Manuscript received August 22, 2011; accepted December 7,2011. Recommended by Editorial Board member Izumi Masubu-chi under the direction of Editor Hyungbo Shim. This work was supported by National Natural Science Founda-tion of China under grants NSFC 61074160, 61021002, and61104193. Kai Liu is with Harbin Institute of Technology, China and Pur-due University, USA (e-mail: [email protected]). Yu Yao and Baoqing Yang are with the Control and SimulationCenter, Harbin Institute of Technology, China (e-mails: [email protected], [email protected]). Venkataramanan Balakrishnan is with the School of Electricaland Computer Engineering, Purdue University, USA (e-mail: [email protected]). Yang Guo is with Harbin Institute of Technology, China (e-
Kai Liu, Yu Yao, Baoqing Yang, Venkataramanan Balakrishnan, and Yang Guo
204
-max
( ) :Aλ = The eigenvalue of matrix A with largest
absolute value.
2. PPLS AND THEIR PROPERTIES
2.1. Problem statement
We consider a class of contnues-time PPLS represted
by
1 1
2 2
( )
( )( ( )) :
( ) ,M M
A x if x t R
A x if x t Rx f x t
A x if x t R
∈ ∈
= = ∈
�
� (1)
where Ai∈R2×2 and Ri are convex and closed polyhedral
cones of the forms
{ }| 0i i
R x E x= ≥ (2)
such that, ,i j ijR R L=∩2
1
R .
M
i
i
R=
=∪ Here ijL dentoes
the common bounday of Ri and Rj, which is a zeroaxial
ray.
It is noted that for a given initial state, there are
possibly mutiple trjectories of PPLS (1) because of the
muti-value of f (x) on the boundary. To exclude these
cases, we restrict the PPLS treated in this paper is well-
posed, i.e. the PPLS has an unique solution without
sliding motions in the sense ofCaratheodory [14].
Definition 1: ( ; )x t z denotes the solution of PPLS (1)
with the initial state z. The PPLS is called
� stable in the sense of Laypunov if, for each 0,ε >
there exists 0,ε
δ > such that ( ; )z x t zε
δ< ⇒
ε≤ for all [0, ).t∈ ∞
� asymptotically stable if it is stable and all the
solutions ( ; )x t z converge to origin.
� exponentially stable if there exist , 0δ κ > and
(0,1)r∈ such that ( ; ) tz x t z r zδ κ< ⇒ ≤
for all [0, ).t∈ ∞
Our objectives are: (i) present a computable sufficient
and necessary criterion of exponential stability for conti-
nuous-time PPLS (1); (ii) compute the exponential
growth rate with conservatism, i.e., the exact r.
Definition 2: The nonzero vector v∈R2 is called the
eigenvector of the PPLS, if there exist some scalars
Rλ∈ such that
( ) .f v vλ= (3)
Furthermore, such v satisfying 1,v = is called unit-
eigenvector of PPLS.
By introducing the above conception, we present the
algorithm to refine the state-space partition in the
following subsection.
Example 1: Consider four-models PPLS with
Fig. 1. Phase portrait of Example 1.
1 3
0.65 0.15,
0.15 0.65A A
− = − = −
2 4
1 2;
2 0.5A A
− − = − =
1 3
1 0,
0 1E E
= − =
2 4
1 0.
0 1E E
− = − =
Fig. 1 illustrates the phase portrait of the system. It is
clear that system trajectories starting from the interior of
region R1 have different transition directions:
11 41 12 12 1 1( ) , ( ) , .Int R L Int R L L L→ → →
However, if separating the region R1 into subregions
R11 and R12, then all the trajectories starting from
11 12( )( ( ))Int R Int R have the same transition direction.
Note that, the boundary L1 is excatly the real eigenvector
of A1 within region R1, i.e. the eigenvector of PPLS.
Motivated by this analysis, we present the following
refinement algorithm to enable the unique transition
direction 1
( 2 / 2, 2 / 2)v = of traectories within each
region interior.
