Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 1, pp. 31-50, Warsaw 2011

LIE ALGEBRA APPROACH IN THE STUDY OF THE

STABILITY OF STOCHASTIC LINEAR HYBRID SYSTEMS

Ewelina Seroka

Lesław Socha

College of Sciences, Cardinal Stefan Wyszyński University in Warsaw, Faculty of Mathematics and

Natural Sciences, Warsaw, Poland; e-mail: ewelina seroka@o2.pl; leslawsocha@poczta.onet.pl

Theproblemof the stability of a class of stochastic linear hybrid systems
witha special structureofmatrices andamultiplicative excitation is con-
sidered. Sufficient conditions of the exponential p-thmean stability and
the almost sure stability for a class of stochastic linear hybrid systems
with the Markovian switching are derived. Also, sufficient conditions of
the exponential mean-square stability for a class of stochastic linear hy-
brid systems satisfying the Lie algebra conditionswith any switching are
found. The obtained results are illustrated by examples and simulations.

Key words: hybrid systems, nonlinear systems, asymptotic stability

1. Introduction

Starting with 50’s, there has been an increasing interest in the analysis of
switching systems, for instance, Itkis (1983), Loparo andAslanis (1987), Utkin
(1978), andvariable structures systems (Kazakow andArtemiev, 1980). Inme-
chanical engineering, the typical example are vibration systems with impacts
(Dimentberg and Iourtchenko, 2004). It concerns both a deterministic and a
stochastic case.

The generalization of these systems are hybrid systems (Liberzon, 2003;
Boukas, 2005), which are dynamic systems consisting of several structures
describedbydeterministic or stochastic differential equations. In the successive
moments of time their structure can change according to the given switching
rule, thereuponcreates thehybrid system.Oneof themost importantproblems
in the analysis of hybrid systems is the determination of stability conditions.
In the case of stochastic systems, usually the almost sure and the p-mean



32 E. Seroka, L. Socha

stability is considered, see for instance Mao et al. (2008), Boukas (2006). It
is a well known fact that even when all subsystems are stable, the whole
hybrid system can be unstable (Liberzon, 2003). Therefore, researchers are
interested in finding sufficient conditions of the stability of hybrid systems.
Such conditions based onLie algebra properties of systemmatriceswere found
in a particular case for linear deterministic systems in Zhai et al. (2006),
Liberzon (2009).

The Lie algebra approachwas also used in the study of control mechanical
systems such that planar bodies, satellites and underwater vehicles (Bullo et
al., 2000), in the analysis of the controllability of dynamic systemswithnonho-
lonomic constrains (Sussman andLiu, 1993; He andLu, 2006). Unfortunately,
a nonholonomic system can not be stabilized by a continuous time invariant
state feedback control (Brockett, 1983; Bloch andCrouch, 1992). Consistently,
there appears a question if the Lie algebra properties can be used in the study
of stability of hybrid linearmechanical systems. It will be shown in this paper
that the answer in some cases is positive and in others is negative.

The paper is organized as follows. Section 2 describes mathematical preli-
minaries. A class of themulti-dimmensional linear hybrid system that satisfies
Lie algebra conditions is introduced in Section 3, followed by sufficient con-
ditions of the stability for systems with a Markovian switching rule and with
any switching in Section 4 and 5, respectively. In Section 6, the Lee algebra
approach is used in the study of practical stability of stochastic linear hybrid
systems. The obtained results are illustrated by examples and simulations.

2. Mathematical preliminaries

Throughout this paper, we use the following notation. Let | · | and < · >
be the Euclidean norm and the inner product in Rn, respectively. By λ(A)
we denote the eigenvalue of the matrix A, λmin(A) and λmax(A) denotes the
smallest and the biggest eigenvalue of the matrix A, respectively. We mark
R+ = [0,∞). Let (Ω,F,{Ft}t­0,P) be a complete probability space with
a filtration {Ft}t­0 satisfying usual conditions. Let w(t), t ­ 0 be the m-
dimensionalWiener process defined on the probability space. Let r(t), t­ 0,
be a right-continuous Markov chain on the probability space taking values in
a finite state space S = {1,2, . . . ,N}with the generator Γ= [γij]N×N, i.e.

