ℋ_∞ Filter Design with Minimum Entropy for Continuous-Time Linear Systems

Abstract

We deal with the design problem of minimum entropy ℋ_∞ filter in terms of linear matrix inequality (LMI) approach for linear continuous-time systems with a state-space model subject to parameter uncertainty that belongs to a given convex bounded polyhedral domain. Given a stable uncertain linear system, our attention is focused on the design of full-order and reduced-order robust minimum entropy ℋ_∞ filters, which guarantee the filtering error system to be asymptotically stable and are required to minimize the filtering error system entropy (at 𝑠₀=∞) and to satisfy a prescribed ℋ_∞ disturbance attenuation performance. Sufficient conditions for the existence of desired full-order and reduced-order filters are established in terms of LMIs, respectively, and the corresponding filter synthesis is cast into a convex optimization problem which can be efficiently handled by using standard numerical software. Finally, an illustrative example is provided to show the usefulness and effectiveness of the proposed design method.

1. Introduction

Estimation is the process of inferring the value of a quantity of interests from indirect, inaccurate, and uncertain observations [1]. State estimation of dynamic systems with both process and measurement noise inputs is a very important and challenging problem in engineering applications. In the past decades, quite a few attention has been devoted to estimation methods that are based on the minimization of the variance of the estimation error, that is, the well-known Kalman filtering approach [2, 3]. Unfortunately, it is recognized soon that the performance of Kalman filter can deteriorate significantly when the process parameters are subjected to relatively small modeling errors. In order to cope with this problem, over the past few years interest has been devoted to the design of robust estimators. There are essentially two approaches to the robust estimation problem. The first is robust ℋ₂ filtering, which minimizes the upper bound of the estimation error variance for all possible parametric uncertainties under the assumption that the noise processes have known power spectral densities [4, 5]. In many practical situations, however, we may not be able to have exactly known information on the spectral densities of the noise processes. In such cases, an alternative is to reformulate the estimation problem in an ℋ_∞ filtering framework, which has been well recognized to be most appropriate for systems with noise input whose stochastic information is not precisely known. It minimizes the worst-case energy gain from the noise inputs to the estimation error [6–8]. There are many results reported on the problem of ℋ_∞ filtering; for example, it has been addressed for linear systems [9], linear systems with uncertain parameters [10, 11], delay systems [6, 12], and stochastic systems [8]. Although the ℋ_∞ filter is known to be less sensitive to modeling errors than ℋ₂ filter, it is generally so conservative as to lead to a large intolerable estimation error variance when the system is driven by white noise signals.

Similar to ℋ_∞ control problem, in the ℋ_∞ filtering problem [13–17], the family of filters that satisfy a filtering error system with an ℋ_∞-norm bound is characterized by a linear fractional transformation of a “ball in ℋ_∞,” and then a natural question is which element of this ball to choose. One choice that has been considered in a closely related problem in mathematics is to choose that which minimizes an entropy integral; that is, the filter is well selected such that not only the filtering error system is asymptotically stable and the ℋ_∞-norm of its transfer function is below a prescribed level, but also the entropy of the filtering error system at infinity is minimized. This kind of optimal filters is referred to as minimum entropy ℋ_∞ filters in the present paper. In fact, minimum entropy ℋ_∞ filtering provides a means of trading off some of the features of other filtering problems, namely, ℋ₂ filtering and ℋ_∞ filtering. As for the minimum entropy ℋ_∞ control problem, there are many important results that have been reported in the literature; for example, controllers which minimize the entropy of the closed-loop transfer function have been studied extensively for linear time-invariant (LTI) systems, both in the continuous and discrete-time cases [18–20]. Minimum entropy control for time-varying systems has been investigated in [21]. To the best of our knowledge, however, there is not any result reported on minimum entropy ℋ_∞ filtering problem in the literature; research in this area should be important and challenging; this motivates us to carry out the present study.

In this paper, we make an attempt to investigate the design of minimum entropy ℋ_∞ filters by using linear matrix inequality (LMI) approach for linear continuous-time systems with a state-space model subject to parameter uncertainty that belongs to a given convex bounded polyhedral domain. Given a stable uncertain linear system, our attention is focused on the design of full-order and reduced-order robust minimum entropy ℋ_∞ filters, which guarantee the filtering error system to be asymptotically stable and are required to minimize the filtering error system entropy (at 𝑠₀=∞) as well as to satisfy a prescribed ℋ_∞ disturbance attenuation performance. Sufficient conditions for the existence of desired full-order and reduced-order filters are established in terms of LMIs, respectively, and the corresponding filter synthesis is cast into a convex optimization problem which can be efficiently handled by using the well-known interior-point algorithms [22]. A numerical simulation example is provided to show the usefulness and effectiveness of the proposed design method.

