5. Properties of Kalman Filtering Orthogonality Principle

5. Properties of Kalman Filtering

Orthogonality Principle

The Kalman filter is a special case of the mean-square estimator, that is, the Kalman estimates are the conditional mean estimates, viz.,

ˆ_KF { | }

x = E x z . (5-1)

Now, we want to show that the Kalman filter satisfies the orthogonality principle. Let RVs x and z be jointly distributed. Then for any function g(⋅),

{ ( )( { | }) }^T 0 E g z x −E x z = .

That is, any function of z is orthogonal to x once the conditional mean has been subtracted out. To show this, write

{ ( )( { | }) } { ( ) } { ( ) { | }}

{ ( ) } { { ( ) | }}

{ ( ) } { ( ) } 0

T T T

T T

T T

E g z x E x z E g z x E g z E x z E g z x E E g z x z E g z x E g z x

− = −

= −

= − =

We used the fact that

( ) { ^T | } ( ( ) ^T | } g z E x z = E g z x z

for any function since g(z) is deterministic if z is fixed. The orthogonality principle says that the RV

( ) g ⋅ { | x= −x E x z},

which is the estimation error, is orthogonal to all other RVs g(z). It has the following implication for estimation theory. If g( )⋅ is any function (or filter), then

{ | } ( )

E x −E x z ≤E x −g z ,

( ) ( )

( ) ( )

( )

2 2

2 2

2

Proof: ( ) { | } { | } ( )

{ | } { | } ( ) (orthogonality principle applied)

{ | }

E x g z E x E x z E x z g z E x E x z E E x z g z

E x E x z

⎛ − = − + − ⎞

⎜ ⎟

⎜ = − + − ⎟

⎜ ⎟

⎜ ≥ − ⎟

⎝ ⎠

where ⋅ denotes the Euclidean norm. Thus no other function of z provides a “closer approximation to x” in a probabilistic sense than does the Kalman filter which estimates x as E x z{ | }.

White Gaussian Residual

The residual is defined by ) ( ˆ −

−

= _{k k k}

k z H x

r , (5-2)

which can be expressed

k t

t k

k k

k k

k k k k

t

t k

k k k

k k k

k k k

v d

G t

H x

H

x H

v d

G t

x H

x H v

x H r

k k k

k

+ Φ

+ + Φ

=

+ Φ

− + Φ

+ Φ

=

−

− +

=

∫

∫

−

−

−

−

−

−

−

−

−

−

1 1

) ( ) ( ) , ( )

~ (

) ( ˆ ]

) ( ) ( ) , ( [

) ˆ (

1 1

1

1 1 1

1 1

τ β τ τ

τ β τ

τ (5-3)

where β(τ) is the Brownian motion, i.e., dβ τ( )= w( )τ τd .

Let’s investigate the right-hand side terms in Eq. (5-3). Notice that the unconditional density

( )

{ }

1 1 1

1 1 1

1 1

( ) ( ) 1 1

( )| 1 1

1/ 2 1

/ 2 1

( )| 1 1 1

( ) ( , )

( ) ( )

(Note that ( | ) 2 ( ) exp 1 ( )

2

k k k

k k k

k k

k

x x z k k

x z z k k

n T

x z k k k

x

f f z dz

f f z dz

f z P P

f

ξ ξ

ξ

ξ π ξ ξ

− − −

− − −

− −

∞

+ + − −

−∞

∞

+ − −

−∞

− −

+ − − −

=

=

⎡ ⎤

= ⎢⎣ + ⎥⎦ − +

=

∫

∫

_{1 1}

1 1 1

1 1

( )| 1 1

( )| 1 1

( )|

( ) since ( ) is independent of .)

( ) ( )

( )

k

k k k

k k

z k k

x z z k k

x z

P z

f f z dz

f

ξ ξ

ξ

− −

− − −

− −

+ − −

∞

+ − −

−∞

+

+

=

=

∫

This equation indicates that ~ ₁

−

xk

−

zk

is independent of . The second term of the right-hand side of Eq. (5-3) is also independent of since it involves only transition matrix and process noise.

Finally, is independent of because it is white. Therefore, we can say that is independent of . On the other hand, we can notice that is a linear function of . Therefore, we can conclude that is a white sequence.

