Top Banner
Stability of Interconnected Dynamical Systems Sergey Dashkovskiy Institute of Mathematics, University of W urzburg Elgersburg, 26.02.2018 1/29
29

Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Jun 30, 2018

Download

Documents

lykien
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Stability of InterconnectedDynamical Systems

Sergey DashkovskiyInstitute of Mathematics, University of Wurzburg

Elgersburg, 26.02.2018

1/29

Page 2: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Inhalt

Motivationfor infinite networks+

Stability of small ΣOnly two interconnected systems

ISS-Lyapunov functions for small ΣConstruction for 2 and n ∈ N interconneced systems

Back to ∞

2/29

Page 3: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Systems of systems

vehicle platooning

Σ1 Σ2 Σ3. . . Σn

. . .. . . - - - - - -

airplane flight formation

6?. . .

6?. . .

6?

. . .

6?

. . .

-. . .

-. . .

Σ1

Σ2

Σ3

Σn. . .

. . .

. . .. . .

-

-

-

?

6 6

? ?

Flocks of birds, schools of fish, . . .

Large networks n ≈ ∞ are modeled as spatially invariant systems 3/29

Page 4: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Problem statement

Given: Σi , i ∈ N with state xi ∈ Rni , ni ∈ NNeighbours sending inputs to Σi are numbered by indices Ii ⊂ NNeighbours state xi ∈ RNi is composed of vectors xj ∈ Rnj , j ∈ Iiordered by the index j and Ni :=

∑j∈Ii nj .

Σi : xi = fi (xi , xi , ui ),

ui ∈ Lloc∞ ([0,∞);Rmi ), fi : Rni+Ni+m1 → Rni is s.t. ∃! solutions.Assume that for each Σi ∃ radially unbounded V s.t.

Vi (xi ) ≥ maxk∈Iiγik(Vk(xk)), γi (|ui |) ⇒ ∇Vi (xi )·fi (xi , xi , ui )) ≤ −αi (|xi |).

here γik ∈ K∞ and α is pos. def.

Question: is the whole interconnection Σ : x = f (x , u) stable?4/29

Page 5: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Promlem statement

x :=

x1

x2...

, u :=

u1

u2...

, f (x , u) :=

f1(x1, x1, u1)f2(x2, x2, u2)

...

Σ : x = f (x , u), f : `∞ × `∞ → `∞

Definition

V : `∞ → [0,∞) is called an ISS Lyapunov function for Σ if ∃α1, α2, γ ∈ K∞ and pos. def α such that ∀x , u ∈ `∞ holds

α1(|x |∞) ≤ V (x) ≤ α2(|x |∞)

V (x) ≥ γ(|u|∞) ⇒∞∑i=1

∂V

∂xifi (xi , xi , ui ) ≤ −α(|u|∞)

Aim: apply small-gain approach and construct V on the base of given Vi

5/29

Page 6: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Recalling stability of feedback systems

”I did know once, only I’ve sort of forgotten.” (Winnie-the-Pooh)

input u −→ Σ −→ y output

u ∈ U, y ∈ Y , U,Y Banach spaces

U

Y graph of Σ is GΣ := (u, y)| u ∈ U

stability :⇔ ∃γ ∈ K : ||y ||Y ≤ γ(||u||U)

inverse graph is G IΣ := (y , u)| u ∈ U

distance to GΣ: d(x ,GΣ) := infz∈GΣ

||x−z ||

for simplicity: U = Y . 6/29

Page 7: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Stability of a well-defined interconnection Σ

Σ :

(d1

d2

)−→

(y1

y2

)

Graph separation theorem:

Σ is stable ⇔ ∃α ∈ K∞ : x ∈ G IΣ2⇒ ||x || ≤ α(d(x ,GΣ1))

If a point on the inverse graph of Σ2 is close to GΣ1 , it must be small.If x ∈ G I

Σ2is large, then d(x ,GΣ1) must be large.

