# RobustH∞ Filtering of 2D Roesser Discrete Systems: A Polynomial Approach

Authors: Chakir El-Kasri; Abdelaziz Hmamed; Teresa Alvarez; Fernando Tadeo
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The problem of robustH∞ 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 proposed method extends previous results in the quadratic framework and the linearly parameter-dependent framework, thus reducing its conservatism. Performance of the proposed method, in comparison with that of existing methods, is illustrated by two examples.


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## Body

## 1. 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 onH∞ 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 robustH∞ 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.


## 2. Problem Formulation

Consider a 2D discrete system described by the following Roesser’s state-space model:(2.1)[xh(i+1,j)xv(i,j+1)]=A(α)[xh(i,j)xv(i,j)]+B(α)w(i,j),y(i,j)=C1(α)[xh(i,j)xv(i,j)]+D1(α)w(i,j),z(i,j)=C(α)[xh(i,j)xv(i,j)]+D(α)w(i,j),
where xh(i,j)∈ℜnh and xv(i,j)∈ℜnv are the horizontal and vertical states, respectively, with the boundary conditions xh(0,k)=x0h,xv(k,0)=x0vforallk, y(i,j)∈ℜp is the measured output, z(i,j)∈ℜr is the signal to be estimated and w(i,j)∈ℜm is the exogenous input.To describe the uncertainty, all matrices are assumed to be real, belonging to the polytope(2.2)P≜{[A(α)B(α)C1(α)D1(α)C(α)D(α)]=∑i=1Nαi(AiBiC1iD1iCiDi),α∈Ω},
where Ω is the unit simplex, defined by
(2.3)Ω={(α1,α2,…,αN):∑i=1Nαi=1,αi≥0}.
The 2D transfer function from the noise ω(t1,t2) to the estimated output z(t1,t2) is then given by
(2.4)Tzw(z1,z2)=C(α)(I(z1,z2)-A(α))-1B(α)+D(α),
where
(2.5)I(z1,z2)=diag⁡(z1Inh,z2Inv).
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.
TheH∞ norm of the transfer function Tzw(ω1,ω2) of the 2D discrete system (2.1) is given by
(2.6)‖Tzw(z1,z2)‖∞=sup⁡ω1,ω2∈[02π]σmax⁡[Tzw(ejω1,ejω2)].In this paper, the basic objective is to find a filter of the form(2.7)[xfh(i+1,j)xfv(i,j+1)]=Af(α)[xfh(i,j)xfv(i,j)]+Bf(α)y(i,j),zf(i,j)=Cf(α)[xfh(i,j)xfv(i,j)],
in order to estimate the signal z from the measurements of y, where xfh(i,j)∈ℜnh and xfv(i,j)∈ℜnv are, respectively, the horizontal and vertical states of the filter, zf(i,j)∈ℜp is the estimate of z(i,j), and Af(α), Bf(α), and Cf(α), are the filter parameter matrices, to be determined using the technique developed in this paper.We now give a proper definition of theH∞ norm of the filtering error. First, define the augmented state vectors and the filtering error output signal, respectively, by
(2.8)x̃h(i,j)=[xh(i,j)Txfh(i,j)T]T,x̃v(i,j)=[xv(i,j)Txfv(i,j)T]T,z̃(i,j)=z(i,j)-zf(i,j).
Then, combining these definitions with (2.1) and (2.7), the error dynamic equations are just
(2.9)[x̃h(i+1,j)x̃v(i,j+1)]=Ã(α)[x̃h(i,j)x̃v(i,j)]+B̃(α)w(i,j),z̃(i,j)=C̃(α)[x̃h(i,j)x̃v(i,j)]+D̃(α)w(i,j),
where
(2.10)Ã(α)=ΦA¯(α)ΦT,C̃(α)=C¯(α)ΦT,B̃(α)=ΦB¯(α),D̃(α)=D(α),
and the augmented matrices are given by
(2.11)A¯(α)=[A(α)0Bf(α)C1(α)Af(α)],(2.12)B¯(α)=[B(α)Bf(α)D1(α)],(2.13)C¯(α)=[C(α)-Cf(α)],(2.14)Φ=[Inh00000Inh00Inv00000Inv].
The problem addressed in this paper is then defined as follows.Definition 2.2.
The robustH∞ 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
(2.15)Tz̃w=C̃(α)[I(z1,z2)-Ã(α)]-1B̃(α)+D̃(α),
satisfies
(2.16)‖Tz̃w‖∞<γ,
with γ>0 a given real number.


