Abstract and Applied Analysis

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Recent Progress in Differential and Difference Equations

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Volume 2011 |Article ID 970978 | https://doi.org/10.1155/2011/970978

Huihong Zhao, Chenghui Zhang, Guangchen Wang, Guojing Xing, " Estimation for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time Delay", Abstract and Applied Analysis, vol. 2011, Article ID 970978, 22 pages, 2011. https://doi.org/10.1155/2011/970978

𝐻 Estimation for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time Delay

Academic Editor: Elena Braverman
Received27 Dec 2010
Accepted18 May 2011
Published17 Jul 2011

Abstract

The issue of 𝐻 estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating the 𝐻 filtering performance index with the Lipschitz conditions of the nonlinearity, the design of robust estimator is formulated as a positive minimum problem of indefinite quadratic form. Then, by introducing the Krein space model and applying innovation analysis approach, the minimum of the indefinite quadratic form is obtained in terms of innovation sequence. Finally, through guaranteeing the positivity of the minimum, a sufficient condition for the existence of the 𝐻 estimator is proposed and the estimator is derived in terms of Riccati-like difference equations. The proposed algorithm is proved to be effective by a numerical example.

1. Introduction

In control field, nonlinear estimation is considered to be an important task which is also of great challenge, and it has been a very active area of research for decades [17]. Many kinds of methods on estimator design have been proposed for different types of nonlinear dynamical systems. Generally speaking, there are three approaches widely adopted for nonlinear estimation. In the first one, by using an extended (nonexact) linearization of the nonlinear systems, the estimator is designed by employing classical linear observer techniques [1]. The second approach, based on a nonlinear state coordinate transformation which renders the dynamics driven by nonlinear output injection and the output linear on the new coordinates, uses the quasilinear approaches to design the nonlinear estimator [24]. In the last one, methods are developed to design nonlinear estimators for systems which consist of an observable linear part and a locally or globally Lipschitz nonlinear part [57]. In this paper, the problem of 𝐻 estimator design is investigated for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input.

In practice, most nonlinearities can be regarded as Lipschitz, at least locally when they are studied in a given neighborhood [6]. For example, trigonometric nonlinearities occurring in many robotic problems, non-linear softening spring models frequently used in mechanical systems, nonlinearities which are square or cubic in nature, and so forth. Thus, in recent years, increasing attention has been paid to estimator design for Lipschitz nonlinear systems [819]. For the purpose of designing this class of nonlinear estimator, a number of approaches have been developed, such as sliding mode observers [8, 9], 𝐻 optimization techniques [1013], adaptive observers [14, 15], high-gain observers [16], loop transfer recovery observers [17], proportional integral observers [18], and integral quadratic constraints approach [19]. All of the above results are obtained in the assumption that the Lipschitz nonlinear systems are delay free. However, time delay is an inherent characteristic of many physical systems, and it can result in instability and poor performances if it is ignored. The estimator design for time-delay Lipschitz nonlinear systems has become a substantial need. Unfortunately, compared with estimator design for delay-free Lipschitz nonlinear systems, less research has been carried out on the time-delay case. In [20], the linear matrix inequality-(LMI-) based full-order and reduced-order robust 𝐻 observers are proposed for a class of Lipschitz nonlinear discrete-time systems with time delay. In [21], by using Lyapunov stability theory and LMI techniques, a delay-dependent approach to the 𝐻 and 𝐿2𝐿 filtering is proposed for a class of uncertain Lipschitz nonlinear time-delay systems. In [22], by guaranteeing the asymptotic stability of the error dynamics, the robust observer is presented for a class of uncertain discrete-time Lipschitz nonlinear state delayed systems; In [23], based on the sliding mode techniques, a discontinuous observer is designed for a class of Lipschitz nonlinear systems with uncertainty. In [24], an LMI-based convex optimization approach to observer design is developed for both constant-delay and time-varying delay Lipschitz nonlinear systems.

In this paper, the 𝐻 estimation problem is studied for a class of Lipschitz nonlinear discrete time-delay systems with disturbance input. Inspired by the recent study on 𝐻 fault detection for linear discrete time-delay systems in [25], a recursive Kalman-like algorithm in an indefinite metric space, named the Krein space [26], will be developed to the design of 𝐻 estimator for time-delay Lipschitz nonlinear systems. Unlike [20], the delay-free nonlinearities and the delayed nonlinearities in the presented systems are decoupling. For the case presented in [20], the 𝐻 observer design problem, utilizing the technical line of this paper, can be solved by transforming it into a delay-free system through state augmentation. Indeed, the state augmentation results in a higher system dimension and, thus, a much more expensive computational cost. Therefore, this paper based on the presented time-delay Lipschitz nonlinear systems, focuses on the robust estimator design without state augmentation by employing innovation analysis approach in the Krein space. The major contribution of this paper can be summarized as follows: (i) it extends the Krein space linear estimation methodology [26] to the state estimation of the time-delay Lipschitz nonlinear systems and (ii) it develops a recursive Kalman-like robust estimator for time-delay Lipschitz nonlinear systems without state augmentation.

The remainder of this paper is arranged as follows. In Section 2, the interest system, the Lipschitz conditions, and the 𝐻 estimation problem are introduced. In Section 3, a partially equivalent Krein space problem is constructed, the 𝐻 estimator is obtained by computed Riccati-like difference equations, and sufficient existence condition is derived in terms of matrix inequalities. An example is given to show the effect of the proposed algorithm in Section 4. Finally, some concluding remarks are made in Section 5.

In the sequel, the following notation will be used: elements in the Krein space will be denoted by boldface letters, and elements in the Euclidean space of complex numbers will be denoted by normal letters; 𝑛 denotes the real 𝑛-dimensional Euclidean space; denotes the Euclidean norm; 𝜃(𝑘)2[0,𝑁] means 𝑁𝑘=0(𝜃𝑇(𝑘)𝜃(𝑘))<; the superscripts “−1” and “𝑇” stand for the inverse and transpose of a matrix, resp.; 𝐼 is the identity matrix with appropriate dimensions; For a real matrix, 𝑃>0 (𝑃<0, resp.) means that 𝑃 is symmetric and positive (negative, resp.) definite; , denotes the inner product in the Krein space; diag{} denotes a block-diagonal matrix; {} denotes the linear space spanned by sequence {}.

