cramer-rao bound - iaaciaac.technion.ac.il/workshops/2010/kfhandouts/lectkf24.pdf · *taylor, j.h.,...
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![Page 1: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/1.jpg)
24 - 1Fundamentals of Kalman Filtering:A Practical Approach
Cramer-Rao Bound
![Page 2: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/2.jpg)
24 - 2Fundamentals of Kalman Filtering:A Practical Approach
What is the Cramer-Rao Lower Bound (CRLB) andWhat Does it Mean?
According to Bar Shalom*- “The mean square error corresponding to the estimator of a parameter cannot be smaller than a certain quantity related to the likelihood function”- “If an estimatorʼs variance is equal to the CRLB, then the estimator is called efficient”
Formula for CRLB found in many texts
E ˆ x (Z ) ! x0[ ] ˆ x (Z ) ! x
0[ ]T( ) " J
!1
J = E !xln"(x)[ ] !x
ln"(x)[ ]T( )
x=x 0
What does this mean and how do I program it?What does Zarchan say about the utility of the CRLB?
“If an estimatorʼs variance is equal to the CRLB, then perhapsthe estimator is called not practical”
*Bar-Shalom, Y., Li,X. and Kirubarajan, T., “Estimation With Applications to Tracking and Navigation, Theory AlgorithmsAnd Software,” John Wiley & Sons, Inc., New York, 2001, pp 109-110.
![Page 3: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/3.jpg)
24 - 3Fundamentals of Kalman Filtering:A Practical Approach
Cramer-Rao Lower Bound (CRLB) as an AlgorithmIt can be shown* in a more understandable way that according to theCRLB the best a least squares filter can do is given by
Pk
!1
= "Pk!1"
T( )!1
+ HTR
!1
H
Where P is the covariance matrix, Φ is the fundamental matrix, H isthe measurement matrix and R is the measurement noise matrix.P represents the smallest error in the estimate that is possible.The above equation can be improved slightly to make it easier toprogram. Let
Ak
= Pk
!1
Therefore the the CRLB equation can be rewritten as
Ak
= !T( )
"1
Ak"1!
"1
+ HTR
"1
H
The initial condition on the preceding matrix difference equation is
A0
= 0
Since
!Pk"1!
T( )"1
= !T( )
"1
Pk"1
"1
!"1
= !T( )
"1
Ak"1!
"1
*Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear Systems,” IEEE TransactionsOn Automatic Control, Vol. AC-24, No. 2, April 1979, pp 343-344.
![Page 4: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/4.jpg)
24 - 4Fundamentals of Kalman Filtering:A Practical Approach
How Does the Cramer-Rao Lower Bound (CRLB)Relate to Our Recursive Least Squares Filter?
• We have studied recursive least squares filters and found their gains and formulas predicting their performance. • We know that a linear polynomial Kalman filter with zero process noise and infinite initial covariance matrix is identical to the recursive least squares filter. • The recursive least squares filter also represents the best a filter can do.
Does the CRLB formula yield the same answers as can be obtainedby examining the covariance matrix of the recursive least squares filter?
![Page 5: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/5.jpg)
24 - 5Fundamentals of Kalman Filtering:A Practical Approach
Recall Recursive Least Squares Filter Structure andGains For Different Order Systems
1 State
Filter Gains
Resk = xk* - xk-1
xk = xk-1 + K1kResk
K1k = 1
k
2 State Resk = xk* - xk-1 - xk-1Ts
xk = xk-1 + xk-1Ts + K1kResk
xk = xk-1 + K2kResk
K1k =
2(2k-1)
k(k+1)
K2k = 6
k(k+1)Ts
3 StateResk = xk
* - xk-1 - xk-1Ts - .5xk-1Ts2
xk = xk-1 + xk-1Ts + .5xk-1Ts2+ K1k
Resk
xk = xk-1 + xk-1Ts2+ K2k
Resk
xk = xk-1 + K3kResk
K1k =
3(3k2-3k+2)
k(k+1)(k+2)
K2k =
18(2k-1)
k(k+1)(k+2)Ts
K3k = 60
k(k+1)(k+2)Ts2
Note that the above Table tells us directly how to build the filter
k=1,2,3,….
