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

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 [1–7]. 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 [2–4]. 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 [5–7]. 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 [8–19]. 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 [10–13], 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 𝐿₂−𝐿_∞ 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; ℝ^𝑛 denotes the real 𝑛-dimensional Euclidean space; ‖⋅‖ denotes the Euclidean norm; 𝜃(𝑘)∈₂[0,𝑁] means ∑^𝑁_𝑘=0(𝜃^𝑇(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; ℒ{⋯} 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)+𝐴_𝑑𝑥𝑘_𝑑+𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘))+ℎ𝑘,𝐻𝑥𝑘_𝑑,𝑢(𝑘)+𝐵𝑤(𝑘),𝑦(𝑘)=𝐶𝑥(𝑘)+𝑣(𝑘),𝑧(𝑘)=𝐿𝑥(𝑘),(2.1) where 𝑘_𝑑=𝑘−𝑑, and the positive integer 𝑑 denotes the known state delay; 𝑥(𝑘)∈ℝ^𝑛 is the state, 𝑢(𝑘)∈ℝ^𝑝 is the measurable information, 𝑤(𝑘)∈ℝ^𝑞 and 𝑣(𝑘)∈ℝ^𝑚 are the disturbance input belonging to 𝑙₂[0,𝑁], 𝑦(𝑘)∈ℝ^𝑚 is the measurement output, and 𝑧(𝑘)∈ℝ^𝑟 is the signal to be estimated; the initial condition 𝑥₀(𝑠)(𝑠=−𝑑,−𝑑+1,…,0) is unknown; the matrices 𝐴∈ℝ^𝑛×𝑛, 𝐴_𝑑∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑞, 𝐶∈ℝ^𝑚×𝑛 and 𝐿∈ℝ^𝑟×𝑛, are real and known constant matrices.

In addition, 𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘)) and ℎ(𝑘,𝐻𝑥(𝑘_𝑑),𝑢(𝑘)) are assumed to satisfy the following Lipschitz conditions:‖𝑓(𝑘,𝐹𝑥(𝑘),𝑢(𝑘))−𝑓(𝑘,𝐹𝑥̆(𝑘),𝑢(𝑘))‖≤𝛼‖𝐹(𝑥(𝑘)−𝑥̆(𝑘))‖,‖ℎ𝑘,𝐻𝑥𝑘_𝑑,𝑢(𝑘)−ℎ𝑘,𝐻𝑥̆𝑘_𝑑,𝑢(𝑘)‖≤𝛽‖𝐻𝑥𝑘_𝑑−𝑥̆𝑘_𝑑‖,(2.2) for all 𝑘∈{0,1,…,𝑁}, 𝑢(𝑘)∈ℝ^𝑝 and 𝑥(𝑘),𝑥̆(𝑘),𝑥(𝑘_𝑑),𝑥̆(𝑘_𝑑)∈ℝ^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)}^𝑘_𝑗=0, find an estimate 𝑧(̆𝑘∣𝑘) of the signal 𝑧(𝑘), if it exists, such that the following inequality is satisfied:sup_{𝑥0,𝑤,𝑣≠0}∑^𝑁_𝑘=0‖𝑧̆(𝑘∣𝑘)−𝑧(𝑘)‖²∑⁰_𝑘=−𝑑‖‖𝑥₀(𝑘)‖‖²_Π−1(𝑘)+∑^𝑁_𝑘=0‖𝑤(𝑘)‖²+∑^𝑁_𝑘=0‖𝑣(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 𝑥̂₀(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)}^𝑘_𝑖=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 to⁰_𝑘=−𝑑‖‖𝑥₀(𝑘)‖‖²_Π−1(𝑘)+^𝑁_𝑘=0‖𝑤(𝑘)‖²+^𝑁_𝑘=0‖𝑣(𝑘)‖²−𝛾^−2𝑁_𝑘=0‖‖𝑣_𝑧(𝑘)‖‖²_{≜𝐽∗𝑁}>0,∀𝑥₀,𝑤,𝑣≠0,(3.1) where 𝑣_𝑧(𝑘)=𝑧(̆𝑘∣𝑘)−𝑧(𝑘).

Moreover, we denote𝑧_𝑓(𝑘)=𝐹𝑥(𝑘),𝑧_𝑓̆(𝑘∣𝑘)=𝐹𝑥̆(𝑘∣𝑘),𝑧_ℎ𝑘_𝑑=𝐻𝑥𝑘_𝑑,𝑧_ℎ̆𝑘_𝑑∣𝑘=𝐻𝑥̆𝑘_𝑑∣𝑘,(3.2) where 𝑧_𝑓̆(𝑘∣𝑘) and 𝑧_ℎ̆(𝑘_𝑑∣𝑘) denote the optimal estimation of 𝑧_𝑓(𝑘) and 𝑧_ℎ(𝑘_𝑑) based on the observation {𝑦(𝑗)}^𝑘_𝑗=0, respectively. And, let𝑤_𝑓(𝑘)=𝑓𝑘,𝑧_𝑓(𝑘),𝑢(𝑘)−𝑓𝑘,𝑧_𝑓̆(𝑘∣𝑘),𝑢(𝑘),𝑤_ℎ𝑘_𝑑=ℎ𝑘,𝑧_ℎ𝑘_𝑑,𝑢(𝑘)−ℎ𝑘,𝑧_ℎ̆𝑘_𝑑∣𝑘,𝑢(𝑘),𝑣_𝑧𝑓(𝑘)=𝑧_𝑓̆(𝑘∣𝑘)−𝑧_𝑓(𝑘),𝑣_𝑧ℎ𝑘_𝑑=𝑧_ℎ̆𝑘_𝑑∣𝑘−𝑧_ℎ𝑘_𝑑.(3.3) From the Lipschitz conditions (2.2), we derive that𝐽^∗_𝑁+^𝑁_𝑘=0‖‖𝑤_𝑓(𝑘)‖‖²+^𝑁_𝑘=0‖‖𝑤_ℎ𝑘_𝑑‖‖²−𝛼^2𝑁_𝑘=0‖‖𝑣_𝑧𝑓(𝑘)‖‖²−𝛽^2𝑁_𝑘=0‖‖𝑣_𝑧ℎ𝑘_𝑑‖‖²_{≜𝐽𝑁}≤𝐽^∗_𝑁.(3.4) Note that the left side of (3.1) and (3.4), 𝐽_𝑁, can be recast into the form𝐽_𝑁=⁰_𝑘=−𝑑‖‖𝑥₀(𝑘)‖‖²_Π−1(𝑘)+^𝑁_𝑘=0‖‖𝑤(𝑘)‖‖²+^𝑁_𝑘=0‖𝑣(𝑘)‖²−𝛾^−2𝑁_𝑘=0‖‖𝑣_𝑧(𝑘)‖‖²−𝛼^2𝑁_𝑘=0‖‖𝑣_𝑧𝑓(𝑘)‖‖²−𝛽^2𝑁_𝑘=𝑑‖‖𝑣_𝑧ℎ𝑘_𝑑‖‖²,(3.5) where Π(𝑘)=⎧⎪⎨⎪⎩Π⁻¹(𝑘)−𝛽²𝐻^𝑇𝐻⁻¹,𝑘=−𝑑,…,−1,Π(𝑘),𝑘=0,𝑤(𝑘)=𝑤^𝑇(𝑘)𝑤^𝑇_𝑓(𝑘)𝑤^𝑇_ℎ𝑘_𝑑^𝑇.(3.6)

