New results on sampled-data feedback stabilization for autonomous nonlinear systems

Systems & Control Letters 61 (2012) 1032–1040

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New results on sampled-data feedback stabilization for autonomous nonlinear systems J. Tsinias ∗ Department of Mathematics, National Technical University, Zografou Campus 15780, Athens, Greece

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Article history: Received 16 January 2012 Received in revised form 18 June 2012 Accepted 23 July 2012 Available online 6 September 2012 Keywords: Stabilization Sampled-data feedback Nonlinear systems

abstract Sufficient conditions are established for sampled-data feedback global asymptotic stabilization for nonlinear autonomous systems. One of our main results is an extension of the well known Artstein–Sontag theorem on feedback stabilization concerning affine in the control systems. A second aim of the present work is to provide sufficient conditions for sampled-data feedback asymptotic stabilization for two interconnected nonlinear systems. Lie algebraic sufficient conditions are derived for the case of affine in the control interconnected systems without drift terms. © 2012 Elsevier B.V. All rights reserved.

1. Introduction In the recent literature on nonlinear control theory an important area that has received much attention is the stabilization problem by means of sampled-data feedback; see for instance, [1–18] and relative references therein, where sufficient conditions are established for the existence of sampled-data and hybrid feedback controllers exhibiting stabilization. We also mention the recent contributions [19,20,12,21], where, under the presence of Control Lyapunov Functions (CLF), ‘‘triggering’’ techniques are developed for the determination of the set of the sampling time instants for the corresponding sampled-data controller. Particularly, in [20] a ‘‘minimum attention control’’ approach is adopted, exhibiting minimization of the open loop operation of the sampled-data control. In [12] a universal formula is proposed for the sample-data feedback exhibiting ‘‘event-based’’ stabilization of affine in the control systems. The corresponding result constitutes a generalization of Sontag’s well known result in [22]. In the recent author’s works [23–25] the concept of Weak Global Asymptotic Stabilization by Sampled-Data Feedback (SDFWGAS) is introduced and Lyapunov-like sufficient conditions for the existence of a sampled-data feedback stabilizer have been established. These conditions are weaker than those proposed in earlier contributions on the same subject. The present paper constitutes a continuation of the previously mentioned author’s works. The paper is organized as follows:

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0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.07.009

The current section contains the precise definition of the SDF-WGAS, as originally given in [24]. In Section 2 we establish a general result (Proposition 1), which provides Lyapunov characterizations of certain properties relying on asymptotic controllability for general nonlinear systems: x˙ = F (x, u),

( x , u) ∈ R n × R ℓ

(1.1a)

F (0, 0) = 0

(1.1b) n

m

n

where we assume that F : R × R → R is Lipschitz continuous. Sections 3 and 4 are devoted to applications of Proposition 1 for the solvability of the SDF-WGAS problem for a certain class of nonlinear systems. In Section 3 we apply the result of Proposition 1 to derive an extension of the so called Artstein–Sontag theorem (see [26,22] and also [2,3] for recent extensions). Particularly, in Proposition 2 a sufficient condition for the solvability of the SDFWGAS problem for the case of affine in the control systems: x˙ = f (x) + ug (x),

(x, u) ∈ Rn × R

f (0) = 0

(1.2a) (1.2b)

is established. This condition is weaker than the familiar hypothesis imposed in [26] and includes the Lie bracket between the vector fields f and g. In Section 4 we use the result of Proposition 1 to provide a small-gain criterion for the possibility of sampled-data feedback global stabilization for composite systems of the form:

Σ1 : x˙ = f (x, y, u) Σ2 : y˙ = g (x, y, u)

(1.3a)

(x, y, u) ∈ Rn × Rm × Rℓ f (0, 0, 0) = 0, g (0, 0, 0) = 0.

(1.3b)

J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

The corresponding result (Proposition 3) extends the main result in [25] and partially extends the well known results in the literature (see for instance [27,28]) establishing small-gain criteria for composite systems Σ1 , Σ2 with no controls and particularly those in [29–31] (see also relative references therein), where each subsystem does not necessarily satisfy the Input-to-State-Stability (ISS) property. As a consequence of Proposition 3, we provide a partial extension of the well known result in [32] due to Coron, concerning the solvability of the stabilization problem by means of smooth time-varying feedback for the affine in the control systems without drift term. In the present work we consider nonholonomic composite systems of the following form:

ξ˙ = F (ξ , u) :=

ℓ 

Fi (ξ ) :=

ui Fi (ξ )

Ai (ξ ) , Bi (ξ )

Ai (0) = 0,



i = 1, . . . , ℓ, ξ := (x, y) ∈ Rn × Rm

Bi (0) = 0,

i = 1, . . . , ℓ

satisfies properties (1.5a) and (1.5b).

2. A general result Consider the system (1.1) and assume that there exist a closed set D1 ⊂ Rn containing zero 0 ∈ Rn , an open neighborhood D2 of D1 , constants

sup {Φ (x) : x ∈ D1 } ≤ L1 ;

(2.2a)

0 < Φ (x) < L2 ,

(2.2b)

(1.4c)

Φ (x) ≥ L2 , ∀x ∈ Rn \ D2 , provided that Rn \ D2 ̸= ∅; (2.2c) Φ (x) = L2 , ∀x ∈ ∂ D2 and for every sequence {xν ∈ D2 \ D1 }

Stability : ∀ε > 0 ⇒ ∃δ = δ(ε) > 0 : |x0 | ≤ δ

∀t ≥ 0

(1.5a)

lim π (t , 0, x0 , ux0 ) = 0,

∀x0 ∈ Rn .

t →∞

(1.5b)

Definition 2. We say that (1.1) is Weakly Globally Asymptotically Stabilizable by Sampled-Data Feedback (SDF-WGAS), if for any constant σ > 0 there exist a map T : Rn → R+ satisfying T (0) = 0 and

∀x ∈ Rn \ {0}

(1.6) ℓ

and a map ϕ := ϕ(t , s, x) : R × R × R → R with ϕ(·, ·, 0) = 0, being measurable and essentially bounded with respect to the first two variables, such that for every x0 ̸= 0, a sequence of times +

+

t0 := 0 < t1 < t2 < · · · < tν < · · · ;

n

tν → ∞

(1.7)

can be found, in such a way that, if we denote: ux0 (t ) := ϕ(t , ti , π (ti )),

t ∈ [ti , ti+1 ) i = 0, 1, 2, . . .

(1.8a)

π(ti ) := π (ti , ti−1 , π (ti−1 ), ux0 ), i = 1, 2, . . . ; π (t0 ) := x0 i = 0, 1, 2, . . .

∀x ∈ D2 \ D1 ;

with lim |xν | = ∞ it holds lim sup Φ (xν ) = L2 ν→∞

ν→∞

(2.2d)

and a function β ∈ K , such that for every constant σ > 0 and for every x ∈ D2 \ D1 there exists a time τ = τ (x) ∈ (0, σ ] and a control ux (·) : [0, τ ] → Rm satisfying the following properties:

Φ (π (τ , 0, x, ux )) < Φ (x), Φ (π (t , 0, x, ux )) ≤ β(Φ (x)),

(2.3a)

∀t ∈ [0, τ ].

