Abstract

The issue of ðŧ∞ estimation for a class of Lipschitz nonlinear discrete-time systems with time delay and disturbance input is addressed. First, through integrating the ðŧ∞ 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 ðŧ∞ 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 ðŧ∞ 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], ðŧ∞ 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 ðŧ∞ 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 ðŧ∞ and ðŋ2−ðŋ∞ 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 ðŧ∞ estimation problem is studied for a class of Lipschitz nonlinear discrete time-delay systems with disturbance input. Inspired by the recent study on ðŧ∞ 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 ðŧ∞ 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 ðŧ∞ 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 ðŧ∞ estimation problem are introduced. In Section 3, a partially equivalent Krein space problem is constructed, the ðŧ∞ 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; ℝ𝑛 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.; 𝐞 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; âŸĻ∗,∗âŸĐ denotes the inner product in the Krein space; diag{â‹Ŋ} denotes a block-diagonal matrix; ℒ{â‹Ŋ} denotes the linear space spanned by sequence {â‹Ŋ}.

2. System Model and Problem Formulation

Consider a class of nonlinear systems described by the following equations:ð‘Ĩ(𝑘+1)=ðīð‘Ĩ(𝑘)+ðī𝑑ð‘Ĩ𝑘𝑑î€ļ+𝑓(𝑘,ðđð‘Ĩ(𝑘),ð‘Ē(𝑘))+ℎ𝑘,ðŧð‘Ĩ𝑘𝑑î€ļ,ð‘Ē(𝑘)î€ļ+ðĩð‘Ī(𝑘),ð‘Ķ(𝑘)=ðķð‘Ĩ(𝑘)+ð‘Ģ(𝑘),𝑧(𝑘)=ðŋð‘Ĩ(𝑘),(2.1) where 𝑘𝑑=𝑘−𝑑, and the positive integer 𝑑 denotes the known state delay; ð‘Ĩ(𝑘)∈ℝ𝑛 is the state, ð‘Ē(𝑘)∈ℝ𝑝 is the measurable information, ð‘Ī(𝑘)∈ℝ𝑞 and ð‘Ģ(𝑘)∈ℝ𝑚 are the disturbance input belonging to 𝑙2[0,𝑁], ð‘Ķ(𝑘)∈ℝ𝑚 is the measurement output, and 𝑧(𝑘)∈ℝ𝑟 is the signal to be estimated; the initial condition ð‘Ĩ0(𝑠)(𝑠=−𝑑,−𝑑+1,â€Ķ,0) is unknown; the matrices ðī∈ℝ𝑛×𝑛, ðī𝑑∈ℝ𝑛×𝑛, ðĩ∈ℝ𝑛×𝑞, ðķ∈ℝ𝑚×𝑛 and ðŋ∈ℝ𝑟×𝑛, are real and known constant matrices.

In addition, 𝑓(𝑘,ðđð‘Ĩ(𝑘),ð‘Ē(𝑘)) and ℎ(𝑘,ðŧð‘Ĩ(𝑘𝑑),ð‘Ē(𝑘)) are assumed to satisfy the following Lipschitz conditions:‖𝑓(𝑘,ðđð‘Ĩ(𝑘),ð‘Ē(𝑘))−𝑓(𝑘,ðđĖ†ð‘Ĩ(𝑘),ð‘Ē(𝑘))‖â‰Ī𝛞‖ðđ(ð‘Ĩ(𝑘)−Ė†ð‘Ĩ(𝑘))‖,‖ℎ𝑘,ðŧð‘Ĩ𝑘𝑑î€ļ,ð‘Ē(𝑘)î€ļ−ℎ𝑘,ðŧĖ†ð‘Ĩ𝑘𝑑î€ļ,ð‘Ē(𝑘)î€ļ‖â‰Īð›―â€–ðŧð‘Ĩ𝑘𝑑î€ļ−Ė†ð‘Ĩ𝑘𝑑î€ļî€ļ‖,(2.2) for all 𝑘∈{0,1,â€Ķ,𝑁}, ð‘Ē(𝑘)∈ℝ𝑝 and ð‘Ĩ(𝑘),Ė†ð‘Ĩ(𝑘),ð‘Ĩ(𝑘𝑑),Ė†ð‘Ĩ(𝑘𝑑)∈ℝ𝑛. where 𝛞>0 and ð›―>0 are known Lipschitz constants, and ðđ, ðŧ are real matrix with appropriate dimension.

The ðŧ∞ estimation problem under investigation is stated as follows. Given the desired noise attenuation level ð›ū>0 and the observation {ð‘Ķ(𝑗)}𝑘𝑗=0, find an estimate Ė†ð‘§(𝑘âˆĢ𝑘) of the signal 𝑧(𝑘), if it exists, such that the following inequality is satisfied:supð‘Ĩ0,ð‘Ī,ð‘Ģî€ļ≠0∑𝑁𝑘=0‖Ė†ð‘§(𝑘âˆĢ𝑘)−𝑧(𝑘)‖2∑0𝑘=−𝑑‖‖ð‘Ĩ0(𝑘)‖‖2Π−1(𝑘)+∑𝑁𝑘=0‖ð‘Ī(𝑘)‖2+∑𝑁𝑘=0‖ð‘Ģ(𝑘)‖2<ð›ū2,(2.3) where Π(𝑘)(𝑘=−𝑑,−𝑑+1,â€Ķ,0) is a given positive definite matrix function which reflects the relative uncertainty of the initial state ð‘Ĩ0(𝑘)(𝑘=−𝑑,−𝑑+1,â€Ķ,0) to the input and measurement noises.

Remark 2.1. For the sake of simplicity, the initial state estimate Ė‚ð‘Ĩ0(𝑘)(𝑘=−𝑑,−𝑑+1,â€Ķ,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 ðŧ∞ estimator for Lipschitz nonlinear systems. To begin with, the ðŧ∞ 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 {ð‘Ķ(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 ðŧ∞ 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‖‖ð‘Ĩ0(𝑘)‖‖2Π−1(𝑘)+𝑁𝑘=0‖ð‘Ī(𝑘)‖2+𝑁𝑘=0‖ð‘Ģ(𝑘)‖2−ð›ū−2𝑁𝑘=0‖‖ð‘Ģ𝑧(𝑘)‖‖2î„ŋî…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…ƒî…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…Œâ‰œð―âˆ—ð‘>0,∀ð‘Ĩ0,ð‘Ī,ð‘Ģî€ļ≠0,(3.1) where ð‘Ģ𝑧(𝑘)=Ė†ð‘§(𝑘âˆĢ𝑘)−𝑧(𝑘).