Algorithm 1: State-space refinement
1) Given the PPLS with the state-space partition
2
1
R .M
i
i
R
=
=∪
2) Compute all the unit-eigenvectors of PPLS, let
vk, 1[1, ]k M∈ denotes the obtained unit-
eigenvectors.
3) Let 1
, [1, ]k
L k M∈ denotes the ray containing
each vk, combine all Lk and all original boundaries
as the boundaries of new state-space partition.
4) Employing every two adjacent boundaries,
construct each new region Di.
5) Return The new state-space partition R2= .
iD∪
2.2. Properties of PPLS
This subsection establishes some essntial properties of
PPLS with refined state-space partition. To characterize
the transition direction of the system trajectories within
Exponential Stability Analysis of Planar Piecewise-Linear Systems: An Integral Function Approach
205
region Di, we introduce the following concept. Definition 3: The transition function of PPLS is
defined as below
T ( ) ( , ), ,i i ix A x Cx x D= ∈ (4)
where 0 1
.1 0
C−⎡ ⎤
= ⎢ ⎥⎣ ⎦
It is noted that Cx is the positive
normal vector of x. The geometric meaning of Ti (x) can be found in Fig. 2.
If Ti (x) > 0, i.e., the angle between the right-hand vector Aix and the positive normal vector of x is less than / 2.π Therefore, the system trajectory will transit in a counterclockwise direction. Fig. 2(a) illustrates this case, and Fig. 2(b) and Fig. 2(c) are corresponding to the cases when Ti (x) < 0 and Ti (x) = 0 respectively.
Now, we are ready to show the properties of PPLS, the key idea of all the proofs is employing Ti (x) to characterize the transition direction of system trajectories within Di. The first Lemma will show that all the trajectories starting from Int(Di) have an identical transition direction.
Lemma 1: For each region , [1, ],iD i m∈ there exists one of its boundaries Lij such that all the system trajectories starting from Int(Di) will arrive or approach to the boundary Lij infinitely.
Proof: From the well-posedness assumption and Remark 1 we learn that, all the nonzero ix D∈ satisfying Ti (x)=0, have been chosen as the boundary points of new state-partition. Therefore, by the continuity of Ti (x) within Di we have, Ti (x) is always no less(greater) than 0 within Di. In particular, T ( ) ( )i x > < 0, ( ),ix Int D∈ where T ( ) 0, ( )i ix x Int D> ∈ corresponds that the system trajectories will transit in a counterclockwise direction consistently. This implies that all the trajectories will arrive or approach to the boundary Li(i+1) infinitely. The similar analysis can be applied to the case Ti (x) < 0. Thus, the proof is completed.
Without loss of generality, we assume the proposed boundary Lij in Lemma 1 is Li(i+1), i.e., Ti (x) > 0. Then, we get the following results for the possible transition directions of trajectories starting from Li(i+1) and Li(i-1).
Lemma 2: Consider the region Di with Ti (x) 0,≥ for
the boundary Li(i-1), there are two possible transition cases: 1) All the trajectories starting from Li(i-1) enter into Int(Di); 2) All the trajectories starting from Li(i-1) slide along Li(i-1) all the time. For the boundary Li(i+1), there are also two possible transition cases: All the trajectories starting from Li(i+1) enter into Int(Di+1); 2) All the trajectories starting from Li(i+1) slide along Li(i+1) all the time.
Proof: We just need to note Ti (x) ≥ 0, x ∈ Li(i-1), where Ti (x) > 0 is corresponding to the case trajectories enter into Int(Di);Ti (x) = 0 is corresponding to another case that trajectories starting from Li(i-1) slide along Li(i-1) all the time. The analysis for x ∈ Li(i+1) is identical.
