Unified Finite Horizon 𝐻_∞ Fusion Filtering for Networked Dynamical System

Abstract

This paper addresses the 𝐻_∞ fusion filtering problem for networked dynamical systems, where measurements may arrive at fusion center in four different scenes and the fusion center could receive none, one, or multiple measurements in a fusion period. A unified 𝐻_∞ performance criterion function, which is suitable for different measurement arrival scenes, is designed for the filtering process of networked dynamical systems. Then, the 𝐻_∞ performance criterion function is described as an indefinite quadratic inequality and solved by a novel noise projection method in Krein space. On this basis, a unified finite horizon 𝐻_∞ filtering method is proposed for networked dynamical systems. Simulation results are provided to illustrate the correctness and the effectiveness of the theoretical analysis.

1. Introduction

The filtering methods are widely utilized in the fields of signal processing and automatic control for dynamical systems. With the development of computer and information technology, researchers begin to pay more and more attention on networked dynamical systems, such as the open channel networks and networked control systems [1–3]. However, it is inevitable that the measurement data is transmitted in the networked dynamical systems with different time delay.

In networked dynamical systems, the targets of interest are (remotely) observed by various sensors. The sampled measurements may arrive at information processing centers (especially refer to fusion filters, as in this paper) in different scenes through the transport network. Scene 1: the measurement arrives at the fusion filter in time, which is abbreviated to “ITM” in this paper. Scene 2: the measurement which arrives at the fusion filter with some time delay, but still in the sampled sequence, is abbreviated to “ISDM.” Scene 3: the delay measurement arrives at the fusion filter out of the sampled sequence, which is abbreviated to “OOSM.” Scene 4: the sampled measurement is missing in the transmitting process, which is also named as “packet dropout” (abbreviated to “PD”). The traditional filters are mainly proposed for systems with all measurements in Scene 1, such as Kalman filter, 𝐻_∞ filter. For the system with measurements arriving at fusion filter in other scenes, several effective filtering methods have also been proposed, recently.(1)For systems with measurement in Scene 2, some novel filtering methods are proposed based on Kalman filter [4, 5]. And the developed 𝐻_∞ filtering approaches are also deduced for this kind of systems with bounded energy noises [6, 7].(2)For systems with measurement in Scene 3, several OOSM filtering problems are investigated with the help of such technologies as nonstandard smoothing [8], Kalman filter with measurement weighted summation [9, 10], and reorganized innovation [11, 12].(3)For systems with measurement in Scene 4, several filtering approaches are developed based on the traditional Kalman filter [13–15] or 𝐻_∞ filter [16], based on different descriptions of the packet dropout phenomenon, such as the Markovian jump approach [13] and the binary Bernoulli distribution approach [14–16].

Although some recent approaches have considered the systems in multiple measurement arrival scenes [17, 18], most results of them are deduced for the system with single sensor. Few papers address the filtering problems for the networked dynamical system with multiple sensors. In [19, 20], the fusion filtering methods for networked multisensor systems are deduced based on Kalman filter, which requires the system noise to satisfy zero mean Gaussian distribution with known variance, which, however, is usually not available. To the best of the author’s knowledge, the filtering problem for the networked multisensor system with unknown statistic noises has not been fully investigated and still remains challenging.

Motivated by the above discussion, a unified finite horizon 𝐻_∞ filtering method is proposed for the networked dynamical system in this paper in which four different kinds of measurement arrival scenes are dealt with in a unified manner. Because of the complex arrival scenes of networked measurements, the fusion filter for the networked dynamical system could receive none, one, or multiple measurements in a fusion period. The 𝐻_∞ filtering algorithm should be deduced to achieve a 𝐻_∞ performance criterion function. In the traditional 𝐻_∞ performance criterion function, an ideal assumption condition is that the measurement sampling time is the same as the measurement arrival time and the filtering time. However, in networked dynamical systems, the arrival time of a networked measurement mostly is not equal to its sampling time. And the fusion filter deals with the sampled measurement at its arrival time, rather than its sampling time. It means that the traditional 𝐻_∞ performance criterion function cannot be applied to networked dynamical systems. In this paper, a novel unified 𝐻_∞ performance criterion function is built for the different measurement arrival scenes in networked dynamical systems, firstly. Secondly, the 𝐻_∞ performance criterion function is described as an indefinite quadratic inequality. The stationary point of indefinite quadratic form in Hilbert space corresponds to a projection in Krein space. In this paper, the stationary point of the indefinite quadratic inequality is obtained by solving a projection in Krein space. However, because of the random delay of networked measurements, the process of solving the projection with the delay measurement becomes more complex. Thirdly, a noise projection approach in Krein space is proposed to solve the projection corresponding to the stationary point. Then, a unified 𝐻_∞ filtering method is proposed for the networked dynamical system. Finally, the validity and effectiveness of the proposed method are verified in the final simulation.

The remainder of this paper is organized as follows. The problem of the fusion filtering for the networked dynamical system is formulated in Section 2. In Section 3, a unified finite horizon 𝐻_∞ filter is deduced based on a novel performance criterion function for the networked dynamical system in various measurement arrival scenes. An example for illustration is given in Section 4, and we conclude this paper in Section 5.

Notation. The elements in Hilbert space are denoted by bold face letters, such as “𝐱,𝐱,” and the elements in Krein space are denoted by the bold face letters with bar, such as “𝐱,𝐱.” The superscripts “^T” and “⁻¹” mean the transposed matrix and inverse matrix, respectively. 𝜃(𝑘)∈𝑙₂[1,𝑁) is the Euclidean norm; that is, ∑^𝑁_𝑘=1𝜃^𝑇(𝑘)𝜃(𝑘)<∞.

2. Problem Formulation

Consider the following discrete networked dynamical system, which is observed by 𝑁 sensors 𝐱(𝑘)()𝐱()(),𝐲=𝐅𝑘,𝑘−1𝑘−1+𝐰𝑘,𝑘−1_𝑙(𝑘)=𝐇_𝑙(𝑘)𝐱(𝑘)+𝐯_𝑙(𝑘)𝐳(𝑘)(𝑘)𝐱(𝑘),,𝑙=1,2,…,𝑁,=𝐋 (1) where 𝐱(𝑘)∈𝐑^𝑛 is the state vector. 𝐲_𝑙(𝑘)∈𝐑^𝑞 is the measurement output of sensor 𝑙. 𝐳(𝑘) is the signal to be estimated. 𝐅(𝑘,𝑘−1),𝐇_𝑙(𝑘), and 𝐋(𝑘) are the system matrices with compatible dimensions. 𝐰(𝑘,𝑘−1)∈𝑙₂[1,𝑁) is the process noise and 𝐯_𝑙(𝑘)∈𝑙₂[1,𝑁) is the corresponding measurement noise of sensor 𝑙.

