ε-Dependent controllability for two time-scale systems

TSINGHUA SCIENCE AND TECHNOLOGY ISSNll1007-0214ll18/18llpp271-280 Volume 14, Number 2, April 2009

-Dependent Controllability for Two Time-Scale Systems*

HU Yenan ( )1, SUN Fuchun ( )1,**, LIU Huaping ( )1, WU Hao ( )2

1. State Key Laboratory of Intelligent Technology and Systems, National Laboratory for Information Science and Technology, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China;

2. Department of Mathematics and Computer Science, Free University of Berlin, Arnimallee 6, Berlin 14195, Germany

Abstract: This paper investigates the controllability of two time-scale systems using both the time-scale

separation model and the slow-fast order reduction model. This work considers the effect of a singular per-

turbation parameter on the model transformations to improve the criterion precision. The Maclaurin expan-

sion method and homotopy arithmetic are introduced to obtain -dependent controllability criteria. Examples

indicate that the -dependent controllability criteria are more accurate and that the controllability of two

time-scale systems does not change during model transformations with these more accurate forms.

Key words: controllability; singular perturbation method; time-scale separation; slow-fast order reduction;

Maclaurin expansion; homotopy arithmetic

Introduction

Multi-time-scale systems, whose behavior must be described in different time-scales, exist widely in real systems, such as flexible space manipulators, catalytic towers, and complex circuits with inductances and ca-pacitances. “A state-space model of a multi-time-scale system is said to be state separable if the fast parts of some of its states are small compared with their slow parts and with the fast parts of the other states.”[1] When differential equations are introduced to describe multi-time-scale systems involving widely different speeds and interacting dynamic phenomena, the mod-els encounter dimensionality problems and are very stiff numerically. The most representative example of a multi-time-scale system is the two time-scale system.

Over the past forty years, a number of papers have been devoted to the study of analysis and control problems of two time-scale systems. Singular pertur-bation methods, physically motivated tools for order reduction and for time-scale separation, are used for many simplifications for control system analysis and design[2]. In singular perturbation methods, the fast phenomena are first neglected to reduce the system order and then added as a form of boundary layer cor-rection for the time-scale separation. Finally, an as-ymptotic expansion method is introduced to improve the approximation precision. The order reduction based on singular perturbation methods for two time-scale systems includes a time-scale separation step with re-gard to the singular perturbation parameter and then a slow-fast order reduction step. In traditional singular perturbation methods, the singular perturbation parameter is neglected to some degree during the time-scale separation. This usually reduces the precision of the slow-fast order reduction which may influence system analysis problems, such as sys-tem controllability.

“Controllability is the ability to provide a transfer of a control system from a given initial state to a given

Received: 2008-03-31; revised: 2008-11-04

* Supported by the National Science Fund for Distinguished YoungScholars (No. 60625304), the National Natural Science Foundationof China (No. 90716021), and the Specialized Research Fund forthe Doctoral Program of Higher Education of MOE, China (No. 20050003049)

** To whom correspondence should be addressed. E-mail: [email protected]; Tel: 86-10-62796858

Tsinghua Science and Technology, April 2009, 14(2): 271-280

272

terminal state by a proper choice of the control of function”[3]. Thus, the controllability indicates whether the system states can be controlled by the control in-puts or not. When the system is controllable, a state feedback controller can be designed to achieve the de-sired system performance. Previous two time-scale system analyses including controllability analyses have usually been based on order reduction results which neglect the singular perturbation parameter. For in-stance, there are various studies[4-8] on linear time-in-variant systems, linear time variant systems, nonlinear systems, constant systems, and discrete systems. These studies have used only -independent controllability criteria with the controllability analyses of two time-scale systems in the time-scale separation form and the slow-fast order reduction form that perhaps are not very accurate. The effect of the singular perturba-tion parameter has been considered as iterative meth-ods[1,9], but these models do not include control vari-ables. Hence, neither the controllability analysis nor the controller design can be implemented based on these models.

The importance of the system analysis leads to the need for higher accuracy controllability criteria for two time-scale systems. This paper deduces -dependent controllability criteria based on the Maclaurin expan-sion with a controllability analysis of a two time-scale system.

1 Preliminaries on Model Transformation

A two time-scale linear system can be represented as a singularly perturbed system as

11 12 1 0 0

21 22 2 0 0

, ( ) ,, ( )

tt

x = A x + A z B u x xz = A x + A z B u z z

(1)

where ( 0 1) is the singular perturbation pa-rameter, nx R is the slow state variable vector,

mz R is the fast state variable vector, and ru R is the control variable vector.