![Page 6: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/6.jpg)
24 - 6Fundamentals of Kalman Filtering:A Practical Approach
Recall Formulas For Errors in Estimates of DifferentOrder Recursive Least Squares Filters
Note that the covariance expressions in the above Table tells usdirectly the best the filter can perform
k=1,2,3,….
![Page 7: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/7.jpg)
24 - 7Fundamentals of Kalman Filtering:A Practical Approach
Important Matrices for Different Order LinearPolynomial Kalman Filters
States
1
2
3
Above matrices will be used in the CRLB equation
![Page 8: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/8.jpg)
24 - 8Fundamentals of Kalman Filtering:A Practical Approach
One-State ExampleFrom previous slide we see that for a one-state system
! = 1, R = "2
and H = 1
Therefore from the formula for the CRLB we can say that
Ak
= !T( )
"1
Ak"1!
"1+ H
TR
"1H = A
k"1+1
#2
With initial condition
A0
= 0
Therefore by inspection we can see that
A1
= A0
+1
!2
= 0 +1
!2
=1
!2
A2
= A1
+1
!2
=1
!2
+1
!2
=2
!2
A3
= A2
+1
!2
=2
!2
+1
!2
=3
!2
By induction it becomes apparent that
Ak
=k
!2
Which means that
Pk
=!2
k
Therefore CRLB covariance for one-state systemis identical to formula for one-statecovariance of recursive least squares filter!
Since
Pk
= Ak
!1
![Page 9: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/9.jpg)
24 - 9Fundamentals of Kalman Filtering:A Practical Approach
Two-State ExampleFrom “Important Matrices” slide we see that for a two-state system
! =1 T
s
0 1
"
# $
%
& '
R = !2
H = 1 0[ ]
Therefore from the formula for CRLB we can say that
Ak
= !T( )
"1
Ak"1!
"1
+ HTR
"1
H
With initial condition
A0
=0 0
0 0
!
" #
$
% &
Pk
= Ak
!1
and
We will use a computer simulation (with a matrix inverse routine) to compute the diagonal elements of the covariance matrix of CRLB.This method will be compared to a Kalman filter with zero processnoise and infinite initial covariance matrix and to the formulas forthe recursive least squares filter
![Page 10: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/10.jpg)
24 - 10Fundamentals of Kalman Filtering:A Practical Approach
Listing For CRLB Comparison in 2-State System-1IMPLICIT REAL*8(A-H,O-Z)REAL*8 M(2,2),P(2,2),K(2,1),PHI(2,2),H(1,2),R(1,1),PHIT(2,2)REAL*8 PHIP(2,2),HT(2,1),KH(2,2),IKH(2,2),A(2,2)REAL*8 MHT(2,1),HMHT(1,1),HMHTR(1,1),HMHTRINV(1,1),IDN(2,2)REAL*8 PHIINV(2,2),PHITINV(2,2),PHITINVA(2,2),TEMP1(2,2)REAL*8 PP(2,2)REAL*8 K1GM,K2GM,K3GMINTEGER ORDEROPEN(1,STATUS='UNKNOWN',FILE='DATFIL')ORDER=2TS=.5SIGNOISE=3.DO 1000 I=1,ORDERDO 1000 J=1,ORDER
PHI(I,J)=0.P(I,J)=0.IDN(I,J)=0.A(I,J)=0.
