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Robust Mixed

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Filtering of 2-D Systems

Hoang Duong Tuan, Member, IEEE, Pierre Apkarian, Associate Member, IEEE, Truong Q. Nguyen, Senior Member, IEEE, and Tatsuo Narikiyo

Abstract—This paper deals with several challenging problems of robust filtering for two-dimensional (2-D) systems. First of all, new linear matrix inequality (LMI) characterizations for the and 2 norms of 2-D systems are introduced and thoroughly established. Based on these preparatory results, convex (LMI) char, 2 , and robust mixed 2 filacterizations for robust tering are derived. The efficiency and viability of the proposed techniques and tools are demonstrated through a set of numerical examples.

where state; measured output; output to be estimated; noise/disturbance. The system data

Index Terms—Linear matrix inequality (LMI), robust filtering, 2-D system.

I. INTRODUCTION

T

WO dimensional (2-D) filters have been addressed in numerous references (see, e.g., [5], [10], [12], [15], and [18]–[20], and references therein). For the IIR case, the 2-D Kalman ( ) filter [15], [18] is the state estimation process for a linear system involving noise with known statistical properties (like white noise, etc.). A complement to the Kalman filter [5], which does not require a priori filter is the knowledge of statistical properties, except the variance is assumed to be finite. However, in practice, the noise input is often characterized by both known and unknown statistical properties, and thus, one has to handle the more appropriate filtering. Thus far, this filtering problem has mixed been addressed for 1-D systems (see, e.g., [8]). There is a lot filters. to be done even for separated 2-D Kalman and Furthermore, little is known about the robust filtering in the case where the system data are uncertain, despite its importance since system data are hardly exactly known in practical models due to unavoidable measurement inaccuracies. Let us explicitly formulate the problem in the space-state setting. Consider the uncertain 2-D system

are uncertain and obey the polytopic model in (2), shown at the bottom of the next page, where is the unit simplex containing the deterministic unknown parameter (3) The problem at hand is to find a filter in the form

(4) which provides good robust estimation for system (1) under difand norms (these ferent norm specifications such as the , , and will be be defined later). The sought matrices are of dimensions , , and , respectively. With these matrices computed offline, the estimation of the output of system (1) can be easily updated online according to (4). In other words, the problems of interest can be any one among the following forms or their combinations: (5) (6)

(1) Manuscript received May 4, 2001; revised March 22, 2002. Their work was supported in part by Monbu-sho under Grant 12650412. The associate editor coordinating the review of this paper and approving it for publication was Prof. Derong Liu. H. D. Tuan and T. Narikiyo are with the Department of Electrical and Computer Engineering, Toyota Technological Institute, Nagoya, Japan (e-mail: tuantoyota-ti.ac.jp; [email protected]). P. Apkarian is with the ONERA-CERT, Toulouse, France (e-mail: [email protected]). T. Q. Nguyen is with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA 92093-0407 USA (e-mail: [email protected]). Publisher Item Identifier S 1053-587X(02)05655-6.

s.t.

(7)

and are norms inducing and norms where criteria [see (11) and (29)]. It is widely recognized that the norm is useful to handle the error over all frequencies and, thus, is a reasonably adequate estimation criterion when the is random with bounded variance. Similarly, disturbance norm is useful to keep the peak amplitude of below the a certain level and, thus, is an appropriate estimation criterion is of white noise type. Obviously, when the disturbance

