Robust H ∞ Filtering of 2 D Roesser Discrete Systems : A Polynomial Approach

The problem of robust H∞ filtering is investigated for the class of uncertain two-dimensional 2D discrete systems described by a Roesser state-space model. The main contribution is a systematic procedure for generating conditions for the existence of a 2D discrete filter such that, for all admissible uncertainties, the error system is asymptotically stable, and the H∞ norm of the transfer function from the noise signal to the estimation error is below a prespecified level. These conditions are expressed as parameter-dependent linear matrix inequalities. Using homogeneous polynomially parameter-dependent filters of arbitrary degree on the uncertain parameters, the proposedmethod extends previous results in the quadratic framework and the linearly parameterdependent framework, thus reducing its conservatism. Performance of the proposed method, in comparison with that of existing methods, is illustrated by two examples.


Introduction
Many practical systems can be modeled as two-dimensional 2D systems, such as many systems in image data processing and transmission, in thermal processes, in gas absorption and water stream heating, and so forth 1, 2 .Therefore, in recent years much attention has been devoted to the analysis and synthesis problems for 2D discrete systems, and many important results are available in the literature.For example, the stability of 2D systems based on Lyapunov approaches was investigated in 3-8 ; a 2D dynamic output feedback control, based on solving a set of 2D polynomial equations, was investigated in 9 , whereas the model approximation problem for these systems was addressed in 10 .
In the filtering literature, the most popular method is probably the celebrated Kalman filtering approach, which provides an optimal estimation of the state variables, in the sense that the covariance of the estimation error is minimized 11 .For the 2D system filtering problem, there are already a significant number of results based on the Kalman filtering approach 12-16 , using state-space or polynomial approaches.
This paper concentrates on H ∞ filtering, as it makes it possible to consider uncertainty explicitly and provides guaranteed bounds.For 1D systems, H ∞ filtering has been extensively studied: see, for example, 17-21 and the references therein.However, for 2D systems we can just cite 22,23 , where H ∞ filter designs were proposed, by using an LMI approach, for 2D Roesser models and Fornasini-Marchesini second models, respectively.The major difficulty in developing results in this framework is the lack of the bounded real lemma for 2D systems: most of the 1D system H ∞ filtering results are based on this lemma, which relates the H ∞ performance measure to the solution of certain Riccati equations or inequalities, see 24, 25 .Thus, in this paper we propose a solution to the robust H ∞ filtering problem for uncertain 2D systems by using a structured polynomially parameter-dependent method.The idea exploits the positivity of the uncertain parameters belonging to the unit simplex, constructed in such a way that additional free variables are generated when the degree g of the polynomial filter matrices increases.This makes it possible to define a sequence of sufficient LMI conditions of increasing precision and smaller conservatism.For any fixed degree, these LMI conditions are constructed following simple rules derived from the vertices of the polytope.It is also shown that if the LMI conditions are fulfilled for a certain degree, then a feasible solution exists for higher degrees.Moreover, the condition proposed reduces, when g 0, to the filter design method in the quadratic framework given in 26 and are equivalent to the sufficient LMI tests based on an affine parameter-dependent Lyapunov function for g 1, which shows the reduced conservatism of the proposed approach.

Problem Formulation
Consider a 2D discrete system described by the following Roesser's state-space model:

2.1
where x h i, j ∈ n h and x v i, j ∈ n v are the horizontal and vertical states, respectively, with the boundary conditions x h 0, k x h 0 , x v k, 0 x v 0 for all k, y i, j ∈ p is the measured output, z i, j ∈ r is the signal to be estimated and w i, j ∈ m is the exogenous input.
To describe the uncertainty, all matrices are assumed to be real, belonging to the polytope where Ω is the unit simplex, defined by The 2D transfer function from the noise ω t 1 , t 2 to the estimated output z t 1 , t 2 is then given by where This 2D transfer function will be central in the rest of the paper, so the following definition of H ∞ norm is given explicitly for this transfer function.
Definition 2.1.The H ∞ norm of the transfer function T zw ω 1 , ω 2 of the 2D discrete system 2.1 is given by σ max T zw e jω 1 , e jω 2 .

2.6
In this paper, the basic objective is to find a filter of the form in order to estimate the signal z from the measurements of y, where x h f i, j ∈ n h and x v f i, j ∈ n v are, respectively, the horizontal and vertical states of the filter, z f i, j ∈ p is the estimate of z i, j , and A f α , B f α , and C f α , are the filter parameter matrices, to be determined using the technique developed in this paper.

