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The Theory of the LS Filtering and its Applications*
479
THE THEORY OF THE LS FILTERING AND ITS APPLICATIONS· Cheng Zhaolin, Huang Daxiang, Wei Cheng, Heng Zhaobo and Yin Hongting Mathematics Department, Shandong University, Jinan 250100, P.R.China K~Y~Q[g~l
LS filtering, disturbance estimation, convergence analysis.
The convergence properties of the LS fi Itering of the state of I inaer systems wIth measurement noise are investigated. The necessary and sufficient conditions for the filter error converging to zero allost surely or in lean square are given and the convergence rates of the filter error are also obtained. As an application of the LS filtering, a new approach to the precise estilate of the periodic disturbances lixed in dynamic systems is proposed. Ab~l[~~l:
INTRODUCTION In this paper, we investigate the least squares (LS)
filtering for the systel
k=0,l, ...
(1)
where AERn ... n, BERn.r, CER.. n, A, Band C are constant, {ukl is a known control sequence, and {v k } is a measurement noise sequence. COlpared with the Kallann filtering, the advantage of the LS filtering is what it does not require any information concerning with the initial value of the systel. As we know, the discussion on the convergence properties of the filtering is significant and difficult. However, this paper establ ishes the contact of the filtering of the system (1) with the parameter estilatlon of the linear time-varying Model related to the systeE, hence, the study of the convergence properties of the LS fi Itering can be reduced to the investigatIon on the strong or weak consistency of the parameter estimation. In this paper, the following results are obtained: (1) the necessary and sufficient conditions to guarantee that the error of the LS filtering converges to zero allost surely or in mean square; (2) the convergence rates of the filter error. As an appl 'cation of the LS fi Iterlng~ a new approach to the precise estimate of the periodic disturbances mixed in dynamic systels is proposed. LS FILTERING For convenience of the discussion, we lake some assulplions: The coefficient matrix A of the system (1) is nonsingular, and the matrix pair (A,C) is completely observable, where C is the output ~atrix of the systel (1). A~~M~Q1IQn _ l.
* Research
supported by the Nature Sciences Fundation of P.R.China
480
Let VkJ be the jth co.ponent of the leasurelent noise Vk of the system (1). The random variable sequence { VkJ • k=0.1 •.••• j-1.2 •••••• I is an independent Gaussian random sequence with zero lean and the bounded variance: A§§Y~~11Qn_f'
0<
where
0
1
(T 1
0
2
02
<
k-0.1 •...•
00 I
j-1,2 ••.• ,m
(2)
are constant.
Denote that (3) h
Yk-h·Yk-h+C~
J-l
A-JBuk+J_h_l.
Yk-Yk.
h-1.2 •...• k
(4)
(5)
and rewrite the syste.
(1)
as the for. of the linear .odel: k-0. 1••••
(6 )
where H(k) and Xk are called the design .atrlx and the tile-varying regression coefficient of (6) respectively. The Identity substitution of (6) for (1) shows the new approach that the filtering of the state of the systel (1) can be reduced to the estilation of the ti.e-varying regression coefficient of the linear lodel (6). We have Q~flnll1Qn_l. The xLs(k) is called the LS filtering of the state Xk of the systel (1). if it Is the LS estilate of the tile-varying regression coefficient Xk of the linear lode I (6),
The LS fl Iter equations can be expressed as the recursive fori as follows:
(8)
L(k)-AP(k)A·C'[I+CAP(klA'C·]-l.
k-n-l.n ••.•
P(kl z [ I-Uk-l)CIAP(k-llA' II-Uk-llC] '+Uk-llA kL'(k-l>,
(9)
k-n.n+l,
P(n-1)2IH'(n-1)H(n-1)]-'H'(n-1lA(n-1lH(n-1)[H'(n-llH(n-ll)-I.
(10) ( 11 )
where P(k) is the covarlance matrix of the filter error xLs(kl-x k, Rk-E{VkVk'} and R(kl-E{V(klV'(kll. ~~"~_l.