According to the traditional finite horizon 𝐻_∞ filter for single-sensor system, for a given scalar 𝛾>0, 𝐳̂(𝑘∣𝑘) can be obtained as an approximation of 𝐳(𝑘) based on the received measurements {𝐲(𝑖)∣𝑖=1,2,…𝑘} to guarantee the following 𝐻_∞ performance criterion function: sup_{𝐰,𝐯∈𝑙2[)1,𝑁}^𝑘_𝑖=1𝐞^𝑇(𝑖)𝐞(𝑖×_𝑘𝑖=1)𝐯^𝑇(𝑖)𝐯(𝑖)+^𝑘_𝑖=1𝐰^𝑇()𝐰()+𝐱𝑖,𝑖−1𝑖,𝑖−1_𝑇0𝐏₀⁻¹𝐱₀⁻¹<𝛾², (2) where 𝐞_𝑧(𝑘)=𝐳(𝑘)−𝐳̂(𝑘∣𝑘), 𝐱₀=𝐱(0)−𝐱₀, and 𝐱₀ is an initial estimate of 𝐱(0). 𝐏₀ is a given positive definite matrix with compatible dimension.

Remark 1. In the above performance criterion function, the filtering time of the fusion filter is the same as the sampling time of measurements, which, however, is not this case in networked dynamical systems.

In a fusion period, the networked measurement could arrive at the fusion filter in four different scenes considered in this paper; namely, the measurement could be ITM, ISDM, or OOSM. The unified filter for networked dynamical systems would deal with various measurement arrival scenes. The filtering time of networked measurement is its arrival time, rather than its sampling time. This means that the performance criterion function shown in (2) cannot be directly extended to the unified filtering process of networked dynamical systems. In the next section, a novel performance criterion function is built for various measurement arrival scenes in the networked dynamical system, firstly. On this basis, a unified finite horizon 𝐻_∞ fusion filtering method is deduced.

3. Unified Finite Horizon 𝐻_∞ Fusion Filtering for Networked Dynamical System

3.1. Performance Criterion Function

Let 𝜅(𝑖) be a counter, which counts for the number of the measurements received by the fusion filter in the fusion period [𝑖,𝑖+1). At the start of the period, 𝜅(𝑖)=0. Whenever a measurement arrives at the fusion filter, 𝜅(𝑖)=𝜅(𝑖)+1. Denote the 𝑗th measurement received by the fusion filter in the period [𝑖,𝑖+1) as 𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗), in which the notations 𝑗,𝛼^𝑖_𝑗,𝛽^𝑖_𝑗 mean that 𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗) is sampled by sensor 𝛼^𝑖_𝑗 at the sampled time 𝛽^𝑖_𝑗. Here 𝛽^𝑖_𝑗≤𝑖,𝛼^𝑖_𝑗≤𝑁,and𝑗≤𝑖𝑁 are all positive integers.

According to the measurement arrival scenes, for a given 𝛾>0, a novel finite horizon 𝐻_∞ fusion filtering performance criterion function could be given as follows in the fusion period [𝑘,𝑘+1) to obtain 𝐳̂(𝑘∣𝑘) based on the received measurement space {𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗)∣1≤𝑖≤𝑘and𝜅(𝑖)≠0,𝑗=1,…,𝜅(𝑖)}. Consider sup_{𝐰,𝐯∈𝑙2[)1,𝑁}^𝑘_{𝜅(𝑖)𝑖=1≠0}^𝜅(𝑘)_𝑗=1𝐞^𝑇_𝑧,𝑗(𝑖)𝐞_𝑧,𝑗(𝑖)×^𝑘_{𝜅(𝑖)𝑖=1≠0}^𝜅(𝑘)_𝑗=1𝐯^𝑇_𝛼𝑖𝑗𝛽^𝑖_𝑗𝐯_𝛼^𝑖_𝑗𝛽^𝑖_𝑗+^𝑘_𝑖=1𝐰^𝑇()𝐰()+𝐱𝑖,𝑖−1𝑖,𝑖−1_𝑇0𝐏₀⁻¹𝐱₀⁻¹<𝛾², (3) where 𝐞_𝑧,𝑗(𝑖)=𝐳̂_𝑗(𝑖∣𝑖)−𝐳(𝑖). 𝐳̂_𝑗(𝑖∣𝑖) is the 𝑗th estimate of 𝐳(𝑖), updated with the measurement 𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗).

Remark 2. The performance criterion function shown in (3) can be utilized for the finite horizon 𝐻_∞ fusion filtering processes of the dynamical system with measurements which could arrive at fusion filter in the aforementioned four kinds of arrival scenes.(1)When none measurement arrives at the fusion filter in the period [𝑘,𝑘+1), 𝜅(𝑘)=0.(2)When a measurement firstly arrives at the fusion filter in [𝑘,𝑘+1), 𝜅(𝑘)=1. If 𝛽^𝑘₁=𝑘, the measurement is an ITM 𝐲_𝛼^𝑘₁(𝑘). If 𝛽^𝑘₁<𝑘, this measurement is a delay measurement (an ISDM or a OOSM) sampled at 𝛽^𝑘₁ by sensor 𝛼^𝑘₁.(3)When a second (third,…, etc.) measurement arrives in [𝑘,𝑘+1), then let 𝜅(𝑘)=𝜅(𝑘)+1. The new measurement would also be an ITM or a delay measurement.

3.2. Unified Finite Horizon 𝐻_∞ Filter Design

The performance criterion function shown in (3) can also be described as the following indefinite quadratic inequality: 𝐽_𝜅(𝑘)=^𝑘_{𝑖=1𝜅(𝑖)≠0}^𝜅(𝑘)_𝑗=1𝐯^𝑇_𝛼𝑖𝑗𝛽^𝑖_𝑗𝐯_𝛼^𝑖_𝑗𝛽^𝑖_𝑗−𝛾⁻²𝐞^𝑇_𝑧,𝑗(𝑖)𝐞_𝑧,𝑗(𝑖)+^𝑘_𝑖=1𝐰^𝑇(𝑖,𝑖−1)𝐰(𝑖,𝑖−1)+𝐱^𝑇₀𝐏₀⁻¹𝐱₀>0. (4)