## 3. RobustH∞ Filter Design

### 3.1. Preliminaries: ConstantP 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=PT. 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 scalarγ>0, the 2D discrete system (2.9) is asymptotically stable and satisfies the H∞ performance ∥Tz̃w∥<γ if there exists a matrix P̃=diag(P̃h,P̃v)>0, with P̃h∈ℜnh×nh and P̃v∈ℜnv×nv, such that the following LMI holds:
(3.1)(-P̃P̃Ã(α)P̃B̃(α)0*-P̃0C̃(α)T**-γ2ID̃(α)T***-I)<0.Now, we are in a position to present a preliminary solvability condition for the robustH∞ filtering problem.Theorem 3.2.
Given a scalarγ>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(Xh,Xv)>0, and Y=diag(Yh,Yv)>0 with Xh, Yh∈ℝnh×nh and Xv, Yv∈ℝnv×nv such that the following LMIs hold:
(3.2)[-Y-YYA(α)YA(α)YB(α)0*-XXA(α)+Ψ(α)C1(α)+Z(α)XA(α)+Ψ(α)C1(α)M240**-Y-Y0C(α)T-Θ(α)T***-X0C(α)T****-γ2ID(α)T*****-I]<0,(3.3)X-Y>0,
where M24=XB(α)+Ψ(α)D1(α).In this case, a 2D discrete filter in the form of (2.7) is obtained when the parameters are selected as follows:
(3.4)Af(α)=X12-1Z(α)Y-1Y12-T,Bf(α)=X12-1Ψ(α),Cf(α)=Θ(α)Y-1Y12-T,
where
(3.5)X12=[Xh1200Xv12],Y12=[Yh1200Yv12],
in which Xh12, Xv12, Yh12, and Yv12 are nonsingular matrices satisfying
(3.6)X12Y12T=I-XY-1.Proof.
LetY¯h=Yh-1, Y¯v=Yv-1, Y¯=Y-1 then the relations (3.3), can be written as
(3.7)(XIIY¯)>0.
By the Schur complement formula, it follows from (3.7) that
(3.8)Y¯-X-1>0,
which implies I-XY¯ is nonsingular. Therefore, by noting the structure of X and Y, there always exist nonsingular matrices Xh12, Xv12, Yh12, and Yv12 such that (3.6) is satisfied, that is,
(3.9)Xh12Yh12T=I-XhY¯h,Xv12Yv12T=I-XvY¯v.
Set
(3.10)Πh1=[Y¯hIYh12T0],Πv1=[Y¯vIYv12T0],Πh2=[IXh0Xh12T],Πv2=[IXv0Xv12T],Π1=[Πh100Πv1],Π2=[Πh200Πv2].
Then, by some calculations, it can be verified that
(3.11)P̃:=Π2Π1-1=[P̃h00P̃v],
where
(3.12)P̃h=[XhXh12Xh12TXh12T(Xh-Yh)-1Xh12],P̃v=[XvXv12Xv12TXv12T(Xv-Yv)-1Xv12].
Observe that
(3.13)Xh-Xh12[Xh12T(Xh-Yh)-1Xh12]-1Xh12T=Yh>0,Xv-Xv12[Xv12T(Xv-Yv)-1Xv12]-1Xv12T=Yv>0.
Therefore, it is easy to see that P̃h>0 and P̃v>0. Now, before and after multiplying (3.2) by diag{Y¯,I,Y¯,I,I,I}, we obtain
(3.14)[-Y¯-InA(α)Y¯A(α)B(α)0*-XM23XA(α)+X12Bf(α)C1(α)XB(α)+X12Bf(α)D1(α)0**-Y¯-In0Y¯C(α)T-Y12(α)T***-X0C(α)T****-γ2ID(α)T*****-I]<0,
where M23=XA(α)Y¯+X12Bf(α)C1(α)Y¯+X12Af(α)Y12T.
LetAf(α), Bf(α), and Cf(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)[-ΦTΠ1TP̃Π1ΦΦTΠ1TP̃ΦA¯(α)ΦTΠ1ΦΦTΠ1TP̃ΦB¯(α)0ΦTΠ1TΦA¯(α)TΦTP̃Π1Φ-ΦTΠ1TP̃Π1Φ0ΦTΠ1TΦC¯(α)TB¯(α)TΦTP̃Π1Φ0-γ2ID̃(α)T0C¯(α)ΦTΠ1ΦD̃(α)-I]<0.
Pre- and postmultiplying (3.15) by diag(Π1-TΦ-T,Π1-TΦ-T,I,I) and diag(Φ-1Π1-1,Φ-1Π1-1,I,I) result in
(3.16)(-P̃P̃Ã(α)P̃B̃(α)0*-P̃0C̃(α)T**-γ2ID̃(α)T***-I)<0.
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.
Theorem3.2 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.


