ISS Gain of Dissipative Systems - CiteSeerX
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 12-14, 2007
ThB17.2
iISS gain of dissipative systems† Bayu Jayawardhana ⋆, Andrew R. Teel∗, Eugene P. Ryan ♯
Abstract— For a class of dissipative nonlinear systems, it is shown that an associated dissipation inequality determines an iISS gain. The result can be used to show the convergence of the state trajectory whenever the input signal has bounded energy where the energy function is related to the dissipation inequality.
I. INTRODUCTION We consider nonlinear systems P, with input u and output y, of the general form x˙ = f (x, u), x(0) = x0 , f (0, 0) = 0, y = h(x), f : Rn × Rm → Rn , h : Rn → Rl locally Lipschitz.
(1) The concept of integral input-to-state stability (iISS) of P is introduced by Angeli et al. in [1] and is defined by the following property: there exist α ∈ K ∞ , β ∈ KL, and γ ∈ K such that, for every locally bounded input u ∈ L ∞ loc and every initial state x0 ∈ Rn , (1) has a unique solution x defined on [0, ∞) and t γ (u(s))ds ∀ t ≥ 0. (2) x(t) ≤ β (x0 ,t) + α 0
The function γ is called an iISS gain of P. If the input u is such that 0∞ γ (u(s))ds < ∞, then the iISS estimate in (2) also implies the converging-state property: x(t) → 0 as t → ∞. It is shown in [1, Theorem 1 and Remark II.3] that, if the equilibrium solution x = 0 (corresponding to u = 0 and x 0 = 0) of P is globally asymptotically stable (GAS) and there exists a continuously differentiable function H : R n → [0, ∞) such that, for some σ ∈ K, limx=∞ H(x) = ∞ and (3) ∇H(x), f (x, u) ≤ σ (u) ∀(x, u) ∈ Rn × Rm , then P is iISS and so (2) holds for every input signal u ∈ L ∞ loc and for every initial state x 0 ∈ Rn . However, given a locally Lipschitz function f and a function σ such that (3) holds, it is not clear in [1] whether γ (s) ≤ σ (s) holds for all s ∈ [0, ∞) (see also the proof of (4 ⇒ 2) in [1, Theorem 1]): the latter inequality does hold if f in [1] is also assumed to satisfy † This work was supported in part by NSF grant ECS-0622253 and AFOSR grant FA9550-06-1-0134. ⋆ Manchester Interdisciplinary Biocentre, University of Manchester, UK. e-mail: [email protected] ∗ Dept. of ECE, University California of Santa Barbara, USA. e-mail: [email protected] ♯ Dept. of Mathematical Sciences, University of Bath, UK. e-mail: [email protected]
1-4244-1498-9/07/$25.00 ©2007 IEEE.
f (0, v) ≤ cσ (v) for all v ∈ Rm , with c > 0 (see also the proof of [1, Proposition 2.5]). Under certain conditions on f , it is shown in Jayawardhana [4] that, if the equilibrium x = 0 of P (with zero input u = 0) is GAS and there exists a continuously differentiable function H such that (3) holds with σ (s) = s p for all s ≥ 0, then the state x(t) converges to zero as t → ∞ for every initial state x0 and for every L p input u. Note that the argument in [4] utilizes infinite-dimensional systems theory. Related work on the state convergence property with L p inputs can be found in Ryan [7]. Under certain assumptions on f , it is shown in [7] that if, for a given L p input u and an initial state x0 , there exists a solution x defined on [0, ∞) and x has non-empty ω -limit set, then x(t) → 0 as t → ∞. Motivated by the state convergence result from [4], we show that (2) holds with γ = σ under a mild assumption on f which is weaker than those posited in [4], [7] and is also weaker than the condition that, for some c > 0, f (0, v) ≤ cσ (v) for all v ∈ Rm . More precisely, we assume that the locally Lipschitz function f and σ are such that, for every compact set K ⊂ Rn , there exists c > 0 such that f (ξ , v) ≤ c(1 + σ (v)) for all (ξ , v) ∈ K × Rm . In Section V, we will discuss the difference of our conditions on f with those in [1], [4] and [7]. Using the fact that γ = σ , we can recover the state convergence results of [4] via the iISS approach of the present paper. This estimate of iISS gain allows one to analyze the stability of interconnected nonlinear systems that contain dissipative systems via a small-gain type argument (see, for example, Ito & Jiang [3]). II. PRELIMINARIES Notation. Throughout, R + = [0, ∞). For a finitedimensional vector x ∈ R n , we define its norm x = ∑ni=1 x2i . For any finite-dimensional vector space V endowed with a norm · , the space L p (R+ , V), p ∈ [1, ∞), consists of all measurable functions f : R + → V such that ∞ p dt < ∞. The L p -norm of a function f ∈ L p (R , V) f (t) + 0 1 is given by f L p = ( 0∞ f (t) p dt) p . For f ∈ L p (R+ , V) and T > 0, f T denotes the concatenation of the functions f | [0,T ] and 0 given by f (t), t ∈ [0, T ] fT (t) := 0, t ∈ (T, ∞) . p The space Lloc (R+ , V) consists of all measurable functions f : R+ → V such that f T ∈ L p (R+ , V) for all T > 0. The space of measurable essentially bounded functions R + → V
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ThB17.2
is denoted by L ∞ (R+ , V); the space of measurable locally essentially bounded functions L ∞ loc (R+ , V) is the space of measurable functions f : R + → V with the property that fT ∈ L∞ (R+ , V) for all T > 0. The space C(Rl , R p ) (respectively C 1 (Rl , R p )) consists of all continuous (respectively continuously differentiable) functions f : Rl → R p . For any ε ≥ 0, we write
as in (6), then (a) (4) holds with γ = σ , and m 0 n ∞ (b) for ∞ each x ∈ R and measurable u ∈ L loc (R+ R ) such that 0 σ (u(t))dt < ∞, the unique global solution x : R + → Rn of (1) has the converging state property: x(t) → 0 as t → ∞.
Bε := {x ∈ Rn | x ≤ ε }.
III. MAIN RESULT In what follows, we will have occasion to impose the following technical assumption on f and σ . (A) For every compact set K ⊂ R n , there exist c > 0 such that
The set K consists of all continuous strictly-increasing functions α : R+ → R+ with α (0) = 0. The set K∞ consists of all functions α ∈ K such that α (s) → ∞ as s → ∞. The set KL consists of all functions β : R + × R+ → R+ such that β (·,t) ∈ K and β (s, ·) converges to zero. The function H : Rn → R+ is called proper if for all c ≥ 0, the (sub-level) set {x | H(x) ≤ c} is compact or it is equivalent to say that H(x) → ∞ whenever x → ∞. m Definition 2.1: Let u ∈ L ∞ loc (R+ , R ). A solution of (1) is an absolutely continuous function x : [0, ω ) → R n , ω > 0, such that
x(t) − x(0) =
t
f (x(τ ), u(τ ))d τ
∀t ∈ [0, ω ).
0
A solution is maximal if it has no right extension that is also a solution.