−1

zk

−1

zk

vk ₁ r_k

−1

zk r_k z_k

rk

Eq. (5-3) also tells us that since is Gaussian, is Gaussian with the following mean and variance.

zk r_k

0 } { )}

, ( { )}

~ ( { }

{r_k = H_kΦ_k−1E x_k−1 + + H_kE I ⋅⋅ + E v_k =

E (5-4)

k T

k k

k

T k k

k k k k

k k T

k k

R H

P H

v x

x H v x

x H E r

r E

+

−

=

+

−

− +

−

−

=

) (

} ) )]

( ˆ [

)(

)]

( ˆ [

{(

}

{ (5-5)

This verifies that the residual r_k is a white Gaussian sequence. This property is useful:

(1) to verify a design of Kalman filter, (2) to detect a sensor failure or bad data.

Stability

Consider the following time-invariant system and the deterministic asymptotic estimation

x_k+1 = Ax_k +Bu_k (5-6a)

z_k = Hx_k (5-6b)

where state x_k ∈Rⁿ, control input u_k ∈R^m, output z_k ∈R^p; and A, B, and H are known constant matrices of appropriate dimension. All variables are deterministic, so that if initial state x₀ is known then Eq. (5-6) can be solved exactly for x_k, z_k for k ≥0.

Deterministic asymptotic estimation problem: Design an estimator whose output ˆx_k converges with k to the actual state x_k of Eq. (5-6) when the initial state x₀ is unknown, but u_k and z_k are given exactly.

An estimator of observer which solves this problem has the form

( )

ˆ 1 ˆ ˆ

x_k+ = Ax_k +L z_k −Hx_k +Bu_k, (5-7)

as shown in Fig. 5-1.

Figure 5-1 State Observer

B z^-1 H

A

z^-1

A

L B H

measurement

PLANT

residual -

z_k

u

_k ^{xk z}k

estimate ˆx _k ˆx _k

OBSERVER

ˆz_k

To Choose L in Eq. (5-7) so that the estimation error x_k = x_k −xˆ_k goes to zero with k for all x , it is ₀ necessary to examine the dynamics of x_k. Write

( )

( ) ( )

1 1 ˆ 1

x x x

ˆ ˆ

x u x z x u

ˆ ˆ

x x x x

( )x

k k k

k k k k k k

k k k k

k

A B A L H B

A L H H

A LH

+ = + − +

= + −⎡⎣ + − + ⎤⎦

= − − −

= −

It is now apparent that in order that x_k go to zero with k for any x₀, observer gain L must be selected so that (A− LH) is stable. L can be chosen so that x_k →0 if and only if ( ,A H) is detectable which is defined in the sequel.

(1). (A, H) is observable if the poles of (A-LH) can be arbitrarily assigned by appropriate choice of the output injection matrix L.

(2). (A, H) is detectable if (A-LH) can be made asymptotically stable by some matrix L. (If (A, H) is observable, then the pair is detectable; but the reverse is not necessarily true.)

(3). (A, B) is reachable if the poles of (A-BK) can be arbitrarily assigned by appropriate choice of the

feedback matrix K.

(4). (A, B) is stabilizable if (A-BK) can be made asymptotically stable by some matrix K. (If (A, B) is reachable, then (A, B) is stabilizable; but the reverse is not necessarily true.)

Now, consider the following stochastic system

1 0 0 0

x_k+ = Ax_k +Bu_k +Gw , x ~_k N(x ,P ), w ~_k N(0, )Q z_k = Hx_k + v , v ~_{k k} N(0, )R .

The stability of the Kalman filter designed for this system is presented by two theorems.

Theorem 5-1. (Sufficient condition for a Kalman filter to be stable)

Let (A, H) be detectable. Then for every choice of P₀ there is a bounded limiting solution P to

T T

k T

k T

k k

k A P P H HP H R HP A GQG

P₊₁(−) = [ (−)− (−) ( (−) + )⁻¹ (−)] + . (5-8) Furthermore, P is a positive semidefinite solution to the algebraic Riccati equation

T T

T

T HPH R HP A GQG

PH P

A

P = [ − ( + )⁻¹ ] + . (5-9)

(Proof)

(A, H) detectable means that there exists an L that makes (A-LH) asymptotically stable.

Construct a suboptimal xˆ_k using the L

1 ( )ˆ

L L

k k

x ₊ = A LH x− +Bu_k +Lz_k

. (This is a state observer.) The estimation error in this case is given by

[ ]

1 1 1

1 1

ˆ

( )ˆ

( )ˆ

( )

L L

k k k

L

k k k k k k

L

k k k k k k k

k k k

x x x

Ax Bu Gw A LH x Bu Lz

Ax Bu Gw A LH x Bu L Hx v A LH x Gw Lv

+ + +

+ +

= −

= + + − − − −

= + + − − − − +

= − + −

The estimation error covariance is

T T

T k

k A LH S A LH LRL GQG

S ₊₁ = ( − ) ( − ) + + .