(2.3b)

The first two statements of the following proposition generalize the result in [24, Proposition 1] and are used in Section 4 for the derivation of sufficient conditions for SDF-WGAS for the cases (1.3) and (1.4). Proposition 1. For system (1.1) assume that (2.1)–(2.3) are fulfilled. Then the following hold: (i) For any constant σ > 0 there exists a map T : D2 \ D1 → R+ \ {0} satisfying (1.6) and a map ϕ := ϕ(t , s, x) : R+ × R+ × (D2 \ D1 ) → Rℓ with the same regularity properties as those in Definition 2, such that for every x0 ∈ D2 \ D1 there exists an increasing sequence of times {tν }, in such a way that, if we denote ux0 (t ) := ϕ(t , ti , π (ti )), t ∈ [ti , ti+1 ), i = 0, 1, 2, . . . , π (ti ) := π (ti , ti−1 , π (ti−1 ), ux0 ), i = 1, 2, . . . ; π (t0 ) := x0 , then (1.9) holds and, if we consider the resulting system (1.10), then for every neighborhood N ⊂ D2 of D1 there exists a time τ := τ (x0 ) > 0 such that its trajectory initiated from x0 ∈ D2 \ N satisfies

π (τ , 0, x0 , ux0 ) ∈ int N .

(2.4)

(ii) If, in addition to previous assumptions, we assume that D1 = {0},

L1 = 0

(2.5)

then for the trajectories of (1.10) both properties (1.5a), (1.5b) hold, provided that x0 ∈ D2 . (iii) If, in addition to (2.5), we assume that D2 = Rn ,

L2 = ∞

(2.6)

then system (1.1) is SDF-WGAS. (1.8b)

then ti+1 − ti = T (π (ti )),

(2.1)

(1.4b)

Definition 1. We say that (1.1) is Globally Asymptotically Controllable at zero (GAC), if for any x0 ∈ Rn there exists a control ux0 (·) : R+ → Rℓ such that π (t , 0, x0 ; ux0 ) exists for all t ≥ 0 and the following properties hold:

  ⇒ π (t , 0, x0 , ux0 ) ≤ ε,

(1.10)

a continuous function Φ : Rn → R+ with

Notation and definitions: Throughout the paper, we adopt the following notation. By xT we denote the transpose of a given vector x ∈ Rn . By K we denote the set containing all continuous strictly increasing functions φ : R+ → R+ with φ(0) = 0 and K∞ denotes the subset of K that is constituted by all φ ∈ K with φ(t ) → ∞ as t → ∞. We denote by π (·, s, x0 , u) the trajectory of (1.1a) with π (s, s, x0 , u) = x0 corresponding to certain (measurable and essentially bounded) control u : [s, Tmax ) → Rm , where Tmax is the corresponding maximal existing time of the trajectory.

T (x) ≤ σ ,

with ux0 (·) as defined in (1.8)

(1.4a)

and in Proposition 4 we derive a set of Lie algebraic sufficient conditions guaranteeing SDF-WGAS for the case (1.4), being weaker than the accessibility rank condition imposed in [32].

Attractivity :

t ∈ [ti , ti+1 )

0 ≤ L1 < L2 ≤ +∞,

i =1



x˙ (t ) = f (x(t ), ux0 (t )),

1033

(1.9)

and (1.1) is GAC by means of the controller ux0 (·) as defined in (1.8); equivalently, the system

Remark 1. (i) The last statement of Proposition 1 coincides with the main result obtained in [24] and constitutes a generalization of Theorem 17 in [33]. To be precise, Proposition 1 in [24] asserts that system (1.1) is SDF-WGAS, if there exist a continuous, positive definite and proper function Φ : Rn → R+ and a function β ∈ K such that, for every constant σ > 0 and for every x ∈ Rn \ {0}, a

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J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

time τ = τ (x) ∈ (0, σ ] and a control ux (·) : [0, τ ] → Rℓ can be found satisfying:

Φ (π(τ , 0, x, ux )) < Φ (x), Φ (π(t , 0, x, ux )) ≤ β(Φ (x)),

(2.7a)

∀t ∈ [0, τ ].

(2.7b)

Notice that both (2.7a), (2.7b) are fulfilled with β(s) = s, if we assume that for every constant σ > 0 and for every x ∈ Rn \ {0} there exists a time τ = τ (x) ∈ (0, σ ] and a control ux (·) : [0, τ ] → Rℓ satisfying:

Φ (π(τ , 0, x, ux )) < Φ (x),

∀t ∈ (0, τ ].

(2.8)

Obviously, (2.8) is fulfilled, if Φ is a C 1 (global) CLF for (1.1), namely, if we assume that for every x ∈ Rn \{0} there exists a vector u ∈ Rm satisfying DΦ (x)F (x, u) < 0. Of course, the converse claim is not in general true. It turns out, that the above hypotheses (2.7a), (2.7b) guaranteeing SDF-WGAS, are weaker than those adopted in earlier works on the literature, that are mainly based on the existence of a C 1 CLF. (ii) It is worthwhile noticing that, as is shown in [24, Proposition 1], the statement (iii) of Proposition 1 is equivalent to the existence of mappings Φ : Rn → R+ , a1 , a2 ∈ K∞ , Φ being generally discontinuous on Rn \ {0} with a1 (|x|) ≤ Φ (x) ≤ a2 (|x|), ∀x ∈ Rn and a function β ∈ K such that for every constant σ > 0, a pair of lower semi-continuous (LSC) mappings T : Rn \{0} → R+ \{0} and q : Rn → R+ \ {0} can be found in such a way that (1.6) holds and for every x ∈ Rn \ {0} there exists a control ux (·) : [0, T (x)] → Rℓ satisfying

Φ (π(T (x), 0, x, ux )) ≤ Φ (x) − q(x) and Φ (π(t , 0, x, ux )) ≤ β(Φ (x)), ∀t ∈ [0, T (x)].

(2.9a) (2.9b)

for a certain constant q := q(x) > 0. Using (2.9) we can determine a locally finite covering of D2 \ D1 of closed spheres Sν := S [xν , rν ] ⊂ D2 \ D1 , ν = 1, 2 . . . of radius rν centered at xν ∈ D2 \ D1 , times τν , controls uν : [0, τν ] → Rℓ and constants qν > 0, in such a way that both (2.9a), (2.9b) are fulfilled, with x := xν , τ := τν , u := uν andq := qν , for every  y ∈ Sν , t ∈ [0, τν ]. For each x ∈ D2 \ D1 let Sνi , i = 1, . . . , j be the sequence of those spheres above for which x ∈ ∩Sνi for 1 ≤ i ≤ j and x ̸∈ Sν for ν ̸= ν1 , . . . , νj . Then an integer k = k(x) ∈ {ν1 , ν2 , . . . , νj } can be found such that τk := min{τν1 , τν2 , . . . , τνj }. Define ux (·) := uk(x) (·), T (x) := τk(x) and q¯ (x) := qk(x) . Then, because of the local finiteness of covering, it follows by taking into account (2.9) that

Φ (π(T (x), 0, x, ux )) ≤ Φ (x) − q¯ (x), Φ (π(t , 0, x, ux )) ≤ β¯ (Φ (x)) ,

∀t ∈ [0, T (x)]; ∀x ∈ D2 \ D1 .