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the mixed problem (7) does nothing more than minisubject to a bound mize the amplitude peak of the error on its gain over the frequency range. A mixed problem can be formulated analogously. or As mentioned previously, only the filtering problems ( ) for a nominal model (i.e., when all data of (1) are exactly known) have been considered thus far [5], [15], [18]–[20]. Yet, computationally tractable and less conservative formulations for these problems have to be found, and this remains a challenging task. In contrast with the rather complete results for 1-D systems [1], only sufficient (but not necessary) computationally tractable conditions for stability as well as norms conditions of 2-D sysfilter tems are available thus far. For the nominal case, the (4) of [5] has the so-called observer structure [3], that is (8) instead so that the corresponding filter design is to find just ) in (4). It is a crucial trick for simof the triple ( plifying matrix inequalities in [5]. Obviously, with the robust filtering problems considered in this paper, the observer struc, , and in (8) are ture is inappropriate as the matrices subject to uncertainties and, thus, cannot be used to design the filter. The purpose of the paper is twofold. First, we derive new -norm constraints (convex) LMI characterizations of the for two-dimensional (2-D) systems, which is not only simpler and less conservative than that of [5] but can also be applied filtering problem with parameter-dependent to the robust -norm is also Lyapunov functions. A new result for the presented in a similar fashion. Second, we establish new linearizing transformations of the filter variables to achieve a , filtering full LMI program description of the robust filtering problems involving and the robust mixed parameter-dependent Lyapunov variables, which cannot be handled by previously developed approaches such as those in [5]. Specifically, different parameter-dependent Lyapunov and norm functions are associated with each individual specifications. Thus, our approach opens a new way to attack difficult filtering problems for 2-D systems. The structure of the paper is as follows. New LMI characterand norms of 2-D systems are considered izations for in Section II. The latter are the main tools for developing the linearization technique to obtain LMI characterizations for the , robust , as well as the robust mixed robust problems presented in Section III. Finally, numerical examples emphasizing the advantages of our techniques are given in Section IV. The notations used in the paper are rather standard. For inis the transpose of a matrix , and is the stance, and another matrix . The notation Kronecker product of

means is negative definite. Analogously, ( , , rep.) means that is negative semi-definite (positive definite, positive semi-definite, resp.). In long block matrix expressions, we use as an ellipsis for terms that are induced by symmetry. For instance

When there is a possibility of ambiguity, we use, for instance, , to indicate the dimensions of matrices. Boldface capital letters are used to emphasize matrix variables. In addition, the following notations will be used throughout the paper: (9) Basic instruments in the subsequent derivations are as follows. • Congruence transformation of matrices: The matrix is negative definite (positive definite, resp.) if and only is negative definite (positive definite, resp.) as if well for any nonsingular matrix of appropriate dimenis called congruent to via the sion. The matrix congruence transformation . • Schur’s complement:

for any matrices sions. II.

AND

,

, or

of appropriate dimen-

NORMS FOR 2-D SYSTEMS

In this section, we will introduce and analyze and norms of 2-D systems that play a crucial role throughout the paper. Two-dimensional systems are described by the following equations:

(10) , , and have the same meaning as where their counterparts in system (1). Since we are interested only with , , we assume, from now on, that in for The

or

-norm of system (10) is introduced first.

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Lemma 1: Suppose that there is a symmetric matrix satisfying the following conditions:

Norm Characterizations Define

(13)

(14) -norm of system (10) is less Then, (12) holds true, i.e., the than . Moreover, system (10) is internally stable in the sense and that for every initial condition as

(15)

Proof: By (13)

Hence, for all

(16) In view of (14) and (16), we have the first equation at the bottom of the next page. Therefore, summing up the last inequalities and then using (16), one obtains The

-norm of system (10) is then defined as (11)

or equivalently, the and only if

-norm of system (10) is less than

if

(12) For every positive definite matrix

(17) i.e.,

and

, define

The usefulness of for the clarified by the following lemma.

(18) and thus, (12) follows from (18) because of the initial condition and the fact that , . Hence, the -norm of (10) has been shown to be less than . The internal stability of system (10) is proved in a similar in (14) yields manner. Setting

-norm characterization is

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such that (22) (19)

where

Analogously to (18), it follows from (19) that (20)

Proof: With the notations

which yields (23) hence, implying as , i.e., , which includes (15) as a particular case and concludes the proof. Now, the LMI characterization for the -norm of system (10) is described by the following theorem. -norm of system (10) is less than if Theorem 1: The satisfying (13) there is a symmetric matrix of size and, additionally, either one of the following inequalities with holds true, the additional slack variable of dimension as in

(21)

we can rewrite (14) as (24), shown at the bottom of the page, and (24) is equivalent to (21) by a Schur’s complement argument. Therefore, we still need to prove the equivalence between (21) and (22). First, performing the congruence transformation diag in (21), we get the equivalent inequality

(25)

which is obviously a particular case of (22) with the choice . To prove the implication (22) (21), first note that

.. .. .. ...

