Common stabilizers for linear control systems in the presence of actuators outage

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Common stabilizers for linear control systems

in the presence of actuators outage

Yew-Wen Liang, Der-Cherng Liaw

*

Department of Electrical and Control Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30039, Taiwan, ROC

Abstract

Issue concerning the existence of common stabilizers for a linear control system experiences actuators’ outage are pre-sented. The possible outage of actuators examined in this study are not restricted to a pre-specified set. By finding common quadratic-type Lyapunov functions, we obtain sufficient conditions for the existence of common stabilizers. For cases of which all the possible failed actuators belonged to a pre-specified set, the results presented in this paper agree with those obtained by Veillette in 1995. The control gain of common stabilizer for non-nested case is explicitly derived to guarantee system stability. A simplified checking condition for the existence of common stabilizers is also obtained for the extreme case when only single actuator can normally operate.

Keywords: Common stabilizers; Actuator outage; Linear control systems; Lyapunov function

1. Introduction

Recently, the study of reliable controls that can tolerate the failure of actuators or sensors in control sys-tems has attracted much attention (see, e.g.,[1–10]). However, most existing results for reliable control design are limited to systems with failure of actuators within a pre-speciﬁed subset. Among these studies, Veillette

[1]also inspected, in his example, whether the designed controllers could tolerate the outage of actuators out-side the pre-speciﬁed subset. In[2], although Medanic investigated the possible outage of actuators outside a pre-speciﬁed subset, it was restricted to single actuator outage. Zhao and Jiang[3]synthesized a reliable con-troller for dynamic systems with redundant actuators. Though their approach does not involve the construc-tion of Lyapunov funcconstruc-tion, the controlled system _x¼ Ax þ Bu is required to have actuator redundancy with (A, bi) is a controllable pair for each i, with B¼ ðb1; . . . ; bpÞ 2 Rnp. Moreover, the pre-compensator for

transforming the non-uniform redundancy property into uniform property might increase system order and reconﬁgure system structure. In this paper, the authors will extend the reliable stabilization of[1]to sys-tems where the outage of actuators might be outside a pre-speciﬁed subset and the number of failed actuators

* _{Corresponding author.}

E-mail address:ldc@cn.nctu.edu.tw(D.-C. Liaw).

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is not restricted to one. Moreover, the control system is not assumed to possess the controllability property as required in[3]. To tackle the reliable design problem, one might consider the existence of either common or non-common Lyapunov functions with regard to faulty systems. In this paper, the authors will consider the existence of common Lyapunov functions, while an example of seeking non-common Lyapunov func-tions for stabilizing switched systems may be found in [11]. Our approach is to seek a common quadratic-type Lyapunov function whose time derivative is negative for all the directions in which the controls have no contribution. A suﬃcient condition for common stabilizers is derived and the method of its implementa-tion is demonstrated.

The goal of this paper is then to propose and implement a checking condition for the existence of common stabilizers for a control system experiencing the outage of actuators. The idea behind the study is to present a common stabilizer that can tolerate the outage of certain actuators without switching the control law, since switching the control law could require more control elements to sense the outage of actuators. Otherwise, the reliability of additional sensor elements would have to be considered. Potential applications of such a sta-bilizer include space missions or any highly dangerous area where actuators of equipment fail. This issue is important because retrieving satellites is expensive and instability of equipment in highly dangerous areas might result in disaster.

There are two main differences between the paper and those of [1]. First, the paper proposes a unified approach to determine the existence of common stabilizers regardless of whether the outage of actuators are confined within a pre-specified set, while those of[1]did not. Moreover, it is also shown that the obtained results for the existence of common stabilizers agree with those of [1] when the outage of actuators are confined within a pre-specified set. Second, once the common stabilizer is determined to exist by the checking condition proposed in this paper, the control gain of the common stabilizers can be determined from the Routh–Hurwitz criteria to fulfill the task, while the choice of control gain in[1]was fixed to one. An example is also given to demonstrate the importance of the selection of such a control gain.

This paper is organized as follows. Section2introduces the problem. An example of which all the faulty systems are completely controllable does not guarantee the existence of common stabilizers is also given. It is followed by the derivation of the existence of common stabilizers. The procedure for implementing such conditions and determining the control gain that guarantees the stability of the faulty systems is also pro-posed. Section4presents an illustrative example to demonstrate the application of the results. The existence of common stabilizer for the admissible faulty systems of the given example is shown not to be obtainable by Veillette’s design[1]. Finally, Section 5gives concluding remarks.

