Analysis of SDC matrices for successfully implementing the SDRE scheme

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Brief paper

Analysis of SDC matrices for successfully implementing the

SDRE scheme

✩

Yew-Wen Liang

a,1

,

Li-Gang Lin

a,b

a_{Institute of Electrical Control Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 30010, Taiwan} b_{ESAT, Katholieke Universiteit Leuven, Kasteelpark Arenberg 10, B-3001 Leuven-Heverlee, Belgium}

a r t i c l e i n f o Article history:

Received 12 August 2012 Received in revised form 19 February 2013 Accepted 2 July 2013

Available online 12 August 2013 Keywords:

State-dependent Riccati equation Nonlinear control system Stability

State-dependent coefficient matrix

a b s t r a c t

The state-dependent Riccati equation (SDRE) approach for stabilization of nonlinear affine systems was recently reported to be effective in many practical applications; however, there is no guideline on the construction of state-dependent coefficient (SDC) matrix when the SDRE solvability condition is violated, which may result in the SDRE scheme being terminated. In this study, we present several easy checking conditions so that the SDRE scheme can be successfully implemented. Additionally, when the presented checking conditions are satisfied, the sets of all feasible SDC matrices and their structures are explicitly depicted for the planar system.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Recently, the state-dependent Riccati equation (SDRE) ap-proach for nonlinear system stabilization has attracted consider-able attention (Bogdanov & Wan, 2007;Bracci, Innocenti, & Pollini, 2006; Çimen, 2010; Cloutier, D’Souza, & Mracek, 1996; Erdem & Alleyne, 2004; Hammett, Hall, & Ridgely, 1998; Lam, Xin, & Cloutier, 2012;Liang & Lin, 2011;Shamma & Cloutier, 2003; Sz-naier, Cloutier, Hull, Jacques, & Mracek, 2000). The SDRE scheme is known to include the following benefits (Çimen, 2010): (i) the concept is intuitive and simple, and directly adopts the LQR de-sign at every nonzero state; (ii) the dede-sign can directly affect sys-tem performance with predictable results by adjusting the state and the control weightings to specify the performance index (for instance, the engineer may modulate the weighting of the sys-tem state to speed up the response, although at the expense of in-creased control effort); (iii) the scheme possesses an extra design

✩ _{This work was supported by the National Science Council, Taiwan, under Grants}

99-2218-E-009-004, 100-2221-E-009-026-MY2, and 101-2623-E-009-005-D. The material in this paper was partially presented at the 18th IFAC World Congress, August 28–September 2, 2011, Milano, Italy. This paper was recommended for publication in revised form by Associate Editor Constantino M. Lagoa under the direction of Editor Roberto Tempo.

E-mail addresses:ywliang@cn.nctu.edu.tw(Y.-W. Liang),

charleslin19841124@gmail.com(L.-G. Lin). 1 Tel.: +886 3 5712121; fax: +886 3 5715998.

degree of freedom arising from the non-unique state-dependent coefficient (SDC) matrix representation of the nonlinear drift term, which can be utilized to enhance controller performance; and (iv) the approach preserves the essential system nonlinearities because it does not truncate any nonlinear terms. Many practical and mean-ingful applications successfully performed by the SDRE design have been reported (seeÇimen, 2010and the references therein). The first solid theoretical contributions on SDRE control have been pro-vided byCloutier et al.(1996) andMracek and Cloutier(1998). The current study attempts to provide further theoretical support of the SDRE control strategy, as discussed in the recent survey by Çi-men(2012), with rigorous mathematical proofs.