Mathematical Problems in Engineering
We now give a proper definition of the H ∞ norm of the filtering error.First, define the augmented state vectors and the filtering error output signal, respectively, by

2.8
Then, combining these definitions with 2.1 and 2.7 , the error dynamic equations are just

2.9
where and the augmented matrices are given by The problem addressed in this paper is then defined as follows.
Definition 2.2.The robust H ∞ filtering problem consists of finding a filter 2.7 such that the filtering error dynamics 2.9 is asymptotically stable and the transfer function of the error system, given as with γ > 0 a given real number.

Preliminaries: Constant P Matrix
In order to solve the filtering problem presented in the previous section we first introduce a lemma presented in 27 which considers a parameter-independent structure for P α , that is, P α P P T .This still corresponds to a quadratic framework, but it will be the base behind the nonquadratic approach presented in the rest of the paper alternative nonquadratic approaches can be found in 17, 18 .
Lemma 3.1.Given a scalar γ > 0, the 2D discrete system 2.9 is asymptotically stable and satisfies the H ∞ performance T zw < γ if there exists a matrix P diag P h , P v > 0, with P h ∈ n h ×n h and P v ∈ n v ×n v , such that the following LMI holds: Now, we are in a position to present a preliminary solvability condition for the robust H ∞ filtering problem.Theorem 3.2.Given a scalar γ > 0 and the uncertain 2D discrete system 2.1 , then, the robust H ∞ filtering problem is solvable if there exists matrices Z α , Θ α , Ψ α , X diag X h , X v > 0, and where In this case, a 2D discrete filter in the form of 2.7 is obtained when the parameters are selected as follows: where , and Y v12 are nonsingular matrices satisfying then the relations 3.3 , can be written as By the Schur complement formula, it follows from 3.7 that which implies I − XY is nonsingular.Therefore, by noting the structure of X and Y , there always exist nonsingular matrices X h12 , X v12 , Y h12 , and Y v12 such that 3.6 is satisfied, that is,

3.10
Then, by some calculations, it can be verified that where

3.12
Mathematical Problems in Engineering 7 Observe that

3.13
Therefore, it is easy to see that P h > 0 and P v > 0. Now, before and after multiplying 3.2 by diag{Y , I, Y , I, I, I}, we obtain Let A f α , B f α , and C f α are given in 3.4 , Φ is given in 2.14 .By 3.11 , the inequality 3.14 can be rewritten as

3.15
Pre-and postmultiplying 3.15 by diag Π Finally, by Lemma 3.1, it follows that the error system 2.9 is asymptotically stable, and the transfer function of the error system satisfies 2.16 .This completes the proof.
Remark 3.3.Theorem 3.2 provides a method for designing H ∞ for fixed α, which casts the nonlinear matrix inequality in Lemma 3.1 into a linear matrix inequality.It is noted that the condition in Theorem 3.2 is dependent on the parameter α, and therefore the decision variables X and Y cannot be used due to the infinite-dimensional nature of the parameter α.
In what follows, based on Theorem 3.2, we propose a new method for designing robust H ∞ filters via a structured polynomially parameter-dependent approach.

Homogenous Polynomially Parameter-Dependent (HPPD) Matrices
Before presenting the main result, some definitions and preliminaries are needed to represent and to handle products and sums of homogeneous polynomials.First, define the HPPD matrices of arbitrary degree g as follows: The above notations are explained as follows: . ., N are the monomials; Ψ K j g , Θ K j g , and Z K j g , are matrices with the corresponding coefficients; K j g is the jth N-tuple of K g , lexically ordered, with j 1, . . ., J g ; finally, K g is the set of N-tuples obtained as all possible combinations of k Since the number of vertices in the polytope P is equal to N, the number of elements in K g is given by J g N g − 1 !/ g! N − 1 ! .Next, for each i 1, . . ., N define the N-tuples K i j g , that are equal to K j g , but with k i > 0 replaced by k i − 1.Note that these K i j g are defined only when the corresponding k i is positive.Note also that, when applied to the elements of K g 1 , the N-tuples to homogeneous polynomial parameter-dependent matrices of degree g.
Finally, define the scalar constant coefficients To clarify this notation, consider as an example a possible polytope with N 3 vertices.The polynomials of degree g 2 are obtained as follows: First, J 2 6, K 2 {002, 011, 020, 101, 110, 200}, so the polynomials of degree 2 are

3.19
Moreover, the 3 tuples are K

Main Result
Using Lemma 3.1 and the homogeneous polynomials just presented, we can derive our main result.
Theorem 3.4.Given a scalar γ > 0 and the uncertain 2D discrete system 2.1 , then, the robust H ∞ filtering problem given in Definition 2.2 is solvable if there exist matrices , such that for all K l g 1 ∈ K g 1 , l 1, . . ., J g 1 such that the following LMIs hold: where

3.22
then the homogeneous polynomially parameter-dependent matrices given by 3.17 ensure 3.2 for all α ∈ Ω.Moreover, if the LMIs of 3.20 -3.21 are fulfilled for a given degree g, then the LMIs corresponding to any degree g > g are also satisfied.