(Chen, Z. and Z. Cheng, 1985) Let
S(kl-' +OD' + . . . +Ok-l (D' )k-l
(12)
where DE cn.n is a constant lIIatr ix, and O' is of D. Then (ll S-l(k)-- 0, as k- p oo , if and only If on or out the unit Circle , i.e.
the conjugate transpose latr Ix
all eigenvalues of the matrix 0 lie
481
AJ (O) I;" I.
j:zI.2 •...• n
( 13)
where A J
(13)
holds, the convergence rate of S-I(k) is
trace S-I(k) .. O(l/k).
as k ---
( 14)
'X' •
~~~~~_f' (Chen, X. et ai, 1985) Let that the random variable sequence (ek. k=I.2, ... 1 be the independent Gaussian randol sequence with zero lean and the bounded variance:
and the rea I nUlber sequence {aN. k • N:zl. 2 .... , k=I, 2, .... NI sat isfy
( 16)
Then N
I ill N-" N--- OQ
Ih~Q[~~_l.
~
a N.l
for any constant 0 > 0.
a.s.
( 17)
k-l
Subject to Assumption 1 and Assumption 2. then
(I) The filter error of LS filtering of the state Xk of the system (1) converges to zero almost surely or in mean square. if and only if all eigenvalues of the matrix A I ie on or inside the unit circle, i.e. j .. l.2 ....
( 18)
,n
where A J (A) is any e i genva I ue of A; (2) If (18) as 11
holds. the convergence rates of the filter error can be expressed
xLs(k)-Xk
11"
0(k-1/4+"),
a.s.
as
k-·x>.
for any constant
0 > 0
(19)
and (20) (3) The expression (20) cannot
or (19) cannot be Improved to
0
be ilproved any more. and the right-hand side (k-1/2).
Proof. I. The proof of the necessity and sufficiency for the fi Iter error converging to zero in mean square. By Definltion 1 and equations (2)-(6), the covariance matrix of the fi Iter error of the LS filtering of the state of system (1) can be expressed as (21 )
482
where R(k)=E{V(klV'(kll, and 0< (J II"Ck+l>
(k~
k-0, I, .••.
(22 )
n-I) is lonotonlcal ly nonincreaslng with res( 23)
where 'I-I
S(q)-~
(2.4)
(A-Jn) 'A-Jn
J-0
and that q-Ik/nl Is the integral part of kin, a and f3 are the least and largest elgenvalues of H'(n-1)H(n-ll respectively. Obviously, a, f3 >0. The inequality above leans that both P(k) and S-I(q) always simultaneously converge to zero or not, as k-oo. Using Lella I and by (23), we have that P(k)-0, as k-+ oo , if and only if all eigenvalues of the matrix A lie on or inside the unit circle, and the convergence rate of P(k) is expressed by (20). It is clear that the expression (20) cannot be improved any lore. Specially, taking A=C=I, and letting {vkl be an independent Gaussian randol sequence with zero mean and Elvk:Zlal, we have k
P(k) .. E{(~
VJ
' (k+l)2I IC I
/ (k+I) .. O(} / IO,
as k-=.
(25)
J-0
2. The proof of the sufficiency and necessity for the filter error converging to zero allost surely. Sufficiency. As we know, the filter error xLs(k)-Xk can also be expressed as (26)
where H(k), V(k) are given in formulas (3) and (5), respectively. Let (18) be holded. Define (27) (28)
lonotonicity of IH'(k)H(k)I- 1
Denote q-Ik/nl. Obviously, by the
with respect
to k, the following inequal ity follows frol (2.4) and Lella I:
where S(q) is given in (2.4), and
a
is the least
Next ."e invest igate 11 J:z(k) 11 . Not ice definite matrix with rank n. Hence, TCk) E RIIl(k+ 1)•• Ck+') such that
eigenvalue of H'(n-I)H(n-I).
that Q'(k)O(k) is the nonnegative there exists an orthogonal latrix
O'(k)Q(k)aT'(kldiag( A lk ... Ank 0 ... 0)1(k)
( 30)
483
where A Ik •..•• A nk are non-zero eigenvalues of O'O()Q(k) with 0 < A Jk
O'(k)Qu.:)-n.
k·n-l.n •...• j=I.2 •..•• n.