Since 𝐽_𝑁≤𝐽^∗_𝑁, it is natural to see that if 𝐽_𝑁>0 then the 𝐻_∞ estimation problem (2.3) is satisfied, that is, 𝐽^∗_𝑁>0. Hence, the 𝐻_∞ estimation problem (2.3) can be converted into finding the estimate sequence {{𝑧(̆𝑘∣𝑘)}^𝑁_𝑘=0;{𝑧_𝑓̆(𝑘∣𝑘)}^𝑁_𝑘=0;{𝑧_ℎ̆(𝑘_𝑑∣𝑘)}^𝑁_𝑘=𝑑} such that 𝐽_𝑁 has a minimum with respect to {𝑥₀,𝑤} and the minimum of 𝐽_𝑁 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)+𝐴_𝑑𝐱𝑘_𝑑+𝑓𝑘,𝐳_𝑓̆‌(𝑘∣𝑘),𝐮(𝑘)+ℎ𝑘,𝐳_ℎ̆‌𝑘_𝑑∣𝑘,𝐮(𝑘)+𝐵𝐰(𝑘),𝐲(𝑘)=𝐶𝐱(𝑘)+𝐯(𝑘),𝐳_𝑓̆‌(𝑘∣𝑘)=𝐹𝐱(𝑘)+𝐯_𝑧𝑓(𝑘),𝐳̆‌(𝑘∣𝑘)=𝐿𝐱(𝑘)+𝐯_𝑧(𝑘),𝐳_ℎ̆‌𝑘_𝑑∣𝑘=𝐻𝐱𝑘_𝑑+𝐯_𝑧ℎ𝑘_𝑑,𝑘≥𝑑,(3.7) where 𝐵=[𝐵𝐼𝐼]; the initial state 𝐱₀(𝑠)(𝑠=−𝑑,−𝑑+1,…,0) and 𝐰(𝑘), 𝐯(𝑘), 𝐯_𝑧𝑓(𝑘), 𝐯_𝑧(𝑘) and 𝐯_𝑧ℎ(𝑘) are mutually uncorrelated white noises with zero means and known covariance matrices Π(𝑠), 𝑄_𝑤(𝑘)=𝐼, 𝑄_𝑣(𝑘)=𝐼, 𝑄_𝑣𝑧𝑓(𝑘)=−𝛼⁻²𝐼, 𝑄_𝑣𝑧(𝑘)=−𝛾²𝐼, and 𝑄_𝑣𝑧ℎ(𝑘)=−𝛽⁻²𝐼; 𝐳_𝑓̆‌(𝑘∣𝑘), 𝐳(̆‌𝑘∣𝑘) and 𝐳_ℎ̆‌(𝑘_𝑑∣𝑘) are regarded as the imaginary measurement at time 𝑘 for the linear combination 𝐹𝐱(𝑘), 𝐿𝐱(𝑘), and 𝐻𝐱(𝑘_𝑑), respectively.

Let𝐲_𝑧(𝑘)=⎧⎪⎨⎪⎩𝐲^𝑇(𝑘)𝐳^𝑇_𝑚̆‌(𝑘∣𝑘)^𝑇,0≤𝑘<𝑑,𝐲^𝑇(𝑘)𝐳^𝑇_𝑚̆‌(𝑘∣𝑘)𝐳^𝑇_ℎ̆‌𝑘_𝑑∣𝑘^𝑇,𝑘≥𝑑,𝐯_𝑧,𝑎(𝑘)=⎧⎪⎨⎪⎩𝐯^𝑇(𝑘)𝐯^𝑇_𝑧𝑓(𝑘)𝐯^𝑇_𝑧(𝑘)^𝑇,0≤𝑘<𝑑,𝐯^𝑇(𝑘)𝐯^𝑇_𝑧𝑓(𝑘)𝐯^𝑇_𝑧(𝑘)𝐯^𝑇_𝑧ℎ𝑘_𝑑^𝑇,𝑘≥𝑑,𝐳_𝑚̆‌(𝑘∣𝑘)=𝐳^𝑇_𝑓̆‌(𝑘∣𝑘)𝐳^𝑇̆‌(𝑘∣𝑘)^𝑇.(3.8)

Definition 3.1. The estimator 𝐲̂‌(𝑖∣𝑖−1) denotes the optimal estimation of 𝐲(𝑖) given the observation ℒ{{𝐲_𝑧(𝑗)}^𝑖−1_𝑗=0}; the estimator 𝐳_𝑚̂‌(𝑖∣𝑖) denotes the optimal estimation of 𝐳_𝑚̆‌(𝑖∣𝑖) given the observation ℒ{{𝐲_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲(𝑖)}; the estimator 𝐳_ℎ̂‌(𝑖_𝑑∣𝑖) denotes the optimal estimation of 𝐳_ℎ̆‌(𝑖_𝑑∣𝑖) given the observation ℒ{{𝐲_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲(𝑖),𝐳_𝑚̆‌(𝑖∣𝑖)}.

Furthermore, introduce the following stochastic vectors and the corresponding covariance matrices𝐲̃‌(𝑖∣𝑖−1)=𝐲(𝑖)−𝐲̂‌(𝑖𝑖−1),𝑅_̃𝑦(𝑖𝑖−1)=⟨𝐲̃‌(𝑖𝑖−1),𝐲̃‌(𝑖𝑖−1)⟩,𝐳_𝑚̃‌(𝑖∣𝑖)=𝐳_𝑚̆‌(𝑖𝑖)−𝐳_𝑚̂‌(𝑖𝑖),𝑅𝑧_𝑚̃(𝑖𝑖)=⟨𝐳_𝑚̃‌(𝑖𝑖),𝐳_𝑚̃‌(𝑖𝑖)⟩,𝐳_ℎ̃‌𝑖_𝑑∣𝑖=𝐳_ℎ̆‌𝑖_𝑑𝑖−𝐳_ℎ̂‌𝑖_𝑑𝑖,𝑅𝑧_ℎ̃𝑖_𝑑𝑖=𝐳_ℎ̃‌𝑖_𝑑𝑖,𝐳_ℎ̃‌𝑖_𝑑𝑖.(3.9) And, denote𝐲̃‌_𝑧(𝑖)=⎧⎪⎨⎪⎩𝐲̃‌^𝑇(𝑖∣𝑖−1)𝐳^𝑇_𝑚̃‌(𝑖∣𝑖)^𝑇,0≤𝑖<𝑑,𝐲̃‌^𝑇(𝑖∣𝑖−1)𝐳^𝑇_𝑚̃‌(𝑖∣𝑖)𝐳^𝑇_ℎ̃‌(𝑖_𝑑∣𝑖)^𝑇,𝑖≥𝑑,𝑅𝑦_𝑧̃(𝑖)=𝐲̃‌_𝑧(𝑖),𝐲̃‌_𝑧(𝑖).(3.10)

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

Lemma 3.2. {{𝐲̃‌_𝑧(𝑖)}^𝑘_𝑖=0} is the innovation sequence which spans the same linear space as that of ℒ{{𝐲_𝑧(𝑖)}^𝑘_𝑖=0}.