For time-scale separation, assume that z Lx and x M where represents the fast vari-ables and represents the slow variables. Substitut-ing these into Eq. (1) gives the diagonal form of a two time-scale system:

00

22

BAu

BA0

0 (2)

where 0 11 12 ( )A A A L , 2 22 12( )A A L A , 0 1B =B

1 2( ) ( ) ( ) ,M L B M B and 2 2 1( ) .B = B L B

Here, “ ” stands for system matrices or control ma-

trices dependent on . ( )L and ( )M are inter-mediate variables in the transformation of the two time-scale system in Eq. (1) to diagonal form in Eq. (2).

( )L and ( )M satisfy the following equations:

22 21 11 12( ) ( ) ( ) ( )A L A L A L A L 0 (3)

22 12 11 12( ) ( ) ( ) ( )M A A A M A L M

12( ) ( )M L A 0 (4) Furthermore, singular perturbation methods[10] are

used for getting the slow-fast order reduction form for Eq. (1) on the basis of Eq. (2). The slow subsystem of Eq. (1) is

s 0 s 0 sx = A x + B u (5a) where su is the input variable vector for the slow subsystem. Introducing f su u u into the slow-fast reduction gives the fast subsystem of Eq. (1):

f 22 f 2 fx = A x + B u (5b) This paper analyzes the controllability of the two

time-scale system in Eq. (1) and its time-scale separa-tion form in Eq. (2) and the slow-fast order reduction form consisting of the slow subsystem in Eq. (5a) and the fast subsystem in Eq. (5b).

2 Three Controllability Criteria

Criterion 1 The necessary and sufficient condition for controllability for Eq. (1) in the singularly per-turbed form is

1rank[ ]n m n mB A B A B (6)

where 11 12

21 22/ /A A

AA A

and 1

2 /B

BB

.

Criterion 2[2] The necessary and sufficient condi-tion for controllability for Eq. (1) in the time-scale separation form in Eq. (2) is

2 -10 0 0 0 0 0 0

2 12 2 2 2 2 2 2

2 3

rank

n m

n m

n m

n mB A B A B A BB A B A B A B (7)

Criterion 3[10] For Eq. (1) in the slow-fast order

HU Yenan ( ) et al -Dependent Controllability for Two Time-Scale Systems

273

reduction form in Eq. (5), there exists a corresponding controllability theorem if

10 0 0 0 0rank[ ]n nB A B A B (8a)

12 22 2 22 2rank[ ]m mB A B A B (8b)

Then, there is some * 0 which makes Eq. (1) con-trollable for *(0, ] .

Remark 1 According to control system theory, the controllability of Eq. (1), and its time-scale separation form in Eq. (2) and the slow subsystem in Eq. (5a) of the slow-fast order reduction form should not change after a model transformation[11]. Therefore, the con-trollability results obtained from the controllability criteria in Eqs. (6), (7), and (8a) must be the same for the same original system in Eq. (1).

Remark 2 ( )L and ( )M , the solutions of the matrix equations in Eqs. (3) and (4), are contained in

0A , 0B , 2A , and 2B . Therefore, the controllability criteria in Eqs. (7) and (8a) both correlate with ( )L and ( )M .

Since -dependent analytical solutions for matrix equations are very difficult, traditional methods di-rectly neglect the effect of in Eqs. (3) and (4) by letting 0 . Assuming that 22A is an invertible ma-trix, then the solutions of Eqs. (3) and (4) are

122 21L A A (9)

112 22M A A (10)

Comparison with Eq. (2) gives 1

0 0 11 12 22 21A A = A A A A ,

2 2 22ˆA A = A , 1

0 0 1 12 22 2ˆB B =B A A B , and 2 2 2

ˆB B =B . Here, “ ” stands for the system matrices or control matrices obtained by neglecting . Substituting these equations into Eq. (7) gives the approximate form of Criterion 2 as

2 10 0 0 0 0 0 0

2 12 2 2 2 2 2 2

2 3

ˆ ˆ ˆˆ ˆ ˆ ˆrank ˆ ˆ ˆˆ ˆ ˆ ˆ

n m

n m

n m

n mB A B A B A B

B A B A B A B (11)

For Criterion 3, the fast subsystem part in Eq. (8b) is independent of , so the simplification of ( )L and

( )M does not influence the result in Eq. (8b). How-ever, for the slow subsystem part in Eq. (8a), as with Criterion 2, the approximate form of criterion 3 in Eq. (8a) is

10 0 0 0 0

ˆ ˆˆ ˆ ˆrank[ ]n nB A B A B (12) Remark 3 Approximate controllability criterion is

widely used[4,7]. The approximate controllability

criterion in Eq. (11) is obtained by neglecting the effect of on the model transformation. Although this method gives useful results, it may result in a wrong controlla-bility estimate. For instance, if a controllable system has the form in Eq. (1), after the model transformation, the controllability matrix in Eq. (11) may be not a full rank matrix which means Eq. (2) is not controllable. This contradicts the conclusion that the controllability of Eqs. (1) and (2) should be the same[11]. The same result may occur for the slow subsystem in Eq. (5a). Thus, a new approach is needed to get higher accuracy controllability criteria for two time-scale systems.

