I. INTRODUCTION

The linear estimation problem has been an important control and signal processing community research topic since Wiener filtering^[1]. The linear estimation problem is usually studied under two indexes,one of which is $H_2$ performance (also termed as optimal index)^{[2, 3, 4]},the other is $H_{\infty}$ performance^{[5, 6]}. Kalman filter is an important approach to deal with linear estimation under the index of $H_2$^{[3, 4]}. For systems without multiplicative noises,the optimal singular system filtering problem has been well studied^{[7, 8, 9]} in recent years. Particularly,Reference ^[9] studied the singular filtering by state augmentation and standard results of non-singular systems^[7] considered the singular system risk-sensitive estimation problem, where the estimation including filter,predictor and smoother are given in a unified form based on J-spectral factorization.

The singular system optimal filter corrupted by multiplicative noises has also been studied well in recent years^{[10, 11, 12, 13]}. Particularly,in ^[11],the filtering problem for the singular or descriptor systems with multi-channel multiplicative noises is dealt with by augmented approach^[12] considered the optimal filtering based on projection theorem and standard Kalman filtering. In ^[13],singular value decomposition is used to tackle the singular system optimal Kalman filtering problem corrupted by multiplicative noise.

It should be noted that,the above references have not considered the singular multiplicative noise systems with delayed measurement,for which state augmentation approach is usually applied for non-singular systems^{[2, 3]},however,the state augmentation approach used to require large calculation amount. The optimal multi-step predictor for singular systems corrupted by multiplicative noises and with delay-observations has been given in ^[14],and the optimal fixed-small-lag and fixed-point smoothing for the above system has also been presented in ^{[15, 16]}. For the sake of presenting the comprehensive results of estimation,in this paper,we will study the Kalman filtering for time invariant discrete-time singular delayed systems corrupted by multiplicative noise,where the instantaneous measurements and delayed ones can be available. Such problem can be applied in many practical engineering problems such as wireless sensor network,detection of an underwater objection (for submarine),multiple-sensor fusion^[17]. By utilizing standard singular value decomposition,the restricted equivalent delayed system will first be given,then Kalman filters for the above restricted singular system will be presented by using the well-known re-organization of innovation analysis lemma developed in ^[18]. Then Kalman filters for the given singular delayed systems will be given in accordance with the results of the restricted equivalent delayed system. It is worth pointing out that the computation burden of the proposed method is less than that by augmentation especially when the dimension of the restricted equivalent delayed system is high. It will show that Kalman filters for the restricted equivalent delayed systems include two sets of standard Riccati equations for normal systems.

II.PROBLEM STATEMENT

The paper focuses on the linear discrete-time singular system corrupted by multiplicative noise,delayed measurements and known coefficients matrices as

$ F{ x}(t+1) = A { x}(t)+B{ u}(t),~\label{e2.1} $

(1)

$ { y}(t)=C{ x}(t)+D{ w}(t){ x}(t)+{ v}(t),~\label{e2.2} $

(2)

$ { y}_d(t)=C_d{ x}(t(d))+D_{d}{ w}(t){ x}(t(d))+{ v}_d(t),~\label{e2.3} $

(3)

where $t(d)=t-d$,the variables ${ x}(t)\in {\bf R}^{n}$,${ y}(t)\in R^{q}$,${ y}_d(t)$ $\in$ ${\bf R}^{q_d}$,${ u}(t)\in R^{p}$,${ v}(t)\in {\bf R}^{q}$,${ v}_d(t)\in R^{q_d}$,and${ w}(t)\in R^1$ is respectively the state,instantaneous measurement, delayed measurement,and noises (including input,observation noises,and multiplicative noise),and the other noises are also uncorrelated with known variance matrices as in ^{[14, 15, 16, 18, 19]}.

Before giving the problem to be dealt with,two assumptions will first be given similar to ^{[14, 15, 19]} as follows:

Assumption 1. $F$ is singular,and the system plant (1)-(3) is regular.

Assumption 2. The system is impulse free system.

The main problem in this paper can be described as follows:

Given the sequence of measurements ${ y}(s)|_{s=0}^t$ and the sequence of delayed measurements ${ y}_d(s)|_{s=d}^t$,find Kalman filter $\hat { x}(t|t)$ of ${ x}(t)$,where ${\bf \xi}(s)|_{s=0}^t=\xi(0),\xi(1),...,\xi(t)$.

III.MAIN RESULTS A.Restricted Equivalent Delayed System

Under Assumptions 1 and 2,in accordance with the traditional singular system result in ^{[12, 13, 14, 16]},there exist nonsingular matrices $\{J_1,J_2\}\in R^{n\times n}$,${ x}(t)=J_2[{{ x}_1(t)} { x}_2(t)]$,and the restricted equivalent system can be given as follows

$ { x}_1(t+1) =\ A_1 { x}_1(t)+B_1{ u}(t), $

(4)

$ { y}(t)=\ C_1{ x}_1(t)+C_2{ x}_2(t)+D_1{ w}(t){ x}_1(t) $

(5)

$ \begin{array}{l} {F_1}{x_2}(t + 1) = \;{x_2}(t) + {B_2}u(t),\; + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{D_2}w(t){x_2}(t) + v(t), \end{array} $

(6)

$ \begin{array}{l} {y_d}(t) = \;{C_{1d}}{x_1}(t(d)) + {C_{2d}}{x_2}(t(d))\; + \\ \;\;\;\;\;\;\;\;\;{D_{1d}}w(t){x_1}(t(d))\; + \\ \;\;\;\;\;\;\;\;{D_{2d}}w(t){x_2}(t(d)) + {v_d}(t), \end{array} $

(7)

where $F_1,A_1,B_1,B_2,C_1,...,D_{2d}$ can be referred to ^[14],^[16].