Proof. From Definition 3.1 and (3.9), 𝐲̃‌(𝑖∣𝑖−1), 𝐳_𝑚̃‌(𝑖∣𝑖) and 𝐳_ℎ̃‌(𝑖_𝑑∣𝑖) are the linear combination of the observation sequence {{𝐲_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲(𝑖)}, {{𝐲_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲(𝑖),𝐳_𝑚̆‌(𝑖∣𝑖)}, and {{𝐲_𝑧(𝑗)}^𝑖_𝑗=0}, respectively. Conversely, 𝐲(𝑖), 𝐳_𝑚̆‌(𝑖∣𝑖) and 𝐳_ℎ̆‌(𝑖_𝑑∣𝑖) can be given by the linear combination of {{𝐲̃‌_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲̃‌(𝑖∣𝑖−1)},{{𝐲̃‌_𝑧(𝑗)}^𝑖−1_𝑗=0;𝐲̃‌(𝑖∣𝑖−1),𝐳_𝑚̃‌(𝑖∣𝑖)} and {{𝐲̃‌_𝑧(𝑗)}^𝑖_𝑗=0}, respectively. Hence, ℒ𝐲̃‌_𝑧(𝑖)^𝑘_𝑖=0=ℒ𝐲_𝑧(𝑖)^𝑘_𝑖=0.(3.11) It is also shown by (3.9) that 𝐲̃‌(𝑖∣𝑖−1), 𝐳_𝑚̃‌(𝑖∣𝑖) and 𝐳_ℎ̃‌(𝑖_𝑑∣𝑖) satisfy 𝐲̃‌(𝑖∣𝑖−1)⟂ℒ𝐲_𝑧(𝑗)^𝑖−1_𝑗=0,𝐳_𝑚̃‌(𝑖∣𝑖)⟂ℒ𝐲_𝑧(𝑗)^𝑖−1_𝑗=0;𝐲(𝑖),𝐳_ℎ̃‌𝑖_𝑑∣𝑖⟂ℒ𝐲_𝑧(𝑗)^𝑖−1_𝑗=0;𝐲(𝑖),𝐳_𝑚̃‌(𝑖∣𝑖).(3.12) Consequently, 𝐲̃‌(𝑖∣𝑖−1)⟂ℒ𝐲̃‌_𝑧(𝑗)^𝑖−1_𝑗=0,𝐳_𝑚̃‌(𝑖∣𝑖)⟂ℒ𝐲̃‌_𝑧(𝑗)^𝑖−1_𝑗=0;𝐲̃‌(𝑖∣𝑖−1),𝐳_ℎ̃‌𝑖_𝑑∣𝑖⟂ℒ𝐲̃‌_𝑧(𝑗)^𝑖−1_𝑗=0;𝐲̃‌(𝑖∣𝑖−1),𝐳_𝑚̃‌(𝑖∣𝑖).(3.13) This completes the proof.

Now, an existence condition and a solution to the minimum of 𝐽_𝑁 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 𝐽_𝑁 has the minimum if only if 𝑅_̃𝑦(𝑘∣𝑘−1)>0,0≤𝑘≤𝑁,𝑅𝑧_𝑚̃(𝑘∣𝑘)<0,0≤𝑘≤𝑁,𝑅𝑧_ℎ̃𝑘_𝑑∣𝑘<0,𝑑≤𝑘≤𝑁.(3.14) In this case the minimum value of 𝐽_𝑁 is given by min𝐽_𝑁=^𝑁_𝑘=0𝑦^𝑇̃(𝑘∣𝑘−1)𝑅⁻¹_̃𝑦(𝑘∣𝑘−1)𝑦̃(𝑘∣𝑘−1)+^𝑁_𝑘=0𝑧^𝑇_𝑚̃(𝑘∣𝑘)𝑅⁻¹𝑧_𝑚̃(𝑘∣𝑘)𝑧_𝑚̃(𝑘∣𝑘)+^𝑁_𝑘=𝑑𝑧^𝑇_ℎ̃𝑘_𝑑∣𝑘𝑅⁻¹𝑧_ℎ̃𝑘_𝑑∣𝑘𝑧_ℎ̃𝑘_𝑑∣𝑘,(3.15) where 𝑦̃(𝑘∣𝑘−1)=𝑦(𝑘)−𝑦̂(𝑘∣𝑘−1),𝑧_𝑚̃(𝑘∣𝑘)=𝑧_𝑚̆(𝑘∣𝑘)−𝑧_𝑚̂(𝑘∣𝑘),𝑧_ℎ̃𝑘_𝑑∣𝑘=𝑧_ℎ̆𝑘_𝑑∣𝑘−𝑧_ℎ̂𝑘_𝑑∣𝑘,𝑧_𝑚̆(𝑘∣𝑘)=𝑧^𝑇_𝑓̆(𝑘∣𝑘)𝑧^𝑇̆(𝑘∣𝑘)^𝑇,(3.16)𝑦(̂𝑘∣𝑘−1) is obtained from the Krein space projection of 𝐲(𝑘) onto ℒ{{𝐲_𝑧(𝑗)}^𝑘−1_𝑗=0}, 𝑧_𝑚̂(𝑘∣𝑘) is obtained from the Krein space projection of 𝐳_𝑚̆‌(𝑘∣𝑘) onto ℒ{{𝐲_𝑧(𝑗)}^𝑘−1_𝑗=0;𝐲(𝑘)}, and 𝑧_ℎ̂(𝑘_𝑑∣𝑘) is obtained from the Krein space projection of 𝐳_ℎ̆‌(𝑘_𝑑∣𝑘) onto ℒ{{𝐲_𝑧(𝑗)}^𝑘−1_𝑗=0;𝐲(𝑘),𝐳_𝑚̆‌(𝑘∣𝑘)}.