3 -Dependent Controllability Criterion

The key problem is that the effects of the singular per-turbation parameter have been ignored in the control-lability Criteria 2 and 3 in the Eqs. (11) and (12). Thus, a new controllability criterion is developed to get more accurate controllability results. Furthermore, the con-trollability of the different forms of the two time-scale system can be guaranteed to be equivalent. The result depends on the solution of the matrix equations in Eqs. (3) and (4) to get their -dependent analytical solutions. However, the solution of matrix equations is usually very complex due to the non-unique eigenvectors.

From Eqs. (9) and (10), the -dependent solutions of Eqs. (3) and (4) can be expressed as

122 21 ( )LL A A O (13)

112 22 ( )MM A A O (14)

First order or high order expansions relative to of Eqs. (13) and (14) can be used to get -dependent solutions of Eqs. (3) and (4) and then the -dependent controllability criteria of the original system in Eq. (1) for the time-scale separation form in Eq. (2) and for the slow-fast reduction form in Eq. (5). The higher order expansions of ( )LO and ( )MO needed to solve the matrix Eqs. (3) and (4) can be obtained using the Maclaurin expansion method and homotopy arithmetic.

Expanding ( )L in the form of a Maclaurin series and ignoring the second and higher order terms give

0 100

d ( )( ) ( )dLL L L L (15)

where 0 0( )L L and 1

0

d ( )dLL . Taking the


274

derivative of both sides of Eq. (3) gives

22 11 11 12d ( ) d ( )( ) ( ) ( )

d dL LA L A A L A L

12 12d ( ) d ( )( ) ( )

d dL LA L L A 0 (16)

Letting 0 gives 22 1 0 11 0 12 0A L L A L A L 0 (17)

Substituting 0 1( )L L L into Eq. (3) gives

22 0 1 21 0 1 11( ) ( )A L L A L L A

0 1 12 0 1( ) ( )L L A L L 0 (18) With

11 12 1

21 22 2

1 2

0 0 0

0 0 00

0 0 0

n

n

m m mn

l l l

l l l

l l l

L (19)

11 12 1

21 22 2

1 2

1 1 1

1 1 11

1 1 1

n

n

m m mn

l l l

l l l

l l l

L (20)

substituting Eqs. (19) and (20) into Eqs. (17) and (18) and noting that the matrix elements on both sides of the equation are equivalent, give a series of nonlinear equations with 2m n unknowns as

( )f x 0 (21) where

11 1 1 11 10 0 0 0 1 1[n m mn n

l l l l l lx

1

T1 1 ]

m mnl l ,

T1 2 2( ) [ ( ) ( ) ( )]m nf f ff x x x x .

Then, homotopy arithmetic[12] is used to solve the nonlinear equations. First, a homotopy equation is de-signed as

0( , ) ( ) ( 1) ( )H x f x f x (22) Kalaba and Tesfatsion[13] proved that Eq. (22) has a

unique solution ( )x x which is the solution of 1 0

0

( ) [ ( )] ( ) ,(0)

x f x f xx x

(23)

where 0x is an arbitrary initial value. Here, 0x is chosen as a zero vector. ( )f x is

11 11

11 11

11 11

1 1 1 1

0 0 1 1

2 2 2 2

0 0 1 1

2 2 2 2

0 0 1 1 (2 ) (2 )

( )

mn mn

mn mn

mn mn

mn mn mn mn

mn mn

f f f fl l l l

f f f fl l l l

f f f fl l l l

f x

Thus, the homotopy arithmetic has formed the nonlin-ear equations into a numerical integration problem. The solutions of Eq. (21) can be obtained by calculat-ing the numerical solutions of the initial value differ-ential equation problem. Here, the Runge-Kutta

method is used to calculate1

0( )d .x x Then,

0 1( )L L L is obtained based on 110 0, ,

mnl l and

111 1, ,mn

l l . Equation (4) is then solved by expanding ( )M in

the form of a Maclaurin series while neglecting the second and higher order terms