The rest of this paper is organized as follows. In Section 2, the minimum entropy ℋ_∞ filtering problem is formulated. Section 3 presents our main results of the full-order and reduced-order minimum entropy ℋ_∞ filters design. Section 4 provides an illustrative example. Finally, conclusions are drawn in Section 5.

Notations. The notation used here is fairly standard except where otherwise stated. 𝐴^𝑇 represents the transpose of 𝐴; ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. the notation 𝑃>0 means that 𝑃 is a real symmetric and positive definite matrix; trace(𝑃) represents the trace of 𝑃; diag{𝐹₁,𝐹₂,…} stands for a block-diagonal matrix whose diagonal blocks are given by 𝐹₁,𝐹₂,…. 𝐼 and 0 represent identity matrix and zero matrix; |⋅|refers to the Euclidean vector norm and ‖⋅‖₂ denotes the ℒ₂ norm of a differential signal; The signals that are square integrable over [0,∞) are denoted by ℒ₂[0,∞) with the norm ‖⋅‖₂; the symbol ∗ in a matrix means that the corresponding term of the matrix can be obtained by symmetric property.

2. Problem Formulation

Consider the following linear time-invariant (LTI) system (Σ): (Σ)∶𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡)(𝑡),𝑧(𝑡)(𝑡),=𝐴𝑥+𝐵𝜔=𝐶𝑥+𝐷𝜔=𝐿𝑥 (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝑦(𝑡)∈ℝ^𝑚 is the measured output; 𝑧(𝑡)∈ℝ^𝑞 is a linear combination of the state variables to be estimated; 𝜔(𝑡)∈ℝ^𝑙 is the disturbance input which belongs to ℒ₂[0,∞); 𝐴, 𝐵, 𝐶, 𝐷, and 𝐿 are constant real matrices of appropriate dimensions, where 𝐿 is a known matrix and 𝐴, 𝐵, 𝐶, 𝐷 are unknown matrices such that the system matrix 𝒢≜𝐴𝐵𝐶𝐷 (2) belongs to a given polytope 𝒟 described by 𝒟≜𝒢∣𝒢=^𝑛𝑠_𝑖=1𝜆_𝑗𝒢_𝑗;^𝑛𝑠_𝑖=1𝜆_𝑗=1,𝜆_𝑗≥0; (3) that is, any admissible system matrix 𝒢 can be written as an unknown convex combination of 𝑛_𝑠 vertices 𝒢_𝑗, 𝑗=1,2,…,𝑛_𝑠, given by 𝒢_𝑗≜𝐴_𝑗𝐵_𝑗𝐶_𝑗𝐷_𝑗, (4) where 𝐴_𝑗, 𝐵_𝑗, 𝐶_𝑗, and 𝐷_𝑗, 𝑗=1,2,…,𝑛_𝑠, are given matrices. Clearly, 𝑛_𝑠=1 corresponds to the case where the system (Σ) is perfectly known.

Before formulating the problem of this paper, we first give some definitions and existing results of the minimum entropy problem. Consider the following LTI system (Π): (Π)∶𝑥̇(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡),=𝐴𝑥+𝐵𝜔=𝐿𝑥 (5) where 𝑦(𝑡) is the output; the other notations are defined as in (1), and let 𝔾 denote its transfer function.

Definition 1 (entropy at 𝑠₀∈(0,∞)). Let 𝔾∈ℛℒ_∞ and 𝛾>0 be a real scalar such that ‖𝔾‖_∞<𝛾. Then the entropy of 𝔾 at 𝑠₀ is defined by ℐ𝔾;𝛾;𝑠₀𝛾≜−²2𝜋^∞_−∞lndet𝐼−𝛾⁻²𝔾^∗𝔾×𝑠𝑗𝜛𝑗𝜛₂₀𝑠²₀+𝜛²𝑑𝜛. (6)
When 𝑠₀→∞, we obtain the entropy at infinity of the system, which has the following definition.