2. Set up of the problem

Consider a linear control system

_x¼ Ax þ Bu; ð1Þ

where x2 Rn

, u2 Rm

, A2 Rnn _{and B}_{2 R}nm_{. Deﬁne the set of control matrices}

B ¼ fBi2 Rnmj Bi is obtained from B by replacing some columns or no column

of B with zero column vector andðA; BiÞ is stabilizableg. ð2Þ

That is, for each Bi2 B, Bidenotes the control matrix resulting from B experiencing the outage of some

actu-ators. Note that the set B contains a ﬁnite number of matrices.

Recall that the goal of this paper is to determine the existence conditions of common stabilizers for all sys-tem pairs (A, Bi) with Bi2 B. Note that the outage of actuators considered here is not conﬁned to be within a

pre-speciﬁed set.

From linear system theory, it is known that a linear system pair (A, B) is stabilizable if the unstable sub-space of A is contained in the controllability sub-space of (A, B) (see, e.g.,[12]). Using this observation, one might predict that the class of systems (A, Bi) with Bi2 B and B as deﬁned in(2)possess a common stabilizer if the

intersection of the controllability space of all the system pairs (A, Bi) contains the unstable subspace of A.

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Example 1. Consider system (1)with A¼ 1 0 0 2 and B¼ 1 1 1 1 . ð3Þ Let B1¼ 1 0 1 0 and B2¼ 0 1 0 1 . ð4Þ

It is easy to check that all the system pairs (A, B) and (A, Bi) for i = 1, 2 are completely controllable. Thus,

according to the deﬁnition in(2), we have B ¼ fB; B1; B2g.

Suppose that these three system pairs possess a common stabilizer u = Kx, where K¼ k11 k12

k21 k22

. ð5Þ

That is, all the matrices A + BK and A + BiK for i = 1, 2 are Hurwitz. Then, from the Routh–Hurwitz stability

criteria, to provide for the stability of system pair (A, B1) one needs to have tr(A + B1K) = k11 k12+ 3 < 0

and det(A + B1K) = 2k11 k12+ 2 > 0, where tr(Æ) and det(Æ) denote the trace and determinant of a matrix.

This results in k11> 1 and k12> 4. Similarly, for system pair (A, B2) one needs to have tr(A + B2K) =

k21+ k22+ 3 < 0 and det(A + B2K) = 2k21+ k22+ 2 > 0. This means that k21> 1 and k22<4. By direct

calculation, for system pair (A, B) one ﬁnds

detðA þ BKÞ ¼ 2k11ðk22þ 1Þ þ 2k21ð1 k12Þ k12þ ðk22þ 2Þ. ð6Þ

According to the stability conditions for system pairs (A, B1) and (A, B2) discussed above, all the terms in the

right-hand side of(6)are negative. It then follows that det(A + BK) < 0. This contradicts u = Kx as a stabilizer for (A, B). Thus, the three given pairs of control systems do not possess a common stabilizer.

3. Main results

In this section, we will employ the Lyapunov approach to derive a condition for the existence of common stabilizers as given byTheorem 1. Then, we will demonstrate the implementation of the existence condition. Details are given as below.

3.1. Existence condition for common stabilizers

Suppose the class of systems (A, Bi), Bi2 B, possesses a common stabilizer K 2 Rmnand A + BiK shares a

common Lyapunov function V(x) = xTPx. Then xTP(A + BiK)x < 0 for all non-zero x and for all i. This leads

to the following result.

Theorem 1. Consider the class of linear control systems (A, Bi), where Bi2 B and B is defined as in(2). If there

exists a symmetric positive definite matrix P > 0 such that xTPAx <0 for all x2 [

Bi2B

NðBT

iPÞ n f0g; ð7Þ

then the class of systems (A, Bi), Bi2 B, possess a common stabilizer. Here, N(Æ) denotes the null space of a

ma-trix. Moreover, a common stabilizer can be chosen in the form u =a Æ BTPx with a satisfying Condition(9).

Proof. By the application of optimal control design, we choose a common stabilizer candidate in the form of u =a Æ BTPx to meet Condition(7). It is observed that, from the special structure of Bi, BiBT¼ BiBTi for all

Bi2 B. The time derivative of V(x) = xTPx along the trajectories of the system _x¼ Ax þ Biu with

u =a Æ BTPx has the form _

V ¼ 2 ðxT_PAx_{a x}T_PB

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In the following, we will show the existence of a such that _V <0 for all x 5 0 and for all Bi2 B. This will then

imply the existence of common stabilizers.

