Mixed ISS systems

A small gain condition for interconnections
of ISS systems
with mixed ISS characterizations
Sergey Dashkovskiy1
Michael Kosmykov1
Fabian Wirth2
1
Centre of Industrial Mathematics, University of Bremen, Bremen, Germany
2
Institute of Mathematics, University of W¨urzburg, W¨urzburg, Germany
August 24, 2009, ECC’09, Budapest, Hungary

Centre for
Industrial Mathematics
Outline
1 Introduction
2 Main result
3 Examples
4 Conclusions
2 / 21Introduction Main result Examples Conclusions

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Interconnected system
Σ1 : ˙x1 = f1(x1, . . . , xn, u1)
...
Σn : ˙xn = fn(x1, . . . , xn, un)
xi ∈ RNi
, ui ∈ Rmi
, fi : R
n
j=1 Nj +mi
→ RNi
are continuous and for all
r ∈ R are locally Lipschitz continuous in x = (x1
T
, . . . , xn
T
)
T
uniformly
in ui for |ui | ≤ r.

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Σ : ˙x = f (x, u) =



f1(x1, . . . , xn, u1)
...
fn(x1, . . . , xn, un)


, x =



x1
...
xn


 , u =



u1
...
un


 .

Centre for
Σ : ˙x = f (x, u) =



f1(x1, . . . , xn, u1)
...
fn(x1, . . . , xn, un)


, x =



x1
...
xn


 , u =



u1
...
un


 .
Is system Σ stable?

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Notation
γ ∈ K γ ∈ K∞
β ∈ KL
L∞ is the set of measurable functions for which the norm · ∞ is ﬁnite.

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Input-to-state stability in maximization formulation
Deﬁnition
Σ : ˙x = f (x, u) is input-to-state stable (ISS), if ∃β ∈ KL and γ ∈ K,
such that
|x(t)| ≤ max{β(|x(0)|, t), γ( u ∞)}
holds for all x(0) ∈ Rn
, u ∈ L∞(R+, Rm
), t ≥ 0.
β(|x(0)|, t)
γ( u ∞)
t0
|x(0)|
x

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Input-to-state stability in summation formulation
Deﬁnition
Σ : ˙x = f (x, u) is input-to-state stable (ISS), if ∃β ∈ KL and γ ∈ K,
such that
|x(t)| ≤ β(|x(0)|, t) + γ( u ∞)
, u ∈ L∞(R+, Rm
), t ≥ 0.
β(|x(0)|, t) + γ( u ∞)
γ( u ∞)
t0
|x(0)|
x

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Input-to-state stability in mixed formulation
Deﬁnition
Σi : ˙xi = fi (x1, . . . , xn, ui ) is ISS if ∃βi ∈ KL, γij , γi ∈ K∞ ∪ {0} such
that for any initial value xi (0) and input u ∈ Rm
the solution xi (t)
satisﬁes
|xi (t)| ≤ βi (|xi (0)|, t) +
n
j=1
γij ( xj[0,t] ∞
) + γi ( u ∞), t ≥ 0 (1)
or
|xi (t)| ≤ max{βi (|xi (0)|, t), max
j
{γij ( xj[0,t] ∞
)}, γi ( u ∞)}, t ≥ 0 .
(2)

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Input-to-state stability in mixed formulation
Deﬁnition
Σi : ˙xi = fi (x1, . . . , xn, ui ) is ISS if ∃βi ∈ KL, γij , γi ∈ K∞ ∪ {0} such
that for any initial value xi (0) and input u ∈ Rm
the solution xi (t)
satisﬁes
|xi (t)| ≤ βi (|xi (0)|, t) +
n
j=1
γij ( xj[0,t] ∞
) + γi ( u ∞), t ≥ 0 (1)
or
|xi (t)| ≤ max{βi (|xi (0)|, t), max
j
{γij ( xj[0,t] ∞
)}, γi ( u ∞)}, t ≥ 0 .
(2)
Let Σi be ISS with:
(1) for i = 1, . . . , k and
(2) for i = k + 1, . . . , n, where 0 ≤ k ≤ n.

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Problem statement
Let subsystems Σ1, . . . , Σn be ISS. Is their interconnection Σ also ISS?

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Problem statement
Answer:
In general not.

