Research ArticleA Class of Transformation Matrices and Its Applications
Wenhui Liu1 Feiqi Deng1 Jiarong Liang2 and Haijun Liu3
1 College of Automation Science and Engineering South China University of Technology Guangzhou 510641 China2 School of Computer Electronics and Information Guangxi University Nanning 530004 China3 School of Mathematics and Statistics Zhengzhou University Zhengzhou 450001 China
Correspondence should be addressed to Feiqi Deng aufqdengscuteducn
Received 8 November 2013 Revised 13 February 2014 Accepted 13 February 2014 Published 1 April 2014
Academic Editor Turgut Ozis
Copyright copy 2014 Wenhui Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper studies a class of transformation matrices and its applications Firstly we introduce a class of transformation matricesbetween two different vector operators and give some important properties of it Secondly we consider its two applications Thefirst one is to improve Qian Jilingrsquos formula And the second one is to deal with the observability of discrete-time stochastic linearsystems with Markovian jump and multiplicative noises A new necessary and sufficient condition for the weak observability willbe given in the second application
1 Introduction
The vector operator is an important concept in matrix analy-sis and an effective mathematical tool in many applications[1] For the symmetric case we will consider a new vectoroperator We find that there exists a class of transformationmatrices between the two vector operators and they can beused to solve many problems In order to demonstrate theimportance of these transformation matrices this paper willconsider their two applications in control theory
Qian Jilingrsquos formula is an important and interestingformula to compute the Lyapunov functions [2] Our paperwill improve Qian Jilingrsquos formula by the help of thesetransformation matrices
Observability and detectability are two basic concepts incontrol theory [3] These concepts of deterministic systemshad been extended to the stochastic systems over the past fewdecades such as [4ndash11] and references therein Particularlythis paper will focus on the observabilitydetectability fordiscrete-time stochastic linear systems The definition ofthe uniform observabilitydetectability [4] for determin-istic discrete-time time-varying linear systems had beenextended to stochastic linear systems in [5] Actually theweakobservabilitydetectability in [5] and the uniform observabil-itydetectability in [4] are consistent Reference [5] showedthat the weak detectability was weaker than the mean square
detectability Still the weak detectability plays the samerole in the discussion of algebraic Lyapunov and Riccatiequations Under the same framework [6] investigated theobservability (ie the weak observability in [5]) problemsfor a class of discrete-time stochastic linear systems subjectto Markovian jump and multiplicative noises and got somenecessary and sufficient conditions Reference [7] studiedthe equivalence between two different definitions of theobservabilitydetectability for discrete-time stochastic linearsystems These are good works about the observability anddetectability of discrete-time stochastic linear systems How-ever our paper will further study the weak observability fordiscrete-time stochastic linear systems with the help of thesetransformation matrices The system in [6] is more compli-cated than ours but we get some more profound conclusionswhich are not included in [6] Although the systems in [5 7]are two special cases of our system our results are not theirparallel results We need these transformation matrices inorder to get these new results
The outline of this paper is as follows In Section 2we define a class of transformation matrices which existsindependently and study its important properties Section 3considers its applications We use it to improve Qian Jilingrsquosformula in Section 31 We use it to deal with the weakobservability of discrete-time stochastic linear systems withMarkovian jump and multiplicative noises in Section 32
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 742098 11 pageshttpdxdoiorg1011552014742098
2 Abstract and Applied Analysis
A new necessary and sufficient condition for the weakobservability will be givenNotations 119877119899 is 119899-dimensional real Euclidean space 119862119899 is 119899-dimensional complex space 119860119879 |119860| and tr(119860) respectivelydenote the transpose determinant and trace of 119860 119860 gt
(ge)0 means that 119860 is a symmetric positive (semipositive)definite matrix E(sdot) denotes the mathematical expectationof a random variable 119885+ ≜ 0 1 2 119878 ≜ 1 2 119904where 119904 is a positive integer 119877119899times119898 denotes 119899 times 119898 real matrixspace with the inner product ⟨119860 119861⟩ ≜ tr(119860119879119861) 119877119899times119898
119904≜
(1198801 119880
119904) | 119880
119894isin 119877119899times119898
119894 isin 119878 with the inner product⟨119880 119881⟩ ≜ sum
119904
119894=1tr(119880119879119894119881119894) 1119860(119909) ≜
1 119909isinA0 119909notin119860
2 Transformation Matrix and Its Properties
21 Transformation Matrix between Two Vector Operators
Definition 1 (see [1]) 1198711 119862119899times119898
rarr 119862119899119898times1 is called a
vector operator by column if 1198711(119883) = (119909
11 11990921 119909
1198991 11990912
11990922 119909
1198992 119909
1119898 1199092119898 119909
119899119898)119879 for any119883 = (119909
119894119895)119899times119898
Definition 2 1198712 119882119899times119899
rarr 119862(119899(119899+1)2)times1 is called a half
vector operator by column if 1198712(119883) = (119909
11 11990921 119909
1198991
11990922 119909
1198992 119909
119899119899)119879 for any 119883 = (119909
119894119895)119899times119899
where 119882119899times119899 ≜119883 | 119883
119879
= 119883 isin 119862119899times119899
Definition 3 (see [1]) For matrices 119860 = (119886119894119895)119899times119898
and 119861 =
(119887119894119895)119904times119905 one calls matrix 119860 otimes 119861 = (119886
119894119895119861)119899times119898
the Kroneckerproduct of matrices 119860 and 119861
It is easy to find that 1198711and 119871
2have many similar
properties such as the following
(a) 1198711and 119871
2are all linear operators
(b) 119883 = 119884 hArr 1198711(119883) = 119871
1(119884)
(c) 119883 = 119884 hArr 1198712(119883) = 119871
2(119884) where119883119884 isin 119882
119899times119899
For convenience we will adopt the following notations119864119899119894119895
denotes a 119899 times 119899 matrix its (119894 119895)-element is 1 and itis zero elsewhere
119867(119899) denotes a 1198992times(119899(119899+1)2)matrix Its column vectorsare denoted by 119867
11
11986721
1198671198991
11986722
1198671198992
119867119899119899
from left to right and row vectors are denoted by11986711 11986721 119867
1198991 119867
1119899 1198672119899 119867
119899119899from top to bottom
where
119867119894119895
≜
1198711(119864119899119894119895
) 119894 = 119895
1198711(119864119899119894119895
+ 119864119899119895119894
) 119894 gt 119895
(1)
It is easy to prove
119867119894119895=
119871119879
2(119864119899119895119894
) 119894 lt 119895
119871119879
2(119864119899119894119895
) 119894 ge 119895
(2)
We will denote119867 = 119867119899≜ 119867(119899)
119867minus
(119899) denotes a (119899(119899 + 1)2) times 1198992 matrix Its
column vectors are denoted by 119867minus11
119867minus21
119867minus1198991
119867minus1119899
119867minus2119899
119867minus119899119899 from left to right and row vectors are
denoted by 119867minus
11 119867minus
21 119867
minus
1198991 119867minus
22 119867
minus
1198992 119867
minus
119899119899from
top to bottom where 119867minus
119894119895≜ 119871119879
1(119864119899119894119895
) We will denote119867minus
= 119867minus
119899≜ 119867minus
(119899)For example
119867(1) = [1] 119867minus
(1) = [1]
119867 (2) =[[[
[
1 0 0
0 1 0
0 1 0
0 0 1
]]]
]
119867minus
(2) = [
[
1 0 0 0
0 1 0 0
0 0 0 1
]
]
(3)
and so forthConsidering Property 3 we generally call 119867 and 119867
minus
transformation matrices between 1198711and 119871
2
22 Properties of Transformation Matrix
Property 1 Consider
119867minus
119899119867119899= 119868119899(119899+1)2
(4)
Proof For 119896 = 119897
119867minus
119894119895119867119896119897
= 119871119879
1(119864119899119894119895
) 1198711(119864119899119896119896
) = 1 119894 = 119895 = 119896
0 others(5)
For 119896 gt 119897
119867minus
119894119895119867119896119897
= 119871119879
1(119864119899119894119895
) 1198711(119864119899119896119897
+ 119864119899119897119896
) = 1 119894 = 119896 119895 = 119897
0 others(6)
Thus119867minus119899119867119899= 119868119899(119899+1)2
Remark 4 Generally119867119899119867minus
119899= 1198681198992 such as
1198672119867minus
2=[[[
[
1 0 0 0
0 1 0 0
0 1 0 0
0 0 0 1
]]]
]
(7)
Property 2 Consider119867119867minus
119867 = 119867119867minus119867119867minus = 119867minus (119867minus119867)119879 = 119867minus119867
Remark 5 Generally (119867119867minus)119879 =119867119867minus such as119867
2119867minus
2which is
not a symmetric matrixThis indicates that119867minus is the 1 2 3-generalized inverse matrix of 119867 but not the 1 2 3 4-generalized inverse matrix of119867 (see [1])
Property 3 If 119883119879 = 119883 ≜ (119909119894119895)119899times119899
then 1198671198712(119883) = 119871
1(119883)
119867minus
1198711(119883) = 119871
2(119883)
Abstract and Applied Analysis 3
Proof Consider
∵ 1198671198941198951198712(119883) = 119871
119879
2(119864119899119894119895
) 1198712(119883) = 119909
119894119895(119894 ge 119895)
1198671198941198951198712(119883) = 119871
119879
2(119864119899119895119894
) 1198712(119883) = 119909
119895119894= 119909119894119895
(119894 lt 119895)
there4 1198671198712(119883) = 119871
1(119883)
1198712(119883) = 119867
minus
1198711(119883) (by Property 1)
(8)
Property 4 For matrices 119860119899times119899 119861119903times119899
and 119883119879119899times119899
= 119883 we havethe following
(a) 1198711(119860119883 + 119883119860
119879
) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)
(b) 1198711(119861119883119861
119879
) = (119861 otimes 119861)1198711(119883)
(c) 1198712(119860119883 + 119883119860
119879
) = 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
(d) 1198712(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)
Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore
1198712(119860119883 + 119883119860
119879
) = 119867minus
1198711(119860119883 + 119883119860
119879
)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
1198712(119861119883119861
119879
) = 119867minus
1198711(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
(9)
Remark 6 If 1198711and 119871
2are defined by row that is