P{r(t+ δ)= j|r(t)= i}=

{
γijδ+o(δ) if i 6= j

1+γiiδ+o(δ) if i= j
(2.1)



Lie algebra approach in the study of the stability... 33

where δ > 0,γij ­ 0 is the transition rate from i to j if i 6= j,γii =−
∑
i6=jγij.

We assume that theMarkov chain is irreducible i.e. rank(Γ)=N−1, and has
a unique stationary distribution P = [p1,p2, . . . ,pN]

⊤ ∈ RN which can be
determined by solving






PΓ=0

subject to
N∑

i=1

pi =1 and pi > 0 for all i∈ S
(2.2)

We consider a linear hybrid system with multiplicative excitations descri-
bed by the vector Itô differential equation

dz(t)=A(σ(t))z(t) dt+
m∑

k=1

Bk(σ(t))z(t)dwk(t) z(0)= z0 (2.3)

t­ 0, σ(0)= σ0 ∈ S, z ∈ R
n, A,Bk : S → R

n×n, k=1, . . . ,m, z0 ∈ R
n is the

initial condition, the process σ(t) : R+ → S is a switching signal. We assume
that the solution z(t) is everywhere continuous.

Processes wk(t) are normalized standardWiener processes with

E[wk(t)] = 0
(2.4)

E[wk(t)wk(s)] = q
2min(t,s) i=1, . . . ,m

where q is a constant parameter.

Throughout the paper, we assume that the processes wk(t), the process
σ(t) and the initial condition are mutually independent. The processes wk(t)
and σ(t) are {Ft}t­0 adapted.

We use the following definitions of the stability (Mao, 1994):

Definition 1. The null solution to (2.3) is said to be almost surely exponen-
tially stable if there exists a constant α > 0 such that for each pair of
(x0, t0)∈ R

n× R+ there is a finite random variable C such that

|x(t,x0, t0)| ¬C exp{−α(t− t0)} a.s. for all t­ t0 (2.5)

or if

limsup
t→∞

1

t
log |x(t,x0, t0)| ¬−α a.s. (2.6)

The left hand side of (2.6) is called the almost sure Lyapunov exponent
of the solution to (2.3).



34 E. Seroka, L. Socha

Definition 2. Thenull solution of (2.3) is said to be p-thmean exponentially
stable if there exists a pair of positive scalars α, c such that ∀(x0, t0)∈
R
n× R+

E[|x(t,x0, t0)|
p]¬ cE|x0|

pexp{−α(t− t0)} t­ t0 (2.7)

or if

limsup
t→∞

1

t
log(E[|x(t,x0, t0)|

p])¬−α (2.8)

The left hand side of (2.8) is called the p-th mean Lyapunov exponent
of the solution to (2.3). In the case of p = 2, it is usually called the
mean-square exponential stability.

Let P be a non-singular n×n dimensional quadratic matrix such that
matrices

Ã(i) =PA(i)P−1 B̃k(i) =PBk(i)P
−1 k=1, . . . ,m

i=1, . . . ,N
(2.9)

are upper triangular.
Multiplying equation (2.3) by the matrix P and introducing I = P−1P,

we find

d(Pz(t))=PA(σ(t))P−1Pz(t)dt+
m∑

k=1

PBk(σ(t))P
−1
Pz(t)dwk(t) (2.10)

Introducing a new vector variable x = Pz, x ∈ Rn in equations (2.10),
we obtain a system with upper triangular matrices Ã(σ(t)) = PA(σ(t))P−1,
B̃k(σ(t)) =PBk(σ(t))P

−1

dx(t)= Ã(σ(t))x(t)dt+
m∑

k=1

B̃k(σ(t))x(t)dwk(t) (2.11)

on t ­ 0 with the initial condition x(0) = x0 = Pz0 ∈ R
n, σ(0) = σ0 ∈ S,

Ã,B̃k : S → R
n×n, the process σ(t) : R+ → S is the switching signal.

We quote a definition and some basic facts for a Lie algebra:

Definition 3. ALie algebra over afield F is a triple (V,+, [·, ·]) where (V,+)
is a vector space over F and where [·, ·] is a bilinear map from V ×V
into V such that

1. [v1,v2] =−[v2,v1] (antisymmetry)

2. [v1, [v2,v3]]+ [v2, [v3,v1]]+ [v3, [v1,v2]] = 0 (Jacobi identity)



Lie algebra approach in the study of the stability... 35

For example, the vector space of all n×n matrices over the field with
[A,B] = AB−BA is a Lie algebra. We mark by L(A1,A2, . . . ,An) the Lie
algebra generated by A1,A2, . . . ,An.