From Lemmas 1 and 2, all the regions Di with Ti (x) 0≥ can be classified into the following four kinds (See
Fig. 3) in the view of transition direction:
( 1) ( 1) 1
( 1) ( 1) ( 1)
( 1) ( 1) ( 1) 1
( 1) ( 1) ( 1) ( 1)
1) ( ) ( )
2) ( )
3) , ( ) ( )
4) , ( )
i i i i i i
i i i i i i i
i i i i i i i i
i i i i i i i i i
L Int D L Int D
L Int D L L
L L Int D L Int D
L L Int D L L
− + +
− + +
− − + +
− − + +
→ → →
→ → →
→ → →
→ → →
The similar classification can be obtained for the regions with T ( ) 0.i x ≤
Fig. 3. Classification of regions with T ( ) 0.i x ≥
(a) (b) (c)
Fig. 2. Geometric meaning of Ti (x).
Kai Liu, Yu Yao, Baoqing Yang, Venkataramanan Balakrishnan, and Yang Guo
206
3. INTEGRAL FUNCTION OF PPLS AND STABLITY ANALYSIS
In this section, we propose the integral function ap-
proach to the exponential stability analysis of PPLS.
3.1. Definition of integral function Definition 4: We define the integral function ( , )G z⋅
of the PPLS as:
20
( , ) ( ; ) .tG z x t z dtλ λ∞
= ∫ (5)
For each fixed 0,λ ≥ ( , )G zλ is a function of z only:
( ) : ( , ).G z G zλ λ= (6)
Furthermore, we define the supreme of λ yielding the finite integral function as the radius of convergence of integral function, i.e.,
{ }: sup 0 | ( ) .G zλλ λ∗ = ≥ < ∞ (7)
3.2. Properties of integral function
Base on the definition of integral function and Lemma 1,2, the following properties of Gλ(z) are established.
Proposition 1: Gλ(z) has the following properties: 1) (Homogeneity) Gλ(z) is homogeneous of degree
two in z, i.e., 2( ) ( ), 0.G z G zλ λα α α= >
2) (Monotonicity) 2( ) ( ( ; )), 0.TG z G x T z Tλ λλ α≥ > 3) (Quadratic-Bound) For each 0,λ ≥ ( )G zλ < ∞
for z ∈R2 implies that 2( )G z c zλ < for some c. 4) (Vertex-Bound) ( ) , [1, ]iG z i mλ < ∞ ∈ implies
that, ( )G zλ < ∞ for z ∈ R2, where 1{ }mi iz = are
the unit-vertexes of PPLS, i.e., 1( 1) S .i i iz L += ∩
Proof: 1) The homogeneity property follows directly from the observation that ( ; ) ( ; ).x t z x t zα α=
2) It is shown by Definition 4,
20
2 20
2 20 0
20
( ) ( ; )
( ; ) ( ; )
( ; ) ( ; ( ; ))
( ; ) ( ( ; ))
( ( ; )), 0.
t
T t tT
T t t T
T t T
T
G z x t z dt
x t z dt x t z dt
x t z dt x t x T z dt
x t z dt G x T z
G x T z T
λ
λ
λ
λ
λ λ
λ λ
λ λ
λ
∞
∞
∞ +
=
= +
= +
= +
≥ ≥
∫
∫ ∫
∫ ∫
∫
(8) 3) Consider the λ is such that Gλ(z) < ∞ . Motivated
by the classsfication of regions with Ti (x) 0≥ (see Fig. 3), we differentiate four cases:
Case 1: Consider the situation (1) in Fig. 3, all the trajectories starting from the Li (i-1) first enter Int(Di), then go across of Li (i+1) to Int(Di+1).
For each 1S ,iz D∈ ∩ let Tz denotes the arrival time from z to Li (i+1), Mz denotes the amplification factor(see Fig. 4(a)), we have
12( ; ) , S ,z z ix T z M z z D= ∈ ∩ (9)
where 12 ( 1) S .i iz L += ∩
Note that in this case, all the trajectories will arrival Li(i+1) in finite time. Therefore, there exist Ti and Mi as the upper bounds of Tz and Mz, which yileds the following results,
20
220
2 220
( ) ( ; )
( ; ) ( )
( ; ) (1 ) ( ).
z z
i i
t
T Ttz
T Tti
G z x t z dt
x t z dt G M z
x t z dt M G z
λ
λ
λ
λ
λ λ
λ λ
∞=
= +
≤ + +
∫
∫
∫
(10)
By the continuity of trajectories ( ; )x t z on z ∈Di, there exists Ni such that
( ; ) , [0, ].i ix t z N t T≤ ∈ (11)
Subsisting this upper bound, we have
(a) (b) (c)
Fig. 4. Geometric meaning of Ti (x).