Remark 1. The restricted delayed system (4)-(7) is presented by using standard singular value decomposition,and it is equivalent to the singular system (1)-(3),so we can first give the Kalman filters for ${ x}_1(t)$ and ${ x}_2(t)$ in (4)-(7),then the needed Kalman filter $\hat{ x}(t|t)$ can be indirectly yielded from $\hat{ x}_1(t|t)$ and $\hat{ x}_2(t|t)$. For the restricted delayed system,Kalman filtering cannot be used directly due to the existence of delayed measurement and multiplicative noise. Usually,such a problem can be solved by system augmentation^[5],^[13] which will lead to much computation burden,in this paper,we propose a different method to tackle the singular system Kalman filtering problem corrupted by multiplicative noise and with delayed measurement.

B. Re-organization of Two Sequences of Measurements

First,we denote $Y_o(t)$ as the measurements of the system model (4)-(7),thus for $t\geq d$,we have

$ Y_o(t)=\left[\begin{array}{*{20}c}{ y}(t) \\ { y}_d(t)\end{array}\right]. $

(8)

This paper will pay attention to the case of $t\geq d$,when $t$ $<$ $d$,(7) cannot be used,$Y_o(t)$ is just ${ y}(t)$,and the problem will be the Kalman filtering for singular systems without delay which has been dealt with before.

Define

$ {Y_2}(t) = \left[{\begin{array}{*{20}{c}} {y(t)}\\ {{y_d}(t + d)} \end{array}} \right], $

(9)

$ {Y_1}(t) = y(t), $

(10)

then,similar to ^[18] we have ${\cal L}\{{\cal Y}_2(s)|_{s=0}^{t(d)};{\cal Y}_1(s)|_{s=t(d)+1}^t\}=$ ${\cal L}\{Y_o(s)|_{s=0}^t\}.$

(6) and (7) can be rewritten as

$ \begin{array}{l} {Y_i}(t) = \;{C_{1i}}{x_1}(t) + {C_{2i}}{x_2}(t) + {D_{1i}}(w,t){x_1}(t)\\ \;\;\;\;\;\;\;\;\;\; + {D_{2i}}(w,t){x_2}(t) + {V_i}(t),\quad i = 1,2, \end{array} $

(11)

where ${\cal C}_{11},{\cal C}_{12},...,{\cal D}_{22}(w,t)$ can be given as in ^[20]. From (4) and (5),we have

$ { x}_1(t)=A^t_1{ x}_1(0)+\sum_{i=0}^{t-1}A_1^{t-i-1}B_1{ u}(i),\label{e3.16} $

(12)

$ { x}_2(t)=-\sum_{i=0}^{\lambda-1}F_1^iB_2{ u}(t+i). $

(13)

Take (12) and (13) into (11),we have

$ {\cal Y}_i(t)={\cal C}_{1i} { x}_1(t)+{\cal D}_{1i}(w,t) { x}_1(t)+{\bar {\cal V}}_i(t),~~~\label{e3.18} $

(14)

where

$ {\bar {\cal V}}_i(t)=\ {\cal V}_i(t)-{\cal C}_{2i} \sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(t+j) -{\cal D}_{2i}(w,t) \sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(t+j),\quad i=1,2. $

C.Re-organization of Innovation Lemma

The re-organization of innovation lemma for the restricted equivalent delayed system (4)-(7) will be presented in this part,some definitions of $\hat{\xi}(t|s)$ and $\hat{\eta}(\tau,i)$ $(i=$ $1,2)$ can be referred to ^[18]. Some important denotations such as ${ e}_1(t,\cdot),$ $\tilde{{\cal Y}}_i(t),\Pi_1(t),P_1(t,\cdot),\{L,T,R\}(t,s,\cdot)$,$Q_{\bar{\cal Y}_i}(t),\{G,H\}(t,s,\cdot),\{G,H\}_j(t,s(s|l),\cdot)$ $(j=1,2,3)$ can be introduced similarly as in ^[20].

Lemma 1. ${\cal L}\{\tilde{\cal Y}_2(s)|_{s=0}^{t(d)};\tilde{\cal Y}_1(s)|_{s=t(d)+1}^t\}$ is an innovation sequence,and the spanned linear space is equivalent to ${\cal L}\{{\cal Y}_2(s)|_{s=0}^{t(d)};{\cal Y}_1(s)|_{s=t(d)+1}^t\}.$

Proof. The lemma can be given similar to ^[16].

Consider (14) with the projection theorem,we have

$ \hat{\cal Y}_i(t,i)={\cal C}_{1i}\hat{ x}_1(t,i)+\hat{\bar{\cal V}}_i(t,i), $

(15)

thus,combining (14) with (15) yields

$ \tilde{{\cal Y}}_i(t)={\cal C}_{1i}{ e}_1(t,i)+{\cal D}_{1i}(w,t){ x}_1(t)+\tilde{{\bar{\cal V}}}_i(t).\label{e3.30} $

(16)

D. Riccati Equation $P_1(t(d)+k+1,i)$ for ${ x_1(\cdot)}$

In this part,the error covariance matrix $P_1(t(d)+k+1,i)$ of ${ e}_1(t(d)+k+1,i)$ will be given.

By using traditional Kalman filtering with multiplicative noises ^{[11, 13, 14, 16, 17]},we can similarly give the recursive equation of $P_1(t(d)+1,2)=\langle{ e}_1(t(d)+1,2),{ e}_1(t(d)+1,2)\rangle$ with the initial values $G(0,0,2)=T(0,0,2),$ $G_1(0,0,2)=$ $G_3(0,$ $0|-1,2)=H_1(0,0,2)=H_2(0,0|-1,2)=0$,so the main problem is to present the recursive equation of $P_1(t(d)$ $+$ $k+1,1)$.

$P_1(t(d)+k+1,1)$ for $k=1,...,d-1$ can be further given as in Theorem 1 of ^[14] similarly,where $P_1(t+1,1)$ needs to be replaced by $P_1(t(d)+k+1,1)$,and the corresponding variable $t$ needs to be replaced by $t(d)+k$,and we will present the main results of the corresponding revised version together with proof.

