DURING THE last two decades, many authors have

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1 614 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 44, NO 7, JULY 1997 Stability the Lyapunov Equation for -Dimensional Digital Systems Chengshan Xiao, David J Hill, Fellow, IEEE, Pan Agathoklis, Senior Member, IEEE Abstract The discrete-time bounded-real lemma for nonminimal discrete systems is presented Based on this lemma, rigorous necessary sufficient conditions for the existence of positive definite solutions to the Lyapunov equation for n-dimensional (n-d) digital systems are proposed These new conditions can be applied to n-d digital systems with n-d characteristic polynomials involving factor polynomials of any dimension, 1-D to n-d Further, the results in this paper show that the positive definite solutions to the n-d Lyapunov equation of an n-d system with characteristic polynomial involving 1-D factors can be obtained from the solutions of a k-d (0 k n) subsystem m (1 m n) 1-D subsystems This could significantly simplify the complexity of solving the n-d Lyapunov equation for such cases Index Terms Lyapunov equation, multidimensional systems, stability I INTRODUCTION DURING THE last two decades, many authors have studied stability the Lyapunov equation for - dimensional ( -D) systems (see [1] [10] the references therein) The stability of -D systems is essential for the design implementation of such systems the -D Lyapunov equation has many useful applications for -D systems in state-space description [3], [4], [8], [10] The relationship between stability the Lyapunov equation for -dimensional ( -D) digital systems described by a state-space model is very important for the stability analysis, design finite wordlength implementation of such systems It is well known that the existence of positive definite matrices satisfying the -D Lyapunov equation is sufficient for stability of -D systems but not necessary, as it was shown with an example in [6] for the 2-D case Further, in [6] it was shown that necessary sufficient conditions for the existence of positive definite solutions for the 2-D Lyapunov equation can be obtained from the discrete-time strictly bounded real lemma This lemma formulates the strictly bounded real Manuscript received October 25, 1994; revised September 1, 1995 January 1, 1996 The work of C Xiao was supported by the National Natural Science Foundation of China Australian OPRS UPRA at the University of Sydney The work of P Agathoklis was supported by the NSERC Micronet This paper was recommended by Associate Editor H Reddy C Xiao was with the Department of Electrical Engineering, University of Sydney, NSW 2006, Australia He is now with Nortel, Ottawa, Ont K1Y 4H7 Canada ( cxiao@nortelca) D J Hill is with the Department of Electrical Engineering, University of Sydney, NSW 2006, Australia ( davidh@eeusydeduau) P Agathoklis is with the Department of Electrical Computer Engineering, University of Victoria, British Columbia, V8W 3P6 Canada ( pan@eceuvicca) Publisher Item Identifier S (97) property of a minimal realization of a transfer matrix leads, therefore, to conditions involving the requirement of reachability observability of where is the system matrix of the 2-D system Similar results were also established for -D systems in [7] If the reachability /or observability conditions are not satisfied, the results of [6] [7] can not be applied directly For the 2-D case, loss of reachability /or observability means that the characteristic polynomial of the 2-D system contains one (or two) 1-D polynomials It was shown in [6] that in such a case the original 2-D system can be decomposed into two subsystems; a 1-D subsystem a 2-D subsystem which satisfies both the reachability observability conditions Then the theorem in [6] can be used for the Lyapunov equation of the 2-D subsystem The question whether a positive definite solution to the Lyapunov equation of the original 2-D system does exist how it can be found was left open in [6] It was only mentioned that such a solution may not necessarily exist even if positive definite solutions of the Lyapunov equations for both the 1-D the 2-D subsystems exist 2-D systems described with a system matrix, (1), with the reachability /or observability conditions not satisfied, have characteristic polynomials involving 1-D factors, as mentioned above It is interesting to note that this is a more general case than the so-called separable denominator 2-D systems, ie, systems with a characteristic polynomial which is a product of two 1-D polynomials In state space description, such systems are usually defined as systems with a system matrix where either or is zero [12] [14] It is easy to see that this is only a sufficient but not necessary condition The necessary sufficient condition for the 2-D characteristic polynomial of to be product of two 1-D polynomials is that all the reachable parts in [or ] are unobservable in [or ] This follows easily from the discussions in [6] [11] If some of the reachable parts in [or ] are observable in [or ], then the 2-D characteristic polynomial of is a product of a 2-D a 1-D polynomials The relationship between these polynomials the system matrix is discussed in [6] These observations for 2-D systems can be extended to -D systems having characteristic polynomials which are products of -D polynomials with The interest in such systems is due to the fact (1) /97$ IEEE