Theorem 3.1: Consider the system P as in (1). Suppose that the zero state of P, with u = 0, is GAS and there exist functions α1 , α2 ∈ K∞ , σ ∈ K and H ∈ C 1 (Rn , R+ ) such that α1 (ξ ) ≤ H(ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn (8) ∇H(ξ ), f (ξ , v) ≤ σ (v) ∀(ξ , v) ∈ Rn × Rm , If f and σ satisfy (A) then there exist a continuous positivedefinite function α 3 : R+ → R+ , functions α4 , α5 ∈ K∞ and U ∈ C1 (Rn , R+ ) such that
The following is a consequence of the standard theory of ordinary differential equations (see, e.g. [8]). m 0 n Proposition 2.2: For each u ∈ L ∞ loc (R+ , R ) and x ∈ R , the initial-value problem (1) has unique maximal solution x : [0, ω ) → Rn .
∀ (ξ , v) ∈ K × Rm . (7)
f (ξ , v) ≤ c(1 + σ (v))
∇U(ξ ), f (ξ , v) ≤ −α3 (ξ ) + σ (v) ∀(ξ , v) ∈ Rn × Rm
(9)
and
α4 (ξ ) ≤ U(ξ ) ≤ α5 (ξ ) ∀ξ ∈ Rn ,
(10)
For ease of reference, we reiterate the following.
(and so U is an iISS Lyapunov function for P).
Definition 2.3: [1] The system P given by (1) is called integral input-to-state stable (iISS) if there exist functions β ∈ KL, α ∈ K∞ and γ ∈ K such that, for every x(0) ∈ R n m and for every u ∈ L ∞ loc (R+ , R ), the unique maximal solution n x : [0, ω ) → R of (1) is such that ω = ∞ and t γ (u(s))ds ∀ t ≥ 0. x(t) ≤ β (x(0),t) + α
Moreover, for every initial state x 0 ∈ Rn and for every meam ) such that ∞ σ (u(τ )) d τ < (R , R surable input u ∈ L ∞ + 0 loc ∞, the unique maximal solution x : [0, ω ) → R n is such that the following hold: (a) ω = ∞; (b) x satisfies (4) with γ = σ ; (c) x(t) → 0 as t → ∞.
0
(4)
Definition 2.4: [1] A continuously differentiable function U : Rn → R+ is called an iISS-Lyapunov function for system P given by (1) if there exist functions α 1 , α2 ∈ K∞ , σ ∈ K and a continuous positive-definite function α 3 , such that
α1 (ξ ) ≤ U(ξ ) ≤ α2 (ξ )
∀ξ ∈ Rn ,
We preface the proof of Theorem 3.1 with three technical lemmas. Lemma 3.2: Let f ∈ C(Rn × Rm , Rn ). Then, for every compact set K, there exists a function ρ K ∈ K∞ such that
(5)
and ∇U(ξ ), f (ξ , v) ≤ −α3 (ξ ) + σ (v) ∀ (ξ , v) ∈ Rn × Rm . (6) For later convenience, in the following proposition we assemble some results, which are consequences of [1]. Proposition 2.5: The system P, given by (1), is iISS if, and only if, it admits an iISS-Lyapunov function. Furthermore, if U is an iISS-Lyapunov function for P with σ ∈ K
∀(ξ , v) ∈ K × Rm . (11)
f (ξ , v) − f (ξ , 0) ≤ ρK (v)
P ROOF. Let K ⊂ Rn be compact and define ρ˜ K : R+ → R+ by
ρ˜ K (a) := max f (ξ , v) − f (ξ , 0) ξ ∈ K, v ∈ Ba . (12) Clearly, ρ˜ K is non-decreasing and so, a fortiori, is measurable (in fact, it can be shown that ρ˜ K is upper semicontinuous). Therefore, the function ρ K : R+ → R+ is well defined by
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ρK (0) := 0,
ρK (a) := a +
1 a
2a a
ρ˜ K (τ ) d τ
∀a > 0.
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 It is readily verified that ρ K ∈ K∞ . Moreover, ρ K (a) ≥ ρ˜ K (a) 2 for all a ∈ R+ and so (11) holds. n
m
ThB17.2 This establishes that f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v) ∀(ξ , v) ∈ (B1 \{0}) × Rm . (17)
n
Lemma 3.3: Assume that f ∈ C(R × R , R ) and σ ∈ K satisfy (A). Let w ∈ Rn → R+ be continuous and such that, for some α ∈ K∞ ,
α (ξ ) ≤ w(ξ ) ∀ ξ ∈ Rn .
For every k ∈ N, k ≥ 2, let Ck denote the compact set Ck := {ξ ∈ Rn | 1 ≤ w(ξ ) ≤ k}.
(13)
Then, for every continuous function θ : (0, ∞) → (0, ∞), there exist a continuous function δ : (0, ∞) → (0, ∞) such that f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δ (w(ξ ))σ (v), ∀ ξ ∈ Rn \{0} ∀ v ∈ Rm . (14)
By Lemma 3.2, for every k ≥ 2, there exists a function ρ k ∈ K∞ such that f (ξ , v) − f (ξ , 0) ≤ ρk (v) ∀(ξ , v) ∈ Ck × Rm . For every k ≥ 2, let χ k ∈ K∞ denote the inverse of ρ k ∈ K∞ and define the continuous function δ k : [1, k] → (0, ∞) by
P ROOF. Since f satisfies (A), it is readily verified that, for every compact set K ⊂ R n , there exists c > 0 such that f (ξ , v) − f (ξ , 0) ≤ c(1 + σ (v)) ∀ (ξ , v) ∈ K × Rm .
˜ + δn (a) := b(a)
Then an argument analogous to that leading to (17) gives
This implies that there exists a strictly increasing sequence (ck ) in N such that f (ξ , v) − f (ξ , 0) ≤ ck (1 + σ (v)) ∀ (ξ , v) ∈ Bk × Rm . Let b : [0, ∞) → (0, ∞) be the continuous function that linearly interpolates the points c k , k ∈ N, that is, b(λ ) := ck + (ck+1 − ck )(λ + 1 − k) ∀λ ∈ [k − 1, k) ∀k ∈ N. Then, f (ξ , v) − f (ξ , 0) ≤ b(ξ )(1 + σ (v)) ∀ (ξ , v) ∈ Rn × Rm . (15)
f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δk (w(ξ ))σ (v) ∀(ξ , v) ∈ Ck × Rm , k = 2, 3, ... . (18) Now, define
δ1∗ := δ1 (1),
k = 2, 3, . . . .
δ (a) ≥ δ1 ∀a ∈ (0, 1], and δ (a) ≥ δk (a) ∀a ∈ [1, k], k = 2, 3, . . . .