2. System Model and Problem Formulation

Consider a class of nonlinear systems described by the following equations:𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴𝑑𝑥𝑘𝑑+𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘))+𝑘,𝐻𝑥𝑘𝑑,𝑢(𝑘)+𝐵𝑤(𝑘),𝑦(𝑘)=𝐶𝑥(𝑘)+𝑣(𝑘),𝑧(𝑘)=𝐿𝑥(𝑘),(2.1) where 𝑘𝑑=𝑘𝑑, and the positive integer 𝑑 denotes the known state delay; 𝑥(𝑘)𝑛 is the state, 𝑢(𝑘)𝑝 is the measurable information, 𝑤(𝑘)𝑞 and 𝑣(𝑘)𝑚 are the disturbance input belonging to 𝑙2[0,𝑁], 𝑦(𝑘)𝑚 is the measurement output, and 𝑧(𝑘)𝑟 is the signal to be estimated; the initial condition 𝑥0(𝑠)(𝑠=𝑑,𝑑+1,,0) is unknown; the matrices 𝐴𝑛×𝑛, 𝐴𝑑𝑛×𝑛, 𝐵𝑛×𝑞, 𝐶𝑚×𝑛 and 𝐿𝑟×𝑛, are real and known constant matrices.

In addition, 𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘)) and (𝑘,𝐻𝑥(𝑘𝑑),𝑢(𝑘)) are assumed to satisfy the following Lipschitz conditions:𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘))𝑓(𝑘,𝐹̆𝑥(𝑘),𝑢(𝑘))𝛼𝐹(𝑥(𝑘)̆𝑥(𝑘)),𝑘,𝐻𝑥𝑘𝑑,𝑢(𝑘)𝑘,𝐻̆𝑥𝑘𝑑,𝑢(𝑘)𝛽𝐻𝑥𝑘𝑑̆𝑥𝑘𝑑,(2.2) for all 𝑘{0,1,,𝑁}, 𝑢(𝑘)𝑝 and 𝑥(𝑘),̆𝑥(𝑘),𝑥(𝑘𝑑),̆𝑥(𝑘𝑑)𝑛. where 𝛼>0 and 𝛽>0 are known Lipschitz constants, and 𝐹, 𝐻 are real matrix with appropriate dimension.

The 𝐻 estimation problem under investigation is stated as follows. Given the desired noise attenuation level 𝛾>0 and the observation {𝑦(𝑗)}𝑘𝑗=0, find an estimate ̆𝑧(𝑘𝑘) of the signal 𝑧(𝑘), if it exists, such that the following inequality is satisfied:sup𝑥0,𝑤,𝑣0𝑁𝑘=0̆𝑧(𝑘𝑘)𝑧(𝑘)20𝑘=𝑑𝑥0(𝑘)2Π1(𝑘)+𝑁𝑘=0𝑤(𝑘)2+𝑁𝑘=0𝑣(𝑘)2<𝛾2,(2.3) where Π(𝑘)(𝑘=𝑑,𝑑+1,,0) is a given positive definite matrix function which reflects the relative uncertainty of the initial state 𝑥0(𝑘)(𝑘=𝑑,𝑑+1,,0) to the input and measurement noises.

Remark 2.1. For the sake of simplicity, the initial state estimate ̂𝑥0(𝑘)(𝑘=𝑑,𝑑+1,,0) is assumed to be zero in inequality (2.3).

Remark 2.2. Although the system given in [20] is different from the one given in this paper, the problem mentioned in [20] can also be solved by using the presented approach. The resolvent first converts the system given in [20] into a delay-free one by using the classical system augmentation approach, and then designs estimator by employing the similar but easier technical line with our paper.

3. Main Results

In this section, the Krein space-based approach is proposed to design the 𝐻 estimator for Lipschitz nonlinear systems. To begin with, the 𝐻 estimation problem (2.3) and the Lipschitz conditions (2.2) are combined in an indefinite quadratic form, and the nonlinearities are assumed to be obtained by {𝑦(𝑖)}𝑘𝑖=0 at the time step 𝑘. Then, an equivalent Krein space problem is constructed by introducing an imaginary Krein space stochastic system. Finally, based on projection formula and innovation analysis approach in the Krein space, the recursive estimator is derived.

3.1. Construct a Partially Equivalent Krein Space Problem

It is proved in this subsection that the 𝐻 estimation problem can be reduced to a positive minimum problem of indefinite quadratic form, and the minimum can be obtained by using the Krein space-based approach.

Since the denominator of the left side of (2.3) is positive, the inequality (2.3) is equivalent to0𝑘=𝑑𝑥0(𝑘)2Π1(𝑘)+𝑁𝑘=0𝑤(𝑘)2+𝑁𝑘=0𝑣(𝑘)2𝛾2𝑁𝑘=0𝑣𝑧(𝑘)2𝐽𝑁>0,𝑥0,𝑤,𝑣0,(3.1) where 𝑣𝑧(𝑘)=̆𝑧(𝑘𝑘)𝑧(𝑘).

Moreover, we denote𝑧𝑓(𝑘)=𝐹𝑥(𝑘),̆𝑧𝑓(𝑘𝑘)=𝐹̆𝑥(𝑘𝑘),𝑧𝑘𝑑=𝐻𝑥𝑘𝑑,̆𝑧𝑘𝑑𝑘=𝐻̆𝑥𝑘𝑑𝑘,(3.2) where ̆𝑧𝑓(𝑘𝑘) and ̆𝑧(𝑘𝑑𝑘) denote the optimal estimation of 𝑧𝑓(𝑘) and 𝑧(𝑘𝑑) based on the observation {𝑦(𝑗)}𝑘𝑗=0, respectively. And, let𝑤𝑓(𝑘)=𝑓𝑘,𝑧𝑓(𝑘),𝑢(𝑘)𝑓𝑘,̆𝑧𝑓(𝑘𝑘),𝑢(𝑘),𝑤𝑘𝑑=𝑘,𝑧𝑘𝑑,𝑢(𝑘)𝑘,̆𝑧𝑘𝑑𝑘,𝑢(𝑘),𝑣𝑧𝑓(𝑘)=̆𝑧𝑓(𝑘𝑘)𝑧𝑓(𝑘),𝑣𝑧𝑘𝑑=̆𝑧𝑘𝑑𝑘𝑧𝑘𝑑.(3.3) From the Lipschitz conditions (2.2), we derive that𝐽𝑁+𝑁𝑘=0𝑤𝑓(𝑘)2+𝑁𝑘=0𝑤𝑘𝑑2𝛼2𝑁𝑘=0𝑣𝑧𝑓(𝑘)2𝛽2𝑁𝑘=0𝑣𝑧𝑘𝑑2𝐽𝑁𝐽𝑁.(3.4) Note that the left side of (3.1) and (3.4), 𝐽𝑁, can be recast into the form𝐽𝑁=0𝑘=𝑑𝑥0(𝑘)2Π1(𝑘)+𝑁𝑘=0𝑤(𝑘)2+𝑁𝑘=0𝑣(𝑘)2𝛾2𝑁𝑘=0𝑣𝑧(𝑘)2𝛼2𝑁𝑘=0𝑣𝑧𝑓(𝑘)2𝛽2𝑁𝑘=𝑑𝑣𝑧𝑘𝑑2,(3.5) where Π(𝑘)=Π1(𝑘)𝛽2𝐻𝑇𝐻1,𝑘=𝑑,,1,Π(𝑘),𝑘=0,𝑤(𝑘)=𝑤𝑇(𝑘)𝑤𝑇𝑓(𝑘)𝑤𝑇𝑘𝑑𝑇.(3.6)