From (16),since ${ x}_1(t(d)+k)=\hat{ x}_1(t(d)+k,1)+{ e}_1(t(d)$ $+$ $k,1)$,$\hat{ x}_1(t(d)+k,1)\perp \tilde{{\cal Y}}_1(t(d)+k)$,${\tilde{\bar{\cal V}}}_1(t(d)+k)=\bar{\cal V}_1(t(d)$ $+$ $k)-{\hat{\bar{\cal V}}}_1(t(d)+k,1)$ and ${ e}_1(t(d)+k,1)\perp {\hat{\bar{\cal V}}}_1(t(d)+k)$,then we have

$ \begin{array}{l} L(t(d) + k,t(d) + k,1)\\ = \langle {x_1}(t(d) + k),{\widetilde Y_1}(t(d) + k)\rangle \quad \\ = \langle {e_1}(t(d) + k,1),{\widetilde Y_1}(t(d) + k)\rangle \quad \\ = {P_1}(t(d) + k,1)C_{11}^{\rm{T}} + \langle {e_1}(t(d) + k,1),{\widetilde {\overline V }_1}(t(d) + k)\rangle \quad \\ = {P_1}(t(d) + k,1)C_{11}^{\rm{T}} + \langle {e_1}(t(d) + k,1),{\overline V _1}(t(d) + k)\quad \quad \\ - {\widehat {\overline V }_1}(t(d) + k,1)\rangle \\ = {P_1}(t(d) + k,1)C_{11}^{\rm{T}} + H_1^{\rm{T}}(t(d) + k,t(d) + k,1). \end{array} $

(17)

From (16),we have

$ Q_{\tilde{\cal Y}_1}(t(d)+k) \\ \quad =\langle\tilde{\cal Y}_1(t(d)+k),\tilde{\cal Y}_1(t(d)+k)\rangle \\ \quad ={\cal C}_{11}P_1(t(d)+k,1){\cal C}_{11}^{\rm T} \\ \quad\quad +{\cal C}_{11}\langle{ e}_1(t(d)+k,1),\bar{\cal V}_1(t(d)+k)\rangle \\ \quad\quad -{\cal C}_{11}\langle{ e}_1(t(d)+k,1),{\hat{\bar{\cal V}}}_1(t(d)+k,1)\rangle \\ \quad\quad +D_1M\Pi_1(t(d)+k)D_1^{\rm T} \\ \quad\quad +\langle{\bar{\cal V}}_1(t(d)+k),{ e}_1(t(d)+k,1)\rangle{\cal C}_{11}^{\rm T} \\ \quad\quad +R(t(d)+k,t(d)+k,1) \\ \quad\quad -\langle{\hat{\bar{\cal V}}}_1(t(d)+k,1),{ e}_1(t(d)+k,1)\rangle{\cal C}_{11}^{\rm T} \\ \quad\quad -\langle{\bar{\cal V}}_1(t(d)+k),{\hat{\bar{\cal V}}}_1(t(d)+k,1)\rangle \\ \quad\quad -\langle{\hat{\bar{\cal V}}}_1(t(d)+k,1),{\bar{\cal V}}_1(t(d)+k)\rangle \\ \quad\quad +\langle{\hat{\bar{\cal V}}}_1(t(d)+k,1),{\hat{\bar{\cal V}}}_1(t(d)+k,1)\rangle \\ \quad ={\cal C}_{11}P_1(t(d)+k,1){\cal C}_{11}^{\rm T}+D_1M\Pi_1(t(d)+k)D_1^{\rm T} \\ \quad\quad +R(t(d)+k,t(d)+k,1) \\ \quad\quad -H_2(t(d)+k,t(d)+k|t(d)+k-1,1) \\ \quad\quad +{\cal C}_{11}H_1^{\rm T}(t(d)+k,t(d)+k,1) \\ \quad\quad +H_1(t(d)+k,t(d)+k,1){\cal C}_{11}^{\rm T}. \label{e3.56} $

(18)

Combining (4) with $\Pi_1(t(d)+k+1)=\langle{ x}_1(t(d)+k+1),$ ${ x}_1(t(d)+k+1)\rangle$ yields

$ \Pi_1(t(d)+k+1)=A_1\Pi_1(t(d)+k)A_1^{\rm T} +B_1Q_uB_1^{\rm T},\label{e3.57a} $

(19)

and from ^[20] for $\{t,s\}=t(d)+k$,we have

$ \begin{array}{l} R(t(d) + k,t(d) + k,1)\\ = {Q_v} + \sum\limits_{j = 0}^{\lambda - 1} {{C_{21}}} F_1^j{B_2}{Q_u}B_2^{\rm{T}}{(F_1^j)^{\rm{T}}}C_{21}^{\rm{T}} + \\ M\sum\limits_{j = 0}^{\lambda - 1} {{D_2}} F_1^j{B_2}{Q_u}B_2^{\rm{T}}{(F_1^j)^{\rm{T}}}D_2^{\rm{T}}. \end{array} $

(20)

From the well-known projection theorem,

$ \hat{ x}_1(t(d)+k+1,1) \\[1mm] \quad ={\rm Proj}\{{ x}_1(t(d)+k+1)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}; \tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t(d)+k}\} \\[1mm] \quad ={\rm Proj}\{A_1{ x}_1(t(d)+k) \\[1mm] \quad\quad +B_1{ u}(t(d)+k)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}; \tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t(d)+k-1}\} \\[1mm] \quad\quad +{\rm Proj}\{A_1{ x}_1(t(d)+k)+B_1{ u}(t(d)+k) \\[1mm] \quad\quad \times |\tilde{\cal Y}_1(t(d)+k)\} \\[1mm] \quad =A_1\hat{ x}_1(t(d)+k,1)+B_1\hat{ u}(t(d)+k,1) \\[1mm] \quad\quad +A_1{\rm Proj}\{{ x}_1(t(d)+k)|\tilde{\cal Y}_1(t(d)+k)\} \\[1mm] \quad\quad +B_1{\rm Proj}\{{ u}(t(d)+k)|\tilde{\cal Y}_1(t(d)+k)\} \\[1mm] \quad =A_1\hat{ x}_1(t(d)+k,1) \\[1mm] \quad\quad +A_1K(t(d)+k,1)\tilde{\cal Y}_1(t(d)+k) \\[1mm] \quad\quad +B_1\hat{ u}(t(d)+k|t(d)+k,1),\label{e3.59} $

