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

Abstract

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 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.

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 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.

2. Problem Formulation

Consider a 2D discrete system described by the following Roesser’s state-space model: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣(𝛼)𝑥𝑖,𝑗+1=𝐴^ℎ𝑥𝑖,𝑗_𝑣(𝛼)𝑤,𝑦𝑖,𝑗+𝐵𝑖,𝑗𝑖,𝑗=𝐶₁(𝛼)𝑥^ℎ𝑥𝑖,𝑗_𝑣𝑖,𝑗+𝐷₁(𝛼)𝑤,𝑧(𝛼)𝑥𝑖,𝑗𝑖,𝑗=𝐶^ℎ𝑥𝑖,𝑗_𝑣(𝛼)𝑤,𝑖,𝑗+𝐷𝑖,𝑗 (2.1) where 𝑥^ℎ(𝑖,𝑗)∈ℜ^𝑛ℎ and 𝑥^𝑣(𝑖,𝑗)∈ℜ^𝑛𝑣 are the horizontal and vertical states, respectively, with the boundary conditions 𝑥^ℎ(0,𝑘)=𝑥^ℎ₀,𝑥^𝑣(𝑘,0)=𝑥^𝑣₀forall𝑘, 𝑦(𝑖,𝑗)∈ℜ^𝑝 is the measured output, 𝑧(𝑖,𝑗)∈ℜ^𝑟 is the signal to be estimated and 𝑤(𝑖,𝑗)∈ℜ^𝑚 is the exogenous input.

To describe the uncertainty, all matrices are assumed to be real, belonging to the polytope 𝒫≜A(𝛼)B(𝛼)C₁(𝛼)D₁(𝛼)C(𝛼)D(𝛼)=^𝑁_𝑖=1𝛼_𝑖𝐴_𝑖𝐵_𝑖𝐶_1𝑖𝐷_1𝑖𝐶_𝑖𝐷_𝑖,𝛼∈Ω, (2.2) where Ω is the unit simplex, defined by Ω=𝛼₁,𝛼₂,…,𝛼_𝑁∶^𝑁_𝑖=1𝛼_𝑖=1,𝛼_𝑖≥0. (2.3) The 2D transfer function from the noise 𝜔(𝑡₁,𝑡₂) to the estimated output 𝑧(𝑡₁,𝑡₂) is then given by 𝑇_𝑧𝑤𝑧₁,𝑧₂=𝐶(𝛼)𝐼𝑧₁,𝑧₂−𝐴(𝛼)⁻¹𝐵(𝛼)+𝐷(𝛼), (2.4) where 𝐼𝑧₁,𝑧₂=diag𝑧₁𝐼_𝑛ℎ,𝑧₂𝐼_𝑛𝑣. (2.5) 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(𝜔₁,𝜔₂) of the 2D discrete system (2.1) is given by 𝑇_𝑧𝑤𝑧₁,𝑧_2∞=sup_{𝜔1,𝜔2∈[]02𝜋}𝜎_max𝑇_𝑧𝑤𝑒^𝑗𝜔1,𝑒^𝑗𝜔2. (2.6)

In this paper, the basic objective is to find a filter of the form 𝑥^ℎ_𝑓𝑥𝑖+1,𝑗_𝑣𝑓𝑖,𝑗+1=𝐴_𝑓(𝛼)𝑥^ℎ_𝑓𝑥𝑖,𝑗_𝑣𝑓𝑖,𝑗+𝐵_𝑓(𝛼)𝑦,𝑧𝑖,𝑗_𝑓𝑖,𝑗=𝐶_𝑓(𝛼)𝑥^ℎ_𝑓𝑥𝑖,𝑗_𝑣𝑓,𝑖,𝑗 (2.7) in order to estimate the signal 𝑧 from the measurements of 𝑦, where 𝑥^ℎ_𝑓(𝑖,𝑗)∈ℜ^𝑛ℎ and 𝑥^𝑣_𝑓(𝑖,𝑗)∈ℜ^𝑛𝑣 are, respectively, the horizontal and vertical states of the filter, 𝑧_𝑓(𝑖,𝑗)∈ℜ^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.

