814 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006

Applying Discrete-Time Proportional Integral Observersfor State and Disturbance Estimations

Jeang-Lin Chang

Abstract—In this note, we apply the proportional integral observer tosimultaneously estimate system states and unknown disturbances for dis-crete-time nonminimum phase systems. Conditions for providing the exis-tence of such observer are given. When the disturbances do not vary toomuch between two consecutive sampling instances, the proposed methodcan render the estimation errors of system states and disturbances to beconstrained in a small bounded region. Simulation results support the the-oretical developments.

Index Terms—Discrete-time, disturbance observer, nonminimum phase,state estimator.

I. INTRODUCTION

Proportional integral observer (PIO) to be devised using an addi-tionally introduced integral term of the output estimation error in ob-server design can offer certain degrees of freedom. Beale and Shafai [2]make use of this additional freedom in the observer-based controllerdesign, as a result of which PIO becomes less sensitive to parametervariation of the system. Niemann et al. [13] used the PIO to study theloop transfer recovery problem in continuous time systems and Shafaiet al. [16] studied the same problem in discrete time systems. More-over, Busawon and Kabore [3] have demonstrated that the PIO designcan effectively reduce the effect of measurement noises as opposed tothe proportional observer. Out of necessity, some authors extended thistechnique to design the observer for descriptor systems [7] or nonlinearsystems [1], [11]. In particular, Shafai et al. [17] applied the PIO in es-timating actuator and sensor faults. Since the PIO has two feedbackloops, a proportional loop and an integral loop of the output estimationerror, i.e., the PIO presented in this note allows us to estimate not onlythe states but also the unknown inputs.There are various formulations related to the unknown input esti-

mation. Based on the transfer function approach, the disturbance ob-server (DO) is known to be very effective to compensate disturbances[8] and is very popular for robust motion control [10], [14]. However,the transfer function approach cannot be used in multiple-input–mul-tiple-output (MIMO) cases, whereas the state space approach can beemployed. In the state space approach, specific disturbance modelsare augmented in the estimator model to estimate the disturbances [6].Hence, it is impossible to build models for disturbances, when the char-acteristics of the disturbance are unknown. To tackle this issue, theso-called unknown input observer (UIO) was developed to estimatesystem states [4], [5], [15]. For obtaining perfect estimation, the mainconstraint in DO design or in UIO method is that the system should beminimum phase (with respect to the relation between the output andthe disturbance). Upon application in a less restrictive sense, Mita etal. [12] proposed a technique of two-delay output control to overcomethis restriction in digital implementation. With proper design, Changet al. [4] combined this skill with UIO scheme to estimate the systemstates and the unknown disturbances in nonminimum phase systems.

Manuscript received March 18, 2004; revised October 5, 2005. Recom-mended by Associate Editor A. Garulli. This work was supported by the Na-tional Science Council, Taiwan, R.O.C., under Contract NSC 94-2213-161-002.The author is with the Department of Electrical Engineering, Oriental

Institute of Technology, Pan-Chiao, Taipei County 220, Taiwan (e-mail:jlchang@ee.oit.edu.tw).Digital Object Identifier 10.1109/TAC.2006.875019

Another notable form of perturbation observer is the time-delay con-trol (TDC) approach [18]–[20]. Under the assumptions that full statesare given and the disturbances do not vary too much between two con-secutive sampling instances, this method can obtain good performance.Since full system states are necessary for TDC approaches, it cannot bedirectly implemented in the output feedback systems.In this note, we attempt to use a PIO structure to simultaneously

estimate the system states and unknown disturbances in discrete-timesystems. Without utilizing the two-delay method, the proposed PIOcan be directly implemented in certain nonminimum phase systems,which do not have unstable zeros lying at one. Applying the analysistechnique in the TDC, it is demonstrated that the present algorithm indigital implementations can make both the state estimation errors andthe disturbance estimation errors at least to the size ofO T 2 , where Tis the sampling period. Especially for nonminimum phase systems, thePIO is effective not only in loop transfer recovery but also in estimatingunknown disturbances.This note is organized as follows. The problem formulation is given

in Section II. Section III presents the designs of a PIO. Section IV givesa numerical example to demonstrate the effectiveness of the proposedcontroller. Finally, Section V presents the concluding remarks.

