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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 970978, 22 pages
http://dx.doi.org/10.1155/2011/970978
Research Article

๐ปโˆž Estimation for a Class of Lipschitz Nonlinear Discrete-Time Systems with Time Delay

School of Control Science and Engineering, Shandong University, 17923 Jingshi Road, Jinan 250061, China

Received 27 December 2010; Accepted 18 May 2011

Academic Editor: Elenaย Braverman

Copyright ยฉ 2011 Huihong Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The issue of ๐ปโˆž estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating the ๐ปโˆž 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 ๐ปโˆž 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 [1โ€“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 [2โ€“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 [5โ€“7]. In this paper, the problem of ๐ปโˆž 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 [8โ€“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], ๐ปโˆž optimization techniques [10โ€“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 ๐ปโˆž 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 ๐ปโˆž and ๐ฟ2โˆ’๐ฟโˆž 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 ๐ปโˆž estimation problem is studied for a class of Lipschitz nonlinear discrete time-delay systems with disturbance input. Inspired by the recent study on ๐ปโˆž 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 ๐ปโˆž 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 ๐ปโˆž 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 ๐ปโˆž estimation problem are introduced. In Section 3, a partially equivalent Krein space problem is constructed, the ๐ปโˆž 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; โ„๐‘› denotes the real ๐‘›-dimensional Euclidean space; โ€–โ‹…โ€– denotes the Euclidean norm; ๐œƒ(๐‘˜)โˆˆ2[0,๐‘] means โˆ‘๐‘๐‘˜=0(๐œƒ๐‘‡(๐‘˜)๐œƒ(๐‘˜))<โˆž; the superscripts โ€œโˆ’1โ€ and โ€œ๐‘‡โ€ stand for the inverse and transpose of a matrix, resp.; ๐ผ is the identity matrix with appropriate dimensions; For a real matrix, ๐‘ƒ>0 (๐‘ƒ<0, resp.) means that ๐‘ƒ is symmetric and positive (negative, resp.) definite; โŸจโˆ—,โˆ—โŸฉ denotes the inner product in the Krein space; diag{โ‹ฏ} denotes a block-diagonal matrix; โ„’{โ‹ฏ} denotes the linear space spanned by sequence {โ‹ฏ}.

2. System Model and Problem Formulation

Consider a class of nonlinear systems described by the following equations:๐‘ฅ(๐‘˜+1)=๐ด๐‘ฅ(๐‘˜)+๐ด๐‘‘๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ+๐‘“(๐‘˜,๐น๐‘ฅ(๐‘˜),๐‘ข(๐‘˜))+โ„Ž๎€ท๐‘˜,๐ป๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ,๐‘ข(๐‘˜)๎€ธ+๐ต๐‘ค(๐‘˜),๐‘ฆ(๐‘˜)=๐ถ๐‘ฅ(๐‘˜)+๐‘ฃ(๐‘˜),๐‘ง(๐‘˜)=๐ฟ๐‘ฅ(๐‘˜),(2.1) where ๐‘˜๐‘‘=๐‘˜โˆ’๐‘‘, and the positive integer ๐‘‘ denotes the known state delay; ๐‘ฅ(๐‘˜)โˆˆโ„๐‘› is the state, ๐‘ข(๐‘˜)โˆˆโ„๐‘ is the measurable information, ๐‘ค(๐‘˜)โˆˆโ„๐‘ž and ๐‘ฃ(๐‘˜)โˆˆโ„๐‘š are the disturbance input belonging to ๐‘™2[0,๐‘], ๐‘ฆ(๐‘˜)โˆˆโ„๐‘š is the measurement output, and ๐‘ง(๐‘˜)โˆˆโ„๐‘Ÿ is the signal to be estimated; the initial condition ๐‘ฅ0(๐‘ )(๐‘ =โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0) is unknown; the matrices ๐ดโˆˆโ„๐‘›ร—๐‘›, ๐ด๐‘‘โˆˆโ„๐‘›ร—๐‘›, ๐ตโˆˆโ„๐‘›ร—๐‘ž, ๐ถโˆˆโ„๐‘šร—๐‘› and ๐ฟโˆˆโ„๐‘Ÿร—๐‘›, are real and known constant matrices.

In addition, ๐‘“(๐‘˜,๐น๐‘ฅ(๐‘˜),๐‘ข(๐‘˜)) and โ„Ž(๐‘˜,๐ป๐‘ฅ(๐‘˜๐‘‘),๐‘ข(๐‘˜)) are assumed to satisfy the following Lipschitz conditions:โ€–๐‘“(๐‘˜,๐น๐‘ฅ(๐‘˜),๐‘ข(๐‘˜))โˆ’๐‘“(๐‘˜,๐นฬ†๐‘ฅ(๐‘˜),๐‘ข(๐‘˜))โ€–โ‰ค๐›ผโ€–๐น(๐‘ฅ(๐‘˜)โˆ’ฬ†๐‘ฅ(๐‘˜))โ€–,โ€–โ„Ž๎€ท๐‘˜,๐ป๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ,๐‘ข(๐‘˜)๎€ธโˆ’โ„Ž๎€ท๐‘˜,๐ปฬ†๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ,๐‘ข(๐‘˜)๎€ธโ€–โ‰ค๐›ฝโ€–๐ป๎€ท๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธโˆ’ฬ†๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ๎€ธโ€–,(2.2) for all ๐‘˜โˆˆ{0,1,โ€ฆ,๐‘}, ๐‘ข(๐‘˜)โˆˆโ„๐‘ and ๐‘ฅ(๐‘˜),ฬ†๐‘ฅ(๐‘˜),๐‘ฅ(๐‘˜๐‘‘),ฬ†๐‘ฅ(๐‘˜๐‘‘)โˆˆโ„๐‘›. where ๐›ผ>0 and ๐›ฝ>0 are known Lipschitz constants, and ๐น, ๐ป are real matrix with appropriate dimension.

The ๐ปโˆž estimation problem under investigation is stated as follows. Given the desired noise attenuation level ๐›พ>0 and the observation {๐‘ฆ(๐‘—)}๐‘˜๐‘—=0, find an estimate ฬ†๐‘ง(๐‘˜โˆฃ๐‘˜) of the signal ๐‘ง(๐‘˜), if it exists, such that the following inequality is satisfied:sup๎€ท๐‘ฅ0,๐‘ค,๐‘ฃ๎€ธโ‰ 0โˆ‘๐‘๐‘˜=0โ€–ฬ†๐‘ง(๐‘˜โˆฃ๐‘˜)โˆ’๐‘ง(๐‘˜)โ€–2โˆ‘0๐‘˜=โˆ’๐‘‘โ€–โ€–๐‘ฅ0(๐‘˜)โ€–โ€–2ฮ โˆ’1(๐‘˜)+โˆ‘๐‘๐‘˜=0โ€–๐‘ค(๐‘˜)โ€–2+โˆ‘๐‘๐‘˜=0โ€–๐‘ฃ(๐‘˜)โ€–2<๐›พ2,(2.3) where ฮ (๐‘˜)(๐‘˜=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0) is a given positive definite matrix function which reflects the relative uncertainty of the initial state ๐‘ฅ0(๐‘˜)(๐‘˜=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0) to the input and measurement noises.

Remark 2.1. For the sake of simplicity, the initial state estimate ฬ‚๐‘ฅ0(๐‘˜)(๐‘˜=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,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 ๐ปโˆž estimator for Lipschitz nonlinear systems. To begin with, the ๐ปโˆž 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 {๐‘ฆ(๐‘–)}๐‘˜๐‘–=0 at the time step ๐‘˜. 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 ๐ปโˆž 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 to0๎“๐‘˜=โˆ’๐‘‘โ€–โ€–๐‘ฅ0(๐‘˜)โ€–โ€–2ฮ โˆ’1(๐‘˜)+๐‘๎“๐‘˜=0โ€–๐‘ค(๐‘˜)โ€–2+๐‘๎“๐‘˜=0โ€–๐‘ฃ(๐‘˜)โ€–2โˆ’๐›พโˆ’2๐‘๎“๐‘˜=0โ€–โ€–๐‘ฃ๐‘ง(๐‘˜)โ€–โ€–2๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œโ‰œ๐ฝโˆ—๐‘>0,โˆ€๎€ท๐‘ฅ0,๐‘ค,๐‘ฃ๎€ธโ‰ 0,(3.1) where ๐‘ฃ๐‘ง(๐‘˜)=ฬ†๐‘ง(๐‘˜โˆฃ๐‘˜)โˆ’๐‘ง(๐‘˜).