Exponential Stability Analysis of Planar Piecewise-Linear Systems: An Integral Function Approach
207
2
2( ) (1 ) ( ).iT
i i iG z N T M G zλ λ
λ≤ + + ⋅ (12)
Case 2: Consider the situation (2) in Fig. 3, all the
trajectories starting from Li(i+1) sliding along itself all the
time, and the trajectories starting from ( 1)( )i i i
Int D L−
∪
will approach to the boundary Li(i+1) infinitely.
Let z1=1
( 1) Si iL
−
∩ and z2=1
( 1) Si iL
+∩ (see Fig. 4(b)).
Note that for each z∈Li(i+1), the trajectory ( ; )x t z
slides along Li(i+1). Thus, the system dynamic degenerates
as one-dimension case, and there exists constant 1
0λ∗>
such that
1
21 ( 1)(1; ) ( ) , .
i ix z z z Lλ
−∗
+= ∈ (13)
We claidfm that *
1,λ λ
∗≥ because
1
2
2 1 20
1 10
( ) ( ) ( ; )
( ) ( ) .
t
t t
G z x t z dt
dt
λλ
λ λ
∗
∞∗
∞∗ ∗ −
=
= = +∞
∫
∫ (14)
Thus, for all *,λ λ< we have
1.λ λ
∗< Then, by the
continuity of function (1; ) /x z z in \{0},i
D we
learn that, there exists �3 1 2z z z∈ such that
1
21 3 2
(1; ) ( ) , ,x z z z z ozλ−
∗≤ ∈∨ (15)
where 1 1
( ) / 2,λ λ λ∗
= +�1 2z z denotes the circular arc
between z1 and z2; 3 2z z oz∈∨ denotes an 2-dimension
cone with unit-vertexes z2 and z3.
This implies that for all �3 2
,z z z∈
2
0
2
10
1
( ) ( ; )
1( ) .
ln( / )
t
t t
G z x t z dt
z dt
λλ
λ λλ λ
∞
∞−
=
≤ =
∫
∫ (16)
On the other hand, for each �1 3
,z z z∈ let Tz denotes
the arrival time for the system trajectory from z1 to the
ray oz, and the Mz as the amplification factor such that
�1 1 3
( ; ) , .z z
x T z M z z z z= ∈ (17)
It is noted that the system trajectory from z1 will
transit to oz in finite time and enter 2,zoz∨ which
implies that there exists Mi as the lower bound of Mz, and
Ti as the upper bound of Tz. Incorporating with the
monotonicity of Gλ(z), we have
2
1
1
1
( ) ( ) ( ( ; ))
( ) 1(1 ) ( ).
z i
i z z
T T
M G z G M z G x T z
G zG z
λ λ λ
λ
λ
λ λ
≤ =
≤ ≤ +
(18)
This implies that
�1
1 32
(1 ) ( )( ) , .
iT
i
G zG z z z z
M
λ
λ
λ−
+ ⋅≤ ∈ (19)
Thus we have
1
2
1
(1 ) ( )1( ) max , .
ln( / )
iT
i
G zG z
M
λ
λ
λ
λ λ
− + ⋅ ≤
(20)
Case 3: Consider the situation (3) in Fig. 3, all the
trajectories starting from Li(i-1) sliding along itself all the
time, and the trajectories starting from Int(Di) transit to
Li(i+1) in finite time then enter Int(Di+1).
Let z1=1
( 1) Si iL
−
∩ and z2=1
( 1) Si iL
+∩ (see Fig. 4(c)).