(21)

where

$ K(t(d)+k,1)=L(t(d)+k,t(d)+k,1) Q_{\tilde{\cal Y}_1}^{-}(t(d)+k),\label{e3.54} $

(22)

then by considering (4) and (21),we have

$ { e}_1(t(d)+k+1,1) \\[1mm] \quad =A_1{ e}_1(t(d)+k,1)+B_1{ u}(t(d)+k) \\[1mm] \quad\quad -B_1\hat{ u}(t(d)+k|t(d)+k,1)\\[1mm] \quad\quad -A_1K(t(d)+k,1)\tilde{\cal Y}_1(t(d)+k).\label{e3.60} $

(23)

Since ${ e}_1(t(d)+k+1,1)$ is uncorrelated with $\tilde{\cal Y}_1(t(d)+k)$,so from (23) it can be easily yielded as

$ P_1(t(d)+k+1,1)+A_1K(t(d)+k,1) \\[1mm] \quad\quad \times Q_{\tilde{\cal Y}_1}(t(d)+k)K^{\rm T}(t(d)+k,1)A_1^{\rm T}\\[1mm] \quad =A_1P_1(t(d)+k,1)A_1^{\rm T} \\[1mm] \quad\quad +A_1\langle{ e}_1(t(d)+k,1),{ u}(t(d)+k)\rangle B_1^{\rm T} \\[1mm] \quad\quad -A_1\langle{ e}_1(t(d)+k,1),\hat{ u}(t(d)+k|t(d)+k,1)\rangle \\[1mm] \quad\quad \times B_1^{\rm T}+B_1Q_uB_1^{\rm T} \\[1mm] \quad\quad + B_1\langle{ u}(t(d)+k),{ e}_1(t(d)+k,1)\rangle A_1^{\rm T} \\[1mm] \quad\quad -B_1\langle{ u}(t(d)+k),\hat{ u}(t(d)+k|t(d)+k,1)\rangle B_1^{\rm T} \\[1mm] \quad\quad +B_1\langle\hat{ u}(t(d)+k|t(d)+k,1),\\[1mm] \quad\quad\ \hat{ u}(t(d)+k|t(d)+k,1)\rangle B_1^{\rm T} \\[1mm] \quad\quad -B_1\langle\hat{ u}(t(d)+k|t(d)+k,1),{ e}_1(t(d)+k,1)\rangle A_1^{\rm T} \\[1mm] \quad\quad -B_1\langle\hat{ u}(t(d)+k|t(d)+k,1),{ u}(t(d)+k)\rangle B_1^{\rm T} \\[1mm] \quad =A_1P_1(t(d)+k,1)A_1^{\rm T}+B_1[Q_u \\[1mm] \quad\quad -G_3(t(d)+k,t(d)+k|t(d)+k,1)]B_1^{\rm T} +A_1G_1^{\rm T}(t(d)+k,t(d)+k,1)B_1^{\rm T} \\[1mm] +B_1G_1(t(d)+k,t(d)+k,1)A_1^{\rm T} .\label{e3.61} $

(24)

According to the projection theorem,we have

$ \hat{{ u}}(t(d)+k|s,1) \\ \quad ={\rm Proj}\{{ u}(t(d)+k)|\tilde{\cal Y}_2(s_1)|_{s_1=0}^{t(d)};\tilde{\cal Y}_1(s_1)|_{s_1=t(d)+1}^{s}\} \\ \quad ={\rm Proj}\{{ u}(t(d)+k)|\tilde{\cal Y}_2(s_1)|_{s_1=0}^{t(d)};\tilde{\cal Y}_1(s_1)|_{s_1=t(d)+1}^{s-1}\} \\ \quad\quad +{\rm Proj}\{{ u}(t(d)+k)|\tilde{\cal Y}_1(s)\} \\ \quad =\hat{{ u}}(t(d)+k|s-1,1) \\ \quad\quad +\langle{ u}(t(d)+k),\tilde{\cal Y}_1(s)\rangle Q_{\tilde{\cal Y}_1}^{-}(s)\tilde{\cal Y}_1(s) \\ \quad =\hat{{ u}}(t(d)+k|s-1,1) \\ \quad\quad +G(t(d)+k,s,1)Q_{\tilde{\cal Y}_1}^{-}(s)\tilde{\cal Y}_1(s), $

(25)

let $s=t(d)+k$,then

$ \hat{{ u}}(t(d)+k|t(d)+k,1) \\ \quad =\hat{{ u}}(t(d)+k|t(d)+k-1,1)+G(t(d) \\ \quad\quad +k,t(d)+k,1) Q_{\tilde{\cal Y}_1}^{-}(t(d)+k)\tilde{\cal Y}_1(t(d)+k),~~\label{e3.63} $

(26)

since

$ \hat{{ u}}(t(d)+k|t(d)+k-1,1)\in {\cal L}(\tilde{\cal Y}_2(s)|_{s=0}^{t(d)};\tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t(d)+k-1}), $

and

$ {e_1}(t(d) + k,1) \bot L({\widetilde Y_2}(s)|_{s = 0}^{t(d)};{\widetilde Y_1}(s)|_{s = t(d) + 1}^{t(d) + k - 1}), $

that is

$ {e_1}(t(d)+k,1)\perp \hat{{ u}}(t(d)+k|t(d)+k-1,1), $

so

$ \langle{ e}_1(t(d)+k,1),\hat{{ u}}(t(d)+k|t(d)+k,1)\rangle \\ \quad =\langle{ e}_1(t(d)+k,1),\hat{{ u}}(t(d)+k|t(d)+k-1,1) \\ \quad\quad +G(t(d)+k,t(d)+k,1) \\ \quad\quad \times Q_{\tilde{\cal Y}_1}^{-}(t(d)+k)\tilde{\cal Y}_1(t(d)+k)\rangle\\ \quad= \langle{ e}_1(t(d)+k,1),\hat{{ u}}(t(d)+k|t(d)+k-1,1)\rangle\\ \quad\quad +\langle{ e}_1(t(d)+k,1),G(t(d)+k,t(d)+k,1) \\ \quad\quad \times Q_{\tilde{\cal Y}_1}^{-}(t(d)+k)\tilde{\cal Y}_1(t(d)+k)\rangle \\ \quad = \langle{ e}_1(t(d)+k,1),G(t(d)+k,t(d)+k,1) \\ \quad\quad \times Q_{\tilde{\cal Y}_1}^{-}(t(d)+k) \\ \quad\quad \times [{\cal C}_{11}{ e}_1(t(d)+k,1)\\ \quad\quad + D_1{ w}(t(d)+k){ x}_1(t(d)+k) \\ \quad\quad + {\tilde{{\bar{\cal V}}}}_1(t(d)+k,1)]\rangle.\label{e3.64} $