The SDRE design for nonlinear systems can be described as fol-lows. Consider a class of nonlinear control systems and a quadratic-like performance index as(1)–(2)below:

˙

x

=

f

(

x

) +

B

(

x

)

u (1) and J

=

1 2



∞ 0



xTQ

(

x

)

x

+

uTR

(

x

)

u



dt (2) where x

∈

_Rn_{and u}

_∈

Rpdenote the system states and control inputs, respectively, f

(

x

) ∈

_Rn

_,

_B

₍

_x

_{) ∈}

Rn×p

,

f

(

0

) =

0

,

QT

(

x

) =

Q

(

x

) ≥

0, RT

₍

_x

_{) =}

_R

₍

_x

_{) >}

₀

_,

_Q

₍

_x

₎

_{, R}

₍

_x

_{) ∈}

_Ck_{, k}

_≥

_{1, and}

_(·)

T de-notes the transpose of a vector or a matrix. Note that the weighting matrices Q

(

x

)

and R

(

x

)

are in general state-dependent. The pro-cedure of the SDRE scheme is summarized as the following three steps (Çimen, 2010):

0005-1098/$ – see front matter©2013 Elsevier Ltd. All rights reserved.

(i) Factorize f

(

x

)

into the SDC matrix representation as f

(

x

) =

A

(

x

)

x, where A

(

x

) ∈

_Rn×n_.

(ii) Symbolically check the stabilizability of

(

A

(

x

),

B

(

x

))

and the observability (resp., detectability) of

(

A

(

x

),

C

(

x

))

to ensure the existence of a unique positive definite (resp., semi-definite) solution of the following SDRE:

AT

(

x

)

P

(

x

) +

P

(

x

)

A

(

x

) +

Q

(

x

)

−

P

(

x

)

B

(

x

)

R−1

(

x

)

BT

(

x

)

P

(

x

) =

0 (3) where C

(

x

) ∈

_Rq×n _{has full rank and satisfies Q}

₍

_x

_{) =}

CT

₍

_x

₎

_C

₍

_x

₎

_.

(iii) Solve for P

(

x

)

from(3)to produce the SDRE controller u

= −

K

(

x

)

x and K

(

x

) =

R−1

(

x

)

BT

(

x

)

P

(

x

).

(4) It should be noted that the SDRE scheme is performed pointwise in x and the resulting closed-loop SDC matrix ACL

(

x

) :=

A

(

x

) −

B

(

x

)

R−1

(

x

)

BT

(

x

)

P

(

x

)

is pointwise Hurwitz everywhere; however, it does not imply global stability of the origin (Tsiotras, Corless, & Rotea, 1996). In addition, though the SDRE approach provides satisfactory performance in many practical applications, the sym-bolic checking conditions stated in (ii) of the SDRE scheme are generally not easy to implement, especially when the system dy-namics are complicated. Moreover, several authors have provided various guidelines on how to systematically construct SDC matri-ces (Çimen,2010;Cloutier et al.,1996); however, there is no guide-line on the construction of SDC matrices when the SDRE solvability condition is violated, which may result in the SDRE scheme being terminated. For instance, let f

(

x

) = [−

x2

,

x1

]

T

,

B

(

x

) = [

0

,

x2

]

T

,

R

(

x

) =

1 and Q

(

x

) =

I2. Suppose that an SDC matrix

representa-tion is given as a11

(

x

) =

a22

(

x

) =

0

,

a12

(

x

) = −

1 and a21

(

x

) =

1, where aij

(

x

)

denotes the

(

i

,

j

)

-entry of the matrix A

(

x

)

. Then,

(

A

(

x

),

C

(

x

))

is always observable, but

(

A

(

x

),

B

(

x

))

is not stabiliz-able at the nonzero states where x2

=

0. By direct calculation, the

SDRE given by(3)does not have any positive semi-definite solu-tion P

(

x

)

when x2

=

0, in which case the SDRE scheme will fail to

operate. However, it will become clear later (seeTheorem 1) that, at those nonzero states x of x2

=

0, there always exists a feasible

SDC matrix representation that makes the SDRE(3)solvable and the resulting ACL

(

x

)

matrix a Hurwitz matrix.