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 𝑥^ℎ𝑖,𝑗=𝑥^ℎ𝑖,𝑗^𝑇𝑥^ℎ_𝑓𝑖,𝑗^𝑇𝑇,𝑥^𝑣𝑖,𝑗=𝑥^𝑣𝑖,𝑗^𝑇𝑥^𝑣_𝑓𝑖,𝑗^𝑇𝑇,𝑧𝑖,𝑗=𝑧𝑖,𝑗−𝑧_𝑓𝑖,𝑗. (2.8) Then, combining these definitions with (2.1) and (2.7), the error dynamic equations are just 𝑥^ℎ𝑖+1,𝑗𝑥^𝑣=𝐴(𝛼)𝑖,𝑗+1𝑥^ℎ𝑖,𝑗𝑥^𝑣+𝐵(𝛼)𝑤,𝑧=𝐶(𝛼)𝑖,𝑗𝑖,𝑗𝑖,𝑗𝑥^ℎ𝑖,𝑗𝑥^𝑣+𝐷(𝛼)𝑤,𝑖,𝑗𝑖,𝑗 (2.9) where 𝐴(𝛼)=Φ𝐴(𝛼)Φ^𝑇,𝐶(𝛼)=𝐶(𝛼)Φ^𝑇,𝐵(𝛼)=Φ𝐵(𝛼),𝐷(𝛼)=𝐷(𝛼), (2.10) and the augmented matrices are given by 𝐴(𝛼)=𝐴(𝛼)0𝐵_𝑓(𝛼)𝐶₁(𝛼)𝐴_𝑓(𝛼),𝐵(𝛼)=𝐵(𝛼)𝐵_𝑓(𝛼)𝐷₁(𝛼),𝐶(𝛼)=𝐶(𝛼)−𝐶_𝑓(𝛼),𝐼Φ=^𝑛ℎ00000𝐼_𝑛ℎ00𝐼_𝑛𝑣00000𝐼_𝑛𝑣. (2.11)(2.12)(2.13)(2.14) 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 𝑇_𝑧𝑤=𝐶(𝛼)𝐼𝑧₁,𝑧₂−𝐴(𝛼)⁻¹𝐵(𝛼)+𝐷(𝛼), (2.15) satisfies 𝑇_𝑧𝑤∞<𝛾, (2.16) with 𝛾>0 a given real number.

3. Robust 𝐻_∞ Filter Design

3.1. 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∈ℜ^{𝑛ℎ×𝑛ℎ} and 𝑃_𝑣∈ℜ^{𝑛𝑣×𝑛𝑣}, such that the following LMI holds: −𝑃𝑃𝐴(𝛼)𝑃𝐵(𝛼)0∗−𝑃0𝐶(𝛼)^𝑇∗∗−𝛾²𝐼𝐷(𝛼)^𝑇∗∗∗−𝐼<0. (3.1)

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 𝑌=diag(𝑌_ℎ,𝑌_𝑣)>0 with 𝑋_ℎ, 𝑌_ℎ∈ℝ^{𝑛ℎ×𝑛ℎ} and 𝑋_𝑣, 𝑌_𝑣∈ℝ^{𝑛𝑣×𝑛𝑣} such that the following LMIs hold: (𝛼)(𝛼)(𝛼)0(𝛼)(𝛼)𝐶−𝑌−𝑌𝑌𝐴𝑌𝐴𝑌𝐵∗−𝑋𝑋𝐴+Ψ₁(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,𝑋−𝑌>0, (3.2)(3.3) 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: 𝐴_𝑓(𝛼)=𝑋⁻¹₁₂𝑍(𝛼)𝑌⁻¹𝑌^−𝑇₁₂,𝐵_𝑓(𝛼)=𝑋⁻¹₁₂Ψ(𝛼),𝐶_𝑓(𝛼)=Θ(𝛼)𝑌⁻¹𝑌^−𝑇₁₂, (3.4) where 𝑋₁₂=𝑋_ℎ1200𝑋_𝑣12,𝑌₁₂=𝑌_ℎ1200𝑌_𝑣12, (3.5) in which 𝑋_ℎ12, 𝑋_𝑣12, 𝑌_ℎ12, and 𝑌_𝑣12 are nonsingular matrices satisfying 𝑋₁₂𝑌^𝑇₁₂=𝐼−𝑋𝑌⁻¹. (3.6)