Definition 2 (entropy at infinity). Let 𝔾∈ℛℒ_∞ and 𝛾>0 be a real scalar such that ‖𝔾‖_∞<𝛾. Then the entropy of 𝔾 at infinity is defined by ℐ𝛾𝔾;𝛾;∞≜−²2𝜋^∞_−∞lndet𝐼−𝛾⁻²𝔾^∗𝑗𝜛𝔾𝑗𝜛𝑑𝜛. (7)

It is well known that for LTI system (Π) in (5), we have the following results for the ℋ₂ optimization problem: ‖𝔾‖²₂=trace𝐵^𝑇𝑄𝐵, (8) where 𝑄=𝑄^𝑇≥0 is the solution to the Lyapunov equation 𝐴^𝑇𝑄+𝑄𝐴+𝐿^𝑇𝐿=0. (9) In fact, 𝑄 is just the controllability Gramian.

Now, considering the ℋ_∞ optimization problem of LTI system, we knew that ‖𝔾‖_∞<𝛾 if there exists a stabilizing solution 𝑃=𝑃^𝑇≥0 to the algebraic Riccati equation 𝐴^𝑇𝑃+𝑃𝐴+𝛾⁻²𝑃𝐵𝐵^𝑇𝑃+𝐿^𝑇𝐿=0, (10) and it is easy to prove that any 𝑃 solving (10) overbounds the controllability Gramian 𝑄; that is, 𝑃≥𝑄. To evaluate the entropy cost, we define the following auxiliary cost.

Definition 3 (auxiliary cost). Let 𝔾∈ℛℒ_∞ and 𝛾>0 be a real scalar such that ‖𝔾‖_∞<𝛾. Then the auxiliary cost with 𝔾 is defined by 𝒥𝔾;𝛾≜trace𝐵^𝑇𝑃𝐵, (11) where 𝑃 is a positive symmetric matrix with the smallest possible maximum singular value among all solutions of the algebraic Riccati equality (10).

The following lemma gives the equivalence between the auxiliary cost and the entropy defined in (7) and (11), respectively, which plays a key role in deriving our main results subsequently.

Lemma 4 (see [20]). Let 𝔾∈ℛℒ_∞ and 𝛾>0 be a real scalar such that ‖𝔾‖_∞<𝛾. Then the entropy equals the auxiliary cost; that is, ℐ𝔾;𝛾;∞=𝒥𝔾;𝛾. (12)

Moreover, according to the result in [22], the minimum entropy ℋ_∞ optimization problem for the LTI system (Π) can be formulated as follows: min_{𝑃>0,𝑅>0}trace(𝑅), (13) subject to 𝐴^𝑇𝑃+𝑃𝐴𝑃𝐵𝐿^𝑇⋆−𝛾²𝐼0⋆⋆−𝐼<0,−𝑅𝐵^𝑇𝑃⋆−𝑃≤0. (14)

The objective of this paper is to design a full-order (or reduced-order) minimum entropy ℋ_∞ filter the system (Σ) in (1) of the following form: Σ∶𝑥̇(𝑡)=𝐴_𝑓𝑥(𝑡)+𝐵_𝑓𝑦(𝑡),𝑥(0)𝑧(𝑡)=0,=𝐿_𝑓𝑥(𝑡), (15) where 𝑥(𝑡)∈ℝ^𝑘 is the filter state vector, 𝐴_𝑓∈ℝ^𝑘×𝑘, 𝐵_𝑓∈ℝ^𝑘×𝑚, and 𝐶_𝑓∈ℝ^𝑞×𝑘 are filter parameters to be determined later. In the case where 𝑘=𝑛, the filter will be referred to as a full-order filter and as a reduced-order filter when 𝑘<𝑛.

Augmenting the model of (1) to include the state of the filter (Σ), we obtain the filtering error system as Σ∶𝜉̇(𝑡)=(𝑡)+(𝑡),𝑒(𝑡)=(𝑡),𝐴𝜉𝐵𝜔𝐿𝜉 (16) where 𝜉(𝑡)≜[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)]^𝑇, 𝑒(𝑡)≜𝑧(𝑡)−𝑧(𝑡) and 𝐵𝐴≜𝐴0_𝑓𝐶𝐴_𝑓,𝐵𝐵𝐵≜_𝑓𝐷,𝐿≜𝐿−𝐿_𝑓. (17)

Our aim in this paper is to determine the matrices 𝐴_𝑓, 𝐵_𝑓, and 𝐿_𝑓 of the full-order (or reduced-order) minimum entropy ℋ_∞ filter (Σ) in (15) such that the filtering error system (Σ) in (16) is asymptotically stable with a prescribed ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy at infinity; that is, ℐ(𝔼;𝛾;∞) is minimized for a given scalar 𝛾>0 (where 𝔼 is defined as the transfer function of the filtering error system (Σ) in (16)).