2 XIAO et al: STABILITY AND THE LYAPUNOV EQUATION 615 that many symmetries in the frequency response specifications lead to -D digital filter designs with -D characteristic polynomials which include -D factors See [15] for discussions further references In this paper the strictly bounded real lemma for nonminimal realizations of a transfer matrix is presented this lemma is used to formulate new necessary sufficient conditions for the existence of positive definite solutions to the -D Lyapunov equation These conditions do not require reachability observability thus can be applied to a much larger class of -D (including 2-D) digital systems than earlier conditions in [6] [7] They can be applied to -D systems which have characteristic polynomials which involve 1-D polynomial factors It is shown in this paper that such systems can be decomposed into one -D subsystem 1-D subsystems Further, the new conditions can be used to show that if positive definite solutions for the 1-D -D subsystems exist, then positive definite solutions for the original system will also exist An algorithm for finding such solutions based on the positive definite solutions of the subsystems is also presented, indicating that the proposed technique simplifies the solution of the -D Lyapunov equation for such cases The paper is organized as follows In Section II the discretetime strictly bounded real lemma (DTSBRL) for nonminimal realizations of 1-D digital transfer functions is developed In Section III the new necessary sufficient condition for the existence of positive definite solutions to the -D Lyapunov equation are presented This new condition is based on the formulation of the DTSBRL in Section II for nonminimal realizations the implications of these new conditions are discussed In Section IV an example is considered to illustrate the theoretical results II THE DTSBRL FOR NONMINIMAL SYSTEMS Consider a square real rational transfer matrix of a -input -output 1-D digital system Strictly bounded realness is defined as follows [6], [9]: Definition 1: Let be a square real rational transfer matrix Then is called strictly bounded real if only if 1) all poles of each entry of lie in 2) is positive definite for all Let the quadruple be a general (nonminimal) realization of such that given by is positive definite Remark 1: This lemma is an extended version of Lemma 1 in [6] for general realizations including minimal nonminimal realizations of The sufficient part of this lemma for nonminimal realizations was stated by a remark of [6] without proof The proof of both the sufficient the necessary part of this lemma are rather lengthy are included in the Appendix III THE -D LYAPUNOV CONDITION Let us now propose the -D Lyapunov condition compare it with the condition for -D stability based on the strictly bounded real lemma presented in the previous section Consider a linear shift invariant -D digital system described by Roesser s state-space model of the following form [7]: where are the current state variables in the th block of are the inputs outputs, respectively, with The stability of an -D system described by (4) (5) depends on the locations of the zeros of the characteristic polynomial (3) (4) (5) (2) then the discrete-time strictly bounded real lemma can be given: Lemma 1: Let be a square real rational transfer matrix a realization of with the spectral radius of being smaller than unit, ie, Then is strictly bounded real if only if there exists a symmetric positive definite matrix such that the matrix (6) The -D stability condition requires that the characteristic polynomial has no zeros inside the closed -disk, ie, in (7)