Moreover, we denote𝑧𝑓(𝑘)=ðđð‘Ĩ(𝑘),Ė†ð‘§ð‘“(𝑘âˆĢ𝑘)=ðđĖ†ð‘Ĩ(𝑘âˆĢ𝑘),𝑧ℎ𝑘𝑑î€ļ=ðŧð‘Ĩ𝑘𝑑î€ļ,Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ=ðŧĖ†ð‘Ĩ𝑘𝑑âˆĢ𝑘î€ļ,(3.2) where Ė†ð‘§ð‘“(𝑘âˆĢ𝑘) and Ė†ð‘§â„Ž(𝑘𝑑âˆĢ𝑘) denote the optimal estimation of 𝑧𝑓(𝑘) and 𝑧ℎ(𝑘𝑑) based on the observation {ð‘Ķ(𝑗)}𝑘𝑗=0, respectively. And, letð‘Ī𝑓(𝑘)=𝑓𝑘,𝑧𝑓(𝑘),ð‘Ē(𝑘)î€ļ−𝑓𝑘,Ė†ð‘§ð‘“(𝑘âˆĢ𝑘),ð‘Ē(𝑘)î€ļ,ð‘Īℎ𝑘𝑑î€ļ=ℎ𝑘,𝑧ℎ𝑘𝑑î€ļ,ð‘Ē(𝑘)î€ļ−ℎ𝑘,Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ,ð‘Ē(𝑘)î€ļ,ð‘Ģ𝑧𝑓(𝑘)=Ė†ð‘§ð‘“(𝑘âˆĢ𝑘)−𝑧𝑓(𝑘),ð‘Ģ𝑧ℎ𝑘𝑑î€ļ=Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ−𝑧ℎ𝑘𝑑î€ļ.(3.3) From the Lipschitz conditions (2.2), we derive thatð―âˆ—ð‘+𝑁𝑘=0‖‖ð‘Ī𝑓(𝑘)‖‖2+𝑁𝑘=0‖‖ð‘Īℎ𝑘𝑑î€ļ‖‖2−𝛞2𝑁𝑘=0‖‖ð‘Ģ𝑧𝑓(𝑘)‖‖2âˆ’ð›―2𝑁𝑘=0‖‖ð‘Ģ𝑧ℎ𝑘𝑑î€ļ‖‖2î„ŋî…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…ƒî…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…€î…Œâ‰œð―ð‘â‰Īð―âˆ—ð‘.(3.4) Note that the left side of (3.1) and (3.4), ð―ð‘, can be recast into the formð―ð‘=0𝑘=−𝑑‖‖ð‘Ĩ0(𝑘)‖‖2Π−1(𝑘)+𝑁𝑘=0‖‖ð‘Ī(𝑘)‖‖2+𝑁𝑘=0‖ð‘Ģ(𝑘)‖2−ð›ū−2𝑁𝑘=0‖‖ð‘Ģ𝑧(𝑘)‖‖2−𝛞2𝑁𝑘=0‖‖ð‘Ģ𝑧𝑓(𝑘)‖‖2âˆ’ð›―2𝑁𝑘=𝑑‖‖ð‘Ģ𝑧ℎ𝑘𝑑î€ļ‖‖2,(3.5) where Π(𝑘)=⎧⎊âŽĻ⎊âŽĐΠ−1(𝑘)âˆ’ð›―2ðŧ𝑇ðŧî€ļ−1,𝑘=−𝑑,â€Ķ,−1,Π(𝑘),𝑘=0,ð‘Ī(𝑘)=ð‘Ī𝑇(𝑘)ð‘Ī𝑇𝑓(𝑘)ð‘Ī𝑇ℎ𝑘𝑑î€ļî€ŧ𝑇.(3.6)

Since ð―ð‘â‰Īð―âˆ—ð‘, it is natural to see that if ð―ð‘>0 then the ðŧ∞ estimation problem (2.3) is satisfied, that is, ð―âˆ—ð‘>0. Hence, the ðŧ∞ estimation problem (2.3) can be converted into finding the estimate sequence {{Ė†ð‘§(𝑘âˆĢ𝑘)}𝑁𝑘=0;{Ė†ð‘§ð‘“(𝑘âˆĢ𝑘)}𝑁𝑘=0;{Ė†ð‘§â„Ž(𝑘𝑑âˆĢ𝑘)}𝑁𝑘=𝑑} such that ð―ð‘ has a minimum with respect to {ð‘Ĩ0,ð‘Ī} and the minimum of ð―ð‘ is positive. As mentioned in [25, 26], the formulated ðŧ∞ estimation problem can be solved by employing the Krein space approach.

Introduce the following Krein space stochastic systemðą(𝑘+1)=ðīðą(𝑘)+ðīð‘‘ðąî€·ð‘˜ð‘‘î€ļ+𝑓𝑘,Ė†â€Œðģ𝑓(𝑘âˆĢ𝑘),ðŪ(𝑘)î€ļ+ℎ𝑘,Ė†â€Œðģℎ𝑘𝑑âˆĢ𝑘î€ļ,ðŪ(𝑘)î€ļ+ðĩ𝐰(𝑘),ðē(𝑘)=ðķðą(𝑘)+ðŊ(𝑘),Ė†â€Œðģ𝑓(𝑘âˆĢ𝑘)=ðđðą(𝑘)+ðŊ𝑧𝑓(𝑘),Ė†â€Œðģ(𝑘âˆĢ𝑘)=ðŋðą(𝑘)+ðŊ𝑧(𝑘),Ė†â€Œðģℎ𝑘𝑑âˆĢ𝑘î€ļ=ðŧðąî€·ð‘˜ð‘‘î€ļ+ðŊ𝑧ℎ𝑘𝑑î€ļ,𝑘â‰Ĩ𝑑,(3.7) where ðĩ=[ðĩ𝐞𝐞]; the initial state ðą0(𝑠)(𝑠=−𝑑,−𝑑+1,â€Ķ,0) and 𝐰(𝑘), ðŊ(𝑘), ðŊ𝑧𝑓(𝑘), ðŊ𝑧(𝑘) and ðŊ𝑧ℎ(𝑘) are mutually uncorrelated white noises with zero means and known covariance matrices Π(𝑠), 𝑄ð‘Ī(𝑘)=𝐞, 𝑄ð‘Ģ(𝑘)=𝐞, 𝑄ð‘Ģ𝑧𝑓(𝑘)=−𝛞−2𝐞, 𝑄ð‘Ģ𝑧(𝑘)=−ð›ū2𝐞, and 𝑄ð‘Ģ𝑧ℎ(𝑘)=âˆ’ð›―âˆ’2𝐞; Ė†â€Œðģ𝑓(𝑘âˆĢ𝑘), Ė†â€Œðģ(𝑘âˆĢ𝑘) and Ė†â€Œðģℎ(𝑘𝑑âˆĢ𝑘) are regarded as the imaginary measurement at time 𝑘 for the linear combination ðđðą(𝑘), ðŋðą(𝑘), and ðŧðą(𝑘𝑑), respectively.

Letðē𝑧(𝑘)=⎧⎊âŽĻ⎊âŽĐðē𝑇(𝑘)Ė†â€Œðģ𝑇𝑚(𝑘âˆĢ𝑘)î€ŧ𝑇,0â‰Ī𝑘<𝑑,ðē𝑇(𝑘)Ė†â€Œðģ𝑇𝑚(𝑘âˆĢ𝑘)Ė†â€Œðģ𝑇ℎ𝑘𝑑âˆĢ𝑘î€ļî€ŧ𝑇,𝑘â‰Ĩ𝑑,ðŊ𝑧,𝑎(𝑘)=⎧⎊âŽĻ⎊âŽĐðŊ𝑇(𝑘)ðŊ𝑇𝑧𝑓(𝑘)ðŊ𝑇𝑧(𝑘)𝑇,0â‰Ī𝑘<𝑑,ðŊ𝑇(𝑘)ðŊ𝑇𝑧𝑓(𝑘)ðŊ𝑇𝑧(𝑘)ðŊ𝑇𝑧ℎ𝑘𝑑î€ļ𝑇,𝑘â‰Ĩ𝑑,Ė†â€Œðģ𝑚(𝑘âˆĢ𝑘)=Ė†â€Œðģ𝑇𝑓(𝑘âˆĢ𝑘)Ė†â€Œðģ𝑇(𝑘âˆĢ𝑘)î€ŧ𝑇.(3.8)

Definition 3.1. The estimator Ė‚‌ðē(𝑖âˆĢ𝑖−1) denotes the optimal estimation of ðē(𝑖) given the observation ℒ{{ðē𝑧(𝑗)}𝑖−1𝑗=0}; the estimator Ė‚‌ðģ𝑚(𝑖âˆĢ𝑖) denotes the optimal estimation of Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖) given the observation ℒ{{ðē𝑧(𝑗)}𝑖−1𝑗=0;ðē(𝑖)}; the estimator Ė‚‌ðģℎ(𝑖𝑑âˆĢ𝑖) denotes the optimal estimation of Ė†â€Œðģℎ(𝑖𝑑âˆĢ𝑖) given the observation ℒ{{ðē𝑧(𝑗)}𝑖−1𝑗=0;ðē(𝑖),Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖)}.