II. PROBLEM FORMULATION

Consider a continuous MIMO linear system with unknown inputsdescribed by

_x_x_x(t) =HxHxHx(t) +DuDuDu(t) + FdFdFd(t)

yyy(t) =CxCxCx(t) (1)

where xxx 2 n are states of the system, uuu 2m are control inputs,

ddd 2 l are unknown inputs, and yyy 2 p are measurable outputs of thesystem. Suppose that the sampling interval is T and a zero-order-holdis adopted for the aforementioned continuous time model. Denotingxxx(k) = xxx(kT ), yyy(k) = yyy(kT ), uuu(k) = uuu(kT ), and ddd(k) = ddd(kT ),the discrete-time model can be then derived as

xxx(k + 1) =AxAxAx(k) +BuBuBu(k) +EdEdEd(k) +O T2

yyy(k) =CxCxCx(k) (2)

where AAA = exp (HHHT ) 2 n�n, BBB =T

0exp (HHH�)DDDd� 2 n�m,

and EEE =T

0exp (HHH�)FFFd� 2

n�l. Further, the unknown distur-bances are converted into

ddd(k) =T

0

eHHH�

ddd ((k+ 1)T � � )d� (3)

where themagnitude ofddd(k) is on the order ofO(T ) if the disturbancesare bounded and smooth. To design the state estimator and the distur-bance observer, the following assumptions are presumed.Assumption 1 [18], [19]: The sampling interval T is sufficiently

small such that the disturbances do not vary too much between twoconsecutive sampling instances. If the previous assumption holds, onecan obtain

ddd(k + 1)� ddd(k) 2 O T2 (4)

that holds for all k. Moreover, the following relations are effectivelyapproximated as:

O (Tn) +O Tn+1

� O (Tn) 8n 2 @ (5)

0018-9286/$20.00 © 2006 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006 815

where @ is the set consisting of all integers.Assumption 2: System (2) is observable and these matrices AAA, EEE,

and CCC satisfy

rankAAA � IIIn EEE

�CCC 0= n+ l: (6)

Although (2) contains unknown disturbances and may be nonmin-imum phase, the objective of this note is to design an observer suchthat the conditions xxx(k) ! xxx(k) and ddd(k) ! ddd(k) for k ! 1 aremet, where xxx(k) are the estimation states and ddd(k) are the estimationdisturbances.

III. STATE ESTIMATOR AND DISTURBANCE OBSERVER DESIGN

Design a PIO as

xxx(k + 1) =AxAxAx(k) +BuBuBu(k) +LLL1 (yyy(kkk)� yyy(kkk)) +EqEqEq(k)

qqq(k + 1) = qqq(k) +LLL2 (yyy(k)� yyy(k))

yyy(k) =CxCxCx(k) (7)

where LLL1 2 n�p and LLL2 2 l�p are matrices designed by the latter.Define the state estimation errors eee(k) = xxx(k)� xxx(k) and the outputestimation errors ~y~y~y(k) = yyy(k)� yyy(k) = CeCeCe(k). From (2) and (7), wehave

eee(k+ 1) = (AAA�LLL1CCC)eee(k) +EdEdEd(k)�EqEqEq(k) +O T2: (8)

Let �(k) = �qqq(k) + ddd(k) and the dynamics of �(k) can be obtainedas

�(k+ 1) = � qqq(k + 1) + ddd(k + 1)

= �(k)� (qqq(k + 1)� qqq(k)) + (ddd(k + 1)� ddd(k))

= �(k)�LLL2CeCeCe(k) + (ddd(k + 1)� ddd(k)) : (9)

Applying the Assumption 1 and augmenting (8) and (9) yields

eee(k+ 1)

�(k + 1)=

AAA�LLL1CCC EEE

�LLL2CCC III l

eee(k)

�(k)+

O T 2

ddd(k + 1)� ddd(k)

= (MMM �LGLGLG)eee(k)

�(k)+O T

2

~y~y~y(k) = [CCC 0 ]eee(k)

�(k)= GGG

eee(k)