Proof. Let 𝑌_ℎ=𝑌_ℎ⁻¹, 𝑌_𝑣=𝑌_𝑣⁻¹, 𝑌=𝑌⁻¹ then the relations (3.3), can be written as 𝐼𝑋𝐼𝑌>0. (3.7)
By the Schur complement formula, it follows from (3.7) that 𝑌−𝑋⁻¹>0, (3.8) 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, 𝑋_ℎ12𝑌^𝑇_ℎ12=𝐼−𝑋_ℎ𝑌_ℎ,𝑋_𝑣12𝑌^𝑇_𝑣12=𝐼−𝑋_𝑣𝑌_𝑣. (3.9) Set Π_ℎ1=𝑌_ℎ𝐼𝑌^𝑇_ℎ120,Π_𝑣1=𝑌_𝑣𝐼𝑌^𝑇_𝑣120,Π_ℎ2=𝐼𝑋_ℎ0𝑋^𝑇_ℎ12,Π_𝑣2=𝐼𝑋_𝑣0𝑋^𝑇_𝑣12,Π₁=Π_ℎ100Π_𝑣1,Π₂=Π_ℎ200Π_𝑣2. (3.10) Then, by some calculations, it can be verified that 𝑃∶=Π₂Π₁⁻¹=𝑃_ℎ00𝑃_𝑣, (3.11) where 𝑃_ℎ=𝑋_ℎ𝑋_ℎ12𝑋^𝑇_ℎ12𝑋^𝑇_ℎ12𝑋_ℎ−𝑌_ℎ⁻¹𝑋_ℎ12,𝑃_𝑣=𝑋_𝑣𝑋_𝑣12𝑋^𝑇_𝑣12𝑋^𝑇_𝑣12𝑋_𝑣−𝑌_𝑣⁻¹𝑋_𝑣12. (3.12) Observe that 𝑋_ℎ−𝑋_ℎ12𝑋^𝑇_ℎ12𝑋_ℎ−𝑌_ℎ⁻¹𝑋_ℎ12⁻¹𝑋^𝑇_ℎ12=𝑌_ℎ𝑋>0,_𝑣−𝑋_𝑣12𝑋^𝑇_𝑣12𝑋_𝑣−𝑌_𝑣⁻¹𝑋_𝑣12⁻¹𝑋^𝑇_𝑣12=𝑌_𝑣>0. (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 −𝑌−𝐼_𝑛𝐴(𝛼)𝑌𝐴(𝛼)𝐵(𝛼)0∗−𝑋𝑀₂₃𝑋𝐴(𝛼)+𝑋₁₂𝐵_𝑓(𝛼)𝐶₁(𝛼)𝑋𝐵(𝛼)+𝑋₁₂𝐵_𝑓(𝛼)𝐷₁(𝛼)0∗∗−𝑌−𝐼_𝑛0𝑌𝐶(𝛼)^𝑇−𝑌₁₂(𝛼)^𝑇∗∗∗−𝑋0𝐶(𝛼)^𝑇∗∗∗∗−𝛾²𝐼𝐷(𝛼)^𝑇∗∗∗∗∗−𝐼<0, (3.14) where 𝑀₂₃=𝑋𝐴(𝛼)𝑌+𝑋₁₂𝐵_𝑓(𝛼)𝐶₁(𝛼)𝑌+𝑋₁₂𝐴_𝑓(𝛼)𝑌^𝑇₁₂.
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 −Φ^𝑇Π^𝑇₁𝑃Π₁ΦΦ^𝑇Π^𝑇₁𝑃Φ𝐴(𝛼)Φ^𝑇Π₁ΦΦ^𝑇Π^𝑇₁𝑃Φ𝐵(𝛼)0Φ^𝑇Π^𝑇₁Φ𝐴(𝛼)^𝑇Φ^𝑇𝑃Π₁Φ−Φ^𝑇Π^𝑇₁𝑃Π₁Φ0Φ^𝑇Π^𝑇₁Φ𝐶(𝛼)^𝑇𝐵(𝛼)^𝑇Φ^𝑇𝑃Π₁Φ0−𝛾²𝐼𝐷(𝛼)^𝑇0𝐶(𝛼)Φ^𝑇Π₁Φ𝐷(𝛼)−𝐼<0. (3.15) 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 −𝑃𝑃𝐴(𝛼)𝑃𝐵(𝛼)0∗−𝑃0𝐶(𝛼)^𝑇∗∗−𝛾²𝐼𝐷(𝛼)^𝑇∗∗∗−𝐼<0. (3.16) 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.