Furthermore, introduce the following stochastic vectors and the corresponding covariance matricesĖƒâ€Œðē(𝑖âˆĢ𝑖−1)=ðē(𝑖)−Ė‚‌ðē(𝑖𝑖−1),𝑅Ėƒð‘Ķ(𝑖𝑖−1)=âŸĻĖƒâ€Œðē(𝑖𝑖−1),Ėƒâ€Œðē(𝑖𝑖−1)âŸĐ,Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖)=Ė†â€Œðģ𝑚(𝑖𝑖)−Ė‚‌ðģ𝑚(𝑖𝑖),𝑅Ėƒð‘§ð‘š(𝑖𝑖)=âŸĻĖƒâ€Œðģ𝑚(𝑖𝑖),Ėƒâ€Œðģ𝑚(𝑖𝑖)âŸĐ,Ėƒâ€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ=Ė†â€Œðģℎ𝑖𝑑𝑖î€ļ−Ė‚‌ðģℎ𝑖𝑑𝑖î€ļ,𝑅Ėƒð‘§â„Žî€·ð‘–𝑑𝑖î€ļ=îŦĖƒâ€Œðģℎ𝑖𝑑𝑖î€ļ,Ėƒâ€Œðģℎ𝑖𝑑𝑖î€ļ.(3.9) And, denoteĖƒâ€Œðē𝑧(𝑖)=⎧⎊âŽĻ⎊âŽĐĖƒâ€Œðē𝑇(𝑖âˆĢ𝑖−1)Ėƒâ€Œðģ𝑇𝑚(𝑖âˆĢ𝑖)î€ŧ𝑇,0â‰Ī𝑖<𝑑,Ėƒâ€Œðē𝑇(𝑖âˆĢ𝑖−1)Ėƒâ€Œðģ𝑇𝑚(𝑖âˆĢ𝑖)Ėƒâ€Œðģ𝑇ℎ(𝑖𝑑âˆĢ𝑖)î€ŧ𝑇,𝑖â‰Ĩ𝑑,𝑅Ėƒð‘Ķ𝑧(𝑖)=îŦĖƒâ€Œðē𝑧(𝑖),Ėƒâ€Œðē𝑧(𝑖).(3.10)

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

Lemma 3.2. {{Ėƒâ€Œðē𝑧(𝑖)}𝑘𝑖=0} is the innovation sequence which spans the same linear space as that of ℒ{{ðē𝑧(𝑖)}𝑘𝑖=0}.

Proof. From Definition 3.1 and (3.9), Ėƒâ€Œðē(𝑖âˆĢ𝑖−1), Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖) and Ėƒâ€Œðģℎ(𝑖𝑑âˆĢ𝑖) are the linear combination of the observation sequence {{ðē𝑧(𝑗)}𝑖−1𝑗=0;ðē(𝑖)}, {{ðē𝑧(𝑗)}𝑖−1𝑗=0;ðē(𝑖),Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖)}, and {{ðē𝑧(𝑗)}𝑖𝑗=0}, respectively. Conversely, ðē(𝑖), Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖) and Ė†â€Œðģℎ(𝑖𝑑âˆĢ𝑖) can be given by the linear combination of {{Ėƒâ€Œðē𝑧(𝑗)}𝑖−1𝑗=0;Ėƒâ€Œðē(𝑖âˆĢ𝑖−1)},{{Ėƒâ€Œðē𝑧(𝑗)}𝑖−1𝑗=0;Ėƒâ€Œðē(𝑖âˆĢ𝑖−1),Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖)} and {{Ėƒâ€Œðē𝑧(𝑗)}𝑖𝑗=0}, respectively. Hence, â„’î‚†î€―Ėƒâ€Œðē𝑧(𝑖)î€ū𝑘𝑖=0=â„’î‚†î€―ðē𝑧(𝑖)î€ū𝑘𝑖=0.(3.11) It is also shown by (3.9) that Ėƒâ€Œðē(𝑖âˆĢ𝑖−1), Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖) and Ėƒâ€Œðģℎ(𝑖𝑑âˆĢ𝑖) satisfy Ėƒâ€Œðē(𝑖âˆĢ𝑖−1)âŸ‚â„’î‚†î€―ðē𝑧(𝑗)î€ū𝑖−1𝑗=0,Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖)âŸ‚â„’î‚†î€―ðē𝑧(𝑗)î€ū𝑖−1𝑗=0;ðē(𝑖),Ėƒâ€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļâŸ‚â„’î‚†î€―ðē𝑧(𝑗)î€ū𝑖−1𝑗=0;ðē(𝑖),Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖).(3.12) Consequently, Ėƒâ€Œðē(𝑖âˆĢ𝑖−1)âŸ‚â„’î‚†î€―Ėƒâ€Œðē𝑧(𝑗)î€ū𝑖−1𝑗=0,Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖)âŸ‚â„’î‚†î€―Ėƒâ€Œðē𝑧(𝑗)î€ū𝑖−1𝑗=0;Ėƒâ€Œðē(𝑖âˆĢ𝑖−1),Ėƒâ€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļâŸ‚â„’î‚†î€―Ėƒâ€Œðē𝑧(𝑗)î€ū𝑖−1𝑗=0;Ėƒâ€Œðē(𝑖âˆĢ𝑖−1),Ėƒâ€Œðģ𝑚(𝑖âˆĢ𝑖).(3.13) This completes the proof.

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

Theorem 3.3. Consider system (2.1), given a scalar ð›ū>0 and the positive definite matrix Π(k)(𝑘=−𝑑,−𝑑+1,â€Ķ,0), then ð―ð‘ has the minimum if only if 𝑅Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)>0,0â‰Ī𝑘â‰Ī𝑁,𝑅Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)<0,0â‰Ī𝑘â‰Ī𝑁,𝑅Ėƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ<0,𝑑â‰Ī𝑘â‰Ī𝑁.(3.14) In this case the minimum value of ð―ð‘ is given by minð―ð‘=𝑁𝑘=0Ėƒð‘Ķ𝑇(𝑘âˆĢ𝑘−1)𝑅−1Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)+𝑁𝑘=0Ėƒð‘§ð‘‡ð‘š(𝑘âˆĢ𝑘)𝑅−1Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)+𝑁𝑘=𝑑Ėƒð‘§ð‘‡â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ𝑅−1Ėƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļĖƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ,(3.15) where Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)=ð‘Ķ(𝑘)−Ė‚ð‘Ķ(𝑘âˆĢ𝑘−1),Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)=Ė†ð‘§ð‘š(𝑘âˆĢ𝑘)−Ė‚𝑧𝑚(𝑘âˆĢ𝑘),Ėƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ=Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ−Ė‚𝑧ℎ𝑘𝑑âˆĢ𝑘î€ļ,Ė†ð‘§ð‘š(𝑘âˆĢ𝑘)=Ė†ð‘§ð‘‡ð‘“(𝑘âˆĢ𝑘)Ė†ð‘§ð‘‡(𝑘âˆĢ𝑘)î€ŧ𝑇,(3.16)Ė‚ð‘Ķ(𝑘âˆĢ𝑘−1) is obtained from the Krein space projection of ðē(𝑘) onto ℒ{{ðē𝑧(𝑗)}𝑘−1𝑗=0}, Ė‚𝑧𝑚(𝑘âˆĢ𝑘) is obtained from the Krein space projection of Ė†â€Œðģ𝑚(𝑘âˆĢ𝑘) onto ℒ{{ðē𝑧(𝑗)}𝑘−1𝑗=0;ðē(𝑘)}, and Ė‚𝑧ℎ(𝑘𝑑âˆĢ𝑘) is obtained from the Krein space projection of Ė†â€Œðģℎ(𝑘𝑑âˆĢ𝑘) onto ℒ{{ðē𝑧(𝑗)}𝑘−1𝑗=0;ðē(𝑘),Ė†â€Œðģ𝑚(𝑘âˆĢ𝑘)}.