Remark: 1) we do not require that Σi is stable 2) Note: ⇔7/29

Page 8: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Proof of the graph separation theorem (Safonov 1977)

Th: Σ is stable ⇔ ∃α ∈ K∞: x ∈ G IΣ2

⇒ ||x || ≤ α(d(x ,GΣ1)).

Σ :

(d1

d2

)−→

(y1

y2

)

Let x := (y2, y1 + d2) ∈ G IΣ2

and z := (y2 + d1, y1) ∈ GΣ1

Note that: x ∈ G IΣ2, z ∈ GΣ1 ⇒

(x − z) = (−d1, d2),||x − z || = ||(d1, d2)||

”⇒” ∃x ∈ G IΣ2

with large ||x || but small d(x ,GΣ1), i.e., ∃z ∈ GΣ1

with small ||x − z || ⇒ (d1, d2) is small. This contradictsstability.

”⇐” for large x 6 ∃ z close to x ⇒ only large input can yield large x⇒ stability of Σ.

8/29

Page 9: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Remarks on graph separation theorem

Th: Σ is stable ⇔ ∃α ∈ K∞: x ∈ G IΣ2

⇒ ||x || ≤ α(d(x ,GΣ1)).

I Note: if and only if condition for stability

I Safonov used this theorem for robustness margins estimations

I Ideas of the theorem go back to conic relations of Zames 1966

I Replace + with max in Σ, then stability ⇔ GΣ1 ∩ GΣ2 = 0

I How this theorem can be used?

9/29

Page 10: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Classical small-gain theorem

Let Σ1 and Σ2 be finite gain stable, i.e., ∃ γ1, γ2 > 0 s.t.

||y1|| ≤ γ1||u1|| and ||y2|| ≤ γ2||u2||

Theorem:γ1γ2 < 1 ⇒ Σ is stable

Remark: This and the next 3 figures are taken from ”Input-output Stability” by Teel, Georgiou, Praly and Sontag in

”The Control Handbook”, CRC Press 199610/29

Page 11: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Classical passivity theorem

Σ is passive if ∀ (u, y) ∈ GΣ holds 〈u, y〉 :=∫∞

0 uT (t)y(t) dt ≥ 0Σ is strictly passive if ∃ ε > 0 with 〈u, y〉 ≥ ε(||u||22 + ||y ||22)Σ is input strictly passive: 〈u, y〉 ≥ ε(||u||22)Σ is output strictly passive: 〈u, y〉 ≥ ε(||y ||22)

Theorem:If Σ1 is passive andΣ2 scaled by −1 isstrictly passive⇒ Σ is stable wrt || · ||2

11/29

Page 12: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Classical passivity theorem

Σ is passive if ∀ (u, y) ∈ GΣ holds 〈u, y〉 :=∫∞

0 uT (t)y(t) dt ≥ 0Σ is strictly passive if ∃ ε > 0 with 〈u, y〉 ≥ ε(||u||22 + ||y ||22)Σ is input strictly passive: 〈u, y〉 ≥ ε(||u||22)Σ is output strictly passive: 〈u, y〉 ≥ ε(||y ||22)

Theorem:Σ1 strictly input (output)passiveΣ2 scaled by −1strictly input (output)passive⇒ Σ is stable wrt || · ||2

Remark: NL: 〈u, y〉 ≥ ||u||2ρ(||u||2) + ||y ||2ρ(||y ||2), ρ ∈ K∞ 12/29

Page 13: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Nonlinear small-gain theorem

Consider Σi withstability gain function γi ∈ K∞ :

||yi || ≤ γi (||ui ||), i = 1, 2.