1198711(119883) = (119909
11 11990912 119909
1119898 11990921 11990922
1199092119898 119909
1198991 1199091198992 119909
119899119898)119879 for 119883 isin 119862
119899times119898
1198712(119883) = (119909
11 11990912 119909
1119899 11990922
1199092119899 119909
119899119899)119879 for 119883 isin 119882
119899times119899
(10)
we find that Properties 3 and 4 are still true
Property 5 For all 119860 isin 119862119899times119899 then
(a) 120590(119860 otimes 119860) = 120590(119867minus
(119860 otimes 119860)119867) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)
(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860
119879
)119867)
where 120590(sdot) denotes the set of all eigenvalues of matrix
Proof (a) Let
1205901(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205902(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883119860119879
= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205901(119860) = 120590 (119860 otimes 119860)
(11)
If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus
(119860 otimes 119860)1198671198712(119883) =
1205821198712(119883) Thus 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)Next we prove 120590
1(119860) = 120590
2(119860) Obviously 120590
2(119860) sube 120590
1(119860)
so we only need to prove 1205901(119860) sube 120590
2(119860)
For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =
119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1]) Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119910119909119879
= 119860119909119910119879
119860119879
+ 119860119910119909119879
119860119879
= 119860119885119860119879
(12)
If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590
1(119860) = 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590
1(119860119879
otimes 119860119879
) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)
(b) Let
1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883 + 119883119860119879
= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)
(13)
If 119883119879 = 119883 then 119860119883 + 119883119860119879
= 120582119883 hArr 119867minus
(119860 otimes 119868 + 119868 otimes
119860)1198671198712(119883) = 120582119871
2(119883)
Thus 1205904(119860) = 120590(119867
minus
(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590
3(119860) = 120590
4(119860) Obviously 120590
4(119860) sube 120590
3(119860)
so we only need to prove 1205903(119860) sube 120590
4(119860)
For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +
119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1])Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119909119910119879
+ 119910119909119879
+ 119910119909119879
= 119860119909119910119879
+ 119909119910119879
119860119879
+ 119910119909119879
119860119879
+ 119860119910119909119879
= 119860119885 + 119885119860119879
(14)
If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
4 Abstract and Applied Analysis
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
3(119860) = 120590
4(119860) = 120590(119867
minus
(119860 otimes 119868 +
119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
1(119860119879
otimes 119868 + 119868 otimes 119860119879
) =
120590(119867minus
(119860119879
otimes119868+119868otimes119860119879
)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867
minus
(119860119879
otimes 119868 + 119868 otimes 119860119879
)119867)
3 Applications of Transformation Matrix
31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices
Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877
119899times119899 is a Hurwitz matrix) thenone has the following
(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique
solution 119884 isin 119877119899times119899
(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909
119879
119884119909 such that 119860119879119884 + 119884119860 =
minus119863 and the expression is
119881 =1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
(15)
where
Δ = 119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)119867
119883 = (1198831 1198832 119883
119899)
1198831= (1199092
1 211990911199092 2119909
1119909119899)
1198832= (1199092
2 211990921199093 2119909
2119909119899)
119883119899= (1199092
119899)
(16)
Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int
infin
0
119890119905119860119879
119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz
matrix so 119884 gt 0 and
119860119879
119884 + 119884119860 = 119860119879
int
infin
0
119890119905119860119879
119863119890119905119860d119905 + int
infin
0
119890119905119860119879
119863119890119905119860d119905119860
= int
infin
0
d (119890119905119860119879
119863119890119905119860
) = minus119863
(17)
By (a) the quadratic Lyapunov function 119881 = 119909119879
119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique
Next we prove that 119881 can be expressed as (15) Note that119860119879
119884 + 119884119860 = minus119863 then
119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)1198671198712(119884) = minus119871
2(119863) (18)
By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and
119910119894119895=
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|(1 le 119895 le 119894 le 119899) (19)
where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ
and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors
are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)
By expanding the determinant we have
1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
=
119899
sum
119894=1
1199092
119894
10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816
|Δ|
+ sum
1le119895lt119894le119899
2119909119895119909119894
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|
=
119899
sum
119894119895=1
119909119894119910119894119895119909119895= 119881
(20)
For the uniqueness of 119881 (15) is the desired expression
Remark 8 The dimensions of Δ and [0 119883
1198712(119863) Δ
] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus
32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices
321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) + 119861120579(119896)
119906 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(21)
for 119896 isin 119885+ where 119909(119896) isin 119877
119899 119906(119896) isin 119877119898 and 119910(119896) isin
119877119903 respectively denote the system state control input and
measured output 119860119894119895isin 119877119899times119899 119861
119895isin 119877119899times119898 and 119862
119894119895isin 119877119903times119899
120579(119896) 119896 isin 119885+
is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is
Abstract and Applied Analysis 5
119875 = (119901119894119895)119904times119904
and initial distribution is 1205830
= 1205830119894
≜
119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908
119873(119896) isin 119877 are wide sense
stationary processes and independent of each other such that
E (119908119894(119896)) = 0 E (119908
119894(119896) 119908119895(119896)) = 120575
119894119895≜
1 119894 = 119895
0 119894 = 119895(22)
and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain
120579(119896) 119896 isin 119885+
For the stochastic system (21) its solution and output
processes with 119909(0) = 1199090and 120579(0) sim 120583
0are respectively
denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909
0 1205830) We will simply
denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909
0 1205830) 119910(119896) =
119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909
0 1205830)
If 119906(119896) equiv 0 the stochastic system (21) becomes
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(23)
for 119896 isin 119885+For convenience we will use the following notations
119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119894)
119883 (119896) ≜ (1198831(119896) 119883
119904(119896))
119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))
119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1
120579(119896)=119894)
(119896) ≜ (1198841(119896) 119884
119904(119896))
119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))
W0≜ 119883 (0) | 119909
0isin 119877119899
1205830isin Θ119883
119894(0) ≜ 119909
0119909119879
01205830119894 119894 isin 119878
(24)
It is easy to get 1198831(119896) + sdot sdot sdot + 119883
119904(119896) = 119883(119896) 119884
1(119896) +
sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =
sum119904
119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877
119899times119899
119904
Specifically E|119909(0)|2 = |1199090|2
For the system (23) we define several operators in119877119899times119899119904
asfollows
E119894(119880) =
119904
sum
119895=1
119901119894119895119880119895
F119894(119880) =
119873
sum
119897=0
119860119879
119897119894E119894(119880)119860
119897119894=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894119880119895119860119897119894
Flowast
119894(119880) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895119880119895119860119879
119897119895
E (119880) = (E1(119880) E
119904(119880))
F (119880) = (F1(119880) F
119904(119880))
Flowast
(119880) = (Flowast
1(119880) F
lowast
119904(119880))
F119896
(119880)
= F (F119896minus1
(119880)) (119896 = 1 2 ) where F0
(119880) = 119880
Flowast119896
(119880)
= Flowast
(Flowast119896minus1
(119880)) (119896 = 1 2 ) where Flowast0
(119880) = 119880
(25)
Flowast and F are called dual operators for each other (seeLemma 15)
Also we define the operator O119894(119897) = 119862
119894+ F119894(O(119897 minus
1)) for 119894 isin 119878 119897 = 1 2 where
119862119894=
119873
sum
119897=0
119862119879
119897119894119862119897119894
O (0) = 0
O (119897) = (O1(119897) O
119904(119897))
(26)
Then
O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862
119904))
= 119862 +F (119862) +F2
(119862) + sdot sdot sdot +F119897minus1
(119862)
119897 = 1 2
(27)
322 Some Preliminaries and Auxiliary Results Let
119882119897
119894(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119904
sum
119894=1
119882119897
119894(119883 (119905))
for 119905 isin 119885+ 119897 = 1 2
(28)
Lemma 9 For the system (23) one has O119894(119897) ge O
119894(119897 minus 1) for
all 119894 isin 119878 119897 = 1 2
Proof Firstly it is easy to get O119894(1) = 119862
119894= sum119873
119897=0119862119879
119897119894119862119897119894ge 0 =
O119894(0) for all 119894 isin 119878 Secondly we assume thatO
119894(119897) ge O
119894(119897minus1)for
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
A new necessary and sufficient condition for the weakobservability will be givenNotations 119877119899 is 119899-dimensional real Euclidean space 119862119899 is 119899-dimensional complex space 119860119879 |119860| and tr(119860) respectivelydenote the transpose determinant and trace of 119860 119860 gt