1000 CONTINUE IDN(1,1)=1.
IDN(2,2)=1.P(1,1)=99999999999999.P(2,2)=99999999999999.PHI(1,1)=1PHI(1,2)=TSPHI(2,2)=1DO 1100 I=1,ORDER
H(1,I)=0. 1100 CONTINUE
H(1,1)=1CALL MATTRN(H,1,ORDER,HT)R(1,1)=SIGNOISE**2CALL MATTRN(PHI,ORDER,ORDER,PHIT)CALL MTINV(PHI,ORDER,PHIINV)CALL MTINV(PHIT,ORDER,PHITINV)
A, I, P, Φ and H matrices for 2-state system
HT, R, ΦT, Φ-1, (ΦT)-1 matrices
Inputs for comparison
![Page 11: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/11.jpg)
24 - 11Fundamentals of Kalman Filtering:A Practical Approach
Listing For CRLB Comparison in 2-State System -2DO 10 XN=1.,100.CALL MATMUL(PHI,ORDER,ORDER,P,ORDER,ORDER,PHIP)CALL MATMUL(PHIP,ORDER,ORDER,PHIT,ORDER,ORDER,M)CALL MATMUL(M,ORDER,ORDER,HT,ORDER,1,MHT)CALL MATMUL(H,1,ORDER,MHT,ORDER,1,HMHT)HMHTR(1,1)=HMHT(1,1)+R(1,1)HMHTRINV(1,1)=1./HMHTR(1,1)CALL MATMUL(MHT,ORDER,1,HMHTRINV,1,1,K)CALL MATMUL(K,ORDER,1,H,1,ORDER,KH)CALL MATSUB(IDN,ORDER,ORDER,KH,IKH)CALL MATMUL(IKH,ORDER,ORDER,M,ORDER,ORDER,P)IF(XN<2)THEN
P11GM=9999999999.P22GM=9999999999.
ELSEP11GM=2.*(2.*XN-1)*SIGNOISE*SIGNOISE/
1 (XN*(XN+1.))P22GM=12.*SIGNOISE*SIGNOISE/(XN*(XN*XN-1.)
1 *TS*TS)ENDIFCALL MATMUL(PHITINV,ORDER,ORDER,A,ORDER,ORDER,PHITINVA)CALL MATMUL(PHITINVA,ORDER,ORDER,PHIINV,ORDER,ORDER,
1 TEMP1) DO 1001 I=1,ORDER
DO 1001 J=1,ORDERA(I,J)=TEMP1(I,J)
1001 CONTINUE A(1,1)=TEMP1(1,1)+1./SIGNOISE**2
CALL MTINV(A,ORDER,PP)WRITE(9,*)XN,P(1,1),P11GM,PP(1,1),P(2,2),P22GM,PP(2,2)
WRITE(1,*)XN,P(1,1),P11GM,PP(1,1),P(2,2),P22GM,PP(2,2) 10 CONTINUE CLOSE(1)
PAUSEEND
Iterate3 Methods
Ricatti equationswith zero processnoise for 2-statesystem
Recursiveleast squaresfilter for 2-statesystem
CRLB for2-state system
![Page 12: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/12.jpg)
24 - 12Fundamentals of Kalman Filtering:A Practical Approach
All Methods For Finding Best Performance in Two-State System Agree
0.1
2
468
1
2
468
10
10080604020Number of Measurements
All 3 MethodsAgree
0.0001
0.001
0.01
0.1
1
10
10080604020Number of Measurements
All 3 MethodsAgree
CRLB and Kalman filter with zero process noise and infinite initialcovariance matrix results are equivalent
![Page 13: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/13.jpg)
24 - 13Fundamentals of Kalman Filtering:A Practical Approach
Three-State ExampleFrom “Important Matrices” slide we see that for a three-state system
! =
1 Ts0.5T
s
2
0 1 Ts
0 0 1
"
#
$
$
$
%
&
'
'
'
R = !2
H = 1 0 0[ ]
Therefore from the formula for CRLB we can say that
Ak
= !T( )
"1
Ak"1!
"1
+ HTR
"1
H
With initial condition
A0
=
0 0 0
0 0 0
0 0 0
!