It is known that a unique positive definite (resp., semi-definite) solution P

(

x

)

in(3)exists, rendering ACL

(

x

)

pointwise Hurwitz, if (resp., if and only if) both the conditions ‘‘

(

A

(

x

),

B

(

x

))

is stabiliz-able’’ and ‘‘

(

A

(

x

),

C

(

x

))

is observable (resp., has no unobservable mode on the j

ω

-axis)’’ are satisfied (Zhou & Doyle, 1998). To avoid the difficulty of symbolic checking conditions, stated above, of the SDRE approach, in this article we will study the following three problems:

Problem 1. Let x

̸=

0 be given. Denote f

=

f

(

x

),

B

=

B

(

x

)

and

C

=

C

(

x

)

. Explore the existence condition and, if the existence condition is satisfied, present all A

∈

_Rn×n_{that satisfy the} condi-tions that Ax

=

f

, (

A

,

B

)

is stabilizable and

(

A

,

C

)

is observable. Problem 2. Same asProblem 1, except that the condition ‘‘

(

A

,

C

)

is observable’’ is replaced with ‘‘

(

A

,

C

)

is detectable’’.

Problem 3. Same asProblem 1, except that the condition ‘‘

(

A

,

C

)

is observable’’ is replaced with ‘‘

(

A

,

C

)

has no unobservable mode on the j

ω

-axis’’.

From the discussions above, this study may also provide an auxiliary means to successfully continue the SDRE scheme at states in which a specific SDC matrix representation fails to operate, but whereProblems 1,2or3is solvable.

To explore the existence condition ofProblems 1–3and charac-terize their solution matrices, we introduce the notations W⊥_and

W⊥as follows. Let W

∈

Rp×nbe given with p

<

n and rank

(

W

) =

p. We define W⊥

₌

_N

₍

_W

₎

_{, null space of W , and W}

⊥

∈

Rn×(n−p)as a selected constant matrix having orthonormal columns and satis-fying WW⊥

=

0. Clearly, W⊥is a vector space of dimension n

−

p,

and the column vectors of W⊥form an orthonormal basis of W⊥.

Similarly, if W

∈

_Rn×q_{and rank}

₍

_W

_{) =}

_q

_<

_{n, we define W}⊥

₌

{

wT

_|

_w

_∈

_N

₍

_WT

_)}

_{and W}

⊥

∈

R(n−q)×nas a selected constant matrix having orthonormal rows and satisfying W⊥W

=

0.

Addi-tionally, we denote Rn∗

= {

xT

|

x

∈

_Rn

}

, known as the dual space of Rn, and R−as the set of negative real numbers.

The rest of this article is organized as follows: Section2presents the necessary and sufficient existence conditions forProblems 1– 3; Section3includes a description of the parameterization of the solution matrices A for the planar case when the existence condi-tions are satisfied; Section4presents an illustrative example; and Section5provides the conclusions.

2. Necessary and sufficient existence conditions

Necessary and sufficient existence conditions forProblems 1–3 are stated asTheorem 1below:

Theorem 1.

(i) Problem 1is unsolvable if and only if

{

x

,

f

}

are linearly dependent (LD) and C x

=

0.

(ii) Problem 2is unsolvable if and only if f

=

kx for some k

≥

0 and

C x

=

0.

(iii) Problem 3is unsolvable if and only if f

=

0 and C x

=

0. Proof. The proofs of (i) and (ii) can be found fromLiang and Lin (2011), while (iii) is easily derived from the proof of (ii). Details are omitted. 

3. Parameterization of all solution matrices

Given that the existence condition ofProblems 1,2or3is sat-isfied, this section explores their solution matrices. To this end, we denoteAxf

,

Ac

,

As,Ao

,

AdandAias the sets of A such that

Ax

=

f

, (

A

,

B

)

is controllable,

(

A

,

B

)

is stabilizable,

(

A

,

C

)

is observ-able,

(

A

,

C

)

is detectable and

(

A

,

C

)

has no unobservable mode on the j

ω

-axis, respectively. Additionally, we assume hereafter that, without loss of any generality, both B and C have full rank.

3.1. The solution matrices ofProblems 1–3

Define Ap

=

_∥x1_∥2fxT. It is clear that Apx

=

f and

Axf

=



Ap

+

K x⊥

|

K

∈

Rn×(n−1)

 ⊂

Rn×n

.