Since 𝐽𝑁𝐽𝑁, it is natural to see that if 𝐽𝑁>0 then the 𝐻 estimation problem (2.3) is satisfied, that is, 𝐽𝑁>0. Hence, the 𝐻 estimation problem (2.3) can be converted into finding the estimate sequence {{̆𝑧(𝑘𝑘)}𝑁𝑘=0;{̆𝑧𝑓(𝑘𝑘)}𝑁𝑘=0;{̆𝑧(𝑘𝑑𝑘)}𝑁𝑘=𝑑} such that 𝐽𝑁 has a minimum with respect to {𝑥0,𝑤} and the minimum of 𝐽𝑁 is positive. As mentioned in [25, 26], the formulated 𝐻 estimation problem can be solved by employing the Krein space approach.

Introduce the following Krein space stochastic system𝐱(𝑘+1)=𝐴𝐱(𝑘)+𝐴𝑑𝐱𝑘𝑑+𝑓𝑘,̆𝐳𝑓(𝑘𝑘),𝐮(𝑘)+𝑘,̆𝐳𝑘𝑑𝑘,𝐮(𝑘)+𝐵𝐰(𝑘),𝐲(𝑘)=𝐶𝐱(𝑘)+𝐯(𝑘),̆𝐳𝑓(𝑘𝑘)=𝐹𝐱(𝑘)+𝐯𝑧𝑓(𝑘),̆𝐳(𝑘𝑘)=𝐿𝐱(𝑘)+𝐯𝑧(𝑘),̆𝐳𝑘𝑑𝑘=𝐻𝐱𝑘𝑑+𝐯𝑧𝑘𝑑,𝑘𝑑,(3.7) where 𝐵=[𝐵𝐼𝐼]; the initial state 𝐱0(𝑠)(𝑠=𝑑,𝑑+1,,0) and 𝐰(𝑘), 𝐯(𝑘), 𝐯𝑧𝑓(𝑘), 𝐯𝑧(𝑘) and 𝐯𝑧(𝑘) are mutually uncorrelated white noises with zero means and known covariance matrices Π(𝑠), 𝑄𝑤(𝑘)=𝐼, 𝑄𝑣(𝑘)=𝐼, 𝑄𝑣𝑧𝑓(𝑘)=𝛼2𝐼, 𝑄𝑣𝑧(𝑘)=𝛾2𝐼, and 𝑄𝑣𝑧(𝑘)=𝛽2𝐼; ̆𝐳𝑓(𝑘𝑘), ̆𝐳(𝑘𝑘) and ̆𝐳(𝑘𝑑𝑘) are regarded as the imaginary measurement at time 𝑘 for the linear combination 𝐹𝐱(𝑘), 𝐿𝐱(𝑘), and 𝐻𝐱(𝑘𝑑), respectively.

Let𝐲𝑧(𝑘)=𝐲𝑇(𝑘)̆𝐳𝑇𝑚(𝑘𝑘)𝑇,0𝑘<𝑑,𝐲𝑇(𝑘)̆𝐳𝑇𝑚(𝑘𝑘)̆𝐳𝑇𝑘𝑑𝑘𝑇,𝑘𝑑,𝐯𝑧,𝑎(𝑘)=𝐯𝑇(𝑘)𝐯𝑇𝑧𝑓(𝑘)𝐯𝑇𝑧(𝑘)𝑇,0𝑘<𝑑,𝐯𝑇(𝑘)𝐯𝑇𝑧𝑓(𝑘)𝐯𝑇𝑧(𝑘)𝐯𝑇𝑧𝑘𝑑𝑇,𝑘𝑑,̆𝐳𝑚(𝑘𝑘)=̆𝐳𝑇𝑓(𝑘𝑘)̆𝐳𝑇(𝑘𝑘)𝑇.(3.8)

Definition 3.1. The estimator ̂𝐲(𝑖𝑖1) denotes the optimal estimation of 𝐲(𝑖) given the observation {{𝐲𝑧(𝑗)}𝑖1𝑗=0}; the estimator ̂𝐳𝑚(𝑖𝑖) denotes the optimal estimation of ̆𝐳𝑚(𝑖𝑖) given the observation {{𝐲𝑧(𝑗)}𝑖1𝑗=0;𝐲(𝑖)}; the estimator ̂𝐳(𝑖𝑑𝑖) denotes the optimal estimation of ̆𝐳(𝑖𝑑𝑖) given the observation {{𝐲𝑧(𝑗)}𝑖1𝑗=0;𝐲(𝑖),̆𝐳𝑚(𝑖𝑖)}.

Furthermore, introduce the following stochastic vectors and the corresponding covariance matrices̃𝐲(𝑖𝑖1)=𝐲(𝑖)̂𝐲(𝑖𝑖1),𝑅̃𝑦(𝑖𝑖1)=̃𝐲(𝑖𝑖1),̃𝐲(𝑖𝑖1),̃𝐳𝑚(𝑖𝑖)=̆𝐳𝑚(𝑖𝑖)̂𝐳𝑚(𝑖𝑖),𝑅̃𝑧𝑚(𝑖𝑖)=̃𝐳𝑚(𝑖𝑖),̃𝐳𝑚(𝑖𝑖),̃𝐳𝑖𝑑𝑖=̆𝐳𝑖𝑑𝑖̂𝐳𝑖𝑑𝑖,𝑅̃𝑧𝑖𝑑𝑖=̃𝐳𝑖𝑑𝑖,̃𝐳𝑖𝑑𝑖.(3.9) And, denotẽ𝐲𝑧(𝑖)=̃𝐲𝑇(𝑖𝑖1)̃𝐳𝑇𝑚(𝑖𝑖)𝑇,0𝑖<𝑑,̃𝐲𝑇(𝑖𝑖1)̃𝐳𝑇𝑚(𝑖𝑖)̃𝐳𝑇(𝑖𝑑𝑖)𝑇,𝑖𝑑,𝑅̃𝑦𝑧(𝑖)=̃𝐲𝑧(𝑖),̃𝐲𝑧(𝑖).(3.10)

For calculating the minimum of 𝐽𝑁, we present the following Lemma 3.2.

