state-space recursive least squares with adaptive memory college of electrical & mechanical...

State-Space Recursive Least Squares with Adaptive Memory

College of Electrical & Mechanical EngineeringNational University of Sciences & Technology (NUST)

EE-869 Adaptive Filters

3

Introduction A recursive algorithm Built around state-space model of an unforced system Based on least squares approach Does not require process or observation noise

statistics Works for time-invariant & time-variant environment

alike Can handle scalar and vector observations Adapts forgetting factor that may be required due

to Model uncertainty Presence of unknown external disturbances Time-varying nature of observed signal Non-stationary behaviour of observation noise

SSRLS

SSRLS

SSRLSWAM

Preview of SSRLS

5

State-Space Model

process statesoutput signalobservation noisesystem matrix (full rank) observation matrix (full rank)L-step observable

[ ][ ][ ][ ][ ][ ], [ ]

n

m

m

n n

m n

kkk

A kC kA k C k

xyv

[ 1]k x [ ]kx[ ]C k

[ ]kv

[ ]ky

Unforced System

1z I

[ ]A k

SSRLS

6

Batch Processed Least Squares Approach

7

Batch of Observations


1

2

2

1

[ ][ 1] [ 1][ 2] [ 2] [ ]

[ ] [ ] [ ][ 2] [ 2] [ ][ 1] [ 1] [ ]

[ ] [ ] [ ]

p

p

p p p

CA x ky k p Cx k py k p Cx k p CA x k

k k ky k Cx k CA x ky k Cx k CA x k

y k Cx k Cx k

y v v

[ ] [ 1] [ 2] [ 1] [ ] Tp k v k p v k p v k v k v

[ ] [ ] [ ]p p pk H x k k y v

Noise Vector

8

Least Squares Solution


1

2

2

1

p

p

p

CA

CA

HCA

CAC

Full rank for p l

1ˆ[ ] ( ) [ ]T Tp p p px k H H H k yx Batch Processed Least

Squares Solution

1ˆ[ ] ( ) [ ]T Tp p p px k H WH H W k y Batch Processed Weighted

Least Squares Solution

1

2

0 0 0

0 0 0

0 0 00 0 0

pm

pm

m

m

I

IW

II

Weighting Matrix

9

Recursive Algorithm

10

Predict and Correct

Recursive Algorithm

Predicted Statesˆ[ ] [ 1]x k Ax k

ˆ[ ] [ ] [ 1]y k Cx k CAx k Predicted Signal

[ ] [ ] [ ]k y k y k Prediction Error

ˆ[ ] [ ] [ ] [ ]x k x k K k k Predictor Corrector Form

Estimator Gain

11

Recursive Solution

Recursive Algorithm

Based on k+1 observations

Weighting Matrix

1 1[ ]Tk kH k CA CA CA C

[ ] [0] [1] [ 1] [ ] Tk y y y k y k y k+1 observations

1

0 0 0

0 0 0[ ]

0 0 00 0 0

km

km

m

m

I

IW k

II

12

Recursive Solution (‘contd)

Recursive Algorithm

Defined variables

Direct Form of SSRLS

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

T

T

k H k W k H k

k H k W k k

y

1

ˆ[ ] [ ] [ ]

ˆ[ ] [ ] [ ]

k x k k

x k k k

13

Recursive Update of

Recursive Algorithm

[ ] [ ] [ ] [ ]Tk H k W k H k

[ ]k

11 1 1[ ]k kk T T k k T T k T T Tk A C CA A C CA A C CA C C

11 1 1[ 1]kk T T k T T Tk A C CA A C CA C C

1[ ] [ 1]T Tk A k A C C Difference Lyapunov Equation

14

Matrix Inversion Lemma

Recursive Algorithm

Matrix Inversion Lemma

1 1Tn n n n n m m m m nE F G D G

1 1( )T TE F FG D GFG GF

15

Recursive Update of

Recursive Algorithm

1[ ]k

Riccati Equation for SSRLS

1 1 1 2 1

11 1 1

[ ] [ 1] [ 1]

[ 1] [ 1]

T T T

T T T

k A k A A k A C

I CA k A C CA k A

1 1 1 2 1

11 1 1

[ ] [ 1] [ 1]

[ 1] [ 1]

T T T

T T

A k A k k A C

I CA k A C CA k

[ ][ 1]

TE A k AF kG CAD I

Define 1[ ] [ ]P k k

16

Recursive Update of

Recursive Algorithm

[ ]k

Recursive solution

11[ ] [0] [1] [ 1] [ ]k kk T T k T T T T Tk A C y A C y A C y k C y k

11[ 1] [0] [ 2] [ 1]kk T T T T Tk A C y A C y k C y k

[ ] [ 1] [ ]T Tk A k C y k

[ ] [ ] [ ] [ ]Tk H k W k k y

17

Observer Gain

Recursive Algorithm

[ ]K k

1[ ] [ ] TK k k C Defined

18

State-Space Representation of SSRLS

Recursive Solution

[ ] [ 1]w k k Defined

[ ] [ 1] [ ]T Tk A k C y k

[ 1] [ ] [ ]T Tw k A w k C y k

1

1

ˆ[ ] [ ] [ ]