(5) Obviously,Axfis a linear variety (i.e., a subspace through a transla-tion) of dimension n2

−

n and K describes the n2

−

n free

parame-ters. Additionally, Aphas the minimum Frobenius norm among the matrices inAxf. To deriveAc

,

As,Ao

,

AdandAi, we present the following two results which can be used to reduce the dimension of checking the system’s controllability, stabilizability, observability and detectability.

Lemma 2 (Chen, 1999). Let A

¯

=



_A¯ 11 A¯12 ¯ A21 A¯22



andB

¯

=



₀ ¯ B2



, where

¯

B2

∈

Rp×pis a nonsingular matrix,A

¯

11

∈

R(n−p)×(n−p)andA

¯

22

∈

Rp×p. Then,

(¯

A

, ¯

B

)

is controllable (resp., stabilizable)

⇔

(¯

A11

, ¯

A12

)

is

controllable (resp., stabilizable). In particular, when p

<

n andA

¯

12

=

0, then

(¯

A

, ¯

B

)

is uncontrollable, and it is stabilizable

⇔

λ(¯

A11

) ⊂

C−. Corollary 3. LetA be partitioned in the form given by

¯

Lemma 2with

¯

A11

∈

R(n−q)×(n−q)andA

¯

22

∈

Rq×q.C

¯

= [

0

, ¯

C2

]

, whereC

¯

2

∈

Rq×qis

a nonsingular matrix. Then

(i)

(¯

A

, ¯

C

)

is observable (resp., detectable)

⇔

(¯

A11

, ¯

A21

)

is observable

(resp., detectable).

(ii)

(¯

A

, ¯

C

)

has no unobservable mode on the j

ω

-axis

⇔

(¯

A11

, ¯

A21

)

has

In particular, when q

<

n andA

¯

21

=

0, then

(¯

A

, ¯

C

)

is unobservable

and

(iii)

(¯

A

, ¯

C

)

is detectable

⇔

λ(¯

A11

) ⊂

C−.

(iv)

(¯

A

, ¯

C

)

has no unobservable mode on the j

ω

-axis

⇔ ¯

A11has no

eigenvalue on the j

ω

-axis.

To applyLemma 2andCorollary 3, we have to transform

(

A

,

B

)

(resp.,

(

A

,

C

)

) into the form of

(¯

A

, ¯

B

)

(resp.,

(¯

A

, ¯

C

)

) as stated in Lemma 2(resp.,Corollary 3). Such coordinate transformation can be chosen to be orthogonal as in the form of(6)below:

x

=

MBx

¯

(

resp., x

=

MCx

¯

)

(6)

where MBand MCare orthogonal matrices. A candidate of MB(resp.,

MC) can be determined by the QR factorization scheme for B (resp.,

CT_{) and then interchanges the position of the first p (resp., q)} columns with the last n

−

p (resp., n

−

q) columns.

Under the coordinate transformation given by Eq.(6)we have

¯

x

=

MTx and_A

¯

=

MTAM, where M

=

MBor M

=

MC. If we let

¯

f

=

MT_f

_{, ¯}

_x ⊥

=

x⊥M andK

¯

=

MTK , thenx

¯

⊥x

¯

=

0, Ax

=

f

⇔

¯

Ax

¯

= ¯

f, and 1 ¯ xT_x¯

¯

fx

¯

T

+ ¯

Kx

¯

⊥

=

MT



1 xT_xfxT

+

K x⊥



M. That is,A

¯

∈

Ax¯¯_f

⇔

A

∈

Axf. Moreover, because controllability, observability, stabilizability and detectability are invariant under equivalence transformation (Chen, 1999), we obtain the following theorem. Theorem 4. Let MB (resp., MC) be an orthogonal matrix given by