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 degree 𝔤 as follows: Ψ_𝑔(𝛼)=^𝐽(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁Ψ_{𝔎𝑗(𝔤)},Θ_𝑔(𝛼)=^𝐽(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁Θ_{𝔎𝑗(𝔤)},𝑍_𝑔(𝛼)=^𝐽(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁𝑍_{𝔎𝑗(𝔤)}, (3.17) with 𝑘₁𝑘₂⋯𝑘_𝑁=𝔎_𝑗(𝔤). (3.18) The above notations are explained as follows: 𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁, 𝛼∈Ω, 𝑘_𝑖∈ℕ, 𝑖=1,…,𝑁 are the monomials; Ψ_{𝔎𝑗(𝔤)}, Θ_{𝔎𝑗(𝔤)}, and 𝑍_{𝔎𝑗(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 𝑘₁𝑘₂⋯𝑘_𝑁, with 𝑘_𝑖∈ℕ, 𝑖=1,…,𝑁, that fulfill 𝑘₁+𝑘₂+⋯+𝑘_𝑁=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 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 𝔎^𝑖_𝑗(𝔤+1) define subscripts 𝑘₁𝑘₂⋯𝑘_𝑁 of matrices Ψ_{𝑘1𝑘2⋯𝑘𝑁}, Θ_{𝑘1𝑘2⋯𝑘𝑁}, and 𝑍_{𝑘1𝑘2⋯𝑘𝑁} associated to homogeneous polynomial parameter-dependent matrices of degree 𝔤.

Finally, define the scalar constant coefficients 𝛽^𝑖_𝑗(𝑗+1)=𝔤!/(𝑘₁!𝑘₂!⋯𝑘_𝑁!), with 𝑘₁𝑘₂⋯𝑘_𝑁∈𝔎^𝑖_𝑗(𝔤+1).

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 Ψ₂(𝛼)=𝛼²₃Ψ₀₀₂+𝛼₂𝛼₃Ψ₀₁₁+𝛼²₂Ψ₀₂₀+𝛼₁𝛼₃Ψ₁₀₁+𝛼₁𝛼₂Ψ₁₁₀+𝛼²₁Ψ₂₀₀,Θ₂(𝛼)=𝛼²₃Θ₀₀₂+𝛼₂𝛼₃Θ₀₁₁+𝛼²₂Θ₀₂₀+𝛼₁𝛼₃Θ₁₀₁+𝛼₁𝛼₂Θ₁₁₀+𝛼²₁Θ₂₀₀,𝑍₂(𝛼)=𝛼²₃𝑍₀₀₂+𝛼₂𝛼₃𝑍₀₁₁+𝛼²₂𝑍₀₂₀+𝛼₁𝛼₃𝑍₁₀₁+𝛼₁𝛼₂𝑍₁₁₀+𝛼²₁𝑍₂₀₀. (3.19) Moreover, the 3 tuples are 𝔎³₁(2)=001, 𝔎²₂(2)=001, 𝔎³₂(2)=010, 𝔎²₃(2)=010, 𝔎¹₄(2)=001, 𝔎³₄(2)=100, 𝔎¹₅(2)=010, 𝔎²₅(2)=100, and 𝔎¹₆(2)=100: these are the only possible triples (3 tuples) 𝔎^𝑖_𝑗(2), 𝑗=1,…,𝔍(2) associated to 𝔎(2).

