연세댿� 2011년 2학기 박진배 교수님 제어공학 프로젝트2 PI PID controller design

The state space equations from the dynamic equation of DC motor given.
If we modify the dynamic equation in the box, we can obtain two equations. First one is
θ ̈_m=-b/J_m θ ̇_m+K_t/J_m i_a
, and second one is
(i_a ) ̇=-K_e/L_a θ ̇_m-R_a/L_a i_a+1/L_a v_a
Finally, if we just add one equation(θ ̇_m=θ ̇_m), we can set three equation for the state space equations. That is,
[■(θ ̇_m@θ ̈_m@(i_a ) ̇ )]=[■(0&1&0@0&-b/J_m &K_t/J_m @0&-K_e/L_a &-R_a/L_a )][■(θ_m@θ ̇_m@i_a )]+[■(0@0@1/L_a )] v_a
y_1=[■(1&0&0)][■(θ_m@θ ̇_m@i_a )] (output equation for problem number 2)
y_2=[■(0&1&0)][■(θ_m@θ ̇_m@i_a )] (output equation for problem number 3)

Rotate the DC motor 2rad, plot the root locus, find the gain margin, design PD.
If the state space equation has the form below,
x ̇=Ax+Bu
y=Cx+Du
we can obtain the transfer function from the this equation.
G(s)=C〖(sI-A)〗^(-1) B+D
So, after the calculation we can get this transfer function.
G(s)=(K_t/(J_m L_a ))/s[s^2+(R_a/L_a +b/J_m )s+(bR_a+K_t K_e)/(J_m L_a )]
And then, if we substitute the values from the m-file for the equation above, we can get this equation.