Stability of sets for nonlinear systems in cascade Antoine Chaillet and Elena Panteley

Abstract— The stability of (not necessarily compact) sets for nonlinear systems in cascade is addressed. It is proved that if two sets are globally asymptotically stable (GAS) for the subsystems taken separately then their cross-product is GAS for the corresponding cascade, provided that the solutions of the latter are globally bounded. In the case of a suitable growth rate of the interconnection in the state of the driven subsystem, we show that this latter requirement can be relaxed to just forward completeness of the cascade. Our results extend similar results on the stability analysis of cascaded systems and find applications in partial stability analysis and adaptive control.

I. I NTRODUCTION The stability analysis of cascaded systems have proved its usefulness in many applications, both in analysis and control design for complex nonlinear systems. The cascade structure is a particular type of interconnection, where the influence of one subsystem to another is unilateral: x˙ 1

= f1 (x1 ) + g(x1 , x2 )

(1a)

x˙ 2

= f2 (x2 ) .

(1b)

From an analysis point of view, the main advantage of this structure is to allow to study each subsystem separately and to deduce a stability property for the whole cascade. This modularity, that often considerably reduces the difficulty of the analysis for complex systems, is also at the origin of control design techniques such as backstepping. The interest of the control community for this field is not recent. It founds its roots into [16], which presents some theoretical challenges for the stability analysis of nonlinear systems in cascade. While it is well established that the cascade composed of two asymptotically stable systems is itself asymptotically stable [15], and that the input-to-state property is also conserved under a cascade interconnection [13], this statement fails for most other stability properties. Typically, the additional requirement is a property of boundedness of solutions. The fact that the cascade composed of two globally asymptotically stable systems is globally asymptotically stable provided that its solutions are globally bounded was proved in [14], [12] in an autonomous context, and then adapted to the timevarying case in [9]. This boundedness assumption was also used in [5] to show a similar result for practical and semiglobal stability properties in a non-autonomous background. Since then, various tools were developed that replace the requirement of boundedness of solutions by some A. Chaillet is with Universit´e Paris Sud, E. Panteley is with CNRS. LSS, Sup´elec, 3 rue Joliot Curie, 91192 Gif s/Yvette, France,

{chaillet,panteley}@lss.supelec.fr

conditions on the structure of the cascade: see [9], [1] for the preservation of the global asymptotic stability, and [4] for the case of integral input-to-state stability. Most of these works deal with the stability of the origin, or at least of compact sets. However, to the best the authors’knowledge, stability of more general sets in a cascade context was not addressed yet. The motivations for the study of set-stability are numerous. For instance, it includes the analysis of partial stability as a special case [20], which has also proved useful in many control applications. Among them, let us cite adaptive control: we may indeed consider as an extended state the actual state plus the adaptation error variables; in many cases, one desires that the ‘real’ state present a convenient stability property, while the convergence of the parameters estimation is often not required. More generally, many specific plants do not require the whole state to present a stability property for achieving a correct behavior. It may also be interesting to study partial stability in the case that only part of the state can be controlled, while no prescribed behavior is imposed on the rest of variables. The set under consideration may also be a prescribed trajectory, allowing use in path following control. In the present paper, we study the stability property of the cascade (1) by assuming that, for each subsystem taken separately, a given set in globally asymptotically stable. More precisely, similarly to the previously evoked results, we show that, if all the solutions of the cascade are globally bounded, then the cross product of the two original sets is globally asymptotically stable for the overall cascade. We also show that the requirement of boundedness of solutions may be relaxed to just boundedness with respect to a set for a certain class of cascaded systems. Furthermore, we give a sufficient condition, in terms of growth rate restrictions, that allows to relax this assumption to just forward completeness and consequently makes our tool easy to use in many applications. Although not included here for space constraints, the results presented here have already been used to provide an alternative proof of the recently established result [2], where the formation control of surface vessels along a straight line, and with a prescribed velocity, is addressed. In this context, a cascade approach is adopted to synchronize the boats in a parallel motion. The notion of set-stability arises naturally since only the relative position of one ship from another is considered. Please refer to [3, Section 4.3] for details. The rest of the paper is organized as follows: in Section II, we recall some definitions associated to the set-stability and we present a useful integral lemma that establishes the

global asymptotic stability of a given set. In Section III, we present our main results. We present the proofs in Section IV, and we conclude with some remarks in Section V. II. N OTATIONS