(ge)0 means that 119860 is a symmetric positive (semipositive)definite matrix E(sdot) denotes the mathematical expectationof a random variable 119885+ ≜ 0 1 2 119878 ≜ 1 2 119904where 119904 is a positive integer 119877119899times119898 denotes 119899 times 119898 real matrixspace with the inner product ⟨119860 119861⟩ ≜ tr(119860119879119861) 119877119899times119898
119904≜
(1198801 119880
119904) | 119880
119894isin 119877119899times119898
119894 isin 119878 with the inner product⟨119880 119881⟩ ≜ sum
119904
119894=1tr(119880119879119894119881119894) 1119860(119909) ≜
1 119909isinA0 119909notin119860
2 Transformation Matrix and Its Properties
21 Transformation Matrix between Two Vector Operators
Definition 1 (see [1]) 1198711 119862119899times119898
rarr 119862119899119898times1 is called a
vector operator by column if 1198711(119883) = (119909
11 11990921 119909
1198991 11990912
11990922 119909
1198992 119909
1119898 1199092119898 119909
119899119898)119879 for any119883 = (119909
119894119895)119899times119898
Definition 2 1198712 119882119899times119899
rarr 119862(119899(119899+1)2)times1 is called a half
vector operator by column if 1198712(119883) = (119909
11 11990921 119909
1198991
11990922 119909
1198992 119909
119899119899)119879 for any 119883 = (119909
119894119895)119899times119899
where 119882119899times119899 ≜119883 | 119883
119879
= 119883 isin 119862119899times119899
Definition 3 (see [1]) For matrices 119860 = (119886119894119895)119899times119898
and 119861 =
(119887119894119895)119904times119905 one calls matrix 119860 otimes 119861 = (119886
119894119895119861)119899times119898
the Kroneckerproduct of matrices 119860 and 119861
It is easy to find that 1198711and 119871
2have many similar
properties such as the following
(a) 1198711and 119871
2are all linear operators
(b) 119883 = 119884 hArr 1198711(119883) = 119871
1(119884)
(c) 119883 = 119884 hArr 1198712(119883) = 119871
2(119884) where119883119884 isin 119882
119899times119899
For convenience we will adopt the following notations119864119899119894119895
denotes a 119899 times 119899 matrix its (119894 119895)-element is 1 and itis zero elsewhere
119867(119899) denotes a 1198992times(119899(119899+1)2)matrix Its column vectorsare denoted by 119867
11
11986721
1198671198991
11986722
1198671198992
119867119899119899
from left to right and row vectors are denoted by11986711 11986721 119867
1198991 119867
1119899 1198672119899 119867
119899119899from top to bottom
where
119867119894119895
≜
1198711(119864119899119894119895
) 119894 = 119895
1198711(119864119899119894119895
+ 119864119899119895119894
) 119894 gt 119895
(1)
It is easy to prove
119867119894119895=
119871119879
2(119864119899119895119894
) 119894 lt 119895
119871119879
2(119864119899119894119895
) 119894 ge 119895
(2)
We will denote119867 = 119867119899≜ 119867(119899)
119867minus
(119899) denotes a (119899(119899 + 1)2) times 1198992 matrix Its
column vectors are denoted by 119867minus11
119867minus21
119867minus1198991
119867minus1119899
119867minus2119899
119867minus119899119899 from left to right and row vectors are
denoted by 119867minus
11 119867minus
21 119867
minus
1198991 119867minus
22 119867
minus
1198992 119867
minus
119899119899from
top to bottom where 119867minus
119894119895≜ 119871119879
1(119864119899119894119895
) We will denote119867minus
= 119867minus
119899≜ 119867minus
(119899)For example
119867(1) = [1] 119867minus
(1) = [1]
119867 (2) =[[[
[
1 0 0
0 1 0
0 1 0
0 0 1
]]]
]
119867minus
(2) = [
[
1 0 0 0
0 1 0 0
0 0 0 1
]
]
(3)
and so forthConsidering Property 3 we generally call 119867 and 119867
minus
transformation matrices between 1198711and 119871
2
22 Properties of Transformation Matrix
Property 1 Consider
119867minus
119899119867119899= 119868119899(119899+1)2
(4)
Proof For 119896 = 119897
119867minus
119894119895119867119896119897
= 119871119879
1(119864119899119894119895
) 1198711(119864119899119896119896
) = 1 119894 = 119895 = 119896
0 others(5)
For 119896 gt 119897
119867minus
119894119895119867119896119897
= 119871119879
1(119864119899119894119895
) 1198711(119864119899119896119897
+ 119864119899119897119896
) = 1 119894 = 119896 119895 = 119897
0 others(6)
Thus119867minus119899119867119899= 119868119899(119899+1)2
Remark 4 Generally119867119899119867minus
119899= 1198681198992 such as
1198672119867minus
2=[[[
[
1 0 0 0
0 1 0 0
0 1 0 0
0 0 0 1
]]]
]
(7)
Property 2 Consider119867119867minus
119867 = 119867119867minus119867119867minus = 119867minus (119867minus119867)119879 = 119867minus119867
Remark 5 Generally (119867119867minus)119879 =119867119867minus such as119867
2119867minus
2which is
not a symmetric matrixThis indicates that119867minus is the 1 2 3-generalized inverse matrix of 119867 but not the 1 2 3 4-generalized inverse matrix of119867 (see [1])
Property 3 If 119883119879 = 119883 ≜ (119909119894119895)119899times119899
then 1198671198712(119883) = 119871
1(119883)
119867minus
1198711(119883) = 119871
2(119883)
Abstract and Applied Analysis 3
Proof Consider
∵ 1198671198941198951198712(119883) = 119871
119879
2(119864119899119894119895
) 1198712(119883) = 119909
119894119895(119894 ge 119895)
1198671198941198951198712(119883) = 119871
119879
2(119864119899119895119894
) 1198712(119883) = 119909
119895119894= 119909119894119895
(119894 lt 119895)
there4 1198671198712(119883) = 119871
1(119883)
1198712(119883) = 119867
minus
1198711(119883) (by Property 1)
(8)
Property 4 For matrices 119860119899times119899 119861119903times119899
and 119883119879119899times119899
= 119883 we havethe following
(a) 1198711(119860119883 + 119883119860
119879
) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)
(b) 1198711(119861119883119861
119879
) = (119861 otimes 119861)1198711(119883)
(c) 1198712(119860119883 + 119883119860
119879
) = 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
(d) 1198712(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)
Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore
1198712(119860119883 + 119883119860
119879
) = 119867minus
1198711(119860119883 + 119883119860
119879
)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
1198712(119861119883119861
119879
) = 119867minus
1198711(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
(9)
Remark 6 If 1198711and 119871
2are defined by row that is
1198711(119883) = (119909
11 11990912 119909
1119898 11990921 11990922
1199092119898 119909
1198991 1199091198992 119909
119899119898)119879 for 119883 isin 119862
119899times119898
1198712(119883) = (119909
11 11990912 119909
1119899 11990922
1199092119899 119909
119899119899)119879 for 119883 isin 119882
119899times119899
(10)
we find that Properties 3 and 4 are still true
Property 5 For all 119860 isin 119862119899times119899 then
(a) 120590(119860 otimes 119860) = 120590(119867minus
(119860 otimes 119860)119867) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)
(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860
119879
)119867)
where 120590(sdot) denotes the set of all eigenvalues of matrix
Proof (a) Let
1205901(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205902(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883119860119879
= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205901(119860) = 120590 (119860 otimes 119860)
(11)
If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus
(119860 otimes 119860)1198671198712(119883) =
1205821198712(119883) Thus 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)Next we prove 120590
1(119860) = 120590
2(119860) Obviously 120590
2(119860) sube 120590
1(119860)
so we only need to prove 1205901(119860) sube 120590
2(119860)
For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =
119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1]) Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119910119909119879
= 119860119909119910119879
119860119879
+ 119860119910119909119879
119860119879
= 119860119885119860119879
(12)
If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590
1(119860) = 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590
1(119860119879
otimes 119860119879
) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)
(b) Let
1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883 + 119883119860119879
= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)
(13)
If 119883119879 = 119883 then 119860119883 + 119883119860119879
= 120582119883 hArr 119867minus
(119860 otimes 119868 + 119868 otimes
119860)1198671198712(119883) = 120582119871
2(119883)
Thus 1205904(119860) = 120590(119867
minus
(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590
3(119860) = 120590
4(119860) Obviously 120590
4(119860) sube 120590
3(119860)
so we only need to prove 1205903(119860) sube 120590
4(119860)
For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +
119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1])Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119909119910119879
+ 119910119909119879
+ 119910119909119879
= 119860119909119910119879
+ 119909119910119879
119860119879
+ 119910119909119879
119860119879
+ 119860119910119909119879
= 119860119885 + 119885119860119879
(14)
If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
4 Abstract and Applied Analysis
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
3(119860) = 120590
4(119860) = 120590(119867
minus
(119860 otimes 119868 +
119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
1(119860119879
otimes 119868 + 119868 otimes 119860119879
) =
120590(119867minus
(119860119879
otimes119868+119868otimes119860119879
)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867
minus
(119860119879
otimes 119868 + 119868 otimes 119860119879
)119867)
3 Applications of Transformation Matrix
31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices
Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877
119899times119899 is a Hurwitz matrix) thenone has the following
(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique
solution 119884 isin 119877119899times119899
(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909
119879
119884119909 such that 119860119879119884 + 119884119860 =
minus119863 and the expression is
119881 =1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
(15)
where
Δ = 119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)119867
119883 = (1198831 1198832 119883
119899)
1198831= (1199092
1 211990911199092 2119909
1119909119899)
1198832= (1199092
2 211990921199093 2119909
2119909119899)
119883119899= (1199092
119899)
(16)
Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int
infin
0
119890119905119860119879
119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz
matrix so 119884 gt 0 and
119860119879
119884 + 119884119860 = 119860119879
int
infin
0
119890119905119860119879
119863119890119905119860d119905 + int
infin
0
119890119905119860119879
119863119890119905119860d119905119860
= int
infin
0
d (119890119905119860119879
119863119890119905119860
) = minus119863
(17)
By (a) the quadratic Lyapunov function 119881 = 119909119879