Eq.(6)such that B

=

MBB (resp., C

¯

T

=

MCC

¯

T),B (resp.,

¯

C ) is given

¯

byLemma 2(resp.,Corollary 3),x

¯

⊥

=

x⊥MBand K

=

MBK (resp.,

¯

¯

x⊥

=

x⊥MCand K

=

MCK ). Additionally, A

¯

=

∥x1∥2fx T

₊

_{K x} ⊥

∈

Axf andA

¯

=

1 ∥ ¯x∥2

¯

fx

¯

T

_{+ ¯}

_Kx

_¯

⊥

∈

Ax¯¯_f. Then

(i)

(

A

,

B

)

is controllable (resp.,

(

A

,

C

)

is observable)

⇔

(¯

A

, ¯

B

)

is controllable (resp.,

(¯

A

, ¯

C

)

is observable).

(ii)

(

A

,

B

)

is stabilizable (resp.,

(

A

,

C

)

is detectable)

⇔

(¯

A

, ¯

B

)

is stabilizable (resp.,

(¯

A

, ¯

C

)

is detectable).

(iii)

(

A

,

C

)

has no unobservable mode on the j

ω

-axis

⇔

(¯

A

, ¯

C

)

has no unobservable mode on the j

ω

-axis.

After deriving the setsAxf

,

As,Ao

,

AdandAi, it is clear that the solutions ofProblems 1–3areAsoxf

:=

Axf

∩

As

∩

Ao

,

Asdxf

:=

Axf

∩

As

∩

AdandAxfsi

:=

Axf

∩

As

∩

Ai, respectively.

3.2. Implementation of the case n

=

2

The case of n

=

1 is trivial; therefore, we only consider the case of n

=

2. When rank

(

B

) =

2 (resp., rank

(

C

) =

2),

(

A

,

B

)

(resp.,

(

A

,

C

)

) is controllable (resp., observable) andAc

xf

=

Asxf

=

Axf

(resp.,Ao_xf

=

Ad_xf

=

Ai_xf

=

Axf). Remaining to be considered is the case of B

=

b

=

(

b1

,

b2

)

T

∈

R2and C

=

c

=

(

c1

,

c2

) ∈

R1×2. In this case, K

=

k

∈

_R2_andA

xfis a 2-dimensional linear variety. To deriveAc

,

As

,

Ao,AdandAi, we need the following lemma. Lemma 5. Consider the two lines L1

(

k

) : ξ

Tk

=

α

1and L2

(

k

) : ξ

⊥k

=

α

₂, where

α

1

, α

2

∈

R and

ξ ∈

R2

\ {

0

}

. Then

(i) L1

(

k

)

can be parameterized as k

(κ) =

_∥α_ξ∥12

ξ + κξ

T ⊥, where

κ ∈

R.

(ii) L1

(

k

)

and L2

(

k

)

are perpendicular and intersect at the point

where k∗

₌

α1 ∥ξ∥2

ξ +

α2 ∥ξ⊥∥2

ξ

T ⊥.

(iii) The half line

{

k

|

L1

(

k

) =

0 but L2

(

k

) ≥

0

}

can be

parameter-ized as

{

k∗

₊

_κξ

⊥

|

κ ≥

0

}

.

(iv) The half plane

αξ

⊥k

≥

0

, α ∈

R

\ {

0

}

, can be parameterized as k

(κ

1

, κ

2

) = κ

1

ξ

T⊥

+

κ

2

ξ

, where sign

(α) · κ

1

≥

0 and

κ

2

∈

R. Define x⊥

=

_∥1_x∥

[

x2

, −

x1

]

,

b⊥

=

_∥1_b∥

[

b2

, −

b1

]

and c⊥

=

_∥1_c∥

[

c2

, −

c1

]

T. The setsAc

,

As,Ao

,

AdandAi, and their structures are

explicitly described in the next result.