Lemma 3.2. {{̃𝐲𝑧(𝑖)}𝑘𝑖=0} is the innovation sequence which spans the same linear space as that of {{𝐲𝑧(𝑖)}𝑘𝑖=0}.

Proof. From Definition 3.1 and (3.9), ̃𝐲(𝑖𝑖1), ̃𝐳𝑚(𝑖𝑖) and ̃𝐳(𝑖𝑑𝑖) are the linear combination of the observation sequence {{𝐲𝑧(𝑗)}𝑖1𝑗=0;𝐲(𝑖)}, {{𝐲𝑧(𝑗)}𝑖1𝑗=0;𝐲(𝑖),̆𝐳𝑚(𝑖𝑖)}, and {{𝐲𝑧(𝑗)}𝑖𝑗=0}, respectively. Conversely, 𝐲(𝑖), ̆𝐳𝑚(𝑖𝑖) and ̆𝐳(𝑖𝑑𝑖) can be given by the linear combination of {{̃𝐲𝑧(𝑗)}𝑖1𝑗=0;̃𝐲(𝑖𝑖1)},{{̃𝐲𝑧(𝑗)}𝑖1𝑗=0;̃𝐲(𝑖𝑖1),̃𝐳𝑚(𝑖𝑖)} and {{̃𝐲𝑧(𝑗)}𝑖𝑗=0}, respectively. Hence, ̃𝐲𝑧(𝑖)𝑘𝑖=0=𝐲𝑧(𝑖)𝑘𝑖=0.(3.11) It is also shown by (3.9) that ̃𝐲(𝑖𝑖1), ̃𝐳𝑚(𝑖𝑖) and ̃𝐳(𝑖𝑑𝑖) satisfy ̃𝐲(𝑖𝑖1)𝐲𝑧(𝑗)𝑖1𝑗=0,̃𝐳𝑚(𝑖𝑖)𝐲𝑧(𝑗)𝑖1𝑗=0;𝐲(𝑖),̃𝐳𝑖𝑑𝑖𝐲𝑧(𝑗)𝑖1𝑗=0;𝐲(𝑖),̃𝐳𝑚(𝑖𝑖).(3.12) Consequently, ̃𝐲(𝑖𝑖1)̃𝐲𝑧(𝑗)𝑖1𝑗=0,̃𝐳𝑚(𝑖𝑖)̃𝐲𝑧(𝑗)𝑖1𝑗=0;̃𝐲(𝑖𝑖1),̃𝐳𝑖𝑑𝑖̃𝐲𝑧(𝑗)𝑖1𝑗=0;̃𝐲(𝑖𝑖1),̃𝐳𝑚(𝑖𝑖).(3.13) This completes the proof.

Now, an existence condition and a solution to the minimum of 𝐽𝑁 are derived as follows.

Theorem 3.3. Consider system (2.1), given a scalar 𝛾>0 and the positive definite matrix Π(k)(𝑘=𝑑,𝑑+1,,0), then 𝐽𝑁 has the minimum if only if 𝑅̃𝑦(𝑘𝑘1)>0,0𝑘𝑁,𝑅̃𝑧𝑚(𝑘𝑘)<0,0𝑘𝑁,𝑅̃𝑧𝑘𝑑𝑘<0,𝑑𝑘𝑁.(3.14) In this case the minimum value of 𝐽𝑁 is given by min𝐽𝑁=𝑁𝑘=0̃𝑦𝑇(𝑘𝑘1)𝑅1̃𝑦(𝑘𝑘1)̃𝑦(𝑘𝑘1)+𝑁𝑘=0̃𝑧𝑇𝑚(𝑘𝑘)𝑅1̃𝑧𝑚(𝑘𝑘)̃𝑧𝑚(𝑘𝑘)+𝑁𝑘=𝑑̃𝑧𝑇𝑘𝑑𝑘𝑅1̃𝑧𝑘𝑑𝑘̃𝑧𝑘𝑑𝑘,(3.15) where ̃𝑦(𝑘𝑘1)=𝑦(𝑘)̂𝑦(𝑘𝑘1),̃𝑧𝑚(𝑘𝑘)=̆𝑧𝑚(𝑘𝑘)̂𝑧𝑚(𝑘𝑘),̃𝑧𝑘𝑑𝑘=̆𝑧𝑘𝑑𝑘̂𝑧𝑘𝑑𝑘,̆𝑧𝑚(𝑘𝑘)=̆𝑧𝑇𝑓(𝑘𝑘)̆𝑧𝑇(𝑘𝑘)𝑇,(3.16)̂𝑦(𝑘𝑘1) is obtained from the Krein space projection of 𝐲(𝑘) onto {{𝐲𝑧(𝑗)}𝑘1𝑗=0}, ̂𝑧𝑚(𝑘𝑘) is obtained from the Krein space projection of ̆𝐳𝑚(𝑘𝑘) onto {{𝐲𝑧(𝑗)}𝑘1𝑗=0;𝐲(𝑘)}, and ̂𝑧(𝑘𝑑𝑘) is obtained from the Krein space projection of ̆𝐳(𝑘𝑑𝑘) onto {{𝐲𝑧(𝑗)}𝑘1𝑗=0;𝐲(𝑘),̆𝐳𝑚(𝑘𝑘)}.