(24)

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the system data, we are allowed to use a Lyapunov variable depending on the parameter [see (60)] for checking the system performance, which is much less conservative than with a single Lyapunov variable independent on . Such an approach has been used in [2], [4], [8], and [16] in robust control and filtering for 1-D systems.

satisfying (22) must be nonsingular. Furthermore, performing the congruence transformation diag in (22) gives the equivalent inequality

Norm Characterizations

B. For

, we define the

-norm of system (10) as (29)

which implies (21) as well since for as

or equivalently, the and only if

-norm of system (10) is less than , if

(30) Remarks: • It can be seen that any solution the sufficient condition for the a solution to (21) by setting

,

, and satisfying -norm of [5, Th.3] is also

In this aspect, the result in (21) is not only more elegant but less conservative than that of [5, Th.3]. • Note that from (21), a sufficient condition for the internal stability of system (10), we have (26) and (27), shown at the bottom of the page, which not only includes all previously developed results [10], [13], [14] as a particular case but also offers much flexibility for analysis and synthesis. For instance, the sufficient condition [10, Th.1] for the internal stability of system (10)

For simplicity, in what follows, we will use the notation

to characterize the -norm. From (29) and (30), it is obvious that in contrast to the -norm defined by (11) to bound the output energy, the -norm here is to keep the peak amplitude of the introduced output below a certain level. It is a natural extension of the -norm for 1-D systems (see, e.g., [11]). generalized -norm defined by (24) and (29) LMI characterizations for are summarized in the following theorem. -norm of system (10) is less than if Theorem 2: The satisfying (13) there is a symmetric matrix of size and (31)

(28) , is a particular case of (26) by restricting , and . Note that (26) is actually LMI (27) and, thus, is computationally tractable, whereas (28) is a (nonconvex) bilinear matrix inequality (BMI) in the variables , , which is more computationally demanding [17]. • The slack variable in (22) as well as later in (33) is intro, arising in (21) [see duced to get rid of the term (25)] involving the Lyapunov variable . As it will become clear later, since the Lyapunov variable is separated from

and, additionally, either one of the following inequalities with holds true: slack variable of dimension (32)

(33) Proof: Using a Schur’s complement, (32) implies

(26) (27)

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i.e., we have the formula at the bottom of the page. Since with defined by (23), the last inequality yields

(34) Thus, analogously to Lemma 1, we can show that

which, together with (31), guarantees that the -norm of the (see, e.g., 1-D systems (37) and (38) is less than when is a discrete [11]). This also means that when the input and zero mean white noise, the variance of the outputs are less than . Thus, like 1-D systems, our introduced -norm also has a close relationship with the 2-D Kalman filtering [18]. Before closing this section, let us also notice that like the LMI-based approach for 1-D systems, the present approach works mostly for “infinite horizon” filtering problems. filSee [5] for results related to the finite horizon case of tering.

(35) III. ROBUST FILTERING PROBLEMS Meanwhile, again by a Schur’s complement, (31) becomes

Equipped with (convex) LMI characterizations (21) and (22) and norms of 2-D systems, we are and (31)–(33) for the ready to turn back to the filtering problem involving the system (1), (4), whose closed form can be rewritten as

(39)

(36) which together with (35) imply (30). Consequently, the -norm of system (10) is less than . The remaining statement on the equivalence between (32) and (33) can be proved along the lines of Theorem 1. Remark: As , 1, 2, it easily follows from (22) that

where

which can be also expressed as which guarantees that the -norm of the following 1-D subsystems is less than (see, e.g., [6])

