ھد ن ا ید ١

Theorem 3.4:1 Suppose that the switched linear system ∑ MiA )( is consistently stabilizable. Then, there is a Mk ∈ such that:

0)(1

≤∑=

n

iki Aλ

Where niAi ≤≤1,)(λ are the eigenvalues of matrix A. Furthermore, if the system is consistently asymptotically stabilizable, then the inequality is strict, i.e.:

0)(1

<∑=

n

iki Aλ

Proof: A) Let σ be a consistent switching signal that stabilizes the switched system. Suppose that the switching duration sequence of σ is:

{ }L),,(),,( 1100 hihiDS =σ 1) If the sequence is finite, i.e., there involve only finite switches in σ,

then it can be seen that the last active subsystem must be stable (note to the consistently stabilizable property) and the theorem follows immediately.

0)(0)(,...,11

≤⇒≤=∀ ∑=

n

ikisubsystemLasti AAni λλ

2) If the sequence is infinite, it follows from the well-posedness of the system (as the system is stabilizable and by def. 3.2) w.r.t the switching signal σ, that there involve only finite switches in any finite time (no zeno phenomena). As a consequence:

∞→∑=

l

jiih

0 as ∞→l

As the system is consistently stabilizable, according to Definition 3.2, there is a consistent switching signal σ such that the system is well-posed and uniformly stable. Therefore by definition 2.4 we have that the system should be uniformly stable, i.e.:

0),,;(0)(0 0000 ≥≥∀<⇒<>=∃>∀ ttxttxthatsuch εσφδεδδε Note that: 1 - Theorem 3.4, Page , “Switched Linear Systems: Control and Design”, Zhendog Sun and Shunzhi S. Ge

ھد ن ا ید ٢

- σ is not a function of 0x , because the switching signal is consistent. - δ is only function of ε (not a function of 0t ), because the system is

uniformly stable. By setting ε = 1, there exists a δ > 0 such that for 00 =t :

01),,0;( 00 ≥∀≤⇒≤ txtx σφδ As the system is linear:

≤≤

+≤≤≤

=

t

hhth

xeee

xeehtxe

xt

hAhAhA

hAA

tA

sisissi

sisi

si

100

0

0

00

0

0011

0011

0

...

),,0;(M

σφ

In particular: ...,2,1,01... 00

0011 =∈∀≤ sBxxeee hAhAhA sisissi

δ Which, . , stands for vector norm. We have by definition of induced-norm of a matrix:

...,2,1,01... 0011 =≤ seee hAhAhA sisissi

δ

Which, . , stands for matrix norm. As a consequence, all entries of the matrices

0011001100 ...,...,, hAhAhAhAhAhA sisissisisisi eeeeee

must be bounded by δ1 . (Because each entry of a matrix is less than its

norm)

For contrary suppose that: { } 0)(min

1>= ∑

=∈

n

ikiMk

Aλρ Then, we have (by diagonal transformation)

0))(exp())ˆ(exp())ˆ(exp(

)(det)det(.)(det).det()(det)(det)(det

111

ˆ1ˆ1ˆˆ 1

>∈≥===

====

∑∑∏===

−−−

hMkeAhAhAh

ePePPePeeh

n

iki

n

iki

n

iki

hAhAhAhPAPhA kkkkk

ρλλλ

As a result,

ھد ن ا ید ٣

0)det(,

0)det(,0)det(,

11

00

11

00

>≥→==

>≥→==

>≥→==

hhAss

hhA

hhA

eehhik

eehhikeehhik

sis

i

i

ρ

ρ

ρ

M

Multiplying both sides, we have:

∞→∞→≥⇒

≥⇒≥

∑

∑

=

=

saseeee

eeeeeeeeees

jj

sisii

s

jj

sisiissisii

hhAhAhA

hhAhAhAhhhhAhAhA

01100

01100101100

)...det(

)...det(....)det(..).det().det(ρ

ρρρρ

This contradicts the boundedness of entries of the matrices and establishes the former part of the theorem. B) The latter part can be proven in a similar manner: Let σ be a consistent switching signal that stabilizes the switched system. Suppose that the switching duration sequence of σ is:

{ }L),,(),,( 1100 hihiDS =σ 1) If the sequence is finite, i.e., there involve only finite switches in σ,

then it can be seen that the last active subsystem must be asymptotically stable (note to the consistently stabilizable) and the theorem follows immediately.

0)(0)(,...,11

<⇒<=∀ ∑=

n

ikisubsystemLasti AAni λλ

2) If the sequence is infinite, it follows from the well-posedness of the system (as the system is stabilizable and by def. 3.2) w.r.t the switching signal σ, that there involve only finite switches in any finite time (no zeno phenomena). As a consequence:

∞→∑=

l

iih

0 as ∞→l

As the system is consistently asymptotically stabilizable, according to Definition 3.2, there is a consistent switching signal σ such that the system is well-posed and uniformly asymptotically stable. Therefore by

ھد ن ا ید ٤

definition 2.4 we have that the system should be both uniformly stable and uniformly attractive. uniformly stable, i.e.:

0),,;(0)(0 0000 ≥≥∀<⇒<>=∃>∀ ttxttxthatsuch εσφδεδδε Note that:

- σ is not a function of 0x , because the switching signal is consistent. - δ is only function of ε (not a function of 0t ), because the system is

uniformly stable. By setting ε = 1, there exists a δ > 0 such that for 00 =t :

01),,0;( 00 ≥∀≤⇒≤ txtx σφδ By uniformly attractively of the system we have for 00 =t :

∞→→⇒≤ tasxtx 0),,0;( 00 σφδ As the system is linear:

000011...),,0;( xeeext hAhAhA sisissi=σφ

In particular: .0... 00

0011 ∞→∈∀→ tBxxeee hAhAhA sisissi

δ Which, . , stands for vector norm. We have by definition of induced-norm of a matrix:

∞→→ teee hAhAhA sisissi 0... 0011 Which, . , stands for matrix norm. As a consequence, all entries of the matrix

0011... hAhAhA sisissi eee (*) must be tend to zero for ∞→s . (Because each entry of a matrix is less than its norm)

for contrary suppose that: { } 0)(min

1≥= ∑

=∈

n

ikiMk

Aλρ Then, we have (by diagonal transformation)

ھد ن ا ید ٥

01))(exp())ˆ(exp())ˆ(exp(

)(det)det(.)(det).det()(det)(det)(det

111

ˆ1ˆ1ˆˆ 1

>∈≥≥===

====

∑∑∏===

−−−

hMkeAhAhAh

ePePPePeeh

n

iki

n

iki

n

iki

hAhAhAhPAPhA kkkkk

ρλλλ

As a result,

∞→∞→⇒≥⇒≥

∑=

sasoreeeeeeeeeeeee

sisii

s

jj

sisiissisii

hAhAhA

hhAhAhAhhhhAhAhA

1)...det()...det(....)det(..).det().det(

1100

01100101100

ρρρρ

This contradicts the zeroness of entries of the matrix (*) and establishes the latter part of the theorem.