Delay-Dependent 𝐻_∞ Filtering for Singular Time-Delay Systems

Abstract

This paper deals with the problem of delay-dependent 𝐻_∞ filtering for singular time-delay systems. First, a new delay-dependent condition which guarantees that the filter error system has a prescribed 𝐻_∞ performance 𝛾 is given in terms of linear matrix inequalities (LMIs). Then, the sufficient condition is obtained for the existence of the 𝐻_∞ filter, and the explicit expression for the desired 𝐻_∞ filter is presented by using LMIs and the cone complementarity linearization iterative algorithm. A numerical example is provided to illustrate the effectiveness of the proposed method.

1. Introduction

Over the past decades, the filtering problem has been widely studied and has found many applications [1, 2]. Current efforts on this topic can be mainly divided into two classes: the Kalman filtering approach and the 𝐻_∞ filtering approach. The objective of the latter one is to find a filter such that the resulting error system is asymptotically stable and the 𝐿₂-induced norm (for continuous systems) or 𝑙₂-induced norm (for discrete systems) from the disturbance input to the filtering error output satisfies a prescribed 𝐻_∞ performance level. In contrast to the Kalman filtering, the 𝐻_∞ filtering approach does not require the exact knowledge of the statistics of the external noise signals, and it is insensitive to the uncertainties. These features render the 𝐻_∞ filtering attracting much attention, and many efforts have been made on this issue [3–6]. The filtering problem for singular systems has also been investigated by many researchers. For example, a necessary and sufficient condition is obtained in [7] for the solvability of the 𝐻_∞ filtering problem and the designed filter is proper with a McMillan degree no more than the exponential modes of the plant, while, in [8], a linear normal 𝐻_∞ filter is obtained for singular systems. Reduced-order 𝐻_∞ filters are designed in [9] for both continuous and discrete singular systems. In [10], a reduced-order 𝐻_∞ filter design approach is developed for a class of discrete singular systems with lossy measurements.

On the other hand, for many practical control systems, time delays are frequently encountered and they are often the sources of instability and degradation in control performance. So, recently, there has been increasing interest in 𝐻_∞ filtering for time-delay systems. Existing results can be classified into two types: delay-independent ones [11–14] and delay-dependent ones [15–23]; the former do not include any information on the size of delay while the latter employ such information. Generally speaking, delay-dependent results are less conservative than the delay-independent ones, especially when the size of delay is small.

Singular time-delay systems, which are also referred to as implicit time-delay systems, descriptor time-delay systems, or generalized differential-difference equations, often appear in various engineering systems, including aircraft attitude control, flexible arm control of robots, large-scale electric network control, chemical engineering systems, and lossless transmission lines (see, e.g., [24]). Since singular time-delay systems are more general, it is of significance to consider the 𝐻_∞ filtering problem for them. Recently, some delay-dependent [25–27] and delay-dependent [28–31] results about 𝐻_∞ filters for such systems have been obtained. In [28], the delay-independent filter is of the Luenberger observer type and the decomposition and transformation of the system matrices are involved, which would result in some numerical problems. A full-order filter is designed in [29] for singular systems with communication delays, and 𝐻_∞ filtering problems are concerned in [30, 31] for singular systems with time-varying delay in a range.

In this paper, the problem of delay-dependent 𝐻_∞ filtering is investigated for singular time-delay systems. We consider the case of discrete delay which is assumed to be constant and known. First, based on the result in [32], we derive a new delay-dependent condition which guarantees that the filter error system has a prescribed 𝐻_∞ performance 𝛾; and it can be seen that this new condition is more “efficient” than that in [32] since no redundant variables are involved. Then, the sufficient condition for the existence of the full-order 𝐻_∞ filter, which is an admissible singular time-delay system, is obtained and the explicit expression for the desired 𝐻_∞ filter is given by using LMIs and the cone complementarity linearization iterative algorithm.

Notations
𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space and 𝑅^𝑛×𝑚 denotes the set of all 𝑛×𝑚 real matrices, 𝐼_𝑛 is the 𝑛-dimensional identity matrix, and diag{⋯} is a block-diagonal matrix. For real symmetric matrix 𝑋, the notation 𝑋≥0(𝑋>0) means that the matrix 𝑋 is positive-semidefinite (positive-definite). The superscript 𝑇 represents the transpose; the symbol ∗ will be used in some matrix expressions to induce a symmetric structure. 𝐿₂[0,∞) refers to the space of square-integrable vector functions over [0,∞) with norm ‖𝑓‖₂∶=(∫^∞₀‖𝑓(𝑡)‖²𝑑𝑡)^1/2.

2. Problem Statement

Consider the following singular time-delay system:𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝜏𝑥()(𝑡),𝑦(𝑡)(𝑡)𝑡−𝜏+𝐵𝑤=𝐶𝑥+𝐶_𝜏𝑥(𝑡−𝜏)+𝐵₁𝑤(𝑡),𝑧(𝑡)=𝐺𝑥(𝑡)+𝐺_𝜏𝑥(𝑡−𝜏)+𝐵₂𝑤(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0], (2.1) where 𝑥(𝑡)∈𝑅^𝑛 is the state, 𝑤(𝑡)∈𝑅^𝑟 is the external disturbance signal that belongs to 𝐿₂[0,∞),𝑦(𝑡)∈𝑅^𝑚 is the measurement output, and 𝑧(𝑡)∈𝑅^𝑠 is the signal to be estimated. 𝐸,𝐴,𝐴_𝜏,𝐵,𝐶,𝐶_𝜏,𝐵₁,𝐺,𝐺_𝜏, and 𝐵₂ are known real constant matrices with appropriate dimensions and 0<rank𝐸=𝑝<𝑛. 𝜏>0 is the known delay constant and 𝜙(𝑡)∈𝐶_𝑛,𝜏 is a compatible vector-valued initial function.

