Kalman Filter

1
Kalman Filter
2
Kalman Filter(1)

• Kalman filter is used to estimate the state of a
  system by using the (noisy) measurements.
• The measurements and system models are used as
  inputs to a feedback filter (which is again
  represented by a differential equation)
• The expected error between the actual and
  estimated value satisfies the Riccati equation
• dS(t)/dt 2F(t)S(t) G2(t)/D2(t) S2(t)
  C2(t)

3
Kalman Filter(2)

• Kalman Filter uses the Concept of Orthogonal
  projection to estimate the state of a system.
• The orthogonal projection of a vector on a plane
  is the approximation of the vector that has the
  least error

4
Kalman Filter(3)

• The Plane that we want to project the state x of
  the system is a (possibly non-linear) space of
  the measurements and model of the system
• In this talk, we assume a linear system x(t)
  s.t.-
• dxt F(t)xt dt C(t)dUt
• Ut is the brownian motion.
• The measurements may be noisy as well-
• dZt/dt Ht G(t).xt.dt D(t)dVt
• Vt is the brownian motion

5
Kalman Filter(4)
Error between actual and estimated state should
be orthogonal to the measurement space
Space of Measurements and system model
System State
Estimated State
6
Kalman Filter(5)

• Question is, what is the projection (or inner
  product) wrt. a space for stochastic processes?
• It is the joint expectation E(xy)
• If xH is the indicator variable of a space H,
  then E(z.xH) E(z H)
• Question is, what is the expression of such a
  conditional expectation. How do we characterize H
  (its dimension, basis etc.)?

7
Kalman Filter(6)

• If (x,z1,z2,..,zn) are jointly gaussian, then-
• The best estimator of x derived from
  z1,z2,..,zn E(x z1,z2,..,zn) is the same as
  some linear combination PL (z1,z2,..,zn) of
  z1,z2,..,zn.
• For any other distribution, a linear combination
  is worst estimator than E(x z1,z2,..,zn)
• If we can prove that the state and measurements
  are jointly gaussian, then we need to calculate
  the best linear estimator only.

8
Kalman Filter(7)

• For a linear system, the pair (xt,zt) is jointly
  gaussian ( Picards Iteration method).
• Hence, a linear combination of Zt is sufficient
  to get the best estimate for Xt.
• Since, Zt is continuous with time, we will take a
  linear functional of Zt
• The best estimate xest(t) is given as -
• xest(t) E(xt) ? f(t)dzt.
• ? f(t,xt)dzt represents the same space as finite
  linear combinations of Zt.
• Now, we need to find f(t,xt).

9
Kalman Filter(8)

• We look at the differential equation of the
  measurements-
• dZt/dt Ht G(t).xt.dt D(t)dVt
• Then dRt G(t). (xt - xest)dt
  D(t)dVt /D(t) dzt G(t)xest /D(t) is a
  brownian motion wrt. the measurement space.
• It is possible as D(t) is assumed to be bounded
  away from zero.
• xt xest is orthogonal to the measurement space.
• Since, xest is in linear space of zt, dRt also
  spans the linear space of zt

10
Kalman Filter(9)

• Now we can state the following-
• xest(t) E(xt) ?s(t)dRt
• We now need to find s(t,xt)
• Note that xt xest(t) is orthogonal to any
  ?f(t)dRt
• Hence, E(xt xest(t). ?f(t)dRt) 0
• This gives us-
• E(xt. ?f(t)dRt) E(xt).E(?f(t)dRt) E(?s(t)dRt.
  ?f(t)dRt)
• E(xt. ?f(t)dRt) 0 E(?s(t)dRt. ?f(t)dRt)
• Select f(t) I(t lt y) (indicator function) to
  get-
• E(xt. Ry) E(?s(t)dRt. Ry) E(?s(t)dt)
  ?s(t)dt, from 0 to y
• ?/dy( E(xt. Ry) ) s(y)

11
Kalman Filter(10)

• So we get-
• xest(t) E(xt) ?s(y)dRy
• dRt G(t). (xt - xest)dt D(t)dVt /D(t)
  dzt G(t)xest /D(t) (from the model of
  measurements)
• The system equation is solved to get expression
  for xt
• s(y) ?/dy E(xt.Ry)
• It can be proved that the expected error xerr(t)
  satisfies the riccati equation-
• ?xerr/dt 2F(t)xerr(t) - G2(t)/D2(t)
  xerr(t)2 C2(t)

12
Kalman Filter(10)

• Solving explicitly for xt for a linear system, we
  get the following equation for xest-
• ?xest/dt F(t) D-2(t).G2(t).xerr xest.dt
  D-2(t).G(t).xerr dZt
• ?xerr/dt 2F(t)xerr(t) - G2(t)/D2(t)
  xerr(t)2 C2(t) (Riccati equation)
• dxt F(t)xt dt C(t)dUtMM
• xt e?F(y)dy x0 ? e-?F(y)dy .C(y).dUy

13
Kalman Filter(11) Example for a brownian motion
Actual State
Estimated State
14
Kalman Filter(12) Example for a brownian
motion(2)
Actual Squared Error in this case
Expected Theoretical Error