Filter Initialization - Model Mismatch Analysis - Engineering Assignment Help

Assignment Task :

Question 1 (Data simulation) 
 
Let us assume that the radar measurements are taken at a sampling rate of 10Hz. Simulate 100 measurement samples, it, k = 1,2„ , 100, for the following assumptions • Initial state is zero vector, i.e. x(0) = [0, OF

• Use = .001 Plot the following on the same axis:

I. The true distance x(k) between the vehicles

2. The measured distance z(k) Plot the following on separate axes:

3. Velocity i(k) 

Plot the following on separate axes: 3. Velocity i(k) 4. Acceleration 3(k) 

Question 2 (Filter initialization) 
Implement the folloiwng two approaches to initialize the Kalman filter (then compare their performance in KF implementations for each question)

1. Three point initialization (of position, velocity and accelration)

2. Random initialization (that could be far away from the tnie value) 

Question 3 (The Kalman filter implementation) 
Implement a Kalman filter Plot the following quantities against time (see Figure 5.3.2.-1 on page 220 of the textbook for hint)

1. Position (tnie vs. KF)

2. Velocity (true vs. KF)

3. Position variance

4. Velocity variance

5. NIS (along with its performance limits on the same axis)

6. NEES (along with its performance limits on the same axis) 

Question 4 (Kalman filter vs. RLS filter) 
1. Develop an RLS filter to estimate x(k).

2. Use the simulated data above to compare the RLS filter against KF

3. Discuss your comparison results 
 

Question 5 (Model mismatch analysis) 
Explain hove model-mismatth can be spotted in a Kalman filter. Implement the following model mismatched filters to demonstrate your analysis 1. A filter consisting of 2-states (using the WNA model) 2. A filter with incorrect knowledge of the covariance matrices 
 

Question 6 (Modelling relative vs. true vehicle state - optional) 
The model described in (1)-(4) models the relative (position, velocity and acceleration) of B w.r.t. A. In order to find the real values, these quantities need to be translated. For example, the relative velocity 3(k) = 3 km/h needs to be translated (baed on the true velocity of A). The objective of this question is to develop a new model such that the state x(k) contains the true state of the vehicle B based on the same measurement z(k) which is either the relative distance or relative velocity of the vehicle B w.r.t. A. 1. Re-write the state-space model (1)-(4) such that the state x(k) represents the true state of vehicle B

2. Demonstrate a Kalman filter based assuming the parameters used in Question 1 for WNA model 

Question 7 (Joint estimation of both vehicles' states - optional) 
In reality, the vehicle A does not know its true position or velocity - all it has is an estimate of these quantities. 
1. Assuming that A has a "measurement" of its true velocity (e.g., through the rpm of 
 

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