Theorem 6. Let x

,

f

,

b

,

cT

∈

_R2and x

̸=

0. Then (i) Ac

xf

=



Axf\ Acxf¯ if{x,b}are LI;

Axf if{x,b}are LD &{x,f}are LI;

∅ if{x,b}are LD &{x,f}are LD,

whereAc_xf¯

:=



A¯c p

+

κ

bx⊥

|

κ ∈

R & Acp¯

=

1 ∥x∥2fxT

−

(b⊥f)(bT_x) ∥x∥2_(x⊥b) bT ⊥x⊥



is a line inAxfin which

(

A

,

b

)

is uncontrollable. (ii) As xf

=















Axf if ‘‘{x,b}are LI & b⊥f x⊥b< 0’’ or ‘‘{x,b}are LD &{x,f}are LI;’’ Axf\ A¯cxf if{x,b}are LI &

b⊥f

x⊥b ≥0;

Axf\ A¯sxf if{x,b}are LD &{x,f}are LD,

whereA¯s xf

:=



Ap

+

κ

1bT⊥x⊥

+

κ

2bx⊥

|

κ

2

∈

R & sign

(

x⊥bT⊥

) ·

κ

1

≥

0



is a half plane inAxfin which

(

A

,

b

)

is unstabilizable. (iii) Ao

xf

=



_A

xf\ Aoxf¯ if cx̸=0;

Axf if cx=0 &{x,f}are LI;

∅ if cx=0 &{x,f}are LD, whereA¯o_xf

:=



Ao_p¯

+

κ

c⊥x⊥

|

κ ∈

R & Aop¯

=

fxT ∥x∥2

−

(cf)(xT_c⊥) ∥x∥2_·∥c∥2_(x⊥c⊥)cTx⊥



is a line inAxfin which

(

A

,

c

)

is

unobser-vable. (iv) Ad xf

=







Axf\ A ¯ d xf if cx̸=0;

Axf if ‘‘cx=0 &{x,f}are LI’’ or

‘‘cx=0 & f=µx, µ <0;’’ ∅ if cx=0 & f=µx, µ ≥0, whereA¯dxf

:=



Ao¯ p

+

(κ +

cT⊥f

)

c⊥x⊥

|

κ ∈

R &

κ ·

sign

(

x⊥c⊥

) ≥

0



is a half line inA¯o xfin which

(

A

,

c

)

is undetectable. (v) Aixf

=



Axf\ A ¯ i xf if cx̸=0; Axf if cx=0 & f̸=0; ∅ if cx=0 & f=0, whereA¯ixf

:=



Ao_p¯

+

(

cT ⊥f

)

c⊥x⊥



is a point inAdxf¯ in which

(

A

,

c

)

has an unobservable mode on the j

ω

-axis.

Proof. Here, we only derive the setsAc

xfandAsxf. The setsAoxf

,

Adxf andAi

xfcan be similarly derived. Let Mb

= [

bT⊥

...

b

∥b∥

]

. It is clear that

¯

A

=

MT

bAMbandb

¯

=

MbTb are in the form described inLemma 2. By direct calculation,A

¯

12

=

∥1b∥



1

∥x∥2

(

b⊥f

) · (

bTx

) + (

b⊥k

) · (

x⊥b

)

]

andA

¯

11

=

_∥x1_∥2

(

b⊥f

)(

b⊥x

) + (

b⊥k

)(

x⊥bT⊥

)

. FromLemma 2and

Theorem 4,

(

A

,

b

)

is uncontrollable

⇔ ¯

A12

=

0, and

(

A

,

b

)

is

unsta-bilizable

⇔ ¯

A12

=

0 andA

¯

11

≥

0. Now if x⊥b

̸=

0, i.e.,

{

x

,

b

}

are LI,

then the set of k such thatA

¯

12

=

0 can be parameterized using (i)

ofLemma 5with

(ξ, α

1

)

being replaced by



bT ⊥

, −

( b⊥f)(bT_x) ∥x∥2_(x⊥b)



. Com-bining the parameterization of k with the expression ofAxfgives the setA¯c

xf. Consequently,Acxf

=

Axf

\

Acxf¯. Additionally, withinA

¯ c

xf (i.e.,A

¯

12

=

0), k satisfies the relation b⊥k

= −

(b⊥f)(b

T_x)