(27)

Since

$ {\tilde{{\bar{\cal V}}}}_1(t(d)+k)={{\bar{\cal V}}}_1(t(d)+k)-{\hat{{\bar{\cal V}}}}_1(t(d)+k,1),\\ { e}_1(t(d)+k,1)\perp {\hat{{\bar{\cal V}}}}_1(t(d)+k,1) $

and

$ \langle{ e}_1(t(d)+k,1),{{\bar{\cal V}}}_1(t(d)+k,1)\rangle\\ \qquad =H_1^{\rm T}(t(d)+k,t(d)+k,1), $

so (27) can be rewritten as

$ \begin{array}{l} \langle {e_1}(t(d) + k,1),\widehat u(t(d) + k|t(d) + k,1)\rangle \;\\ \; = \langle {e_1}(t(d) + k,1),G(t(d) + k,t(d) + k,1)Q_{{{\widetilde Y}_1}}^ - (t(d) + k)\\ \times \left[ {{C_{11}}{e_1}(t(d) + k,1) + {D_1}w(t(d) + k){x_1}(t(d) + k)} \right.\\ + \left. {{{\widetilde {\overline V }}_1}(t(d) + k)} \right]\rangle \quad \\ = \left[ {{P_1}(t(d) + k,1)C_{11}^{\rm{T}} + H_1^{\rm{T}}(t(d) + k,t(d) + k,1)} \right]\\ \times Q_{{{\widetilde Y}_1}}^ - (t(d) + k){G^{\rm{T}}}(t(d) + k,t(d) + k,1)\\ = L(t(d) + k,t(d) + k,1)Q_{{{\widetilde Y}_1}}^ - (t(d) + k)\\ \times {G^{\rm{T}}}(t(d) + k,t(d) + k,1)\\ = K(t(d) + k,1){G^{\rm{T}}}(t(d) + k,t(d) + k,1). \end{array} $

(28)

In addition,since $\langle\hat{ u}(t(d)+k|t(d)+k,1),\tilde{ u}(t(d)+k|t(d)+k,1)\rangle=0$,

$ -\langle{ u}(t(d)+k),\hat{ u}(t(d)+k|t(d)+k,1)\rangle \\ \quad\quad +\langle\hat{ u}(t(d)+k|t(d)+k,1),\hat{ u}(t(d)+k|t(d)+k,1)\rangle \\ \quad\quad -\langle\hat{ u}(t(d)+k|t(d)+k,1),{ u}(t(d)+k)\rangle \\ \quad =-\langle{ u}(t(d)+k),\hat{ u}(t(d)+k|t(d)+k,1)\rangle \\ \quad\quad +\langle\hat{ u}(t(d)+k|t(d)+k,1),\hat{ u}(t(d)+k|t(d)+k,1)\rangle\\ \quad\quad -\langle\hat{ u}(t(d)+k|t(d)+k,1),\hat{ u}(t(d)+k|t(d)+k,1) \\ \quad\quad +\tilde{ u}(t(d)+k|t(d)+k,1)\rangle \\ \quad =-\langle{ u}(t(d)+k),\hat{ u}(t(d)+k|t(d)+k,1)\rangle \\ \quad =-G_3(t(d)+k,t(d)+k|t(d)+k,1).\label{e3.66} $

(29)

Thus,from (24),(28) and (29),the main Riccati equation can be yielded as

$ P_1(t(d)+k+1,1)\\ \quad =A_1P_1(t(d)+k,1)A_1^{\rm T}\\ \quad\quad +A_1\left[G_1^{\rm T}(t(d)+k,t(d)+k,1)-K(t(d)+k,1)\right. \\ \quad\quad \times\left. G^{\rm T}(t(d)+k,t(d)+k,1)\right]B_1^{\rm T} \\ \quad\quad +B_1\left[G_1(t(d)+k,t(d)+k,1)\right. \\ \quad\quad - \left.G(t(d)+k,t(d)+k,1)K^{\rm T}(t(d)+k,1)\right]A_1^{\rm T}\\ \quad\quad +B_1[Q_u-G_3(t(d)+k,t(d)+k|t(d)+k,1)]B_1^{\rm T} \\ \quad\quad -A_1K(t(d)+k,1)Q_{\tilde{\cal Y}_1}(t(d)+k) \\ \quad\quad \times K^{\rm T}(t(d)+k,1)A_1^{\rm T},\qquad k=1,...,d-1,~\label{e3.53} $

(30)

where $\{G,G_1,G_3,H_1,H_2\}(t(d)+k,\cdot,1)$ are given in Lemma 3.3 in ^[20].

E.Kalman Filter for ${ x_1(t)}$

In this part,based on the error covariance matrix $P_1(t,2)$ and $P_1(t(d)+k,1)$,we will give the Kalman filter for ${x}_1(t)$.