Note that for each z∈ Li(i-1), the trajectory ( ; )x t z
slides along Li(i-1). By the same deduction as case 2 we
have, for any given 1,λ λ λ
∗ ∗< ≤ there exists
3z ∈
�1 2z z such that
1
21 1 3
(1; ) ( ) , ,x z z z z ozλ−
≤ ∈∨ (21)
where 1 1
( ) / 2.λ λ λ∗
= +
This implies that for all �1 3 1
\{ },z z z z∈
2
0
2
10
1
( ) ( ; ) ( ( ; ))
( ) ( ( ; ))
1( ( ; )),
ln( / )
zz
z
z
TTt
z
Tt t
z
T
z
G z x t z dt G x T z
z dt G x T z
G x T z
λ λ
λ
λ
λ λ
λ λ λ
λλ λ
∞−
= +
≤ +
= +
∫
∫ (22)
where Tz denotes the arrival time of trajectory ( ; )x t z
from z to ray oz3.
By applying inequality (21) recursively, we have
2 21 1
( ; ) ( ) ( ) .z z
T T
zx T z zλ λ
− −
≤ = (23)
Subsisting this inequality into (22), we obtain that
� { }
3
1 1
3 1 3 1
1
1( ) ( )
ln( / )
1( ), \ .
ln( / )
zT
G z G z
G z z z z z
λ λ
λ
λ
λ λ λ
λ λ
≤ +
≤ + ∈
(24)
On the other hand, for all �3 2
,z z z∈ applying the
same idea with case 1, we obtain the following result
2
2( ) (1 ) ( ).iT
i i iG z N T M G zλ λ
λ≤ + + ⋅ (25)
Incorporating (24) and (25), we have for all �1 2
z z z∈
1
2
2 1
1( )
ln( / )
(1 ) ( ) ( ).i
i i
T
i
G z N T
M G z G z
λ
λ λ
λ λ
λ
≤ +
+ + ⋅ +
(26)
Case 4: This case is the combination of case 2 and 3.
Kai Liu, Yu Yao, Baoqing Yang, Venkataramanan Balakrishnan, and Yang Guo
208
Therefore, if denoting the upper bounds of Gλ(z) for case
1, 2 and 3 by M1, M2 and M3, respectively. Then we have
for situation (4) in Fig. 3,
{ }2 3( ) max M ,M ,G z
λ≤ (27)
where the expressions of M1, M2 and M3 can be found in
(12), (20) and (26), respectively.
In summary,
{ } 1
1 2 3( ) max M ,M ,M , S .G z z
λ≤ ∈ (28)
By the homogeneity of Gλ(z), we conclude that
2 2( ) , R ,G z C z zλ
≤ ∈ (29)
where { }1 2 3max M ,M ,M ,C = called the quadratic
bound of Gλ(z).
4. It is noted that, for given PPLS (1), ,i
N ,i
M ,iT
1λ∗
are all constants. Thus by inequalities (12), (20) and (26),
the integral function Gλ(z) on unit-vertexes 1
{ }mi iz
= fully
determine its quadratic bound in z∈S1, which proves
the property 4.
3.3. Exponential stability criterion of PPLS
In this subsection, a sufficient and necessary condition
of exponential stability is presented for PPLS (1) via the
proposed integral function.
Theorem 1: Consider the PPLS (1) with the final
state-space partition R2= ,i
D∪ the following statements
are equivalent.
1) The PPLS (1) is exponentially stable.
2) The integral function on unit-vertexes is finite, i.e.,
1( ) , [1, ].
iG z i m< ∞ ∈
3) The integral function G1(z) is finite for all z∈S1.
Proof: We begin by (1)⇒ (2). Suppose the PPLS (1)
is exponentially stable, i.e., there exists constants 1κ ≥
and (0,1)r∈ such that
( ; ) , [0, ).tx t z r z tκ≤ ∈ ∞ (30)
By this condition we have, for all 2Rz∈
2
10
2
2 2
0
( ) ( ; )
.2 ln( )
t
G z x t z dt
r dtr
κ
κ
∞
∞
=
−≤ = < ∞
∫
∫ (31)
Note that (2)⇒ (3) is a special case of property 4 in
Proposition 1 when 1.λ ≥ Thus, the rest work is to
show (3)⇒ (1). By the property 3 in Proposition 1, we
learn that, there exists constant 0C > such that
1
1 1( ) ( ) , S .G z G z C z< ∞⇒ ≤ ∈ (32)
This implies that, for any given 0,ε > there exists
T0 0> such that for all 1Sz∈
1 2
0( ; ) , .
