# Gaussian noise definitions

I have an exercise that asks for us to determine the model for a system that goes like this:

A battery manufacturer for cars is studying how their batteries behave. The measurements are influenced by gaussian noise with center in 0V and standard deviation equal to 3 Volts. Nominal value for the Battery voltage is 14V. Determine the model for the system, as well as initial values to implement a Kalman Filter (in matlab).

It is quite clear what we want done. But I don't really understand how the centered values for the gaussian distributions could be different, and so I have no idea how to treat it. It is most likely that I have some concepts wrong, can someone explain how to deal with a noise that has a mean value different to the nominal value of the object in analysis?

• I'd guess what it means is that the sensor reads $r=V+n$, where $V$ is the true battery voltage and $n$ is a Gaussian RV with zero mean and standard deviation 3. You want to find $V$ but all you have is $r$. – MBaz Oct 17 '16 at 18:08

The text:

The measurements are influenced by gaussian noise with center in 0V and standard deviation equal to 3 Volts.

simply means that the Gaussian noise has $\mu = 0$ (the mean is zero) and $\sigma = 3$ (the standard deviation is 3 volts).

Determine the model for the system, as well as initial values to implement a Kalman Filter

As @MBaz says, one model you could use is: $$r = V + n$$ which has no time-dependent component. Making a Kalman-amenable model out of this might look like: $$V_{k+1} = V_k + m_k\tag{1}$$ $$r_k = V_k + n_k\tag{2}$$ and we'd use the 14V nominal voltage to say $V_0 = 14$.

However, we are given no information about $m_k$ - how the true voltage might vary. We might also want to augment the state (perhaps the variation if $V_k$ depends on the derivative of $V+k$?) and so include a state transition matrix in front of $V_k$ in (1). If we did augment the state, we'd also probably want to add an output matrix on (2).

Could you give me an insight about the initial covariance? Can it be reached from the information given?

Generally, I choose the initial state covariance to be VERY large. This is because I am generally given zero information about what it should be, as in this case. In this case, "VERY" might mean 30 (ten standard deviations).

• I believe the true voltage for this case does not vary, and so mk should be zero. – avio11 Oct 17 '16 at 19:38
• @avio11 : In that case, I truly see zero reason for a Kalman filter. :-) It just adds complexity where none is needed. – Peter K. Oct 17 '16 at 19:44
• It was used as an introductory problem. To make students get used to it, I guess? Could you give me an insight about the initial covariance? Can it be reached from the information given? We are asked to implement a filter in matlab, so we need two initial conditions (initial voltage and initial covariance). – avio11 Oct 17 '16 at 20:06
• @avio11 Amended question. – Peter K. Oct 17 '16 at 20:10
• I really appreciate all the help thus far. I guess I can keep going on my own now. Thanks a lot. – avio11 Oct 17 '16 at 20:15