Robust Filtering for 2-D State-Delayed Systems with NFT Uncertainties

Robust Filtering for 2-D State-Delayed

Systems With NFT Uncertainties

Shyh-Feng Chen and I-Kong Fong, Member, IEEE

Abstract—This paper is concerned with the robust filtering

problem for two-dimensional (2-D) state-delayed systems with uncertainties represented by nonlinear fraction transformation. The authors first establish the stability performance and generalized ₂performance criteria for the system. Based on the results, the authors propose efficient methods to solve the robust filtering, generalized ₂ filtering, and mixed generalized 2 filtering problems by using a parameter-dependent Lyapunov function approach. The methods involve solving linear matrix inequalities. Two examples are given to show the effective-ness of the proposed approach.

Index Terms—Linear matrix inequality (LMI), nonlinear

frac-tion transformafrac-tion, robust filter, time-delay systems, two-dimen-sional (2-D) systems.

I. INTRODUCTION

T

HE filtering problem of two-dimensional (2-D) systems has attracted increasing attentions due to its application as well as theoretical importance in the fields such as multidimen-sional digital filtering, linear image processing, and so on [6], [12]. In these applications, it is usually desirable to estimate the values of state variables from the system measurement data. Var-ious schemes, such as the Kalman filter, the filter, and the mixed filter have been addressed in the literature (see, e.g., [5], [6], [19], and [21], and references cited therein).

For the Kalman filtering scheme [19], it requires a priori information about the statistical properties of external noise. Without such a priori information, the Kalman filtering scheme is not applicable. To handle problems with unknown noise prop-erties, an filtering scheme is proposed in [5] and [6]. Re-cently, the robust mixed filtering for 2-D systems with polytopic uncertainties is also reported in [21] by using a much less conservative parameter-dependent Lyapunov function ap-proach [4]. In practical applications, however, the uncertain pa-rameters may affect the system in a nonlinear fashion. To handle this class of uncertainties, a general uncertainty model, the non-linear fraction transformation (NFT), is first proposed by Tuan

et al. [22]. The NFT model can be transformed into a linear

fraction transformation (LFT) model. However, by comparing the NFT model with other types of uncertainty model, such as the LFT and norm-bounded models, an advantage of the NFT Manuscript received August 6, 2004; revised February 16, 2005. This research is supported by the National Science Council of the Republic of China under Grant NSC 92-2213-E-002-040. The associate editor coordi-nating the review of this manuscript and approving it for publication was Dr. Kenneth E. Barner.

The authors are with the Department of Electrical Engineering, Na-tional Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: ikfong@ cc.ee.ntu.edu.tw).

Digital Object Identifier 10.1109/TSP.2005.861055

model is that it can result in less conservative designs than other models.

As is well known, time delays of signal transmissions are fre-quently encountered in engineering and biological systems. Ex-amples of 2-D systems with time delays include the material rolling process [20] and models described by the delayed lat-tice differential equation [11] and partial difference equations [23], [24]. In addition, certain 2-D systems containing digital processors that need finite numerical computation time [2], [18] display the delay phenomenon. Delays are often a source of in-stability and poor performance. Therefore, for the one-dimen-sional (1-D) state-delayed systems, there have been much lit-erature on the robust filtering that offer various schemes (see, e.g., [8], [9], [13], and [16], and the references cited therein). In contrast, most results for the 2-D filtering problem focus on systems without delays, though for specific stability and con-trol problems of uncertain 2-D discrete state-delayed systems research results [17], [18] start to appear.

In this paper, we propose a complete methodology of robust filter synthesis for 2-D state-delayed systems with uncertainties described by the NFT model. In the systems, it is assumed that time delays appear in both the horizontal and vertical directions. The achievements are summarized as follows. First, we present a computationally tractable sufficient linear matrix inequality (LMI) [1] condition for the stability of 2-D state-delayed sys-tems. This LMI condition plays a crucial role throughout the paper. Second, we develop a less conservative LMI formulation for the and generalized performance of the uncertain 2-D state-delayed systems. Finally, we provide an efficient way to solve the robust , generalized , and mixed general-ized filtering problems by using a parameter-depen-dent Lyapunov function approach, which enables us to obtain less conservative design results.