AND DEFINITIONS

Notation. A continuous function α : R≥0 → R≥0 is of class K (α ∈ K), if it is strictly increasing and α(0) = 0; α ∈ K∞ if, in addition, α(s) → ∞ as s → ∞. A continuous function σ : R≥0 → R≥0 is of class L (σ ∈ L) if it is decreasing and tends to zero as its argument tends to infinity. A function β : R≥0 × R≥0 → R≥0 is said to be a class KL function if, β(·, t) ∈ K for any t ≥ 0, and β(s, ·) ∈ L for any s ≥ 0. We denote by x(·, x0 ) the solutions of the differential equation x˙ = f (x) with initial state x0 . We use |·| for the Euclidean norm of vectors. Given a set A ⊂ Rn , we define |x|A := inf z∈A |x − z|. We designate by N≤N the set of all nonnegative integers less than or equal to N . Given a ∈ [−∞; +∞] and two functions f1 , f2 : R → R, we write f1 (s) = O(f2 (s)) as s → a if there exists a nonnegative constant b such that |f1 (s)| ≤ b|f2 (s)| in a neighborhood of a. When the context is sufficiently explicit, we may omit to write the arguments of a function. We first recall some classical definitions related to setstability for nonlinear autonomous systems of the form x˙ = f (x)

(2)

where x ∈ Rn and f : Rn → Rn is a locally Lipschitz function. In the sequel, A refers to a closed (but not necessarily bounded) set of Rn that contains the origin. The definitions below correspond to the stability with respect to two measures used in e.g. [7], [19]. In our case, the first measure is the distance to the set under consideration, while the second is the Euclidean norm. Definition 1 (GS of a set): A is said to be Globally Stable for (2) if there exists a class K∞ function γ such that, for all x0 ∈ Rn , the solution of (2) satisfies |x(t, x0 )|A ≤ γ(|x0 |) ,

∀t ≥ 0 .

Definition 2 (GA of a set): A is said to be Globally Attractive for (2) if, for all x0 ∈ Rn , the solution of (2) satisfies lim |x(t, x0 )|A = 0 . t→∞

Definition 3 (GAS of a set): A is said to be Globally Asymptotically Stable for (2) if it is both GS and GA. This is equivalent to the following statement: there exists a class KL function β such that, for all x0 ∈ Rn , the solution of (2) satisfies |x(t, x0 )|A ≤ β(|x0 | , t) ,

∀t ≥ 0 .

It is worth pointing out that these definitions are less conservative than their uniform versions, i.e. when both measures are the distance to the set A; see e.g. [8], [21], [18]. For instance, it is not required that the set A be

positively invariant. Such relaxed properties commonly appear in partial stability analysis, notably in adaptive control, when only a “non-strict” Lyapunov function can be derived. See [20], [10] for details. We also introduce the following definition, as it constitutes a crucial requirement in two of our main results. Definition 4 (GB): The solutions of (2) are said to be Globally Bounded with respect to A if there exists a class K∞ functions η and a nonnegative constant µ such that, for all x0 ∈ Rn , they satisfy |x(t, x0 )|A ≤ η(|x0 |) + µ ,

∀t ∈ R≥0 .

For the case that A = {0} we just say that the solutions are Globally Bounded. The following property is an adaptation of [18, Definition 5] to the case when the stability properties are defined through two different measures (namely |·|A and |·|). Definition 5 (GSTS): A is said to be Globally Sliding Time Stable for (2) if there exists class K∞ functions T and ρ such that, for all x0 ∈ Rn , the solution of (2) satisfies |x(t, x0 )|A ≤ ρ(|x0 |) ,

∀t ∈ [0; T (|x0 |)] .