Mathematical Problems in Engineering
In this case, the matrices of the 2D discrete-time HPPD filter are given by

3.23
with

3.25
Condition 3.20 , when imposed for all l 1, . . ., J g 1 , ensure condition 3.2 for all α ∈ Ω, and thus the first part is proved.
Suppose that the LMIs of 3.20 -3.21 are fulfilled for a certain degree g, that is, there exist J g matrices Ψ K j g , Θ K j g , and Z K j g , j 1, . . ., J g , such that Ψ g α , Θ g α , and Z g α are homogeneous polynomially parameter-dependent matrices assuring condition in 3.2 -3.3 .Then, the terms of the polynomial matrices satisfy the LMIs of Theorem 3.4 corresponding to the degree g 1, which can be obtained in this case by a linear combination of the LMIs of Theorem 3.4 for g.Remark 3.5.Note that when g 0, we have Ψ g α Ψ, Θ g α Θ, and Z g α Z, then A f α A f , B f α B f , and C f α C f , which will lead to the standard filtering result in the quadratic framework.In addition, when g 1, they are linearly dependent on the parameter α.This is why we say the polynomial α-dependence encompasses the linear α-dependence as a special case.It is also worth noting that since all coefficients α i , i 1, . . ., N, are such that α ∈ Ω.As the degree g of the polynomial increases, the conditions become less conservative since new variables are added to LMIs.Although the number of LMIs is also increasing, each LMI becomes easier to be fulfilled due to the extra degrees of freedom provided by the new variables.

Illustrative Examples
In this section, some numerical examples are presented to illustrate the proposed technique for robust H ∞ HPPD filters.
Example 4.1.A stationary random field can be modeled as the following 2D system 13 : where η i, j is the state of the random field of spacial coordinate i, j , w 1 i, j is a noise input, a 2 1 < 1, and a 2 2 < 1 as a 1 and a 2 represent, respectively, the vertical and horizontal correlations of the random field.The output is then where w 2 i, j is the measurement noise.The signal to be estimated is z i, j Cη i, j Dw i, j .

4.3
As in 28 , define x h i, j η i, j 1 − a 2 η i, j and x v i, j η i, j .It is easy to see that 4.1 -4.3 can be converted into a 2D Roesser model of the form 2.1 with w i, j w 1 i, j w 2 i, j T and the following system matrices:

4.4
Suppose that the uncertain parameters a 1 and a 2 are bounded by 0.15 ≤ a 1 ≤ 0.45 and 0.35 ≤ a 2 ≤ 0.85, so the above system can be represented by a four-vertex polytopic system.The H ∞ filtering design approach presented in this paper was applied to this system.The results of a comparison with the techniques proposed in 29, 30 are shown in Table 1, which shows the smaller conservativeness of the approach proposed in this paper.For the filter designed when g 2, the actual H ∞ norms calculated at the four vertices are shown in Table 2; the corresponding frequency responses of the error system are given in Figures 1, 2, 3, and 4 for each of the vertices, all of which are clearly below the guaranteed bound 1.5713.Example 4.2.In this example we show that less conservative designs are achieved as the degree of the polynomial grows, when applying the HPPD approach.For this, consider the same system of Example 4.1 but this time with C 1 −1 1.6 − a 1 , 0.15 ≤ a 1 ≤ 0.65.The results are given in Table 3, which also provides a comparison with previous results in the literature.

Conclusions
This paper has studied the robust H ∞ filtering problem for 2D discrete systems described by Roesser state-space model.The proposed method, based on using homogeneous polynomially parameter-dependent matrices of arbitrary degree, is less conservative than previous ones in the literature.Moreover, by increasing the degree of the polynomials involved, the obtained filter gets less conservative, which has been demonstrated by two illustrative examples, which provides comparisons with previous results.

2 Figure 1 :
Figure 1: Frequency response of error system for vertex 1.

2 Figure 2 :
Figure 2: Frequency response of error system for vertex 2.

2 Figure 3 :
Figure 3: Frequency response of error system for vertex 3.

2 Figure 4 :
Figure 4: Frequency response of error system for vertex 4.

Table 1 :
Example 4.1: comparison with previous published results.

Table 2 :
H ∞ norms at the vertices.

Table 3 :
Example 4.2: comparison with previous published results.