Moreover, we denote๐‘ง๐‘“(๐‘˜)=๐น๐‘ฅ(๐‘˜),ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜)=๐นฬ†๐‘ฅ(๐‘˜โˆฃ๐‘˜),๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ=๐ป๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ,ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ=๐ปฬ†๐‘ฅ๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,(3.2) where ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜) and ฬ†๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) denote the optimal estimation of ๐‘ง๐‘“(๐‘˜) and ๐‘งโ„Ž(๐‘˜๐‘‘) based on the observation {๐‘ฆ(๐‘—)}๐‘˜๐‘—=0, respectively. And, let๐‘ค๐‘“(๐‘˜)=๐‘“๎€ท๐‘˜,๐‘ง๐‘“(๐‘˜),๐‘ข(๐‘˜)๎€ธโˆ’๐‘“๎€ท๐‘˜,ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜),๐‘ข(๐‘˜)๎€ธ,๐‘คโ„Ž๎€ท๐‘˜๐‘‘๎€ธ=โ„Ž๎€ท๐‘˜,๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ,๐‘ข(๐‘˜)๎€ธโˆ’โ„Ž๎€ท๐‘˜,ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,๐‘ข(๐‘˜)๎€ธ,๐‘ฃ๐‘ง๐‘“(๐‘˜)=ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜)โˆ’๐‘ง๐‘“(๐‘˜),๐‘ฃ๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ=ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธโˆ’๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ.(3.3) From the Lipschitz conditions (2.2), we derive that๐ฝโˆ—๐‘+๐‘๎“๐‘˜=0โ€–โ€–๐‘ค๐‘“(๐‘˜)โ€–โ€–2+๐‘๎“๐‘˜=0โ€–โ€–๐‘คโ„Ž๎€ท๐‘˜๐‘‘๎€ธโ€–โ€–2โˆ’๐›ผ2๐‘๎“๐‘˜=0โ€–โ€–๐‘ฃ๐‘ง๐‘“(๐‘˜)โ€–โ€–2โˆ’๐›ฝ2๐‘๎“๐‘˜=0โ€–โ€–๐‘ฃ๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธโ€–โ€–2๎„ฟ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…ƒ๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…€๎…Œโ‰œ๐ฝ๐‘โ‰ค๐ฝโˆ—๐‘.(3.4) Note that the left side of (3.1) and (3.4), ๐ฝ๐‘, can be recast into the form๐ฝ๐‘=0๎“๐‘˜=โˆ’๐‘‘โ€–โ€–๐‘ฅ0(๐‘˜)โ€–โ€–2ฮ โˆ’1(๐‘˜)+๐‘๎“๐‘˜=0โ€–โ€–๐‘ค(๐‘˜)โ€–โ€–2+๐‘๎“๐‘˜=0โ€–๐‘ฃ(๐‘˜)โ€–2โˆ’๐›พโˆ’2๐‘๎“๐‘˜=0โ€–โ€–๐‘ฃ๐‘ง(๐‘˜)โ€–โ€–2โˆ’๐›ผ2๐‘๎“๐‘˜=0โ€–โ€–๐‘ฃ๐‘ง๐‘“(๐‘˜)โ€–โ€–2โˆ’๐›ฝ2๐‘๎“๐‘˜=๐‘‘โ€–โ€–๐‘ฃ๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธโ€–โ€–2,(3.5) where ฮ (๐‘˜)=โŽงโŽชโŽจโŽชโŽฉ๎€ทฮ โˆ’1(๐‘˜)โˆ’๐›ฝ2๐ป๐‘‡๐ป๎€ธโˆ’1,๐‘˜=โˆ’๐‘‘,โ€ฆ,โˆ’1,ฮ (๐‘˜),๐‘˜=0,๐‘ค(๐‘˜)=๎€บ๐‘ค๐‘‡(๐‘˜)๐‘ค๐‘‡๐‘“(๐‘˜)๐‘ค๐‘‡โ„Ž๎€ท๐‘˜๐‘‘๎€ธ๎€ป๐‘‡.(3.6)

Since ๐ฝ๐‘โ‰ค๐ฝโˆ—๐‘, it is natural to see that if ๐ฝ๐‘>0 then the ๐ปโˆž estimation problem (2.3) is satisfied, that is, ๐ฝโˆ—๐‘>0. Hence, the ๐ปโˆž estimation problem (2.3) can be converted into finding the estimate sequence {{ฬ†๐‘ง(๐‘˜โˆฃ๐‘˜)}๐‘๐‘˜=0;{ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜)}๐‘๐‘˜=0;{ฬ†๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜)}๐‘๐‘˜=๐‘‘} such that ๐ฝ๐‘ has a minimum with respect to {๐‘ฅ0,๐‘ค} and the minimum of ๐ฝ๐‘ is positive. As mentioned in [25, 26], the formulated ๐ปโˆž estimation problem can be solved by employing the Krein space approach.

Introduce the following Krein space stochastic system๐ฑ(๐‘˜+1)=๐ด๐ฑ(๐‘˜)+๐ด๐‘‘๐ฑ๎€ท๐‘˜๐‘‘๎€ธ+๐‘“๎€ท๐‘˜,ฬ†โ€Œ๐ณ๐‘“(๐‘˜โˆฃ๐‘˜),๐ฎ(๐‘˜)๎€ธ+โ„Ž๎€ท๐‘˜,ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,๐ฎ(๐‘˜)๎€ธ+๐ต๐ฐ(๐‘˜),๐ฒ(๐‘˜)=๐ถ๐ฑ(๐‘˜)+๐ฏ(๐‘˜),ฬ†โ€Œ๐ณ๐‘“(๐‘˜โˆฃ๐‘˜)=๐น๐ฑ(๐‘˜)+๐ฏ๐‘ง๐‘“(๐‘˜),ฬ†โ€Œ๐ณ(๐‘˜โˆฃ๐‘˜)=๐ฟ๐ฑ(๐‘˜)+๐ฏ๐‘ง(๐‘˜),ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ=๐ป๐ฑ๎€ท๐‘˜๐‘‘๎€ธ+๐ฏ๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ,๐‘˜โ‰ฅ๐‘‘,(3.7) where ๐ต=[๐ต๐ผ๐ผ]; the initial state ๐ฑ0(๐‘ )(๐‘ =โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0) and ๐ฐ(๐‘˜), ๐ฏ(๐‘˜), ๐ฏ๐‘ง๐‘“(๐‘˜), ๐ฏ๐‘ง(๐‘˜) and ๐ฏ๐‘งโ„Ž(๐‘˜) are mutually uncorrelated white noises with zero means and known covariance matrices ฮ (๐‘ ), ๐‘„๐‘ค(๐‘˜)=๐ผ, ๐‘„๐‘ฃ(๐‘˜)=๐ผ, ๐‘„๐‘ฃ๐‘ง๐‘“(๐‘˜)=โˆ’๐›ผโˆ’2๐ผ, ๐‘„๐‘ฃ๐‘ง(๐‘˜)=โˆ’๐›พ2๐ผ, and ๐‘„๐‘ฃ๐‘งโ„Ž(๐‘˜)=โˆ’๐›ฝโˆ’2๐ผ; ฬ†โ€Œ๐ณ๐‘“(๐‘˜โˆฃ๐‘˜), ฬ†โ€Œ๐ณ(๐‘˜โˆฃ๐‘˜) and ฬ†โ€Œ๐ณโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) are regarded as the imaginary measurement at time ๐‘˜ for the linear combination ๐น๐ฑ(๐‘˜), ๐ฟ๐ฑ(๐‘˜), and ๐ป๐ฑ(๐‘˜๐‘‘), respectively.

