[ 1. Stability and the Matrix Lyapunov Equation for Discrete 2-Dimensional Systems

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1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 3, MARCH Stability the Matrix Lyapunov Equation for Discrete 2-Dimensional Systems BRIAN D. 0. ANDERSON, FELLOW, IEEE, PANAJOTIS AGATHOKLIS, MEMBER, IEEE, E. I. JURY, FELLOW, IEEE, AND M. MANSOUR, FELLOW, IEEE Abstract -The stability of two-dimensional, linear, discrete systems is examined using the 2-D matrix Lyapunov equation. While the existence of a positive definite solution pair to the 2-D Lyapunov equation is sufficient for stability, the paper proves that such existence is not necessary for stability, disproving a long-sting conjecture. T I. INTRODUCTION HE introduction of state-space models for 2-D discrete systems allows the investigation of stability using the state-space approach. The question then arises as to how one of the stard -D linear stability tools, the Lyapunov matrix equation, could be extended to the 2-D case. One would hope that the extension of the Lyapunov equation to the 2-D case would give necessary sufficient conditions for the characteristic polynomial of a linear discrete 2-D system to be void of zeros in the unit bidisc in terms of properties of the solutions to the 2-D Lyapunov equation. There are essentially two different approaches to this problem. The first consists of developing a 2-D Lyapunov equation with constant coefficients [l], [2], while the second approach is considering a -D Lyapunov equation with coefficients which are functions in a complex variable [3], [4]. In [l], the formulation of the Lyapunov equation for n-dimensional continuous systems was first considered. This formulation is extended to discrete systems in [2] using the multi-dimensional bilinear transformation. Fur- thermore, it has been ass.erted that the existence of positive definite matrices satisfying the 2-D Lyapunov equation is necessary sufficient for the 2-D characteristic polynomial to be void of zeros in the unit bidisc. Although the sufficiency part of this assertion can be easily proven, the nedessity part has not been satisfactorily established yet. The aim of this paper is to establish a necessary sufficient condition for the existence of positive definite solutions of the 2-D Lyapunov equation. This condition is developed based on properties of strictly bounded real matrices. Furthermore, it is shown that the developed condition is stronger than the condition that the characteristic polynomial has no zeros in the unit bidisc. This is demonstrated with an example of a stable 2-D system for which no positive definite solutions of the 2-D Lyapunov equation can be found. The paper is organized as follows: In Section II the 2-D discrete state-space model is introduced some properties of strictly bounded real matrices are briefly discussed. In Section III the necessary sufficient condition for the existence of positive definite solutions of the 2-D Lyapunov equation is developed, compared with the condition that the characteristic polynomial has no zeros in the unit bidisc. Finally, in Section IV the result of Section III is applied to two special classes of 2-D systems. Notation: o2 denotes the closed unit bidisc. u2: {h~2)ll~ll Gl, 22 d}, the n X n unity the direct sum of matrices. II. PRELIMINARIES Linear shift invariant 2-D discrete systems can be represented by the following state space model [5]: Manuscript received October 4, 985. This work was supported by NSERC NSF under Grant ECS B. D. 0. Anderson is with the Department of Systems Engineering, Research School of Physical Sciences, Australian National University, Canberra, ACT 260!, Australia. P. Agathoklis is with the Department of Electrical Engineering, University of Victoria, Victoria, B. C., Canada V8W 2Y2. E. I. Jury is with the Department of Electrical Engineering, University of Miami, Coral Gables, FL M. Mansour is with the Institute of Automatic Control Industrial Electronics, Swiss Federal Institute of Technology, 8039 Zurich, Switzerl. IEEE Log Number %4094/86/ $ IEEE where x EIW, x ER~ represent the horizontal vertical states, respectively, II is the input y the output. The system matrix A is given by A= A [ 4, Al2 2 A22