$\hat{ x}_1(t+1|t,2)$ can be given by traditional Kalman filtering as

$ \hat{ x}_1(t+1,2)=A_1\hat{ x}_1(t,2)+A_1K(t,2) \\ \quad \times\left[{\cal Y}_2(t)-{\cal C}_{12}\hat{ x}_1(t,2)-{\hat{\bar{\cal V}}}_2(t,2)\right]+B_1\hat{ u}(t|t,2), \\ \hat{ x}_1(0,2)=\hat{ x}_1(0|-1,2)=0,~~~\label{e3.67} $

(31)

where

$ \begin{array}{l} \widehat u(t|t,2) = \widehat u(t|t - 1,2) + G(t,t,2)Q_{{{\widetilde Y}_2}}^ - (t)\quad \\ \times \left[{{Y_2}(t) - {C_{12}}{{\hat x}_1}(t,2) - {{\widehat {\overline V }}_2}(t,2)} \right],\\ \widehat u(0| - 1,2) = 0, \end{array} $

(32)

$ {\hat{\bar{\cal V}}}_2(t,2)={\hat{\bar{\cal V}}}_2(t|t-1,2) \\ \quad ={\hat{\bar{\cal V}}}_2(t|t-2,2)+H(t,t-1,2)Q_{\tilde{\cal Y}_2}^{-}(t-1)\\ \quad\quad \times\left[{\cal Y}_2(t-1)-{\cal C}_{12}\hat{ x}_1(t-1,2)-{\hat{\bar{\cal V}}}_2(t-1,2)\right],\\ {\hat{\bar{\cal V}}}_2(0,2)=0.~~\label{e3.69} $

(33)

$\hat{ x}_1(t+1|t+1,1)$ can be given as follows.

Theorem 1. Under Assumptions 1 and 2,for the singular system model (1)-(3) and the corresponding restricted equivalent delayed system model (4)-(6),Kalman filter $\hat{ x}_1(t(d)$ $+$ $k + 1|t(d)+k+1,1)$ can be given as

$ \hat{ x}_1(t(d)+k+1|t(d)+k+1,1) \\ \quad =\hat{ x}_1(t(d)+k+1,1)+K(t(d)+k+1,1) \\ \quad\quad \times\Big[{\cal Y}_1(t(d)+k+1)-{\cal C}_{11}\hat{ x}_1(t(d)+k+1,1)\\ \quad\quad -{\hat{\bar{\cal V}}}_1(t(d)+k+1,1)\Big],\label{e3.70} $

(34)

where

$ \hat{ x}_1(t(d)+k+1,1) \\ \quad =A_1\hat{ x}_1(t(d)+k,1)+A_1K(t(d)+k,1) \\ \quad\quad \times\Big[{\cal Y}_1(t(d)+k)-{\cal C}_{11}\hat{ x}_1(t(d)+k,1) \\ \quad\quad -{\hat{\bar{\cal V}}}_1(t(d)+k,1)\Big]\notag\\ \quad\quad +B_1\hat{ u}(t(d)+k|t(d)+k,1),\\ {\hat{ x}}_1(t(d)+1,1)=\hat{ x}_1(t(d)+1,2),\\ \qquad\qquad\qquad\qquad\qquad\qquad k=0,...,d-1,\label{e3.71} $

(35)

and

$ \hat{{ u}}(t(d)+k|t(d)+k,1) \\ \quad =\hat{{ u}}(t(d)+k|t(d)+k-1,1) \\ \quad\quad +G(t(d)+k,t(d)+k,1)Q_{\tilde{\cal Y}_1}^{-}(t(d)+k) \\ \quad\quad \times\Big[{\cal Y}_1(t(d)+k)-{\cal C}_{11}\hat{ x}_1(t(d)+k,1) \\ \quad\quad -{\hat{\bar{\cal V}}}_1(t(d)+k,1)\Big],\\ \hat{ u}(t(d)+1,1)=\hat{ u}(t(d)+1,2),~\label{e3.72} $

(36)

$ {\hat{\bar{\cal V}}}_1(t(d)+k|t(d)+k-1,1) \\ \quad ={\hat{\bar{\cal V}}}_1(t(d)+k|t(d)+k-2,1) \\ \quad\quad +H(t(d)+k,t(d)+k-1,1)Q_{\tilde{\cal Y}_1}^{-}(t(d)+k-1) \\ \quad\quad \times \Big[{\cal Y}_1(t(d)+k-1)-{\cal C}_{11}\hat{ x}_1(t(d)+k-1,1) \\ \quad\quad - {\hat{\bar{\cal V}}}_1(t(d)+k-1,1)\Big],\\ {\hat{\bar{\cal V}}}_1(t(d)+1,1)={\hat{\bar{\cal V}}}_1(t(d)+1,2),\notag\\ {\hat{\bar{\cal V}}}_1(t(d)+1,2) \\ \quad ={\hat{\bar{\cal V}}}_1(t(d)+1|t(d)-1,2) \\ \quad\quad +H(t(d)+1,t(d),12)Q_{\tilde{\cal Y}_2}^{-}(t(d)) \quad\quad \times\left[{\cal Y}_2(t(d))-{\cal C}_{12}\hat{ x}_1(t(d),2) -{\hat{\bar{\cal V}}}_2(t(d),2)\right],~\label{e3.73a} $

(37)

$K(t(d)+k,1)$ is as in (22),$\hat{ x}_1(t(d)+1,2)$ is from (31),$\{G,$ $H\}(t(d)+k,~\cdot,~1),~H(t(d)+1,t(d),12)$ are as in Lemma 3.4 in ^[20].

Proof.~From the projection theorem,we have

$ \hat{ x}_1(t(d)+k+1|t(d)+k+1,1) \\ \quad = {\rm Proj}\left\{{ x}_1(t(d)+k+1)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}; \tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t(d)+k+1}\right\} \\ \quad =\hat{ x}_1(t(d)+k+1,1) \\ \quad\quad +{\rm Proj}\left\{{ x}_1(t(d)+k+1)|\tilde{\cal Y}_1(t(d)+k+1)\right\} \\ \quad =\hat{ x}_1(t(d)+k+1,1) \\ \quad\quad +\langle{ x}_1(t(d)+k+1),\tilde{\cal Y}_1(t(d)+k+1)\rangle \\ \quad\quad \times Q_{\tilde{\cal Y}_1}^{-}(t(d)+k+1)\tilde{\cal Y}_1(t(d)+k+1) \\ \quad =\hat{ x}_1(t(d)+k+1,1) \\ \quad\quad + K(t(d)+k+1,1)\tilde{\cal Y}_1(t(d)+k+1),\label{e3.74} $

(38)

combining (39) with (15) yields (34).