Definition 4. Let us assign for every Lie algebra L the following sequence

L
0 = L
... (2.12)

L
n+1 = [Ln,Ln] = {[A,B] |A,B∈ Ln} n­ 0

The Lie algebra L is abelian if L(1) = {0} and is solvable if Ln = {0}
for some n.

The following result plays a key rule in further considerations.

Lemma 1. (Samelson, 1969) Amatrix Lie algebra is solvable iff there exists
a nonsingular matrix P such that PMP−1 is upper triangular for all
matrices M in the Lie algebra.

Note that if the Lie algebra L(A,B) is solvable, then eigenvalues of a
linear combination of A and B are equal to the same linear combination of
the corresponding eigenvalues

λj(A+µB)= aj +µbj j=1, . . . ,n (2.13)

where aj and bj are eigenvalues of A and B, respectively; µ is a constant.
In the case of diagonal matrices A, Bk, the stability analysis of system

(2.3) is equivalent to the same properties for n first order systems.
This result can be applied if the matrices A, Bk can be diagonalized by

means of the same transformation matrix P. Furthermore, this result can
be extended to the cases when the matrices A, Bk are upper triangular or
when they can be transformed into such a form by the same transformation
matrix P. In this case, the following result is obtained for the nonhybrid
system (a special case of (2.3) with N =1)

Theorem 1. (Willems, 1976) The null solution is p-th mean exponentially
stable, if themapping Q→L(Q) in the space of symmetric matrices of
order n, defined by

L(Q)=
(
A+
p−2

4

m∑

k=1

B
2
k

)
Q+Q

(
A+
p−2

4

m∑

k=1

B
2
k

)⊤
+
p

2

m∑

k=1

BkQB
⊤
k

(2.14)



36 E. Seroka, L. Socha

has only eigenvalues with negative real parts. If all eigenvalues of matri-
ces Bk are real, then this is equivalent to the Hurwitz character of the
matrix

A+
p−1

2

m∑

k=1

B
2
k

(2.15)

The null solution is then almost surely exponentially stable if thematrix

A−
1

2

m∑

k=1

B
2
k

(2.16)

is Hurwitz.

3. Multi-dimensional linear hybrid systems

Let us consider a multi-dimensional linear hybrid system driven by Wiener
processes of the form

dz(t)=A(σ(t))z(t)dt+
m∑

k=1

Bk(σ(t))z(t)dwk(t) (3.1)

on t­ 0 with the initial condition z(0) = z0 ∈ R
n, σ(0) = σ0 ∈ S, z ∈ R

n,
A,B : S → Rn×n, k=1, . . . ,m, theWiener processes wk(t) satisfy conditions
(2.4)1, the process σ(t) is the switching signal. Let us assume that the Lie
algebra

L(A(i),Bk(i), k=1, . . . ,m, i=1, . . . ,N) (3.2)

is solvable.
The solvability of the Lie algebra L implies (Lemma 1) the existence of a

similarity transformation Pwhich brings matrices A(i), Bk(i), k=1, . . . ,m,
i=, . . . ,N into an upper triangular form, and hybrid system (3.1) can be for
every i∈ S transformed into

dx(t)=




a1(i)
· ∗

0 ·
an(i)



x(t)+

m∑

k=1




bk1(i)
· ∗

0 ·
bkn(i)



x(t)dwk(t)

(3.3)
where x = Pz. Elements above the main diagonal (∗) are not essential for
further analysis, and all elements below the main diagonal vanish. It can be