0 100

d ( )( ) ( )d

MM M M M (25)

where 0 0( )M M and 1

0

d ( )d

MM . Substi-

tuting 0 1( )L L L into Eq. (4) gives

22 12 11 12 0 1( ) ( ) ( ) ( )M A A A M A L L M

0 1 12( )( )M L L A 0 (26) Taking the derivatives of both sides gives

22 11 11 12 0

212 0 12 1 12 1

0 12 0 12 1 12

d ( ) d ( )( ) ( )d d

d ( ) d ( )2 ( )d d

d ( )( ) 2 ( )d

M MA A M A A L M

M MA L A L M A L

MM L A L A M L A

21 12

d ( )d

M L A 0 (27)

Letting 0 gives 1 22 11 0 12 0 0 0 0 12M A A M A L M M L A 0 (28)

In addition, substituting 0 1( )M M M and

0 1( )L L L into Eq. (4) gives

0 1 22 12 11 0 1

12 0 1 0 1

( ) ( )( )( )

M M A A A M MA L L M M

0 1 0 1 12( )( )M M L L A 0 (29) With

11 12 1

21 22 2

1 2

0 0 0

0 0 00

0 0 0

m

m

n n nm

m m m

m m m

m m m

M (30)

11 12 1

21 22 2

1 2

1 1 1

1 1 11

1 1 1

m

m

n n nm

m m m

m m m

m m m

M (31) (24)


275

substituting Eqs. (30) and (31) into Eqs. (28) and (29) gives a series of nonlinear equations with 2n m unknowns. Homotopy arithmetic is again introduced to solve these nonlinear equations using the same solution procedure as for calculating 0L and 1L . Finally, the solution of Eq. (4) is obtained in the form of

0 1( )M M M as

0 11 12 11 12 0 12 1( ) ( )A A A L A A L A L (32) 2

2 12 22 22 0 12 1 12( )A LA A A L A L A (33)

0 1 1 2

1 0 2 0 0 1 1 2

( )( ) ( )

B B MLB MBB M B M L B M B

2 30 1 1 1 0 1 1 1 1( )M L B M L B M L B (34)

22 1 2 2 0 1 1 1( )B LB B B L B L B (35)

Substituting Eqs. (32)-(35) into Eq. (7) gives the -dependent controllability criterion for the original

Eq. (1) in its time-scale separation form in Eq. (2): 1

0 0 0 0 01

2 2 2 2 22

( ) ( ) ( ) ( ) ( )rank ( ) ( ) ( ) ( ) ( )

n m

n m

n m

B A B A BB A B A B

1 2

1 2

rank n m

n m

n m (36)

where 111 12 0 12 1 1 0 2(( ) ) (( )i

i A A L A L B M B 2 3

0 0 1 1 2 0 1 1 1 0 1 1 1 1( ) ( ) )M L B M B M L B M L B M L B ,2 1 2

22 0 12 1 12 2 0 1 1 1( ) ( )i

i i

A L A L A B L B L B,

1, ,i n m . Substituting Eqs. (32)-(35) into Eq. (8) gives the

-dependent controllability criteria for the original system in Eq. (1) for its slow-fast reduction form in Eq. (5):

10 0 0 0 0rank[ ( ) ( ) ( ) ( ) ( )]nB A B A B

1 2rank[ ]n n (37a) 1

2 22 2 22 2rank[ ]m mB A B A B (37b) where 1

11 12 0 12 1 1 0 2(( ) ) (( )jj A A L A L B M B

2 30 0 1 1 2 0 1 1 1 0 1 1 1 1( ) ( ) ) ,M L B M B M L B M L B M L B

1, ,j n . Since only the first-order terms are retained in

Eqs. (15) and (25), these results are first-order -dependent controllability criteria. For the time-scale

separation form in Eq. (2), comparison of Eqs. (36) and (11) shows that first-order -dependent controlla-bility criterion in Eq. (36) is more accurate than the approximate controllability criterion in Eq. (11) which

neglects the effect of . Likewise, for the slow-fast re-duction forms in Eq. (5a), the first-order -dependent controllability criterion in Eq. (37a) is more accurate than the -independent criterion in Eq. (12).

Remark 4 With the Maclaurin expansion method, this result can be easily extended to obtain second-order or high-order -dependent controllability criteria for much higher accuracy results. The second-order

-dependent controllability criteria are given below. Expanding ( )L in the form of a Maclaurin series

and neglecting the third and higher order terms give 2

20

0 0

d ( ) d ( )( ) ( )d dL LL L

20 1 2L L L (38)

Then, taking the derivative of both sides of Eq. (16) gives

2 2

22 11 11 112 2

12 12 12

2

12 12 122

d ( ) d ( ) d ( ) d ( )d d d d

d ( ) d ( ) d ( )( ) ( ) ( )d d d

d ( ) d ( ) d ( ) d ( )( ) ( )d d d d

L L L LA A A A

L L LA L L A A L

L L L LA L A L A

2

12 12 2

d ( ) d ( ) d ( )( )d d dL L LA L A 0 (39)