(3})
Defining lklJ as the (i.j)th elelenl of T(k). we have CD (k+ 1 )
~
t
2
kIJ
k·n-l.n •...•
-I.
J-J
i·I.2 •...• n.
(32)
Rewrite V(k) defined in (S) as (33) where eJ Is the jth co.ponent of V(k). Using (28). (30)-(31) and (33) give
Notice that the randol sequence {ek. k=I.2 •... } is an independent Gaussian sequence with zero lean and the bounded variance. and moreover the sequence { t k I J . k=n-l. n ..... izl.2 ..... n. j=1.2 ..... mCktl)} satisfies (32), Then. using Lemma 2 and (34). we have 11 J2 ( k) 11 =0 ( k 0
).
for any constant 6 > 0
a •s •
Obviously. combining (26)-(29) with (3S) gives the convergence rate of filter error xLs(k)-Xk. i.e. (2S) holds.
(3S) the
Now we prove that the right-hand side of (25) cannot be improved to o (k-1I'2). Use reduction to absurdity. Assume it unture. take A-C=l. and let {v k } be an independendent Gaussian random seqence with zero mean and Efv k2 }=I. then we have k-I
I xLs(k)-Xk I = I E vJ
as k
J-0
Obviously. this is in contradiction
with
the following
- =.
(36)
double-logarithm law
(S t OU t. W. F.. 1914):
k-l lill
k- oo
lE
j=0
VJ
I ... (kloglogkl..,2=2..,2.
a.s.
(31)
Necessity (oaitted).
A PRECISE ESTIMATE OF THE PERIODIC DISTURBANCES The ail of this section is to give a precise estimate of the periodic disturbances mixed in the dynamic systellls by using tile LS ri :tering. Consider the system 1<=0.1 ....
(38)
484
where AE Rn.n, 8 E R,,·r, DE Rn. p , CE R··n are constant, (u .. } is a known control sequence, (w k } is a periodic disturbance sequence with period T and unknown ampl itude, and Iv .. } is a leasurelent noise sequence. Denote that
D
o
0
0
Ip 0
000 Ip 0 0
o . .,
~.J
where AI E R
B.-
r ~ , C.-lC 0 '" L
8 1 E R
B ]
01
Cl E R··(n+Tpl. Rewrite k-0, l, ••..
(40)
the system (.41)
And denote zLs(k) as the LS filter of the state z.. of the systel (41), then ZLS.l(k) as the (n+l)th to (n+p)th components of zLs(k). Applying Theorel 1, we have In~Q[~~_f' Let the coefficient latrix A of the systel (38) be nonsingular, the latrlx pair (A,C) be completely observable, and the leasurelent noise sequence Iv k ! satisfy Assulption 2. Then we have
(42)
and 11 zLs.t(k)-w k II_o(k-1/:Z+
O
),
a.s. as k-- oo ,
for any constant 6 >O. (43)
Evidently, Theorem 2 not only gives the prescise estilate of the periodic disturbance wk , but also gives the convergence rates of the estilate error ZLS. I (k)-w ... To illustrate Theorel 2, we give a silulation exalple below. ~~~mQl~.
Consider the system k-0, I, ...
(44)
where Xk and
Yk are scalars, {wkl Is a periodic disturbance sequence, however, its alpl itude Is unknown, and {Vk! is an independent Gaussian sequence with zero lean and El vk :Z!-I. Obviously, the systel (44) satisfies all conditions of Theorem 2. According to Theorel 2, the estilate error zLs .• (k)-w k converges to zero allost surely or in lean square, and the convergence rates can be characterized by the formulas (42)-(43).
The numerical computation of ZLS.I(k) is shown in Fig.l and Fig.2. In Fig.I, If .. is a step disturbance, however, its alplitude is unknown. In Fig.2, W k Is a rectangular wave, and its amplitude is also unknown. CONCLUSiON 1. This paper introduces the LS fi Itering. Co.pared with the Kailan filtering, the advantage of the LS filtering is what it does not require any Information concerning with the initial value of the systel.