(2.10a) (2.10b)

Furthermore, the mappings T and q¯ take strictly positive values, T is LSC and satisfies (1.6) and q¯ satisfies the following property: lim inf q¯ (x) > 0,

x→a

∀a ∈ D2 \ D1 .

(2.11)

We are in a position to show (2.4). Define:

ϕ(t , s, x) := ux (t − s),

t ∈ [s, s + T (x)), x ∈ D2 \ D1

xν := π (tν , tν−1 , xν−1 , uν ), uν (s) := ϕ(s, tν−1 , xν−1 ) = uν−1 (s − tν−1 ), tν − tν−1 := T (xν−1 ), ν = 1, 2, . . . Φν := Φ (xν ), ν = 0, 1, 2, . . .

s ∈ [tν−1 , tν ),

and

(2.13)

provided that xν ∈ D2 \ D1 , ν = 1, 2, . . . , where xν |ν=0 above, coincides with initial value x0 and t0 := 0. It follows from (2.10) and (2.13) that

Φν ≤ Φν−1 − q¯ (xν−1 ) ¯ Φν−1 ), Φ (π (s, tν−1 , xν−1 , uν )) ≤ β(

(2.14a)

s ∈ [tν−1 , tν ), ν = 1, 2, . . .

(2.14b)

therefore, by taking into account (2.14a), it follows:

Φν ≤ Φ0 −

ν 

q¯ (xi ),

ν = 1, 2, . . . .

(2.15)

i=0

From (2.13), (2.15), (2.2b) and (2.2c) we get 0 ≤ Φν ≤ Φ0 < L2 ,

ν = 1, 2, . . . .

(2.16)

ℓ

Let w = wx0 : R → R be the concatenation of the controls uν : [tν−1 , tν ) → Rℓ above, namely: +

w(t ) := ϕ(t , tν−1 , xν−1 ) = uν (t ),

t ∈ [tν−1 , tν ).

(2.17)

Then by (2.16), (2.17) and (2.14b) it follows:

Proof of Proposition 1. From (2.3) and by taking into account continuity of the map Φ and continuity of the solution of (1.1) with respect to the initial values, it follows that for every x ∈ D2 \ D1 , a time τ = τ (x) ∈ (0, σ ] and a control u = ux : [0, τ ] → Rℓ can be found such that, if we denote β¯ := 2β , the following holds:

Φ (π(τ , 0, y, u)) < Φ (y) − q, Φ (π(t , 0, y, u)) ≤ β¯ (Φ (y)) , ∀t ∈ [0, τ ], for every y near x

where ux and T are determined in (2.10). Let x0 ∈ D2 \ N and consider the corresponding control u0 := ux0 satisfying (2.10) with x := x0 . Define:

(2.12)

¯ Φν−1 ) ≤ β( ¯ Φ0 ), Φ (π (s, tν−1 , xν−1 , w)) ≤ β( s ∈ [tν−1 , tν ), ν = 1, 2, . . .

(2.18)

provided that xν ∈ D2 \ D1 , ν = 1, 2, . . . . In order to establish that (2.4) holds for x0 ∈ D2 outside N, it suffices to show that

∃ integer ν := ν(x0 ) : xν ∈ int N .

(2.19)

Indeed, notice first by virtue of (2.13), (2.16), (2.2b) and (2.2c) that

{xν , ν = 1, 2, . . .} ⊂ D2

(2.20)

and suppose on the contrary that {xν , ν = 1, 2, . . .} ∩ int N = ∅. Then, there would exist a subsequence

 ′

xν ⊂ {xν } ⊂ D2 \ N

(2.21)

such that one of the following holds: x′ν → a

for certain a ∈ D2 ∪ ∂ N

→ a for certain a ∈ ∂ D2   lim x′  = ∞.

x′ν

ν→∞

ν

(2.22a) (2.22b) (2.22c)

To exclude (2.22a), we take ∞ into account (2.15), (2.16) and (2.20), which guarantee that i=0 q¯ (xi ) < ∞. It then follows that q¯ (xν ) → 0 and therefore q¯ (x′ν ) → 0 with x′ν → a ∈ D2 \ D1 , which contradicts (2.11). Suppose next that (2.22b) holds. Then by (2.2d) we may assume that without of generality it holds that Φ (x′ν ) → L2 . On the other hand, by virtue of (2.16), lim supν→∞ Φ (x′ν ) ≤ Φ0 < L2 , a contradiction. Likewise, case (2.22c) is excluded, because of assumption (2.2d). Therefore, (2.4) is established and this completes the proof of statement (i). In order to show statement (ii), we first use the result of statement (i), which in conjunction with (2.5) asserts that (2.4) holds for any neighborhood N ⊂ Rn of zero. The latter implies

J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

1035

(1.5b). In order to show (1.5a), we invoke (2.5) and (2.2b)–(2.2d), which guarantee the existence of a function a1 ∈ K∞ such that

Proof. Let σ > 0 and state x0 ̸= 0. According to our assumption (3.1) we may distinguish three cases.

a1 (|x|) ≤ Φ (x),

Case 1: (g Φ )(x0 ) ̸= 0 and (f Φ )(x0 ) < 0. Then, by considering the constant input

∀x ∈ D2 .

(2.23)

It then follows, by virtue of (2.18), (2.20) and (2.23), that the corresponding trajectory π (·, 0, x0 ; w) of (1.1a) satisfies: 1 ¯ |π(t , 0, x0 , w)| ≤ a− 1 (β(Φ (x0 ))),

∀t ≥ 0, x0 ∈ D2

(2.24)

which establishes (1.5a). The proof of last statement is an immediate consequence of statement (i) and (ii). The details are left to the reader.  The following remark plays an important role in the proof of Proposition 3 in Section 3. Remark 2. The result of statement (i) of Proposition 1 is also valid, if we replace (2.2d) plus (2.3) by the following hypothesis: There exist a continuous and positive definite function Φ : Rn → R+ satisfying (2.2a)–(2.2c), a function β ∈ K and a map q : R+ → R+ \ {0} such that lim inf q(s) > 0,

s→a

∀a ∈ R+ \ {0};

(2.25)

inf {Φ (x), x ∈ D2 \ N } > 0

(2.26)

for any closed neighborhood N ⊂ D2 of D1 and in such a way that, instead of (2.3), the following holds: for every constant σ > 0 and for every x ∈ D2 \ D1 , a time τ = τ (x) ∈ (0, σ ] and a control ux (·) : [0, τ ] → Rℓ can be found satisfying:

Φ (π(τ , 0, x, ux )) < Φ (x), Φ (π(t , 0, x, ux )) ≤ β(Φ (x)) − q(Φ (x)),

(2.27a)

∀t ∈ [0, τ ].