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Problem statement
Answer:
In general not.
Are there any conditions to guarantee ISS of Σ?

Centre for
Problem statement
Answer:
In general not.
Are there any conditions to guarantee ISS of Σ?
Let Γ = (γij )n×n, where γii ≡ 0, i = 1, . . . , n. And deﬁne Γ : Rn
+ → Rn
+
by:
Γ(s) :=










γ12(s2) + · · · + γ1n(sn)
...
γk1(s1) + · · · + γkn(sn)
max{γk+1,1(s1), . . . , γk+1,n(sn)}
...
max{γn1(s1), . . . , γn,n−1(sn−1)}










for s ∈ Rn
+.

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The main result
a ≥ b, a, b ∈ Rn
⇒ ∃ i ∈ {1, . . . , n} : ai < bi .

Centre for
The main result
a ≥ b, a, b ∈ Rn
⇒ ∃ i ∈ {1, . . . , n} : ai < bi .
Consider
D :=
k n−k










id + α . . . 0 0 . . . 0
...
...
...
...
...
...
0 . . . id + α 0 . . . 0
0 . . . 0 id . . . 0
...
...
...
...
...
...
0 . . . 0 0 . . . id










, α ∈ K∞.
Operator D : Rn
+ → Rn
+ is deﬁned by
D(s) := (s1 + α(s1), . . . , sk + α(sk ), sk+1, . . . , sn)T
for any s ∈ Rn
+.
Theorem (Main result)
Assume that each subsystem Σi is ISS. If there exists an α ∈ K∞ such
that Γ ◦ D(s) ≥ s for all s ∈ Rn
+{0}, then the system Σ is ISS.

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Relations to known results
From 2007:
1) If k = 0 (maximization case) then
Γ(s) ≥ s ⇔ Γ ◦





id 0 . . . 0
0 id . . . 0
...
...
...
...
0 0 . . . id





(s) ≥ s, ∀s ∈ Rn
+{0}.
2) If k = n (summation case) then
Γ◦D(s) ≥ s ⇔ Γ◦





id + α 0 . . . 0
0 id + α . . . 0
...
...
...
...
0 0 . . . id + α





(s) ≥ s, ∀s ∈ Rn
+{0}.

Centre for
Relations to known results
From 2007:
1) If k = 0 (maximization case) then
Γ(s) ≥ s ⇔ Γ ◦





id 0 . . . 0
0 id . . . 0
...
...
...
...
0 0 . . . id





(s) ≥ s, ∀s ∈ Rn
+{0}.
2) If k = n (summation case) then
Γ◦D(s) ≥ s ⇔ Γ◦





id + α 0 . . . 0
0 id + α . . . 0
...
...
...
...
0 0 . . . id + α





(s) ≥ s, ∀s ∈ Rn
+{0}.
For the case 0 < k < n (mixed case) we can use
maxi=1,...,n{xi } ≤
n
i=1 xi ≤ n maxi=1,...,n{xi }
to pass to the situation with k = 0 or k = n.
The gains are conservative!

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Global stability
Deﬁnition (GS)
System Σ is globally stable (GS), if there exist functions σ, ˆγ of class K,
such that the inequality
|x(t)| ≤ σ(|x(0)|) + ˆγ( u ∞) (3)
, u ∈ L∞(R+, Rm
), t ≥ 0.
Theorem (Small gain condition for GS)
Assume each Σi is GS and there exists an α ∈ K∞ such that
Γ ◦ D(s) ≥ s for all s ∈ Rn
+{0}. Then the system Σ is GS.

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Asymptotic gain property
Deﬁnition (AG)
System Σ has the asymptotic gain (AG) property, if there exists a
function γ of class K, such that the inequality
lim sup
t→∞
|x(t)| ≤ γ( u ∞) (4)
and u ∈ L∞(R+, Rm
).
Theorem (Small gain condition for AG)
Assume each Σi has the AG property and solutions of Σ exist for all
t > 0 and are uniformly bounded. If there exists an α ∈ K∞ such that
Γ ◦ D(s) ≥ s for all s ∈ Rn
+{0} then Σ satisﬁes the AG property.