The notation used throughout the paper is quite standard. is the set of nonnegative integers, is the -dimensional Euclidean space, and is the set of real matrices.

stands for the transpose of a matrix , and

means that the symmetric matrix is positive definite (neg-ative definite). The boldface characters represent matrix vari-ables, and is the Kronecker product. In symmetric block ma-trices, we use as an ellipsis for the terms that are implied by symmetry, and for block-diagonal matrices. The norm of a 2-D signal is defined and denoted by

, where

and is the Euclidean vector norm. We say a 2-D signal if it has a bounded norm. Finally, we shall need the following definitions:

II. PRELIMINARIES

Consider the uncertain 2-D state-delayed system described by the Fornasini–Marchesini second model [12]

(1) where is the state vector, is the disturbance input vector, is the measured output vector,

is the signal vector to be estimated, and

are introduced to handle the nonlinear parameter dependence of the system, and are positive integers denoting time delays along vertical and horizontal directions, respectively, and

(2) It is assumed that

(3)

where is unknown in the unit simplex

(4)

Note that the equations of system (1) may be expressed by the NFT model

(5) where

(6)

Remark 1: When , the NFT model (5) reduces to a polytopic uncertain system with only linear un-certain parameters. On the other hand, when only depends on in (5), it reduces to an LFT model.

In this paper, the basic objective is to find a filter of the form

(7) for the system (1). Define the augmented state vector

and the filtering error output signal . Then we have the error equations

where

(9) III. ROBUSTSTABILITY ANDPERFORMANCECRITERIA The main purpose of this section is to develop some robust stability and performance criteria for 2-D state-delayed systems. These criteria play important roles in solving the robust filtering problems to be discussed in the next section.

A. Stability Analysis

Consider the autonomous nominal 2-D state-delayed system described by

(10)

where and .

Definition 1: The 2-D state-delayed system (10) is

asymp-totically stable if for every initial condition , where

(11) It is known [12] that the system (10) is asymptotically stable if and only if

(12)

for all , where

(13) The above condition is necessary and sufficient for the asymp-totic stability of system (10). Unfortunately, the condition is frequency-dependent and must be checked on infinitely many points in . In the following theorem, a computationally tractable sufficient condition will be given to guarantee the asymptotic stability of the system (10).

Theorem 1: The 2-D state-delayed system (10) is

asymptoti-cally stable if there exist positive definite matrices , , , and such that

(14)

where .

Proof: Suppose the condition (14) is satisfied but (10) is

unstable. Then

(15)

for some , and there exists a nonzero vector such that

(16)

Thus, from (14) and (16)

(17)

where denotes the complex conjugate transpose of , and

It follows from (17) that

(18)

However , , , , ,

and imply that the right-hand side and the left-hand side of (18) are negative and nonnegative, respectively. This leads to a contradiction and concludes the proof.

Remark 2: With the notational change

and , Theorem 1 coincides with Theorem 3 of [18]. Here, the proof is different, and the formulation is arranged for the easy integration with the subsequent performance criteria. In addition, when in (10), the stability condition in Theorem 1 reduces to the well-established stability conditions [14], [15] for 2-D systems without delays.

Remark 3: Clearly, Theorem 1 is a delay-independent

stability condition, which in general is more conservative than delay-dependent results. For stability judgment and state feedback stabilization problems of 2-D state-delayed systems, some delay-dependent results have been derived [3], and the corresponding filter synthesis problems are currently under investigations.

B. Robust Performance

Definition 2: The -norm of the 2-D state-delayed system (8) is defined as

(19)

By the above definition, the -norm of the 2-D delay system (8) is less than or equal to if and only if

for all , , satisfying (4), and . In the following theorem, a sufficient performance condition for the filtering error dynamics (8) is derived.

Theorem 2: Given a scalar , the performance constraint (20) holds for (8) if there exist matrices

, ,

, and such that

(21) and (22), shown at the bottom of the page, for all in the unit

simplex (4), where .