485
2. As we know, to discuss the convergence properties of the filtering is Significant and difficult. However, this paper establ ishes the contact of the fi Itering of the state of system (I) with the parameter estimat ion of the linear time-varying model related to the system, therefore, the discussion of the convergence properties of the fi Itering can be reduced to the invest igat Ion on the strong or weak consistency of the parameter estimation. 3. In this paper, the following results are obtained: (1) the necessary and sufficient conditions to guarantee that the error of the LS fi Itering converges to zero almost surely or in mean square; (2) the convergence rates of the fi Iter error. 4. As an appl ication of the LS fi Itering, a new approach to the preCise estimate of the periodic disturdances mixed in dynallic systems is proposed. Authors hope that the method suggested here would be benefitial to the design for the disturbance compensation.
(a)
(b)
(a) the measurement noise sequence Iv",}; (b) the est imate ZLS.l(k) of the step disturbance w", (comput Ing steps: N:I--(00).
(a)
(b)
(a) the measurement noise sequence Iv", I ; (b) the est imate ZLS. 1 (kl of the rectangular disturbance (cotlputing steps: N=301---3301.
loll<
486
Anderson, B. D. O. and J. B. Moore, 1979, "Optilal Filtering", Prentice-Hall, Inc., Englewood Cl iffs, New Jersey. Chen Z. and Z. Cheng, 1985, "The Mean Square Convergence of l1arkov Estllation of the Initial State for the Linear System with Random Measurement Noise", J~ _ SYS ~_~~l~ _~nd _H~lb ~_ ~~l~, Vol. 5, pp. 63-72. Ch en X., G. Chen, Q. \lu and L. Zhao, 1985, "Est imat ion Theory of Parameters in Linear Models", Science Press, Beij ing. Goodwin, G. C., and R. L. Payne, 1977, "Dynamic Systel Identification: Experiment Design and Data Analysis", Academic Press, New York. Kalman, R. E., 1960, "A New Approach to Linear Fi Itering and Prediction Problems", J ~ _ 6~sl~ _fng~1 _ I[~n ~ _ A~HE1 _ ~~[i~~_Q, Vol. 82, pp. 35-45. Stout, W. F., 1974, "Almost Sure Convergence", Academic Press, New York.
THE THEORY OF THE LS FILTERING AND ITS APPLICATIONS· Cheng Zhaolin, Huang Daxiang, Wei Cheng, Heng Zhaobo and Yin Hongting Mathematics Department, Shandong University, Jinan 250100, P.R.China K~Y~Q[g~l
LS filtering, disturbance estimation, convergence analysis.
The convergence properties of the LS fi Itering of the state of I inaer systems wIth measurement noise are investigated. The necessary and sufficient conditions for the filter error converging to zero allost surely or in lean square are given and the convergence rates of the filter error are also obtained. As an application of the LS filtering, a new approach to the precise estilate of the periodic disturbances lixed in dynamic systems is proposed. Ab~l[~~l:
INTRODUCTION In this paper, we investigate the least squares (LS)
filtering for the systel
k=0,l, ...
(1)
where AERn ... n, BERn.r, CER.. n, A, Band C are constant, {ukl is a known control sequence, and {v k } is a measurement noise sequence. COlpared with the Kallann filtering, the advantage of the LS filtering is what it does not require any information concerning with the initial value of the systel. As we know, the discussion on the convergence properties of the filtering is significant and difficult. However, this paper establ ishes the contact of the filtering of the system (1) with the parameter estilatlon of the linear time-varying Model related to the systeE, hence, the study of the convergence properties of the LS fi Itering can be reduced to the investigatIon on the strong or weak consistency of the parameter estimation. In this paper, the following results are obtained: (1) the necessary and sufficient conditions to guarantee that the error of the LS filtering converges to zero allost surely or in mean square; (2) the convergence rates of the filter error. As an appl 'cation of the LS fi Iterlng~ a new approach to the precise estimate of the periodic disturbances mixed in dynamic systels is proposed. LS FILTERING For convenience of the discussion, we lake some assulplions: The coefficient matrix A of the system (1) is nonsingular, and the matrix pair (A,C) is completely observable, where C is the output ~atrix of the systel (1). A~~M~Q1IQn _ l.