Remark 3. The quadratic form 𝐽_𝜅(𝑘) satisfies the indefinite quadratic inequality above if and only if (1)  𝐽_𝜅(𝑘) has a stationary point, (2) the value of 𝐽_𝜅(𝑘) at the stationary point is a minimum, and (3) the minimum is positive.
The stationary point of an indefinite quadratic form in Hilbert space corresponds to a projection in Krein space which is solved to obtain the stationary point of 𝐽_𝜅(𝑘) in this paper. A Krein space state-space model associated with the system shown in (1) is introduced as 𝐱(𝑖)=𝐅(𝑖,𝑖−1)𝐱(𝑖−1)+𝐰(𝑖,𝑖−1),𝑖=1,…,𝑘,𝐲_𝛼^𝑖_𝑗𝛽^𝑖_𝑗=𝐇_𝛼^𝑖_𝑗𝛽^𝑖_𝑗𝐱𝛽^𝑖_𝑗+𝐯_𝑦,𝛼^𝑖_𝑗𝛽^𝑖_𝑗,(𝑖)(𝑖),̆1≤𝑖≤𝑘,𝜅≠0,0<𝑗≤𝜅𝐳(𝑖)=𝐋(𝑖)𝐱(𝑖)+𝐯_𝑧,𝑗(𝑖),1≤𝑖≤𝑘,𝜅(𝑖)≠0,0<𝑗≤𝜅(𝑖), (5) with 𝐱(0)−𝐱₀𝐰𝑗₁,𝑗₁−1𝐯_𝑦,𝑗3𝛽^𝑙1_𝑗3𝐯_𝑧,𝑗3𝑙₁,𝐱(0)−𝐱₀𝐰𝑗₂,𝑗₂−1𝐯_𝑦,𝑗4𝛽^𝑙2_𝑗4𝐯_𝑧,𝑗4𝑙₂=𝐏₀𝟎𝟎𝟎𝟎𝐈𝛿_𝑗1,𝑗2𝟎𝟎𝟎𝟎𝐈𝛿_𝑗3,𝑗4𝛿_𝑙1,𝑙2𝟎𝟎𝟎𝟎−𝛾²𝐈𝛿_𝑗3,𝑗4𝛿_𝑙1,𝑙2,1≤𝑗₁,𝑗₂≤𝑘,1≤𝑗₃,𝑗₄≤𝑘,𝜅𝑗₃>1,0<𝑙₁≤𝜅𝑗₃,0<𝑙₂≤𝜅𝑗₄. (6)
Denote 𝐖(𝑘)∶=[𝐰^𝑇(1,0),…,𝐰^𝑇(𝑘,𝑘−1)]^𝑇, 𝜉(𝑘)∶=[𝐱^𝑇(0),𝐖^𝑇(𝑘)]^𝑇. The stationary point of the indefinite quadratic form shown in (4) corresponds to the projection of 𝜉(𝑘) into the Krein subspace Γ_𝜅(𝑘)(𝑘) spanned by {𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗),𝐳̆_𝑗(𝑖)∣1≤𝑖≤𝑘and𝜅(𝑖)≠0,𝑗=1,…,𝜅(𝑖)}.
According to the Krein space state equation shown in (5), we have 𝐱(𝑖−1)=𝐅(𝑖−1,𝑖)(𝐱(𝑖)−𝐰(𝑖,𝑖−1)), and thus 𝐲_𝛼^𝑖_𝑗𝛽^𝑖_𝑗=𝐇_𝛼^𝑖_𝑗𝛽^𝑖_𝑗𝐱𝛽^𝑖_𝑗+𝐯_𝑦,𝛼^𝑖_𝑗𝛽^𝑖_𝑗=𝐇_𝛼^𝑖_𝑗𝛽^𝑖_𝑗𝐅𝛽^𝑖_𝑗,𝑖𝐱(𝑖)+𝐯_𝑦,𝛼^𝑖_𝑗𝛽^𝑖_𝑗−𝐇_𝛼^𝑖_𝑗𝛽^𝑖_𝑗𝐅𝛽^𝑖_𝑗,𝑖𝐰𝑖,𝛽^𝑖_𝑗. (7) Let 𝐲^∗_𝑗(𝑖)=𝐲_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗);𝐇^∗_𝑗(𝑖)=𝐇_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗)𝐅(𝛽^𝑖_𝑗,𝑖);𝐯^∗_𝑦,𝑗(𝑖)=𝐯_𝑦,𝛼^𝑖_𝑗(𝛽^𝑖_𝑗)−𝐇_𝛼^𝑖_𝑗(𝛽^𝑖_𝑗)𝐅(𝛽^𝑖_𝑗,𝑖)𝐰(𝑖,𝛽^𝑖_𝑗); we have 𝐲^∗_𝑗(𝑖)=𝐇^∗_𝑗(𝑖)𝐱(𝑖)+𝐯^∗_𝑦,𝑗(𝑖), (8) with the Gramian matrixes 𝐰(𝑙,𝑙−1),𝐯^∗_𝑦,𝑗(𝑖()𝐅)=𝐐𝑙,𝑙−1^𝑇(𝑖,𝑙+1)𝐇^∗_𝑗(𝑖)^𝑇,𝛽^𝑖_𝑗𝐏≤𝑙≤𝑖−1,_{∗𝑣𝑦𝑦,𝑗}(𝑖)=𝐯^∗_𝑦,𝑗(𝑖),𝐯^∗_𝑦,𝑗(𝑖)=𝐈+𝐇^∗_𝑗(𝑖)^𝑖_{𝑙=𝛽𝑖𝑗+1}𝐅(𝑖,𝑙)𝐅^𝑇(𝑖,𝑙)𝐇^∗_𝑗(𝑖)^𝑇,𝛽^𝑖_𝑗<𝑖, (9) where 𝐅(𝑖,𝑖)=𝐈.
Denote the Krein subspace spanned by {𝐲_𝑙(𝛽^𝜏_𝑙),𝐳̆_𝑙(𝜏)∣1≤𝜏≤𝑖−1and𝜅(𝜏)≠0,𝑙=1,…,𝜅(𝜏)} and {𝐲_𝑙(𝛽^𝑖_𝑙),𝐳̆_𝑙(𝑖)∣𝑙=1,…,𝑗−1} by Γ_𝑗−1(𝑖). The projection of 𝐲^∗_𝑗(𝑖) into Γ_𝑗−1(𝑖) is given by 𝐲^∗_𝑗(𝑖∣𝑖−1)=𝐇^∗_𝑗(𝑖)𝐱_𝑗(𝑖∣𝑖−1)+𝐯^∗_𝑦,𝑗(𝑖∣𝑖−1), (10) in which 𝐱_𝑗(𝑖∣𝑖−1)=𝐱_𝑗−1(𝑖∣𝑖),𝑗>1,𝐱₁(𝑖∣𝑖−1)=𝐅(𝑖,𝑖−1)𝐱_{𝜅(𝑖−1)}(𝑖−1∣𝑖−1),𝐯^∗_𝑦,𝑗(𝑖∣𝑖−1)=−𝐇^∗_𝑗(𝑖)𝐰𝑖,𝛽^𝑖_𝑗∣𝑖−1,𝐳̂‌_𝑗(𝑖∣𝑖−1)=𝐋(𝑖)𝐱_𝑗(𝑖∣𝑖−1). (11)(12)(13)
Let 𝐞^∗_{𝑦𝑧,𝑗}(𝑖)∶=^{𝐲∗𝑗(𝑖)−𝐲∗𝑗}_̆^{(𝑖∣𝑖−1)}_{𝐳𝑗̂(𝑖)−𝐳𝑗(𝑖∣𝑖−1)}=^{𝐇∗𝑗(𝑖)}_𝐋(𝑖)𝐱_𝑗(𝑖∣𝑖−1)+^{𝐯∗𝑦,𝑗(𝑖∣𝑖−1)}_{𝐯𝑧,𝑗(𝑖)},𝐑_{𝑤𝑦𝑧,𝑙}(𝑖,𝛽^𝑖_𝑗,𝑘)∶=⟨𝐰(𝑖,𝛽^𝑖_𝑗),𝐞^∗_{𝑦𝑧,𝑙}(𝑘)⟩, 𝐑^∗_{𝑒𝑦𝑧,𝑗}(𝑖)∶=⟨𝐞^∗_{𝑦𝑧,𝑗}(𝑖),𝐞^∗_{𝑦𝑧,𝑗}(𝑖)⟩. It is obvious that 𝐞^∗_{𝑦𝑧,𝑗−1}(𝑖∣𝑖−1)⟂Γ_𝑗−1(𝑖), and {𝐞^∗_{𝑦𝑧,𝑗}(𝑖∣𝑖−1)∣1≤𝑖≤𝑘and𝜅(𝑖)≠0,𝑗=1,…,𝜅(𝑖)} is an orthogonal basis of Γ_𝜅(𝑘)(𝑘). The projection of 𝜉(𝑘) into Γ_𝜅(𝑘)(𝑘) is given by ̂‌𝜉_𝜅(𝑘)(𝑘∣𝑘)=^𝑘_{𝜅(𝑖)𝑖=1,≠0}^𝜅(𝑖)_𝑗=1𝜉(𝑘),𝐞^∗_{𝑦𝑧,𝑗}(𝑖)𝐑^∗_{𝑒𝑦𝑧,𝑗}(𝑖)⁻¹𝐞^∗_{𝑦𝑧,𝑗}(𝑖). (14)