Lie algebra approach in the study of the stability... 37

proved (Willems and Aeyels, 1976) that the p-th mean exponential stability
and the almost sure exponential stability for system (3.3) are equivalent to
the same properties for n first order systems

dxj(t)= aj(i)xj(t)+
m∑

k=1

bkj(i)xj(t)dwk(t)
j=1,2, . . . ,n
i=1, . . . ,N

(3.4)

where aj(i),b1j(i), . . . ,bmj(i), j = 1, . . . ,n are eigenvalues of matrices A(i),
Bk(i), i=1, . . . ,N,k=1, . . . ,m, t­ 0,with the initial valuesxj(0)=xj0 ∈ R
determined from x0, xj ∈ R.
Consider in place of (3.1) n first order systems

dxj(t)= aj(σ(t))xj(t)+
m∑

k=1

bkj(σ(t))xj(t)dwk(t) j=1,2, . . . ,n (3.5)

where aj(i),b1j(i), . . . ,bmj(i), j = 1, . . . ,n are eigenvalues of matrices A(i),
Bk(i), i=1, . . . ,N,k=1, . . . ,m, t­ 0,with the initial valuesxj(0)=xj0 ∈ R,
xj ∈ R.

4. Hybrid systems with the Markovian switching rule

For the one-dimensional hybrid system of the form

dx(t)= a(r(t))x(t)+ b(r(t))x(t)dw(t) (4.1)

where t­ 0 with the initial value x(0) = x0 ∈ R, r(0) = r0 ∈ S, w(t) is the
standard Wiener process, x∈ R, a,b : S → R, the switching rule is given by
(2.1) and (2.2), we have the following lemma.

Lemma 2. (Mao et al., 2008) The sample Lyapunov exponent of (4.1) is

lim
t→∞

1

t
log(|x(t)|) =

N∑

i=1

pi

(
a(i)−

1

2
b2(i)
)

a.s. (4.2)

for x0 6=0. Hence (4.1) with the switching rule given by (2.1) and (2.2)
is almost surely exponentially stable if

N∑

i=1

pi

(
a(i)−

1

2
b2(i)
)
< 0 (4.3)



38 E. Seroka, L. Socha

Using Lemma 2, we find that the null solution to (3.1) for complex values
aj, bkj is almost surely exponentially stable if for j=1, . . . ,n

N∑

i=1

pi

(
Re(aj(i))−

1

2

m∑

k=1

[Re(bkj(i))]
2
)
< 0 (4.4)

Hence, we obtain the theorem for the vector linear hybrid system.

Theorem 2. The trivial solution of hybrid system (3.1) with switching rule
(2.1) and (2.2) is almost surely exponentially stable if condition (4.4) is
satisfied.

Remark 1. If all eigenvalues of matrices Bk(i) are real i = 1, . . . ,N,
k = 1, . . . ,m condition (4.4) is equivalent to the Hurwitz character of
the matrix

N∑

i=1

pi

(
A(i)−

1

2

m∑

k=1

B
2
k(i)
)

(4.5)

Sufficient conditions of the p-th mean exponential stability for one-
-dimensional hybrid system (4.1) are given by the following lemma.

Lemma 3. (Mao et al., 2008) The p-th mean Lyapunov exponent of (4.1) is

lim
t→∞

1

t
log(E[|x(t)|p]) =

N∑

i=1

pip
(
a(i)+

1

2
(p−1)b2(i)

)
a.s. (4.6)

for x0 6=0.Hence, (4.1) with switching rule (2.1) and (2.2) is p-thmean
exponentially stable if

N∑

i=1

pi

(
a(i)+

1

2
(p−1)b2(i)

)
< 0 (4.7)

Using Lemma 3, we find that the null solution to (3.1) is p-thmean exponen-
tially stable if for j=1, . . . ,n

N∑

i=1

pi

(
Re(aj(i))+

p−1

2

m∑

k=1

[Re(bkj(i))]
2
)
< 0 (4.8)



Lie algebra approach in the study of the stability... 39

and the null solution of (3.1) is p-thmean exponentially stable if themapping
Q→L(Q) in the space of symmetric matrices of order n defined as

L(Q)=

( N∑

i=1

pi

(
A(i)+

p−2

4

m∑

k=1

B
2
k(i)
))
Q+

(4.9)

+Q

( N∑

i=1

pi

(
A(i)+

p−2

4

m∑

k=1

B
2
k(i)
))⊤
++

N∑

i=1

pi

(p
2

m∑

k=1

Bk(i)QB
⊤
k
(i)
)

has only eigenvalues with negative real parts.
From these considerations follows the next theorem.