If (A-LH) is asymptotically stable, has a finite steady-state value according to the Lyapunov theorem.

Sk

Since the Kalman is optimal, P_k(−) < S_k, where P_k(−) satisfies Eq. (5-8).

From the above rationale, if (A, H) is detectable, P

P

P_{k k k}

k_{→∞ +} (−) = lim _→∞ (−) =

lim ₁

and P is the solution of the algebraic Riccati equation, Eq. (5-9).

Theorem 5-2: (Necessary and sufficient condition for a Kalman filter to be stable)

Let Q be a square root of the process noise covariance so that Q = Q Q^T ≥ 0, and let the measurement noise have R>0. Suppose (A,G Q) is reachable. Then (A, H) is detectable if and only if

a. There is a unique positive definite limiting solution P to Eq. (5-8) which is independent of . Furthermore, P is the unique positive definite solution to the algebraic Riccati equation.

P0

b. The steady-state error system defined by Kalman filter, viz.,

k k k

k k

k A I K H x Gw AK v

x ₊ (−) = ( − )~ (−)+ −

~ 1 (5-10)

with steady-state Kalman gain ) 1

( + ⁻

= PH HPH R

K ^{T T} (5-11)

is asymptotically stable.

(Proof)

Define D by R = DD^T .

R>0 implies that D ≠ 0. Then, there exists an M that holds H=DM.

) ,

(A G Q reachable implies n Q G A zI

rank[ − ]= . (5-12)

(Suppose that rank[zI − A G Q]≠ n, then there exists an n-dimensional non-zero vector q such that q[zI − A G Q]= 0, that is, qA=zq and qG Q = 0. These equations may be expanded to

2 2

0

0 qAG Q zqG Q

qA G Q zqAG Q z qG Q

= =

= = =

⋅

⋅ resulting in ⋅

0 ] ,

, ,

[ ) ,

(A G Q = q G Q AG Q ⋅⋅⋅ A ⁻¹G Q =

qC ⁿ .

This contradicts to the assumption that (A, G Q) is reachable.) Eq. (5-12) may be expanded to

].

, ,

[

0 0 0

0 0 ]

, ,

[

] , ,

[

LD Q G LH A

zI rank

I M

I I LD Q G A zI rank

LD Q G A zI rank n

+

−

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

=

−

=

This implies that ((A− LH),[G Q, LD]) is reachable.

To prove necessity, let gain L define a suboptimal estimate as in the proof of Theorem 5-1, with (A-LH) asymptotically stable. By Theorem 5-1, (A, H) detectable implies

ˆ_k^L x

P

P_k(−) → with P bounded and at least positive semidefinite. But E x_k( )− ² ≤E x_k^{L 2} for all L because of the optimality of the Kalman filter. Hence, system A(I-KH) is also asymptotically stable with

) 1

( + ⁻

= PH HPH R

K ^{T T} .

Define L′= AK. Then we can write the algebraic Riccati equation as

. ] ][

[ ) (

) (

) ( )

( ) (

T T

T T T

T T

D L Q G D L Q G H

L A P H L A

G Q Q G L

DD L H

L A P H L A P

′ + ′

− ′

− ′

=

′ + + ′

− ′

− ′

= (5-13)

We know that P is an at least positive semidefinite solution of Eq. (5-13). But Eq. (5-13) is also a Lyapunov equation, for which we know that (A− L′H) is stable (according to the Lyapunov Theorem) and also that the pair ((A−L′H),[G Q, L′D]) is reachable (shown above). So the solution P is also unique and positive definite, and the gain K = PH^T(HPH^T + R)⁻¹ is uniquely defined.

(Lyapunov Theorem: (Ref. Panos J. Antsaklis and Anthony N. Michel, Linear Systems)

If there is a positive definite and symmetric matrix X and a positive definite and symmetric matrix Q satisfying ,

T 0

A XA− + =X Q , (Lyapunov Equation)

then the matrix A is stable. Conversely, if A is stable, then, given any symmetric matrix Q, the above Lyapunov equation has a unique solution, and if Q is positive definite then X is positive definite.)

To show sufficiency, note that if A(I-KH) is asymptotically stable, there is an L=AK for which the

system (A-LH) is stable and hence the pair (A, H) is detectable. Q.E.D.

In summary, Theorem 5-2 says that if the state is reachable by the process noise, so that every mode is excited by , then the Kalman filter is asymptotically stable if (A, H) is detectable.

Thus, we can guarantee a stable filter by selecting the measurement matrix H correctly and ensuring that the process model is sufficiently corrupted by noise!

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