### 3.2. 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 degree𝔤 as follows:
(3.17)Ψg(α)=∑j=1J(g)α1k1α2k2⋯αNkNΨKj(g),Θg(α)=∑j=1J(g)α1k1α2k2⋯αNkNΘKj(g),Zg(α)=∑j=1J(g)α1k1α2k2⋯αNkNZKj(g),
with
(3.18)k1k2⋯kN=Kj(g).
The above notations are explained as follows: α1k1α2k2⋯αNkN, α∈Ω, ki∈ℕ, i=1,…,N are the monomials; Ψ𝔎j(𝔤), Θ𝔎j(𝔤), and Z𝔎j(𝔤), are matrices with the corresponding coefficients; 𝔎j(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 k1k2⋯kN, with ki∈ℕ, i=1,…,N, that fulfill k1+k2+⋯+kN=𝔤. 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 eachi=1,…,N define the N-tuples 𝔎ji(𝔤), that are equal to 𝔎j(𝔤), but with ki>0 replaced by ki-1. Note that these 𝔎ji(𝔤) are defined only when the corresponding ki is positive. Note also that, when applied to the elements of 𝔎(𝔤+1), the N-tuples 𝔎ji(𝔤+1) define subscripts k1k2⋯kN of matrices Ψk1k2⋯kN, Θk1k2⋯kN, and Zk1k2⋯kN associated to homogeneous polynomial parameter-dependent matrices of degree 𝔤.Finally, define the scalar constant coefficientsβji(j+1)=𝔤!/(k1!k2!⋯kN!), with k1k2⋯kN∈𝔎ji(𝔤+1).To clarify this notation, consider as an example a possible polytope withN=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)Ψ2(α)=α32Ψ002+α2α3Ψ011+α22Ψ020+α1α3Ψ101+α1α2Ψ110+α12Ψ200,Θ2(α)=α32Θ002+α2α3Θ011+α22Θ020+α1α3Θ101+α1α2Θ110+α12Θ200,Z2(α)=α32Z002+α2α3Z011+α22Z020+α1α3Z101+α1α2Z110+α12Z200.
Moreover, the 3 tuples are 𝔎13(2)=001, 𝔎22(2)=001, 𝔎23(2)=010, 𝔎32(2)=010, 𝔎41(2)=001, 𝔎43(2)=100, 𝔎51(2)=010, 𝔎52(2)=100, and 𝔎61(2)=100: these are the only possible triples (3 tuples) 𝔎ji(2), j=1,…,𝔍(2) associated to 𝔎(2).