Let๐ฒ๐‘ง(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉ๎€บ๐ฒ๐‘‡(๐‘˜)ฬ†โ€Œ๐ณ๐‘‡๐‘š(๐‘˜โˆฃ๐‘˜)๎€ป๐‘‡,0โ‰ค๐‘˜<๐‘‘,๎€บ๐ฒ๐‘‡(๐‘˜)ฬ†โ€Œ๐ณ๐‘‡๐‘š(๐‘˜โˆฃ๐‘˜)ฬ†โ€Œ๐ณ๐‘‡โ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ๎€ป๐‘‡,๐‘˜โ‰ฅ๐‘‘,๐ฏ๐‘ง,๐‘Ž(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉ๎‚ƒ๐ฏ๐‘‡(๐‘˜)๐ฏ๐‘‡๐‘ง๐‘“(๐‘˜)๐ฏ๐‘‡๐‘ง(๐‘˜)๎‚„๐‘‡,0โ‰ค๐‘˜<๐‘‘,๎‚ƒ๐ฏ๐‘‡(๐‘˜)๐ฏ๐‘‡๐‘ง๐‘“(๐‘˜)๐ฏ๐‘‡๐‘ง(๐‘˜)๐ฏ๐‘‡๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ๎‚„๐‘‡,๐‘˜โ‰ฅ๐‘‘,ฬ†โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜)=๎€บฬ†โ€Œ๐ณ๐‘‡๐‘“(๐‘˜โˆฃ๐‘˜)ฬ†โ€Œ๐ณ๐‘‡(๐‘˜โˆฃ๐‘˜)๎€ป๐‘‡.(3.8)

Definition 3.1. The estimator ฬ‚โ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1) denotes the optimal estimation of ๐ฒ(๐‘–) given the observation โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0}; the estimator ฬ‚โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–) denotes the optimal estimation of ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–) given the observation โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–)}; the estimator ฬ‚โ€Œ๐ณโ„Ž(๐‘–๐‘‘โˆฃ๐‘–) denotes the optimal estimation of ฬ†โ€Œ๐ณโ„Ž(๐‘–๐‘‘โˆฃ๐‘–) given the observation โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–),ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)}.

Furthermore, introduce the following stochastic vectors and the corresponding covariance matricesฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1)=๐ฒ(๐‘–)โˆ’ฬ‚โ€Œ๐ฒ(๐‘–๐‘–โˆ’1),๐‘…ฬƒ๐‘ฆ(๐‘–๐‘–โˆ’1)=โŸจฬƒโ€Œ๐ฒ(๐‘–๐‘–โˆ’1),ฬƒโ€Œ๐ฒ(๐‘–๐‘–โˆ’1)โŸฉ,ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)=ฬ†โ€Œ๐ณ๐‘š(๐‘–๐‘–)โˆ’ฬ‚โ€Œ๐ณ๐‘š(๐‘–๐‘–),๐‘…ฬƒ๐‘ง๐‘š(๐‘–๐‘–)=โŸจฬƒโ€Œ๐ณ๐‘š(๐‘–๐‘–),ฬƒโ€Œ๐ณ๐‘š(๐‘–๐‘–)โŸฉ,ฬƒโ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ=ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘๐‘–๎€ธโˆ’ฬ‚โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘๐‘–๎€ธ,๐‘…ฬƒ๐‘งโ„Ž๎€ท๐‘–๐‘‘๐‘–๎€ธ=๎ซฬƒโ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘๐‘–๎€ธ,ฬƒโ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘๐‘–๎€ธ๎ฌ.(3.9) And, denoteฬƒโ€Œ๐ฒ๐‘ง(๐‘–)=โŽงโŽชโŽจโŽชโŽฉ๎€บฬƒโ€Œ๐ฒ๐‘‡(๐‘–โˆฃ๐‘–โˆ’1)ฬƒโ€Œ๐ณ๐‘‡๐‘š(๐‘–โˆฃ๐‘–)๎€ป๐‘‡,0โ‰ค๐‘–<๐‘‘,๎€บฬƒโ€Œ๐ฒ๐‘‡(๐‘–โˆฃ๐‘–โˆ’1)ฬƒโ€Œ๐ณ๐‘‡๐‘š(๐‘–โˆฃ๐‘–)ฬƒโ€Œ๐ณ๐‘‡โ„Ž(๐‘–๐‘‘โˆฃ๐‘–)๎€ป๐‘‡,๐‘–โ‰ฅ๐‘‘,๐‘…ฬƒ๐‘ฆ๐‘ง(๐‘–)=๎ซฬƒโ€Œ๐ฒ๐‘ง(๐‘–),ฬƒโ€Œ๐ฒ๐‘ง(๐‘–)๎ฌ.(3.10)

For calculating the minimum of ๐ฝ๐‘, we present the following Lemma 3.2.

Lemma 3.2. {{ฬƒโ€Œ๐ฒ๐‘ง(๐‘–)}๐‘˜๐‘–=0} is the innovation sequence which spans the same linear space as that of โ„’{{๐ฒ๐‘ง(๐‘–)}๐‘˜๐‘–=0}.

Proof. From Definition 3.1 and (3.9), ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1), ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–) and ฬƒโ€Œ๐ณโ„Ž(๐‘–๐‘‘โˆฃ๐‘–) are the linear combination of the observation sequence {{๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–)}, {{๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–),ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)}, and {{๐ฒ๐‘ง(๐‘—)}๐‘–๐‘—=0}, respectively. Conversely, ๐ฒ(๐‘–), ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–) and ฬ†โ€Œ๐ณโ„Ž(๐‘–๐‘‘โˆฃ๐‘–) can be given by the linear combination of {{ฬƒโ€Œ๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1)},{{ฬƒโ€Œ๐ฒ๐‘ง(๐‘—)}๐‘–โˆ’1๐‘—=0;ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1),ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)} and {{ฬƒโ€Œ๐ฒ๐‘ง(๐‘—)}๐‘–๐‘—=0}, respectively. Hence, โ„’๎‚†๎€ฝฬƒโ€Œ๐ฒ๐‘ง(๐‘–)๎€พ๐‘˜๐‘–=0๎‚‡=โ„’๎‚†๎€ฝ๐ฒ๐‘ง(๐‘–)๎€พ๐‘˜๐‘–=0๎‚‡.(3.11) It is also shown by (3.9) that ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1), ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–) and ฬƒโ€Œ๐ณโ„Ž(๐‘–๐‘‘โˆฃ๐‘–) satisfy ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1)โŸ‚โ„’๎‚†๎€ฝ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0๎‚‡,ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)โŸ‚โ„’๎‚†๎€ฝ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–)๎‚‡,ฬƒโ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธโŸ‚โ„’๎‚†๎€ฝ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0;๐ฒ(๐‘–),ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)๎‚‡.(3.12) Consequently, ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1)โŸ‚โ„’๎‚†๎€ฝฬƒโ€Œ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0๎‚‡,ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)โŸ‚โ„’๎‚†๎€ฝฬƒโ€Œ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0;ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1)๎‚‡,ฬƒโ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธโŸ‚โ„’๎‚†๎€ฝฬƒโ€Œ๐ฒ๐‘ง(๐‘—)๎€พ๐‘–โˆ’1๐‘—=0;ฬƒโ€Œ๐ฒ(๐‘–โˆฃ๐‘–โˆ’1),ฬƒโ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)๎‚‡.(3.13) This completes the proof.

Now, an existence condition and a solution to the minimum of ๐ฝ๐‘ are derived as follows.

Theorem 3.3. Consider system (2.1), given a scalar ๐›พ>0 and the positive definite matrix ฮ (k)(๐‘˜=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0), then ๐ฝ๐‘ has the minimum if only if ๐‘…ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)>0,0โ‰ค๐‘˜โ‰ค๐‘,๐‘…ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)<0,0โ‰ค๐‘˜โ‰ค๐‘,๐‘…ฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ<0,๐‘‘โ‰ค๐‘˜โ‰ค๐‘.(3.14) In this case the minimum value of ๐ฝ๐‘ is given by min๐ฝ๐‘=๐‘๎“๐‘˜=0ฬƒ๐‘ฆ๐‘‡(๐‘˜โˆฃ๐‘˜โˆ’1)๐‘…โˆ’1ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)+๐‘๎“๐‘˜=0ฬƒ๐‘ง๐‘‡๐‘š(๐‘˜โˆฃ๐‘˜)๐‘…โˆ’1ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)+๐‘๎“๐‘˜=๐‘‘ฬƒ๐‘ง๐‘‡โ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ๐‘…โˆ’1ฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,(3.15) where ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)=๐‘ฆ(๐‘˜)โˆ’ฬ‚๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1),ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)=ฬ†๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)โˆ’ฬ‚๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜),ฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ=ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธโˆ’ฬ‚๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,ฬ†๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)=๎€บฬ†๐‘ง๐‘‡๐‘“(๐‘˜โˆฃ๐‘˜)ฬ†๐‘ง๐‘‡(๐‘˜โˆฃ๐‘˜)๎€ป๐‘‡,(3.16)ฬ‚๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1) is obtained from the Krein space projection of ๐ฒ(๐‘˜) onto โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘˜โˆ’1๐‘—=0}, ฬ‚๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜) is obtained from the Krein space projection of ฬ†โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜) onto โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘˜โˆ’1๐‘—=0;๐ฒ(๐‘˜)}, and ฬ‚๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) is obtained from the Krein space projection of ฬ†โ€Œ๐ณโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) onto โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘˜โˆ’1๐‘—=0;๐ฒ(๐‘˜),ฬ†โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜)}.