For the case when A = {0}, we say, with a slight abuse, that (2) is GSTS. Remark 6: This property is little restrictive. For instance, it can be shown that (2) is GSTS in each of these cases: . the function f in (2) is globally Lipschitz . the solutions of (2) are GB. Next, we present a simple integral criterion for the global asymptotic stability of a given closed (but not necessarily bounded) set. It can be seen as an extension of [18, Theorem 1] and [11, Lemma 2.1] to the case of stability with respect to two measures. See Section IV-A for the proof. Lemma 7: Assume that a given subset A of Rn is GSTS for (2) and that there exists a class K function σ and a class K∞ function δ such that, for all x0 ∈ Rn , the solution of (2) satisfies Z ∞ σ(|x(t, x0 )|A )dt ≤ δ(|x0 |) . (3) 0

Then A is GS. If, in addition, the solutions of (2) are GB1 , then A is GAS for (2).  III. M AIN

RESULTS

In this section, we consider the stability properties of some given set for the cascaded system (1), where x := (x1 , x2 ) ∈ Rn1 × Rn2 , and the functions fi : Rni → Rni , i ∈ {1; 2}, and g : Rn → Rn1 , n := n1 +n2 , are all locally Lipschitz. Our first main result (proved in Section IV-B) states that the cascade of two globally set-stable systems 1 which

implies that A is GSTS in view of Remark 6 since 0 ∈ A.

is itself globally set-stable provided that its solutions are globally bounded. Theorem 8: Let A1 and A2 be some sets of Rn1 and R respectively. Under the Assumptions 9–12 below, the cascade (1) is GAS with respect to A := A1 × A2 . n2

Assumption 9 (GAS of A1 ): A1 is GAS for the driven subsystem x˙ 1 = f1 (x1 ). Assumption 10 (GAS of A2 ): A2 is GAS for the driving subsystem (1b). Assumption 11 (Bound on the interconnection term): There exist a continuous function g1 : Rn → Rn1 and a class K∞ function g2 such that, for all x = (x1 , x2 ) ∈ Rn1 × Rn2 , |g(x1 , x2 )| ≤ |g1 (x)| g2 (|x2 |A2 ) . Assumption 12 (Boundedness of solutions): The solutions of (1) are GB. It is worth noting that no Lyapunov function needs to be explicitly known for this first result. However, in the case when the solutions are only bounded with respect to the set A (and not to the origin anymore), the result still holds provided a stronger requirement on the interconnection term and on the gradient of the (supposedly known) Lyapunov function of the driven subsystem. Corollary 13: Under Assumptions 10, 14, 15 and 16, the cascade (1) is GAS with respect to A = A1 × A2 . Assumption 14 (Lyapunov GAS of A1 ): There exist a continuously differentiable function V1 : R≥0 × Rn1 → R≥0 and class K∞ functions α1 , α1 , α1 such that, for all x1 ∈ Rn1 , α1 (|x1 |A1 ) ≤ V1 (x1 ) ≤ α1 (|x1 |) dV1 (x1 )f1 (x1 ) ≤ −α1 (|x1 |A1 ) . dx1 Assumption 15 (Bound on Lg V1 ): There exists a continuous nondecreasing function g1 and a class K∞ function g2 such that, for all x = (x1 , x2 ) ∈ Rn1 × Rn2 , dV1 dx1 (x1 )g(x) ≤ g1 (|x|A )g2 (|x2 |A2 ) . Assumption 16 (Boundedness of solutions): The solutions of (1) are GB with respect to A.

Proof: This result follows directly from Theorem 8 by noticing that, in the proof of the latter (see Section IVB), c1 (|x|) can then be replaced by g1 (|x|A ). Hence, based on Assumption 16, we see that (14) remains valid and the rest of the proof is exactly the same. In the two previous results, the most difficult requirement to verify is often the global boundedness of the solutions of (1) (with respect to the origin or to the set A). In what follows, we present a result which relaxes this assumption to just forward completeness of (1), provided

a growth rate restriction of the x1 -dependency of the interconnection term with respect to the dissipation function of the driven subsystem. Its proof is presented in Section IV-C. Corollary 17: Under the Assumptions 10, 14, 18 and 19, the cascade (1) is GAS with respect to A = A1 × A2 . Assumption 18 (Bound on Lg V1 ): There exists a continuous nondecreasing function g11 : R≥0 → R≥0 and a class K∞ function g2 such that, for all x = (x1 , x2 ) ∈ Rn1 × Rn2 , dV1 ≤ g11 (|x1 | )g2 (|x2 | ) . (x )g(x) A1 A2 dx1 1 Assumption 19 (Growth restriction): The solutions of (1) are forward complete and it holds that g11 (s) = O(α1 (s)) ,

as s → +∞ .