3. Main Results

3.1. Full-Order Robust Filter Design

In this section, we will first study the design of a full-order filter. In order to pave the way for deriving the robust filter, initially we consider the case where the system matrix 𝒢 is perfectly known; that is, 𝑛_𝑠=1. We first give the following result which will play a key role in deriving our subsequent results. Since the result can be easily obtained according to the analysis in Section 2, we omit the proof.

Theorem 5. The filtering error system (Σ) in (16) is asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy, if there exist matrices 𝑃>0 and 𝑅>0 such that the following optimization problem has feasible solution: min_{𝑃>0,𝑅>0,𝐴𝑓,𝐵𝑓,𝐿𝑓}𝑡𝑟𝑎𝑐𝑒(𝑅), (18) subject to𝐴^𝑇𝑃+𝑃𝐴𝑃𝐵𝐿^𝑇⋆−𝛾²𝐵𝐼0⋆⋆−𝐼<0,−𝑅^𝑇𝑃⋆−𝑃≤0. (19)(20)

Notice that the inequalities constraints of (19)-(20) are not convex on the decision variables 𝑃, 𝐴_𝑓, 𝐵_𝑓, and 𝐿_𝑓. In what follows, we will present the design method of the minimum entropy ℋ_∞ filter, and give the following result.

Theorem 6. Consider the system (Σ) with perfectly known system matrix 𝒢. There exists a full-order minimum entropy ℋ_∞  filter of the form of (15) such that the filtering error system (Σ) in (16) is asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy, if there exist matrices 𝒰>0 and 𝒱>0 and matrices 𝒜_𝑓, ℬ_𝑓, and ℒ_𝑓 such that the following optimization problem has feasible solution: min_{𝒰>0,𝒱>0,𝒜𝑓,ℬ𝑓,ℒ𝑓}𝑡𝑟𝑎𝑐𝑒(𝑅), (21) subject to ^{𝒰𝐴+𝐴𝑇𝒰+ℬ𝑓}−^{𝐶+𝐶𝑇ℬ𝑇𝑓𝒜𝑓+𝐴𝑇𝒱+𝐶𝑇ℬ𝑇𝑓𝒰𝐵+ℬ𝑓𝐷𝐿𝑇⋆𝒜𝑓+𝒜𝑇𝑓𝒱𝐵+ℬ𝑓𝐷−ℒ𝑇𝑓⋆⋆−𝛾2𝐼0⋆⋆⋆−𝐼<0,}𝑅𝐵^𝑇𝒰+𝐷^𝑇ℬ^𝑇_𝑓𝐵^𝑇𝒱+𝐷^𝑇ℬ^𝑇_𝑓⋆−𝒰−𝒱⋆⋆−𝒱≤0. (22)(23) Moreover, a desired full-order minimum entropy ℋ_∞  filter can be computed from 𝐴_𝑓𝐵_𝑓𝐿_𝑓0≜𝒱⁻¹0𝒜0𝐼^𝑓ℬ_𝑓ℒ_𝑓0. (24)