Theorem 3. The trivial solution to hybrid system (3.1) with the switching
rule (2.1) and (2.2) is p-th mean exponentially stable if the mapping
Q → L(Q) in the space of symmetric matrices of order n defined by
(4.9) has only eigenvalues with negative real parts.

Remark 2. If all eigenvalues of matrices Bk(i) are real i = 1, . . . ,N,
k = 1, . . . ,m condition (4.4) is equivalent to the Hurwitz character of
the matrix

N∑

i=1

pi

(
A(i)+

p−1

2

m∑

k=1

B
2
k(i)
)

(4.10)

Example 1. Let us consider a particular case of hybrid system (3.1) with
the switching rule given by (2.1) and (2.2) and two structures: almost
surely exponentially and mean-square exponentially stable (i= 1) and
unstable (i=2), m=1, where

A(1)=

[
−1.5 0.5
1.0 −1.0

]
B1(1)=

[
−0.01 0.05
0.05 −0.01

]
(4.11)

and

A(2)=

[
2 1
2 3

]
B1(2)=

[
0.05 0.01
0.01 0.05

]
(4.12)

Matrices A(1) + 1
2
B
2
1(1) and A(1) −

1
2
B
2
1(1) are Hurwitz while

A(2)+ 1
2
B
2
1(2) and A(2)−

1
2
B
2
1(2) are unstable.

We find that the Lie algebra L(A(1),A(2),B1(1),B1(2)) is solvable with
the matrix P given by

P=

[
1 0
1 1

]
(4.13)



40 E. Seroka, L. Socha

Eigenvalues of A(i) + 1
2
B
2
1(i) are {−1.9982,−0.4992} for i = 1 and

{1.0008,4.0018} for i=2, respectively, and fromcriterion (4.10), hybrid
system (3.1) with matrices given by (4.11) and (4.12) is mean-square
exponentially stable if the probabilities pi, i=1,2 satisfy the following
inequalities






−1.9982p1+1.0008p2 < 0
−0.4992p1+4.0018p2 < 0
p2 =1−p1

⇒

{
p1 > 0.89
p2 =1−p1

(4.14)

Eigenvalues of A(i)− 1
2
B
2
1(i) are {−2.0018,−0.5008} for i = 1 and

{0.9992,3.9982} for i=2, respectively, and from criterion (4.5), hybrid
system (3.1) with matrices given by (4.11) and (4.12) is almost surely
exponentially stable if the probabilities pi, i=1,2 satisfy the following
inequalities






−2.0018p1+0.9992p2 < 0
−0.5008p1+3.9982p2 < 0
p2 =1−p1

⇒

{
p1 > 0.89
p2 =1−p1

(4.15)

Exemplary simulations are shown in Fig.1.

Fig. 1. Exemplary samples for system (3.1) with matrices (4.11) and (4.12). (a) A
stable sample of system (3.1) for parameters: p1 =0.9, p2 =0.1; (b) an unstable

sample of (3.1) for parameters: p1 =0.87, p2 =0.13



Lie algebra approach in the study of the stability... 41

5. Hybrid systems with any switching

Let us consider a particular case of linear hybrid system (3.1) described by

dx(t)=A(σ(t))x(t)dt+B(σ(t))x(t)dw(t) i=1, . . . ,N (5.1)

on t­ 0 with the initial condition x(0) =x0 ∈ R
n, x∈ Rn, σ(0) = σ0 ∈ S,

A,B : S → Rn×n and with any switching rule σ(t), w(t) is a scalar standard
Wiener process.

Definition 5. If there exists a common definite matrix H satisfying

A(i)⊤H+HA(i)+B(i)⊤HB(i)< 0 i=1, . . . ,N (5.2)

then V (x)=x⊤Hx is called the common quadratic Lyapunov function
for all subsystems.

Theorem 4. If there exists a common quadratic Lyapunov function for all
subsystems, then hybrid system (5.1) is exponentially mean-square sta-
ble for any switching σ(t).