Proof: See Appendix I.

Remark 4: Theorem 2 provides a new robust perfor-mance criterion for 2-D state-delayed systems with NFT un-certainties. In the simpler case where there are no uncertain-ties in the system, Theorem 2 reduces to Theorem 5 of [17]. In another simpler case [21] where the 2-D system has only polytopic uncertainties and no state delays, i.e.,

and , Theorem 2 with and

reduces to Theorem 1 of [21] with .

C. Robust Generalized Performance

Definition 3: The generalized -norm of the 2-D state-de-layed system (8) is defined as

(23)

By the above definition, the generalized -norm of the 2-D delay system (8) is less than or equal to if and only if

(24) for all , , satisfying (4), and . In the following theorem, a sufficient performance condition for the filtering error dynamics (8) is derived.

Theorem 3: Given a scalar , the generalized performance constraint (24) holds for (8) if there exist

a scalar and matrices ,

, , , , and such that (25) (26) (27) and (28), shown at the bottom of the page, for all in the unit

simplex (4), where .

Proof: See Appendix II.

Remark 5: Theorem 3 provides a new robust generalized

performance criteria for 2-D state-delayed systems with NFT uncertainties. In the simpler case in which

and , Theorem 3 with and

reduces to Theorem 2 of [21] with .

(22)

IV. SYNTHESIS OFROBUSTFILTERS

In this section, the LMI approach is adopted to develop convex optimization methods for synthesizing robust filters. First, some shorthand notations are brought in as follows:

(29) With these shorthand notations, system matrices of the filtering error dynamics (8) can be rewritten as

(30) where

(31) It is noted that , , , and are affine functions of the filter system matrix variables and , respectively. This fact is useful in our subsequent development.

A. Robust Filter Synthesis

The robust filtering problem addressed in this paper is as follows. Given a scalar , find a filter (7) such that the fil-tering error dynamics (8) is asymptotically stable and the performance constraint (20) is satisfied for all admissible un-certainties. In order to reduce the conservatism of the resultant filter synthesis method, a parameter-dependent Lyapunov func-tion, which is quadratic in the uncertain parameters [10], will be utilized implicitly. At this point, it is noted that (21) and (22) are not LMIs with respect to the variable matrices but can be converted into ones in the following Lemma.

Lemma 1: If there exist matrices ,

, ,

, ,

, and such that

(32) and (33), shown at the bottom of the page, for

, where ,

then , ,

, and

sat-isfy (21) and (22) of Theorem 2 for all in the unit simplex (4).

Proof: Note first that in (32) and in (33) together ensure that is nonsingular.

With and , (32)

implies

(34) for all in the unit simplex (4). Performing the congruence transformation to (34) yields

(35)

Since , one has

(36) Thus, (21) holds for all in the unit simplex (4), pro-vided the conditions of this Lemma are satisfied. The proof of (22) is similar, with the relevant congruence

transformation and

inequal-ities and

. The detail steps are omitted for the sake of brevity.

Although (32) and (33) in the above lemma are LMIs with re-spect to the variable matrices, further transformations are nec-essary to get LMIs from which filter system matrices can be conveniently obtained. This is accomplished in the following Theorem.

Theorem 4: For the system (1), if there exist matrices

and , such that (32) and the LMI (37), shown at bottom of the page, hold for , where

(38) then the robust filtering problem stated at the begin-ning of this subsection is solvable, and the filter system matrices in (7) can be obtained from any feasible and

as

and .

Proof: From the negative definiteness of , , and in (37), it is seen that is nonsingular. Applying the

congruence transformation to

(37), where and , one can

obtain (33) with

(39)

by using the identities

(40) In addition, from (31) and (40), it is easy to check that

and (41)

Thus, the proof is complete.

Remark 6: In Theorem 4, is regarded as given. However, (37) is still an LMI when is also a variable. Thus, it is possible to formulate the following convex optimization problem to find a filter with the smallest norm:

subject to and for (42)

with respect to and the variables stated in Theorem 4.

