of the dynamic system. Observability and controllability are further properties of dynamic systems. Example 1 : A video camera captures an object moving along a straight line. Its centroid (location) is described by coordinate x (on this line), and its move by speed v and a constant acceleration a . We do not consider start or end of this motion. The process state is characterized by vector x = ( x, v, a ) T , and we have that ˙ x = ( v, a, 0) T because of ˙ x = v, ˙ v = a, ˙ a = 0 It follows that ˙ x = v a 0 = 0 1 0 0 0 1 0 0 0 · x v a This defines the 3 × 3 system matrix A . It follows that det( A - λ I ) = - λ 3 , i.e. λ 1 , 2 , 3 = 0 (”very stable”) Page 18 September 2006
. . Subject MI37: Kalman Filter - Intro (D) Goal of the Time-Discrete Filter Given is a sequence of noisy observations y 0 , y 1 , . . . , y t - 1 for a linear dynamic system. The goal is to estimate the (internal) state x t = ( x 1 ,t , x 2 ,t , . . . , x n,t ) of the system such that the estimation error is minimized (i.e., this is a recursive estimator). Standard Discrete Filtering Model We assume • a state transition matrix F t which is applied to the (known) previous state x t - 1 , • a control matrix B t which is applied to a control vector u t , and • a process noise vector w t whose joint distribution is a multivariate Gaussian distribution with variance matrix Q t and μ i,t = E [ w i,t ] = 0 , for i = 1 , 2 , . . . , n . We also assume an • observation vector y t of state x t , • an observation matrix H t , and • an observation noise vector v t , whose joint distribution is also a multivariate Gaussian distribution with variance matrix R t and μ i,t = E [ v i,t ] = 0 , for i = 1 , 2 , . . . , n . Page 19 September 2006
. . Subject MI37: Kalman Filter - Intro Kalman Filter Equations Vectors x 0 , w 1 , . . . , w t , v 1 , . . . , v t are all assumed to be mutually independent. The defining equations of a Kalman filter are as follows: x t = F t x t - 1 + B t u t + w t with F t = e Δ t A = I + ∞ X i =1 Δ t i A i i ! y t = H t x t + v t Note that there is often an i 0 > 0 such that A i equals a matrix having zero in all of its components, for all i ≥ i 0 , thus defining a finite sum only for F t . This model is used for deriving the standard Kalman filter - see below. This model represents the linear system ˙ x = A · x with respect to time. There exist modifications of this model, and related modifications of the Kalman filter (not discussed in these lecture notes). Note that e x = 1 + ∞ X i =1 x i i ! Page 20 September 2006
. . Subject MI37: Kalman Filter - Intro Continuation of Example 1 : We continue with considering linear motion with constant acceleration. We have a system vector x t = [ x t , v t , a t ] T (note: a t = a ) and a state transition matrix F t defined by the following equation: x t +1 = 1 Δ t 1 2 Δ t 2 0 1 Δ t 0 0 1 · x t = x t + Δ t · v t + 1 2 Δ t 2 a v t + Δ t · a a Note that “time t ” is short for time t 0 + t · Δ t , that means, Δ t is the actual time difference between time slots t and t + 1 .
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- Spring '17
- Derya ALTUNAY
- Normal Distribution, Probability theory, probability density function, Kalman filter