(x⊥b)∥x∥2. Inserting

this relation intoA

¯

11yieldsA

¯

11

=

(b⊥f)[(x

T_b)2₊₍ b⊥x)2] (x⊥b)∥x∥2 . Thus,A

¯

11

<

0

⇔

b⊥f x⊥b

<

0. Therefore,A s xf

=

Axfif

{

x

,

b

}

are LI and_x⊥bb⊥f

<

0, andAs

xf

=

Axf

\

Acxf¯ if

{

x

,

b

}

are LI andx⊥bb⊥f

≥

0. We now consider the case of x⊥b

=

0, i.e.,

{

x

,

b

}

are LD. This implies that bTx

̸=

0,

andA

¯

12

=

0

⇔

b⊥f

=

0

⇔ {

x

,

f

}

are LD because x⊥b

=

0. As a

re-sult,As_xf

=

Ac_xf

=

Axfif

{

x

,

f

}

are LI. When

{

x

,

f

}

are LD (i.e.,A

¯

12

=

0), we haveAc_xf

= ∅

andA

¯

11

=

(

x⊥bT⊥

)

b⊥k. By (iv) ofLemma 5,

the set of k forA

¯

11

≥

0 is a half plane and can be parameterized as

k

(κ

₁

, κ

₂

) = κ

₁bT

⊥

+

κ

2b, where sign

(

x⊥bT⊥

)·κ

1

≥

0 and

κ

2

∈

R. In-serting this k

(κ

1

, κ

2

)

intoAxfyieldsA¯sxf. Thus,Asxf

=

Axf

\

A¯sxf.  It is interesting to note from Theorem 6 that the setAs_xf is always non-empty, regardless of what nonzero vector b is given. Moreover, it is easy to see that the results ofTheorem 6agree with those ofTheorem 1. That is,Aso

xf

= ∅ ⇔

cx

=

0 and

{

x

,

f

}

are LD; Asd_xf

= ∅ ⇔

cx

=

0

,

f

=

µ

x and

µ ≥

0; andAsi_xf

= ∅ ⇔

cx

=

0 and f

=

0.

4. An illustrative example Consider the following system

˙

x1

=

x1x2 and x

˙

2

= −

x2

+

u

.

(7)

Clearly, this system is in the form of(1)with x

= [

x1

,

x2

]

T

,

f

(

x

) =

[

x1x2

, −

x2

]

Tand B

(

x

) = [

0

,

1

]

T. System(7)is stabilizable and two

global stabilizers, one using the Sontag formula with the control Lyapunov function V

(

x1

,

x2

) := (

x12e2x2

+

x22

)/

2 (Sontag, 1989) and

the other adopting the backstepping scheme (Khalil, 1996), have the following forms:

uSontag

=

x2₂

−



x4₂

+

(

x2₁e2x2

₊

_x 2

)

4 x2 1e2x2

+

x2 (8) and uBS

=

(

1

−

ψ)

x2

−

(

1

+

ψ)

x21

−

2x 2 1x2

, ψ >

0

.

(9)

To demonstrate the SDRE design, we choose Q

(

x

) =

I2

,

R

(

x

) =

1

and an intuitive SDC matrix A

(

x

)

with a11

(

x

) =

a21

(

x

) =

0, a12

(

x

)

=

x1and a22

(

x

) = −

1. Obviously,

(

A

(

x

),

B

(

x

))

is stabilizable

ev-erywhere except the X2-axis where the SDRE solvability condition

is violated; however, byTheorem 1,Asxfγ

̸= ∅

for

γ =

o

,

d

,

i at every nonzero state because C

(

x

)

x

=

x

̸=

0. When x

= [

0

,

x2

]

T

and x2

̸=

0

,

f

= [

0

, −

x2

]

T

= −

x and, by (ii) ofTheorem 6,Asoxf

=

Asd

xf

=

Asixf

=

Asxf

=

Axf

\

Axfs¯

= {

A

|

a11

<

0

,

a12

=

0

,

a21

∈

R & a22

= −

1

}

. In the following, we will choose a11

= −

1 and

a21

=

0 for the SDC matrix of the SDRE scheme when x

∈

X2-axis.