[ ] [ ] [ ] [ ]T

x k k k

k A w k K k y k

Therefore

Similarly

1, , [ ] , [ ]T T TA C k A K k State-Space Matrices

19

Initializating SSRLS

Recursive Algorithm

[0], [1], [ 2]l Rank Deficient

[0] I [0] TC C Ior 1) Initializing using Regularization Term

0 [0]x 0

2) Initialization using batch processing approach leads to delayed recursion - offers better initialization

20

Steady-State SSRLS

21

Steady-State Solution of SSRLS

Steady-State SSRLS

if

1T TA A C C

1

0lim [ ]

i ii T T

k ik A C C A

Can be written like this

min ( )Eigenvalues A

1 For neutrally stable systems

22

Direct Form of Steady-State SSRLS

Steady-State SSRLS

1ˆ[ ] [ ] [ ]x k k k

1ˆ[ ] [ ]x k k

23

Observer Gain for Steady-State SSRLS

Steady-State SSRLS

1 [ ]K k

24

Transfer Function Representation

Steady-State SSRLS

11

11

( ) T T T

T T T

H z A zI A C K

A zI A I C

25

Initialization of Steady-State SSRLS

Steady-State SSRLS

Initialize only [0]x

[0] 0x Preferable choice if no other estimate is available

26

Memory Length

Steady-State SSRLS

Filter Memory1

1

2 111

Asymptotic result

Model Uncertainty and Unknown External Disturbances

28

Underlying Model

process statesoutputexternal disturbance (bounded, deterministic)observation noisesystem matrixinput matrixobservation matrix

[ ]w k

[ ][ ][ ][ ]

n

m

s

m

n no

n so

m no

x ky kw kv kABC

[ 1]x k [ ]x koC

[ ]v k

[ ]y k1z I

oA

oB


Controllable pair

29

Assumptions about Observation Noise

[ ]E k 0v

2

[ ] [ ]T v if k jE k jotherwise

I0

v v

Zero Mean

White


30

Perturbation Matrices


o

o

A A AC C C

o

o

A A AC C C

31

Estimation Error

where


[ ] [ ] [ ]

( )o

F A KCAk F x k B w k

F A K A C A C AC

ˆ[ ] [ ] [ ]

ˆ[ 1] [ 1] [ 1] ( [ ] [ ]

( ) [ 1] [ ] [ 1][ 1] [ ] [ 1]

o o

e k x k x kA x k B w k Ax k K y k y k

A KCA e k Kv k kFe k Kv k k

White Input Deterministic Input

32

Steady-State Mean Estimation Error


[ ] [ 1] [ ] [ 1]e k Fe k Kv k k

[ ] [ 1] [ 1]E e k F E e k k

Deterministic Input

33

Bounds on Steady-State Mean Estimation Error


34

Steady-State Mean Square Error


[ ] [ ] [ ]TR k E e k e k Estimation Error Correlation Matrix

2[ ] [ 1] [ 1]T TvR k FR k F KK k

[ ] [ ] [ ] [ ] [ ] [ ] [ ]T T T Tk k k FE e k k k E e k F where

[ ] [ ] [ ]

( )o

F A KCAk F x k B w k

F A K A C A C AC

35

Bounds on Steady-State Mean Square Estimation Error


SSRLS with Adaptive Memory (SSRLSWAM)

37

The Cost Function cost function

gradient of costfunction

row vector

where

1[ ] [ ] [ ]2

TJ k E k k

[ ][ ]

[ ] [ ]T

J kk

kE k

[ ] ˆ[ ] [ 1]

[ 1]

k y k CAx k

CA k

ˆ[ ][ ] x kk


38

Gradient of Cost Function[ ] [ 1] [ ]T T Tk E k A C k


Deterministic Gradient

1[ ] [ ]P k kˆ[ ] [ ] [ ] [ ]Tx k x k P k C k

[ ][ ] P kS k

Define

ˆ[ ][ ]

ˆ[ 1] [ ][ ] [ ] [ ]

( [ ] ) [ 1] [ ] [ ]

T T

T

x kk

x k kA S k C k P k C

A K k CA k S k C k

39

Gradient of Cost Function (‘contd)

1 1 1 2 1

11 1 1

[ ] [ 1] [ 1]

[ 1] [ 1]

T T T

T T T

k A k A A k A C

I CA k A C CA k A


11 1( ) ( )( ) ( )X XX X

1

1 1

[ ] [ ] [ 1] [ ]

[ ] [ ] [ ]