IV. P ROOFS A. Proof of Lemma 7 The proof is composed of two steps. We first show that A is globally stable if it is GSTS and (3) holds. Next, we establish its global attractivity under the assumption of global boundedness of the solutions of (2). 1) Proof of GS: The proof of the global stability of A is inspired by the one of [18, Theorem 1]. Let T and ρ be generated by the GSTS of A and let κ be any K∞ function satisfying    1 −1 −1 T (s)σ(s) , ∀s ∈ R≥0 . κ (s) ≤ min s ; δ 2 The existence of such a function is ensured by the fact that σ ∈ K and T, ρ ∈ K∞ . Notice that κ(s) ≥ s for all s ∈ R≥0 and that the following property holds: 1 T ◦ κ(s)σ ◦ κ(s) , 2 We claim that, for any x0 ∈ Rn , δ(s) ≤

|x(t, x0 )|A ≤ ρ ◦ κ(|x0 |) ,

∀s ∈ R≥0 .

(4)

∀t ∈ R≥0 .

(5)

First observe that this holds for x0 = 0 due to (3) and the continuity of x(·, x0 ). For |x0 | > 0, we proceed by contradiction. Assume that the property (5) does not hold. Then, there exists a time t1 ∈ R≥0 such that |x(t1 , x0 )|A > ρ ◦ κ(|x0 |) .

(6)

Note that, without loss of generality, ρ can be assumed to satisfy ρ(s) ≥ s for all s ≥ 0. Therefore |x0 |A ≤ |x0 | ≤ κ(|x0 |) ≤ ρ ◦ κ(|x0 |) . So, invoking again the continuity of x(·, x0 ), there exists a time t2 ∈ [0, t1 ) such that |x(t2 , x0 )|A

= κ(|x0 |)

|x(t, x0 )|A

≥ κ(|x0 |) ,

(7) ∀t ∈ (t2 ; t1 ) .

(8)

Furthermore, the GSTS of A combined with (6) and (7) implies that t1 > t2 + T (κ(|x0 |)). From (3) and (8) it follows that, on one hand, Z t1 Z t2 +T (κ(|x0 |)) σ(|x(t, x0 )|A )dt ≥ σ ◦ κ(|x0 |)dt t2

t2

= T (κ(|x0 |))σ ◦ κ(|x0 |) , and, on the other hand, Z t1 Z σ(|x(t, x0 )|A )dt ≤

0

t2

∞

≥

σ(|x(t, x0 )|A )dt ≤ δ(|x0 |) ,

ti

σ ˜ (|x(t)|A )dt

X 1 ε σ ˜ min{Tm ; T } = +∞ , 2 2 i∈N

B. Proof of Theorem 8

2) Proof of GA: This second step follows along the same proof-lines of Barbalat’s lemma (see e.g. [6]; see also [17] and [11, Lemma 2] for similar approaches). We proceed by contradiction. Assume that limt→∞ |x(t, x0 )|A 6= 0 for some x0 ∈ Rn . Then there exist a positive ε and a sequence (ti )i∈N such that limi→∞ ti = +∞ and ∀i ∈ N .

ti +min{Tm ;T }

which establishes the contradiction.

which contradicts (4).

|x(ti , x0 )|A > ε ,

ti

XZ i∈N

T (κ(|x0 |))σ ◦ κ(|x0 |) ≤ δ(|x0 |) ,

(9)

Notice that the sequence (ti )i∈N can be picked in such a way that ti+1 ≥ ti + Tm , ∀i ∈ N , (10) Tm designating a positive constant. Due to the GB assumption and the continuity of f , we can see that, |x˙ 1 (·, x0 )| is bounded, which implies that x(·, x0 ) is uniformly continuous. This means that, given any positive c, there exists a positive T such that, for all t ∈ R≥0 and all τ ∈ [0, T ], |x(t + τ, x0 ) − x(t, x0 )| < c. Hence, letting  σ(s) if s ∈ R≥0 σ ˜ (s) := −σ(−s) if s ∈ R<0 , and picking2 c as σ ˜ −1 (˜ σ (ε/2)/2), there exists a positive T such that, for all t ∈ R≥0 and all τ ∈ [0, T ],   1 ε |x(t + τ, x0 ) − x(t, x0 )| < σ ˜ −1 σ ˜ . (11) 2 2 Using the properties that |y + z| ≥ |y| − |z| for all y, z ∈ Rn , and σ ˜ (a − b) ≥ σ ˜ (a/2) − σ ˜ (b) for all a, b ∈ R and shortening the notation x(·, x0 ) to just x(·), it follows in view of (9) and (11) that, for all t ∈ [ti , ti + T ]