Proof. Based on the definition (3.2) and (3.3), the state equation in system (2.1) can be rewritten as 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐴𝑑𝑥𝑘𝑑+𝑓𝑘,̆𝑧𝑓(𝑘𝑘),𝑢(𝑘)+𝑘,̆𝑧𝑘𝑑𝑘,𝑢(𝑘)+𝐵𝑤(𝑘).(3.17) In this case, it is assumed that 𝑓(𝑘,̆𝑧𝑓(𝑘𝑘),𝑢(𝑘)) and (𝑘,̆𝑧(𝑘𝑑𝑘),𝑢(𝑘)) are known at time 𝑘. Then, we define 𝑦𝑧(𝑘)=𝑦𝑇(𝑘)̆𝑧𝑇𝑓(𝑘𝑘)̆𝑧𝑇(𝑘𝑘)𝑇,0𝑘<𝑑,𝑦𝑇(𝑘)̆𝑧𝑇𝑓(𝑘𝑘)̆𝑧𝑇(𝑘𝑘)̆𝑧𝑇𝑘𝑑𝑘𝑇,𝑘𝑑.(3.18) By introducing an augmented state 𝑥𝑎(𝑘)=𝑥𝑇(𝑘)𝑥𝑇(𝑘1)𝑥𝑇(𝑘𝑑)𝑇,(3.19) we obtain an augmented state-space model 𝑥𝑎(𝑘+1)=𝐴𝑎𝑥𝑎(𝑘)+𝐵𝑢,𝑎𝑢(𝑘)+𝐵𝑎𝑤(𝑘),𝑦𝑧(𝑘)=𝐶𝑧,𝑎(𝑘)𝑥𝑎(𝑘)+𝑣𝑧,𝑎(𝑘),(3.20) where 𝐴𝑎=𝐴00𝐴𝑑𝐼0000𝐼0000𝐼0,𝐵𝑢,𝑎=𝐼𝐼000000,𝐵𝑎=𝐵000,(3.21)C𝑧,𝑎(𝑘)=𝐶00𝐹00𝐿00,0𝑘<𝑑,𝐶00𝐹00𝐿0000𝐻,𝑘𝑑,𝑣𝑧,𝑎(𝑘)=𝑣𝑇(𝑘)𝑣𝑇𝑧𝑓(𝑘)𝑣𝑇𝑧(𝑘)𝑇,0𝑘<𝑑,𝑣𝑇(𝑘)𝑣𝑇𝑧𝑓(𝑘)𝑣𝑇𝑧(𝑘)𝑣𝑇𝑧𝑘𝑑𝑇,𝑘𝑑,𝑢(𝑘)=𝑓𝑇𝑘,̆𝑧𝑓(𝑘𝑘),𝑢(𝑘)𝑇𝑘,̆𝑧𝑘𝑑𝑘,𝑢(𝑘)𝑇.(1) Additionally, we can rewrite 𝐽𝑁 as 𝐽𝑁=𝑥𝑎(0)𝑤𝑁𝑣𝑧,𝑎𝑁𝑇𝑃𝑎(0)000𝐼000𝑄𝑣𝑧,𝑎𝑁1𝑥𝑎(0)𝑤𝑁𝑣𝑧,𝑎𝑁,(3.22) where 𝑃𝑎(0)=diagΠ(0),Π(1),,Π(𝑑),𝑤𝑁=𝑤𝑇(0)𝑤𝑇(1)𝑤𝑇(𝑁)𝑇,𝑣𝑧,𝑎𝑁=𝑣𝑇𝑧,𝑎(0)𝑣𝑇𝑧,𝑎(1)𝑣𝑇𝑧,𝑎(𝑁)𝑇,𝑄𝑣𝑧,𝑎𝑁=diag𝑄𝑣𝑧,𝑎(0),𝑄𝑣𝑧,𝑎(1),,𝑄𝑣𝑧,𝑎(𝑁),𝑄𝑣𝑧,𝑎(𝑘)=diag𝐼,𝛾2,𝛼2,0𝑘<𝑑,diag𝐼,𝛾2,𝛼2,𝛽2,𝑘𝑑.(3.23) Define the following state transition matrix Φ(𝑘+1,𝑚)=𝐴𝑎Φ(𝑘,𝑚),Φ(𝑚,𝑚)=𝐼,(3.24) and let 𝑦𝑧𝑁=𝑦𝑇𝑧(0)𝑦𝑇𝑧(1)𝑦𝑇𝑧(𝑁)𝑇,𝑢𝑁=𝑢𝑇(0)𝑢𝑇(1)𝑢𝑇(𝑁)𝑇.(3.25) Using (3.20) and (3.24), we have 𝑦𝑧𝑁=Ψ0𝑁𝑥𝑎(0)+Ψ𝑢𝑁𝑢𝑁+Ψ𝑤𝑁𝑤𝑁+𝑣𝑧,𝑎𝑁,(3.26) where Ψ0𝑁=𝐶𝑧,𝑎(0)Φ(0,0)𝐶𝑧,𝑎(1)Φ(1,0)𝐶𝑧,𝑎(𝑁)Φ(𝑁,0),Ψ𝑢𝑁=𝜑00𝜑01𝜑0𝑁𝜑10𝜑11𝜑1𝑁𝜑𝑁0𝜑𝑁1𝜑𝑁𝑁,𝜑𝑖𝑗=𝐶𝑧,𝑎(𝑖)Φ(𝑖,𝑗+1)𝐵𝑢,𝑎,𝑖>𝑗,0,𝑖𝑗.(3.27) The matrix Ψ𝑤𝑁 is derived by replacing 𝐵𝑢,𝑎 in Ψ𝑢𝑁 with 𝐵𝑎.
Thus, 𝐽𝑁 can be reexpressed as 𝐽𝑁=𝑥𝑎(0)𝑤𝑁𝑦𝑧𝑁𝑇Γ𝑁𝑃𝑎(0)000𝐼000𝑄𝑣𝑧,𝑎𝑁Γ𝑇𝑁1𝑥𝑎(0)𝑤𝑁𝑦𝑧𝑁,(3.28) where 𝑦𝑧𝑁=𝑦𝑧𝑁Ψ𝑢𝑁𝑢𝑁,Γ𝑁=𝐼000𝐼0Ψ0𝑁Ψ𝑤𝑁𝐼.(3.29)
Considering the Krein space stochastic system defined by (3.7) and state transition matrix (3.24), we have 𝐲𝑧𝑁=Ψ0𝑁𝐱𝑎(0)+Ψ𝑢𝑁𝐮𝑁+Ψ𝑤𝑁𝐰𝑁+𝐯𝑧,𝑎𝑁,(3.30) where matrices Ψ0𝑁, Ψ𝑢𝑁, and Ψ𝑤𝑁 are the same as given in (3.26), vectors 𝐲𝑧𝑁 and 𝐮𝑁 are, respectively, defined by replacing Euclidean space element 𝑦𝑧 and 𝑢 in 𝑦𝑧𝑁 and 𝑢𝑁 given by (3.25) with the Krein space element 𝐲𝑧 and 𝐮, vectors 𝐰𝑁 and 𝐯𝑧,𝑎𝑁 are also defined by replacing Euclidean space element 𝑤 and 𝑣𝑧,𝑎 in 𝑤𝑁 and 𝑣𝑧,𝑎𝑁 given by (3.23) with the Krein space element 𝐰 and 𝐯𝑧,𝑎, and vector 𝐱𝑎(0) is given by replacing Euclidean space element 𝑥 in 𝑥𝑎(𝑘) given by (3.19) with the Krein space element 𝐱 when 𝑘=0.