Letting 0 gives 22 2 1 11 1 12 0 0 12 12 2 2A L L A L A L L A L 0 (40)

Next, substituting Eq. (38) into Eq. (16) and letting 0 give

22 1 0 11 0 12 0A L L A L A L 0 (41) Thirdly, substituting 2

0 1 2( )L L L L into Eq. (3) gives

2 222 0 1 2 21 0 1 2 11( ) ( )A L L L A L L L A

2 20 1 2 12 0 1 2( ) ( )L L L A L L L 0 (42)

With

11 12 1

21 22 2

1 2

2 2 2

2 2 22

2 2 2

n

n

m m mn

l l l

l l l

l l l

L (43)

substituting Eqs. (19), (20), and (43) into Eqs. (40)-(42) gives a series of nonlinear equations with 3m n un-knowns. Homotopy arithmetic can be used to solve these nonlinear equations to get 0 1( )L L L

22L .

Furthermore, expanding ( )M in the form of a


276

Maclaurin series and neglecting the third and higher order terms gives

22

200 0

d ( ) d ( )( ) ( )d d

M MM M

20 1 2M M M (44)

Then, substituting 20 1 2( )L L L L , where 0L ,

1L , and 2L have already been obtained, into Eq. (4) gives

222 12 11 12 0 1 2( ) ( ) ( ) ( )M A A A M A L L L M

20 1 2 12( )( )M L L L A 0 (45)

Taking the derivative of both sides of Eq. (45) gives

22 11 11 12 0

212 0 12 1 12 1

2 312 2 12 2 0 12

20 12 1 12 1 12

d ( ) d ( )( ) ( )d d

d ( ) d ( )2 ( )d d

d ( )3 ( ) ( )d

d ( ) d ( )2 ( )d d

M MA A M A A L M

M MA L A L M A L

MA L M A L M L A

M ML A M L A L A

2 32 12 2 12

d ( )3 ( )d

MM L A L A 0 (46)

Letting 0 gives 1 22 11 0 12 0 0 0 0 12M A A M A L M M L A 0 (47)

Next, taking the derivative of both sides of Eq. (46) gives

2 2

22 11 11 112 2

2

12 0 12 0 12 0 2

12 1 12 1 12 1

22 2

12 1 12 2 12 22

2 312 2 12 2

d ( ) d ( ) d ( ) d ( )d d d d

d ( ) d ( ) d ( )d d d

d ( ) d ( )2 ( ) 2 2d d

d ( ) d ( )6 ( ) 3d dd ( )3

d

M M M MA A A A

M M MA L A L A L

M MA L M A L A L

M MA L A L M A L

MA L A L2

0 122

2

0 12 0 12 1 122

22

1 12 1 12 1 122

2 22 12 2 12 2 12

d ( ) d ( )d d

d ( ) d ( ) 2 ( )d d

d ( ) d ( ) d ( )2 2d d d

d ( ) d ( )6 ( ) 3 3d d

M M L A

M ML A L A M L A

M M ML A L A L A

M MM L A L A L A

23

2 122

d ( )dM L A 0 (48)

Letting 0 gives 2 22 11 1 12 0 1 1 0 122 2 2M A A M A L M M L A

12 1 0 0 1 122 2A L M M L A 0 (49) Thirdly, substituting 2

0 1 2( )M M M M into Eq. (45) gives

2 20 1 2 22 12 11 0 1 2

2 212 0 1 2 0 1 2

( ) ( )

( )( )

M M M A A A M M M

A L L L M M M

2 20 1 2 0 1 2 12( )( )M M M L L L A 0 (50)

With

11 12 1

21 22 2

1 2

2 2 2

2 2 22

2 2 2

m

m

n n nm

m m m

m m m

m m m

M (51)

substituting Eqs. (30), (31), and (51) into Eqs. (47), (49), and (50) gives a series of nonlinear equations with 3n m unknowns. Homotopy arithmetic can be used to solve these nonlinear equations to get

20 1 2( )M M M M . Then,

20 11 12 11 12 0 12 1 12 2( ) ( )A A A L A A L A L A L

(52) 2 3

2 12 22 22 0 12 1 12 2 12( )A LA A A L A L A L A (53)

0 1 1 2

1 0 2 0 0 1 1 2

20 1 1 1 0 1 2 2

30 2 1 1 1 1 2 0 1

( )( ) ( )

( )

( )

B B MLB MBB M B M L B M B

M L B M L B M B

M L B M L B M L B

4 51 2 1 2 1 1 2 2 1( )M L B M L B M L B (54)

2 32 1 2 2 0 1 1 1 2 1( )B LB B B L B L B L B (55)

Substituting Eqs. (52)-(55) into Eqs. (7) and (8) gives the second-order -dependent controllability cri-teria for the original system in Eq. (1) in its time-scale separation form in Eq. (2) and its slow-fast reduction form in Eq. (5).