Proof. Based on the definition (3.2) and (3.3), the state equation in system (2.1) can be rewritten as ๐‘ฅ(๐‘˜+1)=๐ด๐‘ฅ(๐‘˜)+๐ด๐‘‘๐‘ฅ๎€ท๐‘˜๐‘‘๎€ธ+๐‘“๎€ท๐‘˜,ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜),๐‘ข(๐‘˜)๎€ธ+โ„Ž๎€ท๐‘˜,ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,๐‘ข(๐‘˜)๎€ธ+๐ต๐‘ค(๐‘˜).(3.17) In this case, it is assumed that ๐‘“(๐‘˜,ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜),๐‘ข(๐‘˜)) and โ„Ž(๐‘˜,ฬ†๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜),๐‘ข(๐‘˜)) are known at time ๐‘˜. Then, we define ๐‘ฆ๐‘ง(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉ๎€บ๐‘ฆ๐‘‡(๐‘˜)ฬ†๐‘ง๐‘‡๐‘“(๐‘˜โˆฃ๐‘˜)ฬ†๐‘ง๐‘‡(๐‘˜โˆฃ๐‘˜)๎€ป๐‘‡,0โ‰ค๐‘˜<๐‘‘,๎€บ๐‘ฆ๐‘‡(๐‘˜)ฬ†๐‘ง๐‘‡๐‘“(๐‘˜โˆฃ๐‘˜)ฬ†๐‘ง๐‘‡(๐‘˜โˆฃ๐‘˜)ฬ†๐‘ง๐‘‡โ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ๎€ป๐‘‡,๐‘˜โ‰ฅ๐‘‘.(3.18) By introducing an augmented state ๐‘ฅ๐‘Ž(๐‘˜)=๎€บ๐‘ฅ๐‘‡(๐‘˜)๐‘ฅ๐‘‡(๐‘˜โˆ’1)โ‹ฏ๐‘ฅ๐‘‡(๐‘˜โˆ’๐‘‘)๎€ป๐‘‡,(3.19) we obtain an augmented state-space model ๐‘ฅ๐‘Ž(๐‘˜+1)=๐ด๐‘Ž๐‘ฅ๐‘Ž(๐‘˜)+๐ต๐‘ข,๐‘Ž๐‘ข(๐‘˜)+๐ต๐‘Ž๐‘ค(๐‘˜),๐‘ฆ๐‘ง(๐‘˜)=๐ถ๐‘ง,๐‘Ž(๐‘˜)๐‘ฅ๐‘Ž(๐‘˜)+๐‘ฃ๐‘ง,๐‘Ž(๐‘˜),(3.20) where ๐ด๐‘Ž=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ด0โ‹ฏ0๐ด๐‘‘๐ผ0โ‹ฏ000๐ผโ‹ฏ00โ‹ฎโ‹ฎโ‹ฑโ‹ฎโ‹ฎ00โ‹ฏ๐ผ0โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,๐ต๐‘ข,๐‘Ž=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ผ๐ผ0000โ‹ฎโ‹ฎ00โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,๐ต๐‘Ž=โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ต00โ‹ฎ0โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,(3.21)C๐‘ง,๐‘Ž(๐‘˜)=โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉโŽกโŽขโŽขโŽขโŽฃ๐ถ0โ‹ฏ0๐น0โ‹ฏ0๐ฟ0โ‹ฏ0โŽคโŽฅโŽฅโŽฅโŽฆ,0โ‰ค๐‘˜<๐‘‘,โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ถ0โ‹ฏ0๐น0โ‹ฏ0๐ฟ0โ‹ฏ00โ‹ฏ0๐ปโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,๐‘˜โ‰ฅ๐‘‘,๐‘ฃ๐‘ง,๐‘Ž(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉ๎‚ƒ๐‘ฃ๐‘‡(๐‘˜)๐‘ฃ๐‘‡๐‘ง๐‘“(๐‘˜)๐‘ฃ๐‘‡๐‘ง(๐‘˜)๎‚„๐‘‡,0โ‰ค๐‘˜<๐‘‘,๎‚ƒ๐‘ฃ๐‘‡(๐‘˜)๐‘ฃ๐‘‡๐‘ง๐‘“(๐‘˜)๐‘ฃ๐‘‡๐‘ง(๐‘˜)๐‘ฃ๐‘‡๐‘งโ„Ž๎€ท๐‘˜๐‘‘๎€ธ๎‚„๐‘‡,๐‘˜โ‰ฅ๐‘‘,๐‘ข(๐‘˜)=๎€บ๐‘“๐‘‡๎€ท๐‘˜,ฬ†๐‘ง๐‘“(๐‘˜โˆฃ๐‘˜),๐‘ข(๐‘˜)๎€ธโ„Ž๐‘‡๎€ท๐‘˜,ฬ†๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ,๐‘ข(๐‘˜)๎€ธ๎€ป๐‘‡.(1) Additionally, we can rewrite ๐ฝ๐‘ as ๐ฝ๐‘=โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฃ๐‘ง,๐‘Ž๐‘โŽคโŽฅโŽฅโŽฅโŽฆ๐‘‡โŽกโŽขโŽขโŽขโŽฃ๐‘ƒ๐‘Ž(0)000๐ผ000๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘โŽคโŽฅโŽฅโŽฅโŽฆโˆ’1โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฃ๐‘ง,๐‘Ž๐‘โŽคโŽฅโŽฅโŽฅโŽฆ,(3.22) where ๐‘ƒ๐‘Ž(0)=diag๎‚†ฮ (0),ฮ (โˆ’1),โ€ฆ,ฮ (โˆ’๐‘‘)๎‚‡,๐‘ค๐‘=๎€บ๐‘ค๐‘‡(0)๐‘ค๐‘‡(1)โ‹ฏ๐‘ค๐‘‡(๐‘)๎€ป๐‘‡,๐‘ฃ๐‘ง,๐‘Ž๐‘=๎€บ๐‘ฃ๐‘‡๐‘ง,๐‘Ž(0)๐‘ฃ๐‘‡๐‘ง,๐‘Ž(1)โ‹ฏ๐‘ฃ๐‘‡๐‘ง,๐‘Ž(๐‘)๎€ป๐‘‡,๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘=diag๎‚†๐‘„๐‘ฃ๐‘ง,๐‘Ž(0),๐‘„๐‘ฃ๐‘ง,๐‘Ž(1),โ€ฆ,๐‘„๐‘ฃ๐‘ง,๐‘Ž(๐‘)๎‚‡,๐‘„๐‘ฃ๐‘ง,๐‘Ž(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉdiag๎€ฝ๐ผ,โˆ’๐›พ2,โˆ’๐›ผโˆ’2๎€พ,0โ‰ค๐‘˜<๐‘‘,diag๎€ฝ๐ผ,โˆ’๐›พ2,โˆ’๐›ผโˆ’2,โˆ’๐›ฝโˆ’2๎€พ,๐‘˜โ‰ฅ๐‘‘.(3.23) Define the following state transition matrix ฮฆ(๐‘˜+1,๐‘š)=๐ด๐‘Žฮฆ(๐‘˜,๐‘š),ฮฆ(๐‘š,๐‘š)=๐ผ,(3.24) and let ๐‘ฆ๐‘ง๐‘=๎€บ๐‘ฆ๐‘‡๐‘ง(0)๐‘ฆ๐‘‡๐‘ง(1)โ‹ฏ๐‘ฆ๐‘‡๐‘ง(๐‘)๎€ป๐‘‡,๐‘ข๐‘=๎€บ๐‘ข๐‘‡(0)๐‘ข๐‘‡(1)โ‹ฏ๐‘ข๐‘‡(๐‘)๎€ป๐‘‡.(3.25) Using (3.20) and (3.24), we have ๐‘ฆ๐‘ง๐‘=ฮจ0๐‘๐‘ฅ๐‘Ž(0)+ฮจ๐‘ข๐‘๐‘ข๐‘+ฮจ๐‘ค๐‘๐‘ค๐‘+๐‘ฃ๐‘ง,๐‘Ž๐‘,(3.26) where ฮจ0๐‘=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ๐ถ๐‘ง,๐‘Ž(0)ฮฆ(0,0)๐ถ๐‘ง,๐‘Ž(1)ฮฆ(1,0)โ‹ฎ๐ถ๐‘ง,๐‘Ž(๐‘)ฮฆ(๐‘,0)โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,ฮจ๐‘ข๐‘=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ๐œ‘00๐œ‘01โ‹ฏ๐œ‘0๐‘๐œ‘10๐œ‘11โ‹ฏ๐œ‘1๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐œ‘๐‘0๐œ‘๐‘1โ‹ฏ๐œ‘๐‘๐‘โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,๐œ‘๐‘–๐‘—=โŽงโŽชโŽจโŽชโŽฉ๐ถ๐‘ง,๐‘Ž(๐‘–)ฮฆ(๐‘–,๐‘—+1)๐ต๐‘ข,๐‘Ž,๐‘–>๐‘—,0,๐‘–โ‰ค๐‘—.(3.27) The matrix ฮจ๐‘ค๐‘ is derived by replacing ๐ต๐‘ข,๐‘Ž in ฮจ๐‘ข๐‘ with ๐ต๐‘Ž.
Thus, ๐ฝ๐‘ can be reexpressed as ๐ฝ๐‘=โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฆ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ๐‘‡โŽงโŽชโŽจโŽชโŽฉฮ“๐‘โŽกโŽขโŽขโŽขโŽฃ๐‘ƒ๐‘Ž(0)000๐ผ000๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘โŽคโŽฅโŽฅโŽฅโŽฆฮ“๐‘‡๐‘โŽซโŽชโŽฌโŽชโŽญโˆ’1โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฆ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ,(3.28) where ๐‘ฆ๐‘ง๐‘=๐‘ฆ๐‘ง๐‘โˆ’ฮจ๐‘ข๐‘๐‘ข๐‘,ฮ“๐‘=โŽกโŽขโŽขโŽขโŽฃ๐ผ000๐ผ0ฮจ0๐‘ฮจ๐‘ค๐‘๐ผโŽคโŽฅโŽฅโŽฅโŽฆ.(3.29)
Considering the Krein space stochastic system defined by (3.7) and state transition matrix (3.24), we have ๐ฒ๐‘ง๐‘=ฮจ0๐‘๐ฑ๐‘Ž(0)+ฮจ๐‘ข๐‘๐ฎ๐‘+ฮจ๐‘ค๐‘๐ฐ๐‘+๐ฏ๐‘ง,๐‘Ž๐‘,(3.30) where matrices ฮจ0๐‘, ฮจ๐‘ข๐‘, and ฮจ๐‘ค๐‘ are the same as given in (3.26), vectors ๐ฒ๐‘ง๐‘ and ๐ฎ๐‘ are, respectively, defined by replacing Euclidean space element ๐‘ฆ๐‘ง and ๐‘ข in ๐‘ฆ๐‘ง๐‘ and ๐‘ข๐‘ given by (3.25) with the Krein space element ๐ฒ๐‘ง and ๐ฎ, vectors ๐ฐ๐‘ and ๐ฏ๐‘ง,๐‘Ž๐‘ are also defined by replacing Euclidean space element ๐‘ค and ๐‘ฃ๐‘ง,๐‘Ž in ๐‘ค๐‘ and ๐‘ฃ๐‘ง,๐‘Ž๐‘ given by (3.23) with the Krein space element ๐ฐ and ๐ฏ๐‘ง,๐‘Ž, and vector ๐ฑ๐‘Ž(0) is given by replacing Euclidean space element ๐‘ฅ in ๐‘ฅ๐‘Ž(๐‘˜) given by (3.19) with the Krein space element ๐ฑ when ๐‘˜=0.