Using the stochastic characteristic of 𝐱𝑎(0), 𝐰𝑁 and 𝐯𝑧,𝑎, we have 𝐽𝑁=𝑥𝑎(0)𝑤𝑁𝑦𝑧𝑁𝑇𝐱𝑎(0)𝐰𝑁𝐲𝑧𝑁,𝐱𝑎(0)𝐰𝑁𝐲𝑧𝑁1𝑥𝑎(0)𝑤𝑁𝑦𝑧𝑁,(3.31) where 𝐲𝑧𝑁=𝐲𝑧𝑁Ψ𝑢𝑁𝐮𝑁.
In the light of Theorem  2.4.2 and Lemma  2.4.3 in [26], 𝐽𝑁 has a minimum over {𝑥𝑎(0),𝑤𝑁} if and only if 𝑅𝑦𝑧𝑁=𝐲𝑧𝑁,𝐲𝑧𝑁 and 𝑄𝑣𝑧,𝑎𝑁=𝐯𝑧,𝑎𝑁,𝐯𝑧,𝑎𝑁 have the same inertia. Moreover, the minimum of 𝐽𝑁 is given by min𝐽𝑁=𝑦𝑇𝑧𝑁𝑅1𝑦𝑧𝑁𝑦𝑧𝑁.(3.32)
On the other hand, applying the Krein space projection formula, we have 𝐲𝑧𝑁=Θ𝑁̃𝐲𝑧𝑁,(3.33) where ̃𝐲𝑧𝑁=̃𝐲𝑇𝑧(0)̃𝐲𝑇𝑧(1)̃𝐲𝑇𝑧(𝑁)𝑇,(3.34)Θ𝑁=𝜃00𝜃01𝜃0𝑁𝜃10𝜃11𝜃1𝑁𝜃𝑁0𝜃𝑁1𝜃𝑁𝑁,𝜃𝑖𝑗=𝐲𝑧(𝑖),̃𝐲𝑧(𝑗)𝑅1̃𝑦𝑧(𝑗),𝑖>𝑗0,𝐼0𝑚1𝐼,𝑑>𝑖=𝑗0,𝐼00𝑚1𝐼0𝑚2𝑚3𝐼,𝑖=𝑗𝑑,0,0𝑖<𝑗,𝑚1=̆𝐳𝑚(𝑖𝑖),̃𝐲(𝑗𝑗1)𝑅1̃𝑦(𝑗𝑗1),𝑚2=̆𝐳𝑖𝑑𝑖,̃𝐲(𝑗𝑗1)𝑅1̃𝑦(𝑗𝑗1),𝑚3=̆𝐳𝑖𝑑𝑖,̃𝐳𝑚(𝑗𝑗)𝑅1̃𝑧𝑚(𝑗𝑗),𝐲𝑧(𝑖)=𝐲𝑧(𝑖)𝑁𝑗=0𝜑𝑖𝑗𝐮(𝑗),̆𝐳𝑚(𝑖𝑖)=̆𝐳𝑚(𝑖𝑖)𝑁𝑗=0𝜑𝑚,𝑖𝑗𝐮(𝑗),̆𝐳𝑖𝑑𝑖=̆𝐳𝑖𝑑𝑖𝑁𝑗=0𝜑,𝑖𝑗𝐮(𝑗),(2) where 𝜑𝑚,𝑖𝑗 is derived by replacing 𝐶𝑧,𝑎 in 𝜑𝑖𝑗 with𝐹00𝐿00, 𝜑,𝑖𝑗 is derived by replacing 𝐶𝑧,𝑎 in 𝜑𝑖𝑗 with [00𝐻] Furthermore, it follows from (3.33) that 𝑅𝑦𝑧𝑁=Θ𝑁𝑅̃𝑦𝑧𝑁Θ𝑇𝑁,𝑦𝑧𝑁=Θ𝑁̃𝑦𝑧𝑁,(3.35) where 𝑅̃𝑦𝑧𝑁=̃𝐲𝑧𝑁,̃𝐲𝑧𝑁,̃𝑦𝑧𝑁=̃𝑦𝑇𝑧(0)̃𝑦𝑇𝑧(1)̃𝑦𝑇𝑧(𝑁)𝑇,̃𝑦𝑧(𝑖)=̃𝑦𝑇(𝑖𝑖1)̃𝑧𝑇𝑚(𝑖𝑖)𝑇,0𝑖<𝑑,̃𝑦𝑇(𝑖𝑖1)̃𝑧𝑇𝑚(𝑖𝑖)̃𝑧𝑇𝑖𝑑𝑖𝑇,𝑖𝑑.(3.36) Since matrix Θ𝑁 is nonsingular, it follows from (3.35) that 𝑅𝑦𝑧𝑁 and 𝑅̃𝑦𝑧𝑁 are congruent, which also means that 𝑅𝑦𝑧𝑁 and 𝑅̃𝑦𝑧𝑁 have the same inertia. Note that both 𝑅̃𝑦𝑧𝑁 and 𝑄𝑣𝑧,𝑎𝑁 are block-diagonal matrices, and 𝑅̃𝑦𝑧(𝑘)=diag𝑅̃𝑦(𝑘𝑘1),𝑅̃𝑧𝑚(𝑘𝑘),0𝑘<𝑑,diag𝑅̃𝑦(𝑘𝑘1),𝑅̃𝑧𝑚(𝑘𝑘),𝑅̃𝑧𝑘𝑑𝑘,𝑘𝑑,(3.37)𝑄𝑣𝑧,𝑎(𝑘) is given by (3.23). It follows that 𝑅̃𝑦𝑧𝑁 and 𝑄𝑣𝑧,𝑎𝑁 have the same inertia if and only if 𝑅̃𝑦(𝑘𝑘1)>0(0𝑘𝑁), 𝑅̃𝑧𝑚(𝑘𝑘)<0(0𝑘𝑁) and 𝑅̃𝑧(𝑘𝑑𝑘)<0(𝑑𝑘𝑁).
Therefore, 𝐽𝑁 subject to system (2.1) with Lipschitz conditions (2.2) has the minimum if and only if 𝑅̃𝑦(𝑘𝑘1)>0(0𝑘𝑁), 𝑅̃𝑧𝑚(𝑘𝑘)<0(0𝑘𝑁) and 𝑅̃𝑧(𝑘𝑑𝑘)<0(𝑑𝑘𝑁). Moreover, the minimum value of 𝐽𝑁 can be rewritten as min𝐽𝑁=𝑦𝑇𝑧𝑁𝑅1𝑦𝑧𝑁𝑦𝑧𝑁=̃𝑦𝑇𝑧𝑁𝑅1̃𝑦𝑧𝑁̃𝑦𝑧𝑁=𝑁𝑘=0̃𝑦𝑇(𝑘𝑘1)𝑅1̃𝑦(𝑘𝑘1)̃𝑦(𝑘𝑘1)+𝑁𝑘=0̃𝑧𝑇𝑚(𝑘𝑘)𝑅1̃𝑧𝑚(𝑘𝑘)̃𝑧𝑚(𝑘𝑘)+𝑁𝑘=𝑑̃𝑧𝑇𝑘𝑑𝑘𝑅1̃𝑧𝑘𝑑𝑘̃𝑧𝑘𝑑𝑘.(3.38) The proof is completed.