≤ ρ1 (v) ∀(ξ , v) ∈ B1 × Rm . (16) Let θ : (0, ∞) → (0, ∞) be continuous. Denote by χ 1 ∈ K∞ the inverse of the function ρ 1 ∈ K∞ and write b˜ = b ◦ α −1. Define the continuous function δ 1 : (0, 1] → (0, ∞), ˜ ˜ + b(a) − θ (a) . a → δ1 (a) := b(a) σ (χ1 (θ (a))) If ξ ∈ B1 \{0} and v ≤ χ1 (θ (w(ξ ))) then ρ1 (v) ≤ θ (w(ξ )) and so, by (16),
In view of (17) and (18), it follows that (14) holds. This completes the proof. 2 Lemma 3.4: Let σ ∈ K. Consider the system P as in (1) and assume that f and σ satisfy (A). If the zero state of P (with u = 0) is GAS, then, for every ε > 0, there exists a continuous positive-definite function α : R + → R+ and W ∈ C1 (Rn , R+ ) such that W (0) = 0, W (x) > 0 for x = 0 and W (ξ ), f (ξ , v) ≤ −α (ξ ) + εσ (v) ∀(ξ , v) ∈ Rn × Rm . (19)
f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) ≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v). If ξ ∈ B1 \{0} and v > χ1 (θ (w(ξ ))) then, by (13) and (15),
≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v).
a∈[1,k]
The function δ is continuous, with the properties
f (ξ , v) − f (ξ , 0)
˜ ξ )) − θ (w(ξ )) ≤ θ (w(ξ )) + b(w( ˜ + b(w( ξ ))σ (v)
∗ δk∗ := max{ max δk (a), δk−1 },
The sequence (δ k∗ )k∈N so constructed is non-decreasing. Finally, define the function δ : (0, ∞) → (0, ∞) as follows a ∈ (0, 1] δ1 (a) + δ2∗ − δ1∗ , δ (a) := ∗ ∗ ∗ + (δk+2 − δk+1 )(a − k), a ∈ (k, k + 1], k ∈ N δk+1
By Lemma 3.2, there exists ρ 1 ∈ K∞ such that
f (ξ , v) − f (ξ , 0) ≤ b(ξ )(1 + σ (v)) ˜ ≤ b(w( ξ ))(1 + σ (v))
˜ − θ (a) b(a) . σ (χk (θ (a)))
Remark 3.5: We remark that the function W in Lemma 3.4 is not necessarily proper (that is, its sub-level sets are not necessarily compact). P ROOF. The GAS property implies that there exist a smooth V : Rn → R+ , ∇V (0) = 0 and functions α 1 , α2 , α3 ∈ K∞ such that α1 (ξ ) ≤ V (ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn (20) ∇V (ξ ), f (ξ , 0) ≤ −α3 (ξ )
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ThB17.2
(see, for example, [5]). Define α˜ 4 : R+ → R+ by α˜ 4 (a) = max{∇V (ξ ) ξ ∈ Rn , V (ξ ) ≤ a} ∀a ∈ R+ .
Since ∇W (0), f (0, v) = 0 for all v ∈ R m , we conclude that (24) holds for all (ξ , v) ∈ R n × Rm . The proof is completed by setting
1 α (a) := min κ (V (η ))α3 (η ) η ≤ a, η ∈ Rn . 2 2
The function α˜ 4 is non-decreasing and so we may define a continuous function α 4 : R+ → R+ by 1 2a α˜ 4 (τ ) d τ ∀a > 0. a a Moreover, α 4 is non-decreasing with α 4 (a) ≥ α˜ 4 (a) for all a ∈ R+ and
α4 (0) = 0,
α4 (a) =
∇V (ξ ) ≤ α4 (V (ξ )) ∀ξ ∈ Rn . Now define the continuous function θ : (0, ∞) → (0, ∞) by α3 (α2−1 (a)) θ (a) = min a, ∀a ∈ (0, ∞), 2α4 (a)
Define U : Rn → R+ by 1 U(ξ ) = W (ξ ) + H(ξ ) ∀ξ ∈ Rn . 1+ε
in which case, we have ∇V (ξ )θ (V (ξ )) ≤ α4 (V (ξ ))θ (V (ξ )) 1 ≤ α3 (α2−1 (V (ξ ))) 2 1 ≤ α3 (ξ ) ∀ξ ∈ Rn . (21) 2 By Lemma 3.3, there exists a continuous function δ : (0, ∞) → (0, ∞) such that f (ξ , v) − f (ξ , 0) ≤ θ (V (ξ )) + δ (V (ξ ))σ (v) ∀(ξ , v) ∈ (Rn \{0}) × Rm. (22)
κ (V (ξ ))∇V (ξ )δ (V (ξ ))
0 for all ξ = 0. Invoking (20), (21), (22) and (23), we have ∇W (ξ ), f (ξ , v) = κ (V (ξ ))∇V (ξ ), f (ξ , v) − f (ξ , 0) + f (ξ , 0) ≤ κ (V (ξ ))∇V (ξ ) f (ξ , v) − f (ξ , 0)
Proposition 3.6: Consider the system P as in (1). Assume further that P is zero-state detectable and that there exist functions α1 , α2 ∈ K∞ , α , σ ∈ K and H ∈ C 1 (Rn , R+ ) such that α1 (ξ ) ≤ H(ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn , (25) ∇H(ξ ), f (ξ , v) ≤ −α (h(ξ )) + σ (v) n m ∀(ξ , v) ∈ R × R . If f and σ satisfy (A) then there exists an iISS Lyapunov function U ∈ C 1 (Rn , R+ ) such that (9) and (10) holds.
− κ (V (ξ ))α3 (ξ ) ≤ κ (V (ξ ))∇V (ξ ) [θ (V (ξ )) + δ (V (ξ ))σ (v)] − κ (V (ξ ))α3 (ξ ) 1 ≤ − κ (V (ξ ))α3 (ξ ) + εσ (v) 2 ∀(ξ , v) ∈ Rn \{0} × Rm.
Assertions (a), (b) and (c) now follow from Proposition 2.5. 2
0
It follows that
W (ξ ) =
Setting α3 = 1+1 ε α and invoking (19) together with the second of inequalities (8), we may conclude that (9) holds. For each a ∈ R+ define α˜ 4 (a) := min{W (ξ )| ξ ≤ a} ˜4 1 +α and α˜ 5 (a) := max{W (ξ )| ξ ≤ a}. Then α 4 := α1+ ε and α2 +α˜ 5 α5 := 1+ε are K∞ functions and are such that (10) holds. Therefore, U is an iISS Lyapunov function for P.
u →
Let ε > 0 and define a continuous function κ ∈ K by ε κ (0) = 0, κ (a) = min a , ∀a ∈ (0, ∞). α4 (a)δ (a)
ε ∇V (ξ )δ (V (ξ )) α4 (V (ξ ))δ (V (ξ )) ≤ ε ∀ξ ∈ Rn \{0}.
P ROOF OF T HEOREM 3.1. Let ε > 0 be arbitrary. By Lemma 3.4, there exists a continuous positive-definite function α : R+ → R+ and W ∈ C 1 (Rn , R+ ) such that W (0) = 0, W (x) > 0 for x = 0 and (19) holds.
(24)
Moreover, for every initial state x(0) ∈ R n and for every mea∞ m surable input u ∈ L ∞ loc (R+ , R ) such that 0 σ (u(τ )) d τ < ∞, the unique maximal solution x : [0, ω ) → R + of (1) is such that the following hold: (a) ω = ∞;
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ThB17.2 Plot of x
(b) x satisfies (4) with γ = σ ; (c) x(t) → 0 as t → ∞.