* Research
supported by the Nature Sciences Fundation of P.R.China
480
Let VkJ be the jth co.ponent of the leasurelent noise Vk of the system (1). The random variable sequence { VkJ • k=0.1 •.••• j-1.2 •••••• I is an independent Gaussian random sequence with zero lean and the bounded variance: A§§Y~~11Qn_f'
0<
where
0
1
(T 1
0
2
02
<
k-0.1 •...•
00 I
j-1,2 ••.• ,m
(2)
are constant.
Denote that (3) h
Yk-h·Yk-h+C~
J-l
A-JBuk+J_h_l.
Yk-Yk.
h-1.2 •...• k
(4)
(5)
and rewrite the syste.
(1)
as the for. of the linear .odel: k-0. 1••••
(6 )
where H(k) and Xk are called the design .atrlx and the tile-varying regression coefficient of (6) respectively. The Identity substitution of (6) for (1) shows the new approach that the filtering of the state of the systel (1) can be reduced to the estilation of the ti.e-varying regression coefficient of the linear lodel (6). We have Q~flnll1Qn_l. The xLs(k) is called the LS filtering of the state Xk of the systel (1). if it Is the LS estilate of the tile-varying regression coefficient Xk of the linear lode I (6),
The LS fl Iter equations can be expressed as the recursive fori as follows:
(8)
L(k)-AP(k)A·C'[I+CAP(klA'C·]-l.
k-n-l.n ••.•
P(kl z [ I-Uk-l)CIAP(k-llA' II-Uk-llC] '+Uk-llA kL'(k-l>,
(9)
k-n.n+l,
P(n-1)2IH'(n-1)H(n-1)]-'H'(n-1lA(n-1lH(n-1)[H'(n-llH(n-ll)-I.
(10) ( 11 )
where P(k) is the covarlance matrix of the filter error xLs(kl-x k, Rk-E{VkVk'} and R(kl-E{V(klV'(kll. ~~"~_l.
(Chen, Z. and Z. Cheng, 1985) Let
S(kl-' +OD' + . . . +Ok-l (D' )k-l
(12)
where DE cn.n is a constant lIIatr ix, and O' is of D. Then (ll S-l(k)-- 0, as k- p oo , if and only If on or out the unit Circle , i.e.
the conjugate transpose latr Ix
all eigenvalues of the matrix 0 lie
481
AJ (O) I;" I.
j:zI.2 •...• n
( 13)
where A J
(13)
holds, the convergence rate of S-I(k) is
trace S-I(k) .. O(l/k).
as k ---
( 14)
'X' •
~~~~~_f' (Chen, X. et ai, 1985) Let that the random variable sequence (ek. k=I.2, ... 1 be the independent Gaussian randol sequence with zero lean and the bounded variance:
and the rea I nUlber sequence {aN. k • N:zl. 2 .... , k=I, 2, .... NI sat isfy
( 16)
Then N
I ill N-" N--- OQ
Ih~Q[~~_l.
~
a N.l
for any constant 0 > 0.
a.s.
( 17)
k-l
Subject to Assumption 1 and Assumption 2. then
(I) The filter error of LS filtering of the state Xk of the system (1) converges to zero almost surely or in mean square. if and only if all eigenvalues of the matrix A I ie on or inside the unit circle, i.e. j .. l.2 ....
( 18)
,n
where A J (A) is any e i genva I ue of A; (2) If (18) as 11
holds. the convergence rates of the filter error can be expressed
xLs(k)-Xk
11"
0(k-1/4+"),
a.s.
as
k-·x>.
for any constant
0 > 0
(19)
and (20) (3) The expression (20) cannot
or (19) cannot be Improved to
0
be ilproved any more. and the right-hand side (k-1/2).
Proof. I. The proof of the necessity and sufficiency for the fi Iter error converging to zero in mean square. By Definltion 1 and equations (2)-(6), the covariance matrix of the fi Iter error of the LS filtering of the state of system (1) can be expressed as (21 )
482
where R(k)=E{V(klV'(kll, and 0< (J II"Ck+l>
(k~
k-0, I, .••.