Remark 3.4. Due to the built innovation sequence {{̃𝐲𝑧(𝑖)}𝑘𝑖=0} in Lemma 3.2, the form of the minimum on indefinite quadratic form 𝐽𝑁 is different from the one given in [2628]. It is shown from (3.15) that the estimation errors ̃𝑦(𝑘𝑘1), ̃𝑧𝑚(𝑘𝑘) and ̃𝑧(𝑘𝑑𝑘) are mutually uncorrelated, which will make the design of 𝐻 estimator much easier than the one given in [2628].

3.2. Solution of the 𝐻 Estimation Problem

In this subsection, the Kalman-like recursive 𝐻 estimator is presented by using orthogonal projection in the Krein space.

Denote𝐲0(𝑖)=𝐲(𝑖),𝐲1(𝑖)=𝐲𝑇(𝑖)̆𝐳𝑇𝑚(𝑖𝑖)𝑇,𝐲2(𝑖)=𝐲𝑇(𝑖)̆𝐳𝑇𝑚(𝑖𝑖)̆𝐳𝑇(𝑖𝑖+𝑑)𝑇.(3.39) Observe from (3.8), we have 𝐲𝑧(𝑖)𝑗𝑖=0=𝐲1(𝑖)𝑗𝑖=0,0𝑗<𝑑,𝐲𝑧(𝑖)𝑗𝑖=0=𝐲2(𝑖)𝑗𝑑𝑖=0;𝐲1(𝑖)𝑗𝑖=𝑗𝑑+1,𝑗𝑑.(3.40)

Definition 3.5. Given 𝑘𝑑, the estimator ̂𝜉(𝑖𝑗,2) for 0𝑗<𝑘𝑑 denotes the optimal estimate of 𝜉(𝑖) given the observation {{𝐲2(𝑠)}𝑗𝑠=0}, and the estimator ̂𝜉(𝑖𝑗,1) for 𝑘𝑑𝑗𝑘 denotes the optimal estimate of 𝜉(𝑖) given the observation {{𝐲2(𝑠)}𝑘𝑑1𝑠=0;{𝐲1(𝜏)}𝑗𝜏=𝑘𝑑}. For simplicity, we use ̂𝜉(𝑖,2) to denote ̂𝜉(𝑖𝑖1,2), and use ̂𝜉(𝑖,1) to denote ̂𝜉(𝑖𝑖1,1) throughout the paper.

Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matrices̃𝐲2(𝑖,2)=𝐲2(𝑖)̂𝐲2(𝑖,2),𝑅̃𝑦2(𝑖,2)=̃𝐲2(𝑖,2),̃𝐲2(𝑖,2),̃𝐲1(𝑖,1)=𝐲1(𝑖)̂𝐲1(𝑖,1),𝑅̃𝑦1(𝑖,1)=̃𝐲1(𝑖,1),̃𝐲1(𝑖,1),̃𝐲0(𝑖,0)=𝐲0(𝑖)̂𝐲0(𝑖,1),𝑅̃𝑦0(𝑖,0)=̃𝐲0(i,0),̃𝐲0(𝑖,0).(3.41) Similar to the proof of Lemma  2.2.1 in [27], we can obtain that {̃𝐲2(0,2),,̃𝐲2(𝑘𝑑1,2);̃𝐲1(𝑘𝑑,1),,̃𝐲1(𝑘1,1)} is the innovation sequence which is a mutually uncorrelated white noise sequence and spans the same linear space as {𝐲2(0),,𝐲2(𝑘𝑑1);𝐲1(𝑘𝑙),,𝐲1(𝑘1)} or equivalently {𝐲𝑧(0),,𝐲𝑧(𝑘1)}.

