Designing State Observers

[@mohanadelsamadony4824]

Hello Prof. Richard, Thank you very much for your useful lectures. I have a question, please. At time 23:20, you said that the b^2 will cancel each other. Can you kindly explain this, please? there might be a math error here

[@hariharanramamurthy9946]

hi , objectove of state observer is compute x, xhat, from the real syatem and pole placed system, but you didn't show how to compute x after placing the pole or real system , since C is not squarematrix(most cases) to compute x = c^-1 *y

[@alamazahreh6423]

Hey Richard, thank you so much for your videos, i have a small question, how did you obtain the wc = sqrt(2) in this example? and if i want to design an observer, would it be possible for a spring-damped system by just assuming some numbers for the coefficients of mass, spring and damper and then just simply estimate the position?

[@richard_pates]

Hello! Good question. The wc=sqrt(2) is a bit of an artificial choice, mainly just to illustrate the calculation. Deciding how to place the poles is really quite a tricky issue in general, but the suggestion in the video is that the butterworth configuration often works quite well. In practice though you may need to experiment with a range of different values of wc, or even try out some other configurations if no value of wc gives the performance you want. As for your second question, in a way, what you are proposing is very close to what a state-observer does - but with a small but important difference. The state observer equation is \dot(xhat)=Axhat+L(yhat-y). If we remove this L(yhat-y) term, the observer would match your suggestion. ie you assume that you know the coefficients (that is you know what is in the matrix A), and then just estimate the position based on that (i.e. simulate \dot(xhat)=Axhat). However the state-observe contains this extra term. Note that if your estimate matches your measurement exactly, yhat=y, and then this second term is zero, and we're in the situation from before. However if the two do not match, this extra term acts as a correction factor - by choosing the matrix L in the right way, the observer will drive this error to zero, forcing our estimate to match up with the true value of the state. So this set up makes us robust to inaccuracies and that type of thing - which you often need since you never know the coefficients exactly, and your measurements are always noisy etc.

[@blackboardradio343]

Dear Richard, Thank you very much for your lecture. It is really great. I just have one question on the equation xhat(dot) at time 6:00. Do not you think it should be L(y-y(hat))? Because you wrote then on time 31:17 ...L(y-y(hat)) :) Thank you for your effort. Best regards!

[@richard_pates]

Aha, good spot! That's clearly inconsistent! Actually, L(y(hat)-y) is consistent with having the observer poles match the eigenvalues of A+LC (this is what we derive at 6:00). If instead we use L(y-y(hat)) the observer poles should match the eigenvalues of A-LC. Both schemes are actually completely fine - but in one case you should design the observer matrix to be the negative of the other. I've seen both approaches in text books, so I'm not really sure which to recommend most strongly - but if you understand the derivation (which it certainly seems you do) that is the main thing! Thanks for pointing this out!