2 262 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 3, MARCH 986 with the submatrices A,,, i, j =,2 of appropriate dimensions. In the investigation of stability of a 2-D discrete system realized with a model of the above type, it is required to establish that the zeros of the characteristic polynomial are outside i?j2, i.e., C( z,, z2) = det 4 - ZlAll - Z42 - Z2A2, I, - Z2A22 I #O inu2 (4) Although it is possible to test the stability condition (4) in the frequency domain using one of the known methods for testing stability of 2-D polynomials [6], the 2-D Lyapunov equation could be an important tool in the stability analysis in the design of 2-D systems in the state-space. The n-d Lyapunov equation was first presented in [l] as a condition for the multivariable characteristic polynomial of an n-d continuous system to be strictly Hurwitzian (i.e., to have no zeros in the region Re(s,) > 0, i =,.. *, n). This was extended to the discrete case using the double bilinear transformation in [2]. The,resulting discrete 2-D Lyapunov condition is as follows: 2-D Lyapunov condition: There exist matrices W,(n X n), W,(m X m), W= Wi@W2 Q symmetric positive definite such that ATWA-W=-Q (9 is satisfied. It has been asserted [2], [l] that the above 2-D Lyapunov condition is necessary sufficient for the stability condition (4). In the next section it will be shown that the 2-D Lyapunov condition is in general only sufficient for the stability condition (4) not necessary. To show this the following definitions preliminary results for strictly bounded real matrices will be needed. Definition 2: Let S(z- ) be a square matrix of real rational functions in the complex variable z-l. Then S( z-l) is called strictly bounded real if i) all poles of S(z- ) lie in ]z-l] Xl; ii) I - ST(e-jo)S(e w) is positive definite for all w E P,2~. This definition is given in [7] with the only difference that z is used as the complex variable instead of z -. z - is used here in order to conform with the fact that in 2-D stability theory the stability region is defined as the area outside the unit bidisc. Conditions i) ii) on S(z- ) can be reduced to conditions on the matrices of a minimal state-space realization of S(z-l) using the Bounded Real Lemma [7]. Suppose that S(z- ) has a minimal realization given by the quadruple { F, G, H, J } such that S(z- ) = J+ HT(z- I- F)- G (6) then the strictly Bounded Real Lemma can be formulated: Lemma : Let S(z- ) be a square matrix of real rational functions in z-l { F, G, H, J } a minimal reali- zation of S(z-l). Then S( z-i) is strictly bounded real if, only if, there exists a symmetric positive definite matrix P such that the matrix Q, given by I- Jv-G PG -(F~PG+HJ)~ C7) Q,= [ -FTPG-HJ P-FTPF-HHT I is positive definite. This result was essentially established by Anderson Vongpanitlerd ([7, lemma.2.) except for two minor differences. The first is that in [7] the lemma is stated using a particular minimal realization of S(z- ) with a coordinate basis in which P =. Matrix Q,, (7), results after changing the coordinate basis used in [7, lemma.2.l],by applying a transformation matrix T such that T TT = P. The second difference is that in [7], the case of S(z- ) being only bounded real not strictly bounded real is considered. As a result of this the condition on Q, in [7] is weaker than the one in Lemma. In particular, the condition for S( z-l) to be strictly bounded real is that