Using the stochastic characteristic of ๐ฑ๐‘Ž(0), ๐ฐ๐‘ and ๐ฏ๐‘ง,๐‘Ž, we have ๐ฝ๐‘=โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฆ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ๐‘‡๎„”โŽกโŽขโŽขโŽขโŽฃ๐ฑ๐‘Ž(0)๐ฐ๐‘๐ฒ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ,โŽกโŽขโŽขโŽขโŽฃ๐ฑ๐‘Ž(0)๐ฐ๐‘๐ฒ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ๎„•โˆ’1โŽกโŽขโŽขโŽขโŽฃ๐‘ฅ๐‘Ž(0)๐‘ค๐‘๐‘ฆ๐‘ง๐‘โŽคโŽฅโŽฅโŽฅโŽฆ,(3.31) where ๐ฒ๐‘ง๐‘=๐ฒ๐‘ง๐‘โˆ’ฮจ๐‘ข๐‘๐ฎ๐‘.
In the light of Theorem โ€‰2.4.2 and Lemma โ€‰2.4.3 in [26], ๐ฝ๐‘ has a minimum over {๐‘ฅ๐‘Ž(0),๐‘ค๐‘} if and only if ๐‘…๐‘ฆ๐‘ง๐‘=โŸจ๐ฒ๐‘ง๐‘,๐ฒ๐‘ง๐‘โŸฉ and ๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘=โŸจ๐ฏ๐‘ง,๐‘Ž๐‘,๐ฏ๐‘ง,๐‘Ž๐‘โŸฉ have the same inertia. Moreover, the minimum of ๐ฝ๐‘ is given by min๐ฝ๐‘=๐‘ฆ๐‘‡๐‘ง๐‘๐‘…โˆ’1๐‘ฆ๐‘ง๐‘๐‘ฆ๐‘ง๐‘.(3.32)
On the other hand, applying the Krein space projection formula, we have ๐ฒ๐‘ง๐‘=ฮ˜๐‘ฬƒโ€Œ๐ฒ๐‘ง๐‘,(3.33) where ฬƒโ€Œ๐ฒ๐‘ง๐‘=๎€บฬƒโ€Œ๐ฒ๐‘‡๐‘ง(0)ฬƒโ€Œ๐ฒ๐‘‡๐‘ง(1)โ‹ฏฬƒโ€Œ๐ฒ๐‘‡๐‘ง(๐‘)๎€ป๐‘‡,(3.34)ฮ˜๐‘=โŽกโŽขโŽขโŽขโŽขโŽขโŽฃ๐œƒ00๐œƒ01โ‹ฏ๐œƒ0๐‘๐œƒ10๐œƒ11โ‹ฏ๐œƒ1๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐œƒ๐‘0๐œƒ๐‘1โ‹ฏ๐œƒ๐‘๐‘โŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ,๐œƒ๐‘–๐‘—=โŽงโŽชโŽชโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽชโŽชโŽฉ๎ซ๐ฒ๐‘ง(๐‘–),ฬƒโ€Œ๐ฒ๐‘ง(๐‘—)๎ฌ๐‘…โˆ’1ฬƒ๐‘ฆ๐‘ง(๐‘—),๐‘–>๐‘—โ‰ฅ0,โŽกโŽขโŽฃ๐ผ0๐‘š1๐ผโŽคโŽฅโŽฆ,๐‘‘>๐‘–=๐‘—โ‰ฅ0,โŽกโŽขโŽขโŽขโŽฃ๐ผ00๐‘š1๐ผ0๐‘š2๐‘š3๐ผโŽคโŽฅโŽฅโŽฅโŽฆ,๐‘–=๐‘—โ‰ฅ๐‘‘,0,0โ‰ค๐‘–<๐‘—,๐‘š1=๎‚ฌฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–),ฬƒโ€Œ๐ฒ(๐‘—โˆฃ๐‘—โˆ’1)๎‚ญ๐‘…โˆ’1ฬƒ๐‘ฆ(๐‘—โˆฃ๐‘—โˆ’1),๐‘š2=๎‚ฌฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ,ฬƒโ€Œ๐ฒ(๐‘—โˆฃ๐‘—โˆ’1)๎‚ญ๐‘…โˆ’1ฬƒ๐‘ฆ(๐‘—โˆฃ๐‘—โˆ’1),๐‘š3=๎‚ฌฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ,ฬƒโ€Œ๐ณ๐‘š(๐‘—โˆฃ๐‘—)๎‚ญ๐‘…โˆ’1ฬƒ๐‘ง๐‘š(๐‘—โˆฃ๐‘—),๐ฒ๐‘ง(๐‘–)=๐ฒ๐‘ง(๐‘–)โˆ’๐‘๎“๐‘—=0๐œ‘๐‘–๐‘—๐ฎ(๐‘—),ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)=ฬ†โ€Œ๐ณ๐‘š(๐‘–โˆฃ๐‘–)โˆ’๐‘๎“๐‘—=0๐œ‘๐‘š,๐‘–๐‘—๐ฎ(๐‘—),ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ=ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธโˆ’๐‘๎“๐‘—=0๐œ‘โ„Ž,๐‘–๐‘—๐ฎ(๐‘—),(2) where ๐œ‘๐‘š,๐‘–๐‘— is derived by replacing ๐ถ๐‘ง,๐‘Ž in ๐œ‘๐‘–๐‘— with๎€บ๐น0โ‹ฏ0๐ฟ0โ‹ฏ0๎€ป, ๐œ‘โ„Ž,๐‘–๐‘— is derived by replacing ๐ถ๐‘ง,๐‘Ž in ๐œ‘๐‘–๐‘— with [00โ‹ฏ๐ป] Furthermore, it follows from (3.33) that ๐‘…๐‘ฆ๐‘ง๐‘=ฮ˜๐‘๐‘…ฬƒ๐‘ฆ๐‘ง๐‘ฮ˜๐‘‡๐‘,๐‘ฆ๐‘ง๐‘=ฮ˜๐‘ฬƒ๐‘ฆ๐‘ง๐‘,(3.35) where ๐‘…ฬƒ๐‘ฆ๐‘ง๐‘=๎ซฬƒโ€Œ๐ฒ๐‘ง๐‘,ฬƒโ€Œ๐ฒ๐‘ง๐‘๎ฌ,ฬƒ๐‘ฆ๐‘ง๐‘=๎€บฬƒ๐‘ฆ๐‘‡๐‘ง(0)ฬƒ๐‘ฆ๐‘‡๐‘ง(1)โ‹ฏฬƒ๐‘ฆ๐‘‡๐‘ง(๐‘)๎€ป๐‘‡,ฬƒ๐‘ฆ๐‘ง(๐‘–)=โŽงโŽชโŽจโŽชโŽฉ๎€บฬƒ๐‘ฆ๐‘‡(๐‘–โˆฃ๐‘–โˆ’1)ฬƒ๐‘ง๐‘‡๐‘š(๐‘–โˆฃ๐‘–)๎€ป๐‘‡,0โ‰ค๐‘–<๐‘‘,๎€บฬƒ๐‘ฆ๐‘‡(๐‘–โˆฃ๐‘–โˆ’1)ฬƒ๐‘ง๐‘‡๐‘š(๐‘–โˆฃ๐‘–)ฬƒ๐‘ง๐‘‡โ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ๎€ป๐‘‡,๐‘–โ‰ฅ๐‘‘.(3.36) Since matrix ฮ˜๐‘ is nonsingular, it follows from (3.35) that ๐‘…๐‘ฆ๐‘ง๐‘ and ๐‘…ฬƒ๐‘ฆ๐‘ง๐‘ are congruent, which also means that ๐‘…๐‘ฆ๐‘ง๐‘ and ๐‘…ฬƒ๐‘ฆ๐‘ง๐‘ have the same inertia. Note that both ๐‘…ฬƒ๐‘ฆ๐‘ง๐‘ and ๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘ are block-diagonal matrices, and ๐‘…ฬƒ๐‘ฆ๐‘ง(๐‘˜)=โŽงโŽชโŽจโŽชโŽฉdiag๎€ฝ๐‘…ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1),๐‘…ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)๎€พ,0โ‰ค๐‘˜<๐‘‘,diag๎€ฝ๐‘…ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1),๐‘…ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜),๐‘…ฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ๎€พ,๐‘˜โ‰ค๐‘‘,(3.37)๐‘„๐‘ฃ๐‘ง,๐‘Ž(๐‘˜) is given by (3.23). It follows that ๐‘…ฬƒ๐‘ฆ๐‘ง๐‘ and ๐‘„๐‘ฃ๐‘ง,๐‘Ž๐‘ have the same inertia if and only if ๐‘…ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)>0(0โ‰ค๐‘˜โ‰ค๐‘), ๐‘…ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)<0(0โ‰ค๐‘˜โ‰ค๐‘) and ๐‘…ฬƒ๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜)<0(๐‘‘โ‰ค๐‘˜โ‰ค๐‘).
Therefore, ๐ฝ๐‘ subject to system (2.1) with Lipschitz conditions (2.2) has the minimum if and only if ๐‘…ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)>0(0โ‰ค๐‘˜โ‰ค๐‘), ๐‘…ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)<0(0โ‰ค๐‘˜โ‰ค๐‘) and ๐‘…ฬƒ๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜)<0(๐‘‘โ‰ค๐‘˜โ‰ค๐‘). Moreover, the minimum value of ๐ฝ๐‘ can be rewritten as min๐ฝ๐‘=๐‘ฆ๐‘‡๐‘ง๐‘๐‘…โˆ’1๐‘ฆ๐‘ง๐‘๐‘ฆ๐‘ง๐‘=ฬƒ๐‘ฆ๐‘‡๐‘ง๐‘๐‘…โˆ’1ฬƒ๐‘ฆ๐‘ง๐‘ฬƒ๐‘ฆ๐‘ง๐‘=๐‘๎“๐‘˜=0ฬƒ๐‘ฆ๐‘‡(๐‘˜โˆฃ๐‘˜โˆ’1)๐‘…โˆ’1ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1)+๐‘๎“๐‘˜=0ฬƒ๐‘ง๐‘‡๐‘š(๐‘˜โˆฃ๐‘˜)๐‘…โˆ’1ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜)+๐‘๎“๐‘˜=๐‘‘ฬƒ๐‘ง๐‘‡โ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ๐‘…โˆ’1ฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธฬƒ๐‘งโ„Ž๎€ท๐‘˜๐‘‘โˆฃ๐‘˜๎€ธ.(3.38) The proof is completed.