B. Robust Generalized Filter Synthesis

The robust generalized filtering problem addressed in this paper is as follows. Given a scalar , find a filter (7) such that the filtering error dynamics (8) is asymptotically stable and the generalized performance constraint (24) is satisfied for all admissible uncertainties. Because the ideas and procedures involved in proving the following Lemma 2 and Theorem 5 are similar to the above arguments for Lemma 1 and Theorem 4, respectively, Lemma 2 and Theorem 5 are stated without proof for the sake of brevity.

Lemma 2: If there exist a scalar and matrices

, , , , ,

, , ,

, , and such that

(43) (44)

and (45) and (46), shown at the bottom of the page, for

, where ,

then , , ,

, ,

, and satisfy (25)–(28) of

Theorem 3 for all in the unit simplex (4).

Theorem 5: For the system (1), if there exist a scalar and

matrices ,

, , , ,

, , ,

, , and

such that the LMIs (43) and (44) (see (47) and (48), shown at bottom of the page) hold for , where

(49) then the robust generalized filtering problem stated at the beginning of this subsection is solvable, and the filter system matrices in (7) can be obtained from any feasible and

as and

.

Remark 7: Similar to that explained in Remark 6, it is

pos-sible to formulate the following convex optimization problem to find a filter with the smallest norm, as follows:

subject to (43), (44), (47), and (48) for

(50) with respect to and the variables stated in Theorem 5.

(45)

(46)

(47)

C. Robust Mixed Generalized Filter Synthesis

By integrating the above results, a robust mixed generalized filtering problem can be addressed as follows. Find a filter (7) for (1) to

subject to and

(51)

for all , , and , where is

a preselected weighting constant for the tradeoff between the and generalized performances. An upper bound for the optimal objective function value of this problem may be found by applying the following Theorem, which is a combination of Theorems 4 and 5.

Theorem 6: An upper bound for the objective function (51)

in the robust mixed generalized filtering problem can be obtained by solving the following convex optimization problem:

subject to (32), (37), (43), (44), (47), and (48) (52) with respect to ,

, , and for . The filter system

ma-trices in (7) can be obtained from the optimal and

as and

.

V. TWOEXAMPLES

Example 1: Consider the uncertain 2-D state-delayed system

(53) where (54) (55) and (56) It is noted that the system can be represented by the NFT and LFT models as follows: • NFT model: (57) (58) and (59)

• LFT model:

(60)

(61) and

(62) By using the MATLAB LMI Control Toolbox [7], the opti-mization problems (42), (50), and (52) are solved for the system (53)–(56) in both NFT and LFT models. The results are shown in Table I, where it is seen that better performances are obtained for the NFT model than the LFT model. In fact for the opti-mization problem (52) with the LFT model, no feasible results exist. It is also worthy noting that for every problem, the com-putation time for the NFT model is much shorter than that for the LFT model. For instance, the problem (42) can be computed in less than 12 min for the NFT model using a Pentium IV PC, and in about one and half hours for the LFT model. In Table II, the tradeoff between the and generalized performances is displayed with three different weighting values of . Clearly, larger results in smaller optimal and larger optimal .

Note that when and ,

Theorems 2 and 3 with and reduce,

respectively, to Theorems 1 and 2 of [21] with . For and in the problem (42) of this example, it is found that the minimum ’s from the above method and the method given by [21] are 2.588 and 2.586, re-spectively. Similarly, the minimum ’s for the problem (50) are 5.090 and 5.088, respectively. It is seen that due to the absence of the variable , the proposed method gives a slightly larger

and .

TABLE I

OPTIMALOBJECTIVEFUNCTIONVALUES FORDIFFERENTSYSTEMMODELS

TABLE II

OPTIMALRESULTS OF(52) WITHDIFFERENTFOR THENFT MODEL

Example 2: Consider a heat diffusion system along a line

described by the partial differential equation

(63) where is the spatial variable, is the time variable, is the temperature of the line at and , is the thermal diffusivity depending on an uncertain parameter

vector , is the control input, and is

the noise input. Suppose depends on nonlinearly as (64) and the system is controlled by a “mixed” state feedback law

, where .

