)s(C)s(H)s(R

2.1 Step response:

)s(Hs

1)s(C ss

t4753.032

t4993.02

t526.41 ceA)t7883.2cos(eAeA)t(c

36s2.88s74.31s6.12s1.2

36s6.75s64.8)s(H 234

2

Eigenvalues: p1=-4.526, p2,3=-0.4993±2.7883i, p4=-0.4753

)s(sClim)t(climc0st

ss

Final Value Theorem: 136

36)s(Hlimc

0sss

Stability

Steady-state error: ess=1-css ess=0

52

[r,p,k]=residue(nh,[dh,0])

r = -0.1402 -0.3976 + 0.1912i -0.3976 - 0.1912i -0.0646 1.0000

433i

211 rA,r2eA,rA 2

z=r(3);a2=abs(z),fi2=angle(z)

1e0646.0)6934.2t7883.2cos(e8824.0e1402.0)t(c t4753.0t4993.0t526.4

Design criteria for control systems:

Stability

Steady-state error ess=1-css → 0

Sensitivity [css]d → 0

Overshoot , typical % 5, damping ratio 0.7

Settling time tss (depending on application)

62

Homework Pr. 02-01 (b), Pr. 02-03 (b)

sKs

KK)s(G d

ipc

In first stage P control, stability, ess, [css]d

If necessary PI control, eleminates steady-state errror

If necessary PD control, decreases overshoot

If necessary PID control, meets the all design criteria

Step response of second order systems

s

1

)s2s()s(C

2nn

2

2n

1)tcos(Ae)t(c t

10

cos

ωn : undamped natural frequency ξ: Damping ratio2

n 1 21

1A

n

2

in

.

clc,clearwn=1;ksi=0.2;tp=2*pi/wn;dt=tp/20;tson=tp/ksi;t=0:dt:tson;w=wn*sqrt(1-ksi^2);a=wn/w;sigma=ksi*wn;fi=-acos(ksi)-pi/2;c=a*exp(-sigma*t).*cos(w*t-fi)+1;plot(t,c)

)cost1sin(e1

11)t(c 12

nt

2n

)45(707.0 o Critical damping ratio

72

cmax : Peak value, tmax: Peak time, cmax-css: Maximum overshoot

css : Steady-state value, 1-css : Steady-state error

ts : Settling time (%5)

tr : Rise time

td : Delay time

1

)t(c

t

50.0dt

90.0

10.0

rt

05.195.0

stmaxt

maxc

maxt

21/ssmax ecc

nd

7.01t

n

r5.28.0

t

)5(%

2

Tt 0s )13.0(%

Tt 0ss

82

tan/ssmax ecc cos