Remark 3.4. Due to the built innovation sequence {{ฬƒโ€Œ๐ฒ๐‘ง(๐‘–)}๐‘˜๐‘–=0} in Lemma 3.2, the form of the minimum on indefinite quadratic form ๐ฝ๐‘ is different from the one given in [26โ€“28]. It is shown from (3.15) that the estimation errors ฬƒ๐‘ฆ(๐‘˜โˆฃ๐‘˜โˆ’1), ฬƒ๐‘ง๐‘š(๐‘˜โˆฃ๐‘˜) and ฬƒ๐‘งโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) are mutually uncorrelated, which will make the design of ๐ปโˆž estimator much easier than the one given in [26โ€“28].

3.2. Solution of the ๐ปโˆž Estimation Problem

In this subsection, the Kalman-like recursive ๐ปโˆž estimator is presented by using orthogonal projection in the Krein space.

Denote๐ฒ0(๐‘–)=๐ฒ(๐‘–),๐ฒ1(๐‘–)=๎€บ๐ฒ๐‘‡(๐‘–)ฬ†โ€Œ๐ณ๐‘‡๐‘š(๐‘–โˆฃ๐‘–)๎€ป๐‘‡,๐ฒ2(๐‘–)=๎€บ๐ฒ๐‘‡(๐‘–)ฬ†โ€Œ๐ณ๐‘‡๐‘š(๐‘–โˆฃ๐‘–)ฬ†โ€Œ๐ณ๐‘‡โ„Ž(๐‘–โˆฃ๐‘–+๐‘‘)๎€ป๐‘‡.(3.39) Observe from (3.8), we have โ„’๎€ฝ๎€ฝ๐ฒ๐‘ง(๐‘–)๎€พ๐‘—๐‘–=0๎€พ=โ„’๎€ฝ๎€ฝ๐ฒ1(๐‘–)๎€พ๐‘—๐‘–=0๎€พ,0โ‰ค๐‘—<๐‘‘,โ„’๎€ฝ๎€ฝ๐ฒ๐‘ง(๐‘–)๎€พ๐‘—๐‘–=0๎€พ=โ„’๎‚†๎€ฝ๐ฒ2(๐‘–)๎€พ๐‘—๐‘‘๐‘–=0๎‚‡;๎‚†๎€ฝ๐ฒ1(๐‘–)๎€พ๐‘—๐‘–=๐‘—๐‘‘+1๎‚‡,๐‘—โ‰ฅ๐‘‘.(3.40)

Definition 3.5. Given ๐‘˜โ‰ฅ๐‘‘, the estimator ฬ‚โ€Œ๐œ‰(๐‘–โˆฃ๐‘—,2) for 0โ‰ค๐‘—<๐‘˜๐‘‘ denotes the optimal estimate of ๐œ‰(๐‘–) given the observation โ„’{{๐ฒ2(๐‘ )}๐‘—๐‘ =0}, and the estimator ฬ‚โ€Œ๐œ‰(๐‘–โˆฃ๐‘—,1) for ๐‘˜๐‘‘โ‰ค๐‘—โ‰ค๐‘˜ denotes the optimal estimate of ๐œ‰(๐‘–) given the observation โ„’{{๐ฒ2(๐‘ )}๐‘˜๐‘‘โˆ’1๐‘ =0;{๐ฒ1(๐œ)}๐‘—๐œ=๐‘˜๐‘‘}. For simplicity, we use ฬ‚โ€Œ๐œ‰(๐‘–,2) to denote ฬ‚โ€Œ๐œ‰(๐‘–โˆฃ๐‘–โˆ’1,2), and use ฬ‚โ€Œ๐œ‰(๐‘–,1) to denote ฬ‚โ€Œ๐œ‰(๐‘–โˆฃ๐‘–โˆ’1,1) throughout the paper.

Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matricesฬƒโ€Œ๐ฒ2(๐‘–,2)=๐ฒ2(๐‘–)โˆ’ฬ‚โ€Œ๐ฒ2(๐‘–,2),๐‘…ฬƒ๐‘ฆ2(๐‘–,2)=โŸจฬƒโ€Œ๐ฒ2(๐‘–,2),ฬƒโ€Œ๐ฒ2(๐‘–,2)โŸฉ,ฬƒโ€Œ๐ฒ1(๐‘–,1)=๐ฒ1(๐‘–)โˆ’ฬ‚โ€Œ๐ฒ1(๐‘–,1),๐‘…ฬƒ๐‘ฆ1(๐‘–,1)=โŸจฬƒโ€Œ๐ฒ1(๐‘–,1),ฬƒโ€Œ๐ฒ1(๐‘–,1)โŸฉ,ฬƒโ€Œ๐ฒ0(๐‘–,0)=๐ฒ0(๐‘–)โˆ’ฬ‚โ€Œ๐ฒ0(๐‘–,1),๐‘…ฬƒ๐‘ฆ0(๐‘–,0)=๎ซฬƒโ€Œ๐ฒ0(i,0),ฬƒโ€Œ๐ฒ0(๐‘–,0)๎ฌ.(3.41) Similar to the proof of Lemma โ€‰2.2.1 in [27], we can obtain that {ฬƒโ€Œ๐ฒ2(0,2),โ€ฆ,ฬƒโ€Œ๐ฒ2(๐‘˜๐‘‘โˆ’1,2);ฬƒโ€Œ๐ฒ1(๐‘˜๐‘‘,1),โ€ฆ,ฬƒโ€Œ๐ฒ1(๐‘˜โˆ’1,1)} is the innovation sequence which is a mutually uncorrelated white noise sequence and spans the same linear space as โ„’{๐ฒ2(0),โ€ฆ,๐ฒ2(๐‘˜๐‘‘โˆ’1);๐ฒ1(๐‘˜๐‘™),โ€ฆ,๐ฒ1(๐‘˜โˆ’1)} or equivalently โ„’{๐ฒ๐‘ง(0),โ€ฆ,๐ฒ๐‘ง(๐‘˜โˆ’1)}.

Applying projection formula in the Krein space, ฬ‚โ€Œ๐ฑ(๐‘–,2)(๐‘–=0,1,โ€ฆ,๐‘˜๐‘‘) is computed recursively as(3.42)ฬ‚โ€Œ๐ฑ(๐‘–+1,2)=๐‘–๎“๐‘—=0โŸจ๐ฑ(๐‘–+1),ฬƒโ€Œ๐ฒ2(๐‘—,2)โŸฉ๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2)=๐ดฬ‚โ€Œ๐ฑ(๐‘–โˆฃ๐‘–,2)+๐ด๐‘‘ฬ‚โ€Œ๐ฑ๎€ท๐‘–๐‘‘โˆฃ๐‘–,2๎€ธ+๐‘“๎€ท๐‘–,ฬ†โ€Œ๐ณ๐‘“(๐‘–โˆฃ๐‘–),๐ฎ(๐‘–)๎€ธ+โ„Ž๎€ท๐‘–,ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ,๐ฎ(๐‘–)๎€ธ,๐‘–=0,1,โ€ฆ,๐‘˜๐‘‘โˆ’1,ฬ‚โ€Œ๐ฑ(๐œ,2)=0,(๐œ=โˆ’๐‘‘,โˆ’๐‘‘+1,โ€ฆ,0).(3.43) Note thatฬ‚โ€Œ๐ฑ(๐‘–โˆฃ๐‘–,2)=ฬ‚โ€Œ๐ฑ(๐‘–,2)+๐‘ƒ2(๐‘–,๐‘–)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘–,2)ฬƒโ€Œ๐ฒ2(๐‘–,2),ฬ‚โ€Œ๐ฑ๎€ท๐‘–๐‘‘โˆฃ๐‘–,2๎€ธ=ฬ‚โ€Œ๐ฑ๎€ท๐‘–๐‘‘,2๎€ธ+๐‘–๎“๐‘—=๐‘–๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2),(3.44) where๐ถ2=๎€บ๐ถ๐‘‡๐น๐‘‡๐ฟ๐‘‡๐ป๐‘‡๎€ป๐‘‡,๐‘ƒ2(๐‘–,๐‘—)=โŸจ๐ž(๐‘–,2),๐ž(๐‘—,2)โŸฉ,๐ž(๐‘–,2)=๐ฑ(๐‘–)โˆ’ฬ‚โ€Œ๐ฑ(๐‘–,2),๐‘…ฬƒ๐‘ฆ2(๐‘–,2)=๐ถ2๐‘ƒ2(๐‘–,๐‘–)๐ถ๐‘‡2+๐‘„๐‘ฃ2(๐‘–),๐‘„๐‘ฃ2(๐‘–)=diag๎€ฝ๐ผ,โˆ’๐›ผโˆ’2๐ผ,โˆ’๐›พ2๐ผ,โˆ’๐›ฝโˆ’2๐ผ๎€พ.(3.45) Substituting (3.44) into (3.43), we haveฬ‚โ€Œ๐ฑ(๐‘–+1,2)=๐ดฬ‚โ€Œ๐ฑ(๐‘–,2)+๐ด๐‘‘ฬ‚โ€Œ๐ฑ๎€ท๐‘–๐‘‘,2๎€ธ+๐‘“๎€ท๐‘–,ฬ†โ€Œ๐ณ๐‘“(๐‘–โˆฃ๐‘–),๐ฎ(๐‘–)๎€ธ+โ„Ž๎€ท๐‘–,ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ,๐ฎ(๐‘–)๎€ธ+๐ด๐‘‘๐‘–โˆ’1๎“๐‘—=๐‘–๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2)+๐พ2(๐‘–)ฬƒโ€Œ๐ฒ2(๐‘–,2),๐พ2(๐‘–)=๐ด๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘–๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘–,2)+๐ด๐‘ƒ2(๐‘–,๐‘–)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘–,2).(3.46) Moreover, taking into account (3.7) and (3.46), we obtain๐ž(๐‘–+1,2)=๐ด๐ž(๐‘–,2)+๐ด๐‘‘๐ž๎€ท๐‘–๐‘‘,2๎€ธ+๐ต๐ฐ(๐‘–)โˆ’๐พ2(๐‘–)ฬƒโ€Œ๐ฒ2(๐‘–,2)โˆ’๐ด๐‘‘๐‘–โˆ’1๎“๐‘—=๐‘–๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2),๐‘–=0,1,โ€ฆ,๐‘˜๐‘‘โˆ’1.(3.47) Consequently,๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–+1)=โŸจ๐ž(๐‘–โˆ’๐‘—,2),๐ž(๐‘–+1,2)โŸฉ=๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡2๐พ๐‘‡2(๐‘–)โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘ก,2)๐ถ2๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘,๐‘—=0,1,โ€ฆ,๐‘‘,๐‘ƒ2(๐‘–+1,๐‘–+1)=โŸจ๐ž(๐‘–+1,2),๐ž(๐‘–+1,2)โŸฉ=๐ด๐‘ƒ2(๐‘–,๐‘–+1)+๐ด๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘–+1๎€ธ+๐ต๐‘„๐‘ค(๐‘–)๐ต๐‘‡,(3.48) where ๐‘„๐‘ค(๐‘–)=๐ผ. Thus, ๐‘ƒ2(๐‘–,๐‘–)(๐‘–=0,1,โ€ฆ,๐‘˜๐‘‘) can be computed recursively as๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–+1)=๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡2๐พ๐‘‡2(๐‘–)โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘ก,2)๐ถ2๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘,๐‘ƒ2(๐‘–+1,๐‘–+1)=๐ด๐‘ƒ2(๐‘–,๐‘–+1)+๐ด๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘–+1๎€ธ+๐ต๐‘„๐‘ค(๐‘–)๐ต๐‘‡,๐‘—=0,1,โ€ฆ,๐‘‘.(3.49)