Q,, (7), is positive definite, while in [7] the condition for S(z- ) to be bounded real is that Q, is nonegative definite. Remark: If { F, G, H, J } is not a minimal realization of S(z- ) there exists a positive definite P such that matrix Q, in (7) is positive definite, then S(z- ) is still strictly bounded real. However, the converse cannot be established. II. THE 2-D LYAPUNOV EQUATION In this section the necessary sufficient condition for the existence of positive definite matrices W = W@ W,, Q satisfying the 2-D Lyapunov (5) will be presented. This result is based on the Bounded Real Lemma outlined in the previous section. The condition presented in Theorem will be subsequently compared with the stability condition (4) it will be shown by means of an example that the stability condition (4) is not sufficient for the existence of positive definite matrices W Q satisfying the 2-D Lyapunov equation. Theorem : Consider a 2-D discrete system described by (). If for some nonsingular T, S(z;l) given by S(z~)=T[A,,+A,2(z;-A22)-A2,]T- (8) is strictly bounded real if (A,,, A,,) (A,,, AT,) are, respectively, completely reachable completely observable, then there exists some positive definite matrices W,, W,, Q such that W = W,@ W, Q satisfy the 2-D Lyapunov equation: ATWA - W = - Q. (9) Conversely, if (9) holds for positive definite W = W,@ W2 Q, then there exists a nonsingular matrix T such that S( z; ) is strictly bounded real. Proof: Sufficiency: T nonsingular (A,,, A,,), (A229 AT,) completely reachable observable imply that the quadruple { A22, AZITel, AT2TT, TAllTpl} is a minimal realization of S(z; ). From lemma, it follows that there

3 ANDERSON et ai.: THE MATRIX LYAPUNOV EQUATION 263 exists a positive definite matrix P such that the matrix: (0) I - T-TA;,TTTA,lT-l - T-=A;,PAz,T- - ( A;,PAzlT-l + A,,T=TA,,T- ) = Q= - A;*PA,,T- - A;,T=TA,,T- P - A&PA,, - AT,TTTA,2 is positive definite. Equation (0) gives after premultiplication with (TT@I,) postmultiplication with (T@I,): w which is the 2-D Lyapunov equation (9) with W = TTT@ P Q = (TT@Z,). \X,{S,(ej )}\<l, i=l;~~,nforallwe[0,2n] Necessity: Suppose (9) is satisfied. Set T = Wt/, F = A22, G = A2T-, HT= TA,, J = TA,,T-. Then pre- 07) multiply postmultiply (9) by (T-%,,,) (7) results for any nonsingular T (7) is equivalent to with P=W, Q,=(T- ~l,,,)q(t-l~l,j.thestrictly (Xi{S(ej )}I=IXi{TS,(ei )T- }l<l, bounded property of S(z; ) follows then directly from i=l d.., Lemma. > n for all w E [0,27r] (8) The stability condition (4) will now be compared with The strict bounded real property of S(z; ), required in the condition for the existence of positive definite matrices Theorem, can now be compared to conditions (6) satisfying the 2-D Lyapunov equation given in Theorem. (8), which are equivalent to the stability condition (4). It can be assumed without loss of generality that the In the following lemma it is shown that the strict bounded systems considered have completely reachable observ- real property of S(z; ) implies conditions (6) (8). ableable pairs (A,,, A,,) (A,,, AT,), respectively. For However, as it will be seen later, the converse is not true. in the next section it will be shown that if the reachability Lemma 2: Consider S(z; ), given by (8) with the pairs /or observability condition is not satisfied, then this (A229 A,,) (A,,, AT,) being completely reachable system can be decomposed into two subsystems. The first observable respectively. Then (6) (8) hold if S(z; ) subsystem is a -D system the second is a 2-D one is strictly bounded real. satisfying the observability reachability conditions. Proof: If S( z; ) is strictly bounded real, it follows for The stability condition (4) can be rewritten as the eigenvalues at A22, the system matrix in the minimal realization of S(z; ), that det kzlsdz; ) +o jai{a22}i< fori=l;..,m. [ - Z2A2 I,-~2-422 From the strict bounded real property of S(z;l), it for all (zi, z2) E 0 (2) foows that where I- ST(e-jo)S(eja) > 0 for all w E [0,2n] (9) which implies S,( z; ) = A,, + A,,( z; I,,, - A22)-A2. 