(22 )
n-I) is lonotonlcal ly nonincreaslng with res( 23)
where 'I-I
S(q)-~
(2.4)
(A-Jn) 'A-Jn
J-0
and that q-Ik/nl Is the integral part of kin, a and f3 are the least and largest elgenvalues of H'(n-1)H(n-ll respectively. Obviously, a, f3 >0. The inequality above leans that both P(k) and S-I(q) always simultaneously converge to zero or not, as k-oo. Using Lella I and by (23), we have that P(k)-0, as k-+ oo , if and only if all eigenvalues of the matrix A lie on or inside the unit circle, and the convergence rate of P(k) is expressed by (20). It is clear that the expression (20) cannot be improved any lore. Specially, taking A=C=I, and letting {vkl be an independent Gaussian randol sequence with zero mean and Elvk:Zlal, we have k
P(k) .. E{(~
VJ
' (k+l)2I IC I
/ (k+I) .. O(} / IO,
as k-=.
(25)
J-0
2. The proof of the sufficiency and necessity for the filter error converging to zero allost surely. Sufficiency. As we know, the filter error xLs(k)-Xk can also be expressed as (26)
where H(k), V(k) are given in formulas (3) and (5), respectively. Let (18) be holded. Define (27) (28)
lonotonicity of IH'(k)H(k)I- 1
Denote q-Ik/nl. Obviously, by the
with respect
to k, the following inequal ity follows frol (2.4) and Lella I:
where S(q) is given in (2.4), and
a
is the least
Next ."e invest igate 11 J:z(k) 11 . Not ice definite matrix with rank n. Hence, TCk) E RIIl(k+ 1)•• Ck+') such that
eigenvalue of H'(n-I)H(n-I).
that Q'(k)O(k) is the nonnegative there exists an orthogonal latrix
O'(k)Q(k)aT'(kldiag( A lk ... Ank 0 ... 0)1(k)
( 30)
483
where A Ik •..•• A nk are non-zero eigenvalues of O'O()Q(k) with 0 < A Jk
O'(k)Qu.:)-n.
k·n-l.n •...• j=I.2 •..•• n.
(3})
Defining lklJ as the (i.j)th elelenl of T(k). we have CD (k+ 1 )
~
t
2
kIJ
k·n-l.n •...•
-I.
J-J
i·I.2 •...• n.
(32)
Rewrite V(k) defined in (S) as (33) where eJ Is the jth co.ponent of V(k). Using (28). (30)-(31) and (33) give
Notice that the randol sequence {ek. k=I.2 •... } is an independent Gaussian sequence with zero lean and the bounded variance. and moreover the sequence { t k I J . k=n-l. n ..... izl.2 ..... n. j=1.2 ..... mCktl)} satisfies (32), Then. using Lemma 2 and (34). we have 11 J2 ( k) 11 =0 ( k 0
).
for any constant 6 > 0
a •s •
Obviously. combining (26)-(29) with (3S) gives the convergence rate of filter error xLs(k)-Xk. i.e. (2S) holds.
(3S) the
Now we prove that the right-hand side of (25) cannot be improved to o (k-1I'2). Use reduction to absurdity. Assume it unture. take A-C=l. and let {v k } be an independendent Gaussian random seqence with zero mean and Efv k2 }=I. then we have k-I
I xLs(k)-Xk I = I E vJ
as k
J-0
Obviously. this is in contradiction
with
the following
- =.
(36)
double-logarithm law
(S t OU t. W. F.. 1914):
k-l lill
k- oo
lE
j=0
VJ
I ... (kloglogkl..,2=2..,2.
a.s.
(31)
Necessity (oaitted).
A PRECISE ESTIMATE OF THE PERIODIC DISTURBANCES The ail of this section is to give a precise estimate of the periodic disturbances mixed in the dynamic systellls by using tile LS ri :tering. Consider the system 1<=0.1 ....
(38)
484
where AE Rn.n, 8 E R,,·r, DE Rn. p , CE R··n are constant, (u .. } is a known control sequence, (w k } is a periodic disturbance sequence with period T and unknown ampl itude, and Iv .. } is a leasurelent noise sequence. Denote that
D
o
0
0
Ip 0
000 Ip 0 0
o . .,
~.J
where AI E R
B.-
r ~ , C.-lC 0 '" L
8 1 E R
B ]
01
Cl E R··(n+Tpl. Rewrite k-0, l, ••..