(37) (40) (38) Analogously, (32) particularly implies

The robust A. Robust

filtering is treated first. -Filtering

-norm characterization (13), (22) to system To apply the and the slack (39), we first express of dimension

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With the congruence transformation

as

diag

(41)

(46)

(43) becomes (47), shown at the bottom of the page. Now, in view of (40) and (45) and with simple (but tedious) calculations, we can check (48), shown at the bottom of the page. Thus, with the linearization transformations

Then, by virtue of (40), we get

(42)

(49)

and we can rewrite the inequality characterizations (13) and (22) of system (39) as inequality (13) with defined for by (41) and (43), shown at the bottom of the page. By (22), must be nonsingular; therefore, is nonsingular as well. Therefore, we can define

(50)

(44) which meets the following obvious conditions:

equation (47) is equivalent to the LMI in the variables , which is shown in (51) at the bottom of the next , and page. Meanwhile, satisfies (13) if and only if (52)

(45)

To recap, we have obtained the following result. Lemma 2: The nonlinear matrix inequalities (13) and (43) are feasible in characterizing the error estimation and if and only if LMIs (51) and the variables and . (52) are feasible in the variables

(43)

(47)

(48)

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A solution of system (13) and (43) defining the filter (4) can be found from a solution of LMI system (51) and (52) through the following formulas:

By virtue of the expressions (2) and the definition (3), whenever , we have

(53) ,

Proof: With and

, we see that

,

,

(58)

indeed satis-

in (49). Then, (53) readily follows fies the relation by substituting the value of into (50) and inversely deriving from . According to Lemma 2 above, there is a filter (4) such that the robust estimation (54) holds true for system (1) with all data satisfying (2) if the system of inequalities in (55) and (56), shown at the bottom of the page, , depending on the pais feasible in the function variables , where rameter and the variables

diag (57)

with

diag Therefore, with

(59) of appropriate polytopic structure (60)

(55) and (56) take the form of parameterized LMIs (PLMIs), in (61) and (62), shown at depending on the parameter the bottom of the page. Now, it is already almost trivial to see that PLMIs (61) and (62) are equivalent to the LMIs in (63) and (64), shown at the bottom of the next page. Therefore, we are now in a position to state the main result of this section. Theorem 3: There is a filter (4) such that the robust estimation (54) is fulfilled for system (1) with data satisfying (2) if the LMI constraints (63) and (64) are feasible

(51)

(55)

(56)

(61)

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,

, , and

. defining the sought filter (4) can be found Matrices through (53). filtering Consequently, an upper bound for the robust problem (5) can be found from the following LMI optimization problem: (63) and (64),

(65)

B. Robust

-Filtering

With the characterizations (31) and (33) and arguing simifiltering. larly, we obtain the following result for the robust Theorem 4: There is a filter (4) such that the robust estimation

is fulfilled for system (1) with data satisfying (2) if the LMIs in (67)–(69), shown at the bottom of the page, are feasible with respect to variables

Remark: As mentioned previously, when all matrices , , , , and in (1) are assumed to be exactly known, the filter of [5] bears a special observer structure in (8). In this case, only instead of the triple ( , , must be determined. For such a special structure, one can treat immediately the following error equation:

,

,

and , where , and as before, , , , and are defined by (59). The parameter-dependent Lyapunov variable for checking the estimation error is then (70)

(66) , and . where From (21), without the linear transformation (50) and other complex transformations described previously, it is immediate by adding to recover an LMI characterization for , and furthermore, one more new variable by using the projection lemma in [6], one can even eliminate this variable to get simpler LMIs involving only variable . However, as mentioned in the Introduction, such observer-structure filters are no longer solution candidates in our robust filtering problems.

defining the sought filter (4) can be found Matrices from a solution of (67)–(69) through (53). filtering Consequently, an upper bound for the robust problem (6) can be found from the following LMI optimization problem (67)–(69)

C. Mixed

(71)

Filtering

As an immediate consequence of Theorems 2 and 4, we obtain the following result regarding the mixed robust filtering problem (7).