Proof. According to Theorem 5, 𝑃 is nonsingular if the optimization problem of (18)–(20) has feasible solutions since 𝑃>0. Now, partition 𝑃 as 𝑃𝑃≜¹𝑃₃⋆𝑃₂, (25) where 𝑃₁ and 𝑃₂ are 𝑛×𝑛 symmetric positive definite matrices. Without loss of generality, we assume that 𝑃₃ is nonsingular. To see this, let the matrix 𝑄≜𝑃+𝛼𝑊, where 𝛼 is a positive scalar and 𝑄𝑊≜0𝐼⋆0,𝑄≜¹𝑄₃⋆𝑄₂. (26) Observe that, since 𝑃>0, we have 𝑄>0 for 𝛼>0 in a neighborhood of the origin. Thus, it can be easily verified that there exists an arbitrarily small 𝛼>0 such that 𝑄₃ is nonsingular and inequalities (19)-(20) are feasible with 𝑃 replaced by 𝑄 and such that the objective function of (18) will be increased only by an arbitrarily small quantity. Since 𝑄₃ is nonsingular, we thus conclude that there is no loss of generality to assume the matrix 𝑃₃ to be nonsingular.
Define the following matrices which are all nonsingular: Γ≜𝐼00𝑃₂⁻¹𝑃^𝑇₃,𝒰≜𝑃₁,𝒱≜𝑃₃𝑃₂⁻¹𝑃^𝑇₃𝒜_𝑓,ℬ_𝑓ℒ_𝑓0≜𝑃₃0𝐴0𝐼^𝑓𝐵_𝑓𝐿_𝑓0𝑃₂⁻¹𝑃^𝑇₃00𝐼. (27)(28)
Performing congruence transformations to (19)-(20) by diag(Γ,𝐼,𝐼) and diag(𝐼,Γ), respectively, we have Γ^𝑇𝐴^𝑇𝑃Γ+Γ^𝑇𝑃𝐴ΓΓ^𝑇𝑃𝐵Γ^𝑇𝐿^𝑇⋆−𝛾²𝐵𝐼0⋆⋆−𝐼<0,−𝑅^𝑇𝑃Γ⋆−Γ^𝑇𝑃Γ≤0. (29)(30) Considering (17) and (25)–(28), we have Γ^𝑇𝑃𝐴Γ≜𝒰𝐴+ℬ_𝑓𝐶𝒜_𝑓𝒱𝐴+ℬ_𝑓𝐶𝒜_𝑓,Γ^𝑇𝑃𝐵≜𝒰𝐵+ℬ_𝑓𝐷𝒱𝐵+ℬ_𝑓𝐷,Γ^𝑇𝑃Γ≜,𝒰𝒱⋆𝒱𝐿Γ≜𝐿−ℒ_𝑓. (31)
Substituting (31) into (29)-(30) yields (22)-(23), respectively.
On the other hand, (28) is equivalent to 𝐴_𝑓𝐵_𝑓𝐿_𝑓0=𝑃₃⁻¹0𝒜0𝐼^𝑓ℬ_𝑓ℒ_𝑓0𝑃₃^−𝑇𝑃₂0=𝑃0𝐼_3−𝑇𝑃₂⁻¹𝒱⁻¹0𝒜0𝐼^𝑓ℬ_𝑓ℒ_𝑓0𝑃₃^−𝑇𝑃₂0.0𝐼 (32)
Note that the filter matrices of (15) can be written as (32), which implies that 𝑃₃^−𝑇𝑃₂ can be viewed as a similarity transformation on the state-space realization of the filter and, as such, has no effect on the filter mapping from 𝑦 to 𝑧. Without loss of generality, we can set 𝑃₃^−𝑇𝑃₂=𝐼 and thus obtain (24). This completes the proof.

Now, we will consider the design of a full-order robust filter. To this end, consider the system (Σ) in (1) and the uncertainty domain 𝒟 in (3). According to the previous arguments, we have the following result.

Theorem 7. Consider the system (Σ) with system matrix 𝒢∈𝒟. There exists a full-order minimum entropy ℋ_∞ filter of the form of (15) such that the filtering error system (Σ) in (16) is asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy, if there exist matrices 𝒰>0 and 𝒱>0 and matrices 𝒜_𝑓, ℬ_𝑓, and ℒ_𝑓 such that, for 𝑗=1,2,…,𝑛_𝑠, the following optimization problem has a feasible solution: min_{𝒰>0,𝒱>0,𝒜𝑓,ℬ𝑓,ℒ𝑓}𝑡𝑟𝑎𝑐𝑒(𝑅), (33) subject to𝒰𝐴_𝑗+𝐴^𝑇_𝑗𝒰+ℬ_𝑓𝐶_𝑗+𝐶^𝑇_𝑗ℬ^𝑇_𝑓𝒜_𝑓+𝐴^𝑇_𝑗𝒱+𝐶^𝑇_𝑗ℬ^𝑇_𝑓𝒰𝐵_𝑗+ℬ_𝑓𝐷_𝑗𝐿^𝑇⋆𝒜_𝑓+𝒜^𝑇_𝑓𝒱𝐵_𝑗+ℬ_𝑓𝐷_𝑗−ℒ^𝑇_𝑓⋆⋆−𝛾²𝐼0⋆⋆⋆−𝐼<0,−𝑅𝐵^𝑇_𝑗𝒰+𝐷^𝑇_𝑗ℬ^𝑇_𝑓𝐵^𝑇_𝑗𝒱+𝐷^𝑇_𝑗ℬ^𝑇_𝑓⋆−𝒰−𝒱⋆⋆−𝒱≤0. (34)Moreover, a desired full-order minimum entropy ℋ_∞  filter can be computed from (24).

