This presentation discuss a sufficient and necessary condition for quadratic stability of a class of switched systems including two modes. This result has been published in proceeding of the ECC Conference in Zürich, 2013.

1. A new stability result for switched linear systems
Yashar Kouhi, Naim Bajcinca, J¨org Raisch, and Robert Shorten
Technische Universit¨at Berlin, Germany
Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
IBM Research Ireland
European control conference, Z¨urich, July 18, 2013

2. Outline
Necessary conditions for quadratic stability of switched linear
systems
Symmetric transfer function matrices
Strictly positive real systems
Quadratic stability of the class of rank-m difference switched linear
systems

3. Stability of switched linear systems
Switched linear system model
Arbitrary time switching
Σ : ˙x = Ai(t)x i(t) ∈ {1, 2},
A1 and A2 are in Rn×n and are Hurwitz
x ∈ Rn is the state
i(t) is the arbitrary time switching signal
Quadratic stability
A function V(x) = xTPx with P = PT > 0 is a Common Quadratic
Lyapunov Function (CQLF) for Σ iff
AT
1 P + PA1 < 0 AT
2 P + PA2 < 0
Find conditions on A1 and A2 such that a CQLF for Σ exists

4. Necessary condition for CQLF
1 Step 1: Pre- and post-multiply the second ineq. by A−T
2 and A−1
2
AT
1 P + PA1 < 0, A−T
2 AT
2 P + PA2 A−1
2 < 0
2 Step 2: Multiply the second ineq. resulting from Step 1 by ω2
AT
1 P + PA1 < 0, ω2
A−T
2 P + PA−1
2 < 0
3 Step 3: Add two inequalities resulting from Step 2
A1 + ω2
A−1
2
T
P + P A1 + ω2
A−1
2 < 0 ∀w ∈ R
A1 + ω2
A−1
2 is Hurwitz for all ω ∈ R
det A1 + ω2
A−1
2 = 0 implies that det ω2
I + A1A2 = 0
Necessary condition
A1A2 has no real negative eigenvalue

5. Geometrical interpretation of spectral condition
If A1A2 has a real negative eigenvalue then:
there exists a β ∈ R and v ∈ Rn satisfying
A1A2v = −β2v or A2v = −β2A−1
1 v
at some points the degrees between the vector fields of ˙x = A−1
1 x
and ˙x = A2x are 180◦
a sequence of switchings exists which makes the following
switched system unstable ˙x = ˆAi(t)x ˆAi(t) ∈ {A−1
1 , A2}
−5 −4 −3 −2 −1 0 1 2 3 4
−4
−3
−2
−1
0
1
2
3
4
x1
x2

6. Sufficiency and necessity for CQLF
Is A1A2 having no real negative eigenvalues sufficient for
existence of a CQLF for A1 and A2? (No)
Counter example
A1 =




−1.8 0.5 0.1 0.4
0.1 −1.8 0.1 0.9
0.1 0.7 0.6 −2.1
0.1 0.8 0.9 −1.8



 A2 =




−1.8 0.1 0.9 0.3
0.2 −0.4 −1.7 0.3
−0.4 0.7 1.1 −2
0.4 0.6 0.8 −2.8




A1 and A2 are both Hurwitz
Eigenvalues of A1A2 are at 4.696, 0.6303 ± 1.5505i, .2333
No CQLF for this pair exists
Are there classes of switched systems where this necessary
condition is also sufficient for existence of a CQLF? (Yes)

7. Classes of quadratically stable switched systems
Theorem ( Rank-1 difference switched systems)
[Shorten and Narendra(2003)]. Given A ∈ Rn×n Hurwitz, b ∈ Rn×1,
c1×n ∈ Rn, and d > 0. The switched linear system
˙x = A − σ(t)
bc
d
x σ(t) ∈ {0, 1}
is quadratically stable iff
A1A2 has no real negative eigenvalue, or equivalently
g(s) = c(sI − A)−1b + d is strictly positive real
New class of switched linear systems
We now generalize this result to the class of rank-m difference
switched systems related by a symmetric transfer function matrix

8. Class of rank -m difference switched systems
Theorem
Given two real Hurwitz matrices A and A − BD−1C with (A, B)
controllable and (A, C) observable, satisfying
D = DT > 0
CAiB = (CAiB)T i = 0, 1, . . . , n − 1,
or equivalently G(s) = C(sI − A)−1B + D is symmetric, i.e.
G(s) = GT(s). Then, the switched system
˙x = (A − σ(t)BD−1
C)x σ(t) ∈ {0, 1},
is quadratically stable iff A(A−BD−1C) has no real negative eigenvalue
Proof of sufficiency.
A(A − BD−1
C) has no
real negative eigenvalue
⇒ G(s) is SPR ⇒ quadratic stability of
switched system