Applying projection formula in the Krein space, ̂𝐱(𝑖,2)(𝑖=0,1,,𝑘𝑑) is computed recursively as(3.42)̂𝐱(𝑖+1,2)=𝑖𝑗=0𝐱(𝑖+1),̃𝐲2(𝑗,2)𝑅1̃𝑦2(𝑗,2)̃𝐲2(𝑗,2)=𝐴̂𝐱(𝑖𝑖,2)+𝐴𝑑̂𝐱𝑖𝑑𝑖,2+𝑓𝑖,̆𝐳𝑓(𝑖𝑖),𝐮(𝑖)+𝑖,̆𝐳𝑖𝑑𝑖,𝐮(𝑖),𝑖=0,1,,𝑘𝑑1,̂𝐱(𝜏,2)=0,(𝜏=𝑑,𝑑+1,,0).(3.43) Note that̂𝐱(𝑖𝑖,2)=̂𝐱(𝑖,2)+𝑃2(𝑖,𝑖)𝐶𝑇2𝑅1̃𝑦2(𝑖,2)̃𝐲2(𝑖,2),̂𝐱𝑖𝑑𝑖,2=̂𝐱𝑖𝑑,2+𝑖𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗𝐶𝑇2𝑅1̃𝑦2(𝑗,2)̃𝐲2(𝑗,2),(3.44) where𝐶2=𝐶𝑇𝐹𝑇𝐿𝑇𝐻𝑇𝑇,𝑃2(𝑖,𝑗)=𝐞(𝑖,2),𝐞(𝑗,2),𝐞(𝑖,2)=𝐱(𝑖)̂𝐱(𝑖,2),𝑅̃𝑦2(𝑖,2)=𝐶2𝑃2(𝑖,𝑖)𝐶𝑇2+𝑄𝑣2(𝑖),𝑄𝑣2(𝑖)=diag𝐼,𝛼2𝐼,𝛾2𝐼,𝛽2𝐼.(3.45) Substituting (3.44) into (3.43), we havê𝐱(𝑖+1,2)=𝐴̂𝐱(𝑖,2)+𝐴𝑑̂𝐱𝑖𝑑,2+𝑓𝑖,̆𝐳𝑓(𝑖𝑖),𝐮(𝑖)+𝑖,̆𝐳𝑖𝑑𝑖,𝐮(𝑖)+𝐴𝑑𝑖1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗𝐶𝑇2𝑅1̃𝑦2(𝑗,2)̃𝐲2(𝑗,2)+𝐾2(𝑖)̃𝐲2(𝑖,2),𝐾2(𝑖)=𝐴𝑑𝑃2𝑖𝑑,𝑖𝐶𝑇2𝑅1̃𝑦2(𝑖,2)+𝐴𝑃2(𝑖,𝑖)𝐶𝑇2𝑅1̃𝑦2(𝑖,2).(3.46) Moreover, taking into account (3.7) and (3.46), we obtain𝐞(𝑖+1,2)=𝐴𝐞(𝑖,2)+𝐴𝑑𝐞𝑖𝑑,2+𝐵𝐰(𝑖)𝐾2(𝑖)̃𝐲2(𝑖,2)𝐴𝑑𝑖1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗𝐶𝑇2𝑅1̃𝑦2(𝑗,2)̃𝐲2(𝑗,2),𝑖=0,1,,𝑘𝑑1.(3.47) Consequently,𝑃2(𝑖𝑗,𝑖+1)=𝐞(𝑖𝑗,2),𝐞(𝑖+1,2)=𝑃2(𝑖𝑗,𝑖)𝐴𝑇+𝑃𝑇2𝑖𝑑,𝑖𝑗𝐴𝑇𝑑𝑃2(𝑖𝑗,𝑖)𝐶𝑇2𝐾𝑇2(𝑖)𝑖1𝑡=𝑖𝑗𝑃2(𝑖𝑗,𝑡)𝐶𝑇2𝑅1̃𝑦2(𝑡,2)𝐶2𝑃𝑇2𝑖𝑑,𝑡𝐴𝑇𝑑,𝑗=0,1,,𝑑,𝑃2(𝑖+1,𝑖+1)=𝐞(𝑖+1,2),𝐞(𝑖+1,2)=𝐴𝑃2(𝑖,𝑖+1)+𝐴𝑑𝑃2𝑖𝑑,𝑖+1+𝐵𝑄𝑤(𝑖)𝐵𝑇,(3.48) where 𝑄𝑤(𝑖)=𝐼. Thus, 𝑃2(𝑖,𝑖)(𝑖=0,1,,𝑘𝑑) can be computed recursively as𝑃2(𝑖𝑗,𝑖+1)=𝑃2(𝑖𝑗,𝑖)𝐴𝑇+𝑃𝑇2𝑖𝑑,𝑖𝑗𝐴𝑇𝑑𝑃2(𝑖𝑗,𝑖)𝐶𝑇2𝐾𝑇2(𝑖)𝑖1𝑡=𝑖𝑗𝑃2(𝑖𝑗,𝑡)𝐶𝑇2𝑅1̃𝑦2(𝑡,2)𝐶2𝑃𝑇2𝑖𝑑,𝑡𝐴𝑇𝑑,𝑃2(𝑖+1,𝑖+1)=𝐴𝑃2(𝑖,𝑖+1)+𝐴𝑑𝑃2𝑖𝑑,𝑖+1+𝐵𝑄𝑤(𝑖)𝐵𝑇,𝑗=0,1,,𝑑.(3.49)

Similarly, employing the projection formula in the Krein space, the optimal estimator ̂𝐱(𝑖,1)(𝑖=𝑘𝑑+1,,𝑘) can be computed by ̂𝐱(𝑖+1,1)=𝐴̂𝐱(𝑖,1)+𝐴𝑑̂𝐱𝑖𝑑,2+𝑓𝑖,̆𝐳𝑓(𝑖𝑖),𝐮(𝑖)+𝑖,̆𝐳𝑖𝑑𝑖,𝐮(𝑖)+𝐾1(𝑖)̃𝐲1(𝑖,1)+𝐴𝑑𝑘𝑑1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗𝐶𝑇2𝑅1̃𝑦2(𝑗,2)̃𝐲2(𝑗,2)+𝐴𝑑𝑖1𝑗=𝑘𝑑𝑃1𝑖𝑑,𝑗𝐶𝑇1𝑅1̃𝑦1(𝑗,1)̃𝐲1(𝑗,1),̂𝐱𝑘𝑑,1=̂𝐱𝑘𝑑,2,(3.50) where𝐶1=𝐶𝑇𝐹𝑇𝐿𝑇𝑇,𝑃1(𝑖,𝑗)=𝐞(𝑖,2),𝐞(𝑗,1),𝑖<𝑘𝑑,𝐞(𝑖,1),𝐞(𝑗,1),𝑖𝑘𝑑,𝐞(𝑖,1)=𝐱(𝑖)̂𝐱(𝑖,1),𝑅̃𝑦1(𝑖,1)=𝐶1𝑃1(𝑖,𝑖)𝐶𝑇1+𝑄𝑣1(𝑖),𝑄𝑣1(𝑖)=diag𝐼,𝛼2𝐼,𝛾2𝐼,𝐾1(𝑖)=𝐴𝑃1(𝑖,𝑖)𝐶𝑇1𝑅1̃𝑦1(𝑖,1)+𝐴𝑑𝑃1𝑖𝑑,𝑖𝐶𝑇1𝑅1̃𝑦1(𝑖,1).(3.51) Then, from (3.7) and (3.50), we can yield𝐞(𝑖+1,1)=𝐴𝐞(𝑖,1)+𝐴𝑑𝐞𝑖𝑑,2+𝐵𝐰(𝑖)𝐾1(𝑖)̃𝐲1(