### 3.3. Main Result

Using Lemma3.1 and the homogeneous polynomials just presented, we can derive our main result.Theorem 3.4.
Given a scalarγ>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 Z𝔎𝔧(𝔤), Θ𝔎𝔧(𝔤), Ψ𝔎𝔧(𝔤), 𝔎𝔧(𝔤)∈𝔎(𝔤), j=1,…,𝔍(𝔤), X=diag(Xh,Xv)>0, and Y=diag(Yh,Yv)>0 with Xh, Yh∈ℝnh and Xv, Yv∈ℝnv, such that for all 𝔎l(𝔤+1)∈𝔎(𝔤+1), l=1,…,𝔍(𝔤+1) such that the following LMIs hold:
(3.20)∑i∈Nl(g+1)[-βli(g+1)Y-βli(g+1)Yβli(g+1)YAiβli(g+1)YAiβli(g+1)YBi0*-βli(g+1)XJ23J24J250**-βli(g+1)Y-βli(g+1)Y0J36***-βli(g+1)X0J46****-γ2IJ56*****-I]<0,(3.21)X-Y>0,
where
(3.22)J13=βli(g+1)YAi,J15=βli(g+1)YBi,J23=βli(g+1)XAi+ΨKli(g+1)C1i+ZKli(g+1),J24=βli(g+1)XAi+ΨKli(g+1)C1i,J25=βli(g+1)XBi+ΨKli(g+1)D1i,J36=βli(g+1)CiT-ΘKli(g+1)T,J46=βli(g+1)CiT,J56=βli(g+1)DiT,
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>𝔤̂ are also satisfied.In this case, the matrices of the 2D discrete-time HPPD filter are given by(3.23)Afg(α)=∑j=1J(g)α1k1α2k2⋯αNkNAfKj(g),Bfg(α)=∑j=1J(g)α1k1α2k2⋯αNkNBfKj(g),Cfg(α)=∑j=1J(g)α1k1α2k2⋯αNkNCfKj(g),
with
(3.24)k1k2⋯kN=Kj(g),AfKj(g)=X12-1ZKj(g)Y-1Y12-T,BfKj(g)=X12-1ΨKj(g),CfKj(g)=ΘKj(g)Y-1Y12-T.Proof.
Note that (3.2) for (A(α),B(α),C1(α), D1(α), C(α),D(α))∈𝒫, and Ψ(α),Θ(α),Z(α) given by (3.17) are homogeneous polynomial matrices equations of degree 𝔤+1 that can be written as
(3.25)∑l=1J(g+1)αk{∑i∈Nl(g+1)[-βli(g+1)Y-βli(g+1)YJ13J13J150*-βli(g+1)XJ23J24J250**-βli(g+1)Y-βli(g+1)Y0J36***-βli(g+1)X0J46****-γ2IJ56*****-I]}<0,k1k2⋯kN=Kl(g+1),αk=α1k1α2k2⋯αNkN.
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 𝔤̂, that is, there exist 𝔍(𝔤̂) matrices Ψ𝔎j(ĝ), Θ𝔎j(ĝ), and Z𝔎j(ĝ), j=1,…,𝔍(𝔤̂), such that Ψ𝔤̂(α), Θ𝔤̂(α), and Z𝔤̂(α) are homogeneous polynomially parameter-dependent matrices assuring condition in (3.2)-(3.3). Then, the terms of the polynomial matrices Ψ𝔤̂+1(α)=(α1+⋯+αN)Ψ𝔤̂(α), Θ𝔤̂+1(α)=(α1+⋯+αN)Θ𝔤̂(α) and Z𝔤̂+1(α)=(α1+⋯+αN)Z𝔤̂(α) satisfy the LMIs of Theorem 3.4 corresponding to the degree 𝔤̂+1, which can be obtained in this case by a linear combination of the LMIs of Theorem 3.4 for 𝔤̂.Remark 3.5.
Note that when𝔤=0, we have Ψ𝔤(α)=Ψ, Θ𝔤(α)=Θ, and Z𝔤(α)=Z, then Af(α)=Af, Bf(α)=Bf, and Cf(α)=Cf, 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 αi,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.


