I am working with Matlab to design a compensator capable to make a system attain the design requirements ($\omega_n\geq 0.3$ rad/s & $\xi\geq 0.5$).
In order to do so I have a discrete-time transfer function with a sampling time of 1 second, which is the following:
The functions has already a zero = -1 and a double pole = 1.
I am also using Matlab SISOtool to see where is the feasible region in which I should have the root-locus to fulfill the design requirements. Such region is the following (the white one):
The image shows the root-locus of the plant, G(z). Now I want to add a controller, C(z), that allows me to move the root locus into the white region (at least a part of it).
The controller should be in the form:
And therefore, $C(z)G(z)$ should be evaluated to see that part of the root locus is located into the white area.
I want to find the value of the poles and zeros of the controller $C(z)$ that allow me to have part of the root locus into the white region provided in the image, and then being able to fulfill the design requirements.
This is driving me crazy because I have tried all the possible ways of adding poles and zeros and I am unable to obtain a region of the root-locus inside of the white area. I only need some idea on where to place the zeros/poles or if it is even feasible to attain this design requirements (it should be).
n=[0.5 0.5]; d=[1 -2 1]; G=tf(n,d,1);
After that, to run SISOtool just type
sisotool('rlocus',G) in Matlab console.