(63)

(64)

(67)

(68)

(69)

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Theorem 5: An upper bound for the robust mixed problem (7) can be found from the following LMI optimization problem: (63), (64) (67)–(69) (72) The parameter-dependent Lyapunov variables for checking the and are estimations and defined by (70) and (60), respectively. Matrices and defining the sought filter (4) can be found from a solution of (72) through (53). Remark: From the formulation (7) for the robust mixed filtering, it is clear that there is a tradeoff between the and performances: Decreasing for the norm norm optimal value. constraint in (7) leads to increasing the and norm criteria in Another way to handle both the the robust filtering is the combined formulation (73) and are weightings for the and where criteria, which can be normalized by the condition . Obviously, by increasing (and thus decreasing ), the -tracking performance of the filter can be improved, but performance deteriorates. Accordingly, an upper bound of (73) can be found from the following LMI optimization problem

(63), (64) (67)–(69)

(74)

and performances The inherent tradeoff between the will be clarified and exemplified in the next section.

IV. NUMERICAL EXAMPLES As mentioned, the core benefits of using characterizations (22) and (33) are as follows. • With the used parameter-dependent Lyapunov functions (60) and (70), we can easily get the tractable LMI formuand lations (63) and (64) and (67)–(69) for the robust filters. filtering such as (7) and • For the robust mixed and (73), we can use different Lyapunov functions in (72) and (74) to check each of the and performances individually. In contrast, with characterizations (21) and (31) and (32), LMIs formulations for the just-mentioned filtering problems are possible only with a single Lyapunov function [i.e., in (60) and (70)]. In such a case, the linearization technique developed in Section III-A leads to the following LMI optimization problems. filtering (5), we have (75)–(77), shown • For the robust at the bottom of the page. filtering (6), we have (78)–(80), shown • For the robust at the bottom of the page. filtering problems (7) and • For the robust mixed (73), respectively (76), (77), (79), (80)

(81)

(76), (77), (79), (80)

(82)

The following numerical examples clearly demonstrate the advantages of formulations (63) and (64), (67)–(69), and (72) and (74) over (75)–(77), (78)–(80), and (81) and (82).

(75)

(76)

(77)

(77)

(78)

(79)

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TABLE I PERFORMANCES GIVEN BY DIFFERENT CHARACTERIZATIONS

A. Parameter-Dependent Lyapunov Function versus Common Lyapunov Function Consider system (1) and (2) with lowing data:

and with the fol-

Fig. 1. Tradeoff between

H

and

H

criteria in filtering.

With these data, the optimal values of problems (65), (75)–(77), (71), and (78)–(80) are given in Table I. The value associated with a common Lyapunov function is really conservative compared with those associated with a parameter-dependent Lyapunov function. B. Tradeoff Between the

and

Norm Criteria

and performances To see the tradeoff between the of filters, for simplicity, let us consider just the mixed filtering with the following simple data:

Fig. 1 displays the plot of the optimal value of the perperformance formance versus the constraint for the the of filters in (72). Note that in all these cases, the optimal value of (81) is infinite. The robust tracking performance [see (39)] of the robust found from the and LMI optimization problem (65) when the noises are chosen as follows is given by Fig. 2

(83)

Fig. 2.

Robust tracking performance of the robust

H

filter.

TABLE II COMPUTATIONAL DATA OF FILTERS

Note that it is easy to find examples where the optimal values of (65) and (71) are finite, whereas that given by (75)–(77) and (78)–(80) are infinite. The computational data of the optimal , , and mixed (for ) suboptimal filters found from LMI optimization problems (65), (71), and (72) are given of the by Table II, whereas the tracking performances corresponding system (39) with the noises (83) are given by Figs. 3–5. From these figures, one can see that as expected, the filter actually gives better energy-error performance but worse peak-error performance in comparison with that of the filter, and the mixed filter gives balanced energy-error and peak-error performances.

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TRADEOFF BETWEEN THE

Fig. 3. Tracking performance of the

H

H

TABLE III AND CRITERIA IN ANOTHER SETTING

H

filter.