(2.27b)

The proof of the above claim is precisely the same with that given for the first statement of Proposition 1. 3. An extension of Artstein–Sontag theorem on feedback stabilization The well known Artstein–Sontag theorem on stabilization for the affine in the control case (1.2) asserts that existence of a CLF, namely, of a C 1 , positive definite and proper function Φ : Rn → R+ satisfying the implication (DΦ g )(x) = 0, x ̸= 0 ⇒ (DΦ f )(x) < 0, guarantees existence of a C ∞ (Rn \{0}) feedback law exhibiting global asymptotic stabilization for (1.2). We next apply the result of Section 2 to get a generalization of this result for the case (1.2). Our purpose is to show that, under weaker hypotheses, we may succeed SDF-WGAS for (1.2). Assume that f and g are C 2 and there exists a C 2 , positive definite and proper function Φ : Rn → R+ such that the following implication holds:

(g Φ )(x) = 0,  either (f Φ )(x) < 0, x ̸= 0 ⇒ or (f Φ )(x) = 0; ([f , g ]Φ )(x) ̸= 0

ux0 := (−1 − (f Φ )(x0 ))/(g Φ )(x0 )

(3.2)

there exists a time τ = τ (x0 ) ∈ (0, σ ] such that the trajectory π (·) = π (·, x0 , u) : [0, τ ] → Rn of system (1.2a) with π (0, x0 , u) = x0 satisfies:

Φ (π (t , x0 , u)) < Φ (x0 ),

∀t ∈ (0, τ ].

(3.3)

It turns out from (3.3) and Remark 1(i) that both (2.7a), (2.7b) hold with β(s) = s. Case 2: (g Φ )(x0 ) = 0 and (f Φ )(x0 ) < 0. Likewise for this case, we obtain (3.3), thus (2.7a), (2.7b) hold with β(s) = s and zero input. Case 3: (g Φ )(x0 ) = 0 and (f Φ )(x0 ) = 0. Then, according to our assumption (3.1), it also holds:

([f , g ]Φ )(x0 ) ̸= 0.

(3.4)

Let τ ∈ (0, 12 σ ] and a pair of constants ui , i = 1, 2 yet to be determined. Let X := f + u2 g ,

Y := f + u1 g

(3.5)

and consider the corresponding trajectories Xt (r ) := π (t , r , u2 ), Yt (r ) := π (t , r , u1 ) of systems x˙ = X (x) and y˙ = Y (y), respectively, initiated at time t = 0 from r ∈ Rn . Also, consider the trajectory of (1.2a):



R(t ) := (Xt ◦ Yt )(x0 ),

t ∈ 0,

1 2

σ

 (3.6)

corresponding to the concatenation of u1 and u2 and of initial state x0 . Then we have: R˙ (t ) = X (R(t )) + (DXt Y ) ◦ Yt (x0 )

= X (R(t )) + ((DXt Y ) ◦ X−t ) ◦ R(t )  t = X (R(t )) + Y (R(t )) + (DXs [Y , X ]) ◦ X−s (R(t ))ds

(3.7)

0

and therefore: R¨ (t ) = (X 2 + Y 2 + XY + YX ) ◦ R(t ) + (DX (R(t ))



t

+ DY (R(t ))) (DXs [Y , X ]) ◦ X−s (R(t ))ds 0  t  + D(DXs (R(t ))[Y , X ] ◦ X−s (R(t )))ds R˙ (t ) 0

+ (DXt [Y , X ]) ◦ Yt (x0 ).

(3.8)

It follows from (3.7) and (3.8) that (3.1)

where [·, ·] denotes the Lie bracket operator and we have used the standard notation XY := DYX for any pair of C 1 mappings X : Rn → Rn , Y : Rn → Rn . We also recall the well known property that [X , Y ]Φ = XY Φ − YX Φ for any pair of C 2 mappings X , Y : Rn → Rn and C 2 function Φ : Rn → R. Proposition 2. Under previous hypotheses (3.1), system (1.2) is SDF-WGAS.

R˙ (0) = X (x0 ) + Y (x0 ),

(3.9a)

R¨ (0) = (X + Y + 2YX )(x0 ).

(3.9b)

2

2

Since (g Φ )(x0 ) = 0 and (f Φ )(x0 ) = 0, it follows from (3.5) and (3.9a) that DΦ (x0 )R˙ (x0 ) = DΦ (x0 )(X (x0 ) + Y (x0 )) = 0.

(3.10)

We next define: m(t ) := Φ (R(t )).

(3.11)

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J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

4. SDF-WGAS for composite systems and the small-gain property

Then, by invoking (3.5) and (3.9)–(3.11), we get:

˙ (0) + m(t ) = m(0) + t m

t

2

¨ (0) + 0(t 2 ) m 2 = Φ (x0 ) + t (X Φ + Y Φ )(x0 )

Consider the composite system (1.3), where f , g are Lipschitz continuous. For every (x0 , y0 ) ∈ Rn × Rm we denote in the sequel by (x(t ), y(t )) = (x(t , x0 , y0 , u), y(t , x0 , y0 , u)) the solution of (1.3) with initial (x(0), y(0)) = (x0 , y0 ) ∈ Rn × Rm corresponding to certain control u. The main result of this section extends [25, Proposition 2]. For the subsystems Σ1 , Σ2 in (1.3a) we make the following hypotheses: H1. There exist continuous functions V : Rn → R+ and a1 , a2 ∈ K∞ with

t2

(((X + Y )T D2 Φ (X + Y ))(x0 ) 2 + ((X 2 )Φ + (Y 2 )Φ + 2(YX )Φ )(x0 )) + 0(t 2 ) +

t2

(((2f + (u2 + u1 )g )T D2 Φ 2 × (2f + (u2 + u1 )g ))(x0 ) + DΦ (x0 )

= Φ (x0 ) +

× (4f 2 + (3u2 + u1 )fg + (u2 + 3u1 )gf + (u2 + u1 )2 g 2 )(x0 )) + 0(t 2 )

(3.12)

where limt →0+ (O(t )/t ) = 0. By defining w := u2 + u1 , u := u1 , equality (3.12) is rewritten: m(t ) = Φ (x0 ) +

t2

(((2f + w g ) D Φ (2f + wg ))(x0 ) 2 + DΦ (x0 )(4f 2 + (3w − 2u)fg T

t2 2

+ 2u([g , f ])(x0 ) + w (g )(x0 ))) + 0(t ).

(3.13)

((2f + wg )T D2 Φ (2f + w g ))(x0 ) + DΦ (x0 )(4f 2 (x0 ) + w(gf )(x0 ) + 3w(fg )(x0 ))

Φ (R(t )) < Φ (x0 ) − t > 0 near zero

2

(3.14)

∀t ∈ [0, τ1 ]

(4.3b)

∀t ∈ [0, τ1 ].