119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique
Next we prove that 119881 can be expressed as (15) Note that119860119879
119884 + 119884119860 = minus119863 then
119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)1198671198712(119884) = minus119871
2(119863) (18)
By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and
119910119894119895=
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|(1 le 119895 le 119894 le 119899) (19)
where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ
and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors
are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)
By expanding the determinant we have
1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
=
119899
sum
119894=1
1199092
119894
10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816
|Δ|
+ sum
1le119895lt119894le119899
2119909119895119909119894
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|
=
119899
sum
119894119895=1
119909119894119910119894119895119909119895= 119881
(20)
For the uniqueness of 119881 (15) is the desired expression
Remark 8 The dimensions of Δ and [0 119883
1198712(119863) Δ
] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus
32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices
321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) + 119861120579(119896)
119906 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(21)
for 119896 isin 119885+ where 119909(119896) isin 119877
119899 119906(119896) isin 119877119898 and 119910(119896) isin
119877119903 respectively denote the system state control input and
measured output 119860119894119895isin 119877119899times119899 119861
119895isin 119877119899times119898 and 119862
119894119895isin 119877119903times119899
120579(119896) 119896 isin 119885+
is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is
Abstract and Applied Analysis 5
119875 = (119901119894119895)119904times119904
and initial distribution is 1205830
= 1205830119894
≜
119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908
119873(119896) isin 119877 are wide sense
stationary processes and independent of each other such that
E (119908119894(119896)) = 0 E (119908
119894(119896) 119908119895(119896)) = 120575
119894119895≜
1 119894 = 119895
0 119894 = 119895(22)
and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain
120579(119896) 119896 isin 119885+
For the stochastic system (21) its solution and output
processes with 119909(0) = 1199090and 120579(0) sim 120583
0are respectively
denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909
0 1205830) We will simply
denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909
0 1205830) 119910(119896) =
119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909
0 1205830)
If 119906(119896) equiv 0 the stochastic system (21) becomes
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(23)
for 119896 isin 119885+For convenience we will use the following notations
119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119894)
119883 (119896) ≜ (1198831(119896) 119883
119904(119896))
119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))
119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1
120579(119896)=119894)
(119896) ≜ (1198841(119896) 119884
119904(119896))
119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))
W0≜ 119883 (0) | 119909
0isin 119877119899
1205830isin Θ119883
119894(0) ≜ 119909
0119909119879
01205830119894 119894 isin 119878
(24)
It is easy to get 1198831(119896) + sdot sdot sdot + 119883
119904(119896) = 119883(119896) 119884
1(119896) +
sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =
sum119904
119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877
119899times119899
119904
Specifically E|119909(0)|2 = |1199090|2
For the system (23) we define several operators in119877119899times119899119904
asfollows
E119894(119880) =
119904
sum
119895=1
119901119894119895119880119895
F119894(119880) =
119873
sum
119897=0
119860119879
119897119894E119894(119880)119860
119897119894=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894119880119895119860119897119894
Flowast
119894(119880) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895119880119895119860119879
119897119895
E (119880) = (E1(119880) E
119904(119880))
F (119880) = (F1(119880) F
119904(119880))
Flowast
(119880) = (Flowast
1(119880) F
lowast
119904(119880))
F119896
(119880)
= F (F119896minus1
(119880)) (119896 = 1 2 ) where F0
(119880) = 119880
Flowast119896
(119880)
= Flowast
(Flowast119896minus1
(119880)) (119896 = 1 2 ) where Flowast0
(119880) = 119880
(25)
Flowast and F are called dual operators for each other (seeLemma 15)
Also we define the operator O119894(119897) = 119862
119894+ F119894(O(119897 minus
1)) for 119894 isin 119878 119897 = 1 2 where
119862119894=
119873
sum
119897=0
119862119879
119897119894119862119897119894
O (0) = 0
O (119897) = (O1(119897) O
119904(119897))
(26)
Then
O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862
119904))
= 119862 +F (119862) +F2
(119862) + sdot sdot sdot +F119897minus1
(119862)
119897 = 1 2
(27)
322 Some Preliminaries and Auxiliary Results Let
119882119897
119894(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119904
sum
119894=1
119882119897
119894(119883 (119905))
for 119905 isin 119885+ 119897 = 1 2
(28)
Lemma 9 For the system (23) one has O119894(119897) ge O
119894(119897 minus 1) for
all 119894 isin 119878 119897 = 1 2
Proof Firstly it is easy to get O119894(1) = 119862
119894= sum119873
119897=0119862119879
119897119894119862119897119894ge 0 =
O119894(0) for all 119894 isin 119878 Secondly we assume thatO
119894(119897) ge O
119894(119897minus1)for
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Proof Consider
∵ 1198671198941198951198712(119883) = 119871
119879
2(119864119899119894119895
) 1198712(119883) = 119909
119894119895(119894 ge 119895)
1198671198941198951198712(119883) = 119871
119879
2(119864119899119895119894
) 1198712(119883) = 119909
119895119894= 119909119894119895
(119894 lt 119895)
there4 1198671198712(119883) = 119871
1(119883)
1198712(119883) = 119867
minus
1198711(119883) (by Property 1)
(8)
Property 4 For matrices 119860119899times119899 119861119903times119899
and 119883119879119899times119899
= 119883 we havethe following
(a) 1198711(119860119883 + 119883119860
119879
) = (119868 otimes 119860 + 119860 otimes 119868)1198711(119883)
(b) 1198711(119861119883119861
119879
) = (119861 otimes 119861)1198711(119883)
(c) 1198712(119860119883 + 119883119860
119879
) = 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
(d) 1198712(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
Proof We can get (a) and (b) in [1] so we only need to prove(c) and (d)
Obviously119860119883+119883119860119879 and 119861119883119861119879 are symmetricalThere-fore
1198712(119860119883 + 119883119860
119879
) = 119867minus
1198711(119860119883 + 119883119860
119879
)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868) 1198711(119883)
= 119867minus
(119868 otimes 119860 + 119860 otimes 119868)1198671198712(119883)
1198712(119861119883119861
119879
) = 119867minus
1198711(119861119883119861
119879
) = 119867minus
(119861 otimes 119861)1198671198712(119883)
(9)
Remark 6 If 1198711and 119871
2are defined by row that is
1198711(119883) = (119909
11 11990912 119909
1119898 11990921 11990922
1199092119898 119909
1198991 1199091198992 119909
119899119898)119879 for 119883 isin 119862
119899times119898
1198712(119883) = (119909
11 11990912 119909
1119899 11990922
1199092119899 119909
119899119899)119879 for 119883 isin 119882
119899times119899
(10)
we find that Properties 3 and 4 are still true
Property 5 For all 119860 isin 119862119899times119899 then
(a) 120590(119860 otimes 119860) = 120590(119867minus
(119860 otimes 119860)119867) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)
(b) 120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868+119868otimes119860)119867) = 120590(119867minus(119860119879otimes119868 + 119868 otimes 119860
119879
)119867)
where 120590(sdot) denotes the set of all eigenvalues of matrix
Proof (a) Let
1205901(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205902(119860) ≜ 120582 isin 119862 | 119860119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883119860119879
= 120582119883 lArrrArr (119860 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205901(119860) = 120590 (119860 otimes 119860)
(11)
If 119883119879 = 119883 then 119860119883119860119879 = 120582119883 hArr 119867minus
(119860 otimes 119860)1198671198712(119883) =
1205821198712(119883) Thus 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)Next we prove 120590
1(119860) = 120590
2(119860) Obviously 120590
2(119860) sube 120590
1(119860)
so we only need to prove 1205901(119860) sube 120590
2(119860)
For any 120582 isin 1205901(119860) exist isin 120590(119860) such that 120582 = 119909 =
119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1]) Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119910119909119879
= 119860119909119910119879
119860119879
+ 119860119910119909119879
119860119879
= 119860119885119860119879
(12)
If 119885 = 0 then 120582 isin 1205902(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119860) = 120590
1(119860) = 120590
2(119860) = 120590(119867
minus
(119860 otimes 119860)119867)And because 120590(119860 otimes 119860) = 120590
1(119860119879
otimes 119860119879
) = 120590(119867minus
(119860119879
otimes 119860119879
)119867)therefore 120590(119860 otimes 119860) = 120590(119867minus(119860 otimes 119860)119867) = 120590(119867minus(119860119879 otimes 119860119879)119867)
(b) Let
1205903(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883 = 0 119883 isin 119862119899times119899
1205904(119860) ≜ 120582 isin 119862 | 119860119883 + 119883119860
119879
= 120582119883119883119879
= 119883 = 0119883 isin 119862119899times119899
∵ 119860119883 + 119883119860119879
= 120582119883 lArrrArr (119860 otimes 119868 + 119868 otimes 119860) 1198711(119883) = 120582119871
1(119883)
there4 1205903(119860) = 120590 (119860 otimes 119868 + 119868 otimes 119860)
(13)
If 119883119879 = 119883 then 119860119883 + 119883119860119879
= 120582119883 hArr 119867minus
(119860 otimes 119868 + 119868 otimes
119860)1198671198712(119883) = 120582119871
2(119883)
Thus 1205904(119860) = 120590(119867
minus
(119860 otimes 119868 + 119868 otimes 119860)119867)Next we prove 120590
3(119860) = 120590
4(119860) Obviously 120590
4(119860) sube 120590
3(119860)
so we only need to prove 1205903(119860) sube 120590
4(119860)
For any 120582 isin 1205903(119860) exist isin 120590(119860) such that 120582 = +
119909 = 119860119909 with 0 = 119909 isin 119862119899 119910 = 119860119910 with 0 = 119910 isin 119862
119899 (see [1])Let119883 ≜ 119909119910
119879
119885 ≜ 119883 + 119883119879 then
