ackermans.mp4

given a state system of the form here where the states X are all controllable through you we can use state feedback to do pole placement and we discussed that earlier there are three ways that you can do this calculation one is just to brute force the whole thing out the other is to brute force it out but start with a and controller canonical form in the third way which we'll discuss here is to use Ackerman's formula Ackerman's formula for K is this form here a is a unit vector of the form here P is a

controllability matrix and then you'll need to invert it and feed iured of a is a matrix polynomial where the polynomial is the polynomial of the desired closed loop characteristic equation evaluated at the matrix value a so it has a form that looks like this let's illustrate this by example given the second-order system in state space form I want to create a closed-loop system where the desired poles are at minus 2 plus or minus 2i that will correspond to a settling time of 2 seconds in a damping ratio of 0.707 from the desired poles I can

write down the desired closed loop characteristic equation now I'm ready for Ackerman's formula the controllability matrix is given by this value here remember the system has to be controllable and by definition a controllable system has a controllability matrix which is invertible which is required for Ackerman's formula in our case the system is controllable and so P the controllability matrix is invertible next i'm going to evaluate the desired closed loop characteristic equation with the matrix a substituting a into the desired closed loop characteristic equation i have the matrix equation shown here in green and after doing

the algebra i have the value here now i need to substitute all these values back into Ackerman's form completing the algebra I end up with K is equal to 4 point 6 and minus 6 point 5 let's verify that so I substitute the U is equal to minus KX my state feedback in to the value for you here in the closed-loop system using the value of K of 4.6 in minus 6.5 the equation looks like this the closed-loop system has an a value as shown here and indeed the eigen values of a are -2 plus

or minus 2i now I've cheated here a little bit and I haven't held on to all the significant figures so if you put this in you could end up with a value that's not exactly the same so you can use Ackerman's formula to compute the value for K for a state feedback system where u is equal to minus KX you don't have to have a in a diagonal form or controller canonical form can be in any form you what the catch is that it does require you to take the inverse of a matrix the rest

of it's just a little algebra