## 3.1. Preliminaries: ConstantP 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=PT. 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 scalarγ>0, the 2D discrete system (2.9) is asymptotically stable and satisfies the H∞ performance ∥Tz̃w∥<γ if there exists a matrix P̃=diag(P̃h,P̃v)>0, with P̃h∈ℜnh×nh and P̃v∈ℜnv×nv, such that the following LMI holds:
(3.1)(-P̃P̃Ã(α)P̃B̃(α)0*-P̃0C̃(α)T**-γ2ID̃(α)T***-I)<0.Now, we are in a position to present a preliminary solvability condition for the robustH∞ filtering problem.Theorem 3.2.
Given a scalarγ>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(Xh,Xv)>0, and Y=diag(Yh,Yv)>0 with Xh, Yh∈ℝnh×nh and Xv, Yv∈ℝnv×nv such that the following LMIs hold:
(3.2)[-Y-YYA(α)YA(α)YB(α)0*-XXA(α)+Ψ(α)C1(α)+Z(α)XA(α)+Ψ(α)C1(α)M240**-Y-Y0C(α)T-Θ(α)T***-X0C(α)T****-γ2ID(α)T*****-I]<0,(3.3)X-Y>0,
where M24=XB(α)+Ψ(α)D1(α).In this case, a 2D discrete filter in the form of (2.7) is obtained when the parameters are selected as follows:
(3.4)Af(α)=X12-1Z(α)Y-1Y12-T,Bf(α)=X12-1Ψ(α),Cf(α)=Θ(α)Y-1Y12-T,
where
(3.5)X12=[Xh1200Xv12],Y12=[Yh1200Yv12],
in which Xh12, Xv12, Yh12, and Yv12 are nonsingular matrices satisfying
(3.6)X12Y12T=I-XY-1.Proof.
LetY¯h=Yh-1, Y¯v=Yv-1, Y¯=Y-1 then the relations (3.3), can be written as
(3.7)(XIIY¯)>0.
By the Schur complement formula, it follows from (3.7) that
(3.8)Y¯-X-1>0,
which implies I-XY¯ is nonsingular. Therefore, by noting the structure of X and Y, there always exist nonsingular matrices Xh12, Xv12, Yh12, and Yv12 such that (3.6) is satisfied, that is,
(3.9)Xh12Yh12T=I-XhY¯h,Xv12Yv12T=I-XvY¯v.
Set
(3.10)Πh1=[Y¯hIYh12T0],Πv1=[Y¯vIYv12T0],Πh2=[IXh0Xh12T],Πv2=[IXv0Xv12T],Π1=[Πh100Πv1],Π2=[Πh200Πv2].
Then, by some calculations, it can be verified that
(3.11)P̃:=Π2Π1-1=[P̃h00P̃v],
where
(3.12)P̃h=[XhXh12Xh12TXh12T(Xh-Yh)-1Xh12],P̃v=[XvXv12Xv12TXv12T(Xv-Yv)-1Xv12].
Observe that
(3.13)Xh-Xh12[Xh12T(Xh-Yh)-1Xh12]-1Xh12T=Yh>0,Xv-Xv12[Xv12T(Xv-Yv)-1Xv12]-1Xv12T=Yv>0.
Therefore, it is easy to see that P̃h>0 and P̃v>0. Now, before and after multiplying (3.2) by diag{Y¯,I,Y¯,I,I,I}, we obtain
(3.14)[-Y¯-InA(α)Y¯A(α)B(α)0*-XM23XA(α)+X12Bf(α)C1(α)XB(α)+X12Bf(α)D1(α)0**-Y¯-In0Y¯C(α)T-Y12(α)T***-X0C(α)T****-γ2ID(α)T*****-I]<0,
where M23=XA(α)Y¯+X12Bf(α)C1(α)Y¯+X12Af(α)Y12T.
LetAf(α), Bf(α), and Cf(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)[-ΦTΠ1TP̃Π1ΦΦTΠ1TP̃ΦA¯(α)ΦTΠ1ΦΦTΠ1TP̃ΦB¯(α)0ΦTΠ1TΦA¯(α)TΦTP̃Π1Φ-ΦTΠ1TP̃Π1Φ0ΦTΠ1TΦC¯(α)TB¯(α)TΦTP̃Π1Φ0-γ2ID̃(α)T0C¯(α)ΦTΠ1ΦD̃(α)-I]<0.
Pre- and postmultiplying (3.15) by diag(Π1-TΦ-T,Π1-TΦ-T,I,I) and diag(Φ-1Π1-1,Φ-1Π1-1,I,I) result in
(3.16)(-P̃P̃Ã(α)P̃B̃(α)0*-P̃0C̃(α)T**-γ2ID̃(α)T***-I)<0.
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.
Theorem3.2 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.