Proof. Based on the definition (3.2) and (3.3), the state equation in system (2.1) can be rewritten as ð‘Ĩ(𝑘+1)=ðīð‘Ĩ(𝑘)+ðī𝑑ð‘Ĩ𝑘𝑑î€ļ+𝑓𝑘,Ė†ð‘§ð‘“(𝑘âˆĢ𝑘),ð‘Ē(𝑘)î€ļ+ℎ𝑘,Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ,ð‘Ē(𝑘)î€ļ+ðĩð‘Ī(𝑘).(3.17) In this case, it is assumed that 𝑓(𝑘,Ė†ð‘§ð‘“(𝑘âˆĢ𝑘),ð‘Ē(𝑘)) and ℎ(𝑘,Ė†ð‘§â„Ž(𝑘𝑑âˆĢ𝑘),ð‘Ē(𝑘)) are known at time 𝑘. Then, we define ð‘Ķ𝑧(𝑘)=⎧⎊âŽĻ⎊âŽĐð‘Ķ𝑇(𝑘)Ė†ð‘§ð‘‡ð‘“(𝑘âˆĢ𝑘)Ė†ð‘§ð‘‡(𝑘âˆĢ𝑘)î€ŧ𝑇,0â‰Ī𝑘<𝑑,ð‘Ķ𝑇(𝑘)Ė†ð‘§ð‘‡ð‘“(𝑘âˆĢ𝑘)Ė†ð‘§ð‘‡(𝑘âˆĢ𝑘)Ė†ð‘§ð‘‡â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļî€ŧ𝑇,𝑘â‰Ĩ𝑑.(3.18) By introducing an augmented state ð‘Ĩ𝑎(𝑘)=ð‘Ĩ𝑇(𝑘)ð‘Ĩ𝑇(𝑘−1)â‹Ŋð‘Ĩ𝑇(𝑘−𝑑)î€ŧ𝑇,(3.19) we obtain an augmented state-space model ð‘Ĩ𝑎(𝑘+1)=ðī𝑎ð‘Ĩ𝑎(𝑘)+ðĩð‘Ē,𝑎ð‘Ē(𝑘)+ðĩ𝑎ð‘Ī(𝑘),ð‘Ķ𝑧(𝑘)=ðķ𝑧,𝑎(𝑘)ð‘Ĩ𝑎(𝑘)+ð‘Ģ𝑧,𝑎(𝑘),(3.20) where ðī𝑎=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĢðī0â‹Ŋ0ðī𝑑𝐞0â‹Ŋ000𝐞â‹Ŋ00â‹Ūâ‹Ū⋱â‹Ūâ‹Ū00â‹Ŋ𝐞0âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,ðĩð‘Ē,𝑎=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĢ𝐞𝐞0000â‹Ūâ‹Ū00âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,ðĩ𝑎=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĢðĩ00â‹Ū0âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,(3.21)C𝑧,𝑎(𝑘)=⎧⎊⎊⎊⎊⎊âŽĻ⎊⎊⎊⎊⎊âŽĐ⎡âŽĒâŽĒâŽĒâŽĢðķ0â‹Ŋ0ðđ0â‹Ŋ0ðŋ0â‹Ŋ0âŽĪâŽĨâŽĨâŽĨâŽĶ,0â‰Ī𝑘<𝑑,⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĒâŽĢðķ0â‹Ŋ0ðđ0â‹Ŋ0ðŋ0â‹Ŋ00â‹Ŋ0ðŧâŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,𝑘â‰Ĩ𝑑,ð‘Ģ𝑧,𝑎(𝑘)=⎧⎊âŽĻ⎊âŽĐð‘Ģ𝑇(𝑘)ð‘Ģ𝑇𝑧𝑓(𝑘)ð‘Ģ𝑇𝑧(𝑘)𝑇,0â‰Ī𝑘<𝑑,ð‘Ģ𝑇(𝑘)ð‘Ģ𝑇𝑧𝑓(𝑘)ð‘Ģ𝑇𝑧(𝑘)ð‘Ģ𝑇𝑧ℎ𝑘𝑑î€ļ𝑇,𝑘â‰Ĩ𝑑,ð‘Ē(𝑘)=𝑓𝑇𝑘,Ė†ð‘§ð‘“(𝑘âˆĢ𝑘),ð‘Ē(𝑘)î€ļℎ𝑇𝑘,Ė†ð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ,ð‘Ē(𝑘)î€ļî€ŧ𝑇.(1) Additionally, we can rewrite ð―ð‘ as ð―ð‘=⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ģ𝑧,𝑎𝑁âŽĪâŽĨâŽĨâŽĨâŽĶð‘‡âŽĄâŽĒâŽĒâŽĒâŽĢ𝑃𝑎(0)000𝐞000𝑄ð‘Ģ𝑧,𝑎𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ−1⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ģ𝑧,𝑎𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ,(3.22) where 𝑃𝑎(0)=diagΠ(0),Π(−1),â€Ķ,Π(−𝑑),ð‘Ī𝑁=ð‘Ī𝑇(0)ð‘Ī𝑇(1)â‹Ŋð‘Ī𝑇(𝑁)î€ŧ𝑇,ð‘Ģ𝑧,𝑎𝑁=ð‘Ģ𝑇𝑧,𝑎(0)ð‘Ģ𝑇𝑧,𝑎(1)â‹Ŋð‘Ģ𝑇𝑧,𝑎(𝑁)î€ŧ𝑇,𝑄ð‘Ģ𝑧,𝑎𝑁=diag𝑄ð‘Ģ𝑧,𝑎(0),𝑄ð‘Ģ𝑧,𝑎(1),â€Ķ,𝑄ð‘Ģ𝑧,𝑎(𝑁),𝑄ð‘Ģ𝑧,𝑎(𝑘)=⎧⎊âŽĻ⎊âŽĐdiagî€―ðž,−ð›ū2,−𝛞−2î€ū,0â‰Ī𝑘<𝑑,diagî€―ðž,−ð›ū2,−𝛞−2,âˆ’ð›―âˆ’2î€ū,𝑘â‰Ĩ𝑑.(3.23) Define the following state transition matrix ÎĶ(𝑘+1,𝑚)=ðī𝑎ÎĶ(𝑘,𝑚),ÎĶ(𝑚,𝑚)=𝐞,(3.24) and let ð‘Ķ𝑧𝑁=ð‘Ķ𝑇𝑧(0)ð‘Ķ𝑇𝑧(1)â‹Ŋð‘Ķ𝑇𝑧(𝑁)î€ŧ𝑇,ð‘Ē𝑁=ð‘Ē𝑇(0)ð‘Ē𝑇(1)â‹Ŋð‘Ē𝑇(𝑁)î€ŧ𝑇.(3.25) Using (3.20) and (3.24), we have ð‘Ķ𝑧𝑁=ÎĻ0𝑁ð‘Ĩ𝑎(0)+ÎĻð‘Ē𝑁ð‘Ē𝑁+ÎĻð‘Ī𝑁ð‘Ī𝑁+ð‘Ģ𝑧,𝑎𝑁,(3.26) where ÎĻ0𝑁=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĢðķ𝑧,𝑎(0)ÎĶ(0,0)ðķ𝑧,𝑎(1)ÎĶ(1,0)â‹Ūðķ𝑧,𝑎(𝑁)ÎĶ(𝑁,0)âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,ÎĻð‘Ē𝑁=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĢ𝜑00𝜑01â‹Ŋ𝜑0𝑁𝜑10𝜑11â‹Ŋ𝜑1𝑁â‹Ūâ‹Ū⋱â‹Ū𝜑𝑁0𝜑𝑁1â‹Ŋ𝜑𝑁𝑁âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,𝜑𝑖𝑗=⎧⎊âŽĻ⎊âŽĐðķ𝑧,𝑎(𝑖)ÎĶ(𝑖,𝑗+1)ðĩð‘Ē,𝑎,𝑖>𝑗,0,𝑖â‰Ī𝑗.(3.27) The matrix ÎĻð‘Ī𝑁 is derived by replacing ðĩð‘Ē,𝑎 in ÎĻð‘Ē𝑁 with ðĩ𝑎.
Thus, ð―ð‘ can be reexpressed as ð―ð‘=⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ķ𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ𝑇⎧⎊âŽĻ⎊âŽĐÎ“ð‘âŽĄâŽĒâŽĒâŽĒâŽĢ𝑃𝑎(0)000𝐞000𝑄ð‘Ģ𝑧,𝑎𝑁âŽĪâŽĨâŽĨâŽĨâŽĶΓ𝑇𝑁âŽŦ⎩⎮⎩⎭−1⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ķ𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ,(3.28) where ð‘Ķ𝑧𝑁=ð‘Ķ𝑧𝑁−ÎĻð‘Ē𝑁ð‘Ē𝑁,Γ𝑁=⎡âŽĒâŽĒâŽĒâŽĢ𝐞000𝐞0ÎĻ0𝑁ÎĻð‘Ī𝑁𝐞âŽĪâŽĨâŽĨâŽĨâŽĶ.(3.29)
Considering the Krein space stochastic system defined by (3.7) and state transition matrix (3.24), we have ðē𝑧𝑁=ÎĻ0ð‘ðąð‘Ž(0)+ÎĻð‘Ē𝑁ðŪ𝑁+ÎĻð‘Ī𝑁𝐰𝑁+ðŊ𝑧,𝑎𝑁,(3.30) where matrices ÎĻ0𝑁, ÎĻð‘Ē𝑁, and ÎĻð‘Ī𝑁 are the same as given in (3.26), vectors ðē𝑧𝑁 and ðŪ𝑁 are, respectively, defined by replacing Euclidean space element ð‘Ķ𝑧 and ð‘Ē in ð‘Ķ𝑧𝑁 and ð‘Ē𝑁 given by (3.25) with the Krein space element ðē𝑧 and ðŪ, vectors 𝐰𝑁 and ðŊ𝑧,𝑎𝑁 are also defined by replacing Euclidean space element ð‘Ī and ð‘Ģ𝑧,𝑎 in ð‘Ī𝑁 and ð‘Ģ𝑧,𝑎𝑁 given by (3.23) with the Krein space element 𝐰 and ðŊ𝑧,𝑎, and vector ðąð‘Ž(0) is given by replacing Euclidean space element ð‘Ĩ in ð‘Ĩ𝑎(𝑘) given by (3.19) with the Krein space element ðą when 𝑘=0.