4 Illustrative Examples

Three examples are given to illustrate the various con-trollability criteria.

Example 1 The parameters of a second-order two time-scale controllable system are

4 3 7.5 ,0.1 2 5x x z u

z x z u (56)

Criterion 1 gives 4 3

10 20A ,

7.5,

50B


277

7.5 180rank rank 2.

50 1075B A B

The result shows that the original two time-scale system is controllable for the singularly perturbed form in Eq. (1). For the time-scale separation form in Eq. (2), with the -independent form of Criterion 2 in Eq. (11):

0ˆ 2.5A , 2

ˆ 2A , 0ˆ 0B , 2

ˆ 5,B

0 0 0

2 2 22

ˆˆ ˆ0 0

rank rank 1.ˆˆ ˆ 50 1000

B A B

B A B

Thus, Eq. (11) shows that Eq. (56) is uncontrollable with time-scale separation. This conflicts with the con-trollability invariability of the model transformation. Then, considering Criterion 2 for the first-order -dependent form in Eq. (36), 0L , 1L , 0M , and 1M

are first calculated using homotopy arithmetic as 0 0.5025,L 1 0.6262,L 0 1.4745,M 1M 0.7262. Then,

( ) 0.5651L , ( ) 1.5471M .

0 ( ) 2.3046A , 2 ( ) 2.1695A ,

0 ( ) 0.8913B , 2 ( ) 5.4238B . Substituting these results into the controllability cri-

terion in Eq. (36) gives 0 0 0

2 2 22

( ) ( ) ( ) 0.9 2.1rank rank 2.( ) ( ) ( ) 54.2 1176.7

B A BB A B

This result shows that Eq. (56) in the time-scale separation form is controllable. Thus, the -dependent controllability Criterion gives the proper conclusion when the -independent controllability criterion failed.

For Criterion 3, the -independent criterion in Eq. (12) gives

0ˆrank[ ] rank[0] 0 nB .

Thus, for the slow-fast order reduction, the slow subsystem in Eq. (5a) is uncontrollable by Eq. (12). Therefore, the results based on -independent control-lability Criterion 3 conflicts with the controllability invariability of the model transformation. The first-order -dependent criterion in Eq. (37a) gives

0rank[ ( )] rank[ 0.8913] 1 nB . Thus, the slow subsystem in Eq. (5a) is controllable.

For Criterion 3, the correct conclusion is obtained by the first-order -dependent form.

Example 2 The parameters of a fourth-order two

time-scale controllable system are 1 2 1

2 1 2 2

1 2 1 2

1 1 2

0.5 0.05 ,0.2 0.4 ,

0.05 2 0.1 0.1 ,0.05 0.2 0.4

uu

uu

x = x zx = x x z

z = x z zz = x z

(57)

For Criterion 1, 0 1 0.5 01 1 0 0.20 40 20 220 0 0 4

A ,

0.050.428

B ,

1rank[ ]0.05 0.6 5.15 129.550.4 1.15 4.45 23.1

rank 4.2 8 268 49588 31 112 551

n mB A B A B

The result shows that the two time-scale system in Eq. (57) is controllable for the singularly perturbed form in Eq. (1). For the time-scale separation form in Eq. (2), with the -independent form of Criterion 2 in Eq. (11):

0

0.25 0ˆ0 1

A , 2

1 0.1ˆ0 0.2

A ,

0

0ˆ0

B , 2

0.1ˆ ,0.4

B

10 0 0 0 0

12 2 2 2 2

2

ˆ ˆˆ ˆ ˆrank ˆ ˆˆ ˆ ˆ

0 0 0 00 0 0 0

rank 2.2 24 544 10 6248 32 128 512

n m

n m

n m

B A B A B

B A B A B

Thus, the system judged by Eq. (11) shows that Eq. (57) in the time-scale separation form is uncontrollable. This conflicts with the controllability invariability of the system transformation. Then, considering Criterion 2 for the first-order -dependent form in Eq. (36), get

0L , 1L , 0M , and 1M using homotopy arithmetic.

0

0.4964 2.01065.0366 0.0684

L , 1

0.4874 1.96006.2530 0.2039

L ,

0

0.4976 0.25350.0135 1.0271

M , 1

0.3915 0.44552.5731 3.6723

M .