The projection of 𝐱(𝑘) into Γ_𝜅(𝑘)(𝑘) is 𝐱_𝜅(𝑘)(𝑘∣𝑘)=^𝑘_{𝜅(𝑖)𝑖=1,≠0}^𝜅(𝑖)_𝑗=1𝐱(𝑘),𝐞^∗_{𝑦𝑧,𝑗}(𝑖)𝐑^∗_{𝑒𝑦𝑧,𝑗}(𝑖)⁻¹𝐞^∗_{𝑦𝑧,𝑗}(𝑖)=^𝑘_{𝜅(𝑖)𝑖=1,≠0}^𝜅(𝑖)_𝑗=1𝐱(𝑘),𝐞^∗_{𝑦𝑧,𝑗}(𝑖)𝐑^∗_{𝑒𝑦𝑧,𝑗}(𝑖)⁻¹𝐞^∗_{𝑦𝑧,𝑗}(𝑖)=𝐱_{𝜅(𝑘)−1}(𝑘∣𝑘)+𝐱(𝑘),𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘×𝐑_{∗(𝑘)𝑒𝑦𝑧,𝜅})(𝑘)⁻¹𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘)=𝐱_𝜅(𝑘)(𝑘∣𝑘−1)+𝐱(𝑘),𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘×𝐑_{∗𝑒𝑦𝑧,𝜅(𝑘)})(𝑘)⁻¹𝐞^∗_{𝑦𝑧,𝜅(𝑘)}(𝑘)𝐱₀(𝑘∣𝑘)=𝐅(𝑘,𝑘−1)𝐱_{𝜅(𝑘−1)}(𝑘−1∣𝑘−1), (15) and the noise projection 𝐰(𝑖,𝛽^𝑖_𝑗∣𝑖−1) in (12) is given by 𝐰_𝑗𝑖,𝛽^𝑖_𝑗∣𝑖−1=𝐰_𝑗−1𝑖,𝛽^𝑖_𝑗,𝑖−1+𝐑_{𝑤𝑦𝑧,𝑗−1}𝑖,𝛽^𝑖_𝑗,𝑖𝐑^∗_{𝑒𝑦𝑧,𝑗−1}(𝑖)⁻¹×𝐞^∗_{𝑦𝑧,𝑗−1}(𝑖),𝐰₁𝑖,𝛽^𝑖_𝑗∣𝑖−1=𝐰_{𝜅(𝑖−1)}𝑖,𝛽^𝑖_𝑗∣𝑖−2+𝐑_{𝑤𝑦𝑧,𝜅(𝑖−1)}𝑖,𝛽^𝑖_𝑗×𝐑,𝑖−1_{∗()𝑒𝑦𝑧,𝜅𝑖−1}(𝑖−1)⁻¹×𝐞^∗_{()𝑦𝑧,𝜅𝑖−1}(𝑖−1),𝐰_{𝜅(𝑖−1)}𝑖,𝛽^𝑖_𝑗∣𝑖−2=𝐅(𝑖,𝑖−1)𝐰_{𝜅(𝑖−1)}𝑖−1,𝛽^𝑖_𝑗∣𝑖−2, (16) where 𝐊_𝜅(𝑘)(𝑘)∶=𝐱(𝑘),𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘)𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅}(𝑘)⁻¹∶=𝐑_{𝑥𝑦𝑧,𝜅(𝑘)}(𝑘)𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅}(𝑘)⁻¹,𝐑_{𝑥𝑦𝑧,𝜅(𝑘)}(𝑘)=𝐱(𝑘),𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘=)𝐱(𝑘),𝐱_𝜅(𝑘)(𝑘∣𝑘−1)𝐇^∗_𝜅(𝑘)(𝑘)^𝑇𝐋^𝑇(𝑘)−𝐱(𝑘),𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1𝐇^∗_𝜅(𝑘)(𝑘)^𝑇𝟎∶=𝐏_𝜅(𝑘)(𝑘)𝐇^∗_𝜅(𝑘)(𝑘)^𝑇𝐋^𝑇(𝑘)−𝐑_{(𝑘)𝑥𝑤,𝜅}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1𝐇^∗_𝜅(𝑘)(𝑘)^𝑇𝟎∶=𝐑_{𝑥𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑥𝑧,𝜅(𝑘)}(𝑘),𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅}(𝑘)𝐇∶=^{∗𝜅(𝑘)}(𝑘)𝐋(𝑘)𝐏_𝜅(𝑘)(𝑘)𝐇^∗_𝜅(𝑘)(𝑘)^𝑇𝐋^𝑇(𝑘)+𝐑^∗_{𝑣𝑦𝑧,𝜅(𝑘)}(𝑘)−^{𝐇∗𝜅(𝑘)(𝑘)𝐑𝑇(𝑘)𝑥𝑤,𝜅𝑘,𝛽𝑘𝜅(𝑘)𝐇∣𝑘−1∗𝜅(𝑘)(𝑘)𝑇}_𝟎^𝟎_𝟎−^{𝐇∗𝜅(𝑘)(𝑘)𝐑𝑇(𝑘)𝑥𝑤,𝜅𝑘,𝛽𝑘𝜅(𝑘)𝐇∣𝑘−1∗𝜅(𝑘)(𝑘)𝑇}_𝟎^𝟎_𝟎^𝑇,𝐑^∗_{𝑣𝑦𝑧,𝜅(𝑘)}(𝑘)=𝐑^∗_{𝑣𝑦𝑦,𝜅(𝑘)}(𝑘)𝟎𝟎−𝛾²𝐈𝐑_{𝑥𝑤,𝜅(𝑘)}𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐱(𝑘),𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐱(𝑘),𝐰𝑘,𝛽^𝑘_𝜅(𝑘)−𝐱(𝑘),𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1∶=𝐐𝑘,𝛽^𝑘_𝜅(𝑘)−𝐏_{𝑥𝑤,𝜅(𝑘)}𝑘,𝛽^𝑘_𝜅(𝑘),𝐑∣𝑘−1_{∗(𝑘)𝑣𝑦𝑦,𝜅}(𝑘)=𝐯^∗_{(𝑘)𝑦,𝜅−1}(𝑘∣𝑘−1),𝐯^∗_{(𝑘)𝑦,𝜅−1}(=𝑘∣𝑘−1)𝐯^∗_{(𝑘)𝑦,𝜅−1}(𝑘∣𝑘−1),𝐯^∗_{(𝑘)𝑦,𝜅−1}(−𝑘∣𝑘−1)𝐯^∗_{(𝑘)𝑦,𝜅−1}(𝑘∣𝑘−1),𝐯^∗_{(𝑘)𝑦,𝜅−1}(𝑘∣𝑘−1)=𝐏^∗_{𝑣𝑦𝑦,𝜅(𝑘)}(𝑘)−𝐇^∗_𝜅(𝑘)(𝑘)𝐑^𝑇_{𝑤𝑤,𝜅(𝑘)}𝑘,𝛽^𝑘_𝜅(𝑘)×𝐇∣𝑘−1_{∗𝜅(𝑘)}(𝑘)^𝑇,𝐏_{(𝑘)𝑥𝑤,𝜅}𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐱(𝑘),𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐱(𝑘),𝐰_{𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)+∣𝑘−1𝐱(𝑘),𝐞^∗_{(𝑘)𝑦𝑧,𝜅−1}(𝑘)𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅−1}(𝑘)⁻¹×𝐑^𝑇_{(𝑘)𝑤𝑦𝑧,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘),𝑘=𝐏_{(𝑘)𝑥𝑤,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1+𝐑_{𝑥𝑦𝑧,𝜅(𝑘)−1}(𝑘)×𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅−1}(𝑘)⁻¹𝐑^𝑇_{(𝑘)𝑤𝑦𝑧,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘)(𝑘)𝐏,𝑘,𝜅>1,_𝑥𝑤,1𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐱(𝑘),𝐰₁𝑘,𝛽^𝑘_𝜅(𝑘)(∣𝑘−1=𝐅𝑘,𝑘−1)𝐱(𝑘−1),𝐰_{𝜅(𝑘−1)}𝑘−1,𝛽^𝑘_𝜅(𝑘)∣𝑘−2×𝐅^𝑇(𝑘,𝑘−1)+𝐱(𝑘),𝐞^∗_{()𝑦𝑧,𝜅𝑘−1}(×𝐑𝑘−1)_{∗()𝑒𝑦𝑧,𝜅𝑘−1}(𝑘−1)⁻¹𝐑^𝑇_{()𝑤𝑦𝑧,𝜅𝑘−1}𝑘,𝛽^𝑘_𝜅(𝑘)()𝐏,𝑘−1=𝐅𝑘,𝑘−1_{()𝑥𝑤,𝜅𝑘−1}𝑘−1,𝛽^𝑘_𝜅(𝑘)∣𝑘−2𝐅^𝑇(𝑘,𝑘−1)+𝐑_{𝑥𝑦𝑧,𝜅(𝑘−1)}(𝑘−1)𝐑^∗_{()𝑒𝑦𝑧,𝜅𝑘