V. CONCLUDING REMARKS In this paper, new flexible and less-conservative LMI charand norm constraints of 2-D sysacterizations for the tems have been developed. They allow the use of different parameter-dependent Lyapunov functions to check each individual or estimation errors for the robust mixed IIR robust filter design while preserving computational tractability of the problem. Our numerical examples clearly demonstrate that the potential conservatism in the convex approach to robust filtering is substantially reduced with the proposed new tools and techniques.

REFERENCES Fig. 4. Tracking performance of the

Fig. 5.

H

filter.

Tracking performance of the mixed filter.

On the other hand, the tradeoff between the and performances can be also seen from Table III, which displays the ) in (73) optimal values of and for different values of ( and (82). The benefit of formulation (73) over (82) is also obvious by comparing the values of and there.

[1] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [2] P. Apkarian, P. C. Pellanda, and H. D. Tuan, “Mixed =H multichannel linear parameter-varying control in discrete time,” Syst. Contr. Lett., vol. 41, pp. 336–346, 2000. [3] M. Bisiacco, “On the structure of 2-D observer,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 676–680, 1986. [4] M. C. de Oliveira, J. Bernussou, and J. C. Geromel, “A new discrete-time robust stability condition,” Syst. Contr. Lett., vol. 37, pp. 261–265, 1999. [5] C. Du, L. Xie, and Y. C. Soh, “H filtering of 2-D discrete systems,” IEEE Trans. Signal Processing, vol. 48, pp. 1760–1768, June 2000. [6] P. Gahinet and P. Apkarian, “A linear matrix inequality approach to H control,” Int. J. Robust Nonlinear Contr., vol. 4, pp. 421–448, 1994. [7] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali, LMI Control Toolbox. Natick, MA: The Mathworks, Inc., 1995. [8] J. C. Geromel, M. C. de Oliveira, and J. Bernussou, “Robust filtering of discrete-time linear systems with parameter-dependent Lyapunov functions,” in Proc. CDC, 1999, pp. 570–575. [9] J. C. Geromel, “Optimal linear filtering under parameter uncertainty,” IEEE Trans. Signal Processing, vol. 47, pp. 168–175, Jan. 1999. [10] T. Hinamoto, “2-D Lyapunov equation and filter design based on the Fornasini-Marchesini second model,” IEEE Trans. Circuits Syst. I, vol. 40, pp. 102–110, Feb. 1993. [11] I. Kaminer, P. P. Khargonekar, and M. A. Rotea, “Mixed H =H control for discrete-time systems via convex optimization,” Automatica, vol. 29, pp. 57–70, 1993. [12] W. S. Lu and A. Antoniou, Two-Dimensional Digital Filters. New York: Marcel Dekker, 1992. [13] W. S. Lu, “On a Lyapunov approach to stability analysis of 2-D digital filters,” IEEE Trans. Circuits Syst. I, vol. 41, pp. 665–669, Oct. 1994. [14] T. Ooba, “Matrix conditions for the stability of 2-D dynamics,”, submitted for publication. [15] W. A. Porter and J. L. Aravena, “State estimation in discrete m D systems,” IEEE Trans. Automat. Contr., vol. AC-31, pp. 280–283, 1986.

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[16] H. D. Tuan, P. Apkarian, and T. Q. Nguyen, “Robust and reduced-order filtering: New characterizations and methods,” in Proc. Amer. Contr. Conf., 2000, pp. 1327–1331. [17] H. D. Tuan and P. Apkarian, “Low nonconvexity-rank bilinear matrix inequalities: Algorithms and applications in robust controller and structure designs,” IEEE Trans. Automat. Contr., vol. 45, pp. 2111–2117, Nov. 2000. [18] J. W. Woods and V. K. Ingle, “Kalman filtering in two dimensions: Further results,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 188–197, 1981. [19] B. Xuan and R. Bamberger, “FIR principal component filter bank,” IEEE Trans. Signal Processing, vol. 46, pp. 930–940, Apr. 1998. [20] W. P. Zhu, M. O. Ahmad, and M. N. S. Swamy, “Realization of 2-D linear phase FIR filters by using SVD,” IEEE Trans. on Signal Processing, vol. 47, pp. 1349–1358, 1999.