If xTPAx < 0 for all x 5 0, then A must be a Hurwitz matrix[12]and _V <0 for all x 5 0 and for all Bi2 B

no matter what a > 0 is chosen. On the other hand, if xTPAx P 0 for some x 5 0, then Condition(7)implies that xTPBi50 for all i and for all x2 S, S := {x j xTPAx P 0,kxk = 1}. Since S is a non-empty compact set,

it implies that ci:= minx2SkxTPBik > 0 for all Bi2 B. Thus, c: = minici> 0 exists since B only contains a ﬁnite

number of matrices. From the deﬁnition of c, all the non-zero points x satisfying xTPAx P 0 have the property thatkxT_PB

ik ¼ kx T

kxkPBik kxk P c kxk for all Bi2 B. Choose the control gain a satisfying

a >kA

T_Pk

c2 >0. ð9Þ

It then follows from(8) that, if x is a non-zero point with xTPAx P 0, then _ V <2 xT_PAxkA T_Pk c2 kx T_PB ik 2 6₂ xTPAxkA T_Pk c2 c 2_kxk2 6₀ for all Bi2 B. The conclusion of the theorem is hence provided. h

Remark 1. In general, the control gain a for the common stabilizers as deﬁned in Eq.(9)might not be easy to directly calculate. However, an alternative way for a may be determined by employing Routh–Hurwitz criteria (see, e.g.,[12]).

3.2. Existence of a matrix P satisfying condition(7)

According toTheorem 1, if one can find a symmetric positive definite matrix P which satisfies Condition

(7), a common stabilizer for the class of systems (A, Bi), Bi2 B, can then be determined. In this subsection, we

will derive conditions for the existence of such a matrix P. For this purpose, we deﬁne the terminology of nested subset of B.

A subset B1¼ fB1; . . . ; Bkg of B as deﬁned in (2) is said to be nested if it has the property:

Range(B1) Range(B2) Range(Bk). Under this condition, we say that B1corresponds to the worst

case (i.e., minimum number of actuators under operation) for all system pairs (A, Bi) with Bi2 B1.

First, consider the case in which the outage of actuators is conﬁned within a pre-speciﬁed set as considered by[1,6]. That is, the set B of[1,6]is nested. The existence of a P satisfying Condition(7)can be guaranteed by solving the algebraic Riccati equation (ARE) associated with the worst case of B, say B1, as given below:

ATPþ PA PB 1B

T

1 Pþ H ¼ 0 ð10Þ

for any given H > 0. Indeed, under this case,S_B_i_2BNðBT

iPÞ ¼ N ðB T 1 PÞ and, from(10), 2x T PAx =xT Hx < 0 for all x2 N ðBT

1 PÞ n f0g. This veriﬁes the existence of P that satisﬁes Condition(7)and thus the existence of

common stabilizers is guaranteed by Theorem 1. Note that the derived result agrees with that obtained by Veillette[1].

Next, consider the case in which the outage of actuators are not conﬁned within a pre-speciﬁed set. Moti-vated by the previous case, we divide B, as given by(2), into several nested subsets, say B1; . . . ; Bs. Denote Bj

the worst case of Bjfor 1 6 j 6 s. We can check that Condition(7)ofTheorem 1is equivalent to the following

condition:

xTPAx <0 for all x2[

s

j¼1

NðBT

j PÞ n f0g. ð11Þ

In addition, it is not diﬃcult to check that Condition (11)above is equivalent to Condition (12) below by letting x = Wy and W = P1:

yT_{AWy <}₀ _{for all y} ₂[ s

j¼1

NðBT

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Thus, the checking operation for the existence of a P satisfying Condition(7)can be simpliﬁed to proceed for those worst cases associated with each nested set only.

To obtain a matrix P which meets Condition (11), we can choose a matrix among all the worst cases B₁; . . . ; B_s, say B

1, having minimum rank. Let the rank of B1 be l. That is,

rankðB1Þ ¼ min_16j6srankðB

jÞ ¼ l. ð13Þ

Before proceeding the derivation of checking condition to provide relation(11)or (12), we present the next lemma.