"
#
#
#
$
%
&
&
&
Pk
= Ak
!1
and
We will use a computer simulation (with a matrix inverse routine) to compute the diagonal elements of the covariance matrix of CRLB.This method will be compared to a Kalman filter with zero processnoise and infinite initial covariance matrix and to the formulas forthe recursive least squares filter
![Page 14: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/14.jpg)
24 - 14Fundamentals of Kalman Filtering:A Practical Approach
Listing For CRLB Comparison in 3-State System-1IMPLICIT REAL*8(A-H,O-Z)REAL*8 M(3,3),P(3,3),K(3,1),PHI(3,3),H(1,3),R(1,1),PHIT(3,3)REAL*8 PHIP(3,3),HT(3,1),KH(3,3),IKH(3,3),A(3,3)REAL*8 MHT(3,1),HMHT(1,1),HMHTR(1,1),HMHTRINV(1,1),IDN(3,3)REAL*8 PHIINV(3,3),PHITINV(3,3),PHITINVA(3,3),TEMP1(3,3)REAL*8 PP(3,3)REAL*8 K1GM,K2GM,K3GMINTEGER ORDEROPEN(1,STATUS='UNKNOWN',FILE='DATFIL')ORDER=3TS=.5SIGNOISE=3.DO 1000 I=1,ORDERDO 1000 J=1,ORDER
PHI(I,J)=0.P(I,J)=0.IDN(I,J)=0.A(I,J)=0.
1000 CONTINUE IDN(1,1)=1.
IDN(2,2)=1.IDN(3,3)=1.P(1,1)=99999999999999.P(2,2)=99999999999999.P(3,3)=99999999999999.PHI(1,1)=1PHI(1,2)=TSPHI(1,3)=.5*TS*TSPHI(2,2)=1PHI(2,3)=TSPHI(3,3)=1.DO 1100 I=1,ORDER
H(1,I)=0. 1100 CONTINUE
H(1,1)=1CALL MATTRN(H,1,ORDER,HT)R(1,1)=SIGNOISE**2CALL MATTRN(PHI,ORDER,ORDER,PHIT)CALL MTINV(PHI,ORDER,PHIINV)CALL MTINV(PHIT,ORDER,PHITINV)
A, I, P, Φ and H matrices for 3-state system
HT, R, ΦT, Φ-1, (ΦT)-1 matrices
Inputs for comparison
![Page 15: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/15.jpg)
24 - 15Fundamentals of Kalman Filtering:A Practical Approach
Listing For CRLB Comparison in 3-State System-2DO 10 XN=1.,100.CALL MATMUL(PHI,ORDER,ORDER,P,ORDER,ORDER,PHIP)CALL MATMUL(PHIP,ORDER,ORDER,PHIT,ORDER,ORDER,M)CALL MATMUL(M,ORDER,ORDER,HT,ORDER,1,MHT)CALL MATMUL(H,1,ORDER,MHT,ORDER,1,HMHT)HMHTR(1,1)=HMHT(1,1)+R(1,1)HMHTRINV(1,1)=1./HMHTR(1,1)CALL MATMUL(MHT,ORDER,1,HMHTRINV,1,1,K)CALL MATMUL(K,ORDER,1,H,1,ORDER,KH)CALL MATSUB(IDN,ORDER,ORDER,KH,IKH)CALL MATMUL(IKH,ORDER,ORDER,M,ORDER,ORDER,P)IF(XN<3)THEN
P11GM=9999999999.P22GM=9999999999.P33GM=9999999999.