Using the central and back difference approximations

(65) (66) we obtain a discretized approximation of (63)

(67)

where and is selected to be

equal to . For , (67) can be converted into the Fornasini–Marchesini second model of the form (53) with

The matrices and can be represented respectively by (68) (69) where (70) Note that (68) can be expressed by the NFT model

(71) where

(72) For simplicity, we only consider the robust filter design. By solving the optimization problem (42) in this paper, we can obtain the minimum noise attenuation level bound , and the corresponding filter matrices are

Fig. 1 shows the magnitude plot of the filtering error dy-namics over grid frequencies in the range of for and . It can be seen that the maximum magnitude is below the guaranteed noise attenuation level bound. This is also true for other checked uncertainties

.

Fig. 1. Magnitudes of the filtering error transfer dynamics at different frequencies for = 1 and  = 0.

VI. CONCLUSION

For 2-D systems with state delays and uncertainties described by the NFT model, this paper proposes convex optimization based filter synthesis methods. Sufficient conditions are devel-oped in terms of LMI’s for the stability, performance, and generalized performance of the considered 2-D systems. Then, it is shown how to convert the LMIs so that filter gain ma-trices can be obtained efficiently to satisfy the and/or gen-eralized performance constraints. Two examples are given to illustrate the usage of the proposed methods, as well as the advantages of using the NFT model over the LFT model.

APPENDIX I PROOF OFTHEOREM2

First, the asymptotic stability of system (8) is established. For all in the unit simplex (4), (22) implies that

(73)

By the Schur’s complement, (73) is equivalent to

(74) It follows from Theorem 1 that the system (8) is asymptotically stable.

Next, the performance is considered. By the Schur’s complement, (21) is equivalent to

for all , or

(75) for all and satisfying the last equation of (8). Moreover, (22) implies that

(76)

where ,

,

, , and

. It follows from (8) that

which, together with (75), implies

(77) Now, for any integers , (77) leads to

.. .

(78)

Summing up the above inequalities gives

(79) for . Then, summing up (79) for results in

(80) Clearly, with

(81) for all . Thus, the -norm of the system (8) is no greater than . This completes the proof.

APPENDIX II PROOF OFTHEOREM3 First, (28) implies

(82) which can be used to establish the asymptotic stability of system (8), just like how the stability is established from (73) in the proof of Theorem 2. In addition, imitating the argument from (22) to (77) in the proof of Theorem 2, one can show that (28) leads to

(83) For all and satisfying (8), the same reasoning showing that (21) implies (75) also shows that (26) implies the nonnega-tiveness of the sum of the fourth and fifth terms in (83). Hence,

(84) which, with similar steps from (77) to (80), leads to

for . By the Shur’s complement, (27) is equivalent to

(86) Again, with procedures similar to those adopted above, one gets

(87)

from (86), and from

(25). Then

(88) and

(89) Together with (85), one finally obtains

(90) and completes the proof.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions for improving this paper.

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Shyh-Feng Chen was born in Keelung, Taiwan,

R.O.C., in 1962. He received the B.Sc. degree from the National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1988 and the M.S. degree in electrical engineering from the National Taiwan University, Taipei, Taiwan, R.O.C., in 1991. He is currently working toward the Ph.D. degree at the National Taiwan University.

Since August 1991, he has been an instructor with the Department of Electrical Engineering at the China Institute of Technology, Taipei, Taiwan, R.O.C. His research interests include robust filtering and control of two-dimensional systems and time-delay systems.

I-Kong Fong (M’87) received the B.Sc. and Ph.D.

degrees, both in electrical engineering, from the Na-tional Taiwan University, Taipei, Taiwan, R.O.C., in 1981 and 1986, respectively.

From 1984 to 1986, he was an Instructor with the Department of Electrical Engineering, National Taiwan University. In 1986, he worked as a Research Associate at the University of California, Davis. From 1987 to 1993, he as an Associate Professor at the Department of Electrical Engineering, National Taiwan University, where he has been a Professor since 1993. His current research interests include robust control theory, flight control systems, the Stewart platform, and optimization methods.