Without loss of generality, we assume that 𝐶_𝜏=0,𝐵₁=0,𝐺_𝜏=0, and 𝐵₂=0. Otherwise, system (2.1) can be equivalently changed into𝑥̇(𝑡)𝜁̇(𝑡)=𝐸000𝐴00−𝐼_𝑚+𝑠𝑥(𝑡)𝜁(𝑡)+𝐴_𝜏𝐶∣0−−−−−_𝜏∣0_𝑚+𝑠𝐺_𝜏∣𝑥()𝜁()+𝐵𝐵𝑡−𝜏𝑡−𝜏−−₁𝐵₂𝑤(𝑡),𝑦(𝑡)𝑧(𝑡)=𝐼𝐶∣_𝑚+𝑠𝑥(𝑡)𝜁(𝑡).𝐺∣ (2.2) Then in the sequel, we discuss the system model as follows:𝐸𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝜏𝑥()(𝑡),𝑦(𝑡)(𝑡),𝑧(𝑡)(𝑡),𝑥(𝑡)(𝑡)[].𝑡−𝜏+𝐵𝑤=𝐶𝑥=𝐺𝑥=𝜙,𝑡∈−𝜏,0 (2.3) Throughout this paper, we need the following assumption for system (2.3).

Assumption 2.1. System (2.3) is admissible, that is, when 𝑤(𝑡)≡0, system (2.3) is regular, impulse free, and asymptotically stable.

Remark 2.2. About the definitions of regularity, absence of impulses and asymptotical stability for singular time-delay systems, we refer the readers to [33].

For the estimates of 𝑧(𝑡), we consider the following linear filter with delay:𝐸𝑥̇(𝑡)=𝐴_𝑓𝑥(𝑡)+𝐴_𝜏𝑓𝑥(𝑡−𝜏)+𝐵_𝑓𝑦(𝑡),𝑧(𝑡)=𝐶_𝑓𝑥(𝑡),𝑥(𝑡)=𝜓(𝑡),𝑡∈[−𝜏,0], (2.4) where 𝑥(𝑡)∈𝑅^𝑛 and 𝑧(𝑡)∈𝑅^𝑠 are the state and the output of the filter, respectively. The constant matrices 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓, and 𝐶_𝑓 are filter parameters to be determined.

Letting𝑒(𝑡)∶=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)^𝑇,𝑧(𝑡)∶=𝑧(𝑡)−𝑧(𝑡), (2.5) one obtains the filter error system𝐸𝑒̇(𝑡)=𝐴𝑒(𝑡)+𝐴_𝜏𝑒()+(𝑡),𝑧(𝑡)=(𝑡),𝑒(𝑡)=𝜙𝑡−𝜏𝐵𝑤𝐺𝑒_𝑇(𝑡)𝜓^𝑇(𝑡)^𝑇,𝑡∈[−𝜏,0], (2.6) where ,𝐵𝐸=𝐸00𝐸𝐴=𝐴0_𝑓𝐶𝐴_𝑓,𝐴_𝜏=𝐴_𝜏00𝐴_𝜏𝑓,𝐵0𝐵=,𝐺=𝐺−𝐶_𝑓. (2.7)

Thus, the filtering problem to be addressed is stated as follows.

𝐻_∞ Filtering Problem
For a given 𝛾>0, design a full-order filter with delay of the form of (2.4) such that the filter error system (2.6) has prescribed 𝐻_∞ performance 𝛾, that is,(1)system (2.6) is admissible;(2)under zero initial condition, for any nonzero 𝑤(𝑡)∈𝐿₂[0,∞), the 𝐻_∞ performance ‖𝑧(𝑡)‖₂≤𝛾‖𝑤(𝑡)‖₂ is guaranteed.

Remark 2.3. Similar to [17], it is easy to see that system (2.3) is admissible if the error system (2.6) is admissible. That is why we made Assumption 2.1 on system (2.3).

3. Main Results

At first, we will concentrate our attention on 𝐻_∞ performance analysis for the error system (2.6). The following lemma is useful in the proof of Theorem 3.2.

Lemma 3.1 (see [32]). Given a scalar 𝛾>0, the filter error system (2.6) has a prescribed 𝐻_∞ performance 𝛾 if there exist matrices 𝑄>0,𝑍>0,𝑃,𝑌, and 𝑊 satisfying 𝐸^𝑇𝑃^𝑇=𝑃𝐸≥0,ΦΦ=¹Φ₂𝜏𝑌^𝑇𝑃𝐵𝜏𝐴^𝑇𝑍𝐺^𝑇∗Φ₃𝜏𝑊^𝑇0𝜏𝐴^𝑇_𝜏𝑍0∗∗−𝜏𝑍000∗∗∗−𝛾²𝐼𝜏𝐵^𝑇𝑍0∗∗∗∗−𝜏𝑍0∗∗∗∗∗−𝐼<0, (3.1)(3.2) where Φ₁=𝑃𝐴+𝐴^𝑇𝑃^𝑇+𝑄−𝑌^𝑇𝐸−𝐸^𝑇𝑌,Φ₂=𝑃𝐴_𝜏+𝑌^𝑇𝐸−𝐸^𝑇Φ𝑊,₃=−𝑄+𝑊^𝑇𝐸+𝐸^𝑇𝑊. (3.3)

Based on Lemma 3.1, we will present a new delay-dependent bounded real lemma (BRL) for the performance analysis of system (2.6), which can be shown to be more “efficient” than Lemma 3.1.