## 3.2. 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 degree𝔤 as follows:
(3.17)Ψg(α)=∑j=1J(g)α1k1α2k2⋯αNkNΨKj(g),Θg(α)=∑j=1J(g)α1k1α2k2⋯αNkNΘKj(g),Zg(α)=∑j=1J(g)α1k1α2k2⋯αNkNZKj(g),
with
(3.18)k1k2⋯kN=Kj(g).
The above notations are explained as follows: α1k1α2k2⋯αNkN, α∈Ω, ki∈ℕ, i=1,…,N are the monomials; Ψ𝔎j(𝔤), Θ𝔎j(𝔤), and Z𝔎j(𝔤), are matrices with the corresponding coefficients; 𝔎j(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 k1k2⋯kN, with ki∈ℕ, i=1,…,N, that fulfill k1+k2+⋯+kN=𝔤. 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 eachi=1,…,N define the N-tuples 𝔎ji(𝔤), that are equal to 𝔎j(𝔤), but with ki>0 replaced by ki-1. Note that these 𝔎ji(𝔤) are defined only when the corresponding ki is positive. Note also that, when applied to the elements of 𝔎(𝔤+1), the N-tuples 𝔎ji(𝔤+1) define subscripts k1k2⋯kN of matrices Ψk1k2⋯kN, Θk1k2⋯kN, and Zk1k2⋯kN associated to homogeneous polynomial parameter-dependent matrices of degree 𝔤.Finally, define the scalar constant coefficientsβji(j+1)=𝔤!/(k1!k2!⋯kN!), with k1k2⋯kN∈𝔎ji(𝔤+1).To clarify this notation, consider as an example a possible polytope withN=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)Ψ2(α)=α32Ψ002+α2α3Ψ011+α22Ψ020+α1α3Ψ101+α1α2Ψ110+α12Ψ200,Θ2(α)=α32Θ002+α2α3Θ011+α22Θ020+α1α3Θ101+α1α2Θ110+α12Θ200,Z2(α)=α32Z002+α2α3Z011+α22Z020+α1α3Z101+α1α2Z110+α12Z200.
Moreover, the 3 tuples are 𝔎13(2)=001, 𝔎22(2)=001, 𝔎23(2)=010, 𝔎32(2)=010, 𝔎41(2)=001, 𝔎43(2)=100, 𝔎51(2)=010, 𝔎52(2)=100, and 𝔎61(2)=100: these are the only possible triples (3 tuples) 𝔎ji(2), j=1,…,𝔍(2) associated to 𝔎(2).


## 3.3. Main Result

Using Lemma3.1 and the homogeneous polynomials just presented, we can derive our main result.Theorem 3.4.
Given a scalarγ>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 Z𝔎𝔧(𝔤), Θ𝔎𝔧(𝔤), Ψ𝔎𝔧(𝔤), 𝔎𝔧(𝔤)∈𝔎(𝔤), j=1,…,𝔍(𝔤), X=diag(Xh,Xv)>0, and Y=diag(Yh,Yv)>0 with Xh, Yh∈ℝnh and Xv, Yv∈ℝnv, such that for all 𝔎l(𝔤+1)∈𝔎(𝔤+1), l=1,…,𝔍(𝔤+1) such that the following LMIs hold:
(3.20)∑i∈Nl(g+1)[-βli(g+1)Y-βli(g+1)Yβli(g+1)YAiβli(g+1)YAiβli(g+1)YBi0*-βli(g+1)XJ23J24J250**-βli(g+1)Y-βli(g+1)Y0J36***-βli(g+1)X0J46****-γ2IJ56*****-I]<0,(3.21)X-Y>0,
where
(3.22)J13=βli(g+1)YAi,J15=βli(g+1)YBi,J23=βli(g+1)XAi+ΨKli(g+1)C1i+ZKli(g+1),J24=βli(g+1)XAi+ΨKli(g+1)C1i,J25=βli(g+1)XBi+ΨKli(g+1)D1i,J36=βli(g+1)CiT-ΘKli(g+1)T,J46=βli(g+1)CiT,J56=βli(g+1)DiT,
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>𝔤̂ are also satisfied.In this case, the matrices of the 2D discrete-time HPPD filter are given by(3.23)Afg(α)=∑j=1J(g)α1k1α2k2⋯αNkNAfKj(g),Bfg(α)=∑j=1J(g)α1k1α2k2⋯αNkNBfKj(g),Cfg(α)=∑j=1J(g)α1k1α2k2⋯αNkNCfKj(g),
with
(3.24)k1k2⋯kN=Kj(g),AfKj(g)=X12-1ZKj(g)Y-1Y12-T,BfKj(g)=X12-1ΨKj(g),CfKj(g)=ΘKj(g)Y-1Y12-T.Proof.
Note that (3.2) for (A(α),B(α),C1(α), D1(α), C(α),D(α))∈𝒫, and Ψ(α),Θ(α),Z(α) given by (3.17) are homogeneous polynomial matrices equations of degree 𝔤+1 that can be written as
(3.25)∑l=1J(g+1)αk{∑i∈Nl(g+1)[-βli(g+1)Y-βli(g+1)YJ13J13J150*-βli(g+1)XJ23J24J250**-βli(g+1)Y-βli(g+1)Y0J36***-βli(g+1)X0J46****-γ2IJ56*****-I]}<0,k1k2⋯kN=Kl(g+1),αk=α1k1α2k2⋯αNkN.
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 𝔤̂, that is, there exist 𝔍(𝔤̂) matrices Ψ𝔎j(ĝ), Θ𝔎j(ĝ), and Z𝔎j(ĝ), j=1,…,𝔍(𝔤̂), such that Ψ𝔤̂(α), Θ𝔤̂(α), and Z𝔤̂(α) are homogeneous polynomially parameter-dependent matrices assuring condition in (3.2)-(3.3). Then, the terms of the polynomial matrices Ψ𝔤̂+1(α)=(α1+⋯+αN)Ψ𝔤̂(α), Θ𝔤̂+1(α)=(α1+⋯+αN)Θ𝔤̂(α) and Z𝔤̂+1(α)=(α1+⋯+αN)Z𝔤̂(α) satisfy the LMIs of Theorem 3.4 corresponding to the degree 𝔤̂+1, which can be obtained in this case by a linear combination of the LMIs of Theorem 3.4 for 𝔤̂.Remark 3.5.
Note that when𝔤=0, we have Ψ𝔤(α)=Ψ, Θ𝔤(α)=Θ, and Z𝔤(α)=Z, then Af(α)=Af, Bf(α)=Bf, and Cf(α)=Cf, 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 αi,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.