Then,


278

0.5208 2.1086( )

5.3493 0.0582L ,

0.5172 0.2312( )

0.1152 0.8435M .

0

0.2604 0.0543( )

0.0699 0.9884A ,

2

1.0130 0.0789( )

0.1337 0.1994A ,

0

0.0099( )

0.0814B , 2

0.0565( )

0.3855B .

Substituting these results into the controllability cri-terion in Eq. (36) gives

10 0 0 0 0

12 2 2 2 2

2

( ) ( ) ( ) ( ) ( )rank ( ) ( ) ( ) ( ) ( )

0.0099 0.0018 0.0039 0.00330.0814 0.0811 0.0803 0.0796

rank1.1305 10.7374 261.3001 5074.20277.7093 27.7236 139.2901 143.340

n m

n m

n m

B A B A BB A B A B

4

9

This result shows that Eq. (57) in the time-scale separation form is controllable. Thus, the -dependent controllability criterion gets the correct conclusion when the -independent controllability criterion failed.

For Criterion 3, the -independent criterion in Eq. (12) gives

0 0 0

0 0ˆˆ ˆrank[ ] rank 00 0

nB A B .

Thus, after the slow-fast order reduction, the slow subsystem in Eq. (5a) is uncontrollable by Eq. (12), which conflicts with the controllability invariability of the system transformation. For the first-order -dependent criterion in Eq. (37a),

0 0 0rank[ ( ) ( ) ( )]0.0099 0.0018

rank 2 .0.0814 0.0811

n

B A B

Thus, the slow subsystem in Eq. (5a) is controllable. The correct conclusion is obtained again by the -dependent form.

Remark 5 These examples show that the ap-proximate controllability criteria in Eqs. (11) and (12) obtained by neglecting the effect of on the model transformation give the incorrect conclusions, while the first-order -dependent controllability criteria in

Eqs. (36) and (37) give the correct results. Thus, the -dependent controllability criteria are more accurate

than the approximate -independent controllability criteria. Moreover, the controllability of the two time-scale system in the time-scale separation form and the slow subsystem in the slow-fast order reduc-tion form does not change when using the -dependent controllability criterion.

Example 3 This example shows the limitations of the first-order -dependent controllability criteria. The parameters of the second-order two time-scale control-lable system are

11 0.916 ,0.1 0.1x x z u

z x z u (58)

For Criterion 1, 11 110 1

A , 0.916

,10

B

0.9160 0.0760rank rank 2

10 0.8400B A B .

Thus, the two time-scale system in Eq. (58) is con-trollable for the singularly perturbed form in Eq. (1). Then, for the time-scale separation form in Eq. (2), with the -independent form for Criterion 2 in Eq. (11):

0ˆ 1A , 2

ˆ 0.1A , 0ˆ 10.9160B , 2

ˆ 1,B

0 0 0

2 2 22

ˆˆ ˆ10.9160 10.9160

rank rank 1.ˆˆ ˆ 10 10

B A B

B A B

Thus, the result using Eq. (11) shows that Eq. (58) in the time-scale separation form is uncontrollable, which conflicts with the controllability invariability of the system transformation. Then, for Criterion 2 for the first-order -dependent form in Eq. (36), 0L , 1L , 0M , and 1M by homotopy arithmetic are

0 0.0759L , 1 8.4022L ,

0 0.0695M , 1 7.7560M . Then,

( ) 0.91612L , ( ) 0.8451M .

0 ( ) 11.9161A , 2 ( ) 0.008 388A ,

0 ( ) 0B , 2 ( ) 1.0839B . Substituting these results into the controllability cri-

terion in Eq. (36) gives 0 0 0

2 2 22

( ) ( ) ( )rank ( ) ( ) ( )

B A BB A B


279

0.0000 0.0000rank 1.

10.8392 0.9092

Thus, Eq. (58) in the time-scale separation form is also uncontrollable. Thus, the controllability conclu-sions obtained by the first-order -dependent control-lability criterion are incorrect for this case.

For Criterion 3, the -independent criterion in Eq. (12) has

0ˆrank[ ] rank[ 10.9160] 1 nB .

Therefore, after the slow-fast order reduction, the slow subsystem in Eq. (5a) is controllable. The first-order -dependent criterion in Eq. (37a) has

0rank[ ( )] rank[0] 0 nB . Thus, the slow subsystem in Eq. (5a) is uncontrolla-

ble and the first-order -dependent criterion conflicts with the controllability invariability of the model transformation.

Then, consider controllability Criteria 2 and 3 for the second-order -dependent form. First, calculate 0L ,

1L , 2L , 0M , 1M , and 2M using homotopy arithmetic.