Theorem 3.2. Given a scalar 𝛾>0, the filter error system (2.6) has a prescribed 𝐻_∞ performance 𝛾 if there exist matrices 𝑄>0,𝑍>0 and 𝑃 satisfying (3.1) and ΩΩ=¹Ω₂𝑃𝐵𝜏𝐴^𝑇𝑍𝐺^𝑇∗Ω₃0𝜏𝐴^𝑇_𝜏𝑍0∗∗−𝛾²𝐼𝜏𝐵^𝑇𝑍0∗∗∗−𝜏𝑍0∗∗∗∗−𝐼<0, (3.4) where Ω₁=𝑃𝐴+𝐴^𝑇𝑃^𝑇+1𝑄−𝜏𝐸^𝑇𝑍𝐸,Ω₂=𝑃𝐴_𝜏+1𝜏𝐸^𝑇𝑍𝐸,Ω₃1=−𝑄−𝜏𝐸^𝑇𝑍𝐸. (3.5)

Proof. From Lemma 3.1, if we can prove that the feasibility of Ω<0 for solution (𝑄>0,𝑍>0,𝑃) is equivalent to that of Φ<0 for solution (𝑄>0,𝑍>0,𝑃,𝑌,𝑊), then Theorem 3.2 is proved.
Similar to Lemma 4 of [34], take Ψ=ΠΦΠ^𝑇=Ω₁Ω₂𝜏𝑌^𝑇−𝐸^𝑇𝑍𝑃𝐵𝜏𝐴^𝑇𝑍𝐺^𝑇∗Ω₃𝜏𝑊^𝑇+𝐸^𝑇𝑍0𝜏𝐴^𝑇_𝜏𝑍0∗∗−𝜏𝑍000∗∗∗−𝛾²𝐼𝜏𝐵^𝑇,𝑍0∗∗∗∗−𝜏𝑍0∗∗∗∗∗−𝐼 (3.6) with 1Π=𝐼0𝜏𝐸^𝑇10000𝐼−𝜏𝐸^𝑇.00000𝐼000000𝐼000000𝐼000000𝐼 (3.7) It follows from Schur complement that 𝜏𝑌Φ<0⟺Ψ<0⟺𝑍>0,Ω+^𝑇−𝐸^𝑇𝑍𝜏𝑊^𝑇+𝐸^𝑇𝑍000𝜏𝑍⁻¹𝜏𝑌^𝑇−𝐸^𝑇𝑍𝜏𝑊^𝑇+𝐸^𝑇𝑍000^𝑇<0. (3.8)
If there exist 𝑄>0,𝑍>0, 𝑃,𝑌, and 𝑊 satisfying Φ<0, from (3.8) it is easy to see that the above (𝑄,𝑍,𝑃) is a feasible solution of Ω<0. Conversely, if there exist 𝑄>0,𝑍>0 and 𝑃 such that Ω<0 holds, via taking 𝑌=(1/𝜏)𝑍𝐸 and 𝑊=−(1/𝜏)𝑍𝐸,Φ<0 is also feasible for the above (𝑄,𝑍,𝑃,𝑌,𝑊). This completes the proof.

The following corollary is easy to be obtained from Theorem 3.2.

Corollary 3.3. The filter error system (2.6) is admissible if there exist matrices 𝑄>0,𝑍>0 and 𝑃 satisfying (3.1) and 𝑃𝐴+𝐴^𝑇𝑃^𝑇+1𝑄−𝜏𝐸^𝑇𝑍𝐸𝑃𝐴_𝜏+1𝜏𝐸^𝑇𝑍𝐸𝜏𝐴^𝑇𝑍1∗−𝑄−𝜏𝐸^𝑇𝑍𝐸𝜏𝐴^𝑇_𝜏𝑍∗∗−𝜏𝑍<0. (3.9)

Remark 3.4. Theorem 3.2 can also be proved by employing the relationship of two integral inequalities concluded in [35]. In fact, we can see that Lemma 3.1 is obtained by using the integral inequality (7) in [35], while using the integral inequality (9) in [35] yields Theorem 3.2. As shown by [35], the upper bound provided by (9) in [35] is the least upper bound provided by (7) in [35]; therefore introducing more free matrices cannot reduce the conservativeness. Then, Theorem 3.2 can be obtained from Lemma 3.1, and the introduced slack variables 𝑌 and 𝑊 in Lemma 3.1 are redundant variables. Hence, from the computational point of view, Theorem 3.2 is more “efficient” than Lemma 3.1.

In the sequel, based on Theorem 3.2, we are devoted to the design of the filter parameters 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓, and 𝐶_𝑓. Noticing that (3.4) is nonlinear about the unknown variables 𝐴,𝐴_𝜏,𝑃, and 𝑍, to reduce the number of the unknown variables, we can do as follows.