3 616 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 44, NO 7, JULY 1997 where To proceed further in a compact way, we define some matrices first Let be a submatrix obtained by deleting the th block row th block column of be the th block column of with deleting be the th block row of with deleting, for In the case of, they can be expressed as follows: (8) Remark 2: This theorem is an extended version of Theorem 2 in [7] ( Theorem 1 in [6] for 2-D case) In [7] ( [6]), the reachability observability of respectively is required This is a stronger condition than thus the new condition can be used for a larger class of -D systems Proof: The proof of this theorem is an extension of those in [6] [7], details are given below: Necessity: If is nonsingular, is strictly bounded real, then from Lemma 1 follows that there exists a positive definite matrix such that the matrix in (15) shown at the bottom of the page is positive definite Premultiplication of (15) with postmultiplication with yields (9) Let be a matrix as (16) (10) When or, the first or last block row block column are deleted it is easy to express the resulting matrices in a similar way as in (8) (10) Now let (11) (12) where It is straightforward to generate the above matrices for any any For example, in the 2-D case this would lead to two sets of matrices for to two similar 2-D Lyapunov conditions one with respect to one with respect to The new -D Lyapunov condition can now be given with the following theorem Theorem 1: Consider an -D digital system described by (4) For some nonsingular, see (12), given by (13) is strictly bounded real if only if there exist some positive definite matrices, such that satisfy the -D Lyapunov equation (14) Noting that premultiplication with which is the (17) (8) (12), (16) gives after postmultiplication with -D Lyapunov equation with (18) (19) (20) being positive definite Sufficiency: Suppose (14) is satisfied Premultiplying (14) by postmultiplying by gives (21) with Let,, then premultiplying (21) by postmultiplying by, we can obtain that is strictly bounded real from Lemma 1 (15)

4 XIAO et al: STABILITY AND THE LYAPUNOV EQUATION 617 Theorem 2: Consider an -D digital system described by (4) Then, given by (22) is strictly bounded real for some nonsingular, see (12), if only if there exist some nonsingular, see (12), so that given by (13) is strictly bounded real Proof: Both the sufficiency necessity can be easily established by Theorem 1 Theorems 1 2 present the rigorous necessary sufficient conditions for the -D Lyapunov equation they do not involve reachability observability conditions It was shown in [6] that this means that the characteristic polynomial of the -D system contains 1-D factors In the rest of this section two theorems will be presented which will show that if an -D system does not satisfy the reachability /or observability conditions in some dimensions, then the original -D system can be decomposed into subsystems, one is an irreducible -D subsystem, others are 1-D subsystems The original -D system has a positive definite solution to the -D Lyapunov equation if only if all the 1-D subsystems are stable the irreducible -D subsystem has a positive definite solution to the -D Lyapunov equation These results answer the questions left open in [6] [7] also clarify some inaccurate statements made there Further, based on Theorems 3 4 a simple algorithm is developed to obtain the positive definite solution to the -D Lyapunov equation based on the positive definite solutions of the 1-D the -D systems This simplifies the complexity of solving the -D Lyapunov equation when the -D system does not satisfy the reachability /or observability conditions Theorem 3: If is not reachable /or is not observable, there always exists a nonsingular matrix (23) to decompose the original -D system into subsystems One of them is a -D subsystem which satisfies the reachability observability conditions in those dimensions, others are 1-D subsystems in the dimensions To prove the above theorem, the following proposition will be used Proposition 1: If is not reachable /or is not observable, there always exists a nonsingular matrix (24) to decompose the original -D system into two subsystems One is a 1-D subsystem the other is an -D subsystem or ( -1)-D subsystem Proof: Set, then has the form as (11) Based on the results of [11, pp ], we have the following i) If is not reachable, then there always exists a nonsingular matrix ii) such that (25) (26) where is reachable, or is null is not observable, then there always exists a nonsingular matrix given by (25) such that (27) where is observable, or is null iii) If is not reachable is not observable, then there always exists a nonsingular matrix given by (25) such that (28) where is reachable is observable, or is null Transfer back to, we obtain the desired result immediately Theorem 3 can now be proven based on Proposition 1 as follows Step 1: Apply given by (25) to decompose the original -D system into two subsystems, one is a reduced 1-D subsystem, the other is a reduced -D or ( - 1)-D subsystem which system matrix is (29) Step 2: For the reduced -D or ( -1)-D subsystem,if the reachability observability conditions are not satisfied in one (or more) dimension, then we can decompose this subsystem into two subsubsystems, one is a 1-D subsystem, the other is a -D,or subsystem Step 3: Repeat Steps 1 2 until the reduced -D subsystem can not be decomposed any more (say irreducible -D subsystem) The proof of Theorem 3 is then completed