(40)
the system (.41)
And denote zLs(k) as the LS filter of the state z.. of the systel (41), then ZLS.l(k) as the (n+l)th to (n+p)th components of zLs(k). Applying Theorel 1, we have In~Q[~~_f' Let the coefficient latrix A of the systel (38) be nonsingular, the latrlx pair (A,C) be completely observable, and the leasurelent noise sequence Iv k ! satisfy Assulption 2. Then we have
(42)
and 11 zLs.t(k)-w k II_o(k-1/:Z+
O
),
a.s. as k-- oo ,
for any constant 6 >O. (43)
Evidently, Theorem 2 not only gives the prescise estilate of the periodic disturbance wk , but also gives the convergence rates of the estilate error ZLS. I (k)-w ... To illustrate Theorel 2, we give a silulation exalple below. ~~~mQl~.
Consider the system k-0, I, ...
(44)
where Xk and
Yk are scalars, {wkl Is a periodic disturbance sequence, however, its alpl itude Is unknown, and {Vk! is an independent Gaussian sequence with zero lean and El vk :Z!-I. Obviously, the systel (44) satisfies all conditions of Theorem 2. According to Theorel 2, the estilate error zLs .• (k)-w k converges to zero allost surely or in lean square, and the convergence rates can be characterized by the formulas (42)-(43).
The numerical computation of ZLS.I(k) is shown in Fig.l and Fig.2. In Fig.I, If .. is a step disturbance, however, its alplitude is unknown. In Fig.2, W k Is a rectangular wave, and its amplitude is also unknown. CONCLUSiON 1. This paper introduces the LS fi Itering. Co.pared with the Kailan filtering, the advantage of the LS filtering is what it does not require any Information concerning with the initial value of the systel.
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2. As we know, to discuss the convergence properties of the filtering is Significant and difficult. However, this paper establ ishes the contact of the fi Itering of the state of system (I) with the parameter estimat ion of the linear time-varying model related to the system, therefore, the discussion of the convergence properties of the fi Itering can be reduced to the invest igat Ion on the strong or weak consistency of the parameter estimation. 3. In this paper, the following results are obtained: (1) the necessary and sufficient conditions to guarantee that the error of the LS fi Itering converges to zero almost surely or in mean square; (2) the convergence rates of the fi Iter error. 4. As an appl ication of the LS fi Itering, a new approach to the preCise estimate of the periodic disturdances mixed in dynallic systems is proposed. Authors hope that the method suggested here would be benefitial to the design for the disturbance compensation.
(a)
(b)
(a) the measurement noise sequence Iv",}; (b) the est imate ZLS.l(k) of the step disturbance w", (comput Ing steps: N:I--(00).
(a)
(b)
(a) the measurement noise sequence Iv", I ; (b) the est imate ZLS. 1 (kl of the rectangular disturbance (cotlputing steps: N=301---3301.
loll<
486
Anderson, B. D. O. and J. B. Moore, 1979, "Optilal Filtering", Prentice-Hall, Inc., Englewood Cl iffs, New Jersey. Chen Z. and Z. Cheng, 1985, "The Mean Square Convergence of l1arkov Estllation of the Initial State for the Linear System with Random Measurement Noise", J~ _ SYS ~_~~l~ _~nd _H~lb ~_ ~~l~, Vol. 5, pp. 63-72. Ch en X., G. Chen, Q. \lu and L. Zhao, 1985, "Est imat ion Theory of Parameters in Linear Models", Science Press, Beij ing. Goodwin, G. C., and R. L. Payne, 1977, "Dynamic Systel Identification: Experiment Design and Data Analysis", Academic Press, New York. Kalman, R. E., 1960, "A New Approach to Linear Fi Itering and Prediction Problems", J ~ _ 6~sl~ _fng~1 _ I[~n ~ _ A~HE1 _ ~~[i~~_Q, Vol. 82, pp. 35-45. Stout, W. F., 1974, "Almost Sure Convergence", Academic Press, New York.