Lemma 1. Suppose L2 Rnn _{is a symmetric matrix, M}_{2 R}nm _{and rank(M) = l. Then y}T

Ly < 0 for all y2 N(MT_{)n{0} if and only if (M}?₎T

LM?is a negative definite matrix, where M?is a n· (n l) matrix whose columns form an orthonormal basis for N(MT).

Proof. Note that, (M?)TLM?is a negative deﬁnite matrix if and only if vT(M?)TLM?v < 0 for every non-zero v2 Rnl_{. Moreover, the latter condition is equivalent to that u}T

Lu < 0 for every u = M?v2 N(MT_{)n{0}. The}

conclusion of the lemma is hence implied. h

Now, let L = AW + WAT with W = P1. The next result follows readily from Eq.(10)andLemma 1. Theorem 2. Consider the class of systems (A, Bi), Bi2 B. Suppose B₁satisfies the relation(13)and P = W1> 0

is the solution of Eq.(10). Then P is a matrix satisfying Condition(11)or(12)if and only if for each j = 1, . . . , s, ðBj

?_ÞT

ðAW þ WATÞBj

? _{is a negative definite matrix. Here, B} j

? _{denotes a matrix whose columns form an}

orthonormal basis for NðBj T_Þ.

For the case of which rankðB1Þ ¼ 1 and rankðB

jÞ ¼ 1 for some j 5 1, the checking condition(12)

corre-sponding to B_j as given in(14):

yTAWy <0 for all y2 N ðB j

T_Þ _ð14Þ

can be simpliﬁed by verifying the positivity of a scalar instead of checking negative deﬁniteness of the (n 1) · (n 1) matrix ðBj

?_ÞT

ðAW þ WATÞBj

? _{as given in}_{Theorem 2}_{above. Details are discussed as follows.}

Suppose A is not a Hurwitz matrix. From Eq.(10)and W = P1that AW + WAThas exactly one unstable eigenvalue. The unstable eigenvalue may be zero or a positive real number. If the unstable eigenvalue is zero, then yT_{AWy <}_{0 for all y}_{62 E}

0¼ fz j ðAW þ WATÞz ¼ 0g. Here, E0denotes the eigenspace of AW + WAT

asso-ciated with the zero eigenvalue. Thus, Condition(14)hold if and only if E06 N ðBTj Þ ¼ RðB jÞ

?

. On the other hand, if the unstable eigenvalue is a positive real number, an equivalent condition can be constructed. Details are summarized in the next corollary.

Corollary 1. Suppose rankðB

1Þ ¼ 1, rankðBjÞ ¼ 1 for some j 5 1 and P = W

1

> 0 denotes the solution of Eq.

(10). Let b be a non-zero column of Bj. Then the following two statements hold:

(i) If AW + WATpossesses a zero eigenvalue, then Condition (14)holds if and only if E06 RðBjÞ ?

. That is, bTv 5 0 for v2 E0n{0}.

(ii) If AW + WAThas a positive eigenvalue, then Condition(14)holds if and only if bTðAW þ WAT_Þ1

b >0. ð15Þ

Proof. Statement (i) has been discussed in the preceding paragraph ofCorollary 1. The proof of (ii) is given in

Appendix. h

To summarize the extended reliable design discussed above, a procedure for the construction of common stabilizers for system(1) can be listed as follows.

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Algorithm for ﬁnding common stabilizers

Step 1. Divide all the stabilizable system pairs into diﬀerent nested subsets B1; . . . ; Bs, and pick up one of

the worst cases, say B

12 B1, among those subsets.

Step 2. Attempt a reliable control design using the method of[1]. That is, given H > 0, solve for P in Eq.(10)

and check whether all the matrices A BiB i T

P, B_i 2 Bifor all i 5 1, are Hurwitz. If it fails to provide

the desired reliable properties with respective to outages outside the pre-speciﬁed set of actuators, then continue to Step 3. Otherwise, go to Step 4.

Step 3. Check the suﬃcient condition(11)or(12)by employingTheorem 2orCorollary 1, with P being the solution of the Riccati equation used in Step 2. If the condition holds, then a scaling of the feedback gain matrix from Step 2 is guaranteed to work and continue to Step 4. Otherwise, go back to Step 2 with the choice of another worst case.

Step 4. Use the Routh–Hurwitz stability criteria to determine an appropriate scaling a of the control gain from A aB

iB i

T_P

being Hurwitz for all i = 1, . . . , s.