Graph separation condition is satisfied ifdist. between the curves (s, γ1(s)) and (γ2(r), r) grows unbounded⇔∃ ρ ∈ K∞ s.t. the curves (s, γ1(s) + ρ(s)) and (γ2(r) + ρ(r), r)have only one common point 0⇔(γ1 + ρ) (γ2 + ρ)(r) < r , r > 0 (in max-case γ1 γ2(r) < r) 13/29

Page 14: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Derivation of the nonlinear gain γ

Consider a stable system

x = f (x , u)y = h(x , u)

Let V : Rn :→ R+ be smooth and ∃α1, α2, α3, α4 ∈ K∞ s.t.

α1(|x |) ≤ V (x) ≤ α2(|x |)

V (x) := ∇V · f (x , u) ≤ −α3(|x |) + α4(|u|)

h(0, 0) = 0 and continuity ⇒ ∃φx , φu ∈ K∞ s.t.|h(x , u)| ≤ φx(|x |) + φu(|u|)

Then we can take

γ := φx α−11 α2 α−1

3 α4 + φu

see Sontag 198914/29

Page 15: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

x2 = f2(x1, x2, u)

x1 = f1(x1, x2, u)

--

u

u

x2 x1

V1(x1) ≥ maxgamma12(V2(x2)), γ1(|u|)

⇒ ∇V1(x1)f1(x1, x2, u) ≤ −α1(V1)

V2(x2) ≥ maxγ21(V1(x1)), γ2(|u|)

⇒ ∇V2(x2)f2(x1, x2, u) ≤ −α2(V2)

Theorem (Jiang, Mareels, Wang 1996)

∀r > 0γ12 γ21(r) < r

⇒ x :=

(x1

x2

)•=

(f1(x1, x2, u)f2(x1, x2, u)

)=: f (x , u) ISS

with ISS-Lyapunov function V (x1, x2) = maxσ(V1(x1)),V2(x2)where σ is any K∞- function with γ21 ≤ σ ≤ γ−1

12 15/29

Page 16: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Stability of n coupled systems

Consider

x1 = f1(x1, . . . , xn, u)...

xn = fn(x1, . . . , xn, u)

γ12 γn3γn1

γ2n

γ31

γ13

6?. . .

6?. . .

Σ1

Σ2

Σ3

Σn. . .

. . .

. . .. . .

-

-

6

? ?

@@@R

with

|xi (t)| ≤ maxβi (|xi (0)|, t),

nmaxj=1

γij(||xj ||∞), ηi (||u||∞)

and γij ≡ 0 or γij ∈ K∞, and γii := 0.

16/29

Page 17: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

The gain matrix

Γ := (γij) =

0 γ12 . . . . . . γ1n

γ21 0 γ23 . . . γ2n...

...γn−1,1 . . . γn−1,n−2 0 γn−1,n

γn1 . . . . . . γn,n−1 0

Γmax : Rn+ → Rn

+ Γ(s) =

maxnj=1 γ1j(sj)...

maxnj=1 γnj(sj)

In case of gain summation

Γ∑ : Rn+ → Rn

+ Γ(s) =

∑n

j=1 γ1j(sj)...∑n

j=1 γnj(sj)

17/29

Page 18: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Small-gain theorem

Γ : Rn+ → Rn

+, Γ(s) =

maxnj=1 γ1j(sj)...

maxnj=1 γnj(sj)

Theorem (S.D., B. Ruffer, F. Wirth 2007)

Γ(s) 6≥ s ∀ s ∈ Rn+, s 6= 0 ⇒ x = f (x , u) ISS

Notation: x = (xT1 , . . . , xTn )T and f = (f T1 , . . . , f Tn )T

Remarks: 1) For x , y ∈ Rn holds x 6≥ y ⇔ ∃i with xi < yi .2) Equivalence to cycle condition: all cycles are contractions.3) In case of linear gains: Γ(s) 6≥ s ⇔ ρ(Γ) < 1

18/29

Page 19: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Associate discrete time system

Let Γ be a nonlinear operator as above. Considersk+1 := Γ(sk), s0 ∈ Rn

+ , k = 0, 1, 2, . . . , (*)