T
Tx t z dt T Tε
+
< >∫ (33)
Applying First Mean Value Theorem [25] to (33), we
have, there exists [0,1]t∗∈ such that
1 2( ; ) ( ; ) .
T
Tx t z dt x T t z ε
+
∗= + <∫ (34)
It is noted that, for all [ , 1],t T T∈ +
max1
( ; ) e ( ; ) ,x t z x T t zα ⋅
∗≤ + (35)
where { }max maxmax ( ) .
i iAα λ=
Subsisting (34) into (35), we have that for all 1S ,z∈
max max
0( ; ) e ( ; ) e , .x t z x T t z t T
α α
ε∗
≤ + < > (36)
This is to say, the PPLS is asymptotically stable. By
the equivalence of asymptotic stability and exponential
stability for PPLS [Theorem 6.3 [15]], we obtain the
conclusion.
3.4. Characterization of exponential growth rate
In this subsection, we will show that the convergence
radius of the integral function exactly characterize the
exponential growth rate of PPLS (1). The following
theorem states the relationship of the convergence radius
λ* and the exponential growth rate r*.
Theorem 2: Given the PPLS (1) with a convergence
radius λ*, then for any 1/ 2( ) ,r λ∗ −
> there exists a
constant κr such that ( ; ) ,t
rx t z r zκ≤ t ≥ 0. Further-
more, 1/ 2( )λ∗ − is also the smallest value for the previous
statement to hold. That is to say, the exponential growth
rate 1/ 2( ) .r λ∗ ∗ −=
The proof is similar with (1)⇒ (3) and (3)⇒ (1) in
the proof of Theorem, thus omit.
3.5. Computation of integral function
All the analysis methods proposed in the previous
subsections require to compute the integral functions of
PPLS on the unit-vertexes 1
{ } .m
i iz
= Therefore, we
develop an algorithm for computing the truncation of
integral functions as the approximations of Gλ(z).
Definition 5: For each 0,T > define
2
0( ) ( ; ) .
TT t
G z x t z dtλ
λ= ∫ (37)
The following proposition shows the relationship
between ( )TG zλ
and Gλ(z).
Proposition 2: Consider the PPLS (1), the following
two statements hold.
1) For 0,λ > the integral function Gλ(z) is infinite
for some 2R ,z∈ then ( )T
G zλ
will converge to
infinite with the increase of T.
2) For 0,λ > the integral function Gλ(z) is finite for
all 2R ,z∈ then lim ( ) ( ).T
TG z G zλ λ
=
Exponential Stability Analysis of Planar Piecewise-Linear Systems: An Integral Function Approach
209
Proof: 1) The apagoge is employed. Assume that there
exist some 2R ,z∈ ( )G z
λis infinite, but ( )T
G zλ
is
bounded, i.e., there exists 0M > such that ( )TG zλ
≤
M, 0.T∀ ≥ This implies that,
1 2
0( ; ) , .
T
Tx t z dt M T T
+
< ∀ >∫ (38)
By the similar deduction with the proof for (3) (1)⇒
in Theorem 1, we obtain that, there exists constant
0C > such that
1
2( ; ) , 0.t
x t z C tλ−
≤ ≥ (39)
From Theorem 2, the above condition implies that 1/ 2
1/ 2( ) ,λ λ−
∗ −> which further yiields .λ λ
∗<
However, this is contradicted with the assumption that
Gλ(z) is finite.