Proof. Employing the same arguments as in the proof of Theorem 6, it follows that the optimization problem of (33)-(34) is equivalent to min_{𝑃>0,𝑅>0,𝐴𝑓,𝐵𝑓,𝐿𝑓}trace(𝑅), (35) subject to 𝑗=1,2,…,𝑛_𝑠, 𝐴^𝑇_𝑗𝑃+𝑃𝐴_𝑗𝑃𝐵_𝑗𝐿^𝑇⋆−𝛾²𝐵𝐼0⋆⋆−𝐼<0,−𝑅^𝑇_𝑗𝑃⋆−𝑃≤0, (36) where 𝐴_𝑗 and 𝐵_𝑗 are as in (17) with 𝐴, 𝐵, 𝐶, and 𝐷 replaced by 𝐴_𝑗, 𝐵_𝑗, 𝐶_𝑗, and 𝐷_𝑗, respectively.
Now, in view of the convexity of the uncertainty domain and considering that the inequalities (36) are affine in the matrices 𝐴_𝑗, 𝐵_𝑗, 𝐶_𝑗, and 𝐷_𝑗, we have the result that the optimization problem of (35)–(36) is equivalent to that of (18)–(20) with system matrix 𝒢∈𝒟. This completes the proof.

3.2. Reduced-Order Robust Filter Design

In this section, we will consider the design of reduced-order robust filter, that is, the case where the order of the filter 𝑘 is smaller than the order 𝑛 of the original system model. As the former, we first consider the case where the system matrix 𝒢 is perfectly known, that is, 𝑛_𝑠=1, which have the following result.

Theorem 8. Consider system (Σ) with perfectly known system matrix 𝒢. There exists a reduced-order minimum entropy ℋ_∞ filter of the form of (15) such that the filtering error system (Σ) in (16) is asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy, if there exist matrices ℳ>0 and 𝒩>0 and matrices 𝒜_𝑓, ℬ_𝑓, and ℒ_𝑓 such that the following optimization problem has feasible solution: min_{ℳ>0,𝒩>0,𝒜𝑓,ℬ𝑓,ℒ𝑓}𝑡𝑟𝑎𝑐𝑒(𝑅), (37) subject to ^{Π11𝒦𝒜𝑓+𝐴𝑇𝒦𝒩+𝐶𝑇ℬ𝑇𝑓ℳ𝐵+𝒦ℬ𝑓𝐷𝐿𝑇⋆𝒜𝑓+𝒜𝑇𝑓𝒩𝑇𝒦𝑇𝐵+ℬ𝑓𝐷−ℒ𝑇𝑓⋆⋆−𝛾2𝐼0⋆⋆⋆−𝐼<0,−𝑅𝐵𝑇ℳ+𝐷𝑇ℬ𝑇𝑓𝐵𝑇𝒦𝒩+𝐷𝑇ℬ}≤^𝑇𝑓_{⋆−ℳ−𝒦𝒩⋆⋆−𝒩}0, (38)(39) where 𝒦≜^{𝐼𝑘×𝑘}_{0(𝑛−𝑘)×𝑘} and Π₁₁≜ℳ𝐴+𝐴^𝑇ℳ+𝒦ℬ_𝑓𝐶+𝐶^𝑇ℬ^𝑇_𝑓𝒦^𝑇. Moreover, a desired reduced-order minimum entropy ℋ_∞  filter can be computed from 𝐴_𝑓𝐵_𝑓𝐿_𝑓0≜𝒩⁻¹0𝒜0𝐼^𝑓ℬ_𝑓ℒ_𝑓0. (40)