�(k)(10)

whereMMM =AAA EEE

0 III l, LLL = [LLLT

1 LLLT2 ]T and GGG = [CCC 0 ]. From

(10), we know that, if (MMM;GGG) is observable, the matrixMMM �LGLGLG canbe stabilized.Lemma: If the Assumption 2 is satisfied, then the pair (MMM;GGG) is

observable.Proof: From linear theory, the pair (MMM;GGG) is observable if

rank�IIIn+l �MMM

GGG= n+ l 8 � 2 � (MMM)

where � (MMM) denotes the eigenvalues ofMMM . Since the eigenvalues ofMMM are equivalent to f� (AAA) ; 1g, we will discuss two cases (i) � 6= 1and (ii) � = 1 in the following.

i) When � 6= 1, we have�IIIn+l �MMM

GGG=

�IIIn �AAA �EEE

0 (�� 1) IIIlCCC 0

. Moreover

IIIn EEE (�� 1)�1 0

0 IIIl 0

0 0 IIIp

�IIIn �AAA �EEE

0 (�� 1) IIIlCCC 0

=

�IIIn �AAA 0

0 (�� 1) III lCCC 0

which is equivalent to

rank�IIIn+l �MMM

GGG= rank

�IIIn �AAA

CCC+ l = n+ l:

ii) When � = 1, we have�IIIn+l �MMM

GGG=

IIIn �AAA �EEE

0 0

CCC 0

. Hence, it follows that

rankIIIn+l �MMM

GGG= rank

AAA� IIIn EEE

�CCC 0= n+ l:

From i) and ii), we have rank�IIIn+l �MMM

GGG= n + l 8 � 2

� (MMM). This completes the proof of lemma.The following theorem summarizes the major results.Theorem: Consider the dynamic system (1) and its corresponding

discrete model can be described by (2). If the PIO is designed as (7),then it follows that

eee(k+ 1)

�(k+ 1)= (MMM �LGLGLG)

eee(k)

�(k)+O T

2: (11)

To simplify analysis, we neglect the termsO T 2 in the previous equa-tion. Thus, we have

eee(k + 1)

�(k+ 1)= (MMM �LGLGLG)

eee(k)

�(k): (12)

Since a matrix LLL from lemma 1 should be found such thatMMM �LGLGLG isstable, from (12) we can conclude eee(k)! 0 and �(k)! 0 as k !1.From eee(k) = xxx(k)� xxx(k) and �(k) = �qqq(k) + ddd(k), it follows that

a) xxx(k) ! xxx(k) as k ! 1;b) qqq(k) ! ddd(k) as k ! 1.

Since (11) holds accurately, whereas (12) holds approximately to theorder of O T 2 . Therefore, the above analysis using (12) will be onlyaccurate up to the order of O T 2 . Although the proposed observercannot procure the perfect estimation, it makes the estimation errors beconstrained in the small region of O T 2 .

IV. SIMULATION RESULTS

Consider the system (2) without the control input term as

xxx(k + 1) =AxAxAx(k) +EdEdEd(k)

yyy(k) =CxCxCx(k)

816 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006

where the system matrices are given as

AAA =

0:9630 0:0181 0:0187

0:1808 0:8195 �0:0514

�0:1116 0:0344 0:9586

EEE =

0:0996 0:0213

0:0050 0:1277

0:1510 0:0406

; CCC =1 0 �1

�1 1 1

ddd(t) =0:3 sin(0:1k) + 0:5 cos(0:03k)

0:2 cos(0:1k) + 2:

The triple (CCC;AAA;EEE) has an unstable zero at 1.2562; hence, it is non-minimum phase. By selecting poles at f0:1; 0:2; 0:3� 0:5j; 0:4g, weobtain the system matrices LLL1 and LLL2 in (7) as

LLL1 =

31:7392 19:6384

1:8918 1:7307

29:3767 18:9849

and LLL2 =51:1873 34:5803

�21:3249 �10:6399:

For comparison, the algorithm proposed by Mita’s [12] is used in thefollowing. First, an auxiliary output is employed as

zzz(k) = yyy(kT + iT )

where 0 < i < 1. By selecting i = 0:5, we obtain the followingequation for the auxiliary output:

zzz(k) =AAAmxxx(k) +EEEmddd(k)

whereAAAm =1:0382 �0:009 �0:97

�0:9432 0:9141 0:9422

andEEEm =�0:0278 �0:0098

0:0291 0:0766:

Applying Mita’s approach and choosing poles at {0.1,0.2,0.3}, theirstate estimator and disturbance observer can be designed as

xxx(k+1)= (AAA� FFF 1CACACAm � FFF 2CCC)xxx(k) + FFF 1zzz(k)+FFF 2yyy(k)

ddd(k)=(CECECEm)�1(zzz(k)�CACACAmxxx(k))+FFF 3(CxCxCx(k)�yyy(k))

where FFF 1 =

�3:7971 �0:2092

1:8045 1:8976

�5:6278 �0:1926

FFF 2 =

135:2655 85:9832

0:0240 �0:4253

135:2519 85:2840

and FFF 3 =�41:1458 �4:4916

15:2504 13:6308:

Under initial states xxx(0) = [ 0 1 0 ]T , xxx(0) = [ 0 0 0 ]T andqqq(0) = [ 0 0 ]T , Figs. 1–3 show the response of the errors of esti-mated states in both methods. Clearly, the system states are well esti-mated after k > 20. Hence, the proposed observer is capable of es-timating the states under disturbance consideration. The errors of es-timated disturbances for 1 � k � 25 are shown in Figs. 4 and 5.Figs. 6 and 7 illustrate the responses of the estimated disturbances andthe given disturbances for 50 � k � 200. As can be seen from thesefigures, our method can obtain better transient response in comparisonwith Mita’s method and the disturbance attenuation property of our al-gorithm is evident.

Fig. 1. Estimation error of xxx in both methods.

Fig. 2. Estimation error of xxx in both methods.

Fig. 3. Estimation error of xxx in both methods.

V. CONCLUSION

In this note, a full order PIO is proposed to estimate the system statesand unknown disturbances, for discrete-time nonminimum phase sys-tems (with respect to the relation between the output and the distur-bance). The structure and the estimation capacities of this proposed PIOare discussed and analyzed, and the conditions are given and proved.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006 817

Fig. 4. Estimation error of ddd (k) for 0 � k � 25 in both methods.

Fig. 5. Estimation error of ddd (k) for 0 � k � 25 in both methods.

Fig. 6. True and the estimation values of ddd (k) for 50 � k � 200.

Although a perfect estimation paradigm cannot be obtained, the pro-posed algorithm can effectively control the estimated errors of systemstates and disturbances to the size of O T

2 where T is the samplingperiod. Simulation results show that the presented scheme exhibits rea-sonably good estimations with the underlying system having an un-stable zero.

Fig. 7. True and the estimation values of ddd (k) for 50 � k � 200.

REFERENCES

[1] R. Aguilar-Lopez and R. Maya-Yescas, “State estimation for nonlinearsystems under model uncertainties: A class of sliding-mode observer,”J. Process Control, vol. 15, no. 3, pp. 363–370, 2005.

[2] S. Beale and B. Shafai, “Robust control design with a proportional inte-gral observer,” Int. J. Control, vol. 50, no. 1, pp. 97–111, 1989.

[3] K. K. Busawon and P. Kabore, “Disturbance attenuation using propor-tional integral observers,” Int. J. Control, vol. 74, no. 6, pp. 618–627,2001.

[4] S. K. Chang, W. T. You, and P. L. Hsu, “Design of general structureobservers for linear systems with unknown inputs,” J. Franklin Inst., vol.334, no. 2, pp. 213–232, 1997.

[5] M. Darouach, M. Zasadzinski, and S. J. Xu, “Full-order observers forlinear systems,” IEEETrans. Autom. Control, vol. 39, no. 3, pp. 606–609,Mar. 1994.

[6] G. Franklin, J. Powell, and M. Workman, Digital Control of DynamicsSystem. Reading, MA: Addison-Wesley, 1998.

[7] D. Koenig, “Unknown input proportional multiple-integer observer de-sign for linear descriptor system,” IEEE Trans. Autom. Control, vol. 50,no. 2, pp. 212–217, Feb. 2005.

[8] Y. Hori, “Robust motion control based on a two-degree-of-freedom ser-vosystem,” Adv. Robot., vol. 7, no. 6, pp. 363–368, 1993.

[9] M. Hou and P. C. Muller, “Design of observers for linear systemswith unknown inputs,” IEEE Trans. Autom. Control, vol. 37, no. 6, pp.871–875, Jun. 1992.