From (3.4) we know that𝑃𝐴+𝐴^𝑇𝑃^𝑇+1𝑄−𝜏𝐸^𝑇𝑍𝐸𝑃𝐴_𝜏+1𝜏𝐸^𝑇𝑍𝐸1∗−𝑄−𝜏𝐸^𝑇Z𝐸<0. (3.10) Multiplying (3.10) by [𝐼𝐼] from the left and by [𝐼𝐼]^𝑇 from the right results in𝑃𝐴+𝐴_𝜏+𝐴+𝐴_𝜏^𝑇𝑃^𝑇<0, (3.11) which implies that 𝑃 is nonsingular. Let𝑃=𝑃𝑃₂𝑃₃𝑃₄,𝑃∈𝑅^𝑛×𝑛,𝑃_𝑖∈𝑅^𝑛×𝑛,𝑖=2,3,4. (3.12) Without loss of generality, we can assume that 𝑃,𝑃_𝑖,𝑖=2,3,4, are all nonsingular [36]. Then, from (3.1), we have that𝐸^𝑇𝑃^𝑇=𝑃𝐸,𝐸^𝑇𝑃^𝑇₃=𝑃₂𝐸,𝐸^𝑇𝑃^𝑇₄=𝑃₄𝐸. (3.13) Taking𝑇₁=𝐼00𝑃𝑃₃⁻¹,𝑇₂=𝐼00𝑃₂⁻¹𝑃,𝑇₃=diag𝑇₁,𝑇₁,𝐼,𝑇^𝑇₂,𝐼 (3.14) and combining with (2.7) and (3.12), we obtain 𝐸=𝑇₂⁻¹𝐸𝑇^𝑇₁=𝐸00𝑃⁻¹𝑃₂𝐸𝑃₃^−𝑇𝑃^𝑇=𝐸00𝑃⁻¹𝐸^𝑇𝑃^𝑇₃𝑃₃^−𝑇𝑃^𝑇=,𝐸00𝐸𝑃=𝑇₁𝑃𝑇₂=𝑃𝑃𝑃𝑃𝑃₃⁻¹𝑃₄𝑃₂⁻¹𝑃,𝐴=𝑇₂⁻¹𝐴𝑇^𝑇₁=𝑃𝐴0₋₁𝑃₂𝐵_𝑓𝐶𝑃⁻¹𝑃₂𝐴_𝑓𝑃₃^−𝑇𝑃^𝑇=𝐴0𝐵_𝑓𝐶𝐴_𝑓,𝐴_𝜏=𝑇₂⁻¹𝐴_𝜏𝑇^𝑇₁=𝐴_𝜏00𝑃⁻¹𝑃₂𝐴_𝜏𝑓𝑃₃^−𝑇𝑃^𝑇=𝐴_𝜏00𝐴_𝜏𝑓,𝐵=𝑇₂⁻¹𝐵0𝐵=,𝐺=𝐺𝑇^T₁=𝐺−𝐶_𝑓𝑃₃^−𝑇𝑃^𝑇=𝐺−𝐶_𝑓, (3.15) where𝐴_𝑓=𝑃⁻¹𝑃₂𝐴_𝑓𝑃₃^−𝑇𝑃^𝑇,𝐴_𝜏𝑓=𝑃⁻¹𝑃₂𝐴_𝜏𝑓𝑃₃^−𝑇𝑃^𝑇,𝐵_𝑓=𝑃⁻¹𝑃₂𝐵_𝑓,𝐶_𝑓=𝐶_𝑓𝑃₃^−𝑇𝑃^𝑇, (3.16) and denote𝑄=𝑇₁𝑄𝑇^𝑇₁,𝑍=𝑇^𝑇₂𝑍𝑇₂. (3.17) Premultiplying by 𝑇₁ and postmultiplying by 𝑇^𝑇₁ on both sides of (3.1), we have that 𝑇₁𝐸^𝑇𝑇₂^−𝑇𝑇^𝑇₂𝑃^𝑇𝑇^𝑇₁=𝑇₁𝑃𝑇₂𝑇₂⁻¹𝐸𝑇^𝑇₁≥0, (3.18) that is,𝐸^𝑇𝑃^𝑇=𝑃𝐸≥0. (3.19) Multiplying (3.4) by 𝑇₃ from the left and by 𝑇^𝑇₃ from the right yieldsΩ=𝑃𝐴+𝐴^𝑇𝑃^𝑇+1𝑄−𝜏𝐸^𝑇𝑍𝐸𝑃𝐴_𝜏+1𝜏𝐸^𝑇𝑍𝐸𝑃𝐵𝜏𝐴^𝑇𝑍𝐺^𝑇∗−1𝑄−𝜏𝐸^𝑇𝑍𝐸0𝜏𝐴^𝑇_𝜏𝑍0∗∗−𝛾²𝐼𝜏𝐵^𝑇𝑍0∗∗∗−𝜏𝑍0∗∗∗∗−𝐼<0. (3.20)

It can be seen that the systems (𝐸,𝐴,𝐴_𝜏,𝐵,𝐺) and (𝐸,𝐴,𝐴_𝜏,𝐵,𝐺) are algebraically equivalent under the r.s.e. (restricted system equivalence) transformation, where 𝑇₂⁻¹ and 𝑇^𝑇₁ are taken as the row full rank transformation matrix and the coordinate full rank transformation matrix, respectively, and comparing the coefficient matrices of the two systems, we can see that the difference between them is just the filter parameters 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓,𝐶_𝑓, and 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓,𝐶_𝑓. Moreover, in the r.s.e. transformation, the state and the equation of the filter change while the state and the equation of system (2.3) do not change. So, in the design of the filter, we can directly substitute 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓,𝐶_𝑓 for 𝐴_𝑓,𝐴_𝜏𝑓,𝐵_𝑓,𝐶_𝑓. Noticing that𝐼0−𝑃₃𝑃⁻¹𝐼𝑃𝑃₂𝑃₃𝑃₄𝐼−𝑃⁻¹𝑃₂=0𝐼𝑃00𝑃₄−𝑃₃𝑃⁻¹𝑃₂, (3.21) then 𝑃₄−𝑃₃𝑃⁻¹𝑃₂ is nonsingular. Let𝑃𝑃₃⁻¹𝑃₄𝑃₂⁻¹𝑃−𝑃=𝑃𝑃₃⁻¹𝑃₄−𝑃₃𝑃⁻¹𝑃₂𝑃₂⁻¹𝑃=𝑆⁻¹; (3.22) then 𝑃 can be written as𝑃=𝑃𝑃𝑃𝑃+𝑆⁻¹. (3.23)