03) Ai{S(eiw)} <l foralloe[0,28]i=l;..,n. Using Huang s condition [9], [4], it follows that condition (2) is equivalent to the following two conditions: The converse of the above lemma is not true. In other words, the stability condition (4) does not, imply that det(l-z,a,,) ~0, for all lz2 gl (4) S(z; ) is strict bounded real, hence, there exist sysdet(l-z,s,(z,-)) ~0, for all ~~~ It2 =l. terns with a stable characteristic polynomial a S(z;l) which is not strictly bounded real. From Theorem it 05) follows that for such a system positive definite matrices These two conditions require that the eigenvalues of A,, W = W,CB W, Q satisfying the 2-D Lyapunov equation S,(e-j ), denoted by X,{A,,} hi { &(e-j?}, do not exist. In example such a system is discussed. satisfy Example: Consider the 2-D discrete system described by IAi{A2,} ~, i=l;..,m (6) the following system matrix: 0.5 ; ; o :---o---:---d-----i------o o-- ; 0.5 ; A= 0 IO IO I ; OI 0 IO : 0 IO IO IO (20)

4 264 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 3, MARCH 986 where xh E R2 x E R6. Claim 2: There is no nonsingular matrix T for which Claim I: The 2-D characteristic polynomial of matrix S(z, ) = TS,(Z, )T-~ satisfies A, (20), has no zeros in u2. Proof: The characteristic polynomial of A can be I- ST(e-jW)*S(eiW) > 0, for all w E [0,277] (28) given by Proof: Let us assume that there exists a nonsingular T such that (28) is satisfied we will show that this results det I - zls,(z, ) in a contradiction. Let - (2) I Z2A2, I, - Z2A22 P=T=T (29) where then, (28) can be rewritten as S,(z, ) = A,,+ A,,(z; l- A2,)-A,, (22) P - ST( e-j ) PS,( ej ) > 0, for all 0 E [0,27r]. which gives It will be shown that there exists no positive definite P such that (23) P - ST(l) P&(l) > 0 with 0.028~;~ ; P-s,T(-l)P&(-l)>O s,( z;l) = (24) z ~;~ ,-l are satisfied. From (23) to (25) we obtain s,( z;l) = 0.028~;~ ; zy3 +2.8zT2 $2.6572, (25) W)= [, (jy5] (30) (3) (32) sd-l)= [ ; 04;] (33) In order to prove that the characteristic polynomial has no zeros in u2, we have to show that conditions (6) where (Y = p = P is positive definite without loss of generality we Ih,{A22} ~, for i=;..,6 can assume that pl =, which implies the following condi (7) tion: Ihj{ &(ej )} l<l, for i =,2 all w E [0,27r] IPI y62. (34) are satisfied. Consider (30) P - ST(l) P&(l) = cxp2 - (Y2p22-0.5p - ( c$)p,, +0.5ap,, I -0.5p -(0.25- C@)p,, +0.5ap,, 0.75P22 -PP2-P2. The first condition can be easily proven by computing The positive definiteness condition requires the eigenvalues of the A,, (or the poles of S,( z; ) S,( z; )). For the second condition consider 0.75P22 - PP2 - b2 0 det [, - zis,( ej )] or =z : [( Zl p-0.5)2-sa(ej )-S,(ej )]. (26) 0.75P22 P2 - BP2 It can easily be verified by evaluating ls,(e(jw).s,(ejo)l which gives using (34) for all w E [0,2n] that >P2 - IPI!/iG ISa(ejo)~S,(eia) I< 0.02 (27) or which implies that the values of z; for which (26) is zero, are all close to 0.5. These values are precisely the eigenval- (&2+:IPI)2-w>o ues of S,, so that which implies JXi{Sl(eiw)} I< for oe [0,27r], i=l,2 consequently the characteristic polynomial of A has no zeros in U2. Consider next (3) (35) (36) \IE)22 $PI = 0. (37) P-s,T(-l)P&(-l)= PP2 -P2P22-0.5a-(0.25- a/3)p2+0.5pp22-0.5~~ - (0.25- np)p2 +0.5Pp22 o.75p2, - api - a2 I (38)

