Applying a Sigma-Point Kalman Filter to State of Charge Estimation

I’ve found a research paper that would allow me to implement a Kalman filter to estimate state of charge for a LiPo battery, but cannot make sense of some of the symbols as I’m a first year. Can anybody give a rough description of the process used (for the Sigma Point Kalman Filter detailed) and what some of the symbols in the augmented state covariance mean. We have some understanding but do not fully understand the meaning of some of the positive and negative symbols in the superscript and subscript mean.

Need to get an overview so we can finish develop a working model in Matlab.

Here is a link to the paper: https://www.researchgate.net/publication/308672015_State-of-charge_estimation_based_on_microcontroller-implemented_sigma-point_Kalman_filter_in_a_modular_cell_balancing_system_for_Lithium-Ion_battery_packs

Cheers.

Bayesian parameter estimation

I am generally not that knowledgeable for math, so if my question is too broad or inaccurate, please let me know.

I am currently reading a paragraph of one paper (https://www.fil.ion.ucl.ac.uk/spm/doc/papers/karl_bayes_dyn.pdf), which is about Bayesian parameter estimation of nonlinear dynamical system. Here is the link for screenshoted paragraph (link).

I have been going through more or less well entire paragraphs but there is one particular transition that I could not figure out, which is about Equation (13). How I can get specified equations for ‘parameter covariance’ and ‘conditional mean parameters given a data’ as stated in the paper?

University year 1, chapter 7 (estimation)

A random sample of 25 female students is chosen from students at higher education establishments in a particular area of a country, and it is found that their mean height is 163 centimeters with a sample variance of 64.

a. Assuming that the distribution of the heights of the students may be regarded as normally distributed, calculate a 97% confidence interval for the mean height of female students.

b. You are asked to obtain a 97% confidence interval for that mean height of width 2 centimeters. What sample size would be needed in order to achieve that degree of accuracy?

c. Suppose that a sample of 15 had been obtained from a single student hostel, in which there were a large number of female students (still with a distribution of heights which was well approximated by the normal distribution). The mean height is found to be 155 centimeters. Calculate a 99% confidence interval for the mean height of female students in the hostel. How do their heights compare with the group in 4(a)(i) ? [(159.3, 166.7), 339, (148.9, 161.1)]

Hi! Are you able to solve this qns?

Understanding the solution to this polynomial estimation problem

Show that there exist $ K, N > 0$ such that for $ x ∈ R$ , $ $ x ≥ N \implies \frac{3x^ 2 − 4x + 8}{ 5x + 6} ≥ Kx.$ $

Solution:

For $ x ≥ 4,$ we have $ 4x ≤ x^ 2$ so that $ $ 3x^ 2 − 4x + 8 ≥ 3x^ 2 − x^ 2 = 2x^ 2.$ $

Similarly, for $ x ≥ 1,$ we have $ $ 5x + 6 ≤ 11x.$ $ Therefore, when $ x ≥ 4,$ we have $ $ \frac{3x^ 2 − 4x + 8}{ 5x + 6} ≥\frac{ 2x^ 2}{ 11x} = \frac{2} {11} x,$ $

and so we can take $ N = 4$ and $ K = \frac{2}{ 11} .$

I’m struggling to follow the logical steps in this solution. I’m not sure why the inequality of $ x ≥ 4$ and $ x ≥ 1$ has been chosen. Can someone please break down this solution and show where the steps come from?

Robust estimation of the mean

Problem:

Suppose we want to estimate the mean µ of a random variable $ X$ from a sample $ X_1 , \dots , X_N$ drawn independently from the distribution of $ X$ . We want an $ \varepsilon$ -accurate estimate, i.e. one that falls in the interval $ (\mu − \varepsilon, \mu + \varepsilon)$ .

Show that a sample of size $ N = O( \log (\delta^{−1} )\, \sigma^2 / \varepsilon^2 )$ is sufficient to compute an $ \varepsilon$ -accurate estimate with probability at least $ 1 −\delta$ .

Hint: Use the median of $ O(log(\delta^{−1}))$ weak estimates.

It is easy to use Chebyshev’s inequality to find a weak estimate of $ N = O( \sigma^2 / \delta \varepsilon^2 )$ .

However, I do not how to find inequality about their median. The wikipedia of median (https://en.wikipedia.org/wiki/Median#The_sample_median) says sample median asymptotically normal but this does not give a bound for specific $ N$ . Any suggestion is welcome.

Asymptotic estimation of $A_n$

Let $ A_n$ represent the number of integers that can be written as the product of two element of $ [[1,n]]$ .

I am looking for an asymptotic estimation of $ A_n$ .

First, I think it’s a good start to look at the exponent $ \alpha$ such that :

$ $ A_n = o(n^\alpha)$ $

I think we have : $ 2 < alpha $ . To prove this lower bound we use the fact that the number of primer numbers $ \leq n$ is about $ \frac{n}{\log n}$ . Hence we have the trivial lower bound (assuming $ n$ is big enough) :

$ $ \frac{n}{\log n} \cdot \binom{ E(\frac{n}{\log n})}{2} = o(n^3)$ $

Now is it possible to get a good asymptotic for $ A_n$ and not just this lower bound ? Is what I’ve done so far correct ?

Thank you !