Proof. The proof is along the same lines as in the proof of Theorem 6. According to Theorem 5, 𝑃 is nonsingular if the optimization problem of (18)–(20) has feasible solutions since 𝑃>0. Now, partition 𝑃 as 𝑃𝑃≜¹𝑃₃⋆𝑃₂,𝑃₃≜𝑃₄0_{(𝑛−𝑘)×𝑘}, (41) where 𝑃₁∈ℝ^𝑛×𝑛 and 𝑃₂∈ℝ^𝑘×𝑘 are symmetric positive definite matrices and 𝑃₄∈ℝ^𝑘×𝑘. Without loss of generality, we assume that 𝑃₄ is nonsingular; to see this, let the matrix 𝒬≜𝑃+𝛼𝒲, where 𝛼 is a positive scalar and 0𝒲≜^𝑛×𝑛𝒦⋆0_𝑘×𝑘𝒬,𝒬≜¹𝒬₃⋆𝒬₂,𝒬₃≜𝒬₄0_{(𝑛−𝑘)×𝑘}. (42) With the same principle as in the proof of Theorem 6, it can be seen that there exists an arbitrarily small 𝛼>0 such that 𝒬₄ is nonsingular and (18)–(20) have feasible solutions with 𝑃 replaced by 𝒬; thus, we can conclude that there is no loss of generality to assume the matrix 𝑃₄ to be nonsingular.
Define the following matrices: Γ≜𝐼00𝑃₂⁻¹𝑃^𝑇₄,ℳ≜𝑃₁,𝒩≜𝑃₄𝑃₂⁻¹𝑃^𝑇₄𝒜_𝑓,ℬ_𝑓ℒ_𝑓0≜𝑃₄0𝐴0𝐼^𝑓𝐵_𝑓𝐿_𝑓0𝑃₂⁻¹𝑃^𝑇₄00𝐼. (43)(44) Performing congruence transformations to (19) and (20) by diag(Γ,𝐼,𝐼) and diag(𝐼,Γ), respectively, we have Γ^𝑇𝐴^𝑇𝑃Γ+Γ^𝑇𝑃𝐴ΓΓ^𝑇𝑃𝐵Γ^𝑇𝐿^𝑇⋆−𝛾²𝐵𝐼0⋆⋆−𝐼<0,−𝑅^𝑇𝑃Γ⋆−Γ^𝑇𝑃Γ≤0. (45)(46) Considering (17) and (41)–(44), we have Γ^𝑇𝑃𝐴Γ≜ℳ𝐴+𝒦ℬ_𝑓𝐶𝒦𝒜_𝑓𝒩^𝑇𝒦^𝑇𝐴+ℬ_𝑓𝐶𝒜_𝑓,Γ^𝑇𝑃𝐵≜ℳ𝐵+𝒦ℬ_𝑓𝐷𝒩^𝑇𝒦^𝑇𝐵+ℬ_𝑓𝐷,Γ^𝑇𝑃Γ≜,𝐿ℳ𝒦𝒩⋆𝒩Γ≜𝐿−ℒ_𝑓. (47) Substituting (47) into (45)-(46), we obtain (38)-(39), respectively.
The remainder of the proof follows along the same lines as in the proof of Theorem 6.

Now, considering the design of reduced-order robust filter, we give the following result without proof, which can be obtained by employing the same techniques used as those in Theorems 7 and 8.

Theorem 9. Consider the system (Σ) with system matrix 𝒢∈𝒟. There exists a reduced-order minimum entropy ℋ_∞  filter of the form of (15) such that the filtering error system (Σ) in (16) is asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾>0 and a guaranteed minimum entropy, if there exist matrices ℳ>0 and 𝒩>0 and matrices 𝒜_𝑓, ℬ_𝑓, and ℒ_𝑓 such that, for 𝑗=1,2,…,𝑛_𝑠, the following optimization problem has feasible solution: min_{ℳ>0,𝒩>0,𝒜𝑓,ℬ𝑓,ℒ𝑓}𝑡𝑟𝑎𝑐𝑒(𝑅), (48) subject to ^{Π11𝑗𝒦𝒜𝑓+𝐴𝑇𝑗𝒦𝒩+𝐶𝑇𝑗ℬ𝑇𝑓ℳ𝐵𝑗+𝒦ℬ𝑓𝐷𝑗𝐿𝑇⋆𝒜𝑓+𝒜𝑇𝑓𝒩𝑇𝒦𝑇𝐵𝑗+ℬ𝑓𝐷𝑗−ℒ𝑇𝑓⋆⋆−𝛾2𝐼0⋆⋆⋆−𝐼≤0,−𝑅𝐵𝑇𝑗ℳ+𝐷𝑇𝑗ℬ𝑇𝑓𝒦𝑇𝐵𝑇𝑗𝒦𝒩+𝐷𝑇𝑗ℬ𝑇}<^𝑓_{⋆−ℳ−𝒦𝒩⋆⋆−𝒩}0, (49) where Π_11𝑗≜ℳ𝐴_𝑗+𝐴^𝑇_𝑗ℳ+𝒦ℬ_𝑓𝐶_𝑗+𝐶^𝑇_𝑗ℬ^𝑇_𝑓𝒦^𝑇 and 𝒦 is defined in (38). Moreover, a desired reduced-order minimum entropy ℋ_∞  filter can be computed from (40).