3.3. 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 𝑍_{𝔎𝔧(𝔤)}, Θ_{𝔎𝔧(𝔤)}, Ψ_{𝔎𝔧(𝔤)}, 𝔎_𝔧(𝔤)∈𝔎(𝔤), 𝑗=1,…,𝔍(𝔤), 𝑋=diag(𝑋_ℎ,𝑋_𝑣)>0, and 𝑌=diag(𝑌_ℎ,𝑌_𝑣)>0 with 𝑋_ℎ, 𝑌_ℎ∈ℝ^𝑛ℎ and 𝑋_𝑣, 𝑌_𝑣∈ℝ^𝑛𝑣, such that for all 𝔎_𝑙(𝔤+1)∈𝔎(𝔤+1), 𝑙=1,…,𝔍(𝔤+1) such that the following LMIs hold: _{𝑖∈𝔑𝑙𝑔+1}−𝛽^𝑖_𝑙(𝔤+1)𝑌−𝛽^𝑖_𝑙(𝔤+1)𝑌𝛽^𝑖_𝑙(𝔤+1)𝑌𝐴_𝑖𝛽^𝑖_𝑙(𝔤+1)𝑌𝐴_𝑖𝛽^𝑖_𝑙(𝔤+1)𝑌𝐵_𝑖0∗−𝛽^𝑖_𝑙(𝔤+1)𝑋𝐽₂₃𝐽₂₄𝐽₂₅0∗∗−𝛽^𝑖_𝑙(𝔤+1)𝑌−𝛽^𝑖_𝑙(𝔤+1)𝑌0𝐽₃₆∗∗∗−𝛽^𝑖_𝑙(𝔤+1)𝑋0𝐽₄₆∗∗∗∗−𝛾²𝐼𝐽₅₆∗∗∗∗∗−𝐼<0, (3.20)𝑋−𝑌>0, (3.21) where 𝐽₁₃=𝛽^𝑖_𝑙(𝔤+1)𝑌𝐴_𝑖,𝐽₁₅=𝛽^𝑖_𝑙(𝔤+1)𝑌𝐵_𝑖,𝐽₂₃=𝛽^𝑖_𝑙(𝔤+1)𝑋𝐴_𝑖+Ψ_𝔎^𝔦_𝔩(𝔤+1)𝐶_1𝑖+𝑍_𝔎^𝔦_𝔩(𝔤+1),𝐽₂₄=𝛽^𝑖_𝑙(𝔤+1)𝑋𝐴_𝑖+Ψ_𝔎^𝔦_𝔩(𝔤+1)𝐶_1𝑖,𝐽₂₅=𝛽^𝑖_𝑙(𝔤+1)𝑋𝐵_𝑖+Ψ_𝔎^𝔦_𝔩(𝔤+1)𝐷_1𝑖,𝐽₃₆=𝛽^𝑖_𝑙(𝔤+1)𝐶^𝑇_𝑖−Θ^𝑇_{𝔎𝔦𝔩(𝔤+1)},𝐽₄₆=𝛽^𝑖_𝑙(𝔤+1)𝐶^𝑇_𝑖,𝐽₅₆=𝛽^𝑖_𝑙(𝔤+1)𝐷^𝑇_𝑖, (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.

In this case, the matrices of the 2D discrete-time HPPD filter are given by 𝐴_𝑓𝔤(𝛼)=^𝔍(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁𝐴_{𝑓𝔎𝑗(𝑔)},𝐵_𝑓𝔤(𝛼)=^𝔍(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁𝐵_{𝑓𝔎𝑗(𝑔)},𝐶_𝑓𝔤(𝛼)=^𝔍(𝔤)_𝑗=1𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁𝐶_{𝑓𝔎𝑗(𝑔)}, (3.23) with 𝑘₁𝑘₂⋯𝑘_𝑁=𝔎_𝑗𝑔,𝐴_{𝑓𝔎𝑗(𝑔)}=𝑋⁻¹₁₂𝑍_{𝔎𝑗(𝑔)}𝑌⁻¹𝑌^−𝑇₁₂,𝐵_{𝑓𝔎𝑗(𝑔)}=𝑋⁻¹₁₂Ψ_{𝔎𝑗(𝑔)},𝐶_{𝑓𝔎𝑗(𝑔)}=Θ_{𝔎𝑗(𝑔)}𝑌⁻¹𝑌^−𝑇₁₂. (3.24)