1
1
P ROOF. The proof of the proposition follows from Theorem 3.1 once we have shown that (with u = 0) the zero state is GAS. By (25), we may infer that the zero state is a stable equilibrium of x˙ = f (x, 0) and, for each x 0 , the unique maximal solution x of the initial-value problem x˙ = f (x, 0), x(0) = x0 , exists on R+ . It remains to show that the zero state is globally attractive. Let x 0 ∈ Rn be arbitrary, let u = 0 and let x : R+ → Rn be the unique maximal solution of the initial-value problem x˙ = f (x, 0), x(0) = x 0 . Then,
1
x (t)
0.5
0
−0.5 0
5
10
15
20
25
30
35
40
time (t)
d H(x(t)) ≤ −α (h(x(t))) ≤ 0 ∀t ∈ [0, ω ), (26) dt By the zero-state-detectability property, {0} is the only invariant subset of Σ := {ξ ∈ R n | α (h(ξ )) = 0} and so, by the LaSalle invariance principle [6], we may conclude that x(t) → 0 as t → ∞. Hence, the zero state (with u = 0) is GAS. 2
(a)
Plot of x2
1.4 1.2
0.8
2
x (t)
1
0.6
IV. EXAMPLE Let the system P be described by x˙ = xx˙˙12 = f (x, u) −x1 (1 + 1/(u p + 1) + u p) − x2p−1 + u = , x1p−1 y = h(x) := x1 ,
0.4 0.2 0 0
(27) (28)
∇H(ξ ), f (ξ , v) = −pξ1p(1 + 1/(v p + 1) + v p) + pξ1p−1v, ≤ −pξ1p + (p − 1)ξ1p + v p
15
20
25
30
35
40
(b) Fig. 1. The state trajectory of the plant P as in (27) with p = 2, an L2 input u as in (29) and initial states x1 (0) = 1,x2 (0) = 1. (a) The plot of x1 . (b) The plot of x2 .
as in (29) and with initial state components x 1 (0) = 1 = x2 (0). Figure 1 shows the convergence to zero of the state components x 1 (t) and x2 (t) as t → ∞.
= −|h(ξ )| p + |v| p.
V. DISCUSSION
It is readily verified that, for every compact set K ⊂ R 2 , there exist c > 0 such that f (ξ , v) ≤ c(1 + v p) for all (ξ , v) ∈ K × R, for example, c = max{5 + 4|ξ1|(p−1) + |ξ2 | p−1 } ξ ∈K
suffices. Hence (A) holds. The system P is also zero-state detectable. Therefore, P satisfies the hypotheses of Proposition 3.6, with α (s) = s p and σ (s) = s p for all s ≥ 0, and so, for every input u ∈ L p (R+ , R) and for every initial state x 0 ∈ R2 , the solution x is defined on R + , satisfies (4) with γ (s) = s p for all s ≥ 0 and lim x(t) = 0. t→∞ For simulation purpose, we take p = 2 and choose the function u : R + → R given by n n − n14 < t ≤ n, n ∈ N (29) u(t) = 0 elsewhere, lim sup |u(t)| = ∞. Using Matlab, we simulate P with u t∈R+
10
time (t)
where p ∈ {2n | n ∈ N}. Defining ξ → H(ξ1 , ξ2 ) := ξ1p + ξ2p and invoking Young’s inequality, we have
∞ 2 so that u ∈ L∞ loc (R+ , R), 0 u (τ ) d τ < ∞ and
5
In view of recent results on L p -input state-convergence, we briefly discuss the various assumptions on f used in [1], [4], [7] and compare these with assumption (A) of the present paper. We focus on the case wherein σ (s) = s p for all s ≥ 0 for p ∈ [1, ∞). It is common to assume that f ∈ C(R n × Rm , Rn ) and f (0, 0) = 0. The article [4] assumes the following on f : (A1) For every compact set K ⊂ R n , there exist c1 , c2 > 0 such that f (ξ1 , v) − f (ξ2 , v) ≤ (c1 + c2 v p)ξ1 − ξ2 ∀ξ1 , ξ2 ∈ K, ∀v ∈ Rm . (30) (A2) For each fixed ξ ∈ R n , there exist c3 , c4 > 0 such that f (ξ , v) ≤ c3 + c4 v p
∀v ∈ Rm .
(31)
It can be shown that (A1) and (A2) imply (A). Indeed, let K ⊂ Rn be compact and let ξ ∈ K. By Assumption (A2),
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 there exist c3 , c4 > 0 such that (31) holds. Using Assumption (A1), there exist c 1 , c2 > 0 such that (30) holds and we have f (η , v) ≤ f (η , v) − f (ξ , v) + f (ξ , v) ≤ (c1 + c2v p )η − ξ + (c3 + c4 v p) ≤ c(1 + v p)
∀(η , v) ∈ K × Rm ,
where c = maxη ∈K {c1 η − ξ + c3 + c2 η − ξ + c4}. However, it is easy to see that (A) does not imply (A1) and (A2). In [7], the function f satisfies the following assumptions (with the state space X = Rn ): (A3) For every compact set K ⊂ R n , there exists k > 0 such that f (ξ , v) − f (ξ , 0) ≤ kv
∀(ξ , v) ∈ K × Rm . (32)
(A4) f (·, 0) is locally Lipschitz. Note that f in [7] is not necessarily locally Lipschitz on both arguments. We will show that (A3) and (A4) imply (A). Let K ⊂ R n be compact. By (A4), there exists L > 0 such that f (ξ , 0) ≤ Lξ for all ξ ∈ K. By (A3), there exists k > 0 such that (32) holds. Therefore, we have f (ξ , v) ≤ f (ξ , v) − f (ξ , 0) + f (ξ , 0) ≤ k + kv p + Lξ ≤ c(1 + v p)
∀(ξ , v) ∈ K × Rm ,
where c = maxξ ∈K {2k + Lξ }. On the other hand, it can be checked that (A) does not imply (A3). As mentioned in the Introduction, by imposing that there exist c > 0 and p ≥ 1 such that f (0, v) ≤ cv p for all v ∈ Rm and by using the technique to prove the main result in [1], we can conclude that γ (s) = s p holds for all s ∈ [0, ∞). However, this assumption excludes many wellstudied nonlinear systems. For example, for affine nonlinear systems, the function f (x, u) := f˜(x)+ g(x)u with continuous f˜ and g, does not satisfy this assumption for p > 1. R EFERENCES [1] D. Angeli, E.D. Sontag, Y. Wang, “A characterization of integral input to state stability,” IEEE Trans. Automatic Control, vol. 45, pp. 1082– 1097, 2000. [2] A. Isidori, Nonlinear Control Systems II, Springer-Verlag, London, 1999. [3] H. Ito, Z-P. Jiang, “Nonlinear small-gain condition covering iISS systems: necessity and sufficiency from a Lyapunov perspective,” Proc. 45th IEEE CDC, San Diego, 2006. [4] B. Jayawardhana, “Remarks on the state convergence of nonlinear systems given any Lp input,” Proc. 45th IEEE CDC, San Diego, 2006. [5] J. Kurzweil, “On the inversion of Lyapunov’s second theorem on stability of motion”, American Mathematical Society Translations, ser. 2, vol. 24, pp. 19–77, 1956. [6] J.P. La Salle, The Stability of Dynamical Systems, with an Appendix by Z. Artstein, SIAM, Philadelphia, Pennsylvania, 1976. [7] E.P. Ryan, “Remarks on the Lp -input converging-state property,” IEEE Trans. Automatic Control, vol. 50, pp. 1051–1052, 2005. [8] E. D. Sontag, Mathematical Control Theory, 2nd Edition, Springer, New York, 1998.