−1}(𝑘−1)⁻¹×𝐑^𝑇_{()𝑤𝑦𝑧,𝜅𝑘−1}𝑘,𝛽^𝑘_𝜅(𝑘),𝐑,𝑘−1_{𝑤𝑦𝑧,𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)=,𝑘𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐞^∗_{𝑦𝑧,𝜅(𝑘)−1}(𝑘=)𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐇^∗_{𝜅(𝑘)−1}(𝑘)𝐋(𝑘)𝐱_{𝜅(𝑘)−1}()+𝑘∣𝑘−1𝐯^∗_{(𝑘)𝑦,𝜅−1}(𝑘∣𝑘−1)𝐯_{(𝑘)𝑧,𝜅−1}(𝑘)=𝐑^𝑇_{(𝑘)𝑥𝑤,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘)𝐇∣𝑘−1^{∗𝜅(𝑘)−1}(𝑘)𝐋(𝑘)^𝑇−𝐑^𝑇_{(𝑘)𝑤𝑤,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘)𝑘−1𝐇^∗_{𝜅(𝑘)−1}(𝑘)^𝑇𝟎,𝐑_{𝑤𝑤,𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1∶=𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰_{𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰𝑘,𝛽^𝑘_𝜅(𝑘)−𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰_{𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1=𝐐𝑘,𝛽^𝑘_𝜅(𝑘)−𝐏_{(𝑘)𝑤𝑤,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘),𝐏∣𝑘−1_{𝑤𝑤,𝜅(𝑘)}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1∶=𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰_{𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)+∣𝑘−1𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐞^∗_{𝑦𝑧,𝜅(𝑘)−1}(𝑘×𝐑_{∗𝑒𝑦𝑧,𝜅(𝑘)−1})(𝑘)⁻¹𝐑^𝑇_{𝑤𝑦𝑧,𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘),𝑘=𝐏_{𝑤𝑤,𝜅(𝑘)−1}𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1+𝐑_{𝑤𝑦𝑧,𝜅(𝑘)−1}(𝑘)×𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅−1}(𝑘)⁻¹𝐑^𝑇_{(𝑘)𝑤𝑦𝑧,𝜅−1}𝑘,𝛽^𝑘_𝜅(𝑘)(𝑘)𝐏,𝑘,𝜅>1,_𝑤𝑤,1𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰₁𝑘,𝛽^𝑘_𝜅(𝑘)=∣𝑘−1𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐰_{𝜅(𝑘−1)}𝑘,𝛽^𝑘_𝜅(𝑘)+∣𝑘−2𝐰𝑘,𝛽^𝑘_𝜅(𝑘),𝐞^∗_{𝑦𝑧,𝜅(𝑘−1)}(×𝐑𝑘−1)_{∗𝑒𝑦𝑧,𝜅(𝑘−1)}(𝑘−1)⁻¹𝐑^𝑇_{𝑤𝑦𝑧,𝜅(𝑘−1)}𝑘,𝛽^𝑘_𝜅(𝑘)()𝐏,𝑘−1=𝐅𝑘,𝑘−1_{𝑤𝑤,𝜅(𝑘−1)}𝑘−1,𝛽^𝑘_𝜅(𝑘)∣𝑘−2𝐅^𝑇(𝑘,𝑘−1)+𝐑_{𝑤𝑦𝑧,𝜅(𝑘−1)}𝑘,𝛽^𝑘_𝜅(𝑘),𝑘−1𝐑^∗_{()𝑒𝑦𝑧,𝜅𝑘−1}(𝑘−1)⁻¹×𝐑^𝑇_{()𝑤𝑦𝑧,𝜅𝑘−1}𝑘,𝛽^𝑘_𝜅(𝑘),𝑘−1, (17)(18) and 𝐐𝑘,𝛽^𝑘_𝜅(𝑘)=𝟎,𝛽^𝑘_𝜅(𝑘)=𝑘;_{𝑘𝑖=𝛽𝑘𝜅(𝑘)+1}𝐅(𝑘,𝑖)𝐅^𝑇(𝑘,𝑖),𝛽^𝑘_𝜅(𝑘)<𝑘. (19)
The projection of 𝜉(𝑘) in (14) corresponds to a stationary point of the indefinite quadratic form 𝐽_𝜅(𝑘) in (4), and the value of 𝐽_𝜅(𝑘) at this stationary point is 𝐉^∗_𝜅(𝑘)(𝑘)=^𝑘_{𝑖=1,𝜅(𝑖)≠0}^𝜅(𝑖)_𝑗=1𝐞^∗_{𝑦𝑧,𝑗}(𝑖)^𝑇𝐑^∗_{𝑒𝑦𝑧,𝑗}(𝑖)⁻¹𝐞^∗_{𝑦𝑧,𝑗}(𝑖)=𝐉^∗_{𝜅(𝑘)−1}(𝑘)+𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘)^𝑇𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅}(𝑘)⁻¹×𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘), (20) where 𝐉^∗₀(𝑘)=𝐉^∗_{𝜅(𝑘−1)}(𝑘−1), 𝐞^∗_{(𝑘)𝑦𝑧,𝜅}(𝑘)=𝐞^∗_{𝑦,𝜅(𝑘)}()𝐞𝑘∣𝑘−1_{𝑧,𝜅(𝑘)}()=𝐲𝑘∣𝑘−1_{𝛼𝑘𝜅(𝑘)}𝛽^𝑘_𝜅(𝑘)−𝐲^∗_𝜅(𝑘)()𝐳̂𝑘∣𝑘−1_𝜅(𝑘)(𝑘∣𝑘)−𝐳̂_𝜅(𝑘)(),𝐲𝑘∣𝑘−1_{∗𝜅(𝑘)}(𝑘∣𝑘−1)=𝐇^∗_𝜅(𝑘)(𝑘)𝐱_𝜅(𝑘)(𝑘∣𝑘−1)−𝐇^∗_𝜅(𝑘)(𝑘)𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘),𝐳̂∣𝑘−1_𝜅(𝑘)(𝑘∣𝑘−1)=𝐋(𝑘)𝐱_𝜅(𝑘)(),𝐰𝑘∣𝑘−1_𝑗𝑖,𝛽^𝑖_𝑗∣𝑖−1=𝐰_𝑗−1𝑖,𝛽^𝑖_𝑗∣𝑖−1+𝐑_{𝑤𝑦𝑧,𝑗−1}𝑖,𝛽^𝑘_𝑗×𝐑,𝑖_{∗𝑒𝑦𝑧,𝑗−1}(𝑖)⁻¹𝐞^∗_{𝑦𝑧,𝑗−1}(𝑖)𝐰,𝑗>1,₁𝑖,𝛽^𝑖_𝑗∣𝑖−1=𝐰_{𝜅(𝑖−1)}𝑖,𝛽^𝑖_𝑗∣𝑖−2+𝐑_{𝑤𝑦𝑧,𝜅(𝑖−1)}𝑖,𝛽^𝑘_𝑗×𝐑,𝑖−1_{∗()𝑒𝑦𝑧,𝜅𝑖−1}(𝑖−1)⁻¹×𝐞^∗_{()𝑦𝑧,𝜅𝑖−1}(),𝐰𝑖−1_{𝜅(𝑖−1)}𝑖,𝛽^𝑖_𝑗∣𝑖−2=𝐅(𝑖,𝑖−1)𝐰_{𝜅(𝑖−1)}𝑖−1,𝛽^𝑖_𝑗,𝐱∣𝑖−2_𝜅(𝑘)(𝑘∣𝑘−1)=𝐱_{𝜅(𝑘)−1}()(𝑘)𝐱𝑘∣𝑘,𝜅>0,₀(𝑘∣𝑘−1)=𝐅(𝑘,𝑘−1)𝐱_{𝜅(𝑘−1)}(𝑘−1∣𝑘−1). (21)