[10] C. J. Kempf and S. Kobayashi, “Disturbance observer and feedforwarddesign for a high-speed direct drive position table,” IEEE Trans. ControlTechnol., vol. 7, no. 5, pp. 513–526, May 1999.

[11] R. Martinez-Guerra and S. Diop, “Diagnosis of nonlinear system usingan unknown-input observer: Aan algebraic and differential approach,”Proc. Inst. Elect. Eng.—Control Theory Appl., vol. 151, no. 1, pp.130–135, 2004.

[12] T. Mita, Y. Chiba, Y. Kaku, and H. Numasato, “Two-delay robust digitalcontrol and its applications—Avoiding the problem on unstable limitingzeros,” IEEE Trans. Autom. Control, vol. 35, no. 8, pp. 962–970, Aug.1990.

[13] H. H. Niemann, J. L. Stoustrup, B. Shafai, and S. Beale, “LTR design ofproportional-integral observers,” Int. J. Robust Nonlinear Control, vol.5, pp. 671–693, 1995.

[14] K. Ohnishi, M. Shibata, and T. Murakami, “Motion control for advancedmechatronics,” IEEE/ASME Trans. Mechatron., vol. 1, no. 1, pp. 56–67,Jan. 1996.

[15] R. H. C. Takahashi and P. L. D. Peres, “Unknown input observer foruncertain systems: A unifying approach,” Eur. J. Control, vol. 5, no. 2,pp. 261–275, 1999.

[16] B. Shafai, S. Beale, H. H. Niemann, and J. L. Stoustrup, “LTR designfor discrete-time proportional-integral observers,” IEEE Trans. Autom.Control, vol. 41, no. 7, pp. 1056–1062, Jul. 1996.

[17] B. Shafai, C. T. Pi, and S. Nork, “Simultaneous disturbance attenuationand fault detection using proportional integral observers,” in Proc. Amer.Control Conf., Anchorage, AK, May 2002, pp. 1647–1649.

818 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006

[18] W. C. Su, S. V. Drakunov, and U. Ozguner, “AnO(T ) boundary layerin sliding mode for sampled–data systems,” IEEE Trans. Autom. Con-trol, vol. 45, no. 3, pp. 482–485, Mar. 2000.

[19] A. Tesfaye and M. Tomizuka, “Robust digital tracking with perturbationestimation via the Euler operator,” Int. J. Control, vol. 63, no. 1, pp.239–256, 1996.

[20] K. Youcef-Toumi and O. Ito, “A time delay controller with unknown dy-namics,” J. Dyna. Syst., Measure., Control, vol. 112, no. 1, pp. 133–142,1990.

Generalized Quadratic Stability for Continuous-TimeSingular Systems With Nonlinear Perturbation

Guoping Lu and Daniel W. C. Ho

Abstract—This note considers the generalized quadratic stabilityproblem for continuous-time singular system with nonlinear perturbation.The perturbation is a function of time and system state and satisfies aLipschitz constraint. In this work, a sufficient condition for the existenceand uniqueness of solution to the singular system is firstly presented.Then by using S-procedure and matrix inequality approach, a necessaryand sufficient condition is presented in terms of linear matrix inequality,under which the maximal perturbation bound is obtained to guarantee thegeneralized quadratic stability of the system. That is, the system remainsexponential stable and the nominal system is regular and impulse free.Furthermore, robust stability for nonsingular systems with perturbationcan be obtained as a special case. Finally, the effectiveness of the developedapproach for both singular and nonsingular systems is illustrated bynumerical examples.

Index Terms—Continuous-time singular system, generalized quadraticstability, linear matrix inequality, perturbation.

I. INTRODUCTION

Robust quadratic stability for linear systems with time-varying non-linear perturbation has received much attention, see [2], [6], [11], [14],and the references therein. The issue of robust quadratic stability forperturbed system is to find a tolerable perturbation bound such thatfor all admissible parameter perturbations, the system is stable and theassociated Lyapunov function is quadratic and deterministic. In [2],the tolerable bound is derived iteratively by adjusting a sequence ofLyapunov matrices. The bound is improved in [6]. In [14], the authorsconsider the problem of stability robustness with respect to a class ofnonlinear time-varying perturbations which are bounded in a compo-nent-wise rather than aggregated manner. It is shown that the boundis greater than those by [2] and [6]. In addition, by S-procedure, [11]