Hoang Duong Tuan (M’92) was born in Hanoi, Vietnam, in 1964. He received the diploma and the Ph.D. degree, both in applied mathematics, from Odessa State University, Odessa, Ukraine, in 1987 and 1991, respectively. From 1991 to 1994, he was a Researcher with the Optimization and Systems Division, Vietnam National Center for Science and Technologies. From 1994 to 1999, he was an Assistant Professor with the Department of Electronic-Mechanical Engineering, Nagoya University, Nagoya, Japan. He joined the Toyota Technological Institute, Nagoya, in 1999, where he is an Associate Professor with the Department of Electrical and Computer Engineering. His research interests include theoretical developments and applications of optimization-based methods in broad areas of control, signal processing, and communication.

Pierre Apkarian (A’94) received the Engineer’s degree from the Ecole Supérieure d’Informatique, Electronique, Automatique de Paris, Paris, France, in 1985, the M.S. degree and “Diplôme d’Etudes Appronfondies” degree in mathematics from the University of Paris VII in 1985 and 1986, and the Ph.D. degree in control engineering from the “Ecole Nationale Supérieure de l’Aéronautique et de l’Espace” (ENSAE), in 1988. He was qualified as a Professor from the University of Paul Sabatier, Toulouse, France, in both control engineering and applied mathematics in 1999 and 2001, respectively. Since 1988, he has been a Research Scientist at ONERA-CERT, Toulouse, an Associate Professor with ENSAE, and with the Mathematics Department of Paul Sabatier University. His research interests include robust and gain-scheduling control theory, linear matrix inequality techniques, mathematical programming, and applications in aeronautics. Dr. Apkarian has served as an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL since 2001.

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Truong Q. Nguyen (SM’95) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the California Institute of Technology (Caltech), Pasadena, in 1985, 1986, and 1989, respectively. He was with the Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, from June 1989 to July 1994, as a member of technical staff. From August 1994 to July 1998, he was with the Electrical and Computer Engineering Department, University of Wisconsin, Madison. He was with Boston University, Boston, MA, from 1996 to 2001, and he is currently with the Electrical and Computer Engineering Department at the University of California at San Diego, La Jolla. His research interests are in the theory of wavelets and filterbanks and applications in image and video compression, telecommunications, bioinformatics, medical imaging and enhancement, and analog/digital conversion. He is the coauthor (with Prof. G. Strang) of a popular textbook Wavelets & Filter Banks (Wellesley, MA: Wellesley-Cambridge Press, 1997) and the author of several matlab-based toolboxes on image compression, electrocardiogram compression, and filterbank design. He also holds a patent on an efficient design method for wavelets and filterbanks and several patents on wavelet applications including compression and signal analysis. He is currently the Series Editor (Digital Signal Processing) for Academic Press. Prof. Nguyen received the IEEE TRANSACTIONS ON SIGNAL PROCESSING Paper Award (Image and Multidimensional Processing area) for the paper he co-wrote with Prof. P. P. Vaidyanathan on linear-phase perfect-reconstruction filter banks in 1992. He received the NSF Career Award in 1995 and served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1994 to 1996 and for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1996 to 1997.

Tatsuo Narikiyo was born in 1952 in Fukuoka, Japan. He received the B.E. degree in applied physics from Nagoya University, Nagoya, Japan, in 1978 and the Dr. Eng. degree in control engineering from the same university in 1984. He was a Research Scientist with the Government Industrial Research Institute, Kyushu, Japan, from 1983 to 1990. Starting in April 1990, he was an Associate Professor with the Department of Information and Control Engineering, Toyota Technological Institute, Nagoya. Since April 1998, he has been a Professor at the same Institute. His main research interests are in the control system design for linear and nonlinear mechanical system.