Numerical results for initial states x

(

0

) = [

1

,

1

]

Tare summa-rized inFig. 1andTable 1, where we have adopted the following three controllers: uSontag(labeled Sontag), uBSwith

ψ =

2 (labeled

BS) and the SDRE controller (labeled SDRE). It is observed from Fig. 1that all of the system states of the three schemes converge to zero and, fromTable 1, the SDRE scheme has better performances than the other two schemes in the performance indices that are listed in the table, where

∥

u

∥

∞

:=

maxt

∥

u

∥

denotes the maxi-mum control magnitude that required during the control period and the integration is evaluated from t

=

0 to t

=

1000.

It is noted that the solution trajectories of the three schemes remain on the X2-axis if they start from there because x

˙

1

=

x1x2

|

x1=0

=

0. Thus, the trajectories of the three schemes will

never reach the X2-axis unless they start from there. By direct

cal-culation, uSontag

=

uSDRE

=

(

1

−

√

2

)

x2and uBS

=

(

1

−

ψ)

x2if

the system state starts from the X2-axis. The resulting closed-loop

dynamics for x2arex

˙

2

= −

ψ

x2for the BS design andx

˙

2

= −

√

2x2

for both the Sontag and SDRE schemes. It is interesting to note that, when x

∈

X2-axis, uSDRE remains unchanged regardless of

the choice of A

(

x

) ∈

As_xf; however, if the weighting matrices are changed to be Q

(

x

) =

diag

(

q1

,

q2

) >

0 and R

(

x

) =

r

>

0, then

uSDRE

=

(

1

−

√

1

+

q2

/

r

)

x2and the resulting closed-loop

dynam-ics for x2becomesx

˙

2

= −

√

1

+

q2

/

r

·

x2, both are independent of

q1. Moreover, uSDRE

≈

0

=

uBS

|

ψ=1when r

≫

q2, which implies

that the control effort should be reduced as much as possible. 5. Conclusions

This article has presented necessary and sufficient conditions for the existence of SDC matrices in a nonlinear system such that the SDRE scheme can be successfully implemented. These exis-tence conditions are easy to verify, and when they are satisfied, all of the feasible SDC matrices are explicitly parameterized for the planar case. An example is also given to demonstrate the use of the main results. Nevertheless, the application of this study in SDRE design for better system performance, including optimal control recovery and basin of attraction estimation, needs further investi-gation.

a

_b

c

_d

Fig. 1. Time history of the system states and control inputs.

Table 1

Performances of the three schemes.

Final time of xT_x=0.01  (xT_x+u2₎  u2 _∥u∥ ∞ Sontag 3.2×103 _{13.6 3.4 8.3} BS 8.3×102 _{9.7 5.8 6} SDRE 86.3 6.1 2.2 2 References

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Yew-Wen Liang (M’02) was born in Taiwan in 1960. He

received the B.S. degree in Mathematics from the Tung Hai University, Taichung, Taiwan, Republic of China, in 1982, the M.S. degree in Applied Mathematics in 1984 and the Ph.D. degree in Electrical and Computer Engineering in 1998 from the National Chiao Tung University, Hshinchu, Taiwan, Republic of China. Since August 1987, he has been with the National Chiao Tung University, where he is currently an Associate Professor of Electrical and Control Engineering. His research interests include nonlinear control systems, reliable control, and fault detection and diagnosis issues.

Li-Gang Lin received the B.S. degree and M.S. degree in

Electrical Control Engineering (ECE) from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2008 and in 2010, respectively. He is currently working toward his joint/double Ph.D. degrees from ESAT of Katholieke Universiteit Leuven, Belgium, and institute of ECE in NCTU. His research interests include nonlinear control systems, state-dependent (differential/difference) Riccati equation, reliable and robust control.