In (20), 𝐑^∗_{𝑒𝑦𝑧,𝜅(𝑘)}(𝑘) can also be expressed as follows: 𝐑^∗_{(𝑘)𝑒𝑦𝑧,𝜅}(𝑘)𝐑∶=^{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑦𝑧,𝜅(𝑘)}(𝑘)𝐑_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑧𝑧,𝜅(𝑘)}(𝑘); (22) then, 𝐑^∗_{𝑒𝑦𝑧,𝜅(𝑘)}(𝑘)⁻¹=𝐑_{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑦𝑧,𝜅(𝑘)}𝐑(𝑘)_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑧𝑧,𝜅(𝑘)}(𝑘)⁻¹=𝐈−𝐑⁻¹_{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐑_{𝑦𝑧,𝜅(𝑘)}(𝑘)×𝟎𝐈_{𝐑𝑦𝑦,𝜅(𝑘)(𝑘)𝟎𝟎𝐑𝑧𝑧,𝜅(𝑘)(𝑘)−𝐑𝑧𝑦,𝜅(𝑘)(𝑘)𝐑−1𝑦𝑦,𝜅(𝑘)(𝑘)𝐑𝑦𝑧,𝜅(𝑘)(𝑘)−1}×𝐈𝟎−𝐑_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑_(𝑘)⁻¹_{𝑦𝑦,𝜅}(𝑘)𝐈. (23)
Letting 𝐳̂^∗_𝜅(𝑘)(𝑘∣𝑘)=𝐳̂_𝜅(𝑘)(𝑘∣𝑘−1)−𝐑_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑⁻¹_{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐞^∗_{𝑦,𝜅(𝑘)}(𝑘∣𝑘−1), the last term in (20) is 𝐞^∗_{𝑦𝑧,𝜅(𝑘)}(𝑘)𝐑^∗_{𝑒𝑦𝑧,𝜅(𝑘)}(𝑘)⁻¹𝐞^∗_{𝑦𝑧,𝜅(𝑘)}(𝑘)=𝐞^∗_{𝑦,𝜅(𝑘)}(𝑘∣𝑘−1)^𝑇𝐑⁻¹_{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐞^∗_{𝑦,𝜅(𝑘)}()+𝐳̂𝑘∣𝑘−1_𝜅(𝑘)(𝑘∣𝑘)−𝐳̂^∗_𝜅(𝑘)(𝑘∣𝑘)^𝑇×𝐑_{𝑧𝑧,𝜅(𝑘)}(𝑘)−𝐑_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑_(𝑘)⁻¹_{𝑦𝑦,𝜅}(𝑘)𝐑_{𝑦𝑧,𝜅(𝑘)}(𝑘)⁻¹×𝐳̂_𝜅(𝑘)(𝑘∣𝑘)−𝐳̂^∗_𝜅(𝑘)(𝑘∣𝑘). (24)
According to Lemma  12 in [21], 𝐉^∗_𝜅(𝑘)(𝑘) is the minimum of 𝐉_𝜅(𝑘)(𝑘) if and only if 𝐑^∗_{𝑒𝑦𝑧,𝜅(𝑘)}(𝑘) and 𝐑^∗_{𝑣𝑦𝑧,𝜅(𝑘)}(𝑘) have the same inertia. Considering the block triangular factorization of 𝐑^∗_{𝑒𝑦𝑧,𝜅(𝑘)}(𝑘) as shown in (23), the minimum condition can also be given by 𝐑_{𝑧𝑧,𝜅(𝑘)}(𝑘)−𝐑_{𝑧𝑦,𝜅(𝑘)}(𝑘)𝐑_(𝑘)⁻¹_{𝑦𝑦,𝜅}(𝑘)𝐑_{𝑦𝑧,𝜅(𝑘)}(𝑘)<𝟎. (25)
Therefore, a choice of 𝐳̂_𝜅(𝑘)(𝑘∣𝑘) to guarantee 𝐉^∗_𝜅(𝑘)(𝑘)>0 is 𝐳̂_𝜅(𝑘)(𝑘∣𝑘)=𝐳̂^∗_𝜅(𝑘)(𝑘∣𝑘), and the minimum of 𝐉_𝜅(𝑘)(𝑘) is 𝐉^∗_𝜅(𝑘)(𝑘)=𝐉^∗_{𝜅(𝑘)−1}(𝑘)+𝐞^∗_{𝑦,𝜅(𝑘)}()×𝐑𝑘∣𝑘−1_{𝑦𝑦,𝜅(𝑘)}(𝑘)⁻¹𝐞^∗_{𝑦,𝜅(𝑘)}(𝑘∣𝑘−1). (26)
Then, the estimation of the signal to be estimated is 𝐳̂(𝑘∣𝑘)=𝐋(𝑘)𝐱_𝜅(𝑘)(𝑘∣𝑘) (27) in which 𝐱_𝜅(𝑘)(𝑘∣𝑘)=𝐱_𝜅(𝑘)(𝑘∣𝑘−1)+𝐊_{𝑦,𝜅(𝑘)}(𝑘)𝐞^∗_{𝑦,𝜅(𝑘)}(𝑘∣𝑘−1),𝐊_{(𝑘)𝑦,𝜅}(𝑘)=𝐑_{𝑥𝑦,𝜅(𝑘)}(𝑘)𝐑_(𝑘)⁻¹_{𝑦𝑦,𝜅}(𝑘). (28)(29) The parameters 𝐑_{𝑥𝑦,𝜅(𝑘)}(𝑘)and𝐑⁻¹_{𝑦𝑦,𝜅(𝑘)}(𝑘) in (29) can be obtained by iterating the equations in (18).
In summary, the unified finite horizon 𝐻_∞ fusion filtering algorithm is given by 𝐳̂(𝑘∣𝑘)=𝐋(𝑘)𝐱_𝜅(𝑘)()𝐱𝑘∣𝑘_𝜅(𝑘)(𝑘∣𝑘)=𝐱_{𝜅(𝑘)−1}(𝑘∣𝑘)+𝐊_{𝑦,𝜅(𝑘)}(𝑘)𝐞^∗_{𝑦,𝜅(𝑘)}()𝐱𝑘∣𝑘−1₀(𝑘∣𝑘)=𝐅(𝑘,𝑘−1)𝐱_{𝜅(𝑘−1)}(𝑘−1∣𝑘−1), (30) where 𝐊_{𝑦,𝜅(𝑘)}(𝑘)=𝐑_{𝑥𝑦,𝜅(𝑘)}(𝑘)𝐑⁻¹_{𝑦𝑦,𝜅(𝑘)}(𝑘)𝐞^∗_{𝑦,𝜅(𝑘)}(𝑘∣𝑘−1)=𝐲_𝛼^𝑘_𝜅(𝑘)𝛽^𝑘_𝜅(𝑘)−𝐲^∗_𝜅(𝑘)(𝑘∣𝑘−1)=𝐲_𝛼^𝑘_𝜅(𝑘)𝛽^𝑘_𝜅(𝑘)−𝐇^∗_𝜅(𝑘)(𝑘)𝐱_𝜅(𝑘)(𝑘∣𝑘−1)−𝐇^∗_𝜅(𝑘)(𝑘)𝐰_𝜅(𝑘)𝑘,𝛽^𝑘_𝜅(𝑘)∣𝑘−1. (31)
The Riccati equations are given as follows.(1)If the next measurement arrives at fusion filter in the fusion period [𝑘,𝑘+1), the Ricatti equation is 𝐏_𝜅(𝑘)+1(𝑘)=𝐏_𝜅(𝑘)(𝑘)−𝐊_𝜅(𝑘)(𝑘)𝐑^𝑇_{(𝑘)𝑥𝑦𝑧,𝜅}(𝑘). (32)(2)Otherwise, the next measurement arrives at fusion filter in the fusion period [𝑘+1,𝑘+2). The Ricatti equation is 𝐏₁(𝑘+1)=𝐅(𝑘+1,𝑘)𝐏_𝜅(𝑘)(𝑘)−𝐊_𝜅(𝑘)(𝑘)𝐑^𝑇_{𝑥𝑦,𝜅(𝑘)}(𝑘)×𝐅^𝑇(𝑘+1,𝑘)+𝐈. (33)