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)+𝐴_𝑑𝑥𝑘_𝑑+𝑓𝑘,𝑧_𝑓̆(𝑘∣𝑘),𝑢(𝑘)+ℎ𝑘,𝑧_ℎ̆𝑘_𝑑∣𝑘,𝑢(𝑘)+𝐵𝑤(𝑘).(3.17) In this case, it is assumed that 𝑓(𝑘,𝑧_𝑓̆(𝑘∣𝑘),𝑢(𝑘)) and ℎ(𝑘,𝑧_ℎ̆(𝑘_𝑑∣𝑘),𝑢(𝑘)) are known at time 𝑘. Then, we define 𝑦_𝑧(𝑘)=⎧⎪⎨⎪⎩𝑦^𝑇(𝑘)𝑧^𝑇_𝑓̆(𝑘∣𝑘)𝑧^𝑇̆(𝑘∣𝑘)^𝑇,0≤𝑘<𝑑,𝑦^𝑇(𝑘)𝑧^𝑇_𝑓̆(𝑘∣𝑘)𝑧^𝑇̆(𝑘∣𝑘)𝑧^𝑇_ℎ̆𝑘_𝑑∣𝑘^𝑇,𝑘≥𝑑.(3.18) By introducing an augmented state 𝑥_𝑎(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−1)⋯𝑥^𝑇(𝑘−𝑑)^𝑇,(3.19) we obtain an augmented state-space model 𝑥_𝑎(𝑘+1)=𝐴_𝑎𝑥_𝑎(𝑘)+𝐵_𝑢,𝑎𝑢(𝑘)+𝐵_𝑎𝑤(𝑘),𝑦_𝑧(𝑘)=𝐶_𝑧,𝑎(𝑘)𝑥_𝑎(𝑘)+𝑣_𝑧,𝑎(𝑘),(3.20) where 𝐴_𝑎=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐴0⋯0𝐴_𝑑𝐼0⋯000𝐼⋯00⋮⋮⋱⋮⋮00⋯𝐼0⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐵_𝑢,𝑎=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐼𝐼0000⋮⋮00⎤⎥⎥⎥⎥⎥⎥⎥⎦,𝐵_𝑎=⎡⎢⎢⎢⎢⎢⎢⎢⎣𝐵00⋮0⎤⎥⎥⎥⎥⎥⎥⎥⎦,(3.21)C_𝑧,𝑎(𝑘)=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩⎡⎢⎢⎢⎣𝐶0⋯0𝐹0⋯0𝐿0⋯0⎤⎥⎥⎥⎦,0≤𝑘<𝑑,⎡⎢⎢⎢⎢⎢⎢⎣𝐶0⋯0𝐹0⋯0𝐿0⋯00⋯0𝐻⎤⎥⎥⎥⎥⎥⎥⎦,𝑘≥𝑑,𝑣_𝑧,𝑎(𝑘)=⎧⎪⎨⎪⎩𝑣^𝑇(𝑘)𝑣^𝑇_𝑧𝑓(𝑘)𝑣^𝑇_𝑧(𝑘)^𝑇,0≤𝑘<𝑑,𝑣^𝑇(𝑘)𝑣^𝑇_𝑧𝑓(𝑘)𝑣^𝑇_𝑧(𝑘)𝑣^𝑇_𝑧ℎ𝑘_𝑑^𝑇,𝑘≥𝑑,𝑢(𝑘)=𝑓^𝑇𝑘,𝑧_𝑓̆(𝑘∣𝑘),𝑢(𝑘)ℎ^𝑇𝑘,𝑧_ℎ̆𝑘_𝑑∣𝑘,𝑢(𝑘)^𝑇.(1) Additionally, we can rewrite 𝐽_𝑁 as 𝐽_𝑁=⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑣_{𝑧,𝑎𝑁}⎤⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝑃_𝑎(0)000𝐼000𝑄_{𝑣𝑧,𝑎𝑁}⎤⎥⎥⎥⎦⁻¹⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑣_{𝑧,𝑎𝑁}⎤⎥⎥⎥⎦,(3.22) where 𝑃_𝑎(0)=diagΠ(0),Π(−1),…,Π(−𝑑),𝑤_𝑁=𝑤^𝑇(0)𝑤^𝑇(1)⋯𝑤^𝑇(𝑁)^𝑇,𝑣_{𝑧,𝑎𝑁}=𝑣^𝑇_𝑧,𝑎(0)𝑣^𝑇_𝑧,𝑎(1)⋯𝑣^𝑇_𝑧,𝑎(𝑁)^𝑇,𝑄_{𝑣𝑧,𝑎𝑁}=diag𝑄_{𝑣𝑧,𝑎}(0),𝑄_{𝑣𝑧,𝑎}(1),…,𝑄_{𝑣𝑧,𝑎}(𝑁),𝑄_{𝑣𝑧,𝑎}(𝑘)=⎧⎪⎨⎪⎩diag𝐼,−𝛾²,−𝛼⁻²,0≤𝑘<𝑑,diag𝐼,−𝛾²,−𝛼⁻²,−𝛽⁻²,𝑘≥𝑑.(3.23) Define the following state transition matrix Φ(𝑘+1,𝑚)=𝐴_𝑎Φ(𝑘,𝑚),Φ(𝑚,𝑚)=𝐼,(3.24) and let 𝑦_𝑧𝑁=𝑦^𝑇_𝑧(0)𝑦^𝑇_𝑧(1)⋯𝑦^𝑇_𝑧(𝑁)^𝑇,𝑢_𝑁=𝑢^𝑇(0)𝑢^𝑇(1)⋯𝑢^𝑇(𝑁)^𝑇.(3.25) Using (3.20) and (3.24), we have 𝑦_𝑧𝑁=Ψ_0𝑁𝑥_𝑎(0)+Ψ_𝑢𝑁𝑢_𝑁+Ψ_𝑤𝑁𝑤_𝑁+𝑣_{𝑧,𝑎𝑁},(3.26) where Ψ_0𝑁=⎡⎢⎢⎢⎢⎢⎣𝐶_𝑧,𝑎(0)Φ(0,0)𝐶_𝑧,𝑎(1)Φ(1,0)⋮𝐶_𝑧,𝑎(𝑁)Φ(𝑁,0)⎤⎥⎥⎥⎥⎥⎦,Ψ_𝑢𝑁=⎡⎢⎢⎢⎢⎢⎣𝜑₀₀𝜑₀₁⋯𝜑_0𝑁𝜑₁₀𝜑₁₁⋯𝜑_1𝑁⋮⋮⋱⋮𝜑_𝑁0𝜑_𝑁1⋯𝜑_𝑁𝑁⎤⎥⎥⎥⎥⎥⎦,𝜑_𝑖𝑗=⎧⎪⎨⎪⎩𝐶_𝑧,𝑎(𝑖)Φ(𝑖,𝑗+1)𝐵_𝑢,𝑎,𝑖>𝑗,0,𝑖≤𝑗.(3.27) The matrix Ψ_𝑤𝑁 is derived by replacing 𝐵_𝑢,𝑎 in Ψ_𝑢𝑁 with 𝐵_𝑎.
Thus, 𝐽_𝑁 can be reexpressed as 𝐽_𝑁=⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑦_𝑧𝑁⎤⎥⎥⎥⎦^𝑇⎧⎪⎨⎪⎩Γ_𝑁⎡⎢⎢⎢⎣𝑃_𝑎(0)000𝐼000𝑄_{𝑣𝑧,𝑎𝑁}⎤⎥⎥⎥⎦Γ^𝑇_𝑁⎫⎪⎬⎪⎭⁻¹⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑦_𝑧𝑁⎤⎥⎥⎥⎦,(3.28) where 𝑦_𝑧𝑁=𝑦_𝑧𝑁−Ψ_𝑢𝑁𝑢_𝑁,Γ_𝑁=⎡⎢⎢⎢⎣𝐼000𝐼0Ψ_0𝑁Ψ_𝑤𝑁𝐼⎤⎥⎥⎥⎦.(3.29)
Considering the Krein space stochastic system defined by (3.7) and state transition matrix (3.24), we have 𝐲_𝑧𝑁=Ψ_0𝑁𝐱_𝑎(0)+Ψ_𝑢𝑁𝐮_𝑁+Ψ_𝑤𝑁𝐰_𝑁+𝐯_{𝑧,𝑎𝑁},(3.30) where matrices Ψ_0𝑁, Ψ_𝑢𝑁, and Ψ_𝑤𝑁 are the same as given in (3.26), vectors 𝐲_𝑧𝑁 and 𝐮_𝑁 are, respectively, defined by replacing Euclidean space element 𝑦_𝑧 and 𝑢 in 𝑦_𝑧𝑁 and 𝑢_𝑁 given by (3.25) with the Krein space element 𝐲_𝑧 and 𝐮, vectors 𝐰_𝑁 and 𝐯_{𝑧,𝑎𝑁} are also defined by replacing Euclidean space element 𝑤 and 𝑣_𝑧,𝑎 in 𝑤_𝑁 and 𝑣_{𝑧,𝑎𝑁} given by (3.23) with the Krein space element 𝐰 and 𝐯_𝑧,𝑎, and vector 𝐱_𝑎(0) is given by replacing Euclidean space element 𝑥 in 𝑥_𝑎(𝑘) given by (3.19) with the Krein space element 𝐱 when 𝑘=0.