120582119885 = 119909119910119879
+ 119909119910119879
+ 119910119909119879
+ 119910119909119879
= 119860119909119910119879
+ 119909119910119879
119860119879
+ 119910119909119879
119860119879
+ 119860119910119909119879
= 119860119885 + 119885119860119879
(14)
If 119885 = 0 then 120582 isin 1205904(119860) Actually we always have 119885 = 0
Suppose that 119885 = 0 then 119911119894119894= 0 for 119894 = 1 2 119899 (ie
119909119894119910119894= 0 for 119894 = 1 2 119899) Without loss of generality let
4 Abstract and Applied Analysis
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
3(119860) = 120590
4(119860) = 120590(119867
minus
(119860 otimes 119868 +
119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
1(119860119879
otimes 119868 + 119868 otimes 119860119879
) =
120590(119867minus
(119860119879
otimes119868+119868otimes119860119879
)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867
minus
(119860119879
otimes 119868 + 119868 otimes 119860119879
)119867)
3 Applications of Transformation Matrix
31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices
Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877
119899times119899 is a Hurwitz matrix) thenone has the following
(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique
solution 119884 isin 119877119899times119899
(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909
119879
119884119909 such that 119860119879119884 + 119884119860 =
minus119863 and the expression is
119881 =1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
(15)
where
Δ = 119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)119867
119883 = (1198831 1198832 119883
119899)
1198831= (1199092
1 211990911199092 2119909
1119909119899)
1198832= (1199092
2 211990921199093 2119909
2119909119899)
119883119899= (1199092
119899)
(16)
Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int
infin
0
119890119905119860119879
119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz
matrix so 119884 gt 0 and
119860119879
119884 + 119884119860 = 119860119879
int
infin
0
119890119905119860119879
119863119890119905119860d119905 + int
infin
0
119890119905119860119879
119863119890119905119860d119905119860
= int
infin
0
d (119890119905119860119879
119863119890119905119860
) = minus119863
(17)
By (a) the quadratic Lyapunov function 119881 = 119909119879
119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique
Next we prove that 119881 can be expressed as (15) Note that119860119879
119884 + 119884119860 = minus119863 then
119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)1198671198712(119884) = minus119871
2(119863) (18)
By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and
119910119894119895=
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|(1 le 119895 le 119894 le 119899) (19)
where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ
and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors
are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)
By expanding the determinant we have
1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
=
119899
sum
119894=1
1199092
119894
10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816
|Δ|
+ sum
1le119895lt119894le119899
2119909119895119909119894
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|
=
119899
sum
119894119895=1
119909119894119910119894119895119909119895= 119881
(20)
For the uniqueness of 119881 (15) is the desired expression
Remark 8 The dimensions of Δ and [0 119883
1198712(119863) Δ
] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus
32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices
321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) + 119861120579(119896)
119906 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(21)
for 119896 isin 119885+ where 119909(119896) isin 119877
119899 119906(119896) isin 119877119898 and 119910(119896) isin
119877119903 respectively denote the system state control input and
measured output 119860119894119895isin 119877119899times119899 119861
119895isin 119877119899times119898 and 119862
119894119895isin 119877119903times119899
120579(119896) 119896 isin 119885+
is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is
Abstract and Applied Analysis 5
119875 = (119901119894119895)119904times119904
and initial distribution is 1205830
= 1205830119894
≜
119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908
119873(119896) isin 119877 are wide sense
stationary processes and independent of each other such that
E (119908119894(119896)) = 0 E (119908
119894(119896) 119908119895(119896)) = 120575
119894119895≜
1 119894 = 119895
0 119894 = 119895(22)
and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain
120579(119896) 119896 isin 119885+
For the stochastic system (21) its solution and output
processes with 119909(0) = 1199090and 120579(0) sim 120583
0are respectively
denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909
0 1205830) We will simply
denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909
0 1205830) 119910(119896) =
119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909
0 1205830)
If 119906(119896) equiv 0 the stochastic system (21) becomes
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(23)
for 119896 isin 119885+For convenience we will use the following notations
119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119894)
119883 (119896) ≜ (1198831(119896) 119883
119904(119896))
119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))
119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1
120579(119896)=119894)
(119896) ≜ (1198841(119896) 119884
119904(119896))
119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))
W0≜ 119883 (0) | 119909
0isin 119877119899
1205830isin Θ119883
119894(0) ≜ 119909
0119909119879
01205830119894 119894 isin 119878
(24)
It is easy to get 1198831(119896) + sdot sdot sdot + 119883
119904(119896) = 119883(119896) 119884
1(119896) +
sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =
sum119904
119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877
119899times119899
119904
Specifically E|119909(0)|2 = |1199090|2
For the system (23) we define several operators in119877119899times119899119904
asfollows
E119894(119880) =
119904
sum
119895=1
119901119894119895119880119895
F119894(119880) =
119873
sum
119897=0
119860119879
119897119894E119894(119880)119860
119897119894=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894119880119895119860119897119894
Flowast
119894(119880) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895119880119895119860119879
119897119895
E (119880) = (E1(119880) E
119904(119880))
F (119880) = (F1(119880) F
119904(119880))
Flowast
(119880) = (Flowast
1(119880) F
lowast
119904(119880))
F119896
(119880)
= F (F119896minus1
(119880)) (119896 = 1 2 ) where F0
(119880) = 119880
Flowast119896
(119880)
= Flowast
(Flowast119896minus1
(119880)) (119896 = 1 2 ) where Flowast0
(119880) = 119880
(25)
Flowast and F are called dual operators for each other (seeLemma 15)
Also we define the operator O119894(119897) = 119862
119894+ F119894(O(119897 minus
1)) for 119894 isin 119878 119897 = 1 2 where
119862119894=
119873
sum
119897=0
119862119879
119897119894119862119897119894
O (0) = 0
O (119897) = (O1(119897) O
119904(119897))
(26)
Then
O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862
119904))
= 119862 +F (119862) +F2
(119862) + sdot sdot sdot +F119897minus1
(119862)
119897 = 1 2
(27)
322 Some Preliminaries and Auxiliary Results Let
119882119897
119894(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119904
sum
119894=1
119882119897
119894(119883 (119905))
for 119905 isin 119885+ 119897 = 1 2
(28)
Lemma 9 For the system (23) one has O119894(119897) ge O
119894(119897 minus 1) for
all 119894 isin 119878 119897 = 1 2
Proof Firstly it is easy to get O119894(1) = 119862
119894= sum119873
119897=0119862119879
119897119894119862119897119894ge 0 =
O119894(0) for all 119894 isin 119878 Secondly we assume thatO
119894(119897) ge O
119894(119897minus1)for
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
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4 Abstract and Applied Analysis
119909119894= 0 119910119895= 0 then 119910
119894= 0 119909
119895= 0 119911
119894119895= 119909119894119910119895+ 119910119894119909119895= 119909119894119910119895= 0
This contradicts 119885 = 0 Then 119885 = 0Thus 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
3(119860) = 120590
4(119860) = 120590(119867
minus
(119860 otimes 119868 +
119868 otimes 119860)119867)And because 120590(119860 otimes 119868 + 119868 otimes 119860) = 120590
1(119860119879
otimes 119868 + 119868 otimes 119860119879
) =
120590(119867minus
(119860119879
otimes119868+119868otimes119860119879
)119867) therefore120590(119860otimes119868+119868otimes119860) = 120590(119867minus(119860otimes119868 + 119868 otimes 119860)119867) = 120590(119867
minus
(119860119879
otimes 119868 + 119868 otimes 119860119879
)119867)
3 Applications of Transformation Matrix
31 Application 1 The Improvement of Qian Jilingrsquos FormulaQian Jilingrsquos formula is an important and interesting formulato compute the Lyapunov functions [2]However the formulacan be improved with the help of the above transformationmatrices
Theorem 7 Assume that the system (119905) = 119860119909(119905) isasymptotically stable (ie119860 isin 119877
119899times119899 is a Hurwitz matrix) thenone has the following
(a) for any119863 isin 119877119899times119899119860119879119884+119884119860 = 119863 there exists a unique
solution 119884 isin 119877119899times119899
(b) for any 119863 gt 0 there exists a unique quadraticLyapunov function 119881 = 119909
119879
119884119909 such that 119860119879119884 + 119884119860 =
minus119863 and the expression is
119881 =1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
(15)
where
Δ = 119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)119867
119883 = (1198831 1198832 119883
119899)
1198831= (1199092
1 211990911199092 2119909
1119909119899)
1198832= (1199092
2 211990921199093 2119909
2119909119899)
119883119899= (1199092
119899)
(16)
Proof We can get (a) in [2] so we only need to prove (b)Let119884 = int
infin
0
119890119905119860119879
119863119890119905119860d119905 Because119863 gt 0 and119860 is aHurwitz
matrix so 119884 gt 0 and
119860119879
119884 + 119884119860 = 119860119879
int
infin
0
119890119905119860119879
119863119890119905119860d119905 + int
infin
0
119890119905119860119879
119863119890119905119860d119905119860
= int
infin
0
d (119890119905119860119879
119863119890119905119860
) = minus119863
(17)
By (a) the quadratic Lyapunov function 119881 = 119909119879
119884119909 whichsatisfies 119860119879119884 + 119884119860 = minus119863 is unique
Next we prove that 119881 can be expressed as (15) Note that119860119879
119884 + 119884119860 = minus119863 then
119867minus