3840
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ThB17.2
iISS gain of dissipative systems† Bayu Jayawardhana ⋆, Andrew R. Teel∗, Eugene P. Ryan ♯
Abstract— For a class of dissipative nonlinear systems, it is shown that an associated dissipation inequality determines an iISS gain. The result can be used to show the convergence of the state trajectory whenever the input signal has bounded energy where the energy function is related to the dissipation inequality.
I. INTRODUCTION We consider nonlinear systems P, with input u and output y, of the general form x˙ = f (x, u), x(0) = x0 , f (0, 0) = 0, y = h(x), f : Rn × Rm → Rn , h : Rn → Rl locally Lipschitz.
(1) The concept of integral input-to-state stability (iISS) of P is introduced by Angeli et al. in [1] and is defined by the following property: there exist α ∈ K ∞ , β ∈ KL, and γ ∈ K such that, for every locally bounded input u ∈ L ∞ loc and every initial state x0 ∈ Rn , (1) has a unique solution x defined on [0, ∞) and t γ (u(s))ds ∀ t ≥ 0. (2) x(t) ≤ β (x0 ,t) + α 0
The function γ is called an iISS gain of P. If the input u is such that 0∞ γ (u(s))ds < ∞, then the iISS estimate in (2) also implies the converging-state property: x(t) → 0 as t → ∞. It is shown in [1, Theorem 1 and Remark II.3] that, if the equilibrium solution x = 0 (corresponding to u = 0 and x 0 = 0) of P is globally asymptotically stable (GAS) and there exists a continuously differentiable function H : R n → [0, ∞) such that, for some σ ∈ K, limx=∞ H(x) = ∞ and (3) ∇H(x), f (x, u) ≤ σ (u) ∀(x, u) ∈ Rn × Rm , then P is iISS and so (2) holds for every input signal u ∈ L ∞ loc and for every initial state x 0 ∈ Rn . However, given a locally Lipschitz function f and a function σ such that (3) holds, it is not clear in [1] whether γ (s) ≤ σ (s) holds for all s ∈ [0, ∞) (see also the proof of (4 ⇒ 2) in [1, Theorem 1]): the latter inequality does hold if f in [1] is also assumed to satisfy † This work was supported in part by NSF grant ECS-0622253 and AFOSR grant FA9550-06-1-0134. ⋆ Manchester Interdisciplinary Biocentre, University of Manchester, UK. e-mail: [email protected] ∗ Dept. of ECE, University California of Santa Barbara, USA. e-mail: [email protected] ♯ Dept. of Mathematical Sciences, University of Bath, UK. e-mail: [email protected]
1-4244-1498-9/07/$25.00 ©2007 IEEE.
f (0, v) ≤ cσ (v) for all v ∈ Rm , with c > 0 (see also the proof of [1, Proposition 2.5]). Under certain conditions on f , it is shown in Jayawardhana [4] that, if the equilibrium x = 0 of P (with zero input u = 0) is GAS and there exists a continuously differentiable function H such that (3) holds with σ (s) = s p for all s ≥ 0, then the state x(t) converges to zero as t → ∞ for every initial state x0 and for every L p input u. Note that the argument in [4] utilizes infinite-dimensional systems theory. Related work on the state convergence property with L p inputs can be found in Ryan [7]. Under certain assumptions on f , it is shown in [7] that if, for a given L p input u and an initial state x0 , there exists a solution x defined on [0, ∞) and x has non-empty ω -limit set, then x(t) → 0 as t → ∞. Motivated by the state convergence result from [4], we show that (2) holds with γ = σ under a mild assumption on f which is weaker than those posited in [4], [7] and is also weaker than the condition that, for some c > 0, f (0, v) ≤ cσ (v) for all v ∈ Rm . More precisely, we assume that the locally Lipschitz function f and σ are such that, for every compact set K ⊂ Rn , there exists c > 0 such that f (ξ , v) ≤ c(1 + σ (v)) for all (ξ , v) ∈ K × Rm . In Section V, we will discuss the difference of our conditions on f with those in [1], [4] and [7]. Using the fact that γ = σ , we can recover the state convergence results of [4] via the iISS approach of the present paper. This estimate of iISS gain allows one to analyze the stability of interconnected nonlinear systems that contain dissipative systems via a small-gain type argument (see, for example, Ito & Jiang [3]). II. PRELIMINARIES Notation. Throughout, R + = [0, ∞). For a finitedimensional vector x ∈ R n , we define its norm x = ∑ni=1 x2i . For any finite-dimensional vector space V endowed with a norm · , the space L p (R+ , V), p ∈ [1, ∞), consists of all measurable functions f : R + → V such that ∞ p dt < ∞. The L p -norm of a function f ∈ L p (R , V) f (t) + 0 1 is given by f L p = ( 0∞ f (t) p dt) p . For f ∈ L p (R+ , V) and T > 0, f T denotes the concatenation of the functions f | [0,T ] and 0 given by f (t), t ∈ [0, T ] fT (t) := 0, t ∈ (T, ∞) . p The space Lloc (R+ , V) consists of all measurable functions f : R+ → V such that f T ∈ L p (R+ , V) for all T > 0. The space of measurable essentially bounded functions R + → V
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ThB17.2
is denoted by L ∞ (R+ , V); the space of measurable locally essentially bounded functions L ∞ loc (R+ , V) is the space of measurable functions f : R + → V with the property that fT ∈ L∞ (R+ , V) for all T > 0. The space C(Rl , R p ) (respectively C 1 (Rl , R p )) consists of all continuous (respectively continuously differentiable) functions f : Rl → R p . For any ε ≥ 0, we write
as in (6), then (a) (4) holds with γ = σ , and m 0 n ∞ (b) for ∞ each x ∈ R and measurable u ∈ L loc (R+ R ) such that 0 σ (u(t))dt < ∞, the unique global solution x : R + → Rn of (1) has the converging state property: x(t) → 0 as t → ∞.
Bε := {x ∈ Rn | x ≤ ε }.
III. MAIN RESULT In what follows, we will have occasion to impose the following technical assumption on f and σ . (A) For every compact set K ⊂ R n , there exist c > 0 such that
The set K consists of all continuous strictly-increasing functions α : R+ → R+ with α (0) = 0. The set K∞ consists of all functions α ∈ K such that α (s) → ∞ as s → ∞. The set KL consists of all functions β : R + × R+ → R+ such that β (·,t) ∈ K and β (s, ·) converges to zero. The function H : Rn → R+ is called proper if for all c ≥ 0, the (sub-level) set {x | H(x) ≤ c} is compact or it is equivalent to say that H(x) → ∞ whenever x → ∞. m Definition 2.1: Let u ∈ L ∞ loc (R+ , R ). A solution of (1) is an absolutely continuous function x : [0, ω ) → R n , ω > 0, such that
x(t) − x(0) =
t
f (x(τ ), u(τ ))d τ
∀t ∈ [0, ω ).
0
A solution is maximal if it has no right extension that is also a solution.
Theorem 3.1: Consider the system P as in (1). Suppose that the zero state of P, with u = 0, is GAS and there exist functions α1 , α2 ∈ K∞ , σ ∈ K and H ∈ C 1 (Rn , R+ ) such that α1 (ξ ) ≤ H(ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn (8) ∇H(ξ ), f (ξ , v) ≤ σ (v) ∀(ξ , v) ∈ Rn × Rm , If f and σ satisfy (A) then there exist a continuous positivedefinite function α 3 : R+ → R+ , functions α4 , α5 ∈ K∞ and U ∈ C1 (Rn , R+ ) such that
The following is a consequence of the standard theory of ordinary differential equations (see, e.g. [8]). m 0 n Proposition 2.2: For each u ∈ L ∞ loc (R+ , R ) and x ∈ R , the initial-value problem (1) has unique maximal solution x : [0, ω ) → Rn .