4. Numerical Example

In this section, we present an illustrative example to demonstrate the effectiveness of the proposed algorithm. Consider the linear continuous-time system (Σ) in (1) with parameter matrix 𝒢 belonging to polyhedral domain 𝒟, and assume 𝑛_𝑠=3, then the system data 𝒢_𝑗, (𝑗=1,2,3) are given as follows: 𝐴₁=−2.30.2−0.3−0.4−0.60.00.00.5−1.3,𝐵₁=,𝐶0.60.3−0.9₁=1.00.31.0,𝐷₁𝐴=0.2,₂=−2.50.00.60.0−1.3−0.2−0.20.5−1.6,𝐵₂=,𝐶−0.40.6−0.3₂=0.41.2−0.7,𝐷₂𝐴=0.8,₃=−1.60.5−0.20.3−1.60.20.20.0−0.6,𝐵₃=,𝐶−0.1−0.30.6₃=0.3−1.50.8,𝐷₃.=−0.5,𝐿=0.30.50.8 (50) First, we consider the full-order filter design; solving the LMIs condition in Theorem 7 by applying the well-developed LMI Toolbox in the MATLAB environment directly, we obtain that the minimum 𝛾 is 𝛾^∗=0.4666, the minimum entropy of the filtering error system is ℐ(𝔼;𝛾^∗;∞)=0.3618 (where 𝔼 denotes the transfer function of filtering error system), and 𝐴_𝑓=,𝐵−0.3015−0.0948−0.0196−0.3450−1.0629−0.6874−0.2279−0.4217−0.4166_𝑓=,𝐿0.0125−0.5696−0.2754_𝑓=−0.3007−0.5002−0.8002. (51) With 𝜆₁=0.7, 𝜆₂=0.1, and 𝜆₃=0.2 in (3), the states of the full-order designed filter are given in Figure 1, where the initial condition is 1.0−0.5−1.0^𝑇, and the exogenous disturbance input 𝜔(𝑡) is given as 𝜔(𝑡)=1/(0.5+1.8𝑡), 𝑡≥0. Figure 2 shows the error response of 𝑒(𝑡).

Figure 1 

States of the designed full-order filter.

Figure 2 

Filtering error of the full-order filtering.

Now, we consider the reduced-order filter design, and two cases of such filters are considered.

Case 1. Set 𝑘=2; that is, the order of the reduced filter is 𝑘=2; solving the LMIs condition in Theorem 9, we obtain that the minimum 𝛾 is 𝛾^∗=0.5069, the minimum entropy of the filtering error system is ℐ(𝔼;𝛾^∗;∞)=0.9401, and 𝐴_𝑓=−1.94830.53540.5302−0.1546,𝐵_𝑓=,𝐿0.6662−0.1850_𝑓=1.1058−0.3003. (52) Under the same conditions as in the full-order filter design, the states of the designed reduced-order filter are given in Figure 3. Figure 4 shows the error response of 𝑒(𝑡).

Figure 3 

States of the designed reduced-order filter (with the order of 𝑘=2).

Figure 4 

Filtering error of the reduced-order filtering (with the order of 𝑘=2).

Case 2. Set 𝑘=1; that is, the order of the reduced filter is 𝑘=1; by solving the LMIs condition in Theorem 9, we obtain that the minimum 𝛾 is 𝛾^∗=0.5148, the minimum entropy of the filtering error system is ℐ(𝔼;𝛾^∗;∞)=0.8677, and 𝐴_𝑓=−1.8368,𝐵_𝑓=0.4853,𝐿_𝑓=1.2617. (53) The states and the filtering error of the designed reduced-order filter are given in Figures 5 and 6, respectively.

Figure 5 

States of the designed reduced-order filter (with the order of 𝑘=1).

Figure 6 

Filtering error of the reduced-order filtering (with the order of 𝑘=1).

5. Conclusion

In this paper, the robust minimum entropy ℋ_∞ filter has been designed for linear continuous-time systems with polytopic parameter uncertainty. Sufficient conditions have been established for the existence of general full- and reduced-order minimum entropy ℋ_∞ filters in terms of LMIs, which guarantee the filtering error system to be robustly asymptotically stable and to have a prescribed ℋ_∞ performance as well as a guaranteed minimum entropy. The filler design could be cast into a convex optimization problem and a numerical example has been provided to demonstrate the effectiveness of the proposed design method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work has been supported by the National Natural Science Foundation of China (Grant no. 61104109), the Natural Science Foundation of Jiangsu Province of China (Grant no. BK2011703), the Support of Science and Technology and Independent Innovation Foundation of Jiangsu Province of China (Grant no. BE2012178), and the Doctoral Fund of Ministry of Education of China (Grant no. 20113219110027).