Proof: For V (x)=x⊤Hx, we have

λmin(H)|x|
2 ¬V (x)¬λmax(H)|x|

2 (5.3)

To show the exponential stability of hybrid system (5.1), we first find positive
scalars κi such that

A(i)⊤H+HA(i)+B(i)⊤HB(i)<−κiH (5.4)

hold for all i. Then, from (5.4), we find that

LiV (x)¬−κiV (x) (5.5)

where

LiV (x)=
∂V

∂t
+(A(i)x(t))⊤

∂V

∂x
+
1

2

〈
B(i)x(t),

∂V

∂x

〉2
(5.6)

Hence, we obtain
d

dt
E[V (x)]¬−κE[V (x)] (5.7)

where κ=mini{κi} and

E[V (x)]¬V (x0)exp{−κ(t− t0)} (5.8)



42 E. Seroka, L. Socha

From (5.3), we have

E
[
|x(t)|2

]
¬E
[
|x(0)|2

]λmax(H)
λmin(H)

exp{−κ(t− t0)} (5.9)

Theorem 4 with use of the common quadratic Lyapunov function establishes
sufficient conditions for the mean-square exponentially stability for linear hy-
brid systems with parametric excitations for any switching.

Another sufficient condition of themean-square exponentially stability for
hybrid systems can be proposed for linear systems with a special structure of
matrices defined by the Lie algebra.

Theorem 5. If the Lie algebra L(A(i),B(i), i = 1, . . . ,N) is solvable and
furthermore

2Re(λj(A(i)))+ |λj(B(i))|
2 < 0

j=1, . . . ,n
i=1, . . . ,N

(5.10)

then hybrid system (5.1) is mean-square exponentially stable for any
switching σ(t).

Proof: According to Theorem 4, the proof is reduced to finding a common
definite matrix H for all subsystems of (5.1). If the Lie algebra L(A(i),B(i),
i=1, . . . ,N) is solvable,wecanfindanonsingular complexmatrix P such that
for every i∈ S thematrices A(i) = [ajl(i)]j,l=1,...,n and B(i) = [bjl(i)]j,l=1,...,n,
i=1,2, . . . ,N have the following form

Ã(i)=PA(i)P−1 B̃(i) =PB(i)P−1 (5.11)

where the complex matrices Ã(i) = [ãjl(i)]j,l=1,...,n, B̃(i) = [̃bjl(i)]j,l=1,...,n,
i = 1, . . . ,N are upper triangular. First, we show that there exists a real
positive definite matrix H̃ such that ∀i=1, . . . ,N

Ã(i)∗H̃+ H̃Ã(i)+ B̃(i)∗H̃B̃(i)< 0 H̃= diag{h̃j}j=1,...,n (5.12)

Thematrices Ã(i)∗H̃+H̃Ã(i)+B̃(i)∗H̃B̃(i) areHermitian for every i∈ S, for
instance, for n=3 have the following form

Ã(i)∗H̃+ H̃Ã(i)+ B̃(i)∗H̃B̃(i) = [qijl] (5.13)



Lie algebra approach in the study of the stability... 43

where

qi11 = h̃1
(
2Re(ã11(i))+ |̃b11(i)|

2
)

qi12 = h̃1
(
ã12(i)+ b̃11(i)b̃12(i)

)

qi13 = h̃1
(
ã13(i)+ b̃11(i)b̃13(i)

)

(5.14)
qi22 = h̃2

(
2Re(ã22(i))+ |̃b22(i)|

2
)
+ h̃1(|̃b12(i)|

2)

qi23 = h̃2
(
ã23(i)+ b̃22(i)b̃23(i)

)
+ h̃1(b̃12(i)b̃13(i))

qi33 = h̃3
(
2Re(ã33(i))+ |̃b33(i)|

2
)
+ h̃1(|̃b13(i)|

2)+ h̃2(|̃b23(i)|
2)

From the assumption of Theorem 5, we have

(
2Re(ãjj(i))+ |̃bjj(i)|

2
)
< 0

j=1, . . . ,n
i=1, . . . ,N

(5.15)

Hence we can find sufficiently large positive scalars h̃j, j = 1,2,3, such that
matrix (5.13) is negative definite.