2 Note

i∈N

≥

Combining these two bounds, we obtain that

σ ˜ (|x(t)|A )

Based on what precedes and (10), we then have that Z ∞ Z ∞ σ(|x(t)|A )dt = σ ˜ (|x(t)|A )dt 0 0 X Z ti+1 ≥ σ ˜ (|x(t)|A )dt

≥ σ ˜ (|x(ti )|A − |x(t) − x(ti )|A )   1 |x(ti )|A − σ ˜ (|x(t) − x(ti )|A ) ≥ σ ˜ 2 ε 1 ε 1 ε ≥ σ ˜ − σ ˜ = σ ˜ . 2 2 2 2 2

that, even though σ may not be a class K∞ function, σ ˜ (ε/2)/2 necessarily belongs to the domain of invertibility of σ ˜ by construction.

We start by invoking [19, Corollary 1] to generate a Lyapunov function for each of the two subsystems, based on Assumptions 9 and 10. More precisely, for each i ∈ {1, 2}, there exist a smooth function Vi : Rni → R≥0 and class K∞ functions αi and αi such that, for all xi ∈ Rni , αi (|xi |Ai ) ≤ Vi (xi ) ≤ αi (|xi |)

(12)

dVi (xi )fi (xi ) ≤ −Vi (xi ) ≤ −α(|xi |Ai ) . dxi

(13)

In view of Assumption 11, the derivative of V1 along the trajectories of (1) yields, for all x = (x1 , x2 ) ∈ Rn , dV1 ˙ V1 (x1 ) ≤ −α1 (|x1 |A1 ) + (x1 ) g1 (x)g2 (|x2 |A2 ) . dx1 Let c1 : R≥0 → R≥0 be the function defined as dV1 ∀s ∈ R≥0 . c1 (s) := max (x1 ) g1 (x) , |x|≤s dx1

Due to the smoothness of V1 and the continuity of g1 , it can be seen that c1 is a continuous nondecreasing function, and we have that, for all t ∈ R≥0 , V˙ 1 (x1 (t)) ≤ −α1 (|x1 (t)|A1 ) + c1 (|x(t)|)g2 (|x2 (t)|A2 ) , where we denote x1 (·, x0 ) by just x1 (·) and similarly for x2 . From Assumption 12, there exists a class K∞ function η and a nonnegative constant µ such that |x(t, x0 )| ≤ η(|x0 |) + µ , so we obtain that V˙ 1 (x1 (t)) ≤ −α1 (|x1 (t)|A1 ) + c˜1 (|x0 |)g2 (|x2 (t)|A2 ) , (14) where c˜1 (·) := c1 (η(·) + µ). Let a1 and a2 be positive numbers such that α2 (a1 ) < g2 (a2 ) and let g˜2 : R≥0 → R≥0 be any continuous increasing function satisfying, for all s ∈ R≥0 ,  α2 (s) if s ∈ [0, a1 ] g˜2 (s) = g2 (s) if s ≥ a2 . Note that g˜2 can always be completed on the interval (a1 , a2 ) in order to be an increasing function since α2 (a1 ) < g2 (a2 ) and both α2 and g2 are class K∞

functions. Then, it can be seen that g˜2 is a class K∞ function that satisfies

continuously differentiable function V˜2 such that, for all x2 ∈ Rn2 ,

g˜2 (s) = O(α2 (s))

as s → 0+

(15a)

˜ 2 (|x2 |) α ˜ 2 (|x2 |A2 ) ≤ V˜2 (x2 ) ≤ α

g2 (s) = O(˜ g2 (s))

as s → +∞ .