2) For ,λ λ∗
< let 1
( ) / 2,λ λ λ∗
= + then we have,
1
2( ) , R .G z zλ
< ∞ ∈ It can be learned from Theorem 2,
there exists constant 0κ > such that
1
22
1( ; ) , 0.
t
x t z z tκλ
−
≤ ≥ (39)
With the help of this inequality, we have
2
2
1
2
1 1
( ) ( ) ( ; )
.ln( / )
T t
T
t t
T
T
G z G z x t z dt
z dt
z
λ λλ
κ λ λ
κ λ
λ λ λ
∞
∞−
− =
≤
=
∫
∫ (40)
Note that 1
0 ,λ λ< < let T converges to infinity, we
have, for each z∈R2,
lim ( ) ( ).T
TG z G zλ λ→∞
= (41)
From the above proposition, approximation of the
integral function Gλ(z) are provided by ( )TG zλ
for T
large enough. Furthermore, given 0λ > and 0,T > to
compute the ( ),T
iG zλ
simulations of PPLS (1) from
0t = to t T= is first carried out with (0)i
x z= by
the existing software PWLTOOL [16]. The ( )T
iG zλ
can
be estimated by the Simpson Complex Formula.
The above idea is summarized as Algorithm 2.
Algorithm 2: Computation of ( )T
iG zλ
1) Let 1
{ }mi iz
= be the unit-vertexes of PPLS (1);
2) Initialize / 21, 0
TT g
λ= = and 0.01ε = ;
3) Repeat 2T T← ;
4) for each , [1, ]iz i m∈ , do
5) { }2 1
1( ) pwlsim( , ,[0. ]);
N
j ij
T x t PPLS z T+
=
=
6) 2 1 2 2 1
1
( ) ( ) 4 ( ) ( )6
NT
i j j j
j
TG z x t x t x t
Nλ − +
=
= + + ∑
7) end for
8) Set { }max ( )T Ti ig G z
λ λ= ;
9) Until / 2T T
g gλ λ
ε− < ;
10) Return { }1
( )m
T
ii
G zλ
=
and Tgλ
.
By increasing simulation time T and decreasing the
step-length (T/N), we can obtain the underestimates of
{ }1
( )m
T
ii
G zλ
=
and Tgλ
with any precision are permitted
by the numerical computation errors. By checking gλ for
a increasing sequence of λ, the λ* can also be obtained.
4. NUMERICAL EXAMPLES
In this section, we will demonstrate the proposed
approach through two numerical examples.
Example 2 [3]: Consider the PPLS (1) with
1 3
0.1 5,
1 0.1A A
− = = − −
2 4
0.1 1;
5 0.1A A
− = = −
1 3
1 0,
0 1E E
= − =
2 4
1 0.
0 1E E
− = − =
By computation, there is no real eigenvectors for any
system matrix Ai, which implies the PPLS does not have
eigenvectors from Definition 2. Thus, the further
refinement is not necessary. That is to say, and the unit-
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Kai Liu received his B.S. degree in Mathematic from Jilin University, China in 2007. He is currently a joint Ph.D. student of Automatic Control in Harbin Institute of Technology, China and Pur-due University, USA. His research inter-ests include hybrid systems, piecewise-linear systems and robust control.
Yu Yao received his B.S., M.S. and Ph.D. degrees in Automatic Control, in 1983, 1986 and 1990, respectively, all from Harbin Institute of Technology, China. He is currently a professor in Control and Simulation Center, in Harbin Institute of Technology, China. His research interests include robust control, nonlinear systems and flight control.
Baoqing Yang received his B.S., M.S. and Ph.D. degrees in Automatic Control in 2003, 2005 and 2009, respectively, from Harbin Institute of Technology, China. He is currently a professor in Control and Simulation Center, in Harbin Institute of Technology, China. His research interests include predictive control and flight control.
Venkataramanan Balakrishnan re-ceived his B.S. degree in Electrical Engineering from Indian Institute of Technology, India in 1985, and his Ph.D. degree in Electrical Engineering from Stanford University, USA in 1992. He is currently a professor and head of School of Electrical and Computer Engineering, Purdue University, USA. His research
interests include robust control, convex optimization and robotics.
Yang Guo received his B.S. and M.S. degrees in Xi’an Research Institute of High-Tech, China, in 2005 and 2008, respectively. Now he is a Ph.D. candidate in Harbin Institute of Technology. His research interests include Finite time Stabilization, Control.