9. Symmetric transfer function matrix
Lemma
Given A, A2 ∈ Rn×n. A symmetric transfer function matrix
G(s) = C(sI − A)−1B + D with D = DT > 0, can be associated with A
and A2 such that
A2 = A − BD−1
C,
with B ∈ Cn×m, C ∈ Cm×n, and D ∈ Cm×m iff
E := In ⊗ A − A ⊗ In, E2 := In ⊗ A2 − A2 ⊗ In,
share a common eigenvector corresponding to a zero eigenvalue, say
vec(Y) = [y11 . . . yn1 y12 . . . yn2, . . . , y1n . . . ynn]T such that the matrix
Y = [yij]n×n is symmetric invertible
⊗ denotes the Kronecker Product

10. Strictly positive real systems (SPR)
An m × m rational transfer function matrix G(s) = C(sI − A)−1B + D is
said to be SPR if
1 there exists an α > 0 such that if G(s) is analytic for all s for which
Re(s) −α
2 G(jω − α) + GT(−jω − α) 0 for all ω ∈ R
Lemma ([Corless and Shorten(2010)] D non-singular)
Given a Hurwitz matrix A, B ∈ Cn×m and C ∈ Cm×n, and D ∈ Cn×n with
D > 0. The rational transfer function matrix G(s) = C(sI − A)−1B + D is
SPR iff
G(jω) + GT
(−jω) > 0 ∀ω ∈ R

11. Sufficiency
G(s) being SPR is concluded by:
1 Step 1: A(A − BD−1C) has no real negative eigenvalue
det(ω2
I + A(A − BD−1
C)) > 0 ⇒ det(D − C(ω2
I + A2
)−1
AB) > 0
2 Step 2: Symmetry of G(s)
D − C(ω2
I + A2
)−1
AB =
1
2
G(jω) + G(−jω)T
⇒
det
1
2
G(jω) + G(−jω)T
> 0
3 Step 3: From D > 0 and continuity of G(jω) w.r.t ω everywhere
G(jω) + GT
(−jω) > 0 ∀ω ∈ R

12. Kalman-Yakubovic-Popov (KYP) lemma
Lemma (KYP lemma)
Let (A, B) be controllable and (A, C) be observable, then
G(s) = C(sI − A)−1B + D is SPR iff there exist matrices P = P∗ > 0, Q
and W, and a number α > 0 satisfying
1 A∗P + PA = −Q∗Q − αP
2 B∗P − C = W∗Q
3 D + D∗ = W∗W
Item 1 of KYP lemma implies
A∗
P + PA < 0
Items 1,2, and 3 of KYP lemma imply
(A − BD−1
C)∗
P + P(A − BD−1
C) =
= −αP − (Q − WD−1
C)∗
(Q − WD−1
C) < 0

13. SPR system and CQLF for the switched system
Lemma (KYP lemma)
Let (A, B) be controllable and (A, C) be observable, then
G(s) = C(sI − A)−1B + D is SPR iff there exist matrices P = P∗ > 0, Q
and W, and a number α > 0 satisfying
1 A∗P + PA = −Q∗Q − αP
2 B∗P − C = W∗Q
3 D + D∗ = W∗W
A∗
P + PA < 0, (A − BD−1
C)∗
P + P(A − BD−1
C) < 0
Define: P = Re(P)
V(x) = xTPx for x ∈ Rn is the CQLF for A and A2 := A − BD−1C

14. Conclusions and Future work
We presented
necessary spectral conditions for quadratic stability of switched
linear systems
necessary and sufficient conditions on pair of matrices that a
symmetric transfer function matrices for them exists
necessary and sufficient conditions for a symmetric transfer
function matrix to be strictly positive real
necessary and sufficient conditions for quadratic stability of class
of rank-m difference switched linear systems related by a
symmetric transfer function matrix
Future work:
Strictly positive real systems with singular matrix D and symmetric
transfer function matrix
Weak quadratic stability of switched linear systems

15. M. Corless and R. Shorten.
On the characterization of strict positive realness for general
matrix transfer functions.
IEEE Transactions on Automatic Control, (8):1899 –1904, 2010.
R. N. Shorten and K. S. Narendra.
On common quadratic Lyapunov functions for pairs of stable LTI
systems whose system matrices are in companion form.
IEEE Transactions on Automatic Control, (4):618 – 621, 2003.