stability of interconnected dynamical systems - tu … · motivation stability of small...
TRANSCRIPT
Stability of InterconnectedDynamical Systems
Sergey DashkovskiyInstitute of Mathematics, University of Wurzburg
Elgersburg, 26.02.2018
1/29
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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