As mentioned above, Kalman filtering relies on two successive updates, namely the prediction step and the...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
As mentioned above, Kalman filtering relies on two successive updates, namely the prediction step and the data update (also referred to as correction step). These steps are detailed below. During these updates, the goal is to update the mean m and covariance matrix of the state vector p (position and velocity here). First, let us introduce some variables mt-1: Expected value of p at time t, before observing z(t) (only 2(t-1)= [(1),..., 2(t-1)] have been observed). • Ett-1: Covariance matrix of p at time t, before observing z(t) (only 2(t-1) = [z(1),..., 2(t-1)] have been observed). mtt: Expected value of p at time t after having observed z(t). • Ett: Covariance matrix of p at time t, after having observed z(t). In other words, • mt-1 and Ett-1 are respectively the mean and covariance matrix of f(pt) 2(t-1)). met and Ett are respectively the mean and covariance matrix associated with f(pt) |z(t)). To compute these quantities sequentially, we will use the Bayes' rule and compute marginal distributions. Let's assume that before any measurement, we model the prior distribution of p(0), denoted by f(p)), by a bivariate normal distribution with known mean mojo and known covariance matrix 20|0- First we will predict the value of p(¹) (before observing z(¹)) using the result of Question 3. Question 5: using f(p¹)) = f f(p¹) p)f(p0)dp(0), verify that f(p(1)) is the density of a bivariate Gaussian distribution and compute its mean and covariance matrix. Detail how you obtain the final result. (4 Marks) The prediction of p(t) (before observing z(t) but after having observed z(t-1)) can be per- formed in the similar fashion, as we will see next. Question 6: If we assume the f(p(t-1) 2(t-1)) is a Gaussian probability density function and using f(pt) 2(t-1)) = f(pt) |p(t-1))f (p(t-1) 2(t-1))dp(t-1), verify that f(pt) |z(t-1)) is As mentioned above, Kalman filtering relies on two successive updates, namely the prediction step and the data update (also referred to as correction step). These steps are detailed below. During these updates, the goal is to update the mean m and covariance matrix of the state vector p (position and velocity here). First, let us introduce some variables mt-1: Expected value of p at time t, before observing z(t) (only 2(t-1)= [(1),..., 2(t-1)] have been observed). • Ett-1: Covariance matrix of p at time t, before observing z(t) (only 2(t-1) = [z(1),..., 2(t-1)] have been observed). mtt: Expected value of p at time t after having observed z(t). • Ett: Covariance matrix of p at time t, after having observed z(t). In other words, • mt-1 and Ett-1 are respectively the mean and covariance matrix of f(pt) 2(t-1)). met and Ett are respectively the mean and covariance matrix associated with f(pt) |z(t)). To compute these quantities sequentially, we will use the Bayes' rule and compute marginal distributions. Let's assume that before any measurement, we model the prior distribution of p(0), denoted by f(p)), by a bivariate normal distribution with known mean mojo and known covariance matrix 20|0- First we will predict the value of p(¹) (before observing z(¹)) using the result of Question 3. Question 5: using f(p¹)) = f f(p¹) p)f(p0)dp(0), verify that f(p(1)) is the density of a bivariate Gaussian distribution and compute its mean and covariance matrix. Detail how you obtain the final result. (4 Marks) The prediction of p(t) (before observing z(t) but after having observed z(t-1)) can be per- formed in the similar fashion, as we will see next. Question 6: If we assume the f(p(t-1) 2(t-1)) is a Gaussian probability density function and using f(pt) 2(t-1)) = f(pt) |p(t-1))f (p(t-1) 2(t-1))dp(t-1), verify that f(pt) |z(t-1)) is
Expert Answer:
Related Book For
Fundamentals of Hydraulic Engineering Systems
ISBN: 978-0136016380
4th edition
Authors: Robert J. Houghtalen, A. Osman H. Akan, Ned H. C. Hwang
Posted Date:
Students also viewed these mathematics questions
-
The density of a uniform cylinder s determined by measuring its mass m, length I and diameter d. Calculate the density (in kg m-3) and its error from the following values m = 47.36 + 0.01 g. 1- 15.28...