From (21) and (15),

$ \hat{ x}_1(t(d)+k+1,1) \\ \quad =A_1\hat{ x}_1(t(d)+k,1) \\ \quad\quad +A_1K(t(d)+k,1)\tilde{\cal Y}_1(t(d)+k) \\ \quad\quad +B_1\hat{ u}(t(d)+k|t(d)+k,1) \\ \quad = A_1\hat{ x}_1(t(d)+k,1)+B_1\hat{ u}(t(d)+k|t(d)+k,1)\\ \quad\quad +A_1K(t(d)+k,1)\Big[{\cal Y}_1(t(d)+k) \\ \quad\quad -{\cal C}_{11}{\hat{ x}}_1(t(d)+k,1)-{\hat{\bar{\cal V}}_1}(t(d)+k,1)\Big], $

(39)

which is (35).

From (26) and (15),

$ \hat{{ u}}(t(d)+k|t(d)+k,1) \\ \quad =\hat{{ u}}(t(d)+k|t(d)+k-1,1) \\ \quad\quad +G(t(d)+k,t(d)+k,1) \\ \quad\quad \times Q_{\tilde{\cal Y}_1}^{-}(t(d)+k)\tilde{\cal Y}_1(t(d)+k)\\ \quad =\hat{{ u}}(t(d)+k|t(d)+k-1,1) \\ \quad\quad +G(t(d)+k,t(d)+k,1)Q_{\tilde{\cal Y}_1}^{-}(t(d)+k) \\ \quad\quad \times\Big[{\cal Y}_1(t(d)+k)-{\cal C}_{11}\hat{ x}_1(t(d)+k,1) \\ \quad\quad -{\hat{\bar{\cal V}}}_1(t(d)+k,1)\Big],\label{e3.76} $

(40)

which is (36).

From the projection theorem,we have

$ \begin{array}{l} {\widehat {\overline V }_1}(t(d) + k|s,1) = \\ \;\;\;{\rm{Proj}}\{ {\overline V _1}(t(d) + k)|{\widetilde Y_2}(s)|_{s = 0}^{t(d)};\quad \\ \;\;\;\;{\widetilde Y_1}(s)|_{s = t(d) + 1}^s\} \\ = {\rm{Proj}}\{ {\overline V _1}(t(d) + k)|{\widetilde Y_2}(s)|_{s = 0}^{t(d)};\quad \\ \;\;\;\;\;{\widetilde Y_1}(s)|_{s = t(d) + 1}^{s - 1}\} \quad \\ \;\;\;\;\; + {\rm{Proj}}\{ {\overline V _1}(t(d) + k)|{\widetilde Y_1}(s)\} = \\ \;\;\;\;\;{\widehat {\overline V }_1}(t(d) + k|s - 1,1)\\ \;\;\;\; + \langle {\overline V _1}(t(d) + k),{\widetilde Y_1}(s)\rangle Q_{{{\widetilde Y}_1}}^ - (s){\widetilde Y_1}(s)\\ \;\;\;\;\;\; = {\widehat {\overline V }_1}(t(d) + k|s - 1,1)\\ \;\;\;\;\; + H(t(d) + k,s,1)Q_{{{\widetilde Y}_1}}^ - (s){\widetilde Y_1}(s), \end{array} $

(41)

then combine (42) with (15),(37) can be given when $s=t(d)$ $+$ $k-1$.

Similarly,

$ {\hat{\bar{\cal V}}}_1(t(d)+1,2) \\ \quad ={\rm Proj}\left\{\bar{\cal V}_1(t(d)+1)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}\right\} \\ \quad ={\rm Proj}\left\{\bar{\cal V}_1(t(d)+1)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)-1}\right\} \\ \quad\quad +{\rm Proj}\left\{{\bar{\cal V}}_1(t(d)+1)|\tilde{\cal Y}_2(t(d))\right\} \\ \quad ={\hat{\bar{\cal V}}}_1(t(d)+1|t(d)-1,2) \\ \quad\quad +H(t(d)+1,t(d),12)Q_{\tilde{\cal Y}_2}^{-}(t(d)) \\ \quad\quad \times \left[{\cal Y}_2(t(d))-{\cal C}_{12}\hat{ x}_1(t(d),2) -{\hat{\bar{\cal V}}}_2(t(d),2)\right],~~\label{e3.77a} $

(42)

which is (38),then the proof is over.

F.Kalman Filter for ${ x_2(t)}$

Kalman filter $\hat{ x}_2(t|t)$ for the second restricted state variable can be given as in Theorem 2.

Theorem 2. Given the initial value for the system model (5)-(7) with ${ x}_2(0)=-\sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(j)$,$\hat{ x}_2(t|t)=\hat{ x}_2(t|t,1)$ can be given as

$ \hat{ x}_2(t|t,1)=-\sum_{j=0}^{\lambda-1}F_1^jB_2\hat{ u}(t+j|t,1),~\label{e3.94} $

(43)

where

$ \hat{ u}(t+j|t,1)=\hat{ u}(t+j|t-1,1) \\ \quad\ + G(t+j,t,1)Q_{\tilde{\cal Y}_1}^{-}(t)\tilde{\cal Y}_1(t),\label{e3.95} $

(44)

with $\hat{ u}(t+j|t(d),1)=\hat{ u}(t+j|t(d),2)$ being given as

$ \hat{ u}(t+j|t(d),2)=\hat{ u}(t+j|t(d)-1,2) \\ \quad\ +G(t+j,t(d),2)Q_{\tilde{\cal Y}_2}^{-}(t(d)) \tilde{\cal Y}_2(t(d)),\label{e3.95a} $

(45)

with

$ {\tilde{\cal Y}}_i(t)={\cal Y}_i(t)-{\cal C}_{1i}{\hat{ x}}_1(t,i)+{\hat{\bar{\cal V}}}_i(t,i),~~i=1,2,~\label{e3.96} $

(46)

and $\hat{ x}_1(t,1)$ is from (35),and $\hat{ x}_1(t(d),2)$ is from (31).