Using the stochastic characteristic of ðąð‘Ž(0), 𝐰𝑁 and ðŊ𝑧,𝑎, we have ð―ð‘=⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ķ𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶð‘‡î„”âŽĄâŽĒâŽĒâŽĒâŽĢðąð‘Ž(0)𝐰𝑁ðē𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ,⎡âŽĒâŽĒâŽĒâŽĢðąð‘Ž(0)𝐰𝑁ðē𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ−1⎡âŽĒâŽĒâŽĒâŽĢð‘Ĩ𝑎(0)ð‘Ī𝑁ð‘Ķ𝑧𝑁âŽĪâŽĨâŽĨâŽĨâŽĶ,(3.31) where ðē𝑧𝑁=ðē𝑧𝑁−ÎĻð‘Ē𝑁ðŪ𝑁.
In the light of Theorem  2.4.2 and Lemma  2.4.3 in [26], ð―ð‘ has a minimum over {ð‘Ĩ𝑎(0),ð‘Ī𝑁} if and only if 𝑅ð‘Ķ𝑧𝑁=âŸĻðē𝑧𝑁,ðē𝑧𝑁âŸĐ and 𝑄ð‘Ģ𝑧,𝑎𝑁=âŸĻðŊ𝑧,𝑎𝑁,ðŊ𝑧,𝑎𝑁âŸĐ have the same inertia. Moreover, the minimum of ð―ð‘ is given by minð―ð‘=ð‘Ķ𝑇𝑧𝑁𝑅−1ð‘Ķ𝑧𝑁ð‘Ķ𝑧𝑁.(3.32)
On the other hand, applying the Krein space projection formula, we have ðē𝑧𝑁=Θ𝑁Ėƒâ€Œðē𝑧𝑁,(3.33) where Ėƒâ€Œðē𝑧𝑁=Ėƒâ€Œðē𝑇𝑧(0)Ėƒâ€Œðē𝑇𝑧(1)â‹ŊĖƒâ€Œðē𝑇𝑧(𝑁)î€ŧ𝑇,(3.34)Θ𝑁=⎡âŽĒâŽĒâŽĒâŽĒâŽĒâŽĢ𝜃00𝜃01â‹Ŋ𝜃0𝑁𝜃10𝜃11â‹Ŋ𝜃1𝑁â‹Ūâ‹Ū⋱â‹Ū𝜃𝑁0𝜃𝑁1â‹Ŋ𝜃𝑁𝑁âŽĪâŽĨâŽĨâŽĨâŽĨâŽĨâŽĶ,𝜃𝑖𝑗=⎧⎊⎊⎊⎊⎊âŽĻ⎊⎊⎊⎊⎊âŽĐîŦðē𝑧(𝑖),Ėƒâ€Œðē𝑧(𝑗)𝑅−1Ėƒð‘Ķ𝑧(𝑗),𝑖>𝑗â‰Ĩ0,⎡âŽĒâŽĢ𝐞0𝑚1𝐞âŽĪâŽĨâŽĶ,𝑑>𝑖=𝑗â‰Ĩ0,⎡âŽĒâŽĒâŽĒâŽĢ𝐞00𝑚1𝐞0𝑚2𝑚3𝐞âŽĪâŽĨâŽĨâŽĨâŽĶ,𝑖=𝑗â‰Ĩ𝑑,0,0â‰Ī𝑖<𝑗,𝑚1=Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖),Ėƒâ€Œðē(𝑗âˆĢ𝑗−1)𝑅−1Ėƒð‘Ķ(𝑗âˆĢ𝑗−1),𝑚2=Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ,Ėƒâ€Œðē(𝑗âˆĢ𝑗−1)𝑅−1Ėƒð‘Ķ(𝑗âˆĢ𝑗−1),𝑚3=Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ,Ėƒâ€Œðģ𝑚(𝑗âˆĢ𝑗)𝑅−1Ėƒð‘§ð‘š(𝑗âˆĢ𝑗),ðē𝑧(𝑖)=ðē𝑧(𝑖)−𝑁𝑗=0𝜑𝑖𝑗ðŪ(𝑗),Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖)=Ė†â€Œðģ𝑚(𝑖âˆĢ𝑖)−𝑁𝑗=0𝜑𝑚,𝑖𝑗ðŪ(𝑗),Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ=Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ−𝑁𝑗=0𝜑ℎ,𝑖𝑗ðŪ(𝑗),(2) where 𝜑𝑚,𝑖𝑗 is derived by replacing ðķ𝑧,𝑎 in 𝜑𝑖𝑗 withðđ0â‹Ŋ0ðŋ0â‹Ŋ0î€ŧ, 𝜑ℎ,𝑖𝑗 is derived by replacing ðķ𝑧,𝑎 in 𝜑𝑖𝑗 with [00â‹Ŋðŧ] Furthermore, it follows from (3.33) that 𝑅ð‘Ķ𝑧𝑁=Θ𝑁𝑅Ėƒð‘Ķ𝑧𝑁Θ𝑇𝑁,ð‘Ķ𝑧𝑁=Θ𝑁Ėƒð‘Ķ𝑧𝑁,(3.35) where 𝑅Ėƒð‘Ķ𝑧𝑁=îŦĖƒâ€Œðē𝑧𝑁,Ėƒâ€Œðē𝑧𝑁,Ėƒð‘Ķ𝑧𝑁=Ėƒð‘Ķ𝑇𝑧(0)Ėƒð‘Ķ𝑇𝑧(1)â‹ŊĖƒð‘Ķ𝑇𝑧(𝑁)î€ŧ𝑇,Ėƒð‘Ķ𝑧(𝑖)=⎧⎊âŽĻ⎊âŽĐĖƒð‘Ķ𝑇(𝑖âˆĢ𝑖−1)Ėƒð‘§ð‘‡ð‘š(𝑖âˆĢ𝑖)î€ŧ𝑇,0â‰Ī𝑖<𝑑,Ėƒð‘Ķ𝑇(𝑖âˆĢ𝑖−1)Ėƒð‘§ð‘‡ð‘š(𝑖âˆĢ𝑖)Ėƒð‘§ð‘‡â„Žî€·ð‘–𝑑âˆĢ𝑖î€ļî€ŧ𝑇,𝑖â‰Ĩ𝑑.(3.36) Since matrix Θ𝑁 is nonsingular, it follows from (3.35) that 𝑅ð‘Ķ𝑧𝑁 and 𝑅Ėƒð‘Ķ𝑧𝑁 are congruent, which also means that 𝑅ð‘Ķ𝑧𝑁 and 𝑅Ėƒð‘Ķ𝑧𝑁 have the same inertia. Note that both 𝑅Ėƒð‘Ķ𝑧𝑁 and 𝑄ð‘Ģ𝑧,𝑎𝑁 are block-diagonal matrices, and 𝑅Ėƒð‘Ķ𝑧(𝑘)=⎧⎊âŽĻ⎊âŽĐdiagî€―ð‘…Ėƒð‘Ķ(𝑘âˆĢ𝑘−1),𝑅Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)î€ū,0â‰Ī𝑘<𝑑,diagî€―ð‘…Ėƒð‘Ķ(𝑘âˆĢ𝑘−1),𝑅Ėƒð‘§ð‘š(𝑘âˆĢ𝑘),𝑅Ėƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļî€ū,𝑘â‰Ī𝑑,(3.37)𝑄ð‘Ģ𝑧,𝑎(𝑘) is given by (3.23). It follows that 𝑅Ėƒð‘Ķ𝑧𝑁 and 𝑄ð‘Ģ𝑧,𝑎𝑁 have the same inertia if and only if 𝑅Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)>0(0â‰Ī𝑘â‰Ī𝑁), 𝑅Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)<0(0â‰Ī𝑘â‰Ī𝑁) and 𝑅Ėƒð‘§â„Ž(𝑘𝑑âˆĢ𝑘)<0(𝑑â‰Ī𝑘â‰Ī𝑁).
Therefore, ð―ð‘ subject to system (2.1) with Lipschitz conditions (2.2) has the minimum if and only if 𝑅Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)>0(0â‰Ī𝑘â‰Ī𝑁), 𝑅Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)<0(0â‰Ī𝑘â‰Ī𝑁) and 𝑅Ėƒð‘§â„Ž(𝑘𝑑âˆĢ𝑘)<0(𝑑â‰Ī𝑘â‰Ī𝑁). Moreover, the minimum value of ð―ð‘ can be rewritten as minð―ð‘=ð‘Ķ𝑇𝑧𝑁𝑅−1ð‘Ķ𝑧𝑁ð‘Ķ𝑧𝑁=Ėƒð‘Ķ𝑇𝑧𝑁𝑅−1Ėƒð‘Ķ𝑧𝑁Ėƒð‘Ķ𝑧𝑁=𝑁𝑘=0Ėƒð‘Ķ𝑇(𝑘âˆĢ𝑘−1)𝑅−1Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)Ėƒð‘Ķ(𝑘âˆĢ𝑘−1)+𝑁𝑘=0Ėƒð‘§ð‘‡ð‘š(𝑘âˆĢ𝑘)𝑅−1Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)Ėƒð‘§ð‘š(𝑘âˆĢ𝑘)+𝑁𝑘=𝑑Ėƒð‘§ð‘‡â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ𝑅−1Ėƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļĖƒð‘§â„Žî€·ð‘˜ð‘‘âˆĢ𝑘î€ļ.(3.38) The proof is completed.

