EE 495 Modern Navigation Systems
State Augmentation
Mon, March 28
EE 495 Modern Navigation Systems
Slide 1 of 10
State Augmentation
Case 1. Colored State Noise
• Given a continuous state-space system
x1 (t )  F1 (t ) x1 (t )  G1 (t ) w1 (t )
z (t )  H1 (t ) x1 (t )  v (t )
 where the state noise 𝑤1 is non-white (e.g., Gauss Markov)
 We can model the non-white noise as
x2 (t )  F2 (t ) x2 (t )  G2 (t ) w2 (t )
w1 (t )  H 2 (t ) x2 (t )
 where 𝑤2 is now a white noise process!!
Mon, March 28
EE 495 Modern Navigation Systems
Slide 2 of 10
State Augmentation
Case 1. Colored State Noise
• Considering the augmented state vector
 x1 (t ) 
x (t )  

x
(
t
)
 2 
 we now have
x1 (t )  F1 (t ) x1 (t )  G1 (t ) w1 (t )
z (t )  H1 (t ) x1 (t )  v (t )
x2 (t )  F2 (t ) x2 (t )  G2 (t ) w2 (t )
w1 (t )  H 2 (t ) x2 (t )
 x1 (t ) 
 F1 (t ) G1 (t ) H 2 (t )   x1 (t )  0 
x (t )  
x (t )  
w2 (t )
 



F2?(t )   x2 (t )  G2 (t ) 
 0?
 x2 (t ) 
 and
z (t )   H1 (t ) 0 x (t )  v (t )
The augmented system is now driven by white noise
Mon, March 28
EE 495 Modern Navigation Systems
Slide 3 of 10
State Augmentation
Case 1. Colored State Noise
• An example
 Given the ECEF error mechanization
 ebe 
 e 
  veb 
  rebe 


 ebe 
 e 
  veb 
  rebe 



e


033
ie


ˆ
 [Cˆ be f ibb ] 2iee


 033
I 33
033
2 g 0 ( Lˆb )rˆebe ˆ e
reb
2
e ˆ
e
r ( L ) rˆ
𝐵1

  e  033
eb
 e
T
e 
   veb    Cˆ b
 e  
   reb  033

 
eS
b
eb
033
 ebe 


F1   vebe   B1  f ibb  B2 ibb
  rebe 


𝐵2
Cˆ be 
  f b 
033   ibb 

033   ib 

B1 , B2  9 x 3
 where the gyro error is due to an uncompensated bias
instability term and the accel error is zero (  fibb  0 )
o
Mon, March 28
Assume that we have calibrated all of the other
measurement error terms (or they are ~0)
EE 495 Modern Navigation Systems
Slide 4 of 10
State Augmentation
Case 1. Colored State Noise
 Hence,
ibb  bg   I  M g  ibb  Gg f ibb  wg

 ibb  bg  M g ibb  Gg f ibb  wg
ibb  ibb  ibb

 ibb  ibb
 Recalling the relationship between meas errors and 
ibb  ˆibb  ibb
bBI
 If the BI term is a Gauss-Markov process, then
 1 
bBI (t )  
 bBI (t )  w(t )
  BI 
Mon, March 28
EE 495 Modern Navigation Systems
Slide 5 of 10
State Augmentation
Case 1. Colored State Noise
• An Example
 Recalling the ECEF error mechanization


v
r

e
eb
e
eb
e
eb







F1   v
r

e
eb
e
eb
e
eb
ibb   bBI
0


b
b

B

f

B

2
ib
 1 ib


 Augmenting the state vector
e
 xINS



 bBI 
Mon, March 28
e
F1  xINS
 B2 bBI
 1 
bBI (t )  
 bBI (t )  w(t )
  BI 
e e
e ee
xINS


x


   09 x 3 

x

F

B
1 B2 2 INS
F1INS  
xxINS
INS
     
   
w(t )



? bbBIbBI   I 3 x 3 
 b0BI3bxBI9 I 3?x 3 /  BI
 BI 
EE 495 Modern Navigation Systems
Slide 6 of 10
State Augmentation
Case 2. Colored Measurement Noise
• Given a continuous state-space system
x1 (t )  F1 (t ) x1 (t )  G1 (t ) w(t )
z1 (t )  H1 (t ) x1 (t )  v1 (t )
 where the measurement noise 𝑣1 is non-white
 We can model the non-white noise as
x2 (t )  F2 (t ) x2 (t )  G2 (t ) v2 (t )
v1 (t )  H 2 (t ) x2 (t )
 where 𝑣2 is now a white noise process!!
Mon, March 28
EE 495 Modern Navigation Systems
Slide 7 of 10
State Augmentation
Case 2. Colored Measurement Noise
• Considering the augmented state vector
 x1 (t ) 
x (t )  

x
(
t
)
 2 
 we now have
 x1 (t ) 
x (t )  

 x2 (t ) 
x1 (t )  F1 (t ) x1 (t )  G1 (t ) w(t )
z (t )  H1 (t ) x1 (t )  v1 (t )
x2 (t )  F2 (t ) x2 (t )  G2 (t ) v2 (t )
v1 (t )  H 2 (t ) x2 (t )
 F11(t ) 0 0   G1(tG) 1 (t0)   w0(t )   w(t ) 

x (t )x(t)  
 v (t ) 

 G
0
F
(
t
)
0
v
(
(
t
t
)
)
0
0
0
  22   2 

2
  
 and
z (t )   H1 (t ) H 2 (t )  x (t )
The augmented system is now driven by white noise
Mon, March 28
EE 495 Modern Navigation Systems
Slide 8 of 10
State Augmentation
Continuous to Discrete State-Space Representation
• State-space from continuous  discrete time:
 Given the state and measurement equations
x (t )  A x (t )  B u (t )
z (t )  H x (t )  v (t )
 The solution to the state equation can be shown to be
t
x (t )   (t , t0 ) x (t0 )  B   (t   ) u ( )d
t0
t
 e A(t t0 ) x (t0 )  B  e A(t  ) u ( )d
t0
 By considering t = k T and t0 = (k-1) T, we can write
Mon, March 28
EE 495 Modern Navigation Systems
Slide 9 of 10
State Augmentation
Continuous to Discrete State-Space Representation
• Continuing
t
x (t )  e A(t t0 ) x (t0 )  B  e A(t  ) u ( )d
t0
t = k T and t0 = (k-1) T
t
x (k )  e A( T ) x (k  1)  B  e A(t  ) u ( )d
t0
 Thus,
 t A(t  ) 
F x (k  1)   B  e
d  u (k )
 t0

x (k )
If u changes “slowly” wrt T
F x (k  1)  G u (k )
 Also, evaluating the measurement eqn at t = k T , gives
z (k )  H x (k )  v (k )
Mon, March 28
EE 495 Modern Navigation Systems
Slide 10 of 10