Using the stochastic characteristic of 𝐱_𝑎(0), 𝐰_𝑁 and 𝐯_𝑧,𝑎, we have 𝐽_𝑁=⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑦_𝑧𝑁⎤⎥⎥⎥⎦^𝑇⎡⎢⎢⎢⎣𝐱_𝑎(0)𝐰_𝑁𝐲_𝑧𝑁⎤⎥⎥⎥⎦,⎡⎢⎢⎢⎣𝐱_𝑎(0)𝐰_𝑁𝐲_𝑧𝑁⎤⎥⎥⎥⎦⁻¹⎡⎢⎢⎢⎣𝑥_𝑎(0)𝑤_𝑁𝑦_𝑧𝑁⎤⎥⎥⎥⎦,(3.31) where 𝐲_𝑧𝑁=𝐲_𝑧𝑁−Ψ_𝑢𝑁𝐮_𝑁.
In the light of Theorem  2.4.2 and Lemma  2.4.3 in [26], 𝐽_𝑁 has a minimum over {𝑥_𝑎(0),𝑤_𝑁} if and only if 𝑅_𝑦𝑧𝑁=⟨𝐲_𝑧𝑁,𝐲_𝑧𝑁⟩ and 𝑄_{𝑣𝑧,𝑎𝑁}=⟨𝐯_{𝑧,𝑎𝑁},𝐯_{𝑧,𝑎𝑁}⟩ have the same inertia. Moreover, the minimum of 𝐽_𝑁 is given by min𝐽_𝑁=𝑦^𝑇_𝑧𝑁𝑅⁻¹_𝑦𝑧𝑁𝑦_𝑧𝑁.(3.32)
On the other hand, applying the Krein space projection formula, we have 𝐲_𝑧𝑁=Θ_𝑁𝐲̃‌_𝑧𝑁,(3.33) where 𝐲̃‌_𝑧𝑁=𝐲̃‌^𝑇_𝑧(0)𝐲̃‌^𝑇_𝑧(1)⋯𝐲̃‌^𝑇_𝑧(𝑁)^𝑇,(3.34)Θ_𝑁=⎡⎢⎢⎢⎢⎢⎣𝜃₀₀𝜃₀₁⋯𝜃_0𝑁𝜃₁₀𝜃₁₁⋯𝜃_1𝑁⋮⋮⋱⋮𝜃_𝑁0𝜃_𝑁1⋯𝜃_𝑁𝑁⎤⎥⎥⎥⎥⎥⎦,𝜃_𝑖𝑗=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝐲_𝑧(𝑖),𝐲̃‌_𝑧(𝑗)𝑅𝑦⁻¹_𝑧̃(𝑗),𝑖>𝑗≥0,⎡⎢⎣𝐼0𝑚₁𝐼⎤⎥⎦,𝑑>𝑖=𝑗≥0,⎡⎢⎢⎢⎣𝐼00𝑚₁𝐼0𝑚₂𝑚₃𝐼⎤⎥⎥⎥⎦,𝑖=𝑗≥𝑑,0,0≤𝑖<𝑗,𝑚₁=𝐳_𝑚̆‌(𝑖∣𝑖),𝐲̃‌(𝑗∣𝑗−1)𝑅⁻¹_̃𝑦(𝑗∣𝑗−1),𝑚₂=𝐳_ℎ̆‌𝑖_𝑑∣𝑖,𝐲̃‌(𝑗∣𝑗−1)𝑅⁻¹_̃𝑦(𝑗∣𝑗−1),𝑚₃=𝐳_ℎ̆‌𝑖_𝑑∣𝑖,𝐳_𝑚̃‌(𝑗∣𝑗)𝑅⁻¹𝑧_𝑚̃(𝑗∣𝑗),𝐲_𝑧(𝑖)=𝐲_𝑧(𝑖)−^𝑁_𝑗=0𝜑_𝑖𝑗𝐮(𝑗),𝐳_𝑚̆‌(𝑖∣𝑖)=𝐳_𝑚̆‌(𝑖∣𝑖)−^𝑁_𝑗=0𝜑_{𝑚,𝑖𝑗}𝐮(𝑗),𝐳_ℎ̆‌𝑖_𝑑∣𝑖=𝐳_ℎ̆‌𝑖_𝑑∣𝑖−^𝑁_𝑗=0𝜑_ℎ,𝑖𝑗𝐮(𝑗),(2) where 𝜑_{𝑚,𝑖𝑗} is derived by replacing 𝐶_𝑧,𝑎 in 𝜑_𝑖𝑗 with^𝐹0⋯0_𝐿0⋯0, 𝜑_ℎ,𝑖𝑗 is derived by replacing 𝐶_𝑧,𝑎 in 𝜑_𝑖𝑗 with [00⋯𝐻] Furthermore, it follows from (3.33) that 𝑅_𝑦𝑧𝑁=Θ_𝑁𝑅𝑦̃_𝑧𝑁Θ^𝑇_𝑁,𝑦_𝑧𝑁=Θ_𝑁𝑦̃_𝑧𝑁,(3.35) where 𝑅𝑦̃_𝑧𝑁=𝐲̃‌_𝑧𝑁,𝐲̃‌_𝑧𝑁,𝑦̃_𝑧𝑁=𝑦^𝑇_𝑧̃(0)𝑦^𝑇_𝑧̃(1)⋯𝑦^𝑇_𝑧̃(𝑁)^𝑇,𝑦_𝑧̃(𝑖)=⎧⎪⎨⎪⎩𝑦^𝑇̃(𝑖∣𝑖−1)𝑧^𝑇_𝑚̃(𝑖∣𝑖)^𝑇,0≤𝑖<𝑑,𝑦^𝑇̃(𝑖∣𝑖−1)𝑧^𝑇_𝑚̃(𝑖∣𝑖)𝑧^𝑇_ℎ̃𝑖_𝑑∣𝑖^𝑇,𝑖≥𝑑.(3.36) Since matrix Θ_𝑁 is nonsingular, it follows from (3.35) that 𝑅_𝑦𝑧𝑁 and 𝑅𝑦̃_𝑧𝑁 are congruent, which also means that 𝑅_𝑦𝑧𝑁 and 𝑅𝑦̃_𝑧𝑁 have the same inertia. Note that both 𝑅𝑦̃_𝑧𝑁 and 𝑄_{𝑣𝑧,𝑎𝑁} are block-diagonal matrices, and 𝑅𝑦_𝑧̃(𝑘)=⎧⎪⎨⎪⎩diag𝑅_̃𝑦(𝑘∣𝑘−1),𝑅𝑧_𝑚̃(𝑘∣𝑘),0≤𝑘<𝑑,diag𝑅_̃𝑦(𝑘∣𝑘−1),𝑅𝑧_𝑚̃(𝑘∣𝑘),𝑅𝑧_ℎ̃𝑘_𝑑∣𝑘,𝑘≤𝑑,(3.37)𝑄_{𝑣𝑧,𝑎}(𝑘) is given by (3.23). It follows that 𝑅𝑦̃_𝑧𝑁 and 𝑄_{𝑣𝑧,𝑎𝑁} have the same inertia if and only if 𝑅_̃𝑦(𝑘∣𝑘−1)>0(0≤𝑘≤𝑁), 𝑅𝑧_𝑚̃(𝑘∣𝑘)<0(0≤𝑘≤𝑁) and 𝑅𝑧_ℎ̃(𝑘_𝑑∣𝑘)<0(𝑑≤𝑘≤𝑁).
Therefore, 𝐽_𝑁 subject to system (2.1) with Lipschitz conditions (2.2) has the minimum if and only if 𝑅_̃𝑦(𝑘∣𝑘−1)>0(0≤𝑘≤𝑁), 𝑅𝑧_𝑚̃(𝑘∣𝑘)<0(0≤𝑘≤𝑁) and 𝑅𝑧_ℎ̃(𝑘_𝑑∣𝑘)<0(𝑑≤𝑘≤𝑁). Moreover, the minimum value of 𝐽_𝑁 can be rewritten as min𝐽_𝑁=𝑦^𝑇_𝑧𝑁𝑅⁻¹_𝑦𝑧𝑁𝑦_𝑧𝑁=𝑦̃^𝑇_𝑧𝑁𝑅⁻¹𝑦̃_𝑧𝑁𝑦̃_𝑧𝑁=^𝑁_𝑘=0𝑦^𝑇̃(𝑘∣𝑘−1)𝑅⁻¹_̃𝑦(𝑘∣𝑘−1)𝑦̃(𝑘∣𝑘−1)+^𝑁_𝑘=0𝑧^𝑇_𝑚̃(𝑘∣𝑘)𝑅⁻¹𝑧_𝑚̃(𝑘∣𝑘)𝑧_𝑚̃(𝑘∣𝑘)+^𝑁_𝑘=𝑑𝑧^𝑇_ℎ̃𝑘_𝑑∣𝑘𝑅⁻¹𝑧_ℎ̃𝑘_𝑑∣𝑘𝑧_ℎ̃𝑘_𝑑∣𝑘.(3.38) The proof is completed.