(119868 otimes 119860119879
+ 119860119879
otimes 119868)1198671198712(119884) = minus119871
2(119863) (18)
By (a) we have |119868 otimes 119860119879 + 119860119879 otimes 119868| = 0By Property 5 we have |119867minus(119868 otimes 119860119879 + 119860119879 otimes 119868)119867| = 0By the Cramer rule (18) has a unique solution and
119910119894119895=
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|(1 le 119895 le 119894 le 119899) (19)
where we use minus1198712(119863) to replace Δrsquos (119894 119895)-column in Δ
and denote this new matrix by Δlowast119894119895 (Δrsquos column vectors
are respectively called the (1 1) (2 1) (119899 1) (2 2) (119899 2) (119899 119899)-column vector from left to right)
By expanding the determinant we have
1
|Δ|
10038161003816100381610038161003816100381610038161003816
0 119883
1198712(119863) Δ
10038161003816100381610038161003816100381610038161003816
=
119899
sum
119894=1
1199092
119894
10038161003816100381610038161003816Δlowast11989411989410038161003816100381610038161003816
|Δ|
+ sum
1le119895lt119894le119899
2119909119895119909119894
10038161003816100381610038161003816Δlowast11989411989510038161003816100381610038161003816
|Δ|
=
119899
sum
119894119895=1
119909119894119910119894119895119909119895= 119881
(20)
For the uniqueness of 119881 (15) is the desired expression
Remark 8 The dimensions of Δ and [0 119883
1198712(119863) Δ
] in (15) aresignificantly smaller than those in [2] due to the applicationof the transformation matrices119867 and119867minus
32 Application 2NewResults for theObservability of Stochas-tic Linear Systems This subsection considers the observabil-ity of discrete-time stochastic linear systems Anewnecessaryand sufficient condition for the weak observability will begiven with the help of these transformation matrices
321 Description of the Stochastic Linear Systems This sub-section considers the following discrete-time stochastic linearsystem with Markovian jump and multiplicative noises
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) + 119861120579(119896)
119906 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(21)
for 119896 isin 119885+ where 119909(119896) isin 119877
119899 119906(119896) isin 119877119898 and 119910(119896) isin
119877119903 respectively denote the system state control input and
measured output 119860119894119895isin 119877119899times119899 119861
119895isin 119877119899times119898 and 119862
119894119895isin 119877119903times119899
120579(119896) 119896 isin 119885+
is a discrete-time homogeneous Markovianchain Its state space is 119878 transition probability matrix is
Abstract and Applied Analysis 5
119875 = (119901119894119895)119904times119904
and initial distribution is 1205830
= 1205830119894
≜
119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908
119873(119896) isin 119877 are wide sense
stationary processes and independent of each other such that
E (119908119894(119896)) = 0 E (119908
119894(119896) 119908119895(119896)) = 120575
119894119895≜
1 119894 = 119895
0 119894 = 119895(22)
and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain
120579(119896) 119896 isin 119885+
For the stochastic system (21) its solution and output
processes with 119909(0) = 1199090and 120579(0) sim 120583
0are respectively
denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909
0 1205830) We will simply
denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909
0 1205830) 119910(119896) =
119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909
0 1205830)
If 119906(119896) equiv 0 the stochastic system (21) becomes
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(23)
for 119896 isin 119885+For convenience we will use the following notations
119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119894)
119883 (119896) ≜ (1198831(119896) 119883
119904(119896))
119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))
119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1
120579(119896)=119894)
(119896) ≜ (1198841(119896) 119884
119904(119896))
119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))
W0≜ 119883 (0) | 119909
0isin 119877119899
1205830isin Θ119883
119894(0) ≜ 119909
0119909119879
01205830119894 119894 isin 119878
(24)
It is easy to get 1198831(119896) + sdot sdot sdot + 119883
119904(119896) = 119883(119896) 119884
1(119896) +
sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =
sum119904
119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877
119899times119899
119904
Specifically E|119909(0)|2 = |1199090|2
For the system (23) we define several operators in119877119899times119899119904
asfollows
E119894(119880) =
119904
sum
119895=1
119901119894119895119880119895
F119894(119880) =
119873
sum
119897=0
119860119879
119897119894E119894(119880)119860
119897119894=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894119880119895119860119897119894
Flowast
119894(119880) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895119880119895119860119879
119897119895
E (119880) = (E1(119880) E
119904(119880))
F (119880) = (F1(119880) F
119904(119880))
Flowast
(119880) = (Flowast
1(119880) F
lowast
119904(119880))
F119896
(119880)
= F (F119896minus1
(119880)) (119896 = 1 2 ) where F0
(119880) = 119880
Flowast119896
(119880)
= Flowast
(Flowast119896minus1
(119880)) (119896 = 1 2 ) where Flowast0
(119880) = 119880
(25)
Flowast and F are called dual operators for each other (seeLemma 15)
Also we define the operator O119894(119897) = 119862
119894+ F119894(O(119897 minus
1)) for 119894 isin 119878 119897 = 1 2 where
119862119894=
119873
sum
119897=0
119862119879
119897119894119862119897119894
O (0) = 0
O (119897) = (O1(119897) O
119904(119897))
(26)
Then
O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862
119904))
= 119862 +F (119862) +F2
(119862) + sdot sdot sdot +F119897minus1
(119862)
119897 = 1 2
(27)
322 Some Preliminaries and Auxiliary Results Let
119882119897
119894(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119904
sum
119894=1
119882119897
119894(119883 (119905))
for 119905 isin 119885+ 119897 = 1 2
(28)
Lemma 9 For the system (23) one has O119894(119897) ge O
119894(119897 minus 1) for
all 119894 isin 119878 119897 = 1 2
Proof Firstly it is easy to get O119894(1) = 119862
119894= sum119873
119897=0119862119879
119897119894119862119897119894ge 0 =
O119894(0) for all 119894 isin 119878 Secondly we assume thatO
119894(119897) ge O
119894(119897minus1)for
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
119875 = (119901119894119895)119904times119904
and initial distribution is 1205830
= 1205830119894
≜
119875(120579(0) = 119894)119894isin119878 1199081(119896) 119908
119873(119896) isin 119877 are wide sense
stationary processes and independent of each other such that
E (119908119894(119896)) = 0 E (119908
119894(119896) 119908119895(119896)) = 120575
119894119895≜
1 119894 = 119895
0 119894 = 119895(22)
and all independent of 120579(119896) 119896 isin 119885+Let Θ ≜ 120583 | 120583 be initial distribution of Markovian chain
120579(119896) 119896 isin 119885+
For the stochastic system (21) its solution and output
processes with 119909(0) = 1199090and 120579(0) sim 120583
0are respectively
denoted by 119909(119896 120596 1199090 1205830) and 119910(119896 120596 119909
0 1205830) We will simply
denote 119909(119896) = 119909(119896 1199090 1205830) ≜ 119909(119896 120596 119909
0 1205830) 119910(119896) =
119910(119896 1199090 1205830) ≜ 119910(119896 120596 119909
0 1205830)
If 119906(119896) equiv 0 the stochastic system (21) becomes
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)
119910 (119896) = 1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)
(23)
for 119896 isin 119885+For convenience we will use the following notations
119883119894(119896) ≜ E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119894)
119883 (119896) ≜ (1198831(119896) 119883
119904(119896))
119883 (119896) ≜ E (119909 (119896) 119909119879 (119896))
119884119894(119896) ≜ E (119910 (119896) 119910119879 (119896) 1
120579(119896)=119894)
(119896) ≜ (1198841(119896) 119884
119904(119896))
119884 (119896) ≜ E (119910 (119896) 119910119879 (119896))
W0≜ 119883 (0) | 119909
0isin 119877119899
1205830isin Θ119883
119894(0) ≜ 119909
0119909119879
01205830119894 119894 isin 119878
(24)
It is easy to get 1198831(119896) + sdot sdot sdot + 119883
119904(119896) = 119883(119896) 119884
1(119896) +
sdot sdot sdot + 119884119904(119896) = 119884(119896) E|119909(119896)|2 = E(119909119879(119896)119909(119896)) = tr(119883(119896)) =
sum119904
119894=1tr(119883119894(119896)) = ⟨119883(119896) 119868⟩ where 119868 ≜ [119868 119868] isin 119877
119899times119899
119904
Specifically E|119909(0)|2 = |1199090|2
For the system (23) we define several operators in119877119899times119899119904
asfollows
E119894(119880) =
119904
sum
119895=1
119901119894119895119880119895
F119894(119880) =
119873
sum
119897=0
119860119879
119897119894E119894(119880)119860
119897119894=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894119880119895119860119897119894
Flowast
119894(119880) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895119880119895119860119879
119897119895
E (119880) = (E1(119880) E
119904(119880))
F (119880) = (F1(119880) F
119904(119880))
Flowast
(119880) = (Flowast
1(119880) F
lowast
119904(119880))
F119896
(119880)
= F (F119896minus1
(119880)) (119896 = 1 2 ) where F0
(119880) = 119880
Flowast119896
(119880)
= Flowast
(Flowast119896minus1
(119880)) (119896 = 1 2 ) where Flowast0
(119880) = 119880
(25)
Flowast and F are called dual operators for each other (seeLemma 15)
Also we define the operator O119894(119897) = 119862
119894+ F119894(O(119897 minus
1)) for 119894 isin 119878 119897 = 1 2 where
119862119894=
119873
sum
119897=0
119862119879
119897119894119862119897119894
O (0) = 0
O (119897) = (O1(119897) O
119904(119897))
(26)
Then
O (119897) = 119862 +F (O (119897 minus 1)) (119862 ≜ (1198621 119862
119904))
= 119862 +F (119862) +F2
(119862) + sdot sdot sdot +F119897minus1
(119862)
119897 = 1 2
(27)
322 Some Preliminaries and Auxiliary Results Let
119882119897
119894(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) ≜
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119904
sum
119894=1
119882119897
119894(119883 (119905))
for 119905 isin 119885+ 119897 = 1 2
(28)
Lemma 9 For the system (23) one has O119894(119897) ge O
119894(119897 minus 1) for
all 119894 isin 119878 119897 = 1 2
Proof Firstly it is easy to get O119894(1) = 119862
119894= sum119873
119897=0119862119879
119897119894119862119897119894ge 0 =
O119894(0) for all 119894 isin 119878 Secondly we assume thatO
119894(119897) ge O
119894(119897minus1)for
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
all 119894 isin 119878 119897 = 1 2 119898 Then for 119897 = 119898 + 1 for all 119894 isin 119878 wehave
O119894(119898 + 1) minus O