Denote𝐽^𝑇=𝑆+𝑃⁻¹,𝑇₄=𝐽^𝑇−𝑆𝐼0,𝑇₅=𝑇₄𝑃𝑍⁻¹. (3.24) Since 𝑃(𝑆+𝑃⁻¹)=(𝑃+𝑆⁻¹)𝑆=𝑃𝑃₃⁻¹𝑃₄𝑃₂⁻¹𝑃𝑆 is nonsingular, 𝐽 is also a nonsingular matrix.

From (3.19), we have that,𝑇₄𝐸^𝑇𝑃^𝑇𝑇^𝑇₄=𝑇₄𝑃𝐸𝑇^𝑇₄≥0. (3.25) Noticing (3.23) and (3.24), we derive 𝑇₄,𝑇𝑃=𝐼0𝑃𝑃₄𝑃𝐸𝑇^𝑇₄=𝐸𝐽𝐸𝑃𝐸𝐽−𝑃𝐸𝑆^𝑇=𝑃𝐸𝐸𝐽𝐸𝑃𝐸𝑃^−𝑇=𝑃𝐸𝐸𝐸𝐽𝐸_𝑇,𝑇𝑃𝐸₄𝐸^𝑇𝑃^𝑇𝑇^𝑇₄=𝐽^𝑇𝐸^T𝐸𝐸^𝑇𝐸^𝑇𝑃^𝑇; (3.26) then (3.25) is just𝐸𝐽=𝐽^𝑇𝐸^𝑇,𝑃𝐸=𝐸^𝑇𝑃^𝑇,𝐸𝐸𝐽𝐸_𝑇𝑃𝐸≥0. (3.27)

Premultiplying by diag{𝑇₄,𝑇₄,𝐼,𝑇₅,𝐼} and postmultiplying by diag{𝑇^𝑇₄,𝑇^𝑇₄,𝐼,𝑇^𝑇₅,𝐼} on both sides of (3.20), we have𝑇₄𝐴^𝑇𝑃^𝑇𝑇^𝑇₄+𝑇₄𝑃𝐴𝑇^𝑇₄+𝑇₄𝑄𝑇^𝑇₄−1𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄𝑇₄𝑃𝐴_𝜏𝑇^𝑇₄+1𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄𝑇₄𝑃𝐵𝜏𝑇₄𝐴^𝑇𝑍𝑇^𝑇₅𝑇₄𝐺^𝑇∗−𝑇₄𝑄𝑇^𝑇₄−1𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄0𝜏𝑇₄𝐴^𝑇_𝜏𝑍𝑇^𝑇₅0∗∗−𝛾²𝐼𝜏𝐵^𝑇𝑍𝑇^𝑇₅0∗∗∗−𝜏𝑇₅𝑍𝑇^𝑇₅0∗∗∗∗−𝐼<0. (3.28) Noticing that 𝑇₄𝑃𝐴𝑇^𝑇₄=𝐴𝐽𝐴𝑃𝐴𝐽+𝑃𝐵_𝑓𝐶𝐽−𝑃𝐴_𝑓𝑆^𝑇𝑃𝐴+𝑃𝐵_𝑓𝐶,𝑇₄𝑃𝐴_𝜏𝑇^𝑇₄=𝐴_𝜏𝐽𝐴_𝜏𝑃𝐴_𝜏𝐽−𝑃𝐴_𝜏𝑓𝑆^𝑇𝑃𝐴_𝜏,𝑇₄𝑃𝐵𝐵=𝑃𝐵,𝑇₄𝐺^𝑇=𝐽^𝑇𝐺^𝑇+𝑆𝐶^𝑇_𝑓𝐺^𝑇,𝑇₄𝐴^𝑇𝑍𝑇^𝑇₅=𝑇₄𝐴^𝑇𝑍𝑍⁻¹𝑃^𝑇𝑇^𝑇₄=𝑇₄𝐴^𝑇𝑃^𝑇𝑇^𝑇₄,𝑇₄𝐴^𝑇_𝜏𝑍𝑇^𝑇₅=𝑇₄𝐴^𝑇_𝜏𝑍𝑍⁻¹𝑃^𝑇𝑇^𝑇₄=𝑇₄𝐴^𝑇_𝜏𝑃^𝑇𝑇^𝑇₄,𝐵^𝑇𝑍𝑇^𝑇₅=𝐵^𝑇𝑍𝑍⁻¹𝑃^𝑇𝑇^𝑇₄=𝐵^𝑇𝑃^𝑇𝑇^𝑇₄,𝑇₅𝑍𝑇^𝑇₅=𝑇₄𝑃𝑍⁻¹𝑍𝑍⁻¹𝑃^𝑇𝑇^𝑇₄=𝑇₄𝑃𝑍⁻¹𝑃^𝑇𝑇^𝑇₄, (3.29) denote𝑄=𝑇₄𝑄𝑇^𝑇₄=𝑄₁𝑄₂𝑄^𝑇₂𝑄₃,𝑍=𝑇₄𝑃𝑍⁻¹𝑃^𝑇𝑇^𝑇₄=𝑍₁𝑍₂𝑍^𝑇₂𝑍₃,𝐿=𝑃𝐴𝐽+𝑃𝐵_𝑓𝐶𝐽−𝑃𝐴_𝑓𝑆^𝑇,𝐿_𝜏=𝑃𝐴_𝜏𝐽−𝑃𝐴_𝜏𝑓𝑆^𝑇,𝑊_𝐵=𝑃𝐵_𝑓,𝑊_𝐶=𝐶_𝑓𝑆^𝑇. (3.30)(3.31)(3.32)(3.33)