Remark 3.4. Due to the built innovation sequence {{𝐲̃‌_𝑧(𝑖)}^𝑘_𝑖=0} in Lemma 3.2, the form of the minimum on indefinite quadratic form 𝐽_𝑁 is different from the one given in [26–28]. It is shown from (3.15) that the estimation errors 𝑦(̃𝑘∣𝑘−1), 𝑧_𝑚̃(𝑘∣𝑘) and 𝑧_ℎ̃(𝑘_𝑑∣𝑘) are mutually uncorrelated, which will make the design of 𝐻_∞ estimator much easier than the one given in [26–28].

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)=𝐲^𝑇(𝑖)𝐳^𝑇_𝑚̆‌(𝑖∣𝑖)^𝑇,𝐲₂(𝑖)=𝐲^𝑇(𝑖)𝐳^𝑇_𝑚̆‌(𝑖∣𝑖)𝐳^𝑇_ℎ̆‌(𝑖∣𝑖+𝑑)^𝑇.(3.39) Observe from (3.8), we have ℒ𝐲_𝑧(𝑖)^𝑗_𝑖=0=ℒ𝐲₁(𝑖)^𝑗_𝑖=0,0≤𝑗<𝑑,ℒ𝐲_𝑧(𝑖)^𝑗_𝑖=0=ℒ𝐲₂(𝑖)^𝑗𝑑_𝑖=0;𝐲₁(𝑖)^𝑗_{𝑖=𝑗𝑑+1},𝑗≥𝑑.(3.40)

Definition 3.5. Given 𝑘≥𝑑, the estimator 𝜉̂‌(𝑖∣𝑗,2) for 0≤𝑗<𝑘_𝑑 denotes the optimal estimate of 𝜉(𝑖) given the observation ℒ{{𝐲₂(𝑠)}^𝑗_𝑠=0}, and the estimator 𝜉̂‌(𝑖∣𝑗,1) for 𝑘_𝑑≤𝑗≤𝑘 denotes the optimal estimate of 𝜉(𝑖) given the observation ℒ{{𝐲₂(𝑠)}^𝑘𝑑−1_𝑠=0;{𝐲₁(𝜏)}^𝑗_{𝜏=𝑘𝑑}}. For simplicity, we use 𝜉̂‌(𝑖,2) to denote 𝜉̂‌(𝑖∣𝑖−1,2), and use 𝜉̂‌(𝑖,1) to denote 𝜉̂‌(𝑖∣𝑖−1,1) throughout the paper.

Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matrices𝐲̃‌₂(𝑖,2)=𝐲₂(𝑖)−𝐲̂‌₂(𝑖,2),𝑅𝑦₂̃(𝑖,2)=⟨𝐲̃‌₂(𝑖,2),𝐲̃‌₂(𝑖,2)⟩,𝐲̃‌₁(𝑖,1)=𝐲₁(𝑖)−𝐲̂‌₁(𝑖,1),𝑅𝑦₁̃(𝑖,1)=⟨𝐲̃‌₁(𝑖,1),𝐲̃‌₁(𝑖,1)⟩,𝐲̃‌₀(𝑖,0)=𝐲₀(𝑖)−𝐲̂‌₀(𝑖,1),𝑅𝑦₀̃(𝑖,0)=𝐲̃‌₀(i,0),𝐲̃‌₀(𝑖,0).(3.41) Similar to the proof of Lemma  2.2.1 in [27], we can obtain that {𝐲̃‌₂(0,2),…,𝐲̃‌₂(𝑘_𝑑−1,2);𝐲̃‌₁(𝑘_𝑑,1),…,𝐲̃‌₁(𝑘−1,1)} is the innovation sequence which is a mutually uncorrelated white noise sequence and spans the same linear space as ℒ{𝐲₂(0),…,𝐲₂(𝑘_𝑑−1);𝐲₁(𝑘_𝑙),…,𝐲₁(𝑘−1)} or equivalently ℒ{𝐲_𝑧(0),…,𝐲_𝑧(𝑘−1)}.

Applying projection formula in the Krein space, 𝐱̂‌(𝑖,2)(𝑖=0,1,…,𝑘_𝑑) is computed recursively as(3.42)𝐱̂‌(𝑖+1,2)=^𝑖_𝑗=0⟨𝐱(𝑖+1),𝐲̃‌₂(𝑗,2)⟩𝑅⁻¹𝑦₂̃(𝑗,2)𝐲̃‌₂(𝑗,2)=𝐴𝐱̂‌(𝑖∣𝑖,2)+𝐴_𝑑𝐱̂‌𝑖_𝑑∣𝑖,2+𝑓𝑖,𝐳_𝑓̆‌(𝑖∣𝑖),𝐮(𝑖)+ℎ𝑖,𝐳_ℎ̆‌𝑖_𝑑∣𝑖,𝐮(𝑖),𝑖=0,1,…,𝑘_𝑑−1,𝐱̂‌(𝜏,2)=0,(𝜏=−𝑑,−𝑑+1,…,0).(3.43) Note that𝐱̂‌(𝑖∣𝑖,2)=𝐱̂‌(𝑖,2)+𝑃₂(𝑖,𝑖)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑖,2)𝐲̃‌₂(𝑖,2),𝐱̂‌𝑖_𝑑∣𝑖,2=𝐱̂‌𝑖_𝑑,2+^𝑖_{𝑗=𝑖𝑑}𝑃₂𝑖_𝑑,𝑗𝐶^𝑇₂𝑅𝑦⁻¹₂̃(𝑗,2)𝐲̃‌₂(𝑗,2),(3.44) where𝐶₂=𝐶^𝑇𝐹^𝑇𝐿^𝑇𝐻^𝑇^𝑇,𝑃₂(𝑖,𝑗)=⟨𝐞(𝑖,2),𝐞(𝑗,2)⟩,𝐞(𝑖,2)=𝐱(𝑖)−𝐱̂‌(𝑖,2),𝑅𝑦₂̃(𝑖,2)=𝐶₂𝑃₂(𝑖,𝑖)𝐶^𝑇₂+𝑄_𝑣2(𝑖),𝑄_𝑣2(𝑖)=diag𝐼,−𝛼⁻²𝐼,−𝛾²𝐼,−𝛽⁻²𝐼.(3.45) Substituting (3.44) into (3.43), we have𝐱̂‌(𝑖+1,2)=𝐴𝐱̂‌(𝑖,2)+𝐴_𝑑𝐱̂‌𝑖_𝑑,2+𝑓𝑖,𝐳_𝑓̆‌(𝑖∣𝑖),𝐮(𝑖)+ℎ𝑖,𝐳_ℎ̆‌𝑖_𝑑∣𝑖,𝐮(𝑖)+𝐴^𝑑𝑖−1_{𝑗=𝑖𝑑}𝑃₂𝑖_𝑑,𝑗𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑗,2)𝐲̃‌₂(𝑗,2)+𝐾₂(𝑖)𝐲̃‌₂(𝑖,2),𝐾₂(𝑖)=𝐴_𝑑𝑃₂𝑖_𝑑,𝑖𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑖,2)+𝐴𝑃₂(𝑖,𝑖)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑖,2).(3.46) Moreover, taking into account (3.7) and (3.46), we obtain𝐞(𝑖+1,2)=𝐴𝐞(𝑖,2)+𝐴_𝑑𝐞𝑖_𝑑,2+𝐵𝐰(𝑖)−𝐾₂(𝑖)𝐲̃‌₂(𝑖,2)−𝐴^𝑑𝑖−1_{𝑗=𝑖𝑑}𝑃₂𝑖_𝑑,𝑗𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑗,2)𝐲̃‌₂(𝑗,2),𝑖=0,1,…,𝑘_𝑑−1.(3.47) Consequently,𝑃₂(𝑖−𝑗,𝑖+1)=⟨𝐞(𝑖−𝑗,2),𝐞(𝑖+1,2)⟩=𝑃₂(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₂𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₂(𝑖−𝑗,𝑖)𝐶^𝑇₂𝐾^𝑇₂(𝑖)−^𝑖−1_{𝑡=𝑖−𝑗}𝑃₂(𝑖−𝑗,𝑡)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑡,2)𝐶₂𝑃^𝑇₂𝑖_𝑑,𝑡𝐴^𝑇_𝑑,𝑗=0,1,…,𝑑,𝑃₂(𝑖+1,𝑖+1)=⟨𝐞(𝑖+1,2),𝐞(𝑖+1,2)⟩=𝐴𝑃₂(𝑖,𝑖+1)+𝐴_𝑑𝑃₂𝑖_𝑑,𝑖+1+𝐵𝑄_𝑤(𝑖)𝐵^𝑇,(3.48) where 𝑄_𝑤(𝑖)=𝐼. Thus, 𝑃₂(𝑖,𝑖)(𝑖=0,1,…,𝑘_𝑑) can be computed recursively as𝑃₂(𝑖−𝑗,𝑖+1)=𝑃₂(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₂𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₂(𝑖−𝑗,𝑖)𝐶^𝑇₂𝐾^𝑇₂(𝑖)−^𝑖−1_{𝑡=𝑖−𝑗}𝑃₂(𝑖−𝑗,𝑡)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑡,2)𝐶₂𝑃^𝑇₂𝑖_𝑑,𝑡𝐴^𝑇_𝑑,𝑃₂(𝑖+1,𝑖+1)=𝐴𝑃₂(𝑖,𝑖+1)+𝐴_𝑑𝑃₂𝑖_𝑑,𝑖+1+𝐵𝑄_𝑤(𝑖)𝐵^𝑇,𝑗=0,1,…,𝑑.(3.49)