Similarly, employing the projection formula in the Krein space, the optimal estimator ฬ‚โ€Œ๐ฑ(๐‘–,1)(๐‘–=๐‘˜๐‘‘+1,โ€ฆ,๐‘˜) can be computed by ฬ‚โ€Œ๐ฑ(๐‘–+1,1)=๐ดฬ‚โ€Œ๐ฑ(๐‘–,1)+๐ด๐‘‘ฬ‚โ€Œ๐ฑ๎€ท๐‘–๐‘‘,2๎€ธ+๐‘“๎€ท๐‘–,ฬ†โ€Œ๐ณ๐‘“(๐‘–โˆฃ๐‘–),๐ฎ(๐‘–)๎€ธ+โ„Ž๎€ท๐‘–,ฬ†โ€Œ๐ณโ„Ž๎€ท๐‘–๐‘‘โˆฃ๐‘–๎€ธ,๐ฎ(๐‘–)๎€ธ+๐พ1(๐‘–)ฬƒโ€Œ๐ฒ1(๐‘–,1)+๐ด๐‘‘๐‘˜๐‘‘โˆ’1๎“๐‘—=๐‘–๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2)+๐ด๐‘‘๐‘–โˆ’1๎“๐‘—=๐‘˜๐‘‘๐‘ƒ1๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘—,1)ฬƒโ€Œ๐ฒ1(๐‘—,1),ฬ‚โ€Œ๐ฑ๎€ท๐‘˜๐‘‘,1๎€ธ=ฬ‚โ€Œ๐ฑ๎€ท๐‘˜๐‘‘,2๎€ธ,(3.50) where๐ถ1=๎€บ๐ถ๐‘‡๐น๐‘‡๐ฟ๐‘‡๎€ป๐‘‡,๐‘ƒ1(๐‘–,๐‘—)=โŽงโŽชโŽจโŽชโŽฉโŸจ๐ž(๐‘–,2),๐ž(๐‘—,1)โŸฉ,๐‘–<๐‘˜๐‘‘,โŸจ๐ž(๐‘–,1),๐ž(๐‘—,1)โŸฉ,๐‘–โ‰ฅ๐‘˜๐‘‘,๐ž(๐‘–,1)=๐ฑ(๐‘–)โˆ’ฬ‚โ€Œ๐ฑ(๐‘–,1),๐‘…ฬƒ๐‘ฆ1(๐‘–,1)=๐ถ1๐‘ƒ1(๐‘–,๐‘–)๐ถ๐‘‡1+๐‘„๐‘ฃ1(๐‘–),๐‘„๐‘ฃ1(๐‘–)=diag๎€ฝ๐ผ,โˆ’๐›ผโˆ’2๐ผ,โˆ’๐›พ2๐ผ๎€พ,๐พ1(๐‘–)=๐ด๐‘ƒ1(๐‘–,๐‘–)๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘–,1)+๐ด๐‘‘๐‘ƒ1๎€ท๐‘–๐‘‘,๐‘–๎€ธ๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘–,1).(3.51) Then, from (3.7) and (3.50), we can yield๐ž(๐‘–+1,1)=๐ด๐ž(๐‘–,1)+๐ด๐‘‘๐ž๎€ท๐‘–๐‘‘,2๎€ธ+๐ต๐ฐ(๐‘–)โˆ’๐พ1(๐‘–)ฬƒโ€Œ๐ฒ1(๐‘–,1)โˆ’๐ด๐‘‘๐‘˜๐‘‘โˆ’1๎“๐‘—=๐‘–๐‘‘๐‘ƒ2๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘—,2)ฬƒโ€Œ๐ฒ2(๐‘—,2)โˆ’๐ด๐‘‘๐‘–โˆ’1๎“๐‘—=๐‘˜๐‘‘๐‘ƒ1๎€ท๐‘–๐‘‘,๐‘—๎€ธ๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘—,1)ฬƒโ€Œ๐ฒ1(๐‘—,1).(3.52) Thus, we obtain that(1)if ๐‘–โˆ’๐‘—โ‰ฅ๐‘˜๐‘‘, we have ๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–+1)=โŸจ๐ž(๐‘–โˆ’๐‘—,1),๐ž(๐‘–+1,1)โŸฉ=๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡1๐พ๐‘‡1(๐‘–)โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘ก,1)๐ถ1๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘,(3.53)(2)if ๐‘–โˆ’๐‘—<๐‘˜๐‘‘, we have๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–+1)=โŸจ๐ž(๐‘–โˆ’๐‘—,2),๐ž(๐‘–+1,1)โŸฉ=๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡1๐พ๐‘‡1(๐‘–)โˆ’๐‘˜๐‘‘โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘ก,2)๐ถ2๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘˜๐‘‘๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘ก,1)๐ถ1๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘,(3.54)๐‘ƒ1(๐‘–+1,๐‘–+1)=โŸจ๐ž(๐‘–โˆ’๐‘—,2),๐ž(๐‘–+1,1)โŸฉ=๐ด๐‘ƒ1(๐‘–,๐‘–+1)+๐ด๐‘‘๐‘ƒ1๎€ท๐‘–๐‘‘,๐‘–+1๎€ธ+๐ต๐‘„๐‘ค(๐‘–)๐ต๐‘‡.(3.55) It follows from (3.53), (3.54), and (3.55) that ๐‘ƒ1(๐‘–,๐‘–)(๐‘–=๐‘˜๐‘‘+1,โ€ฆ,๐‘˜) can be computed by๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–+1)=๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡1๐พ๐‘‡1(๐‘–)โˆ’๐‘˜๐‘‘โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ2(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡2๐‘…โˆ’1ฬƒ๐‘ฆ2(๐‘ก,2)๐ถ2๐‘ƒ๐‘‡2๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘˜๐‘‘๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘ก,1)๐ถ1๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ด๐‘‡๐‘‘,๐‘–โˆ’๐‘—<๐‘˜๐‘‘,๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–+1)=๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ด๐‘‡+๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘–โˆ’๐‘—๎€ธ๐ด๐‘‡๐‘‘โˆ’๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘–)๐ถ๐‘‡1๐พ๐‘‡1(๐‘–)โˆ’๐‘–โˆ’1๎“๐‘ก=๐‘–โˆ’๐‘—๐‘ƒ1(๐‘–โˆ’๐‘—,๐‘ก)๐ถ๐‘‡1๐‘…โˆ’1ฬƒ๐‘ฆ1(๐‘ก,1)๐ถ1๐‘ƒ๐‘‡1๎€ท๐‘–๐‘‘,๐‘ก๎€ธ๐ดT๐‘‘,๐‘–โˆ’๐‘—โ‰ฅ๐‘˜๐‘‘,๐‘ƒ1(๐‘–+1,๐‘–+1)=๐ด๐‘ƒ1(๐‘–,๐‘–+1)+๐ด๐‘‘๐‘ƒ1๎€ท๐‘–๐‘‘,๐‘–+1๎€ธ+๐ต๐‘„๐‘ค(๐‘–)๐ต๐‘‡,๐‘—=0,1,โ€ฆ,๐‘‘.(3.56)

Next, according to the above analysis, ฬ‚โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜) as the Krein space projections of ฬ†โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜) onto โ„’{{๐ฒ๐‘ง(๐‘—)}๐‘˜โˆ’1๐‘—=0;๐ฒ0(๐‘˜)} can be computed by the following formulaฬ‚โ€Œ๐ณ๐‘š(๐‘˜โˆฃ๐‘˜)=๐ถ๐‘šฬ‚โ€Œ๐ฑ(๐‘˜,1)+๐ถ๐‘š๐‘ƒ1(๐‘˜,๐‘˜)๐ถ๐‘‡๐‘…โˆ’1ฬƒ๐‘ฆ0(๐‘˜,0)ฬƒโ€Œ๐ฒ0(๐‘˜,0),(3.57) where๐ถ๐‘š=๎€บ๐น๐‘‡๐ฟ๐‘‡๎€ป๐‘‡,๐‘…ฬƒ๐‘ฆ0(๐‘˜,0)=๐ถ๐‘ƒ1(๐‘˜,๐‘˜)๐ถ๐‘‡+๐‘„๐‘ฃ(๐‘˜).(3.58) And, ฬ‚โ€Œ๐ณโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) as the Krein space projections of ฬ†โ€Œ๐ณโ„Ž(๐‘˜๐‘‘โˆฃ๐‘˜) onto โ„’{{๐ฒ๐‘ง(๐‘—