5 618 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 44, NO 7, JULY 1997 Theorem 4: If is not reachable /or is not observable, then the positive definite solutions to the Lyapunov equations for the reduced -D subsystem the reduced 1-D subsystems exist, if only if some positive definite solutions to the -D Lyapunov equation for the original -D system exist To prove the above theorem, the following proposition will be used Proposition 2: If is not reachable /or is not observable, then positive definite solutions to the Lyapunov equations for both the reduced 1-D the reduced -D or ( -1)-D subsystems exist, if only if some positive definite solutions to the -D Lyapunov equation for the original -D system exist Proof: Set again given by (11) (17) Consider the following cases: i) If is not reachable, is observable Sufficiency: Suppose (14) is satisfied We have which is the same as (21) with As we know, there exists a nonsingular matrix by (25) to transfer to given by (26) (30) given (31) with Premultiplying postmultiplying (30) by, respectively, yields The top left submatrix of is given to be (32) principal (33) According to any principle submatrix of a positive definite matrix is positive definite [16, p 397] together with (25), (26), (32), we can conclude that there exist some positive definite matrices to satisfy the -D Lyapunov equation for the reduced -D system which satisfies the reachability observability conditions Furthermore, from (14) (26), we have, therefore, there always exists a positive definite matrix such that is positive definite [16, p 410] (34) Necessity: Suppose (33) (34) are satisfied, this means that the positive definite solutions to the Lyapunov equations for both the reduced 1-D the reduced -D subsystems exist Let with given by (25) From (26) (30), we have with (35) If is large enough, given by (36) is positive definite (36) Based on being positive definite [16, Theorem 776], we conclude that given by (35) is positive definite, thus given by (14) is positive definite if is large enough Similarly, we can prove this theorem: ii) if is reachable is not observable, iii) if is not reachable is not observable At this stage, it is not hard to prove Theorem 4 based on Theorem 3 Proposition 2 Details are omitted here for brevity For completeness the lemma proposed in [7] for comparing the stability condition (7) with the condition for the existence of positive definite solutions to the -D Lyapunov equation, can be extended to: Lemma 2: Consider an -D digital system described by (4) If, given by (13), is strictly bounded real, then the characteristic polynomial of system (4) has no zeros in the unit -disk Proof: It is easy to prove this lemma based on Lemma 1 of [7], Propositions 1 2 in this paper Combining Theorem 1 Lemma 2, we obtain the following result Corollary 1: Consider an -D digital system described by (4), if there exist some positive definite solutions to the -D Lyapunov equation (14), then the characteristic polynomial of system (4) has no zeros in the unit -disk IV ILLUSTRATIVE EXAMPLE To demonstrate the applicability of the present results compare them with previous results, we now consider a 4-D (3, 2, 2, 3)th-order digital systems with given by (37) at the bottom of the next page It is easy to check that there is not any state in dimension, which satisfies both the reachability observability conditions thus, we can treat the 3-D subsystem separately For the subsystem, the third state is unreachable in dimension, the second state in dimension is unreachable the first state in dimension is unobservable Therefore, we can get an irreducible 3-D (2,

6 XIAO et al: STABILITY AND THE LYAPUNOV EQUATION 619 1, 2)th-order subsystem from (38) Note that all the unreachable /or unobservable states of are stable, therefore, we only need to test the stability of the 3-D (2, 1, 2)th-order subsystem Using the method of [10], we obtain a positive definite matrix such that (39) (40) is positive definite Hence the 4-D system given by (37) is stable Moreover, based on given by (39), adjusting (see the proofs of Proposition 2 Lemma 1), we obtain a positive definite matrix (41) V CONCLUSION In this paper, the discrete-time strictly bounded-real lemma for nonminimal realizations of 1-D digital systems is presented Based on this lemma, new conditions for the existence of positive definite solutions to the -D Lyapunov equation have been presented the relationship between stability these new conditions is established These new conditions do not involve reachability observability requirements thus can be applied to -D system which have characteristic polynomials with 1-D factor polynomials Theorem 2 shows that if there are no nonsingular matrices, given by (12), so that, given by (13), is strictly bounded