Note that, if the above procedure fails to construct a common stabilizer, one might attempt to ﬁnd a new matrix P by the use of diﬀerent weighting matrices H in the Riccati equation(10).

4. Illustrative example

This section presents an example to determine the application of the main results as summarize in the pro-cedure above given in Section3. As given inExample 2, the existence of common stabilizers for all admissible faulty systems cannot be provided by using Veillett’s design[1]when both weighting matrices Q and R are identity matrices.

Example 2. Consider system(1) with

A¼ 1 2 1 0 0 0 0 1 1 0 B @ 1 C A and B ¼ 1 0:1 10 0:05 9 0:01 0 B @ 1 C A. ð16Þ

Let B1and B2be derived from B which correspond to the failure of the second and ﬁrst actuators, respectively.

That is, B1¼ 1 0 10 0 9 0 0 B @ 1 C A and B2¼ 0 0:1 0 0:05 0 0:01 0 B @ 1 C A. ð17Þ

It is easy to check that both (A, B1) and (A, B2) are stabilizable. This leads to B ¼ fB; B1; B2g, which is not

nested. Clearly, B contains two nested subsets B1¼ fB1; Bg and B2¼ fB2; Bg. The two worst cases associated

with B1 and B2 are B1¼ B1 and B2¼ B2, respectively.

According to Veillette’s method[1], the ﬁrst thing to do is to solve the following ARE: AT_M

iþ MiA MiBiR1BTiMiþ Q ¼ 0; Q >0 ð18Þ

for i = 1. Then, verify if the matrix A B2R1BTM1is stable. If it is not, redo this process for i = 2 and check

if the matrix A B1R1BTM2is stable. Unfortunately, the method proposed by Veillette does not work in this

example for both R and Q being the identity matrix. Indeed, for i = 1, the eigenvalues of A B2R1BTM1are

{0.921, 0.026,1.00}; and for i = 2, the eigenvalues of A B1R1BTM2are {0.1,1, 9.7 · 103}. Although

Veillette’s method might work for the construction of common stabilizers for this example by a suitable choice of weighting matrices Q and R, however, no guideline of choosing matrices Q and R has been proposed in[1]

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To employ the proposed methodology, we ﬁrst solve the ARE (10)for H being the identity matrix. The unique solution is calculated to be

P¼ 3:685 2:143 2:469 2:143 1:670 1:823 2:469 1:823 2:158 0 B @ 1 C A. ð19Þ

Then, by direct calculation, the index as given in(15)is bT(AW + WAT)1b = 6.639· 104> 0, where b is the non-zero column of B₂and W = P1. According toCorollary 1, matrix P = W1as in(19)satisﬁes Condition

(11). The common stabilizer can hence be obtained fromTheorem 1in the form of

u¼ a BT_Px _{for some a > 0.} _ð20Þ

By applying Routh–Hurwitz criteria on A a Æ B2BTP, this matrix is veriﬁed to be Hurwitz if a > 19.671. By

direct calculation, the eigenvalues of A a Æ BBTP, A a Æ B1BTP and A a Æ B2BTP with a = 25 are found to

be {392.522, 2.202, 1.076}, {392.276, 1.224, 0.902} and {0.128 ± 0.719j, 1.142}, respectively. These verify the reliable stabilization of the system.

Example 3. Consider system (1)with

A¼ 1 0 0 0 2 0 0 0 2 0 B @ 1 C A and B ¼ 1 0 0 0 1 0 1 0 1 0 B @ 1 C A. ð21Þ

Since rank(A 2I) = 1, if follows that the control system cannot be stabilizable if any two actuators fail. For cases in which only one actuator fails, it is easy to check that the control system is stabilizable when the ﬁrst or the third actuator fails. The system is not stabilizable if the second actuator fails, however. This means that there are two worst cases as deﬁned below:

B₁¼ 0 0 0 0 1 0 0 0 1 0 B @ 1 C A and B2¼ 1 0 0 0 1 0 1 0 0 0 B @ 1 C A. ð22Þ

The two worst cases are found not to be inclusive.