119894(119898)
= F119894(O (119898)) minusF
119894(O (119898 minus 1))
=
119873
sum
119897=0
119904
sum
119895=1
119901119894119895119860119879
119897119894(O119895(119898) minus O
119895(119898 minus 1))119860
119897119894
ge 0
(29)
By mathematical induction it is right
Lemma 10 For the system (23) one has F119897+1(O(119898 minus 1)) =
F119897(O(119898)) minusF119897(119862) for119898 = 1 2 119897 = 0 1
Proof It is easy to get the result by mathematical inductionso we will omit the details
Lemma 11 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
(ie for all 119883(0) = 1198830isinW0) one has
119883 (119896 + 1) = Flowast
(119883 (119896)) for 119896 isin 119885+
119883 (119896 + 119905) = Flowast119905
(119883 (119896)) for 119896 119905 isin 119885+(30)
Proof For all 119894 isin 119878 we have
Flowast
119894(119883 (119896)) =
119873
sum
119897=0
119904
sum
119895=1
119901119895119894119860119897119895E (119909 (119896) 119909119879 (119896) 1
120579(119896)=119895)119860119879
119897119895
=
119873
sum
119897=0
119904
sum
119895=1
119901119895119894E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896)=119895
)
=
119873
sum
119897=0
E (119860119897120579(119896)
119909 (119896) 119909119879
(119896) 119860119879
119897120579(119896)1120579(119896+1)=119894
)
=E
[
[
1198600120579(119896)
119909 (119896) +
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
times [
[
1198600120579(119896)
119909 (119896)
+
119873
sum
119895=1
119860119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896+1)=119894
=E 119909 (119896 + 1) 119909119879 (119896 + 1) 1120579(119896+1)=119894
= 119883119894(119896 + 1) 119896 isin 119885
+
(31)
then119883(119896 + 1) = Flowast(119883(119896)) for 119896 isin 119885+By induction we have 119883(119896 + 119905) = Flowast119905(119883(119896)) for 119896 119905 isin
119885+
Lemma 12 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
for 119905 isin 119885+ 119894 isin 119878 119897 = 1 2
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
for 119905 isin 119885+ 119897 = 1 2
(32)
Proof Because
119884119894(119896) = E
[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
times[
[
1198620120579(119896)
119909 (119896) +
119873
sum
119895=1
119862119895120579(119896)
119909 (119896)119908119895(119896)]
]
119879
1120579(119896)=119894
=
119873
sum
119897=0
119862119897119894119883119894(119896) 119862119879
119897119894
(33)
therefore
⟨119883119894(119896) 119862
119894⟩ = tr[119883
119894(119896)(
119873
sum
119897=0
119862119879
119897119894119862119897119894)] = tr (119884
119894(119896))
119882119897
119894(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883119894(119905 + 119896) 119862
119894⟩ =
119897minus1
sum
119896=0
tr (119884119894(119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896) 1120579(119905+119896)=119894
]
⟨119883 (119896) 119862⟩ =
119904
sum
119894=1
tr (119883119894(119896) 119862119894) = tr (119884 (119896))
119882119897
(119883 (119905)) =
119897minus1
sum
119896=0
⟨119883 (119905 + 119896) 119862⟩ =
119897minus1
sum
119896=0
tr (119884 (119905 + 119896))
=
119897minus1
sum
119896=0
E [119910119879 (119905 + 119896) 119910 (119905 + 119896)]
(34)
Remark 13 There is a physical interpretation119882119897(119883(119905)) is theaccumulated energy of the output process119910(119896) on the interval119905 le 119896 le 119905 + 119897 minus 1 The 119894th modal is119882119897
119894(119883(119905))
Lemma 14 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has119882119897(119883(119905)) = ⟨119883(119905)O(119897)⟩ for 119897 = 1 2
Proof It is easy to get the result by mathematical inductionso we will omit the details
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
Lemma 15 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119898 isin 119885+
119905 isin 119885+ one has
⟨119883 (119905) F119897
(O (119898))⟩ = ⟨Flowast119897
(119883 (119905)) O (119898)⟩ 119897 isin 119885+
(35)
Proof Firstly it is easy to get
⟨119883 (119905) F0
(O (119898))⟩
= ⟨119883 (119905) O (119898)⟩
= ⟨Flowast0
(119883 (119905)) O (119898)⟩ forall119898 119905 isin 119885+
(36)
Secondly we assume that ⟨119883(119905)F119897(O(119898))⟩ = ⟨Flowast119897(119883(119905))O(119898)⟩ for all119898 119905 isin 119885+ 119897 = 1 2 119902 Then for 119897 = 119902 + 1 forall119898 119905 isin 119885+ we have
⟨119883 (119905) F119902+1
(O (119898))⟩ = ⟨119883 (119905) F119902
(O (119898 + 1))⟩
minus ⟨119883 (119905) F119902
(119862)⟩
= ⟨Flowast119902
(119883 (119905)) O (119898 + 1)⟩
minus ⟨Flowast119902
(119883 (119905)) 119862⟩
= ⟨119883 (119905 + 119902) O (119898 + 1)⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= 119882119898+1
(119883 (119905 + 119902)) minus ⟨119883 (119905 + 119902) 119862⟩
=
119898
sum
119896=0
⟨119883 (119905 + 119902 + 119896) 119862⟩
minus ⟨119883 (119905 + 119902) 119862⟩
= ⟨119883 (119905 + 119902 + 1) O (119898)⟩
= ⟨Flowast119902+1
(119883 (119905)) O (119898)⟩
(37)
By mathematical induction it is right
Corollary 16 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ 119905 isin 119885+ one has
⟨119883 (119905) F119897
(119862)⟩ = E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] (38)
Proof Consider
⟨119883 (119905) F119897
(119862)⟩ = ⟨119883 (119905) F119897
(O (1))⟩
= ⟨Flowast119897
(119883 (119905)) O (1)⟩
= ⟨119883 (119905 + 119897) 119862⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)]
(39)
In particular we have ⟨119883(0)F119897(119862)⟩ = E[119910119879(119897)119910(119897)]when119905 = 0119898 = 1
Lemma 17 (see [12]) For any random variable 119910 isin 119877119899 one
has
119910 = 0 aslArrrArr E (119910119910119879) = 0 lArrrArr E (119910119879119910) = 0 (40)
Lemma 18 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
one has
lim119896rarrinfin
E [119909 (119896) 119909119879 (119896)] = 0 lArrrArr lim119896rarrinfin
E [119909119879 (119896) 119909 (119896)] = 0(41)
Proof This proof is omitted
Lemma 19 (see [1]) If 0 le 119883 isin 119877119899times119899
119903 ≜ rank(119883) then thereexist nonzero vectors 119909
1 119909
119903isin 119877119899 such that 119883 = 119909
1119909119879
1+
sdot sdot sdot + 119909119903119909119879
119903
323 A Useful Formula Define two operators L1(119880) =
[
1198711(1198801)
1198711(119880119904)
] for 119880119894isin 119877119899times119899 andL
2(119880) = [
1198712(1198801)
1198712(119880119904)
] for 119880119894isin 119883 |
119883119879
= 119883 isin 119877119899times119899
in 119877119899times119899119904
For convenience we naturally think 119884 isin 119883 | 119883
119879
= 119883 isin
119877119899times119899
when we use 1198712(119884) in this subsection
Theorem 20 For the system (23) for all 119880119881 isin 119877119899times119899
119904 one has
L1(F (119880)) = AL
1(119880)
L2(F (119880)) = BL
2(119880)
⟨119880 119881⟩ = L119879
1(119880)L
1(119881)
⟨119880 119881⟩ = L119879
1(119880) (119868 otimes 119867)L
2(119881)
(42)
where
A ≜
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971 119860
119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
B ≜
119873
sum
119897=0
diag (119867minus (1198601198791198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
(43)
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
Proof When 119880119879119894= 119880119894for 119894 isin 119878 then
L2(F (119880)) =
[[[[[[[[[
[
1198712(
119873
sum
119897=0
119904
sum
119895=1
1199011119895119860119879
11989711198801198951198601198971)
1198712(
119873
sum
119897=0
119904
sum
119895=1
119901119904119895119860119879
119897119904119880119895119860119897119904)
]]]]]]]]]
]
=
119873
sum
119897=0
[[[[[[[[
[
119867minus
(119860119879
1198971otimes 119860119879
1198971)119867
119904
sum
119895=1
11990111198951198712(119880119895)
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867
119904
sum
119895=1
1199011199041198951198712(119880119895)
]]]]]]]]
]
= Σ119873
119897=0diag (119867minus (119860119879
1198971otimes 119860119879
1198971)119867
119867minus
(119860119879
119897119904otimes 119860119879
119897119904)119867) (119875 otimes 119868)L
2(119880)
=BL2(119880)
(44)
By the same way it is easy to get
L1(F (119880)) =
119873
sum
119897=0
diag (1198601198791198971otimes 119860119879
1198971
119860119879
119897119904otimes 119860119879
119897119904) (119875 otimes 119868)L
1(119880)
= AL1(119880)
(45)
Consider
⟨119880 119881⟩ =
119904
sum
119894=1
tr (119880119879119894119881119894) =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=L119879
1(119880)L
1(119881)
⟨119880 119881⟩ =
119904
sum
119894=1
119871119879
1(119880119894) 1198711(119881119894)
=
119904
sum
119894=1
119871119879
1(119880119894)1198671198712(119881119894)
=L119879
1(119880) (119868 otimes 119867)L
2(119881)
(46)
Let 119889 ≜ 119904(119899(119899 + 1)2)
Remark 21 For the symmetric case L2(sdot) is more suit-
able than L1(sdot) in applications (such as Lemma 22 and
Theorem 26) Without the transformation matrices it isvery difficult to obtain these results Thus we say that thetransformation matrices are an effective mathematical tool
Lemma 22 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin Θ
119905 isin 119885+ one has
119882d(119883 (119905)) = 0 lArrrArr 119882
119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(47)
Proof By Lemma 12 it is easy to get
119882119897
(119883 (119896)) = 0 (119897 ge 1 119896 ge 119905)
lArrrArr 119864(119910119879
(119896) 119910 (119896)) = 0 (119896 ge 119905)
(48)
Obviously119882119897(119883(119896)) = 0 (119897 ge 1 119896 ge 119905) rArr 119882119889
(119883(119905)) = 0We only need to prove 119882119889(119883(119905)) = 0 rArr 119882
119897
(119883(119896)) =
0 (119897 ge 1 119896 ge 119905)
∵L2(F (119880)) =BL
2(119880) (by Theorem 20)
O (119897) = 119862 +F (O (119897 minus 1))
there4L2(O (119897)) =L
2(119862) +BL
2(O (119897 minus 1))
= 119902 +B [119902 +BL2(O (119897 minus 2))]
where 119902 ≜L2(119862)
= 119902 +B119902 + sdot sdot sdot +B119897minus1
119902 (119897 = 1 2 )
∵ ⟨119883 (119905) F119897
(119862)⟩
= E [119910119879 (119905 + 119897) 119910 (119905 + 119897)] ge 0 (by Corollary 16)
there4 ⟨119883 (119905) F119897
(119862)⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)L
2(F119897
(119862)) (by Theorem 20)
=L119879
1(119883 (119905)) (119868 otimes 119867)BL
2(F119897minus1
(119862))
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119897
119902 ge 0 (119897 isin 119885+
)
(49)
If119882119889(119883(119905)) = 0 then
⟨119883 (119905) O (119889)⟩ =L119879
1(119883 (119905)) (119868 otimes 119867)L
2(O (119889))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (119902 +B119902 + sdot sdot sdot +B119889minus1
119902)
= 0
(50)
ThusL1198791(119883(119905))(119868 otimes 119867)B119896119902 = 0 for 119896 = 0 1 119889 minus 1
For B isin 119877119889times119889 there exist 119886
0(119897) 1198861(119897) 119886