real, then it is impossible for, given by (22),, to be strictly bounded real thus, no positive definite solutions to the - D Lyapunov equation exist Further, in Theorems 3 4 it is shown that the positive definite solutions to the -D Lyapunov equation can be obtained from the positive definite solutions of the Lyapunov equations for an irreducible -D subsystem 1-D subsystems These results could significantly simplify the complexity of solving the -D Lyapunov equation for such cases The applications of them to get improved lower bounds for the stability margin of -D digital systems, to get tighter lower bounds for coefficient wordlengths for finite wordlength implementations of -D systems with guaranteed stability were presented in [17] Moreover, the application of the new results to present the discrete-time lossless bounded real lemma for digital systems is currently under preparation -dimensional APPENDIX PROOF OF LEMMA 1 Proof: If is a minimal realization of, then this Lemma is identical to Lemma 1 of [6] Therefore, we consider that is a nonminimal realization of, which has three cases as follows: i) is not reachable is observable In this case, there always exists a nonsingular matrix such that [11, p 130] (A1) such that is positive definite (A2) (37)

7 620 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL 44, NO 7, JULY 1997 Let with is reachable Based on (A1) (A3) together with (2), given (A3) can be (A4) (A10) then (A10) gives after some manipulations together with (A1) (A3) (see A11) at the bottom of the page) There is always a large enough, so that given by (A12) is positive definite Sufficiency: There exists a positive definite matrix such that given by (3) is positive definite Premultiplying postmultiplying (3) by, respectively, yields where (A12) (A13) (A5) where Equation (A5) gives after some algebraic manipulations (A6) Based on being positive definite Theorem 776 of [16], we can conclude that given by (A11) is positive definite, hence given by (A10) is positive if is large enough ii) is reachable is not observable In this case there always exist [11, p 132] a nonsingular matrix such that (A14) where is the top left principal submatrix of denotes any matrix Using the fact that any principle submatrix of a positive definite matrix is positive definite [16, p 397], then there exists a positive definite such that the matrix (A15) (A16) (A7) is positive definite Therefore, is strictly bounded real [6] Since, we have Necessity: is strictly bounded real There exist two positive definite matrices such that the matrices with is observable The proof in this case is essentially as case i) iii) is not reachable is not observable In this case there always exist [11, p 132] a nonsingular matrix such that (A17) are positive definite (A8) (A9) (A18) with is reachable is observable Based on the discussions of i) ii), it is not difficult to prove this case At this stage, the desired result is completely proven (A11)

8 XIAO et al: STABILITY AND THE LYAPUNOV EQUATION 621 REFERENCES [1] E I Jury, Stability of multidimensional systems other related problems, in Multidimensional Systems: Techniques Applications, S G Tzafestas, Ed New York: Marcel Dekker, 1986 [2] M S Piekarski, Algebraic characterization of matrices whose multivariable characteristic polynomial is Hurwitzian, in Proc Int Symp Operator Theory, Lubbock, TX, 1977, pp [3] N G El-Agizi M M Fahmy, Two-dimensional digital filters with no overflow oscillations, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-27, pp , 1979 [4] J H Lodge M M Fahmy, Stability overflow oscillations in 2-D state-space digital filters, IEEE Trans Acoust, Speech, Signal Processing, vol ASSP-29, pp , 1981 [5] W-S Lu E B Lee, Stability analysis for two-dimensional systems via a Lyapunov approach, IEEE Trans Circuits Syst, vol CAS-32, pp p 1196, 1985 [6] B D O Anderson, P Agathoklis, E I Jury, M Mansour, Stability the matrix Lyapunov equation for discrete 2-dimensional systems, IEEE TransCircuits Syst, vol CAS-33, pp , 1986 [7] P Agathoklis, The Lyapunov equation for n-dimensional discrete systems, IEEE Trans Circuits Syst, vol 35, pp , 1988 [8], Lower bounds for the stability margin of discrete twodimensional systems based on the two-dimensional Lyapunov equation, IEEE Trans Circuits Syst, vol 35, pp , 1988 [9] P Agathoklis, E I Jury, M Mansour, The discrete-time strictly bounded-real lemma the computation of positive definite solutions to the 2-D Lyapunov equation, IEEE Trans Circuits Syst, vol 36, pp , 1989 [10] D Liu A N Michel, Stability analysis of state-space realizations for two-dimensional filters with overflow nonlinearities, IEEE Trans Circuits Syst I, vol 41, pp , 1994 [11] T Kailath, Linear Systems Englewood Cliffs, NJ: Prentice-Hall, 1980 [12] S-Y Kung, B C Levy, M Morf, T Kailath, New results in 2- D systems theory, part II: 2-D state-space