Remark 4. In ideal communication networks, all the measurements arrive at the fusion filter in time; namely, the measurements are all ITMs. In this case, the above unified finite horizon 𝐻_∞ fusion filtering algorithm will be simplified into the following sequential finite horizon 𝐻_∞ fusion filtering algorithm, which can deal with the ITMs in real time according to their sampled sequence 𝐳̂(𝑘∣𝑘)=𝐋(𝑘)𝐱_𝑁(𝑘∣𝑘), (34) in which 𝐱_𝑙(𝑘∣𝑘)=𝐱_𝑙−1(𝑘∣𝑘)+𝐊_𝑦,𝑙(𝑘)𝐞_𝑦,𝑙()𝐱𝑘∣𝑘−1,𝑙=1,2,…,𝑁,₀(𝑘∣𝑘)=𝐅(𝑘,𝑘−1)𝐱_𝑁(),𝐊𝑘−1∣𝑘−1_𝑦,𝑙(𝑘)=𝐇_𝑙(𝑘)𝐏_𝑙(𝑘)𝐇_𝑙(𝑘)𝐏_𝑙(𝑘)𝐇^𝑇_𝑙(𝑘)+𝐈⁻¹,𝐞_𝑦,𝑙(𝑘∣𝑘−1)=𝐲_𝑙(𝑘)−𝐇_𝑙(𝑘)𝐱_𝑙−1(𝑘∣𝑘−1). (35) The corresponding Riccati equations are 𝐏₁(𝑘)()×𝐏=𝐅𝑘,𝑘−1_𝑁(𝑘−1)−𝐑_{𝑥𝑦𝑧,𝑙}(𝑘)𝐑⁻¹_{𝑒𝑦𝑧,𝑙}(𝑘)×𝐑^𝑇_{𝑥𝑦𝑧,𝑁}(𝑘−1)×𝐅^𝑇()𝐏𝑘,𝑘−1+𝐈_𝑙+1(𝑘)=𝐏_𝑙(𝑘)−𝐑_{𝑥𝑦𝑧,𝑙}(𝑘)𝐑⁻¹_{𝑒𝑦𝑧,𝑙}(𝑘)𝐑^𝑇_{𝑥𝑦𝑧,𝑙}(𝑘),𝐑_{𝑥𝑦𝑧,𝑙}(𝑘)=𝐏_𝑙(𝑘)𝐇_𝑙(𝑘)^𝑇𝐋^𝑇(𝑘),𝐑_{𝑒𝑦𝑧,𝑙}(𝑘)𝐇∶=^𝑙(𝑘)𝐋(𝑘)𝐏_𝑙(𝑘)×𝐇_𝑙(𝑘)^𝑇𝐋^𝑇(𝑘)+𝐼𝟎𝟎−𝛾²𝐈. (36)