5 ANDERSON et al.: THE MATRIX LYAPUNOV EQUATION 265 The condition for positive definiteness requires PP2 - P2P22 0 (3% which, after similar calculations as in the previous case, yields the condition Condition (40) contradicts condition (37), therefore, there is no nonsingular matrix T- such that (28) is satisfied. The implication of Claim 2 Theorem is that for this 2-D system there exist no positive definite matrices W = WI@ W, Q such that the 2-D Lyapunov equation is satisfied. This same system has a characteristic polynomial which has no zeros in e2 as shown in Claim. Consequently, the assertion that the stability of the characteristic polynomial implies the existence of positive definite matrices W = WI@ W, Q satisfying the 2-D Lyapunov equation is not correct. In other words, the necessity part of the 2-D Lyapunov condition is different than in the -D case. This establishes another case where -D results cannot be extended to the higher-dimensional case. Furthermore, it should be noted that these results can easily be extended to the continuous case. IV. SPECIAL CASES In this section Theorem is applied to the following two classes of 2-D discrete systems. i) 2-D realizations with a nonreachable pair (A,,, A,,); ii) 2-D systems with a characteristic polynomial which is first order in at least one variable. Consider the 2-D discrete system described with () where the pair (A,,, A,,) is not reachable. This implies that the quadruple {A,,, A2, AT2, A,,} is not a minimal realization of S,(z; ), therefore, theorem cannot be applied. However, it is well known that a minimal realization of S( z; ) can be obtained by separating out the reachable part of { A22, A,,, AT2, A,,}. There exists a nonsingular matrix R such that (4) (42) ; m-q 42R= [A42 ; 422. (43) with the following characteristic polynomial, L - ZlAll - ~242, C(z,, z2) = det - z A 2 q - Z2A22,.det[I,-,-z2A223]. (44) The structure of the system matrix A in the new coordinate basis allows the decomposition of the original 2-D system into two subsystems. The first is one with a -D characteristic polynomial, the second has a 2-D characteristic polynomial, in addition { A22, A29 Al,,7 A,,} is a minimal realization of S(z;l). Theorem can now be applied to the reduced 2-D system if S( z; ) is strictly bounded real, there exist positive definite solutions of the 2-D Lyapunov equation for the reduced 2-D system. It should be noticed, however, that even if positive definite solutions of the Lyapunov equations for both the -D the 2-D subsystems exist, it is not necessary that positive definite solutions to the 2-D Lyapunov equation for the original 2-D system exist. Obviously, similar observations can be made in the case where the pair (A,,, AT,) is not observable. Consider now the second special class of 2-D systems, namely those with a characteristic polynomial which is first order in at least one of the variables zi, z2. For these systems Corollary can be given. Corollary I: Consider a 2-D discrete system described by () with n =. If the characteristic polynomial of this system has no zeros in u2 if the pairs (A,,, A,,) ( Ali, AT,) are completely reachable observable, respectively, then there exist some positive definite matrices W = WI@ W, Q such that the 2-D Lyapunov equation A=WA-W=-Q (45) is satisfied. Conversely, if (45) is satisfied, then the characteristic polynomial of the system has no zeros in f2. Proof: The proof of this theorem follows from theorem the fact that for n =, S(z; ) is scalar function in z;. Consequently, the two scalar conditions IA,{S(ej )}I<l, l- S(e-j )S(ejw) > 0, foralloe[0,27r] for all 0 E [O,~P] are equivalent. A similar corollary can be given for the case m =. In the case m = n =l tij (i, j=,2) become scalar, therefore, the reachabthty observability conditions are not needed, which yields the same condition given in Theorem 2 in [6]. Further, the pair (A22, A,,,) is completely reachable. V. CONCLUSIONS Applying the coordinate transformation (I,@ R) to the 2-D system described by (l), the system yields The necessary sufficient condition for the existence of positive definite matrices satisfying the 2-D Lyapunov equation has been developed. It has been shown that the existence of such solution pairs implies that the 2-D characteristic polynomial has no zeros in the unit bidisc but the converse is not true. A stable 2-D system is given for which