0 0.0036L , 1 0.3965L , 2 87.2829L ,

0 0.0033M , 1 0.3656M , 2 80.5278M . Then,

( ) 0.9161L , ( ) 0.8451M .

0 ( ) 11.9161A , 2 ( ) 0.008 392A ,

0 ( ) 0.0001B , 2 ( ) 1.0839B . Substituting these results into the second-order

-dependent controllability Criterion 2 gives

0 0 0

2 2 22

( ) ( ) ( )rank ( ) ( ) ( )

0.0001 0.0007rank 2.

10.8391 0.9096

B A BB A B

This shows that Eq. (58) in the time-scale separation form is controllable. The second-order -dependent criterion 3 has

0rank[ ( )] rank[0.0001] 1 .nB Thus, the slow subsystem in Eq. (5a) is controllable

which is correct. Remark 6 Example 3 shows that when neither the

-independent controllability criteria nor the first-order -dependent controllability criteria are accurate enough,

the right conclusions are obtained by the second-order -dependent controllability criteria. These results indi-

cate that the second-order -dependent controllability criteria are more accurate than the approximate -independent controllability criteria and the first-order -dependent controllability criteria. In addition, the

controllability of the two time-scale system does not change in the time-scale separation form or the slow-fast order reduction form when the second-order -dependent controllability criteria are used.

5 Conclusions

-dependent controllability criteria based on the Maclaurin expansion method and homotopy arithmetic are developed for the time-scale separation model and the slow-fast order reduction model for two time-scale systems. Examples illustrate that the correct results are obtained by the -dependent controllability criteria, while previous -independent controllability criteria usually lead to incorrect conclusions. The -dependent controllability criteria given here maintain the control-lability invariability of the two time-scale system in the time-scale separation form and in the slow-fast order reduction form. These results are important for con-troller designs for two time-scale systems. This analy-sis can be easily extended to second or higher-order -dependent controllability criteria for more accurate

results.

References

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[2] Kokotovic P V, Omalley R E, Sannuti P. Singular perturba-tions and order reduction in control theory An overview. Automatica, 1976, 12(2): 123-132.

[3] Glizer V Y. Controllability of nonstandard singularly per-turbed systems with small state delay. IEEE Trans. on Automatic Control, 2003, 48(7): 1280-1285.

[4] Sannuti P. On the controllability of singularly perturbed systems. IEEE Trans. on Automatic Control, 1977, 22(4): 622-624.

[5] O’Reilly J. Two time scale feedback stabilization of linear time varying singularly perturbed system. Journal of the Franklin Institute Engineering and Applied Mathematics, 1979, 308(5): 465-474.

[6] Sannuti P. On the controllability of some singularly perturbed nonlinear systems. Journal of Mathematical


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Analysis and Applications, 1978, 64(3): 579-591. [7] Phillips R G. A two stage design of linear feedback controls.

IEEE Trans. on Automatic Control, 1980, 25(6): 1220- 1223.

[8] Phillips R G. Reduced order modeling and control of two time scale discrete systems. International Journal of Con-trol, 1980, 31(4): 765-780.

[9] Kokotovic P V, Allemong J J, Winkelman J R, et al. Singu-lar perturbation and iterative separation of time scales. Automatica, 1980, 16(1): 23-33.

[10] Saksena V R, O’Reilly J, Kokotovic P V. Singular pertur-bations and time-scale methods in control theory: Survey 1976-1983. Automatica, 1984, 20(3): 273-293.

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University of Southern California Professor Kai Hwang Delivers Tsinghua Global Vision Lecture

University of Southern California (USC) Professor Kai Hwang delivered the Tsinghua Global Vision Lecture “Global Advances in Computer and Information Technology: Can China Take the Lead?” in the FIT Lecture Hall on January 6, 2009. During his lecture, Professor Hwang shared his ideas about several global advances in com-puter, networking, and information technology and discussed the great challenges yet to be confronted in informa-tion and technology in academia and in industry and the building of world-class universities. In his address, he stressed that we need to learn from successful examples of development in order to achieve a broader impact on the world economy and to build a harmonious society.

Dr. Hwang is a Professor of Electrical Engineering and Computer Science and Director of the Internet and P2P/Grid Computing Laboratory at USC. He was elected an IEEE Fellow in 1986 for his significant contributions in computer architecture, digital arithmetic, and parallel processing. He currently serves as an EMC Endowed Chair Professor in Computer Science in the Department of Computer Science and Technology at Tsinghua University and a Visiting Chief Scientist at the Institute of Computing Technology of the Chinese Academy of Sciences in Beijing.

(From http://news.tsinghua.edu.cn, 2009-01-08)

ε-Dependent controllability for two time-scale systems

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