Remark 3.4. Due to the built innovation sequence {{Ėƒâ€Œðē𝑧(𝑖)}𝑘𝑖=0} in Lemma 3.2, the form of the minimum on indefinite quadratic form ð―ð‘ is different from the one given in [26–28]. It is shown from (3.15) that the estimation errors Ėƒð‘Ķ(𝑘âˆĢ𝑘−1), Ėƒð‘§ð‘š(𝑘âˆĢ𝑘) and Ėƒð‘§â„Ž(𝑘𝑑âˆĢ𝑘) are mutually uncorrelated, which will make the design of ðŧ∞ estimator much easier than the one given in [26–28].

3.2. Solution of the ðŧ∞ Estimation Problem

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

Denoteðē0(𝑖)=ðē(𝑖),ðē1(𝑖)=ðē𝑇(𝑖)Ė†â€Œðģ𝑇𝑚(𝑖âˆĢ𝑖)î€ŧ𝑇,ðē2(𝑖)=ðē𝑇(𝑖)Ė†â€Œðģ𝑇𝑚(𝑖âˆĢ𝑖)Ė†â€Œðģ𝑇ℎ(𝑖âˆĢ𝑖+𝑑)î€ŧ𝑇.(3.39) Observe from (3.8), we have â„’î€―î€―ðē𝑧(𝑖)î€ū𝑗𝑖=0î€ū=â„’î€―î€―ðē1(𝑖)î€ū𝑗𝑖=0î€ū,0â‰Ī𝑗<𝑑,â„’î€―î€―ðē𝑧(𝑖)î€ū𝑗𝑖=0î€ū=â„’î‚†î€―ðē2(𝑖)î€ū𝑗𝑑𝑖=0;î‚†î€―ðē1(𝑖)î€ū𝑗𝑖=𝑗𝑑+1,𝑗â‰Ĩ𝑑.(3.40)

Definition 3.5. Given 𝑘â‰Ĩ𝑑, the estimator Ė‚‌𝜉(𝑖âˆĢ𝑗,2) for 0â‰Ī𝑗<𝑘𝑑 denotes the optimal estimate of 𝜉(𝑖) given the observation ℒ{{ðē2(𝑠)}𝑗𝑠=0}, and the estimator Ė‚‌𝜉(𝑖âˆĢ𝑗,1) for 𝑘𝑑â‰Ī𝑗â‰Ī𝑘 denotes the optimal estimate of 𝜉(𝑖) given the observation ℒ{{ðē2(𝑠)}𝑘𝑑−1𝑠=0;{ðē1(𝜏)}𝑗𝜏=𝑘𝑑}. For simplicity, we use Ė‚‌𝜉(𝑖,2) to denote Ė‚‌𝜉(𝑖âˆĢ𝑖−1,2), and use Ė‚‌𝜉(𝑖,1) to denote Ė‚‌𝜉(𝑖âˆĢ𝑖−1,1) throughout the paper.