## 4. Illustrative Examples

In this section, some numerical examples are presented to illustrate the proposed technique for robustH∞ HPPD filters.Example 4.1.
A stationary random field can be modeled as the following 2D system [13]:
(4.1)η(i+1,j+1)=a1η(i,j+1)+a2η(i+1,j)-a1a2η(i,j)+w1(i,j),
where η(i,j) is the state of the random field of spacial coordinate (i,j), w1(i,j) is a noise input, a12<1, and a22<1 as a1 and a2 represent, respectively, the vertical and horizontal correlations of the random field. The output is then
(4.2)y(i,j)=a1η(i,j+1)+(1-a1a2)η(i,j)+w2(i,j),
where w2(i,j) is the measurement noise. The signal to be estimated is
(4.3)z(i,j)=Cη(i,j)+Dw(i,j).
As in [28], define xh(i,j)=η(i,j+1)-a2η(i,j) and xv(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)=[w1(i,j)w2(i,j)]T and the following system matrices:
(4.4)A=[a101a2],B=[1000],C1=[a10],D1=[01],C=[01],D=[00].
Suppose that the uncertain parameters a1 and a2 are bounded by 0.15≤a1≤0.45 and 0.35≤a2≤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.Table 1
Example4.1: comparison with previous published results.
gγ in this paperγ in [29]γ in [30]02.43422.43733.870911.57131.86272.545021.57131.82902.5028Table 2
H∞ norms at the vertices.
a10.150.150.450.45a20.350.850.350.85∥Tz̃w∥∞1.16821.33261.56941.4070Figure 1
Frequency response of error system for vertex 1.Figure 2
Frequency response of error system for vertex 2.Figure 3
Frequency response of error system for vertex 3.Figure 4
Frequency response of error system for vertex 4.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 Example4.1 but this time with C1=[-11.6-a1],0.15≤a1≤0.65. The results are given in Table 3, which also provides a comparison with previous results in the literature.Table 3
Example4.2: comparison with previous published results.
gγ in this paperγ in [29]γ in [30]07.59817.598111.415715.52165.88868.232125.48845.76948.110935.48705.76368.105045.48705.76368.1050


## 5. Conclusions

This paper has studied the robustH∞ 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.



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*Source: 521675-2012-03-14.xml*