ELSEP11GM=3.*(3.*XN*XN-3.*XN+2.)*SIGNOISE*SIGNOISE/
1 (XN*(XN+1.)*(XN+2.))P22GM=12.*(16.*XN*XN-30.*XN+11.)*SIGNOISE*
1 SIGNOISE/(XN*(XN*XN-1.)*(XN*XN-4.) 1 *TS*TS) P33GM=720.*SIGNOISE*SIGNOISE/(XN*(XN*XN-1.)* 1 (XN*XN-4.)*TS*TS*TS*TS)
ENDIFCALL MATMUL(PHITINV,ORDER,ORDER,A,ORDER,ORDER,PHITINVA)CALL MATMUL(PHITINVA,ORDER,ORDER,PHIINV,ORDER,ORDER,
1 TEMP1) DO 1001 I=1,ORDER
DO 1001 J=1,ORDERA(I,J)=TEMP1(I,J)
1001 CONTINUE A(1,1)=TEMP1(1,1)+1./SIGNOISE**2
CALL MTINV(A,ORDER,PP)WRITE(9,*)XN,P(1,1),P11GM,PP(1,1),P(2,2),P22GM,PP(2,2),
1 P(3,3),P33GM,PP(3,3) WRITE(1,*)XN,P(1,1),P11GM,PP(1,1),P(2,2),P22GM,PP(2,2), 1 P(3,3),P33GM,PP(3,3) 10 CONTINUE CLOSE(1)
PAUSEEND
Iterate3 Methods
Ricatti equationswith zero processnoise for 3-statesystem
Recursiveleast squaresfilter for 3-statesystem
CRLB for3-state system
![Page 16: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/16.jpg)
24 - 16Fundamentals of Kalman Filtering:A Practical Approach
All Methods For Finding Best Performance in Three-State System Agree-1
0.1
2
468
1
2
468
10
10080604020Number of Measurements
All 3 MethodsAgree
0.0001
0.001
0.01
0.1
1
10
10080604020Number of Measurements
All 3 MethodsAgree
CRLB and Kalman filter with zero process noise and infinite initialcovariance matrix results are equivalent
![Page 17: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/17.jpg)
24 - 17Fundamentals of Kalman Filtering:A Practical Approach
All Methods For Finding Best Performance in Three-State System Agree-2
10-5
10-4
10-3
10-2
10-1
100
101
10080604020Number of Measurements
All 3 MethodsAgree
![Page 18: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/18.jpg)
24 - 18Fundamentals of Kalman Filtering:A Practical Approach
Observations
• The Cramer-Rao Lower Bound (CRLB) tells us the best a least squares filter can do
- But so can a recursive least squares filter or the Kalman filter Ricatti equations with zero process noise and infinite initial covariance matrix
• Knowing the best a filter can do does not tell us how to build the filter so that it will work in the real world• Generally, building a filter with zero process noise is a bad idea because the filter stops paying attention to the measurements
- Numerous examples have been presented in the course demonstrating how a filter can fall apart with zero process noise
![Page 19: Cramer-Rao Bound - IAACiaac.technion.ac.il/workshops/2010/KFhandouts/LectKF24.pdf · *Taylor, J.H., “The Cramer-Rao Estimation Error Lower Bound Computation for Deterministic Nonlinear](https://reader034.vdocuments.us/reader034/viewer/2022052609/5b5381647f8b9a575f8bcc69/html5/thumbnails/19.jpg)
24 - 19Fundamentals of Kalman Filtering:A Practical Approach
Simple Derivation the CRLBFrom Ricatti Equations
P = (I ! KH )M
K = MHT
HMHT
+ R( )!1
Therefore Substitution Yields
P = I ! MHT
HMHT
+ R( )!1
H[ ]M = M ! MHT
HMHT
+ R( )!1
HM
We Want to Prove That
P!1
= M!1
+ HT
R!1
H
For Preceding Equation to be True
I = PP!1
I = M ! MHT
HMHT
+ R( )!1
HM[ ] M!1
+ HT
R!1
H[ ]
Multiplying Terms Out and Combining
I = I + MHT
R!1
! HMHT
+ R( )!1
I + HMHT
R!1( )[ ]H
But
I + HMHT
R!1( ) = R + HMH
T( )R!1
Therefore
I = I + MHT
R!1
! HMHT
+ R( )!1
R + HMHT( )R!1[ ]H = I + MH
T
R!1
! IR!1[ ]H = I + 0 = I
or
P!1
= M!1
+ HTR
!1
H = "P"T
+ Q( )!1
+ HTR
!1
H = "P"T( )
!1
+ HTR
!1
H if Q=0