Question

Exercise: Given the mass-damper-spring network below: x(t) flt) m- 1kg; X(s) F(s) (s2 +10s + 20) b-10N-m/s 20N/m; f(t)-1 N Show how each of the controller gain, Kp, Kd and Ki contributes to obtain Fast rise time Minimum overshoot i. No steady state error MATLAB code S-tf('s') Sys 1/(sA2+10*s+20) Step Proportional Controller: Kp 300 % for faster reponse Gpspid(Кр) sys_p-feedback(sys Gp, 1) t-0:0.01:2 step(sys, sys p) Proportional-Derivative Controller: Kp 300 Kd-10 Gpdspid(Kp,0,Kd) sys pd feedback(Gpd sys, 1) step( sys, sys_p, sys_ pd) Proportional-Integral Controller Kp 300 Ki-70 Gpi-pid(Kp,Ki) sys_pi feedback(Gpi sys, 1) stepl sys, sys_p, sys_pd, sys_pi) Proportional-Integral-Derivative Kp 350 Ki-300 Kd-50 Gpispid(Kp,Ki, Kd) sys_pid-feedback(Gpid*sys, 1) step( sys, sys_p, sys_pd, sys_pi, sys_pid) Use simulink PID(s) +10s+2 Step PID Contaller Transfer Fon Scope

Answer 1

For proportional controller:

Time Series Plot 0.8 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (seconds)

For proportional derivative controller:

Time Series Plot: 06 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Time (seconds)

For proportional-integral controller:

Time Series Plot: 0.2 0.3 0.4 0.6 0.7 0.8 0.9 Time (seconds)

For proportional-integral-derivative controller:

Time Series Plot 0.6 0.4 0.2 0.05 0.1 0.15 Time (seconds) 0.2 0.25 0.3

From the results it is clearly observable that:

Kp reduces rise time and increases overshoot.

Ki reduces steadystate error.

Kd reduces overshoot.