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 ^𝐽(𝑔+1)_𝑙=1𝛼^k_{𝑖∈𝔑𝑙𝑔+1}−𝛽^𝑖_𝑙(𝔤+1)𝑌−𝛽^𝑖_𝑙(𝔤+1)𝑌𝐽₁₃𝐽₁₃𝐽₁₅0∗−𝛽^𝑖_𝑙(𝔤+1)𝑋𝐽₂₃𝐽₂₄𝐽₂₅0∗∗−𝛽^𝑖_𝑙(𝔤+1)𝑌−𝛽^𝑖_𝑙(𝔤+1)𝑌0𝐽₃₆∗∗∗−𝛽^𝑖_𝑙(𝔤+1)𝑋0𝐽₄₆∗∗∗∗−𝛾²𝐼𝐽₅₆𝑘∗∗∗∗∗−𝐼<0,₁𝑘₂⋯𝑘_𝑁=𝔎_𝑙(),𝛼𝔤+1_𝑘=𝛼^𝑘1₁𝛼^𝑘2₂⋯𝛼^𝑘𝑁_𝑁. (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_𝔎𝑗(𝑔₎, Θ_𝔎𝑗(𝑔₎, and 𝑍_𝔎𝑗(𝑔₎, 𝑗=1,…,𝔍(𝔤), such that Ψ𝔤(𝛼), Θ𝔤(𝛼), and 𝑍𝔤(𝛼) are homogeneous polynomially parameter-dependent matrices assuring condition in (3.2)-(3.3). Then, the terms of the polynomial matrices Ψ𝔤₊₁(𝛼)=(𝛼₁+⋯+𝛼_𝑁)Ψ𝔤(𝛼), Θ𝔤₊₁(𝛼)=(𝛼₁+⋯+𝛼_𝑁)Θ𝔤(𝛼) and 𝑍𝔤₊₁(𝛼)=(𝛼₁+⋯+𝛼_𝑁)𝑍𝔤(𝛼) 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.

4. 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]: 𝜂𝑖+1,𝑗+1=𝑎₁𝜂𝑖,𝑗+1+𝑎₂𝜂𝑖+1,𝑗−𝑎₁𝑎₂𝜂𝑖,𝑗+𝑤₁𝑖,𝑗, (4.1) where 𝜂(𝑖,𝑗) is the state of the random field of spacial coordinate (𝑖,𝑗), 𝑤₁(𝑖,𝑗) is a noise input, 𝑎²₁<1, and 𝑎²₂<1 as 𝑎₁ and 𝑎₂ represent, respectively, the vertical and horizontal correlations of the random field. The output is then 𝑦𝑖,𝑗=𝑎₁𝜂𝑖,𝑗+1+1−𝑎₁𝑎₂𝜂𝑖,𝑗+𝑤₂𝑖,𝑗, (4.2) where 𝑤₂(𝑖,𝑗) is the measurement noise. The signal to be estimated is 𝑧𝑖,𝑗=𝐶𝜂𝑖,𝑗+𝐷𝑤𝑖,𝑗. (4.3) As in [28], define 𝑥^ℎ(𝑖,𝑗)=𝜂(𝑖,𝑗+1)−𝑎₂𝜂(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)=[𝑤₁(𝑖,𝑗)𝑤₂(𝑖,𝑗)]^𝑇 and the following system matrices: 𝑎𝐴=¹01𝑎₂,𝐵=1000,𝐶₁=𝑎₁0,𝐷₁=01,𝐶=01,𝐷=00. (4.4) 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 

Example 4.1: comparison with previous published results.

Table 2 

𝐻_∞ norms at the vertices.

Figure 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 Example 4.1 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 

Example 4.2: comparison with previous published results.

5. 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.

Acknowlegments

This work is funded by AECI (A/030426/10) and MiCInn (DPI2010-21589-c05).