Using the obtained H̃, and substituting (5.11) into (5.12), we obtain

(P−1)∗A(i)⊤P∗H̃+ H̃PA(i)P−1+(P−1)∗B(i)⊤P∗H̃PB(i)P−1 < 0 (5.16)

Multiplying equation (5.16) fromthe left-hand sideby thematrix P∗ and from
the right-hand side by the matrix P, we have

A(i)⊤P∗H̃P+P∗H̃PA(i)+B(i)⊤P∗H̃PB(i)< 0 (5.17)

Hence, we obtain

A(i)⊤H+HA(i)+B(i)⊤HB(i)< 0 for H=P∗H̃P (5.18)

Wewrite the complexmatrix Has H=Re(H)+iIm(H). Since H isHermitian,
Im(H) is skew-symmetric and x⊤Hx=x⊤Re(H)x> 0, x 6= 0. Thus, Re(H)
is a real positive definite matrix and

A(i)⊤Re(H)+Re(H)A(i)+B(i)⊤Re(H)B(i)< 0 (5.19)

which implies that Re(H) is the common Lyapunov matrix we want to com-
pute.



44 E. Seroka, L. Socha

Remark 3. If all eigenvalues of matrices A(i) and B(i) are real, condition
(5.10) is equivalent to the Hurwitz character of matrices 2A(i)+B2(i),
i=1, . . . ,N.

Example 2. Let us consider hybrid system (5.1) with two exponentially
mean-square stable structures, where

A(1)=

[
−2 −1
0.5 −0.5

]
B(1)=

[
−0.01 0.05
0.05 −0.01

]
(5.20)

and

A(2)=

[
−3 2
1 −4

]
B(2)=

[
0.05 0.01
0.01 0.05

]
(5.21)

The Lie algebra L(A(1),A(2),B(1),B(2)) is solvable with thematrix P
given by

P=

[
1 0
1 1

]
(5.22)

Thematrices A(i)+ 1
2
B
2(i), i=1,2 are Hurwitz, and fromTheorem 5,

hybrid system (5.1) with matrices given by (5.20) and (5.21) is mean-
square exponentially stable for any switching (Fig.2). Theorem5 cannot
be applied to the system considered in Example 1 because we cannot
use the random switching rule in it, and to make this system mean-
square exponentially stable we have to apply limitations concerning the
switching rule of this system.

Fig. 2. An exemplary simulation for system (5.1) with matrices (5.20) and (5.21).
(a) A stable sample for system (5.1), (b) the mean-square of the solution to

system (5.1)



Lie algebra approach in the study of the stability... 45

Example 3. Let us consider a special case of hybrid system (5.1) with two
unstable structures, where

A(1)=

[
0.3 0.2
0.3 0.4

]
B(1)=

[
−0.1 0.5
0.5 −0.1

]
(5.23)

and

A(2)=

[
0.4 0.1
0.2 0.5

]
B(2)=

[
0.5 0.1
0.1 0.5

]
(5.24)

The Lie algebra L(A(1),A(2),B(1),B(2)) is solvable with thematrix P
given by

P=

[
1 0
1 1

]
(5.25)

If both structures are unstable andmatrices (5.23), (5.24) create the solvable
Lie algebra, hybrid system (5.1) is always unstable regardless of the switching
(Fig.3).

Fig. 3. An exemplary simulation for system (5.1) with matrices (5.23) and (5.24).
(a) An unstable sample for system (5.1), (b) the mean-square of the solution to

system (5.1)

6. Limitation of the applicability of the Lie algebra approach to

mechanical hybrid systems

In what follows, we show that for an oscillator hybrid system the Lie algebra
sufficient conditions of stability are not satisfied, and one can consider both
unstable a hybrid system consisting of stable subsystems and stable a hybrid
system consisting of unstable subsystems.



46 E. Seroka, L. Socha

Example 4. Let us consider a particular case of hybrid system (3.1)with two
exponentially mean-square stable structures with matrices

A(1)=

[
0 1
−0.25 −0.01

]
B(1)=

[
0 0

−0.125 0

]
(6.1)

and

A(2)=

[
0 1
−4 −0.01

]
B(2)=

[
0 0
−8 0

]
(6.2)

Condition (5.10) is satisfied, but a nonsingularmatrix which bringsma-
trices (6.1) and (6.2) into an upper triangular form does not exists, and
hencematrices (6.1) and (6.2) donot create a solvable Lie algebra.Thus,
assumptionsofTheorem5arenot satisfied, andwecanfindthe switching
rule that hybrid system (5.1) with this special switching is unstable. An
exemplary simulation is shown in Fig.4.

