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Write observer ordered

welcome to lecture number 22 for ece 463 modern control reduced order observers now as a recap we've looked at full order observers a full order observer is where i have a plant with a known input a known output but i can't measure all the states in that case i can build a full order observer that's basically an exact duplicate of the plant with some feedback gain h that allows you to estimate all the states the resulting system is order 2n i've got n states for the plant second end states for the observer well here's a question for you do i have to estimate all n states i'm already measuring a couple states call it m states why do i have to measure or estimate the states that i'm measuring can't i just estimate the states that i'm not measuring if you could do that that would be called a reduced order observer and the answer is yes you can the derivation is a little bit convoluted but it does work for completeness we're going to cover that turns out reduced order observers are more sensitive to modeling errors more sensitive to noise so they're typically not used it's also much more complicated having an observer that you understand has more robustness for modeling variations more robustness for noise is a nice advantage so typically observers are full order observers but again for completeness we'll cover the reduced order observers now the derivation is as follows assume you've got a plant at the form x dot equals ax plus bu y equals cx and of course the plant has to be observable now let's separate the state's x into two parts x1 and x2 x1 is measured directly c1 is invertible so i know what x1 is just by taking c inverse c1 inverse times y i know x1 the first set of m states x2 are the n minus m states the ones that i don't measure directly the ones they have to estimate so take the a matrix when you rearrange it this way split it up into four terms the b matrix has two terms and the c matrix again the second term is 0 because i'm not measuring x 2. the full order observer is as follows i'm going to estimate states x1 and x2 that'll be in the same form as the plant plus i'll have the speedback gain h times the difference between the measured output and the estimated output now since c1 is invertible by assumption i can tell you what x1e is directly so we'll just substitute that right in then the pull order observer is just the second set of equations sx2e is a21 times x1e which is c1 inverse y plus a22 times x2e plus b2 times u if i let x2e to be some matrix l times y plus z z is my dummy state that will come from the dynamics of the the reduced...