Similarly, employing the projection formula in the Krein space, the optimal estimator 𝐱̂‌(𝑖,1)(𝑖=𝑘_𝑑+1,…,𝑘) can be computed by 𝐱̂‌(𝑖+1,1)=𝐴𝐱̂‌(𝑖,1)+𝐴_𝑑𝐱̂‌𝑖_𝑑,2+𝑓𝑖,𝐳_𝑓̆‌(𝑖∣𝑖),𝐮(𝑖)+ℎ𝑖,𝐳_ℎ̆‌𝑖_𝑑∣𝑖,𝐮(𝑖)+𝐾₁(𝑖)𝐲̃‌₁(𝑖,1)+𝐴^{𝑑𝑘𝑑−1}_{𝑗=𝑖𝑑}𝑃₂𝑖_𝑑,𝑗𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑗,2)𝐲̃‌₂(𝑗,2)+𝐴^𝑑𝑖−1_{𝑗=𝑘𝑑}𝑃₁𝑖_𝑑,𝑗𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑗,1)𝐲̃‌₁(𝑗,1),𝐱̂‌𝑘_𝑑,1=𝐱̂‌𝑘_𝑑,2,(3.50) where𝐶₁=𝐶^𝑇𝐹^𝑇𝐿^𝑇^𝑇,𝑃₁(𝑖,𝑗)=⎧⎪⎨⎪⎩⟨𝐞(𝑖,2),𝐞(𝑗,1)⟩,𝑖<𝑘_𝑑,⟨𝐞(𝑖,1),𝐞(𝑗,1)⟩,𝑖≥𝑘_𝑑,𝐞(𝑖,1)=𝐱(𝑖)−𝐱̂‌(𝑖,1),𝑅𝑦₁̃(𝑖,1)=𝐶₁𝑃₁(𝑖,𝑖)𝐶^𝑇₁+𝑄_𝑣1(𝑖),𝑄_𝑣1(𝑖)=diag𝐼,−𝛼⁻²𝐼,−𝛾²𝐼,𝐾₁(𝑖)=𝐴𝑃₁(𝑖,𝑖)𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑖,1)+𝐴_𝑑𝑃₁𝑖_𝑑,𝑖𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑖,1).(3.51) Then, from (3.7) and (3.50), we can yield𝐞(𝑖+1,1)=𝐴𝐞(𝑖,1)+𝐴_𝑑𝐞𝑖_𝑑,2+𝐵𝐰(𝑖)−𝐾₁(𝑖)𝐲̃‌₁(𝑖,1)−𝐴^{𝑑𝑘𝑑−1}_{𝑗=𝑖𝑑}𝑃₂𝑖_𝑑,𝑗𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑗,2)𝐲̃‌₂(𝑗,2)−𝐴^𝑑𝑖−1_{𝑗=𝑘𝑑}𝑃₁𝑖_𝑑,𝑗𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑗,1)𝐲̃‌₁(𝑗,1).(3.52) Thus, we obtain that(1)if 𝑖−𝑗≥𝑘_𝑑, we have 𝑃₁(𝑖−𝑗,𝑖+1)=⟨𝐞(𝑖−𝑗,1),𝐞(𝑖+1,1)⟩=𝑃₁(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₁𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₁(𝑖−𝑗,𝑖)𝐶^𝑇₁𝐾^𝑇₁(𝑖)−^𝑖−1_{𝑡=𝑖−𝑗}𝑃₁(𝑖−𝑗,𝑡)𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑡,1)𝐶₁𝑃^𝑇₁𝑖_𝑑,𝑡𝐴^𝑇_𝑑,(3.53)(2)if 𝑖−𝑗<𝑘_𝑑, we have𝑃₁(𝑖−𝑗,𝑖+1)=⟨𝐞(𝑖−𝑗,2),𝐞(𝑖+1,1)⟩=𝑃₁(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₂𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₁(𝑖−𝑗,𝑖)𝐶^𝑇₁𝐾^𝑇₁(𝑖)−^𝑘𝑑−1_{𝑡=𝑖−𝑗}𝑃₂(𝑖−𝑗,𝑡)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑡,2)𝐶₂𝑃^𝑇₂𝑖_𝑑,𝑡𝐴^𝑇_𝑑−^𝑖−1_{𝑡=𝑘𝑑}𝑃₁(𝑖−𝑗,𝑡)𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑡,1)𝐶₁𝑃^𝑇₁𝑖_𝑑,𝑡𝐴^𝑇_𝑑,(3.54)𝑃₁(𝑖+1,𝑖+1)=⟨𝐞(𝑖−𝑗,2),𝐞(𝑖+1,1)⟩=𝐴𝑃₁(𝑖,𝑖+1)+𝐴_𝑑𝑃₁𝑖_𝑑,𝑖+1+𝐵𝑄_𝑤(𝑖)𝐵^𝑇.(3.55) It follows from (3.53), (3.54), and (3.55) that 𝑃₁(𝑖,𝑖)(𝑖=𝑘_𝑑+1,…,𝑘) can be computed by𝑃₁(𝑖−𝑗,𝑖+1)=𝑃₁(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₂𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₁(𝑖−𝑗,𝑖)𝐶^𝑇₁𝐾^𝑇₁(𝑖)−^𝑘𝑑−1_{𝑡=𝑖−𝑗}𝑃₂(𝑖−𝑗,𝑡)𝐶^𝑇₂𝑅⁻¹𝑦₂̃(𝑡,2)𝐶₂𝑃^𝑇₂𝑖_𝑑,𝑡𝐴^𝑇_𝑑−^𝑖−1_{𝑡=𝑘𝑑}𝑃₁(𝑖−𝑗,𝑡)𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑡,1)𝐶₁𝑃^𝑇₁𝑖_𝑑,𝑡𝐴^𝑇_𝑑,𝑖−𝑗<𝑘_𝑑,𝑃₁(𝑖−𝑗,𝑖+1)=𝑃₁(𝑖−𝑗,𝑖)𝐴^𝑇+𝑃^𝑇₁𝑖_𝑑,𝑖−𝑗𝐴^𝑇_𝑑−𝑃₁(𝑖−𝑗,𝑖)𝐶^𝑇₁𝐾^𝑇₁(𝑖)−^𝑖−1_{𝑡=𝑖−𝑗}𝑃₁(𝑖−𝑗,𝑡)𝐶^𝑇₁𝑅⁻¹𝑦₁̃(𝑡,1)𝐶₁𝑃^𝑇₁𝑖_𝑑,𝑡𝐴^T_𝑑,𝑖−𝑗≥𝑘_𝑑,𝑃₁(𝑖+1,𝑖+1)=𝐴𝑃₁(𝑖,𝑖+1)+𝐴_𝑑𝑃₁𝑖_𝑑,𝑖+1+𝐵𝑄_𝑤(