Centre for
Remarks on the proofs
µ : Rn
+ × Rn
+ → Rn
+ is deﬁned by
µ(w, v) :=










w1 + v1
...
wk + vk
max{wk+1, vk+1}
...
max{wn, vn}










, w ∈ Rn
+, v ∈ Rn
+.
Lemma
Let Γ satisfy Γ ◦ D(s) ≥ s for any s ∈ Rn
+{0}. Then there exists a
φ ∈ K∞ such that for all w, v ∈ Rn
+,
w ≤ µ(Γ(w), v)
implies w ≤ φ( v ).

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Remarks on the proof of the main theorem
Theorem (1986)
A system is ISS if and only if it is GS and has the AG property.

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Remarks on the proof of the main theorem
Theorem (1986)
A system is ISS if and only if it is GS and has the AG property.
Proof of the main result.
ISS of subsystems Σi ⇒ GS and AG of subsystems Σi .
GS and AG of Σi and Γ ◦ D(s) ≥ s ⇒ GS and AG of system Σ.
GS and AG of system Σ ⇒ ISS of Σ.

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Application of the main result
|x1(t)| ≤ β1(|x(0)|) + γ13(||x3[0,t]||∞) + γ1( u ∞)
|x2(t)| ≤ max{β2(|x(0)|), γ21(||x1[0,t]||∞),
γ23(||x3[0,t]||∞), γ2( u ∞)}
|x3(t)| ≤ max{β3(|x(0)|), γ32(||x2[0,t]||∞), γ3( u ∞)}
(5)
γ13(t) = (id + ρ)−1
(t), ρ ∈ K∞, γ21(t) = t, γ23(t) = t and
γ32(t) = t(1 − e−t
), t ≥ 0.
The small gain condition reads as


γ13(s3)
max{γ21 ◦ (id + α)(s1), γ23(s3)}
γ32(s2)

 ≥


s1
s2
s3

 , ∀s ∈ Rn
+{0}.

Centre for
Application of the main result
The small gain condition is equivalent to: ∃α ∈ K∞ such that
(id + α) ◦ γ13 ◦ γ32 ◦ γ21(t) < t
γ23 ◦ γ32(t) < t
for all t > 0.
Take α = ρ ⇒ the first inequality is satisfied:
(id + ρ) ◦ (id + ρ)−1
◦ (t(1 − e−t
)) = t(1 − e−t
) < t.
The second inequality is also satisfied:
t(1 − e−t
) < t.
Thus by the main result system Σ is ISS.

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An attempt to apply known results
Estimating maximums by sums in (5) we get:
|x1(t)| ≤ β1(|x(0)|) + γ13(||x3[0,t]||∞) + γ1( u ∞)
|x2(t)| ≤ β2(|x(0)|) + γ21(||x1[0,t]||∞)
+γ23(||x3[0,t]||∞) + γ2( u ∞)
|x3(t)| ≤ β3(|x(0)|) + γ32(||x2[0,t]||∞) + γ3( u ∞)
The small gain condition from 2007 is: ∃α ∈ K∞, i = 1, 2, 3 such that
γ13 ◦ (id + α)(s3)
γ21 ◦ (id + α)(s1) + γ23 ◦ (id + α)(s3)
γ32 ◦ (id + α)(s2)
≥
s1
s2
s3
, ∀s ∈ Rn
+{0}
for all s ∈ R3
+{0}.

Centre for
Attempt to apply known results
Suppose the small gain condition holds. Take
s =


0
t
γ32 ◦ (id + α)(t)

 and t ≥ 0 ⇒
γ23 ◦ (id + α) ◦ γ32(t) < t, i.e.
γ23◦(id+α)◦γ32(t) = (id+α)(t(1−e−t
)) = (t(1−e−t
))+α(t(1−e−t
)) < t
⇒ α(t(1 − e−t
)) < te−t
for all t ≥ 0.
This leads to a contradiction, since lim
t→∞
α(t(1 − e−t
)) = +∞ and
lim
t→∞
te−t
= 0.
Thus we cannot conclude whether Σ is ISS.

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Conclusions
1. Interconnection of subsystems with diﬀerent ISS formulations was
considered.
2. A small gain condition establishing ISS of such an interconnection was
presented.
3. This condition applies to a larger class of interconnections of ISS
systems.
4. It is less conservative in the case of mixed ISS systems.

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Thank you for your attention!

Mixed ISS systems

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