Since (3.31) implies that 𝑍=𝑃^𝑇𝑇^𝑇₄𝑍⁻¹𝑇₄𝑃, then𝑇₄𝐸^𝑇𝑍𝐸𝑇^T₄=𝑇₄𝐸^𝑇𝑃^𝑇𝑇^𝑇₄𝑍⁻¹𝑇₄𝑃𝐸𝑇^𝑇₄=𝐸𝐸𝐽𝐸_𝑇𝑍𝑃𝐸⁻¹𝐸𝐸𝐽𝐸_𝑇.𝑃𝐸 (3.34)

Introduce matrix 𝑊=^𝑊1𝑊2_{𝑊𝑇2𝑊3}≥0 satisfying𝐸𝜏𝑊≤𝐸𝐽𝐸_𝑇𝑍𝑃𝐸⁻¹𝐸𝐸𝐽𝐸_𝑇,𝑃𝐸 (3.35) then−1𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄1𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄1∗−𝜏𝑇₄𝐸^𝑇𝑍𝐸𝑇^𝑇₄≤;−𝑊𝑊∗−𝑊 (3.36) Obviously, if there exist matrices 𝑄₁>0,𝑄₃>0,𝑊₁≥0,𝑊₃≥0,𝑍₁>0,𝑍₃>0,𝑃,𝐽,𝑊_𝐵,𝑊_𝐶,𝐿,𝐿_𝜏,𝑄₂,𝑊₂, and 𝑍₂ with 𝑃,𝐽 being nonsingular, satisfying (3.35) andΞ₁₁Ξ₁₂𝐴_𝜏𝐽+𝑊₁𝐴_𝜏+𝑊₂𝐵𝜏𝐽^𝑇𝐴^𝑇𝜏𝐿^𝑇𝐽^𝑇𝐺^𝑇+𝑊^𝑇_𝑐∗Ξ₂₂𝐿_𝜏+𝑊^𝑇₂𝑃𝐴_𝜏+𝑊₃𝑃𝐵𝜏𝐴^𝑇𝜏𝐴^𝑇𝑃^𝑇+𝜏𝐶^𝑇𝑊^𝑇_𝐵𝐺^𝑇∗∗−𝑄₁−𝑊₁−𝑄₂−𝑊₂0𝜏𝐽^𝑇𝐴^𝑇_𝜏𝜏𝐿^𝑇_𝜏0∗∗∗−𝑄₃−𝑊₃0𝜏𝐴^𝑇_𝜏𝜏𝐴^𝑇_𝜏𝑃^𝑇0∗∗∗∗−𝛾²𝐼𝜏𝐵^𝑇𝜏𝐵^𝑇𝑃^𝑇0∗∗∗∗∗−𝜏𝑍₁−𝜏𝑍₂0∗∗∗∗∗∗−𝜏𝑍₃0∗∗∗∗∗∗∗−𝐼<0 (3.37) with Ξ₁₁=𝐴𝐽+𝐽^𝑇𝐴^𝑇+𝑄₁−𝑊₁,Ξ₁₂=𝐴+𝐿^𝑇+𝑄₂−𝑊₂,Ξ₂₂=𝑃𝐴+𝐴^𝑇𝑃^𝑇+𝑊_𝐵𝐶+𝐶^𝑇𝑊^𝑇_𝐵+𝑄₃−𝑊₃, (3.38) then taking𝑆=𝐽^𝑇−𝑃⁻¹,𝐵_𝑓=𝑃⁻¹𝑊_𝐵,𝐶_𝑓=𝑊_𝐶𝑆^−𝑇,𝐴_𝑓=𝑃⁻¹𝑃𝐴𝐽+𝑊_𝐵𝐶𝐽−𝐿𝑆^−𝑇,𝐴_𝜏𝑓=𝑃⁻¹𝑃𝐴_𝜏𝐽−𝐿_𝜏𝑆^−𝑇, (3.39) one obtains that there are solutions 𝑄>0,𝑍>0, and 𝑃 to (3.20).

Hence we get the following theorem for the design of the filter (2.4).

Theorem 3.5. Given a scalar 𝛾>0, if there are matrices 𝑄₁>0,𝑄₃>0,𝑊₁≥0,𝑊₃≥0,𝑍₁>0,𝑍₃>0,𝑃,𝐽,𝑊_𝐵,𝑊_𝐶,𝐿,𝐿_𝜏,𝑄₂,𝑊₂,𝑍₂ with 𝑃,𝐽 being nonsingular, satisfying (3.27), (3.35), and (3.37), then the 𝐻_∞ filter of the form of (2.4) exists and the parameters are given by (3.39).