(15b)

dV˜2 (x2 )f2 (x2 ) ≤ −˜ g2 (|x2 |A2 ) , dx2

Next, we need the following result. Claim 20: Let A1 and A2 be two given sets of Rn and Rm respectively. Let c be a nonnegative constant and V : Rn → R≥0 be a continuously differentiable function satisfying, for all x ∈ Rn and all u ∈ Rm ,

˜ 2 are class K∞ functions. Integrating the where α ˜ 2 and α last inequality, we obtain that Z ∞ ˜ 2 (|x20 |) . g˜2 (|x2 (t, x20 )|A2 )dt ≤ α (18) 0

α(|x|A1 ) ≤ V (x) ≤ α(|x|) dV (x)f (x, u) ≤ −α(|x|A1 ) + cγ(|u|A2 ) , dx where α, α, α and γ are class K∞ functions. Let α ˜ (resp. γ˜ ) be a class K∞ function satisfying α ˜ (s) = O(α(s)) resp. γ(s) = O(˜ γ (s))

as s → 0+
as s → +∞ .

If α, α, α, γ and V are independent of c, there exist a continuously differentiable V˜ and class K∞ functions γ˜ ˜ independent of c, such that, for all (resp. α), ˜ α ˜ and α, n x ∈ R and all u ∈ Rm , ˜ α ˜ (|x|A1 ) ≤ V˜ (x) ≤ α(|x|) dV˜ (x)f (x, u) ≤ −α ˜ (|x|A1 ) + c˜ γ (|u|A2 ) . dx The proof of the above claim follows with minor modifications that of [13, Theorem 2]. In view of (15b) and noticing that |x1 |A1 ≤ |x|, we can apply Claim 20 to V1 with γ˜ = g˜2 and c = c˜1 (|x0 |) to obtain that there exists a continuously differentiable function V˜1 such that, for all x1 ∈ Rn1 , ˜ 1 (|x1 |) α ˜ 1 (|x1 |A1 ) ≤ V˜1 (x1 ) ≤ α

(16)

and, for all t ∈ R≥0 , V˜˙ 1 (x1 (t)) ≤ −α ˜ 1 (|x1 (t)|A1 ) + c˜1 (|x0 |)˜ g2 (|x2 (t)|A2 ) , ˜ 1 ∈ K∞ . In addition, the functions α ˜ 1, α where α ˜1, α ˜ 1, α ˜1 ˜ and V1 are all independent of c˜1 (|x0 |) and consequently of x0 . Integrating the previous differential inequality, we obtain that Z ∞ Z ∞ α ˜1 (|x1 (t)|A1 )dt ≤ V˜1 (x10 )+˜ c1 (|x0 |) g˜2 (|x2 (t)|A2 )dt . 0

0

Substituting this bound into (17) then yields Z ∞ ˜ 1 (|x10 |) + c˜1 (|x0 |)α ˜ 2 (|x20 |) . α ˜1 (|x1 (t)|A1 )dt ≤ α 0

(19)

Thus, defining the following class K∞ function n p  p o σ(s) := min α ˜1 s/2 ; g˜2 s/2 ,

we get from (18) and (19) that Z ∞  σ(2 |x1 (t)|2A1 ) + σ(2 |x2 (t)|2A2 ) dt ≤ 0

˜ 2 (|x20 |) . ˜ 1 (|x10 |) + c˜1 (|x0 |)α α

Since σ is an increasing function, we have that σ(a + b) ≤ σ(2a) + σ(2b) for all a, b ∈ R≥0 . Therefore, using the fact 2 2 2 that |x|A = |x1 |A1 + |x2 |A2 , we get that Z ∞ ˜ 1 (|x0 |) + c˜1 (|x0 |)α ˜ 2 (|x0 |) . σ(|x(t)|2A )dt ≤ α 0

The GAS of A then follows from Lemma 7. C. Proof of Corollary 17 In view of Theorem 13, and noticing that Assumption 18 implies Assumption 15, it is enough to show that solutions are GB with respect to A. The proof is based on similar arguments as the one of [9, Theorem 3]. First, from the forward completeness assumption, there exists a continuous nondecreasing function ϑ : R≥0 × R≥0 → R≥0 such that, for all x0 ∈ Rn , the solution of (1) satisfies |x(t, x0 )|A ≤ |x(t, x0 )| ≤ ϑ(|x0 | , t) ,

∀t ≥ 0 . (20)

Next, in view of Assumptions 14 and 18, the derivative of V1 along the trajectories of (1) satisfies, for all x = (x1 , x2 ) ∈ Rn , V˙ 1 ≤ −α1 (|x1 |A1 ) + g11 (|x1 |A1 )g2 (|x2 |A2 ) .