Theorem

Γ(s) 6≥ s for all s ∈ Rn+ \ 0 ⇔ (*) is GAS

Remarks:

I The system (*) can be used as a comparison system

I Note the dimension reduction compared with original system

19/29

Page 20: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Topological property

The above observations help to prove:

Theorem (S.D., B. Ruffer, F. Wirth 2010)

Γ(s) 6≥ s ∀ s ∈ Rn+ \ 0 ⇒ ∃σ1, . . . , σn ∈ K∞ :

∀ t > 0 : Γ(σ(t)) < σ(t), σ(t) = (σ1(t), . . . , σn(t))T

σ : [0,∞)→ Rn+

is called Ω-path

20/29

Page 21: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Geometric interpretationCones for inverse graphs

Ωi =

x ∈ Rn : si >∑j 6=i

γij(sj)

.

Ωi =

x ∈ RN : |xi | >∑j 6=i

γij(|xj |)

.

Γ(s) 6≥ s ∀s 6= 0, s ≥ 0 is equivalent to

In⋃

i=1Ωi = RN \ 0 and

In⋂

i=1Ωi 6= ∅.

Ω3

Ω2

Ω1

On Ωi there exist ISSLyapunov functions Vi

withVi (x) = ∇Vi (x) · fi (x) < 0if x ∈ Ωi .

The ISS-Lyapunov function for the network is V (x) = maxiσ−1i (Vi (xi )).

21/29

Page 22: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

ISS-Lyapunov function for networks

Let Vi be an ISS-Lyapunov function for the i-th system:

ψi1(|xi |) ≤ Vi (xi ) ≤ ψi2(|xi |), xi ∈ RNi

Vi (xi ) ≥ maxj=1,...,n

γij(Vj(xj)), γi (|ui |) ⇒ Vi (x) ≤ −αi (Vi (xi ))

Define Γ = (γij)i ,j=1,...,n, Γ : Rn+ → Rn

+.

Theorem (S.D., B. Ruffer, F. Wirth (2010))

Γ(s) 6≥ s ∀ s ∈ Rn+ \ 0 ⇒ V (x) = max

iσ−1

i (Vi (xi ))

is ISS-Lyapunov function for x = f (x , u).

22/29

Page 23: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Small-gain theory for other types of systems

These small gain results were recently extended to other classes ofsystems:

I discrete time systems

I switched systems

I impulsive systems

I hybrid systems

I infinite dimensional systems

Additional poprties or conditions are sometimes needed(dwell-time, uniformity, Zeno solutions).

23/29

Page 24: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Example 1Cascade of finite number of ISS Σi is ISS, but for u, xi ∈ R, i ∈ N

Σ1 Σ2 Σ3. . . Σn

. . .. . . - - - - - -

Σ1 : x1 = −x1 + uΣ2 : x2 = −x2 + 2x1

. . .Σk : xk+1 = −xk+1 + 2xk. . .

each Σi is ISS, however the cascade is not ISS:take u = 1 and xi (0) = 1, i ∈ N then∀ t ≥ 0 x2 > 0, x3 > 0, . . . ⇒ |x |`∞ grows at any time,i.e.

the solution grows unbounded to the constant point given byx1 = 1, x2 = 2, . . . , xk = 2k−1, . . . and in particularlimt→∞ |x(t)|∞ =∞ contradicting ISS

24/29

Page 25: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Example 2: we make the gains smaller

In the next example all gains are identities:x1 = −x1 + ux2 = −x2 + x1

. . .xk+1 = −xk+1 + xk. . .

Taking zero input u = 0 and initial state xi (0) = 1, i ∈ N the

solution is given by xi (t) = e−t(∑i

k=0tk

k!

), i ∈ N for which

limt→∞ ||x ||∞ = 1 6= 0 contradicting the ISS property.