T T T

T

S k I K k C AS k A I C K k

P k K k K k

1[ ] [ ]P k k[ ][ ] P kS k

40

Tuning Forgetting Factor

[ ] [ 1] [ ]k k k

ˆ [ ] [ 1] [ ]T T Tk k A C k


Stochastic Gradient

Update using Stochastic Gradient Method

ˆ[ ] [ 1] [ ]

[ 1] [ 1] [ ]T T T

k k k

k k A C k

41

SSRLSWAM – Complete Algorithm


1

11

1

1

1

[ ] [ 1] [ 1]

[ 1] [ 1]

ˆ[ ] [ ] [ 1]ˆ ˆ[ ] [ 1] [ ] [ ]

[ ] [ 1] [ 1]

[ 1] [ ] [ 1]

[ ] [ 1] [ 1] [ ]

[ ] [ ] [ ] [ 1

T T

T T

T

T

T T T

K k k AP k A C

I k CAP k A C

k y k CAx kx k Ax k K k k

P k k AP k A

k K k CAP k A

k k k A C k

S k k I K k C AS k

1 1

] [ ]

[ ] [ ] [ ] [ ] [ ]

[ ] ( [ ] ) [ 1] [ ] [ ]

T T T

T

T

A I C K k

k P k k K k K k

k A K k CA k S k C k

42

Initializing SSRLSWAM

[0]

2) Initialization using batch processing approach leads to delayed recursion - offers better initialization


[0] 0

Some suitable value < 1

[0]P

ˆ[0]x

Approximate Solution

44

Approximate Solution using Symbolic Computations


Discrete Lyapunov Equation for S4RLS

Can be computed Symbolically, Off-line

1[ ] T Tk A A C C

( [ ]) ( )k 1( ) ( )P

( )[ ] ( ) PS k S

45

Approximate Solution using Symbolic Computations (‘contd)


1

11

1

ˆ[ ] [ ] [ 1]

[ ] [ 1] [ 1] [ ]

[ ] [ ]

[ ] [ ]

[ ] [ 1]

[ 1] [ 1]

[ 1] [ 1]

[ ] ( [ ])ˆ ˆ[ ] [ 1] [ ] [ ]

[ ] ( [ ] ) [ 1]

T T T

T T

T T

T T

k y k CAx k

k k k A C k

P k P k

S k S k

K k K k

k AP k A C

I k CAP k A C

P k C k Cx k Ax k K k k

k A K k CA k

[ ] [ ]TS k C k

46

Approximate Solution using Symbolic Computations (‘contd)


( ) ( )

( ) ( ) T

F A K CA

G S C

ˆ[ ] [ ] [ 1]

[ ] [ 1] [ 1] [ ]

ˆ ˆ[ ] [ 1] ( [ ]) [ ][ ] ( [ ]) [ 1] ( [ ]) [ ]

T T T

k y k CAx k

k k k A C k

x k Ax k K k kk F k k G k k

Define

Simplified Algorithm

47

A Special Case (Constant Acceleration)

48

A Special Case (Constant Acceleration)


System Matrices

21 20 10 0 1

1 0 0

T TA T

C

3

2

3

2

1

3(1 ) (1 )( )2

(1 )

KT

T

Symbolic Computation

49

A Special Case (Constant Acceleration) – ‘Continued


2 33 3

2 2 2

3 3 3

2

23(1 ) (1 ) 3(1 ) (1 ) 3 (1 ) (1 )( ) 1

2 2 4(1 ) (1 ) (1 )1

2

TT

TF T

T

TT

Symbolic Computation

2

2

2

(1 )(1 3 )( ) 32

(1 )

GT

T

Computational Complexity

51

Computational Complexities: Standard Algorithms

52

Computational Complexities: SSRLSWAM and Variants

Example of Tracking a Noisy Chirp

54

Chirp Signal


Sinusoid whose frequency drifts with time

Model Used by Tracker

2( ) sin 0.0001 3y t t

21 20 10 0 1

1 0 0

T TA T

C

Model Mismatch

Actual Model is Chirped Sinusoid Model

55

Performance of SSRLSWAM

2

0.00050.1

0.1[0] 0.98[0]

v

T

0

Simulation Parameters


56

Tuning of Forgetting Factor


2

0.00050.1

0.1[0] 0.98[0]

v

T

0


57

Performance of SSRLS

2

0.00050.1

0.1[0] 0.98[0]

v

T

0



58

Conclusion

SSRLSWAM is a combination of SSRLS and stochastic gradient method

S4RLSWAM alleviates computational burden of SSRLSWAM

Suitable for time-varying scenario Compensates for model uncertainty to some extent

Conclusion

59

References Mohammad Bilal Malik, “State-space recursive least

squares with adaptive memory”, Signal Processing Journal, Vol. 86, pp 1365-1374, 2006

References

state-space recursive least squares with adaptive memory college of electrical & mechanical...

Documents