The solution of Eq.(10)with H being chosen to be the identity matrix is found to be

P¼ 0:5 0 0 0 4:236 0 0 0 4:236 0 B @ 1 C A. ð23Þ

By direct calculation, the matrixðB 1 ?_ÞT ðAW þ WAT_ÞB 1 ? _and_ðB 2 ?_ÞT ðAW þ WAT_ÞB 2

? _{are solved to be}_{4 and}

1.528, respectively, which are negative definite. According toTheorem 2, matrix P = W1as in(23)hence satisfies Condition(11).Theorem 1then implies the existence of common stabilizers. By applying Routh–Hurwitz criteria on A a Æ B2BTP, this matrix is verified to be Hurwitz if a > 0.62. With a = 1, the eigenvalues of A a Æ BBTP,

A a Æ B1BTP and A a Æ B2BTP are calculated to be {1.105, 2.236, 6.867}, {1, 2.236, 2.236} and

{0.367, 3.369, 2.236} respectively, which veriﬁes the reliable stabilization of the system.

5. Conclusions

In this paper, we has employed the Lyapunov approach to study the existence conditions of common sta-bilizers for linear control systems. The control systems considered in this paper result from an actual system where some actuators failed. The unique aspect of this study, compared with earlier studies, is that the possible outage of the actuators is not confined within a pre-specified set. In this paper, we have obtained a sufficient condition for the existence of common stabilizers and provided a procedure to implement such a condition.

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When the possible outage of actuators are confined within a pre-specified set, the obtained results agree with those previous findings[1].

Acknowledgement

This research was supported by the National Science Council, Taiwan, ROC under Grant NSC 92-2213-E-009-124 and NSC 91-2213-E-009-034.

Appendix. Proof of (ii) forCorollary 1

Suppose AW + WATis in diagonal form. Let AW þ WAT_{¼ diagðk}2 1;k

2

2; . . . ;k 2

nÞ. We now show that

Con-dition(15)implies Condition (14). Denote b = (b1, . . . , bn)T and y¼ ðy1; . . . ; ynÞ T 2 N ðB j T_{Þ n f0g. Condition} (15)then becomes Xn i¼2 b2_i=k2_i < b2₁=k2₁. ðA:1Þ

This implies that b150. From the structure of AW + WAT, if y1= 0, we then have yT(AW + WAT)y < 0. For

y150, bTy¼ b1y1þ

Pn

i¼2biyi¼ 0, which implies that

1 ¼X n i¼2 biyi b1y1 ¼X n i¼2 bi b1 k1 ki ki k1 y_i y₁ . ðA:2Þ

By employing Cauchy–Schwartz inequality from(A.2)and the inequality from(A.1), we have

1 6 X n i¼2 b2_ik21 b2₁k2_i ! Xn i¼2 k2_iy2 i k2₁y2 1 ! <X n i¼2 k2_iy2 i k2₁y2 1 ; ðA:3Þ

which leads toPn_i¼2k2_iy2 i >k 2 1y 2 1 and y T_{ðAW þ WA}T_{Þy ¼ k}2 1y 2 1 Pn i¼2k 2 iy 2 i <0.

Next, we show that Condition(14)implies (15)by contradiction. Suppose there exists a non-zero vector y2 N ðB

j

T_{Þ such that Condition}

(15)does not hold. Thus, we have b2₁=k2₁Pn_i¼2b2_i=k2_i 6_{0. This implies that} (b2, . . . , bn)

T

is a non-zero vector since b is a non-zero vector. Choose y¼ ðPn_i¼2b2_i=k_i2;b1b2=k22; . . . ;

b1bn=k 2 nÞ

T

. It is clear that y is a non-zero vector and y2 N ðBT

j Þ. That is, b T y = 0. By direct calculation, we have yT_{ðAW þ WA}T_{Þy ¼} k2₁X n i¼2 b2_i k2_i ! Xn i¼2 b2_i k2_i b2₁ k2₁ ! P0.

For the case of which AW + WATis not a diagonal matrix, a similarity transformation can be pre-applied to fulﬁll the proof. Since AW + WAT is a symmetric matrix, there exists an orthogonal matrix U such that UTðAW þ WAT_{ÞU ¼ diag ðk}2

1;k 2

2; . . . ;k 2

nÞ, where ki> 0 for all i = 1, . . . , n. Let z = Uy and D = diagðk21;

k2

2; . . . ;k 2

nÞ. It is clear that z T

Dz < 0 for all z2 N ððUB jÞ

T

Þ and Condition(15)becomes (Ub)TD1(Ub) > 0. The rest of the proof is similar to the one given above and is hence omitted. The conclusion ofCorollary 1is hence implied. h

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