H2. There exist continuous functions W : R b1 , b2 ∈ K∞ with b1 (|y|) ≤ W (y) ≤ b2 (|y|),

∀y ∈ Rm

(4.4)

and functions β2 , Γ2 ∈ K , such that for any σ > 0 and for every (x0 , y0 ) ∈ Rn × Rm with (4.5)

there exist a time τ2 = τ2 (x0 , y0 ) ∈ (0, σ ] and a control u2 = u2(x0 ,y0 ) (t ) : [0, τ2 ] → Rℓ satisfying: (4.6a)

W (y(t , x0 , y0 , u2 )) ≤ β2 (W (y0 )),

2

(3.15)

where R(t ) = (Xt ◦ Yt )(x0 ); Xt (r ) = π (t , r , w − u), Yt (r ) = π(t , r , u) with arbitrary constant input w and constant control u as defined in (3.14). As in the Cases 1 and 2, it follows from (3.15) that there exists a time τ = τ (x0 ) ∈ (0, σ ] such that the trajectory π(·) = π (·, x0 , u) : [0, τ ] → Rn of system (1.2a) satisfies (3.3), hence, we again conclude that (2.7a), (2.7b) hold with β(s) = s. It follows, by invoking Remark 1(i) and statement (iii) of Proposition 1, that system (1.2) is SDF-WGAS.  Remark 3. The result of Proposition 2 can directly be extended to multi-input affine in the control systems. Also, we note that, by considering higher derivatives of the map t → Φ ((Xt ◦ Yt )(x0 )), we can get more general algebraic conditions in terms of f , g and Φ for SDF-WGAS, provided that that dynamics of (1.2) and the function Φ are smooth enough. Example 1. We illustrate the nature of Proposition 2 by considering the elementary planar case (1.2) with dynamics f (x) := (a(x1 , x2 ), a(x1 , x2 ))T , g (x) := (x1 , −x2 )T , x := (x1 , x2 ) ∈ R2 , where a : R → R+ is C 1 and satisfies a(0, 0) = 0 and x1 a(x1 , x1 ) < 0 and a(x1 , −x1 ) ̸= 0 for every x1 ̸= 0. Then it can be easily verified that system above satisfies (3.1) with V = 12 (x21 + x22 ), hence, it is SDF-WGAS.

(4.3c) +

→ R and

W (y(τ2 , x0 , y0 , u2 )) < W (y0 )

t + O(t ) < Φ (x0 ), 2

(4.3a)

Γ2 (|x0 |) < |y0 |

and therefore from (3.13) and (3.14) we get: c

(4.2)

m

By taking into account (3.4), it follows that every pair of constants w and c > 0, a constant u can be found such that

+ 2u([g , f ]Φ )(x0 ) + w2 (g 2 )(x0 ) ≤ −c

|y0 | < γ1 (|x0 |)

V (x(t , x0 , y0 , u1 )) ≤ β1 (V (x0 )),

+ DΦ (x0 )((4f )(x0 ) + w(gf )(x0 ) + 3w(fg )(x0 ) 2

and functions β1 , γ1 ∈ K such that for any σ > 0 and for every (x0 , y0 ) ∈ Rn × Rm with

|y(t , x0 , y0 , u1 )| < γ1 (|x(t , x0 , y0 , u1 )|),

2

2

(4.1)

V (x(τ1 , x0 , y0 , u1 )) < V (x0 )

(((2f + w g )T D2 Φ (2f + wg ))(x0 ) 2

∀x ∈ R n

there exist a time τ1 = τ1 (x0 , y0 ) ∈ (0, σ ] and a control u1 = u1(x0 ,y0 ) (t ) : [0, τ1 ] → Rℓ satisfying:

2

+ (w + 2u)gf + w2 g 2 )(x0 )) + 0(t 2 ) = Φ ( x0 ) +

a1 (|x|) ≤ V (x) ≤ a2 (|x|),

∀t ∈ [0, τ2 ]

Γ2 (|x(t , x0 , y0 , u2 )|) < |y(t , x0 , y0 , u2 )|,

∀t ∈ [0, τ2 ].

(4.6b) (4.6c)

In addition to H1 and H2, the following hold: H3a. Small-gain property: 1 −1 (b1 ◦ γ1 ◦ a− 2 )(s) > (b2 ◦ Γ2 ◦ a1 )(s),

∀s > 0.

(4.7)

H3b. For any σ > 0 and for every nonzero (x0 , y0 ) ∈ Rn × Rm with

|y0 | < γ1 (|x0 |); Γ2 (|x0 |) < |y0 |

(4.8)

there exist a time τ = τ (x0 , y0 ) ∈ (0, σ ] and a control u = u(x0 ,y0 ) (t ) : [0, τ ] → Rℓ satisfying: V (x(τ , x0 , y0 , u)) < V (x0 ) V (x(t , x0 , y0 , u)) ≤ β1 (V (x0 )),

(4.9a)

∀t ∈ [0, τ )

W (y(τ , x0 , y0 , u)) < W (y0 ) W (y(t , x0 , y0 , u)) ≤ β2 (W (y0 )),

(4.9b) (4.9c)

∀t ∈ [0, τ ).

(4.9d)

The main result in [25] guarantees that, under H1–H3, the composite system (1.3) is SDF-WGAS provided that γ1 ∈ K∞ . The following proposition establishes that, when γ1 is bounded, system (1.3) is SDF-WGAS under H1–H3 plus some more appropriate hypotheses. Proposition 3. For the subsystems system Σ1 , Σ2 of (1.3) assume that H1, H2 and H3 are fulfilled. Then

J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

(i) The composite system (1.3) is SDF-WGAS, provided that γ1 ∈ K∞ . (ii) If γ1 is bounded, then system (1.3) is SDF-WGAS, provided that, in addition to (4.7), it holds: 1 −1 lim (b1 ◦ γ1 ◦ a− 2 )(s) > r := lim (b2 ◦ Γ2 ◦ a1 )(s)

s→∞

s→∞

(4.10)

and there exists a map q : R+ → R+ \ {0} satisfying (2.25) and such that, for any σ > 0 and for every (x0 , y0 ) ∈ Rn × Rm with W (y0 ) ≥ r

(4.11a)

1037

for certain τ2 ∈ (0, σ ] and control u2 ;

ℓi (V (x0 )) = W (y0 ), (x0 , y0 ) ̸= 0 ℓ (V (x(τ , x , y , u))) < ℓ (V (x )) i 0 0 i 0    ℓi (V (x(t , x0 , y0 , u))) ≤ β2 (ℓi (V (y0 ))),   ∀t ∈ [0, τ ] ⇒ W (y(τ , x0 , u0 , u)) < W (y0 )     W (y(t , x0 , y0 , u)) ≤ β2 (W (y0 )), ∀t ∈ [0, τ ]

(4.18)

there exist a time τ2 = τ2 (x0 , y0 ) ∈ (0, σ ] and a control u2 = u2(x0 ,y0 ) (t ) : [0, τ2 ] → Rℓ satisfying (4.6b) and simultaneously, instead of (4.6a), the following stronger property holds:

for certain τ ∈ (0, σ ] and control  u.  Next, define B(s) := 2 max β¯ 1 (s), β2 (s) and consider the pair of continuous functions:

W (y(τ2 , x0 , y0 , u2 )) < W (y0 ) − q(W (y0 )).