Fig. 4. An exemplary simulation for system (5.1) with matrices (6.1) and (6.2).
(a) An unstable sample for system (5.1), (b) the mean-square of the solution to

system (5.1)

Example 5. Let us consider a special case of hybrid system (3.1) with two
unstable structures with matrices

A(1)=

[
0 1
−0.25 −0.01

]
B(1)=

[
0 0
−0.05 0

]
(6.3)

and

A(2)=

[
0 1
−4 −0.01

]
B(2)=

[
0 0
−0.2 0

]
(6.4)



Lie algebra approach in the study of the stability... 47

Similar topreviousCounterexample4, anonsingularmatrixwhichbrings
matrices (6.3) and (6.4) into an upper triangular form does not exists,
and hence matrices (6.3) and (6.4) do not create a solvable Lie algebra.
Despite ofboth structures areunstable, hybrid system(5.1) canbe stable
with the special switching. An exemplary simulation is shown in Fig.5.

Fig. 5. An exemplary simulation for system (5.1) with matrices (6.3) and (6.4).
(a) A stable sample for system (5.1), (b) the mean-square of the solution to

system (5.1)

Now we consider the case of stochastic second order simple nonholonomic
hybrid system that satisfies Lie algebra conditions butmatrices A(i) have not
negative real parts of their eigenvalues.

Example 6. Let us consider a special case of hybrid system (3.1) with two
structures with matrices

A(1)=

[
0 1
0 −2

]
B(1)=

[
0 0
0 0.05

]
(6.5)

and

A(2)=

[
0 1
0 −0.2

]
B(2)=

[
0 0
0 0.5

]
(6.6)

An exemplary simulation is shown in Fig.6.

From the structures of matrices (6.5) and (6.6), it follows that the both
subsystems are not exponentially stable (one eigenvalue in both structurema-
trices A(i), i=1,2 is equal to zero). Therefore, the sufficient conditions deri-
ved in the previous section are not satisfied. However, one can show that the
hybrid system is practically mean–square stable in the sense of the following
definition (Yin et al., 2008).



48 E. Seroka, L. Socha

Fig. 6. An exemplary simulation for system (5.1) with matrices (6.5) and (6.6).
(a) A stable sample for system (5.1), (b) the mean-square of the solution to

system (5.1)

Definition 6. System (2.3) is said to be practically mean-square stable if
there exists a pair of positive scalars α, β such that α < β that
∀(x0, t0)∈ R

n× R+ the condition

E[|x0|
2]¬α (6.7)

implies

E[|x(t,x0, t0)|
2])¬β (6.8)

7. Conclusions

In this paper, a class of linear hybrid systems have been analyzed from the
point of view concerning their stability.

We have considered linear stochastic hybrid systems with a special case of
matrices creating the solvable Lie algebra. We have analyzed hybrid systems
parametrically excited by a white noise consisted of both stable and unstable
structures described by Itô stochastic differential equations. Two cases of the
switching rules have been studied:Markovian andany switching.Theobtained
sufficient conditions of the almost sure and the p-mean stability have been
illustratedbyexamples and simulations. It hasbeen shownthat theLie algebra
sufficient conditions of stability can not be found for oscillator hybrid systems,
andonly in somecases of secondorder systems sufficient conditions of practical
stability are satisfied.



Lie algebra approach in the study of the stability... 49

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Metody Lie algebr w badaniu stabilności stochastycznych układów

hybrydowych

Streszczenie

W pracy rozważony został problem stabilności klasy liniowych stochastycznych
układówhybrydowych ze szczególną strukturąmacierzy imultiplikatywnym szumem.
Znalezione zostały warunki wystarczające dla eksponencjalnej p-średniej stabilności
i eksponencjalnej stabilności prawie na pewno dla klasy stochastycznych liniowych
układówhybrydowych zMarkowskimprzełączaniem.Dodatkowo podane zostaływa-
runki wystarczające dla eksponencjalnej stabilności średnio-kwadratowej dla klasy
stochastycznych liniowych układów hybrydowych z dowolnym przełączaniem speł-
niających warunki Lie algebry. Otrzymane wyniki zilustrowane zostały przykładami
i symulacjami.

Manuscript received November 4, 2009; accepted for print April 23, 2010