6 266 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, VOL. CAS-33, NO. 3, MARCH 986 no positive definite matrices satisfying the 2-D Lyapunov equation exist. PI PI [3 [4 [5 WI [7 PI [9 REFERENCES M. S. Piekarski, Algebraic characterization of matrices whose multivariable characteristic polynomial is Hurwitzian, in Proc. Int. Symp. Operator Theory, Lubbock, TX, Aug. 977, pp J. H. Lodge M. M. Fahmy, Stability overflow oscillations in 2-D state-space digital filters, IEEE Trans. Acourt., Speech, Signal Processin., vol. ASSP-29, pp. 6-7, 98. E. Fornasini an cf G. Marchesini, Stability analysis of 2-D systems, IEEE Trans. Circuits Syst., vol. CAS-27, pp , Dec Wu Sheng Lu E. B. Lee, Stability analysis for two-dimensional systems via a Lyapunov approach, IEEE Trans. Circuits Syst., vol. CAS-32, pp. 6-68, Jan R. P. Roesser, A discrete state-space model for linear image rg5essing, IEEE Trans. Automat. Contr., vol. AC-20, pp. l-0, E. I. Jury, Stability of multidimensional scalar matrix polynomials. Proc. IEEE. vol. 66. un B. D. 6,. Anderson S. Vongpanitlerd, ketwork Analysis Synthesis, A Modern Systems Theory Approach. Englewood Cliffs, NJ: Prentice-Hall, 973. P. Agathoklis, E. I. Jury, M. Mansour, Criteria for the absence of limit cycles in two-dimensional discrete systems, IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-32, pp , Apr T. S. Huang, Stability of two-dimensional recursive filters, IEEE Trans. Audio Electroacoust, vol. AU-20, pp , 972. m.,._- Brian D. 0. Anderson was born in Sydney, Australia, in 94. He received the B.S. degrees in pure mathematics electrical engineering from the University of Sydney, Sydney, Australia, the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 966. He is currently Professor Head of Department of Systems Engineering at the Australian National University; from 967 through 98 he was Professor of Electrical Engineering at the University of Newcastle. He has also held appointments as Visiting Professor at a number of universities in the U.S., Australasia, Europe. He is co-author of several books his research interests are in control signal processing. He was editor of Automatica is currently a Vice-President of IFAC.Dr. Anderson is a Fellow of the Australian Academy of Science,?Australian Academy of Technological Sciences an Honorary Fellow of the Institute of Engineers, Australia. Victoria, B.C., Canada, as a Visiting Assistant Professor from 984 as NSERC University Research Fellow adjunct Assistant Professor. His fields of interest are the stability design of 2-D digital filters control systems. m E. I. Jury (M 54-SM 57-F 68) was born in Baghdad. Iraa. He received the E.E. degree from &Israel Insiitute of Technology, Hai?a, Israel, where he was a Goldberg Scholar, in 947, the M.S. degree in electrical engineering from Harvard University, Cambridge, MA, in 949, the Sc.D. degree in engineering science from Columbia University, New York, NY, where he was a Higgins Fellow, in 953. In 982, the degree of Dr. Sc.Techn. (honoris causa) was conferred on him by the Swiss Federal Institute of Technology, Zurich, Switzerl. After a six months stay at the Electronics Research Laboratory of Columbia University in 953, he joined the Department of Electrical Engineering Computer Science, University of California, Berkeley, as an Instructor in 954. He was appointed Professor of Electrical Engineering in 964. During , he was a Visiting Professor at the University of Paris, Paris, France, at the Swiss Federal Institute of Technology, Zurich, Switzerl. During , he was a Visiting Professor Senior NSF Post-Doctoral Fellow at the Imperial College of Science Technology, London University, London, Engl. In 