Ψi (x, y) :=

(4.11b)

Proof. The proof of statement (i) has been given in [25]. Suppose next that γ1 is bounded. Notice that (4.7) and the boundedness of γ1 implies that Γ2 is also bounded and therefore r < ∞, as the latter defined in (4.10). To establish our claim we generalize the analysis adopted in [25]. First, by (4.7) and (4.10) we can find a pair of bounded functions ℓi ∈ K ∩ C 1 (R+ \ {0}), i = 1, 2 with

(b2 ◦ Γ2 ◦ α1−1 )(s) < ℓ1 (s) < ℓ2 (s) < (b1 ◦ γ1 ◦ α2−1 )(s), ∀s > 0

(4.12a)

and in such a way that, if we define: Ri := lim ℓi (t ), t →∞

i = 1, 2

(4.12b)

then R1 > R2 > r .

(4.12c)

Obviously, since each ℓi is strictly increasing and bounded, we have: Ri > ℓi (s),

∀s ≥ 0, i = 1, 2

(4.13)

and the latter in conjunction with (4.1) implies:

ℓi (V (x)) < Ri for all x ∈ Rn , i = 1, 2.

(4.14)

Next, by using (4.1), (4.4) and (4.12a), we can easily verify that for each i = 1, 2 it holds: W (y0 ) < ℓi (V (x0 )),

(x0 , y0 ) ̸= 0 ⇒ (4.2)

(4.15a)

W (y0 ) > ℓi (V (x0 )),

(x0 , y0 ) ̸= 0 ⇒ (4.5)

(4.15b)

W (y0 ) = ℓi (V (x0 )),

(x0 , y0 ) ̸= 0 ⇒ (4.8).

(4.15c)

Let β¯ 1 ∈ K satisfying (ℓi ◦β1 )(s) ≤ (β¯ 1 ◦ℓi )(s), ∀s > 0, i = 1, 2. Then, by virtue of (4.3)–(4.6), (4.8), (4.9) and (4.15), the following properties hold for each i = 1, 2: W (y0 ) < ℓi (V (x0 )),

(x0 , y0 ) ̸= 0  ℓi (V (x(τ1 , x0 , y0 , u1 ))) < ℓi (V (x0 ))   ℓ (V (x(t , x , y , u ))) ≤ β¯ (ℓ (V (x ))),  i 0 0 1 1 i 0 ⇒ ∀t ∈ [0, τ1 ]    W (y(t , x0 , y0 , u1 )) < ℓi (V (x(t , x0 , y0 , u1 ))), ∀t ∈ [0, τ1 ]



W (y), for W (y) ≥ ℓi (V (x)) ℓi (V (x)), for ℓi (V (x)) > W (y),

i = 1, 2.

(4.19)

By exploiting (4.16)–(4.19) for i = 1 and applying the same arguments with those in proof of Proposition 2 in [25], we can establish the following property: P1. For each σ > 0 we can determine a map T : (Rn × Rm ) \ {0} → R+ \{0} satisfying T (x, y) ≤ σ for all (x, y) ∈ (Rn ×Rm )\{0} and in such a way that for every (x0 , y0 ) ∈ (Rn × Rm ) \ {0}, there exists a control u = u(x0 ,y0 ) : [0, T (x0 , y0 )] → Rℓ satisfying:

Ψ1 (x(T (x0 , y0 ), x0 , y0 , u), y(T (x0 , y0 ), x0 , y0 , u)) < Ψ1 (x0 , y0 ) Ψ1 (x(t , x0 , y0 , u), y(t , x0 , y0 , u)) ≤ B(Ψ1 (x0 , y0 )), ∀t ∈ [0, T (x0 , y0 )].

(4.20a) (4.20b)

For i = 2, we consider the function Ψ2 (·, ·) as defined by (4.19) and we exploit our additional assumption that (4.11a) implies both (4.11b) and (4.6b). Notice that, when W (y) > R2 , it follows from (4.14) and (4.12c) that W (y) > r > ℓ2 (V (x)) thus, by (4.19) we have Ψ2 (x, y) = W (y). The latter in conjunction with second inequality in (4.17) for i = 2, (which is a consequence of (4.6b) and definition of B(·)), together with implication (4.11a) ⇒ (4.11b), imply the following property: P2. For every σ > 0 there exists a map T : (Rn × Rm ) \ {0} → + R \ {0} satisfying T (x, y) ≤ σ for all (x, y) ∈ (Rn × Rm ) \ {0} such that for every (x0 , y0 ) ∈ (Rn × Rm ) \ {0} with W (y0 ) ≥ R2 , a control u = u(x0 ,y0 ) : [0, T (x0 , y0 )] → Rℓ can be found satisfying:

Ψ2 (x(T (x0 , y0 ), x0 , y0 , u), y(T (x0 , y0 ), x0 , y0 , u)) < Ψ2 (x0 , y0 ) − q(Ψ2 (x0 , y0 )); Ψ2 (x(t , x0 , y0 , u), y(t , x0 , y0 , u)) ≤ B(Ψ2 (x0 , y0 )), ∀t ∈ [0, T (x0 , y0 )].

(4.21a) (4.21b)

We are in a position, by exploiting Properties P1 and P2 and the results of Proposition 1 and Remark 2, to establish that (1.3) is SDF-WGAS. We distinguish two cases: Case 1: (x0 , y0 ) ∈ Rn × Rm : W (y0 ) < R1 , as the latter is defined by (4.12b). We define:

(4.16)

Φ (x, y) := Ψ1 (x, y), L1 := 0,

β := B,

L2 := R1 ;

(4.22a)

for certain τ1 ∈ (0, σ ] and control u1 ;

D1 := {(0, 0) ∈ Rn × Rm };

(4.22b)

ℓi (V (x0 )) < W (y0 ), (x0 , y0 ) ̸= 0  W (y(τ2 , x0 , y0 , u2 )) < W (y0 )   W (y(t , x0 , y0 , u2 )) ≤ β2 (W (y0 )),  ∀t ∈ [0, τ2 ] ⇒   ℓ i (V (x(t , x0 , u0 , u2 )))   < W (y(t , x0 , y0 , u2 )), ∀t ∈ [0, τ2 ]

D2 := {(x, y) ∈ Rn × Rm : R1 > W (y) ≥ 0}.