Remark 3.6. It is worth noting that (3.35) is not an LMI. In order to use the LMI Toolbox in MATLAB to get the solutions, we can do as follows.
Assume that 𝐸=^𝐼𝑝0₀₀; otherwise, we can find nonsingular matrices 𝑀 and 𝑁 such that 𝑀𝐸𝑁=^𝐼𝑝0₀₀. It is worth noting that the feasibility of (3.27), (3.35), and (3.37) is not affected by the selection of 𝑀 and 𝑁. Then, the matrices 𝑃,𝐽 satisfying (3.27) are of the forms 𝑃𝑃=¹¹𝑃₁₂0𝑃₂₂𝐽,𝐽=¹¹0𝐽₂₁𝐽₂₂,𝑃₁₁∈𝑅^𝑝×𝑝,𝐽₁₁∈𝑅^𝑝×𝑝 (3.40) with 𝐽₁₁𝐼𝐼𝑃₁₁≥0. (3.41) Introduce another variable 𝑈>0; then (3.35) can be replaced by 𝐸𝜏𝑊≤𝐸𝐽𝐸_𝑇𝑈𝑃𝐸𝐸𝐸𝐽𝐸_𝑇,𝑃𝐸𝑈𝑍=𝐼. (3.42)(3.43) Write 𝑈 as 𝑈𝑈=¹¹𝑈₁₂𝑈₁₃𝑈₁₄𝑈^𝑇₁₂𝑈₂₂𝑈₂₃𝑈₂₄𝑈^𝑇₁₃𝑈^𝑇₂₃𝑈₃₃𝑈₃₄𝑈^𝑇₁₄𝑈^𝑇₂₄𝑈^𝑇₃₄𝑈₄₄>0, (3.44) where 𝑈₁₁∈𝑅^𝑝×𝑝,𝑈₂₂∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)},𝑈₃₃∈𝑅^𝑝×𝑝,𝑈₄₄∈𝑅^{(𝑛−𝑝)×(𝑛−𝑝)}. (3.45) Noticing that 𝐸𝐸𝐽𝐸_𝑇𝑈𝑃𝐸𝐸𝐸𝐽𝐸_𝑇=Π𝑃𝐸¹¹0Π₁₃0∗000∗∗Π₃₃0,∗∗∗0 (3.46) where Π₁₁=𝐽₁₁𝑈₁₁𝐽₁₁+𝑈^𝑇₁₃𝐽₁₁+𝐽₁₁𝑈₁₃+𝑈₃₃,Π₁₃=𝐽₁₁𝑈₁₁+𝑈^𝑇₁₃+𝐽₁₁𝑈₁₃𝑃₁₁+𝑈₃₃𝑃₁₁,Π₃₃=𝑈₁₁+𝑃₁₁𝑈^𝑇₁₃+𝑈₁₃𝑃₁₁+𝑃₁₁𝑈₃₃𝑃₁₁, (3.47) we can assume that 𝑊𝑊=¹¹0𝑊₂₁0𝑊0000_𝑇210𝑊₃₁0𝑊0000≥0,¹¹𝑊₂₁𝑊^𝑇₂₁𝑊₃₁>0. (3.48) Then (3.42) is just 𝜏𝑊₁₁𝑊₂₁𝑊^𝑇₂₁𝑊₃₁≤𝐽₁₁𝐼𝐼𝑃₁₁𝑈₁₁𝑈₁₃𝑈^𝑇₁₃𝑈₃₃𝐽₁₁𝐼𝐼𝑃₁₁. (3.49) Invoking Schur complement again, we have that (3.49) is equivalent to 𝑈₁₁𝑈₁₃𝑈^𝑇₁₃𝑈₃₃𝐽₁₁𝐼𝐼𝑃₁₁⁻¹𝐽₁₁𝐼𝐼𝑃₁₁⁻¹𝜏𝑊₁₁𝑊₂₁𝑊^𝑇₂₁𝑊₃₁⁻¹≥0. (3.50) Introducing 𝛼𝛼=¹𝛼₂𝛼^𝑇₂𝛼₃𝜃>0,𝜃=¹𝜃₂𝜃^𝑇₂𝜃₃>0, (3.51) then (3.50) can be replaced by 𝑈₁₁𝑈₁₃𝜏𝛼₁𝜏𝛼₂∗𝑈₃₃𝜏𝛼^𝑇₂𝜏𝛼₃∗∗𝜏𝜃₁𝜏𝜃₂∗∗∗𝜏𝜃₃≥0,𝐽₁₁𝐼𝐼𝑃₁₁𝛼₁𝛼₂𝛼^𝑇₂𝛼₃𝑊=𝐼,¹¹𝑊₂₁𝑊^𝑇₂₁𝑊₃₁𝜃₁𝜃₂𝜃^𝑇₂𝜃₃=𝐼. (3.52)(3.53) Therefore, one can consider the 𝐻_∞ filter design problem as the following cone complementary problems: Minimize()𝐽tr𝑈𝑍+tr¹¹𝐼𝐼𝑃₁₁𝛼₁𝛼₂𝛼^𝑇₂𝛼₃+𝑊₁₁𝑊₂₁𝑊^𝑇₂₁𝑊₃₁𝜃₁𝜃₂𝜃^𝑇₂𝜃₃ (3.54) subject to LMIs (3.30), (3.31), (3.37), (3.40), (3.41), (3.44), (3.48), (3.51), (3.52), and 𝛼𝑄>0,𝑍>0,𝑈𝐼𝐼𝑍≥0,¹𝛼₂𝛼𝐼0_𝑇2𝛼₃0𝐼𝐼0𝐽₁₁𝐼0𝐼𝐼𝑃₁₁𝜃≥0,¹𝜃₂𝜃𝐼0_𝑇2𝜃₃0𝐼𝐼0𝑊₁₁𝑊₂₁0𝐼𝑊^𝑇₂₁𝑊₃₁≥0. (3.55) Then the filer (2.4) can be solved by using the iterative algorithm as [37], in the interests of economy, which is omitted here.