∀ (ξ , v) ∈ K × Rm . (7)
f (ξ , v) ≤ c(1 + σ (v))
∇U(ξ ), f (ξ , v) ≤ −α3 (ξ ) + σ (v) ∀(ξ , v) ∈ Rn × Rm
(9)
and
α4 (ξ ) ≤ U(ξ ) ≤ α5 (ξ ) ∀ξ ∈ Rn ,
(10)
For ease of reference, we reiterate the following.
(and so U is an iISS Lyapunov function for P).
Definition 2.3: [1] The system P given by (1) is called integral input-to-state stable (iISS) if there exist functions β ∈ KL, α ∈ K∞ and γ ∈ K such that, for every x(0) ∈ R n m and for every u ∈ L ∞ loc (R+ , R ), the unique maximal solution n x : [0, ω ) → R of (1) is such that ω = ∞ and t γ (u(s))ds ∀ t ≥ 0. x(t) ≤ β (x(0),t) + α
Moreover, for every initial state x 0 ∈ Rn and for every meam ) such that ∞ σ (u(τ )) d τ < (R , R surable input u ∈ L ∞ + 0 loc ∞, the unique maximal solution x : [0, ω ) → R n is such that the following hold: (a) ω = ∞; (b) x satisfies (4) with γ = σ ; (c) x(t) → 0 as t → ∞.
0
(4)
Definition 2.4: [1] A continuously differentiable function U : Rn → R+ is called an iISS-Lyapunov function for system P given by (1) if there exist functions α 1 , α2 ∈ K∞ , σ ∈ K and a continuous positive-definite function α 3 , such that
α1 (ξ ) ≤ U(ξ ) ≤ α2 (ξ )
∀ξ ∈ Rn ,
We preface the proof of Theorem 3.1 with three technical lemmas. Lemma 3.2: Let f ∈ C(Rn × Rm , Rn ). Then, for every compact set K, there exists a function ρ K ∈ K∞ such that
(5)
and ∇U(ξ ), f (ξ , v) ≤ −α3 (ξ ) + σ (v) ∀ (ξ , v) ∈ Rn × Rm . (6) For later convenience, in the following proposition we assemble some results, which are consequences of [1]. Proposition 2.5: The system P, given by (1), is iISS if, and only if, it admits an iISS-Lyapunov function. Furthermore, if U is an iISS-Lyapunov function for P with σ ∈ K
∀(ξ , v) ∈ K × Rm . (11)
f (ξ , v) − f (ξ , 0) ≤ ρK (v)
P ROOF. Let K ⊂ Rn be compact and define ρ˜ K : R+ → R+ by
ρ˜ K (a) := max f (ξ , v) − f (ξ , 0) ξ ∈ K, v ∈ Ba . (12) Clearly, ρ˜ K is non-decreasing and so, a fortiori, is measurable (in fact, it can be shown that ρ˜ K is upper semicontinuous). Therefore, the function ρ K : R+ → R+ is well defined by
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ρK (0) := 0,
ρK (a) := a +
1 a
2a a
ρ˜ K (τ ) d τ
∀a > 0.
46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 It is readily verified that ρ K ∈ K∞ . Moreover, ρ K (a) ≥ ρ˜ K (a) 2 for all a ∈ R+ and so (11) holds. n
m
ThB17.2 This establishes that f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v) ∀(ξ , v) ∈ (B1 \{0}) × Rm . (17)
n
Lemma 3.3: Assume that f ∈ C(R × R , R ) and σ ∈ K satisfy (A). Let w ∈ Rn → R+ be continuous and such that, for some α ∈ K∞ ,
α (ξ ) ≤ w(ξ ) ∀ ξ ∈ Rn .
For every k ∈ N, k ≥ 2, let Ck denote the compact set Ck := {ξ ∈ Rn | 1 ≤ w(ξ ) ≤ k}.
(13)
Then, for every continuous function θ : (0, ∞) → (0, ∞), there exist a continuous function δ : (0, ∞) → (0, ∞) such that f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δ (w(ξ ))σ (v), ∀ ξ ∈ Rn \{0} ∀ v ∈ Rm . (14)
By Lemma 3.2, for every k ≥ 2, there exists a function ρ k ∈ K∞ such that f (ξ , v) − f (ξ , 0) ≤ ρk (v) ∀(ξ , v) ∈ Ck × Rm . For every k ≥ 2, let χ k ∈ K∞ denote the inverse of ρ k ∈ K∞ and define the continuous function δ k : [1, k] → (0, ∞) by
P ROOF. Since f satisfies (A), it is readily verified that, for every compact set K ⊂ R n , there exists c > 0 such that f (ξ , v) − f (ξ , 0) ≤ c(1 + σ (v)) ∀ (ξ , v) ∈ K × Rm .
˜ + δn (a) := b(a)
Then an argument analogous to that leading to (17) gives
This implies that there exists a strictly increasing sequence (ck ) in N such that f (ξ , v) − f (ξ , 0) ≤ ck (1 + σ (v)) ∀ (ξ , v) ∈ Bk × Rm . Let b : [0, ∞) → (0, ∞) be the continuous function that linearly interpolates the points c k , k ∈ N, that is, b(λ ) := ck + (ck+1 − ck )(λ + 1 − k) ∀λ ∈ [k − 1, k) ∀k ∈ N. Then, f (ξ , v) − f (ξ , 0) ≤ b(ξ )(1 + σ (v)) ∀ (ξ , v) ∈ Rn × Rm . (15)
f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) + δk (w(ξ ))σ (v) ∀(ξ , v) ∈ Ck × Rm , k = 2, 3, ... . (18) Now, define
δ1∗ := δ1 (1),
k = 2, 3, . . . .
δ (a) ≥ δ1 ∀a ∈ (0, 1], and δ (a) ≥ δk (a) ∀a ∈ [1, k], k = 2, 3, . . . .