(4.22c)

(4.17)

Then, we can verify that all conditions of second statement of Proposition 1 are fulfilled with Φ , β, L1 , L2 , D1 and D2 as above. Indeed, note that, for (x, y) ∈ D2 \ {0}, we have, due to (4.19) and (4.22a), (4.22c), that Ψ1 (x, y) = W (y) ≤ R1 (=L2 ) for the case ℓ1 (V (x)) ≤ W (y) and, due to (4.14), (4.19) and (4.22a), (4.22c),

1038

J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

it holds that Ψ1 (x, y) = ℓ1 (V (x)) ≤ R1 (=L2 ) when ℓ1 (V (x)) ≥ W (y), hence, (2.2b) is satisfied. Also, for (x, y) ̸∈ D2 we have, due to (4.14), that W (y) ≥ R1 > ℓ1 (V (x)), hence, by (4.19) it follows that Ψ1 (x, y) = W (y) ≥ R1 (=L2 ), therefore (2.2c) holds as well. We next show (2.2d). Obviously, by (4.22c) we have W (a2 ) = R1 for every a := (a1 , a2 ) ∈ ∂ D2 , which, in conjunction with (4.14) and (4.19), implies that Ψ1 (a) = W (a2 ) = R1 . Consider next a sequence {(xν , yν ) ∈ D2 } with limν→∞ |(xν , yν )| = ∞. We show that lim supν→∞ Ψ1 (xν , yν ) = R1 . Because of (4.4) and (4.22c), the sequence {yν } is bounded, hence, limν→∞ |xν | = ∞. Then, by (4.1) and (4.12b) we have: lim ℓi (V (xν )) = R1 .

xν →∞

(4.23)

Since R1 > W (yν ), ν = 1, 2, . . . , we may distinguish two ′ ′ cases.  ′ The  first is R1 > W (yν ) ≥ ℓ1 (V (xν )) for certain subsequence ′ (xν , yν ) ⊂ {(xν , yν )} and therefore by (4.19), R1 > Ψ1 (x′ν , y′ν ) ≥ ℓ1 (V (x′ν )). The latter in conjunction with (4.23) implies: lim Ψ1 (x′ν , y′ν ) = R1 .

ν→∞

(4.24)

′ ′ The other  case is R1 > ℓ1 (V (xν )) > W (yν ) for certain subsequence (x′ν , y′ν ) ⊂ {(xν , yν )}. Then by (4.19) it follows that Ψ1 (x′ν , y′ν ) = ℓ1 (V (x′ν )), hence, by (4.23) we again obtain (4.24). The above discussion asserts that (2.2d) is fulfilled. Finally, (2.3a), (2.3b) is a consequence of Property P1 and definitions (4.22). We can therefore apply the result of Proposition 1(ii) to establish the existence of a sampled-data controller u = u(x0 ,y0 ) (·), (x0 , y0 ) ∈ D2 , such that the corresponding trajectory (x(·, x0 , y0 , u(x0 ,y0 ) ), y(·, x0 , y0 , u(x0 ,y0 ) )) of (1.3a) is tending to zero (0, 0) ∈ Rn × Rm as t → ∞. Also, according to the second statement of Proposition 1, we have:

∀ε > 0 ⇒ ∃δ = δ(ε) > 0 : |(x0 , y0 )| ≤ δ   ⇒ x(t , x0 , y0 , u(x0 ,y0 ) ), y(t , x0 , y0 , u(x0 ,y0 ) ) ≤ ε, ∀t ≥ 0.

(4.25)

Case 2: (x0 , y0 ) ∈ (Rn × Rm ) : W (y0 ) ≥ R1 . Let R¯ be a constant such that R1 > R¯ > R2 and define:

Φ (x, y) := Ψ2 (x, y), β := B, L1 := R2 , L2 := ∞;

(4.26a)

D1 := {(x, y) ∈ Rn × Rm : W (y) ≤ R2 };

(4.26b)

N := {(x, y) ∈ Rn × Rm : W (y) ≤ R¯ };

(4.26c)

D2 := Rn × Rm .

(4.26d)

According to definitions (4.26) and by taking into account Property P2, we can easily verify that all conditions of Remark 2 are satisfied with Φ , β, L1 , L2 , D1 and D2 as above. (For completeness, we note that (2.26) and (2.27) are consequences of (4.4), (4.19), (4.21) and (4.26).) Therefore, according to the Remark 2, there exists a sampled-data controller u = u(x0 ,y0 ) (·), (x0 , y0 ) ∈ (Rn × Rm ) \ D1 such that for (x0 , y0 ) ∈ (Rn × Rm ) \ N the corresponding trajectory (x(·, x0 , y0 , u(x0 ,y0 ) ), y(·, x0 , y0 , u(x0 ,y0 ) )) of (1.3a) enters int N after some finite time, say τ1 ; particularly, if we denote:

(x1 , y1 ) := (x(τ1 , x0 , y0 , u(x0 ,y0 ) ), y(τ1 , x0 , y0 , u(x0 ,y0 ) ))

(4.27)

then we have:

Ψ2 (x1 , y1 ) = W (y1 ) < R¯

(4.28)

therefore W (y1 ) < R1 . We then apply the arguments used for the Case 1 with (x0 , y0 ) := (x1 , y1 ) as above and find a sampled-data controller u¯ = u¯ (x1 ,y1 ) (·) such that the trajectory (x(·, x1 , y1 , u¯ (x1 ,y1 ) ), y(·, x1 , y1 , u¯ (x1 ,y1 ) )) of (1.3a) is tending to zero (0, 0) ∈ Rn × Rm as t → ∞. It turns out, by considering

the concatenation w = w(x0 ,y0 ) (·) of the sampled-data controls u = u(x0 ,y0 ) (·), t ∈ [0, τ1 ) and u¯ = u¯ (x1 ,y1 ) (·), t ∈ [τ1 , ∞), that the corresponding trajectory (x(·, x1 , y1 , w(x0 ,y0 ) ), y(·, x1 , y1 , w(x0 ,y0 ) )) of (1.3a) is tending to zero (0, 0) ∈ Rn × Rm as t → ∞. The latter in conjunction with (4.25) asserts that the composite system (1.3) is SDF-WGAS.  Remark 4. (i) It can be easily verified that implication (4.11a) ⇒ ((4.6b) plus (4.11b)) is fulfilled, if W is C 1 and there exists a function d ∈ K , such that for every (x, y) ∈ Rn × Rm with W (y) ≥ r there exists a vector u ∈ Rℓ such that DW (y)G(x, y, u) ≤ −d(W (y)). (ii) As is pointed out in [25], the sufficient conditions of Proposition 3 are simplified when, either f , or g is independent of u. Particularly, for the case of systems Σ1 : x˙ = f (x, y); Σ2 : y˙ = g (x, y, u) assume that, instead of H1 for Σ1 , there exist continuous functions V : Rn → R+ and a1 , a2 ∈ K∞ , γ1 , β1 ∈ K such that (4.1) holds and in such a way that for every (x0 , y0 ) ∈ Rn × Rm , y0 ̸= 0 for which (4.2) holds and for any σ > 0 there exists a time τ1 ∈ (0, σ ] satisfying V (x(τ1 , x0 , y0 )) < V (x0 ) V (x(t , x0 , y0 )) ≤ β1 (V (x0 )),

(4.29a)

∀t ∈ [0, τ1 ]

