The first figure shows the response of the "ideal" system along with that measured experimentally. The latter is the same transient response plot you generated above. The "ideal" system response can be generated by applying the average of the numerically computed K and to the Matlab function lsim fed with the time and input vectors used to generate the first plot. For details on how to use lsim refer to http://www.mathworks.com/help/control/ref/lsim.html