119889minus1(119897) isin 119877 by
the 119862119886119910119897119890119910-119867119886119898119894119897119905119900119899 theorem such that
B119897
= 1198860(119897)B0
+ 1198861(119897)B1
+ sdot sdot sdot + 119886119889minus1
(119897)B119889minus1
(119897 isin 119885+
)
(51)
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
For 119897 ge 1 119896 ge 119905 we have
119882119897
(119883 (119896)) = ⟨119883 (119905) F119896minus119905
(O (119897))⟩
=L119879
1(119883 (119905)) (119868 otimes 119867)B
119896minus119905
L2(O (119897))
=L119879
1(119883 (119905)) (119868 otimes 119867)
times (B119896minus119905
+B119896minus119905+1
+ sdot sdot sdot +B119896minus119905+119897minus1
) 119902
=L119879
1(119883 (119905)) (119868 otimes 119867)
times [1198870(])B0 + 119887
1(])B1
+ sdot sdot sdot + 119887119889minus1
(])B119889minus1] 119902
= 1198870(])L119879
1(119883 (119905)) (119868 otimes 119867)B
0
119902
+ 1198871(])L119879
1(119883 (119905)) (119868 otimes 119867)B
1
119902
+ sdot sdot sdot + 119887119889minus1
(])L1198791(119883 (119905)) (119868 otimes 119867)B
119889minus1
119902
= 0
(52)
where 1198870(]) 1198871(]) 119887
119889minus1(]) isin 119877
Corollary 23 For the system (23) for all 1199090isin 119877119899 for all 120583
0isin
Θ one has119882119889(1198830) = 0 hArr E(119910119879(119896)119910(119896)) = 0(119896 ge 0)
324 Main Results for the Weak Observability
Definition 24 The stochastic linear system (23) is weakobservable (119882-observable) if for all 120583
0isin Θ one has
119910 (119896) = 0 as forall119896 isin 119885+ 997904rArr 1199090= 0 (53)
Remark 25 By Lemma 22 (53) is equivalent to
119882119889
(1198830) = 0 997904rArr 119909
0= 0 (54)
Theorem 26 The system (23) is 119882-observable if and only ifO119894(119889) gt 0 for all 119894 isin 119878
Proof (Sufficiency) For all 1205830isin Θ if 119910(119896) = 0 as (119896 isin 119885+)
by Lemmas 14 and 22 we have
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
tr (119883119894(0)O119894(119889))
=
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090= 0
(55)
Because O119894(119889) gt 0 for all 119894 isin 119878 so 119909
0= 0
That is the system (23) is119882-observable(Necessity) Suppose that the system (23) is119882-observable
then for all 1205830isin Θ we have
1199090= 0 997904rArr exist119896
0≜ 1198960(1205830 1199090) such that 119875 (119910 (119896
0) = 0) = 1
(56)
By Corollary 23 if 119882119889(1198830) = 0 then E(119910119879(119896)119910(119896)) = 0 for
119896 isin 119885+ Thus we can consistently select 0 le 119896
0le 119889 for above
1199090 such that
119882119889+1
(1198830) =
119889
sum
119896=0
E (119910119879 (119896) 119910 (119896)) ge E (119910119879 (1198960) 119910 (1198960)) gt 0
(57)
Assume that there exists 119894 isin 119878 such that O119894(119889) is strictly
semipositive definite (ie exist0 = V isin 119877119899 such that V119879O119894(119889)V =
0) Select 1205830isin Θ such that 120583
0119895= 0 (119895 = 119894) Let 119883
0=
(VV1198791205831199001 VV119879120583
119900119904) then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119889) V = 0 (58)
By Lemma 22119882119889+1(1198830) = 0 It contradicts119882119889+1(119883
0) gt 0
Thus O119894(119889) gt 0 for 119894 isin 119878
Theorem27 For the system (23) ifexist119896 isin 119885+ such thatO119894(119896) gt
0 for any 119894 isin 119878 then O119894(119896 + 1) gt 0 for any 119894 isin 119878
Proof Assume that there exists 119894 isin 119878 such that O119894(119896 + 1)
is strictly semipositive definite (ie exist0 = V isin 119877119899 such that
V119879O119894(119896 + 1)V = 0) Select 120583
0isin Θ to satisfy 120583
0119895= 0 (119895 = 119894)
Let1198830= (VV119879120583
1199001 VV119879120583
119900119904) then
119882119896+1
(1198830) = ⟨119883
0O (119896 + 1)⟩ =
119904
sum
119895=1
1205830119895V119879O119895(119896 + 1) V = 0
(59)
It contradicts119882119896+1(1198830) ge 119882
119896
(1198830) = ⟨119883
0O(119896)⟩ gt 0 Thus
O119894(119896 + 1) gt 0 for 119894 isin 119878
Theorem28 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
Proof (a) rArr (b) The system (23) is 119882-observable thenO119894(119889) gt 0 for 119894 isin 119878 Select119873
119889= 119889 then
119882119889
(1198830) = ⟨119883
0O (119889)⟩ =
119904
sum
119894=1
1205830119894119909119879
0O119894(119889) 1199090
ge
119904
sum
119894=1
[1205830119894120582min (O119894 (119889))]
100381610038161003816100381611990901003816100381610038161003816
2
ge
119904
sum
119894=1
[1205830119894120574]100381610038161003816100381611990901003816100381610038161003816
2
= 120574100381610038161003816100381611990901003816100381610038161003816
2
(60)
where 120574 = min120582min(O1(119889)) 120582min(O119904(119889)) gt 0(b)rArr (a) For all 120583
0isin Θ if 119910(119896) = 0 as(119896 isin 119885+) then
119882119897
(1198830) =
119897minus1
sum
119896=0
E (119910119879 (119896) 119910 (119896)) = 0 (61)
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
for 119897 = 1 2 Because 0 = 119882119873119889(1198830) ge 120574|119909
0|2 so 119909
0= 0
That is the system (23) is119882-observable
By Theorem 28 and [6] it is easy to get the followingtheorem
Theorem29 For the system (23) the following two statementsare equivalent
(a) the system (23) is119882-observable(b) exist119896 isin 119885+ such that O
119894(119896) gt 0 for all 119894 isin 119878
Remark 30 Based on the above theorems we can determinethat the119882-observability in this paper is essentially consistentwith [5] We believe that Definition 24 in this paper is morealigned with the intuitive physical meanings FurthermoreTheorem 26 gives a new necessary and sufficient conditionfor the119882-observability
If we do not consider the noises that is 119860119895119894= 119862119895119894= 0
for 119895 = 1 119873 119894 isin 119878 (23) becomes the followingMarkovianjump linear system
119909 (119896 + 1) = 1198600120579(119896)
119909 (119896)
119910 (119896) = 1198620120579(119896)
119909 (119896)
(62)
If the Markovian chain degenerates into an ordinarysituation that is 119878 = 1 (23) becomes the followingstochastic linear system
119909 (119896 + 1) = 1198600119909 (119896) +
119873
sum
119895=1
119860119895119909 (119896)119908
119895(119896)
119910 (119896) = 1198620119909 (119896) +
119873
sum
119895=1
119862119895119909 (119896)119908
119895(119896)
(63)
where 119860119895≜ 1198601198951 119862119895≜ 1198621198951
for 119895 = 0 1 119873For the above two special situations it is easy to get the
following theorems
Theorem 31 For the Markovian jump linear system (62) thefollowing statements are equivalent
(a) the Markovian jump linear system (62) is 119882-observable
(b) exist119873119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 and 120583
0isin Θ one has
119882119873119889(1198830) ge 120574|119909
0|2
(c) exist119896 isin 119885+ such that O119894(119896) gt 0 for all 119894 isin 119878
(d) O119894(119889) gt 0 for all 119894 isin 119878
Theorem 32 For the stochastic linear system (63) the follow-ing statements are equivalent
(a) the stochastic linear system (63) is119882-observable(b) exist119873
119889ge 1 120574 gt 0 for all 119909
0isin 119877119899 one has119882119873119889(119909
0119909119879
0) ge
120574|1199090|2
(c) exist119896 isin 119885+ such that O(119896) gt 0(d) O(119889) gt 0
Remark 33 Theorems 31 and 32 have essential improvementscompared with those theorems in [5ndash7] (because 119889 = 119904(119899(119899 +1)2) le 119904119899
2) Furthermore O119894(119889)119894isin119878
are the characteristicmatrices for the observability of discrete-time stochasticlinear system (such as (23) (62) and (63))
4 Conclusions
This paper studies a class of important transformationmatrices between two different vector operators gives someimportant properties of it and demonstrates its applicationswith two examples One is to improve Qian Jilingrsquos formulaAnother is to give a new necessary and sufficient conditionfor the 119882-observability of stochastic linear systems Weemphasize that the transformation matrices are not onlyused to study these two problems but also used to studyother problems Actually the transformation matrices are aneffective mathematical tool in applications
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Thiswork is supported by theNational Natural Science Foun-dation under Grants 61273126 61363002 and 61374104 theNatural Science Foundation of Guangdong Province underGrants 10251064101000008 S2012010009675 the ResearchFoundation for the Doctoral Program of Higher Educationof China under Grant 20130172110027 and the Foundationof Key Laboratory of Autonomous Systems and NetworkedControl under Grant 2013A01
References
[1] R C Shi and F Wei Matrix Analysis Beijing Institute ofTechnology Press Beijing China 3rd edition 2010
[2] J L Qian and X X Liao ldquoA new solutionmethod for thematrixequation 119860119879119883+119883119861 = 119862 and its applicationrdquo Journal of CentralChina Normal University vol 21 no 2 pp 159ndash165 1987
[3] D Z Zheng Linear System Theory Tsinghua University PressBeijing China 2nd edition 2002
[4] B D O Anderson and J B Moore ldquoDetectability and stabiliz-ability of time-varying discrete-time linear systemsrdquo Journal onControl and Optimization vol 19 no 1 pp 20ndash32 1981
[5] E F Costa and J B R do Val ldquoOn the detectability andobservability of discrete-time Markov jump linear systemsrdquoSystems amp Control Letters vol 44 no 2 pp 135ndash145 2001
[6] V Dragan and T Morozan ldquoObservability and detectability ofa class of discrete-time stochastic linear systemsrdquo IMA Journalof Mathematical Control and Information vol 23 no 3 pp 371ndash394 2006
[7] Z Y Li Y Wang B Zhou and G R Duan ldquoDetectabilityand observability of discrete-time stochastic systems and theirapplicationsrdquo Automatica vol 45 no 5 pp 1340ndash1346 2009
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
[8] W Zhang and B-S Chen ldquoOn stabilizability and exact observ-ability of stochastic systems with their applicationsrdquo Automat-ica vol 40 no 1 pp 87ndash94 2004
[9] T Damm ldquoOn detectability of stochastic systemsrdquo Automaticavol 43 no 5 pp 928ndash933 2007
[10] W Zhang H Zhang and B-S Chen ldquoGeneralized Lyapunovequation approach to state-dependent stochastic stabiliza-tiondetectability criterionrdquo IEEE Transactions on AutomaticControl vol 53 no 7 pp 1630ndash1642 2008
[11] JW Lee and P P Khargonekar ldquoDetectability and stabilizabilityof discrete-time switched linear systemsrdquo IEEE Transactions onAutomatic Control vol 54 no 3 pp 424ndash437 2009
[12] S S Mao Y M Cheng and X L Pu Probability Theory andMathematical Statistics Higher education press Beijing China2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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