25/29

Page 26: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

Example 3: we make the gains smaller again

In the next example all gains are smaller than the identityx1 = −2x1 + x2 + u1

x2 = −32x2 + x3 + u2

. . .

xk = −k+1k xk + xk+1 + uk

. . .

x1 x2 x3 x4 . . . xk . . . . . .γ12 = 1

2 γ23 = 23 γ34 = 3

4 . . . . . . γk,k+1 = k+1k . . . . . . . . .

This is an infinite cascade with the interconnection gains

γij = 0 ⇔ j 6= i + 1 and γk,k+1 = kk+1 < id, k ∈ N

26/29

Page 27: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

The matrix A of the system has an unbounded inverse A−1

A =

−2 1 0 0 0 · · ·0 − 3

2 1 0 0 · · ·0 0 − 4

3 1 0 · · ·0 0 0 − 5

4 1 · · ·0 0 0 0 − 5

4 · · ·...

......

......

. . .

,A−1 =

− 12 − 1

3 − 14 − 1

5 − 16 · · ·

0 − 23 − 1

2 − 25 − 1

3 · · ·0 0 − 3

4 − 35 − 1

2 · · ·0 0 0 − 4

5 − 23 · · ·

0 0 0 0 − 56 · · ·

......

......

.... . .

⇒ λ = 0 ∈ σ(A) ⇒ the system is not 0-GAS, hence it is notISS.

Γ =

0 12 0 0 0 · · ·

0 0 23 0 0 · · ·

0 0 0 34 0 · · ·

0 0 0 0 45 · · ·

0 0 0 0 0. . .

......

......

.... . .

⇒ ∀s ∈ `+

∞ \ 0 Γ(s) < s.

Γ satisfies the usual SGC, but Σ is not ISS

27/29

Page 28: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

This example shows: previously known SGC is not enough.Moreover in spite of Γ(s) < s we have 1 ∈ σ(Γ), because (id−Γ)−1

=

1 −12 0 0 0 · · ·

0 1 −23 0 0 · · ·

0 0 1 −34 0 · · ·

0 0 0 1 −45 · · ·

0 0 0 0 1. . .

......

......

.... . .

=

1 12

13

14

15 · · ·

0 1 23

12

25 · · ·

0 0 1 34

35 · · ·

0 0 0 1 45 · · ·

0 0 0 0 1 · · ·...

......

......

. . .

is unbounded. The spectral radius of Γ does not satisfy r(Γ) < 1,which is different from the finite dimensional case.

xk = Γ(xk−1) is GAS 6⇔ Γ(s) 6≥ s for all s ∈ Rn+ \ 0

28/29

Page 29: Stability of Interconnected Dynamical Systems - tu … · Motivation Stability of small ISS-Lyapunov functions for small Back to 1 Systems of systems vehicle platooning-1 2 3::: n:::

Motivation Stability of small Σ ISS-Lyapunov functions for small Σ Back to∞

HypothesisFor i ∈ N consider Σi with gains γij ∈ K∞ ∪ 0, j ∈ N

I The number of neigbours of any Σi is uniformly bounded

I Any cycle (i1 = ik) is a contraction:γi1i2 γi2i3 · · · γik−1ik (r) < r , r > 0

I ∃c ∈ (0, 1), M ∈ N ∀ i1, i2 . . . , ik with k ≥ Mγi1i2 γi2i3 · · · γik−1ik (r) < cr , r > 0

Then

I Q(x) := supx , Γ(x), Γ2(x), . . . is a well defined mapQ : Rn

+ → Rn+ satisfying Γ(Q(x)) ≤ Q(x), x ∈ Rn

+

I σi (r) := [Q(1)]i is a K∞ function

I V (x) := supjσ−1j (Vj(xj)) satisfies

V (x) ≥ γ(|u|∞) ⇒∞∑i=1

∂V

∂xifi (xi , xi , ui ) ≤ −α(|u|∞)

29/29