models-realization the notions of controllability, observability, minimality, Proc IEEE, vol 65, pp , 1977 [13] T Hinamoto S Maekawa, Canonic form state space realization of two dimensional transfer functions having separable denominator, Int J Syst Sci, vol 13, pp , 1982 [14] M Kawamata T Higuchi, Synthesis of 2-D separable denominator digital filters with minimum roundoff noise no overflow oscillations, IEEE Trans Circuits Syst, vol CAS-33, pp , 1986 [15] V Rajaravivarma, P K Rajan, H C Reddy, Planar symmetries in 3-D filter response their application in 3-D filter design, IEEE Trans Circuits Syst II, vol 39, pp , 1992 [16] R A Horn C R Johnson, Matrix Analysis Cambridge, UK: Cambridge Univ Press, 1985 [17] C Xiao D J Hill, Stability results for decomposable multidimensional digital systems based on the Lyapunov equation, Multidimen Syst Signal Processing, vol 7, pp , 1996 David J Hill (S 72 M 76 SM 91 F 93 ) received the BE degree in electrical engineering BSc degree in mathematics from the University of Queensl, Australia, in , respectively In 1976 he received the PhD degree in electrical engineering from the University of Newcastle, Australia Since 1994, he has held the Chair in Electrical Engineering at the University of Sydney Previous appointments include research positions in the Electronics Research Laboratory, University of California Berkeley, from 1978 to 1980 a Queen Elizabeth II Research Fellowship with the Department of Electrical Computer Engineering, University of Newcastle from 1980 to 1982 During 1986 he was a Guest Professor with the Department of Automatic Control, Lund Institute of Technology, Sweden From 1982 to 1993 he held various academic positions at the University of Newcastle, including Assistant Director of the Centre for Industrial Control Science during , Pacific Power Chair was Assistant Dean of Engineering (Graduate Studies) from 1991 to 1993 His research interests are mainly in nonlinear systems control, stability theory power system dynamics security His recent applied work consists of various projects in power system stabilization power plant control carried out in collaboration with utilities in Australia Sweden Dr Hill is a Communicating Editor for the Journal of Mathematical Systems, the Estimation Control Guest Editor for the PROCEEDINGS OF THE IEEE Special Issue on Nonlinear Phenomena in Power Systems He is a Fellow of the IEAust Pan Agathoklis (M 81 SM 89) received the Dipl Ing degree in electrical engineering the Dr Sc Tech degree from the Swiss Federal Institute of Technology, Zurich, Switzerl, in , respectively From 1981 until 1983, he was with the University of Calgary as a Post-Doctoral Fellow part-time Instructor Since 1983, he has been with the Department of Electrical Computer Engineering, University of Victoria, BC, Canada, where he is currently a Professor His fields of interest are in digital signal processing, control theory, stability robustness analysis, image processing, scientific visualization He has received a NSERC University Research Fellowship ( ) Visiting Fellowships from the Swiss Federal Institute of Technology (1982, 1984, 1986, 1993) from the Australian National University (1987) Dr Agathoklis was an Associate Editor for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1990 to 1993 has served in Grant Selection Committees for NSERC BC Science Council He has been a member of the Technical Program Committee of many international conferences has served as the program Chair of the 1991 IEEE Pacific Rim Conference on Communications, Computers Signal Processing Chengshan Xiao received the BE degree in electronic engineering from the University of Electronic Science Technology of China (UESTC), Chengdu, China, in 1987, the ME degree in electronic engineering from Tsinghua University, Beijing, China, in 1989, PhD degree in electrical engineering from the University of Sydney, Australia, in 1997 From 1989 to 1993, he was a Research Assistant, then a Lecturer with Tsinghua University He held a few visiting positions with Australian Canadian Universities from 1993 to 1996 He was conducting research work at the University of Toronto, Canada, from 1996 to 1997 He is now a member of scientific staff with the Wireless Communication Department, Nortel, Ottawa, Canada His research interests include multidimensional system theory filter design, image signal processing, wireless communications nonlinear control He had more than 20 papers published or accepted by international journals during his three-year PhD program Dr Xiao received the Best Bachelor s Thesis Award from UESTC in 1987, the Guanghua Fellowship in 1989 the Excellent Young Teacher Award in 1991, both from Tsinghua University He received Australian OPRS, UPRA NIP scholarships from 1993 to 1996 He also received a Canadian NSERC Postdoctoral Fellowship in 1997