(21)

Hence, in view of (16), we have that In addition, we know from Assumption 19 that there exist Z ∞ Z ∞ positive constants s0 and λ such that ˜ 1 (|x10 |)+˜ c1 (|x0 |) g˜2 (|x2 (t)|A2 )dt . α ˜1 (|x1 (t)|A1 )dt ≤ α 0 0 g11 (s) ≤ λα1 (s) , ∀s ≥ s0 . (22) (17) In order to estimate the integral in the right-hand side term of the previous bound, we follow a similar procedure. Based on (12), (13) and (15a), we apply Claim 20 to V2 with α ˜ = g˜2 and c = 0. We obtain that there exists a

Furthermore, Assumption 10 ensures that there exists a KL function β2 such that, for all x20 ∈ Rn2 , |x2 (t, x20 )|A2 ≤ β2 (|x20 | , t) ,

∀t ≥ 0 .

(23)

Using the fact that g2 is a class K∞ function, we get that, for any x20 ∈ Rn2 , there exists a nonnegative time T (|x20 |) such that 1 , ∀t ≥ T (|x20 |) . λ Note that, without loss of generality, T (·) can be picked as a continuous nondecreasing function. From (21), (22) and the previous inequality, we obtain that, for all t ≥ T (|x20 |), g2 (|x2 (t, x20 )|A2 ) ≤

|x1 (t, x0 )|A1 ≥ s0

⇒

V˙ 1 (x1 (t, x0 )) ≤ 0 .

Using a direct extension of [21, Theorem 10.2], the previous implication ensures the boundedness of |x1 (t, x0 )|A1 (and consequently, in view of (23), of |x(t, x0 )|A ) for all t ≥ T (|x20 |). In other words, there exists η ∈ K∞ and µ > 0 such that, for all x0 ∈ Rn , |x(t, x0 )|A ≤ η(|x0 |) + µ ,

∀t ≥ T (|x20 |) .

Thus, in view of (20) and recalling that T (·) is continuous and increasing, we obtain that |x(t, x0 )|A ≤ η˜(|x0 |) + µ ˜,

∀t ≥ 0 ,

where, for all s ∈ R≥0 , η˜(s) := η(s) + ϑ(s, T (s)) − ϑ(0, T (0)) µ ˜ := µ + ϑ(0, T (0)) . The conclusion follows by observing that η˜ is a class K∞ function. V. C ONCLUSION We have presented several results for the analysis of global set-stability for systems in cascade. The first one establishes the conservation of this property provided that the solutions of the cascade are globally bounded. The second one relaxes this latter requirement to just global boundedness of the solutions with respect to the set under consideration, but imposes a stronger property on the interconnection term. As for the third result, it requires only that the cascade be forward complete, but restricts the growth rate of the interconnection term with respect to the dissipation function of its Lyapunov function. One major domain of applications for this study concerns partial stability; cf. [3] for an example in marine control. R EFERENCES [1] M. Arcak, D. Angeli, and E. Sontag. A unifying integral ISS framework for stability of nonlinear cascades. SIAM J. on Contr. and Opt., 40:888–1904, 2002. [2] E. Børhaug, A. Pavlov, and K. Y. Pettersen. Croos-track formation control of underactuated surface vessels. In Proc. 45th. IEEE Conf. Decision Contr., San Diego, USA, 2006. [3] A. Chaillet. On stability and robustness of nonlinear systems – Applications to cascaded systems. PhD thesis, Universit´e Paris Sud - LSS - Sup´elec, Orsay, France, 2006. English version available at ftp://ftp.lss.supelec.fr/pub/users/chaillet/Chaillet-PhD-thesis.pdf. [4] A. Chaillet and D. Angeli. Integral input to state stable systems in cascade. Submitted to Systems and Control Letters, 2005. [5] A. Chaillet and A. Lor´ıa. Uniform semiglobal practical asymptotic stability for non-autonomous cascaded systems and applications. Submitted to Systems and Control Letters, 2005.

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