Based on the above definition, we introduce the following stochastic sequence and the corresponding covariance matricesĖƒâ€Œðē2(𝑖,2)=ðē2(𝑖)−Ė‚‌ðē2(𝑖,2),𝑅Ėƒð‘Ķ2(𝑖,2)=âŸĻĖƒâ€Œðē2(𝑖,2),Ėƒâ€Œðē2(𝑖,2)âŸĐ,Ėƒâ€Œðē1(𝑖,1)=ðē1(𝑖)−Ė‚‌ðē1(𝑖,1),𝑅Ėƒð‘Ķ1(𝑖,1)=âŸĻĖƒâ€Œðē1(𝑖,1),Ėƒâ€Œðē1(𝑖,1)âŸĐ,Ėƒâ€Œðē0(𝑖,0)=ðē0(𝑖)−Ė‚‌ðē0(𝑖,1),𝑅Ėƒð‘Ķ0(𝑖,0)=îŦĖƒâ€Œðē0(i,0),Ėƒâ€Œðē0(𝑖,0).(3.41) Similar to the proof of Lemma  2.2.1 in [27], we can obtain that {Ėƒâ€Œðē2(0,2),â€Ķ,Ėƒâ€Œðē2(𝑘𝑑−1,2);Ėƒâ€Œðē1(𝑘𝑑,1),â€Ķ,Ėƒâ€Œðē1(𝑘−1,1)} is the innovation sequence which is a mutually uncorrelated white noise sequence and spans the same linear space as ℒ{ðē2(0),â€Ķ,ðē2(𝑘𝑑−1);ðē1(𝑘𝑙),â€Ķ,ðē1(𝑘−1)} or equivalently ℒ{ðē𝑧(0),â€Ķ,ðē𝑧(𝑘−1)}.

Applying projection formula in the Krein space, Ė‚â€Œðą(𝑖,2)(𝑖=0,1,â€Ķ,𝑘𝑑) is computed recursively as(3.42)Ė‚â€Œðą(𝑖+1,2)=𝑖𝑗=0âŸĻðą(𝑖+1),Ėƒâ€Œðē2(𝑗,2)âŸĐ𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2)=ðīĖ‚â€Œðą(𝑖âˆĢ𝑖,2)+ðī𝑑Ė‚â€Œðąî€·ð‘–ð‘‘âˆĢ𝑖,2î€ļ+𝑓𝑖,Ė†â€Œðģ𝑓(𝑖âˆĢ𝑖),ðŪ(𝑖)î€ļ+ℎ𝑖,Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ,ðŪ(𝑖)î€ļ,𝑖=0,1,â€Ķ,𝑘𝑑−1,Ė‚â€Œðą(𝜏,2)=0,(𝜏=−𝑑,−𝑑+1,â€Ķ,0).(3.43) Note thatĖ‚â€Œðą(𝑖âˆĢ𝑖,2)=Ė‚â€Œðą(𝑖,2)+𝑃2(𝑖,𝑖)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑖,2)Ėƒâ€Œðē2(𝑖,2),Ė‚â€Œðąî€·ð‘–ð‘‘âˆĢ𝑖,2î€ļ=Ė‚â€Œðąî€·ð‘–ð‘‘,2î€ļ+𝑖𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2),(3.44) whereðķ2=ðķ𝑇ðđ𝑇ðŋ𝑇ðŧ𝑇î€ŧ𝑇,𝑃2(𝑖,𝑗)=âŸĻ𝐞(𝑖,2),𝐞(𝑗,2)âŸĐ,𝐞(𝑖,2)=ðą(𝑖)−Ė‚â€Œðą(𝑖,2),𝑅Ėƒð‘Ķ2(𝑖,2)=ðķ2𝑃2(𝑖,𝑖)ðķ𝑇2+𝑄ð‘Ģ2(𝑖),𝑄ð‘Ģ2(𝑖)=diagî€―ðž,−𝛞−2𝐞,−ð›ū2𝐞,âˆ’ð›―âˆ’2𝐞î€ū.(3.45) Substituting (3.44) into (3.43), we haveĖ‚â€Œðą(𝑖+1,2)=ðīĖ‚â€Œðą(𝑖,2)+ðī𝑑Ė‚â€Œðąî€·ð‘–ð‘‘,2î€ļ+𝑓𝑖,Ė†â€Œðģ𝑓(𝑖âˆĢ𝑖),ðŪ(𝑖)î€ļ+ℎ𝑖,Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ,ðŪ(𝑖)î€ļ+ðī𝑑𝑖−1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2)+ðū2(𝑖)Ėƒâ€Œðē2(𝑖,2),ðū2(𝑖)=ðī𝑑𝑃2𝑖𝑑,𝑖î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑖,2)+ðī𝑃2(𝑖,𝑖)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑖,2).(3.46) Moreover, taking into account (3.7) and (3.46), we obtain𝐞(𝑖+1,2)=ðī𝐞(𝑖,2)+ðī𝑑𝐞𝑖𝑑,2î€ļ+ðĩ𝐰(𝑖)−ðū2(𝑖)Ėƒâ€Œðē2(𝑖,2)−ðī𝑑𝑖−1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2),𝑖=0,1,â€Ķ,𝑘𝑑−1.(3.47) Consequently,𝑃2(𝑖−𝑗,𝑖+1)=âŸĻ𝐞(𝑖−𝑗,2),𝐞(𝑖+1,2)âŸĐ=𝑃2(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇2𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃2(𝑖−𝑗,𝑖)ðķ𝑇2ðū𝑇2(𝑖)−𝑖−1î“ð‘Ą=𝑖−𝑗𝑃2(𝑖−𝑗,ð‘Ą)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(ð‘Ą,2)ðķ2𝑃𝑇2𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑,𝑗=0,1,â€Ķ,𝑑,𝑃2(𝑖+1,𝑖+1)=âŸĻ𝐞(𝑖+1,2),𝐞(𝑖+1,2)âŸĐ=ðī𝑃2(𝑖,𝑖+1)+ðī𝑑𝑃2𝑖𝑑,𝑖+1î€ļ+ðĩ𝑄ð‘Ī(𝑖)ðĩ𝑇,(3.48) where 𝑄ð‘Ī(𝑖)=𝐞. Thus, 𝑃2(𝑖,𝑖)(𝑖=0,1,â€Ķ,𝑘𝑑) can be computed recursively as𝑃2(𝑖−𝑗,𝑖+1)=𝑃2(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇2𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃2(𝑖−𝑗,𝑖)ðķ𝑇2ðū𝑇2(𝑖)−𝑖−1î“ð‘Ą=𝑖−𝑗𝑃2(𝑖−𝑗,ð‘Ą)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(ð‘Ą,2)ðķ2𝑃𝑇2𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑,𝑃2(𝑖+1,𝑖+1)=ðī𝑃2(𝑖,𝑖+1)+ðī𝑑𝑃2𝑖𝑑,𝑖+1î€ļ+ðĩ𝑄ð‘Ī(𝑖)ðĩ𝑇,𝑗=0,1,â€Ķ,𝑑.(3.49)

