o Problem 02-01 Reconsider Problem 01-01. a) Find the transfer function H(s) of the closed loop control system (Kp is a symbol) b) Determine the value of Kp to obtain steady state error of %2 to a unit step input. c) Find the step response of the control system (Use Kp=65.3333) d) Find the settling time of the closed loop control system. e) Find the sensitivity of steady state error to a disturbance. o K1=7500 K2=0.05 R=0.002 C=2000

Solutions of Week 2: Problem 02-01: Open-loop system Closed-loop system Transfer function between disturbance and output. Transfer function between Reference and Output

c) Step response of closed loop system v2(t) Solutions of Week 2: v2ss c) Step response of closed loop system v2(t) Residue Theorem

d) The settling time of the closed loop control system ts Solutions of Week 2: The coefficients A and B can be determined by applying Residue Theorem. d) The settling time of the closed loop control system ts The eigenvalue of the closed loop control system is s=-12.5. Then the time constant is calculated as e) Sensitivity of the steady state error to disturbance (The fraction of the steady state error due to disturbance in the form of a unit step function) For unit step disturbance ess=0.001 Sensitivity %0.1 disturbance

Problem 02-02 (Kuo, s.195) Reconsider Problem 01-02. a) Find the transfer function of the open loop system [H1(s)]. b) Find the eigenvalues of the open loop system. c) Find the transfer function of the closed loop control system H(s) (Kp is a symbol). d) Find the eigenvalues of the closed loop control system (Use Kp=9). e) Write the form of the step response of the closed loop control system. f) Determine the steady state error ess. g) Find the damping ratio, undamped frequency, time increment Δt and settling time.

>>roots([10 12 32.59]) s1,2=-0.6±1.7026i Solutions of Week 2: Open system Closed system Problem 02-02: a) Transfer function of open loop system b) Eigenvalues of open loop system c) Transfer function of closed loop system c) Eigenvalues of closed loop system for Kp=9. >>roots([10 12 32.59]) s1,2=-0.6±1.7026i

ω0 Im 1.7026 α -0.6 Re Solutions of Week 2: e) The form of the response of the closed loop system to a unit step input. e) Steady state error : f) Damping ratio, undamped natural frequency, time step (increment) and settling time. Re Im -0.6 1.7026 ω0 α

Problem 02-03 Reconsider Problem 01-03. a) Find the transfer function of the closed loop control system H(s). (Gc=Kp is a sembol) b) Determine the value of Kp in order to obtain the sensitivity to disturbance as % 4.

Open system Closed System Solutions of Week 2: K1=0.0034 K2=5.34 Problem 02-03: K1=0.0034 K2=5.34 Closed System Open system a) Transfer function of closed loop system for Gc(s)=Kp

b) Kp value for the sensitivity of 4%.

Problem 02-06 Answer: Problem 02-07 The step response of a system is given in the figure. Determine a 2nd order tranfer function for this system. Answer: Problem 02-07 Determine the K and Kt values for the system whose block diagram is given in the figure a) To obtain the maximum overshoot as 6% and the steady state error as 4.8%. b) Write the Matlab code in order to plot the step response of the system.

Solutions of Week 2: Problem 02-07: K and Kt values for maximum overshoot 6% and steady state arror 4.8%.

Solutions of Week 2:

clc;clear K=0.534;Kt=0.027; num=[25*K]; den=[1 5 25*(K+Kt)]; %step(num,den) hs=tf(num,den); dt=0.01;ts=3; t=0:dt:ts; [c,t]=step(hs,t); plot(t,c,[0 ts],[1 1],'--') xlabel('Time [s]') ylabel ('c(t)') title('Unit Step Reference Output c(t)') mpeak=max(c) %maximum value css=c(length(c)) %steady-state value ess=1-css %steady-state error ov=(mpeak-css)/css %overshoot