Proof. Similar to the above projection formulae and (13),we have

$ \hat{ x}_2(t|t,1)={\rm Proj}\left\{{ x}_2(t)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}; \tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t}\right\}\\ \quad ={\rm Proj}\Big\{-\sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(t+j)|\tilde{\cal Y}_2(s)|_{s=0}^{t(d)}; \\ \quad\quad \tilde{\cal Y}_1(s)|_{s=t(d)+1}^{t}\Big\} \\ \quad =-\sum_{j=0}^{\lambda-1}F_1^jB_2\hat{ u}(t+j|t,1),\label{e3.97} $

(47)

which is (44).

(45) can be obtained from (25),(46) can be obtained similar to (45),and (47) can be obtained from (15).

G.Main Results

Having given Kalman filters $\hat{ x}_1(t|t,1)$ and $\hat{ x}_2(t|t,1)$,Kalman filter for the delayed singular system corrupted by multiplicative noises will be proposed without proof,

Theorem 3. For (1)-(3) under Assumptions 1 and 2,if the initial condition $[{0 \ I}]J_2^{-1}{ x}(0)= -\sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(j)$ or ${ x}_2(0)$ $=$ $-\sum_{j=0}^{\lambda-1}F_1^jB_2{ u}(j)$ is satisfied,then Kalman filter $\hat{ x}(t|t)=$ $\hat{ x}(t|t,1)$ can be given by

$ \hat{ x}(t|t)=J_2\left[\begin{array}{*{20}c}\hat{ x}_1(t|t,1)\\ \hat{ x}_2(t|t,1)\end{array}\right],\label{e3.98} $

(48)

where $\hat{ x}_1(t|t,1)$ is from (34),$\hat{ x}_2(t|t,1)$ is from (44).

IV.NUMERICAL EXAMPLE

In this part,a numerical example of petroleum seismic exploration will be given. As is well known,during the process of petroleum seismic exploration,there exist many uncertain factors,such as time-variant,uncertainties,the spreading loss,transmission of the source wavelet,and so on. So,only additive noise can not describe this case,so we use multiplicative noise in the system model to simulate this case. In addition,in this paper,to show the comparison between Kalman filter and the multiple-step predictor,the system model is same as in ^[14]. By computing, there exists

$ {J_1} = \left[{\begin{array}{*{20}{c}} 0&1&0\\ 1&{ - 1}&0\\ 0&0&2 \end{array}} \right],{J_2} = \left[{\begin{array}{*{20}{c}} 1&0&0\\ 1&1&0\\ 0&0&1 \end{array}} \right], $

(49)

and

$ \begin{array}{l} {A_1} = \left[ {\begin{array}{*{20}{c}} {0.4}&0\\ 0&{ - 0.4} \end{array}} \right],{B_1} = \left[ {\begin{array}{*{20}{c}} 1\\ 0 \end{array}} \right]\\ N = 0,{F_1} = 0,\lambda = 1,{B_2} = 4,\\ {C_1} = \left[ {\begin{array}{*{20}{c}} {2\;\;\;\;1} \end{array}} \right],{C_2} = 1,{C_{1d}} = \left[ {3\;\;\;\;\;2} \right],{C_{2d}} = 1,\\ {D_1} = \left[ {\begin{array}{*{20}{c}} 3&2 \end{array}} \right],{D_2} = 2,{D_{1d}} = \left[ {3\;\;\;\;\;\;1} \right],{D_{2d}} = 1. \end{array} $

Computing as in Theorems 1-3,we have

$ \hat{ x}(t|t)=J_2 \left[\begin{array}{*{20}c}\hat{ x}_1(t|t,1)\\ \hat{ x}_2(t|t,1)\end{array}\right] \\[1mm] \quad =\left[\begin{array}{*{20}c}\hat{ x}_{11}(t|t,1) \\ \hat{ x}_{11}(t|t,1)+\hat{ x}_{12}(t|t,1)\\ \hat{ x}_2(t|t,1)\end{array}\right]=\left[\begin{array}{*{20}c}\hat{ x}_{F}(t|t)\\ \hat{ x}_{S}(t|t)\\\hat{ x}_T(t|t)\end{array}\right], $

(50)

then the origin,its optimal filter $\hat{ x}(t|t)$ and 3-step predictor $\hat{ x}(t$ $+$ $3|t)$ can be drawn,the first sub-state filter $\hat{ x}_{F}(t|t)=\hat{ x}_{11}(t|t,$ $1)$ and predictor $\hat{ x}_{F}(t+3|t)=\hat{ x}_{11}(t+3|t,1)$ are as in Fig.1,the second sub-state $\hat{ x}_{S}(t|t)=\hat{ x}_{11}(t|t,1)+\hat{ x}_{12}(t|t,1)$ and predictor $\hat{ x}_{S}(t+3|t)=\hat{ x}_{12}(t+3|t,1)$ are drawn in Fig.2,and the third sub-state $\hat{ x}_{T}(t|t)=\hat{ x}_2(t|t,1)$ and predictor $\hat{ x}_{T}(t+3|t)$ $=\hat{ x}_{2}(t+3|t,1)$ are as in Fig.3. In Fig.4,the filter error variance and the predictor error variance are given.

From Figs. 1-3,the optimal filters $\hat{ x}_{F}(t|t)$, $\hat{ x}_{S}(t|t)$,$\hat{ x}_{T}(t|t)$ of $\hat{ x}(t|t)$ for the original system (1)-(3) can give good estimation, and they are better than the 3-step predictor,which can also be verified in Fig.4.

V.CONCLUSION

Kalman filtering problem for discrete-time singular system corrupted by multiplicative noises is dealt with,and the system model includes two measurements,one of which is delayed measurement. By utilizing the method of singular value decomposition and the method of re-organization of innovation analysis lemma,Kalman filters and the corresponding filters for the restricted equivalent delayed system and the original singular system are given. The proposed Kalman filtering result extends the series of the estimation for singular system with multiplicative noises and delayed measurement ^{[14, 15, 16, 20]},where ^[14] has studied the optimal multi-step predictor for singular systems corrupted by multiplicative noises and delayed observations,and the optimal FSL (fixed-small-lag) and FP (fixed-point) smoothing for the above system has also been presented in ^{[15, 16]}. From the results of the proposed figs in the numerical example,the presented approach can estimate the original state well,and it estimates better than $3$-step predictor.