Remark 3.7. Since the filter (2.4) is designed with parameters (3.39) such that inequality (3.20) holds, we have 𝑃𝐴+𝐴^𝑇𝑃^𝑇+1𝑄−𝜏𝐸^𝑇𝑍𝐸𝑃𝐴_𝜏+1𝜏𝐸^𝑇𝑍𝐸𝜏𝐴^𝑇𝑍∗−1𝑄−𝜏𝐸^𝑇𝑍𝐸𝜏𝐴^𝑇_𝜏𝑍∗∗−𝜏𝑍<0. (3.56) By (3.15), (3.16), (3.17), (3.23) and letting 𝑃_𝑓∶=𝑃+𝑆⁻¹, 𝑄=^𝑄1𝑄2_∗𝑄3, and 𝑍=^𝑍1𝑍2_∗𝑍3 with 𝑄₁∈𝑅^𝑛×𝑛 and 𝑍₁∈𝑅^𝑛×𝑛, we can conclude from (3.56) that 𝑃_𝑓𝐴_𝑓+𝐴^𝑇_𝑓𝑃^𝑇_𝑓+𝑄₃−1𝜏𝐸^𝑇𝑍₃𝐸𝑃_𝑓𝐴_𝜏𝑓+1𝜏𝐸^𝑇𝑍₃𝐸𝜏𝐴^𝑇_𝑓𝑍₃∗−𝑄₃−1𝜏𝐸^𝑇𝑍₃𝐸𝜏𝐴^𝑇_𝜏𝑓𝑍₃∗∗−𝜏𝑍₃<0. (3.57) In addition, (3.19) implies that 𝐸^𝑇𝑃^𝑇_𝑓=𝑃_𝑓𝐸≥0. (3.58) Invoking Corollary 3.3, it is obtained that the designed filter (2.4) is admissible, and then it is proper and can be realized in practice.

4. Numerical Examples

Example 4.1. Consider the singular time-delay system given in [25] without uncertainties and distributed delay and with 𝐸=100010000,𝐴=−200.50.1−0.90.200.50.3,𝐴_𝜏=,.0.20.100.200.150.1−0.230.1𝐵=0.20000.20000.2,𝐶=100,𝐺=10.70.8 (4.1) By Theorem 3.2, for 𝜏=2 and 𝛾=1, after 10 iterations, the corresponding filter is obtained with the following parameters: 𝐴_𝑓=−0.9429−0.01020.3206−0.1042−0.84930.19040.37500.48700.3369,𝐴_𝜏𝑓=,𝐵0.09830.09600.00460.1248−0.00710.14610.0749−0.24380.0966_𝑓=0.6224−0.20770.4488,𝐶_𝑓=−0.8379−0.7382−0.9940. (4.2) With this filter, Figures 1, 2, and 3 show the state responses 𝑥(𝑡) of the original system, the state responses 𝑥(𝑡) of the filter system, and the error estimation signal 𝑧(𝑡)=𝑧(𝑡)−𝑧(𝑡) with the initial condition 𝜙(𝑡)=[11−1.425]^𝑇, 𝜓(𝑡)=[11−2.6341]^𝑇 for 𝑡∈[−2,0] and the exogenous disturbance input 𝑤(𝑡)=diag{𝑒^−0.5𝑡,𝑒^−0.5𝑡,𝑒^−0.5𝑡}. By connecting the filter to the original system, the singular value curve of the resulting filtering error system is also plotted in Figure 4. We can see that all the maximum singular values are less than 1, which illustrate the effectiveness of the proposed method in this paper.

Figure 1 

State responses 𝑥(𝑡) of the original system.

Figure 2 

State responses 𝑥(𝑡) of the filter system.

Figure 3 

Error estimation signal 𝑧(𝑡) with the designed filter.

Figure 4 

Singular value curve of the filtering error system.

5. Conclusions and Future Works

In this paper, we have studied the 𝐻_∞ filtering problem for singular system with a constant discrete delay. Based on an improved BRL, a delay-dependent sufficient condition for the existence of the 𝐻_∞ filter with delay is obtained. Then, by using LMIs and the cone complementarity linearization iterative algorithm, the 𝐻_∞ filter is designed, which guarantees that the resulting error system is regular, impulse-free, internally stable, and the 𝐿₂-induced norm from the disturbance input to the filtering error output satisfies a prescribed 𝐻_∞ performance level. It can be seen that the designed filter in this paper is a full-order filter, that is, the finite mode of the filter is equal to rank 𝐸. To study the delay-dependent reduced-order 𝐻_∞ filtering problem for singular time-delay systems is the key research in the future.

Acknowledgments

The authors would like to thank the Editor, the Associate Editor, and the anonymous reviewers very much for the valuable comments and good suggestions. This work was supported by National Natural Science Foundation of P. R. China (61004011).