Manuscript received January 29, 2003; revised September 9, 2003, May 18,2004, April 8, 2005, and February 1, 2006. Recommended by Associate EditorL. Glielmo. This work was supported by the National Natural Science Foun-dation of China under Grants 60474076 and 60574006, by the Natural ScienceFoundation under Grant 04KJB510105 from the Jiangsu Provincial Departmentfor Education, and by the Hong Kong Research Grant Council under GrantsCityU 101103 and 101004.G. Lu is with the College of Electrical Engineering, Nantong University,

Jiangsu 226007, China (e-mail: lu.gp@ntu.edu.cn).D. W. C. Ho is with the Department of Mathematics, City University of Hong

Kong, Hong Kong (e-mail: madaniel@cityu.edu.hk).Digital Object Identifier 10.1109/TAC.2006.875017

discusses the robust stability for perturbed discrete-time systems, thetolerable perturbation bound is presented by means of a convex opti-mization.

Recently, there has been a growing interest in singular systems fortheir extensive application in control theory, circuits, economics, me-chanical systems and other areas, see [3]–[5], [7], [9], and [12]. One ofthe important issues for singular systems is robust stability. The robuststability for singular system is more complicated than that of nonsin-gular systems for not only asymptotic stability has to be considered, inaddition, the system regularity and impulse elimination are also neededto be addressed. In [5], robust stability for linear continuous-time sin-gular system with structured parameter perturbations is investigated,where the perturbations are represented by a time-invariant structureduncertain matrix. The matrix is bounded by its modulus matrix. In [3],the robust stability for the same system is considered, the presentedcriterion is shown to be less conservative than [5]. Motivated by theaforementioned robust stability results for both nonsingular and sin-gular systems, it is natural to ask whether or not we can get any robuststability result for continuous-time singular system with time-varyingnonlinear perturbations (CSSP), that is, a counterpart of [2], [6], [14],and the references therein. In other words, it is known that the general-ized quadratic stability implies the global asymptotic stability for lineartime-invariant singular systems. Does generalized quadratic stabilityimply the global asymptotic stability for this time-varying CSSP? Un-fortunately, there has not been any answer to this question so far. Oneof the difficulties lies in that the existence and uniqueness of the solu-tion for nonlinear singular systems is still an open problem and has notbeen fully investigated (see Remark 2.2). In addition, the standard Lya-punov stability theory cannot be applied to CSSP directly (see Remark2.3), and the open problems remains to be important and challenging.This note considers the robust stability for CSSP. Initially, the open

problem is addressed, and a sufficient condition of the existence anduniqueness of solution for CSSP is presented. A notion of generalizedquadratic stability is then introduced for CSSP. By using S-procedureand some matrix inequalities, it is shown that generalized quadraticstability for CSSP implies that the system remains exponentially stableand the nominal system is regular and impulse free. Hence, it followsthat a necessary and sufficient condition is obtained in terms of a convexoptimization, under which the maximal perturbation bound is obtainedto ensure generalized quadratic stability for CSSP. Furthermore, robuststability results for nonsingular systems with perturbation can be ob-tained as a special case. It is worth pointing out that the approach de-veloped in this note is different from those in the literature, see [2], [3],[5], [6], and [14], since the standard Lyapunov stability theory approachcannot be applied to CSSP (see Remark 2.3). The developed approachfor both singular and nonsingular systems in this note are also shownto be effective by some numerical examples.

A. Notation W T

Transpose of matrix W 2 Rn�m; kWk: [�max(WTW )]1=2, i.e.,the square root of the maximal eigenvalue ofW TW ;X�T : transposeof matrix X�1; I (Ir): identity matrix of appropriate dimensions (ofRr�r); kxk =

pxTx, kxk1 = maxfjxij; 1 � i � ng, where

x = (x1 x2 � � � xn )T 2 Rn; Throughout this note, for sym-

metric matrices X and Y , X � Y respectively, X > Y ): X � Y ispositive semi-definite (respectively, positive definite); X � Y respec-tively,X < Y ):X�Y is negative semidefinite (respectively, negativedefinite). Matrices, if not explicitly stated, are assumed to have com-patible dimensions.

0018-9286/$20.00 © 2006 IEEE