Similarly, employing the projection formula in the Krein space, the optimal estimator Ė‚â€Œðą(𝑖,1)(𝑖=𝑘𝑑+1,â€Ķ,𝑘) can be computed by Ė‚â€Œðą(𝑖+1,1)=ðīĖ‚â€Œðą(𝑖,1)+ðī𝑑Ė‚â€Œðąî€·ð‘–ð‘‘,2î€ļ+𝑓𝑖,Ė†â€Œðģ𝑓(𝑖âˆĢ𝑖),ðŪ(𝑖)î€ļ+ℎ𝑖,Ė†â€Œðģℎ𝑖𝑑âˆĢ𝑖î€ļ,ðŪ(𝑖)î€ļ+ðū1(𝑖)Ėƒâ€Œðē1(𝑖,1)+ðī𝑑𝑘𝑑−1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2)+ðī𝑑𝑖−1𝑗=𝑘𝑑𝑃1𝑖𝑑,𝑗î€ļðķ𝑇1𝑅−1Ėƒð‘Ķ1(𝑗,1)Ėƒâ€Œðē1(𝑗,1),Ė‚â€Œðąî€·ð‘˜ð‘‘,1î€ļ=Ė‚â€Œðąî€·ð‘˜ð‘‘,2î€ļ,(3.50) whereðķ1=ðķ𝑇ðđ𝑇ðŋ𝑇î€ŧ𝑇,𝑃1(𝑖,𝑗)=⎧⎊âŽĻ⎊âŽĐâŸĻ𝐞(𝑖,2),𝐞(𝑗,1)âŸĐ,𝑖<𝑘𝑑,âŸĻ𝐞(𝑖,1),𝐞(𝑗,1)âŸĐ,𝑖â‰Ĩ𝑘𝑑,𝐞(𝑖,1)=ðą(𝑖)−Ė‚â€Œðą(𝑖,1),𝑅Ėƒð‘Ķ1(𝑖,1)=ðķ1𝑃1(𝑖,𝑖)ðķ𝑇1+𝑄ð‘Ģ1(𝑖),𝑄ð‘Ģ1(𝑖)=diagî€―ðž,−𝛞−2𝐞,−ð›ū2𝐞î€ū,ðū1(𝑖)=ðī𝑃1(𝑖,𝑖)ðķ𝑇1𝑅−1Ėƒð‘Ķ1(𝑖,1)+ðī𝑑𝑃1𝑖𝑑,𝑖î€ļðķ𝑇1𝑅−1Ėƒð‘Ķ1(𝑖,1).(3.51) Then, from (3.7) and (3.50), we can yield𝐞(𝑖+1,1)=ðī𝐞(𝑖,1)+ðī𝑑𝐞𝑖𝑑,2î€ļ+ðĩ𝐰(𝑖)−ðū1(𝑖)Ėƒâ€Œðē1(𝑖,1)−ðī𝑑𝑘𝑑−1𝑗=𝑖𝑑𝑃2𝑖𝑑,𝑗î€ļðķ𝑇2𝑅−1Ėƒð‘Ķ2(𝑗,2)Ėƒâ€Œðē2(𝑗,2)−ðī𝑑𝑖−1𝑗=𝑘𝑑𝑃1𝑖𝑑,𝑗î€ļðķ𝑇1𝑅−1Ėƒð‘Ķ1(𝑗,1)Ėƒâ€Œðē1(𝑗,1).(3.52) Thus, we obtain that(1)if 𝑖−𝑗â‰Ĩ𝑘𝑑, we have 𝑃1(𝑖−𝑗,𝑖+1)=âŸĻ𝐞(𝑖−𝑗,1),𝐞(𝑖+1,1)âŸĐ=𝑃1(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇1𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃1(𝑖−𝑗,𝑖)ðķ𝑇1ðū𝑇1(𝑖)−𝑖−1î“ð‘Ą=𝑖−𝑗𝑃1(𝑖−𝑗,ð‘Ą)ðķ𝑇1𝑅−1Ėƒð‘Ķ1(ð‘Ą,1)ðķ1𝑃𝑇1𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑,(3.53)(2)if 𝑖−𝑗<𝑘𝑑, we have𝑃1(𝑖−𝑗,𝑖+1)=âŸĻ𝐞(𝑖−𝑗,2),𝐞(𝑖+1,1)âŸĐ=𝑃1(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇2𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃1(𝑖−𝑗,𝑖)ðķ𝑇1ðū𝑇1(𝑖)−𝑘𝑑−1î“ð‘Ą=𝑖−𝑗𝑃2(𝑖−𝑗,ð‘Ą)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(ð‘Ą,2)ðķ2𝑃𝑇2𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑−𝑖−1î“ð‘Ą=𝑘𝑑𝑃1(𝑖−𝑗,ð‘Ą)ðķ𝑇1𝑅−1Ėƒð‘Ķ1(ð‘Ą,1)ðķ1𝑃𝑇1𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑,(3.54)𝑃1(𝑖+1,𝑖+1)=âŸĻ𝐞(𝑖−𝑗,2),𝐞(𝑖+1,1)âŸĐ=ðī𝑃1(𝑖,𝑖+1)+ðī𝑑𝑃1𝑖𝑑,𝑖+1î€ļ+ðĩ𝑄ð‘Ī(𝑖)ðĩ𝑇.(3.55) It follows from (3.53), (3.54), and (3.55) that 𝑃1(𝑖,𝑖)(𝑖=𝑘𝑑+1,â€Ķ,𝑘) can be computed by𝑃1(𝑖−𝑗,𝑖+1)=𝑃1(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇2𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃1(𝑖−𝑗,𝑖)ðķ𝑇1ðū𝑇1(𝑖)−𝑘𝑑−1î“ð‘Ą=𝑖−𝑗𝑃2(𝑖−𝑗,ð‘Ą)ðķ𝑇2𝑅−1Ėƒð‘Ķ2(ð‘Ą,2)ðķ2𝑃𝑇2𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑−𝑖−1î“ð‘Ą=𝑘𝑑𝑃1(𝑖−𝑗,ð‘Ą)ðķ𝑇1𝑅−1Ėƒð‘Ķ1(ð‘Ą,1)ðķ1𝑃𝑇1𝑖𝑑,ð‘Ąî€ļðī𝑇𝑑,𝑖−𝑗<𝑘𝑑,𝑃1(𝑖−𝑗,𝑖+1)=𝑃1(𝑖−𝑗,𝑖)ðī𝑇+𝑃𝑇1𝑖𝑑,𝑖−𝑗î€ļðī𝑇𝑑−𝑃1(𝑖−𝑗,𝑖)ðķ𝑇1ðū𝑇1(𝑖)−𝑖−1î“ð‘Ą=𝑖−𝑗𝑃1(𝑖−𝑗,ð‘Ą)ðķ𝑇1𝑅−1Ėƒð‘Ķ1(ð‘Ą,1)ðķ1𝑃𝑇1𝑖𝑑,ð‘Ąî€ļðīT𝑑,𝑖−𝑗â‰Ĩ𝑘𝑑,𝑃1(𝑖+1,𝑖+1)=ðī𝑃1(𝑖,𝑖+1)+ðī𝑑𝑃1𝑖𝑑,𝑖+1î€ļ+ðĩ𝑄ð‘Ī(