-
The density of a rock will be measured by placing it into a graduated cylinder partially filled with water, and then measuring the volume of water displaced. The density D is given by D = m/(V1 V0),...
-
You need to determine the density of a ceramic statue. If you suspend it from a spring scale, the scale reads 28.4 N. If you then lower the statue into a tub of water, so that it is completely...
-
Solve the polynomial inequality. (a) Symbolically and (b) Graphically. Use interval notation to write the solution set. x-x>0
-
A solution is made by mixing 33.8 g of CH3OH in 50.0 g of H2O. What is the mole fraction of CH3OH in the solution?
-
On 1 October 2022, the Australian company Pilkari Ltd enters a loan agreement with the Bank of Scotland to borrow 2 000 000 for a period of 5 years. The interest on the borrowings is payable...
-
(a) Complete a steady-state simulation of the vinyl-chloride process in Figure 2.6. First, create a simulation flowsheet. Assume that: Cooling water is heated from 30 to \(50^{\circ} \mathrm{C}\)...
-
In 2008, 3 years after it began operations, the Pearce Corporation decided to change from the direct write-off method of recording bad debts to estimating bad debts. The following information is...
-
The world long jump record is 9.95 m (Mike Powell, USA, 1991).Treated as a projectile, what is the maximum range, in meters, obtainable by a person if he or she has a take-off speed of 8.9 m/s?...
-
Part 2: Presentation and Data Work (90 marks) You have been asked to collect company data and to create a final written presentation based on your analysis of the data and the performance measures...
-
In 1626, the Indians sold the island of Manhattan for $20. If this money were deposited in a bank and the annual growth was N%, what would the capital be this year? Use the current year, 2022, for...
-
An increase in the price level would shift the: A. IS curve. B. LM curve. C. aggregate demand curve.
-
If an excise tax is paid by the buyer instead of the seller, which of the following statements is most likely to be true? A. The price paid will be higher than if the seller had paid the tax. B. The...
-
An increase in the nominal money supply would shift the: A. IS curve and the LM curve. B. IS curve and the aggregate demand curve. C. LM curve and the aggregate demand curve.
-
An increase in government spending would shift the: A. IS curve and the LM curve. B. IS curve and the aggregate demand curve. C. LM curve and the aggregate demand curve.
-
Which of the following would be included in Canadian GDP for a given year? The market value of: A. wine grown in Canada by U.S. citizens. B. electronics made in Japan and sold in Canada. C. movies...
-
10. (RSA Public Key Cryptosystem) Let us pick two prime numbers p = 41,q = 67. Then, n pq 41 67 = 2747 (n) (p 1)(q-1) = 40 66 = 2640 We pick a small prime numbers, with gcd (s, (n)) = 1. s = 13....
-
Apply Jacobis method to the given system. Take the zero vector as the initial approximation and work with four-significant-digit accuracy until two successive iterates agree within 0.001 in each...
-
A solid brass sphere of 30-cm diameter is used to hold a cylindrical buoy in place (Figure P2.8.3 l in seawater (sp. gr. = 1.03). The buoy (sp. gr. = 0.45) has a height of 2 m and is tied to the...
-
A model is built to study the surface tension phenomena in a reservoir. Determine the conversion ratios between the model and the prototype for the following quantities if the model is built with a...
-
Determine the discharge in each pipe in Figure P5.7.3 if EA= 100 ft, E8 = 80 ft, and Ee = 120 ft. The pipe and pump characteristics are shown in the following tables. Pipe Characteristics Pump...
-
Vertical analysis would rarely be performed on which of the following statements or schedules? a. Income statement b. Adjusting entry worksheet c. Balance sheet d. All of the above are common targets...
-
A statement that lists the assets, liabilities, and stockholders equity of a company in percentages only with no dollar amounts is a a. common-size income statement. b. benchmarking analysis. c....
-
In vertical analysis, the base used for comparison on the income statement is a. total expenses. b. total assets. c. net sales. d. gross profit.
Study smarter with the SolutionInn App