establishes that V is differentiable over Rn \ {0} and, in turn, establishes the bound (24) with any continuous function c : Rn → R≥0 satisfying   k1 xξ (T (ξ), ξ) , c(ξ) ≥ sup |µ0 (s)| c1 (ξ)T (ξ) + qm

s∈[0;η(|ξ|)]

∀ξ ∈ Rn .

5.1.4. Upper bound on V˙ Finally, we exhibit the bound on the total derivative of V along the trajectories. Let ξ ∈ Rn and t ≥ 0, we then have that Z ∞ V (ξ) = µ(|x(τ, ξ)|)dτ 0

Z

t

= 0

Z

t

= 0

Z

= 0

t

µ(|x(τ, ξ)|)dτ +

Z

µ(|x(τ, ξ)|)dτ +

Z

µ(|x(τ, ξ)|)dτ +

Z

∞

t

∞

t

0

∞

µ(|x(τ, ξ)|)dτ

5.2. Proof of Lemma 14 The proof we present here consists in slight modifications of the one of [2, Proposition II.5]. We first establish the following result, which should be seen as an adaptation of [2, Lemma IV.10]. Proposition 15. Under the assumptions of Lemma 14, the function V is such that, for all x ∈ Rn \ {0} and all u ∈ Rm ,

∂V (x)f (x, u) ≤ −µ(|x|) + ν(|x|)δ(|u|), ∂x where δ is a class K function and ν is a positive continuous increasing function. Proof. Consider x 6= 0 and compute the total derivative of V along the trajectories of the system with input u:

∂V f (x, u) ∂x ∂V ∂V ∂V = f (x, 0) + f (0, u) [f (x, u) − f (x, 0) − f (0, u)] + ∂x ∂x ∂x ∂V ∂V ≤ −µ(|x|) + |f (x, u) − f (x, 0) − f (0, u)| + |f (0, u)| . ∂x ∂x Define the function γ(r, s) := r + s + max|x|≤r,|u|≤s |f (x, u) − f (x, 0) − f (0, u)|. Then, γ is of class K in each of its two arguments. So, by [2, Corollary IV.5], there exists a class K function σ such that γ(r, s) ≤ σ(r)σ(s). It follows that ∂V ∂V ∂V σ(|x|)σ(|u|) + |f (0, u)| . f (x, u) ≤ −µ(|x|) +

∂x

∂x

∂x

Define next, for all r > 0, κ(r) := r + sup0<|x|≤r ∂∂Vx (x) . Note that

κ is well defined for all positive r since V is differentiable over Rn \ {0} and lim sup|x|→0 ∂∂Vx (x) ≤ c(0) < ∞. If, in addition, we let κ(0) := 0, then κ is a positive definite nondecreasing function, continuous on R>0 . Hence, there exists a continuous increasing function κ˜ such that κ˜ (·) ≥ κ(·). Thus we get that, for x 6= 0, ∂V ˜ ˜ ∂x (x)f (x, u) ≤ −µ(|x|) + κ(|x|)σ(|x|)σ(|u|) + κ(|x|) |f (0, u)|. By the local Lipschitz continuity of f , there exists a class K function χ such that |f (0, u)| ≤ χ(|u|). This final observation establishes the result with the functions ν(·) = κ˜ (·)(1 + σ(·)) and δ(·) := σ(·) + χ(·).  Let’s now go back to the R r proof of Lemma 14. Define the following function π(r) := 0 ds/(1 + ν ◦ α−1 (s)), where ν is the positive continuous increasing function generated by the previous proposition. Notice that, since ν ◦ α−1 is a nonnegative function, π belongs to class K . Letting W := π ◦ V , it follows that W is positive definite and differentiable out of the origin and, for all x 6= 0,

∂W ∂V 1 −µ(|x|) f (x, u) = f (x, u) ≤ ∂x ∂x 1 + ν ◦ α−1 (V ) 1 + ν ◦ α−1 ◦ α(|x|) ν(|x|)δ(|u|) + . 1 + ν(|x|) Define ρ(·) :=

µ(·)

1+ν◦α−1 ◦α(·)

. Then ρ is a class P D function. In

addition, lims→0 µ(s)/ρ(s) = lims→0 1 + ν ◦ α−1 ◦ α(s) = 1, which establishes the result. 6. Concluding remarks

µ (|x(τ − t, x(t, ξ))|) dτ µ(|x(τ, x(t, ξ))|)dτ.

We thus get that, for all ξ ∈ Rn and all t ≥ 0, V (x(t, ξ)) − V (ξ) = R − 0t µ(|x(τ, ξ)|)dτ . The bound (23) follows by differentiating this equality with respect to t.

Three main results are presented in this paper. The first one gives a sufficient condition for the cascade composed of an iISS system driven by a GAS one to be GAS. This condition is given in terms of the dissipation rate and the supply function provided by the Lyapunov-like functions of each subsystem. In this sense, it extends the main result in [3], which states this condition based on the estimates of the trajectories. This extension may be useful in

A. Chaillet, D. Angeli / Systems & Control Letters 57 (2008) 519–527

practice since both GAS and iISS are often established through the study of a Lyapunov function, and as it naturally adapts to the case of multiple cascaded systems. The two other results study the cascade interconnection of two iISS systems. They provide sufficient conditions for the iISS property to be preserved, in terms of the iISS Lyapunov-like functions and in terms of the estimate of the trajectories of each disconnected subsystem. It is notably shown that, if the driven subsystem has a locally Lipschitz iISS gain and the driving subsystem is iISS and 0-LES, then the cascade is iISS. Through an example it was also shown that the conjecture that the cascade has the same iISS gain as its driving subsystem is in general wrong.

[5]

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Acknowledgments [11]

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