≤ ρ1 (v) ∀(ξ , v) ∈ B1 × Rm . (16) Let θ : (0, ∞) → (0, ∞) be continuous. Denote by χ 1 ∈ K∞ the inverse of the function ρ 1 ∈ K∞ and write b˜ = b ◦ α −1. Define the continuous function δ 1 : (0, 1] → (0, ∞), ˜ ˜ + b(a) − θ (a) . a → δ1 (a) := b(a) σ (χ1 (θ (a))) If ξ ∈ B1 \{0} and v ≤ χ1 (θ (w(ξ ))) then ρ1 (v) ≤ θ (w(ξ )) and so, by (16),
In view of (17) and (18), it follows that (14) holds. This completes the proof. 2 Lemma 3.4: Let σ ∈ K. Consider the system P as in (1) and assume that f and σ satisfy (A). If the zero state of P (with u = 0) is GAS, then, for every ε > 0, there exists a continuous positive-definite function α : R + → R+ and W ∈ C1 (Rn , R+ ) such that W (0) = 0, W (x) > 0 for x = 0 and W (ξ ), f (ξ , v) ≤ −α (ξ ) + εσ (v) ∀(ξ , v) ∈ Rn × Rm . (19)
f (ξ , v) − f (ξ , 0) ≤ θ (w(ξ )) ≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v). If ξ ∈ B1 \{0} and v > χ1 (θ (w(ξ ))) then, by (13) and (15),
≤ θ (w(ξ )) + δ1 (w(ξ ))σ (v).
a∈[1,k]
The function δ is continuous, with the properties
f (ξ , v) − f (ξ , 0)
˜ ξ )) − θ (w(ξ )) ≤ θ (w(ξ )) + b(w( ˜ + b(w( ξ ))σ (v)
∗ δk∗ := max{ max δk (a), δk−1 },
The sequence (δ k∗ )k∈N so constructed is non-decreasing. Finally, define the function δ : (0, ∞) → (0, ∞) as follows a ∈ (0, 1] δ1 (a) + δ2∗ − δ1∗ , δ (a) := ∗ ∗ ∗ + (δk+2 − δk+1 )(a − k), a ∈ (k, k + 1], k ∈ N δk+1
By Lemma 3.2, there exists ρ 1 ∈ K∞ such that
f (ξ , v) − f (ξ , 0) ≤ b(ξ )(1 + σ (v)) ˜ ≤ b(w( ξ ))(1 + σ (v))
˜ − θ (a) b(a) . σ (χk (θ (a)))
Remark 3.5: We remark that the function W in Lemma 3.4 is not necessarily proper (that is, its sub-level sets are not necessarily compact). P ROOF. The GAS property implies that there exist a smooth V : Rn → R+ , ∇V (0) = 0 and functions α 1 , α2 , α3 ∈ K∞ such that α1 (ξ ) ≤ V (ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn (20) ∇V (ξ ), f (ξ , 0) ≤ −α3 (ξ )
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ThB17.2
(see, for example, [5]). Define α˜ 4 : R+ → R+ by α˜ 4 (a) = max{∇V (ξ ) ξ ∈ Rn , V (ξ ) ≤ a} ∀a ∈ R+ .
Since ∇W (0), f (0, v) = 0 for all v ∈ R m , we conclude that (24) holds for all (ξ , v) ∈ R n × Rm . The proof is completed by setting
1 α (a) := min κ (V (η ))α3 (η ) η ≤ a, η ∈ Rn . 2 2
The function α˜ 4 is non-decreasing and so we may define a continuous function α 4 : R+ → R+ by 1 2a α˜ 4 (τ ) d τ ∀a > 0. a a Moreover, α 4 is non-decreasing with α 4 (a) ≥ α˜ 4 (a) for all a ∈ R+ and
α4 (0) = 0,
α4 (a) =
∇V (ξ ) ≤ α4 (V (ξ )) ∀ξ ∈ Rn . Now define the continuous function θ : (0, ∞) → (0, ∞) by α3 (α2−1 (a)) θ (a) = min a, ∀a ∈ (0, ∞), 2α4 (a)
Define U : Rn → R+ by 1 U(ξ ) = W (ξ ) + H(ξ ) ∀ξ ∈ Rn . 1+ε
in which case, we have ∇V (ξ )θ (V (ξ )) ≤ α4 (V (ξ ))θ (V (ξ )) 1 ≤ α3 (α2−1 (V (ξ ))) 2 1 ≤ α3 (ξ ) ∀ξ ∈ Rn . (21) 2 By Lemma 3.3, there exists a continuous function δ : (0, ∞) → (0, ∞) such that f (ξ , v) − f (ξ , 0) ≤ θ (V (ξ )) + δ (V (ξ ))σ (v) ∀(ξ , v) ∈ (Rn \{0}) × Rm. (22)
κ (V (ξ ))∇V (ξ )δ (V (ξ ))
0 for all ξ = 0. Invoking (20), (21), (22) and (23), we have ∇W (ξ ), f (ξ , v) = κ (V (ξ ))∇V (ξ ), f (ξ , v) − f (ξ , 0) + f (ξ , 0) ≤ κ (V (ξ ))∇V (ξ ) f (ξ , v) − f (ξ , 0)
Proposition 3.6: Consider the system P as in (1). Assume further that P is zero-state detectable and that there exist functions α1 , α2 ∈ K∞ , α , σ ∈ K and H ∈ C 1 (Rn , R+ ) such that α1 (ξ ) ≤ H(ξ ) ≤ α2 (ξ ) ∀ξ ∈ Rn , (25) ∇H(ξ ), f (ξ , v) ≤ −α (h(ξ )) + σ (v) n m ∀(ξ , v) ∈ R × R . If f and σ satisfy (A) then there exists an iISS Lyapunov function U ∈ C 1 (Rn , R+ ) such that (9) and (10) holds.
− κ (V (ξ ))α3 (ξ ) ≤ κ (V (ξ ))∇V (ξ ) [θ (V (ξ )) + δ (V (ξ ))σ (v)] − κ (V (ξ ))α3 (ξ ) 1 ≤ − κ (V (ξ ))α3 (ξ ) + εσ (v) 2 ∀(ξ , v) ∈ Rn \{0} × Rm.
Assertions (a), (b) and (c) now follow from Proposition 2.5. 2
0
It follows that
W (ξ ) =
Setting α3 = 1+1 ε α and invoking (19) together with the second of inequalities (8), we may conclude that (9) holds. For each a ∈ R+ define α˜ 4 (a) := min{W (ξ )| ξ ≤ a} ˜4 1 +α and α˜ 5 (a) := max{W (ξ )| ξ ≤ a}. Then α 4 := α1+ ε and α2 +α˜ 5 α5 := 1+ε are K∞ functions and are such that (10) holds. Therefore, U is an iISS Lyapunov function for P.
u →
Let ε > 0 and define a continuous function κ ∈ K by ε κ (0) = 0, κ (a) = min a , ∀a ∈ (0, ∞). α4 (a)δ (a)
ε ∇V (ξ )δ (V (ξ )) α4 (V (ξ ))δ (V (ξ )) ≤ ε ∀ξ ∈ Rn \{0}.
P ROOF OF T HEOREM 3.1. Let ε > 0 be arbitrary. By Lemma 3.4, there exists a continuous positive-definite function α : R+ → R+ and W ∈ C 1 (Rn , R+ ) such that W (0) = 0, W (x) > 0 for x = 0 and (19) holds.
(24)
Moreover, for every initial state x(0) ∈ R n and for every mea∞ m surable input u ∈ L ∞ loc (R+ , R ) such that 0 σ (u(τ )) d τ < ∞, the unique maximal solution x : [0, ω ) → R + of (1) is such that the following hold: (a) ω = ∞;
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ThB17.2 Plot of x
(b) x satisfies (4) with γ = σ ; (c) x(t) → 0 as t → ∞.