4. Simulation

In order to illustrate the viability and the effectiveness of the proposed method, the discrete dynamical system as shown in (1) is considered in this section, in which 𝐅(𝑘,𝑘−1)=¹¹₀₁, 𝐇_𝑖=[1,1],(𝑖=1,2), 𝐿=[0.1,0], 𝛾=1.5. Moreover, the initial value is selected as 𝐱₀=[1,5], and 𝐏₀=^11/2_1/22.

The measurement arrival scenes are designed as follows. All the measurements are sampled in time. For Sensor 1, all the measurements arrive at the fusion filter in time. For Sensor 2, the measurements sampled at the moments with indexes modulo 3 equal to 1 or 2 arrive at the fusion filter with one-step delay, and other measurements arrive in time, as shown in Figure 1.

Figure 1 

The measurement arrival scenes.

In this simulation, two simulation results are compared. The first one is the result of the proposed method in the above measurement arrival scene, which is for short marked as “Algorithm 1” in this section. The other one is the simulation result of the sequential fusion filtering method in the scene that all the measurements arrive in time, as shown in Remark 4, which is marked as “Algorithm 2.”

According to the simulation results shown in Table 1, Figures 1 and 2, the following performances of the proposed algorithm are illustrated.(1)The proposed algorithm could deal with different kinds of arrived measurements: ITMs (the measurements sampled by Sensor 1 and the ones sampled by Sensor 2 at the sampled moments with indexes divided by 3 exactly), ISDMs (the measurements sampled by Sensor 2 at the sampled moments with indexes modulo 3 equal to 1), and OOSMs (the measurements sampled by Sensor 2 at the sampled moments with indexes modulo 3 equal to 2).(2)For Algorithm 1, at the fusion time with the indexes divided by 3 exactly, the delay measurements can all arrive at the fusion center. At this fusion time, the fusion filtering results of Algorithm 1are better than the ones of Algorithm 2. The mean absolute estimation errors at these fusion times of Algorithm 1 are 0.3787, while the one of Algorithm 2 is 0.3832. It is because more amount of information is applied to update the estimate at this fusion time in Algorithm 1 (Figure 3).(3)It is implied that the proposed algorithm could deal with the delay measurements effectively.

Table 1 

The mean absolute estimation errors at filtering time with the indexes are divided by 5 exactly.

Figure 2 

The estimate curves.

Figure 3 

The absolute estimation error curves at the fusion time with the indexes are divided by 3 exactly.

5. Conclusion

In this paper, a unified finite horizon 𝐻_∞ filtering method is proposed for general networked dynamical systems, the fusion filter of which could receive none, one, or multiple measurements in a fusion period. According to the complex arrival scenes of networked measurements, a novel 𝐻_∞ performance criterion function is built to restrain the 𝐻_∞ filtering process. Based on the projection method in Krein space, a novel 𝐻_∞ filtering method is proposed to uniformly deal with various delay measurements and ITMs in the centralized fusion frame. Otherwise, there are several interesting future directions along the line of this work:(1)how to deal with the networked measurements in the distributed fusion frame in a uniform manner,(2)how to deal with various quantified delay measurements for networked multisensor systems.

Conflict of Interests

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

Acknowledgment

The paper was supported by the National Natural Science Foundation of China under Grants 61333005, 61172133, 61174112, 61304258, and 61273075.