Chapter 7
State Estimation
What is in this Chapter?
Kalman Filtering plays a central role in parameter estimation. The basic problem is to nd the
best estimate of the state of a plant at time k given
Probability Theory
1 Preliminaries
2 Probability Spaces
3 Conditioning and Independence
4 Random Variables
5 Distribution and Density Functions
6 Expectation
7 Examples
8 Random Vectors
9 Functions of
E 231
Solutions # 9
1 A simple calculation reveals that
E [Y ] = 0.25E [2X] = 0
E Y2
= 0.25E 4X 2 = 1
Thus X and Y have identical second order statistics. Typical sample paths of these random
variable
E 231
Solutions # 6
1 There are only four possible events A through D, which can be conveniently depicted in the table
below.
rain
A
C
forecast of rain
forecast of no rain
no rain
B
D
Notice that thes
E 231
Solutions # 7
1 (a) Since X, Y are independent, the joint density function factors as
pXY (x, y) = pX(x) pY (y) =
1
2
2
exp x
2
0
y [2, 3]
else
(b) The conditional density function is easily com
E 231
Assignment # 10
Issued: November 03, 2014
Due: November 10, 2014
1 The Newsboy Problem
A paperboy can buy newspapers at the wholesale price of p per paper.
He sells the newspapers at a retail pr
E 231
Assignment # 8
Issued: October 27, 2014
Due: November 03, 2014
1 Sample Paths
Let X N (0, 1) and let
Y =
0
with probability 0.75
2X with probability 0.25
(a) Find E [Y ] and E Y 2 .
(b) From par
E 231
Assignment # 8
Issued: October 15, 2014
Due: October 23, 2014
1 Let X, Y be independent random variables with X N (0, 1) and Y U [2, 3].
(a) Find the joint density function pXY (x, y).
(b) Find
E 231
Assignment # 6
Issued: October 08, 2014
Due: October 15, 2014
1 According to statistical data, it rains in Berkeley two out of three weekends. Forecasters predict
the weather correctly with prob
E231 HW 4, due Wednesday, October 1, 6:00 PM
1. Suppose A Cmn and rank(A) = k. Let F Cmm be an invertible matrix
that transforms A into it row echelon form (so Aref = F A). Let cfw_c1 , c2 , . . . , c
HW4
l. (a) Let W 6 CM" be the invertible matrix which transforms A=k into (A*)wf.
The chain of cquivalenccs makes it clear.
{.12 E C" : An: = ()m} {m E C" : :1:'A" =01xm}
{1: E C : :rWlWA = ()lel}
{m
E231 HW 5 (part a), due Wednesday, October 8, 6:00 PM
1. Recall that C is an eigenvalue of A Cnn if there is a v Cn , v = 0n such
that Av = v. We showed that is an eigenvalue if and only if det (In A)
E231 HW 5 (part B), due Wednesday, October 8, 6:00 PM
1. Suppose x is a vector-valued function of a scalar real variable, so x(t) Cn for each
t R. Let A Cnn , and the evolution of x is governed by the
HW3
l.
(a)
(C)
Addition and scalar multiplication in 8 are inherited from R3. The main
axioms (conmmtativity, associativity and distributivity) hold since 5 is a
subset of the set of all n. X 7;, matr
E231 HW 3, due Wednesday, September 24, 6:00 PM
1. Let S nn be dened as
S 33 := A R33 : A = AT
(a) Show that S 33 (with the eld R) is a vector space, with addition and scalar
multiplication as dened f
E231 HW 2, due Wednesday, Sept 17, 6:00 PM
Capital letters denote matrices. If dimensions are not specied, assume that the dimensions are arbitrary, and if several matrices are involved, the dimension
A.
HWI
1. Here are a few examples that work:
1 l
-1]
. 00
A~[10]
2. Since A is invertible, and B is the common left and right inverse, it must be
that AB = BA = I. Taking transposes of this relation
HWZ
l.
(a)
(b)
The equation AI: = (l, for
0 1 2 () 0 2 4
0 0 0 1 0 () 1
0 0 0 0 1 1 6
0 0 0 0 0 0 0
A:
has many solutions, and since A is in row-echelon form, determining the
general form of solutions
E231 HW 1, due Wednesday, September 10, 6:00 PM
Current due-date is next Wednesday. I will extend by a few days, if necessary.
Capital letters denote matrices. If dimensions are not specied, assume th