1
1
P ROOF. The proof of the proposition follows from Theorem 3.1 once we have shown that (with u = 0) the zero state is GAS. By (25), we may infer that the zero state is a stable equilibrium of x˙ = f (x, 0) and, for each x 0 , the unique maximal solution x of the initial-value problem x˙ = f (x, 0), x(0) = x0 , exists on R+ . It remains to show that the zero state is globally attractive. Let x 0 ∈ Rn be arbitrary, let u = 0 and let x : R+ → Rn be the unique maximal solution of the initial-value problem x˙ = f (x, 0), x(0) = x 0 . Then,
1
x (t)
0.5
0
−0.5 0
5
10
15
20
25
30
35
40
time (t)
d H(x(t)) ≤ −α (h(x(t))) ≤ 0 ∀t ∈ [0, ω ), (26) dt By the zero-state-detectability property, {0} is the only invariant subset of Σ := {ξ ∈ R n | α (h(ξ )) = 0} and so, by the LaSalle invariance principle [6], we may conclude that x(t) → 0 as t → ∞. Hence, the zero state (with u = 0) is GAS. 2
(a)
Plot of x2
1.4 1.2
0.8
2
x (t)
1
0.6
IV. EXAMPLE Let the system P be described by x˙ = xx˙˙12 = f (x, u) −x1 (1 + 1/(u p + 1) + u p) − x2p−1 + u = , x1p−1 y = h(x) := x1 ,
0.4 0.2 0 0
(27) (28)
∇H(ξ ), f (ξ , v) = −pξ1p(1 + 1/(v p + 1) + v p) + pξ1p−1v, ≤ −pξ1p + (p − 1)ξ1p + v p
15
20
25
30
35
40
(b) Fig. 1. The state trajectory of the plant P as in (27) with p = 2, an L2 input u as in (29) and initial states x1 (0) = 1,x2 (0) = 1. (a) The plot of x1 . (b) The plot of x2 .
as in (29) and with initial state components x 1 (0) = 1 = x2 (0). Figure 1 shows the convergence to zero of the state components x 1 (t) and x2 (t) as t → ∞.
= −|h(ξ )| p + |v| p.
V. DISCUSSION
It is readily verified that, for every compact set K ⊂ R 2 , there exist c > 0 such that f (ξ , v) ≤ c(1 + v p) for all (ξ , v) ∈ K × R, for example, c = max{5 + 4|ξ1|(p−1) + |ξ2 | p−1 } ξ ∈K
suffices. Hence (A) holds. The system P is also zero-state detectable. Therefore, P satisfies the hypotheses of Proposition 3.6, with α (s) = s p and σ (s) = s p for all s ≥ 0, and so, for every input u ∈ L p (R+ , R) and for every initial state x 0 ∈ R2 , the solution x is defined on R + , satisfies (4) with γ (s) = s p for all s ≥ 0 and lim x(t) = 0. t→∞ For simulation purpose, we take p = 2 and choose the function u : R + → R given by n n − n14 < t ≤ n, n ∈ N (29) u(t) = 0 elsewhere, lim sup |u(t)| = ∞. Using Matlab, we simulate P with u t∈R+
10
time (t)
where p ∈ {2n | n ∈ N}. Defining ξ → H(ξ1 , ξ2 ) := ξ1p + ξ2p and invoking Young’s inequality, we have
∞ 2 so that u ∈ L∞ loc (R+ , R), 0 u (τ ) d τ < ∞ and
5
In view of recent results on L p -input state-convergence, we briefly discuss the various assumptions on f used in [1], [4], [7] and compare these with assumption (A) of the present paper. We focus on the case wherein σ (s) = s p for all s ≥ 0 for p ∈ [1, ∞). It is common to assume that f ∈ C(R n × Rm , Rn ) and f (0, 0) = 0. The article [4] assumes the following on f : (A1) For every compact set K ⊂ R n , there exist c1 , c2 > 0 such that f (ξ1 , v) − f (ξ2 , v) ≤ (c1 + c2 v p)ξ1 − ξ2 ∀ξ1 , ξ2 ∈ K, ∀v ∈ Rm . (30) (A2) For each fixed ξ ∈ R n , there exist c3 , c4 > 0 such that f (ξ , v) ≤ c3 + c4 v p
∀v ∈ Rm .
(31)
It can be shown that (A1) and (A2) imply (A). Indeed, let K ⊂ Rn be compact and let ξ ∈ K. By Assumption (A2),
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46th IEEE CDC, New Orleans, USA, Dec. 12-14, 2007 there exist c3 , c4 > 0 such that (31) holds. Using Assumption (A1), there exist c 1 , c2 > 0 such that (30) holds and we have f (η , v) ≤ f (η , v) − f (ξ , v) + f (ξ , v) ≤ (c1 + c2v p )η − ξ + (c3 + c4 v p) ≤ c(1 + v p)
∀(η , v) ∈ K × Rm ,
where c = maxη ∈K {c1 η − ξ + c3 + c2 η − ξ + c4}. However, it is easy to see that (A) does not imply (A1) and (A2). In [7], the function f satisfies the following assumptions (with the state space X = Rn ): (A3) For every compact set K ⊂ R n , there exists k > 0 such that f (ξ , v) − f (ξ , 0) ≤ kv
∀(ξ , v) ∈ K × Rm . (32)
(A4) f (·, 0) is locally Lipschitz. Note that f in [7] is not necessarily locally Lipschitz on both arguments. We will show that (A3) and (A4) imply (A). Let K ⊂ R n be compact. By (A4), there exists L > 0 such that f (ξ , 0) ≤ Lξ for all ξ ∈ K. By (A3), there exists k > 0 such that (32) holds. Therefore, we have f (ξ , v) ≤ f (ξ , v) − f (ξ , 0) + f (ξ , 0) ≤ k + kv p + Lξ ≤ c(1 + v p)
∀(ξ , v) ∈ K × Rm ,
where c = maxξ ∈K {2k + Lξ }. On the other hand, it can be checked that (A) does not imply (A3). As mentioned in the Introduction, by imposing that there exist c > 0 and p ≥ 1 such that f (0, v) ≤ cv p for all v ∈ Rm and by using the technique to prove the main result in [1], we can conclude that γ (s) = s p holds for all s ∈ [0, ∞). However, this assumption excludes many wellstudied nonlinear systems. For example, for affine nonlinear systems, the function f (x, u) := f˜(x)+ g(x)u with continuous f˜ and g, does not satisfy this assumption for p > 1. R EFERENCES [1] D. Angeli, E.D. Sontag, Y. Wang, “A characterization of integral input to state stability,” IEEE Trans. Automatic Control, vol. 45, pp. 1082– 1097, 2000. [2] A. Isidori, Nonlinear Control Systems II, Springer-Verlag, London, 1999. [3] H. Ito, Z-P. Jiang, “Nonlinear small-gain condition covering iISS systems: necessity and sufficiency from a Lyapunov perspective,” Proc. 45th IEEE CDC, San Diego, 2006. [4] B. Jayawardhana, “Remarks on the state convergence of nonlinear systems given any Lp input,” Proc. 45th IEEE CDC, San Diego, 2006. [5] J. Kurzweil, “On the inversion of Lyapunov’s second theorem on stability of motion”, American Mathematical Society Translations, ser. 2, vol. 24, pp. 19–77, 1956. [6] J.P. La Salle, The Stability of Dynamical Systems, with an Appendix by Z. Artstein, SIAM, Philadelphia, Pennsylvania, 1976. [7] E.P. Ryan, “Remarks on the Lp -input converging-state property,” IEEE Trans. Automatic Control, vol. 50, pp. 1051–1052, 2005. [8] E. D. Sontag, Mathematical Control Theory, 2nd Edition, Springer, New York, 1998.
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