970 he was a Visiting Scientist at DFVLR, the Institute of Dynamical Systems in Oberpfaffenhoven, Germany, During the Fall of 973, he was Visiting Professor at the University of Rome, Rome, Italy. In the Fall of 974, he was a Senior Post-Doctoral Felo.w Visiting Professor at the University of Newcastle, New Castle, Australia. In September 979, he was a senior Fullbright-Hays Fellow, lecturing at Kiev Polytechnic Institute, Kiev,, USSR. In 98, he received the title Professor Emeritus, University of California, Berkeley. Presentely, he is Research Professor at the University of Miami, Coral Glades, FL. He is the author of Sampled- Data Control Systems (New York: Wiley, 958, which was translated into French, Japanese Russian), Theory Application of the z-transform Method (New York: Wiley, 964, which was translated into Polish), Inners Stability of Dynamic Systems (New York: Wiley, 974, which was translated into Russian), contributed to Multidimensional Systems, Techniques Applications, (edited by S. G. Tsafestas, Marcel Dekker,) Nov His fields of interest are control systems, circuit theory, operational methods, digital filters bioengineering. He has published numerous articles in various scientific journals. Dr. Jun, is a member of the Harvard Engineering Society, the American Association for the Advancement of Science, the New York Academy of Sciences, Tau Beta Pi, Eta Kappa Nu, an Honorary Member of Sigma Xi. He is the recipient of the ASME Centennial Medal, 980. Calgary, Alta, Canada with the Department Panajotis Agathoklis (M 8) received the Dipl.el.Ing. degree in electrical engineering the Dr.Sc.Tech. degree from the Swiss Federal Institute of Technology (ETH), Zurich, Switzerl, in , respectively. From 976 until 980, he was with the Institute of Automatic Control Industrial Electronics at ETH, initially as an Assistant, from 979, as a Research Assistant. From 980 until 983, he was with the Department of Electrical Engineering at the University of Calgary, as a Post-Doctoral Fellow. Since 983, he has been of Electrical Engineering, University of Victoria, M. Mansour (SM 77-F 85) was born in Damietta, Eevut. in August 928. He received the B.Sc. MSc. degrees in electrical engineering from the University of Alexria, Egypt, in , respectively, the Dr.Sc.Techn. degree in electrical engineering from ETH-Zlirich, Switzerl, in 965, when he was awarded the silver medal of ETH. He was Assistant Professor in Electrical Engineering at Queen s University, Canada, from 967 to 968. He has been Professor Head of

7 ANDERSON et al. : THE MATRIX LYAPUNOV EQUATION 267 the Department of Automatic Control at ETH-Zurich since 968, Dean of Electrical Engineering , Director of the Institute of Automatic Control Industrial Electronics, ETH-Zurich, during , , since 984. He was Visiting Professor from September to December 974 at IBM Research Laboratory, San Jose, CA; from January to March 975 at the University of Florida, Gainesville; from August to December 98 at the University of Illinois, Urbana; from January to March 983 at the University of California, Berkeley. He is also President of the Swiss Federation of Automatic Control, member of the Council Treasurer of IFAC, President of the 4th IFAC/IFIP Conference on Digital Computer Applications for Process Control 074, Chairman of the International Program Committee of the IFAC Symposium on Commuter Aided Design of-control Systems, 979, Chairm&of the Internat?onal Program Committee of the 4th IFAC/IFORS Svmnosium on Lame Scale Systems: Theory Applications, 986, Vice-Chairman of they IFAC Education Committee , member of the Senate of the Swiss Academy of Natural Sciences, , Delegate of IFAC to the United Nations in Geneva. His fields of interest are control systems, especially stability theory digital control, stability of power systems, digital filters.