(4.29b)

where x(·, x0 , y0 ) denotes the solution of Σ1 . Also, assume that Σ2 satisfies H2 and, either γ1 is unbounded, or property (4.10) and implication (4.11a) ⇒ (4.11b) plus (4.6b) are fulfilled for the case where γ1 is bounded. Then, under H3a the system above is SDF-WGAS by means of a static sampled-data feedback stabilizer; (the additional condition H3b is not required). Implication (4.2) ⇒ (4.29a), (4.29b) is fulfilled with β1 (s) = s, if we assume that Σ1 satisfies the Lyapunov characterization of the version of the ISS property, with input y and state x, which is originally introduced in [34]. As a consequence of Proposition 3(i) we provide the following result concerning the case (1.4): Proposition 4. For system (1.4) we assume that Ai , Bi are C ∞ and there exist functions γ1 , Γ2 ∈ K∞ with

γ1 (s) > Γ2 (s),

∀s > 0

(4.30)

such that

• dim Lie {Ai (·, y), i = 1, . . . , ℓ} |x = n, ∀(x, y) ∈ Rn × Rm , x ̸= 0 with |y| < γ1 (|x|) • dim Lie {Bi (x, ·), i = 1, . . . , ℓ} |y = m,

(4.31)

(x, y) ∈ Rn × Rm , y ̸= 0 with |y| > Γ2 (|x|) • dim Lie {Fi , i = 1, . . . , ℓ} |ξ = n + m, ∀ξ = (x, y) ∈ Rn × Rm with |y| < γ1 (|x|) and

(4.32)

|y| > Γ2 (|x|).

(4.33)

Then (1.4) is SDF-WGAS. Remark 5. If the assumptions of Proposition 4 are substituted by the stronger accessibility rank condition for the whole system (1.4a), (1.4b), namely, dim Lie{Fi , i = 1, . . . , ℓ}|ξ = n+m for every nonzero ξ = (x, y) ∈ Rn × Rm , then the result of Proposition 4 is strengthened; particularly, according to the well known result in [32], this condition guarantees that the system (1.4) is globally stabilizable by means of a smooth time-varying feedback. A variant of this condition has been used in [24, Corollary 1] to establish SDF-WGAS for the nonholonomic case (1.4).

J. Tsinias / Systems & Control Letters 61 (2012) 1032–1040

1039

Proof of Proposition 4. Consider the case (x0 , y0 ) ∈ Rn × Rm with

(A1 )x A2 |(x,y) ̸= (A2 )x A1 |(x,y) ,

∀(x, y) ∈ R2 , x ̸= 0

(4.41a)

|y0 | < γ1 (|x0 |)

(B1 )y B2 |(x,y) ̸= (B2 )y B1 |(x,y) ,

∀(x, y) ∈ R , y ̸= 0

(4.41b)

(4.34)

and for the given y0 above consider the auxiliary system: X˙ = f (X , y0 , u) :=

ℓ 

A1 B2 |(x,y) ̸= A2 B1 |(x,y) ,

ui Ai (X , y0 , u)

(4.35a)

ui Bi (X , y0 , u)

(4.35b)

i=1

Y˙ = g (X , y0 , u) :=

ℓ 

2

∀(x, y) ∈ R2 , x ̸= 0, y ̸= 0.

(4.42)

By taking arbitrary γ1 , Γ2 ∈ K∞ satisfying (4.30), we can easily verify that all conditions of Proposition 4 are fulfilled, therefore (4.39) is SDF-WGAS. For completeness we note that (4.41a), (4.41b) imply (4.31) and (4.32), respectively and (4.42) implies (4.33). 

i =1

and let us denote by (X (·), Y (·)) = (X (·, x0 , y0 , u), Y (·, x0 , y0 , u)) its trajectory initiated from (x0 , y0 ) ∈ Rn × Rm at time t = 0 and corresponding to some control u. Also define V (x) := |x|2 . Then using (4.31) and applying the result of Corollary 1 in [24] for the subsystem (4.35a) it follows there exist β1 ∈ K such that for any σ > 0 there exists a time τ1 ≤ σ and a control u1 = u1(x0 ,y0 ) (t ) : [0, τ1 ] → Rℓ such that for (x0 , y0 ) ∈ Rn × Rm satisfying (4.34), the corresponding trajectory

(X (·), Y (·)) = (X (·, x0 , y0 , u1 ), Y (·, x0 , y0 , u1 )) : [0, τ1 ] → Rℓ of (4.35) satisfies: V (X (τ1 , x0 , y0 , u1 )) < V (x0 )

(4.36a)

V (X (t , x0 , y0 , u1 )) ≤ β1 (V (x0 )),

∀t ∈ [0, τ1 ]

|Y (t , x0 , y0 , u1 )| < γ1 (|X (t , x0 , y0 , u1 )|),

(4.36b)

∀t ∈ [0, τ1 ].

(4.36c)

It turns out, by taking into account (4.36) and using elementary continuity arguments, that there exists a time τ1′ ≤ τ1 (≤ σ ) such that V (x(τ1′ , x0 , y0 , u1 )) < V (x0 )

(4.37a)

V (x(t , x0 , y0 , u1 )) ≤ 2β1 (V (x0 )),

∀t ∈ [0, τ1 ]

|y(t , x0 , y0 , u1 )| < γ1 (|x(t , x0 , y0 , u1 )|),

′

(4.37b)

∀t ∈ [0, τ1 ] ′

(4.37c)

where (x(·), y(·)) = (x(·, x0 , y0 , u), y(·, x0 , y0 , u)) denotes the solution of the original system (1.4a), (1.4b) with initial (x(0), y(0)) = (x0 , y0 ). We conclude that Hypothesis H1 of Proposition 3 is satisfied with V (x) := |x|2 . Likewise, for the case (x0 , y0 ) ∈ Rn × Rm with |y0 | > Γ2 (|x0 |) we consider for this x0 the auxiliary system: X˙ = f (x0 , Y , u) :=

ℓ 

ui A i ( x 0 , Y , u)

(4.38a)

ui Bi (x0 , Y , u).

(4.38b)

i=1

Y˙ = g (x0 , Y , u) :=

ℓ  i =1

Then by invoking (4.32), defining W (y) := |y|2 and by again applying Corollary 1 in [24] for the subsystem (4.38b) we can verify as above that Hypothesis H2 of Proposition 3 is satisfied as well. Likewise, by exploiting (4.33), we can verify that H3a and H3b hold for (1.4) with V and W as above, a1 = a2 ≡ s2 , b1 = b2 ≡ s2 and γ1 , Γ2 ∈ K as given in our statement. We conclude, according to the result of Proposition 3, that system (1.4) is SDF-WGAS. Example 2. We illustrate the nature of Proposition 4 by considering the planar case:

      x˙ A 1 ( x, y ) A2 (x, y) = u1 + u2 , y˙ B1 (x, y) B2 (x, y)

(x, y) ∈ R2

(4.39)

where Ai , Bi are C 1 and satisfy: A1 (0, ·) = A2 (0, ·) = 0,

B1 (·, 0) = A2 (·, 0) = 0

(4.40)

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