control engineering project submission

8/13/2019 Control Engineering Project Submission

http://slidepdf.com/reader/full/control-engineering-project-submission 1/19

F = kθθ



G(s)

GA(s) = (s + 10)

s(s + 90)

℄

℄



1(s+100+i10)(s+100−i10)

(s+300)(s+40)

(s+10)s(s+20+i100)(s+20−i100)

(s+10)(s+150)

(s+8+i60)(s+8−i60)(s+1+i10)(s+1−i10)



GA(s)

jω

ζ ∼ 0.1 ωn ∼ 10 M p = e−ζΠ√ 1−ζ2 = 73.04%

Lims→0sG(s)1s

= 0.01

1(s+1−i10)(s+1+i10)

s+300s+25



s−10(s+20+i20)(s+20−i20)

1(s+5)(s+2+i20)(s+2−i20)

(s+40)(s+300)

s(s+1+i10)(s+1−i10)



Gc(s)G p(s) = H 1(s) = (s + α)

s3 + (1 + α)s2 + (α− 1)s + (1− α)

α < ±10%

K p

Steady StateE rror = 1

1 + K p≤ 0.1

=⇒ K p ≥ 9

K p = lims→0 H 1(s) = lims→0(s+α)

s3+(1+α)s2+(α−1)s+(1−α) = α1−α

≥ 9 =⇒ α ≥ 0.9

α

H 1(s) s3+(1+α)s2+(α−1)s+(1−α) = (s3+s2−s+1)+α(s2+s−1)

H 2(s) = 1 + α(s2+s−1)(s3+s2−s+1) H 1(s)

H 2(s)

H 2(s)

α

α



H (s) = H 1(s)1+H 1(s) .

1+H 1(s) = 0 =⇒ 1+ (s+α)

s3+(1+α)s2+(α−1)s+(1−α) = 0 =⇒ s3+(1+α)s2+αs+1 = 0

α (s3+s2+1)+α(s2+s) = 0

H 3(s) = s2+ss3+s2+1 1 + H 3(s) = 0

−0.469 α = 4.48

α = 4.48

2nd 2 = 4

ξωn

ξωn α = 4.48



α = 4.48

℄

℄ ℄

α



α > 4.48

αjust < 4.48

3.8

11.3s

α = 3.82

α < 3.82

α

G(s) = 1(s+1)3

H (s) = KG(s)1+KG(s) = 1

(s+1)3+K

Characteristic Equation = s3 + 3s2 + 3s + (K + 1)

s3

s2

s 8−K 3

1

0 < K < 8



0 < K < 8 K = 8

−0.124

G( jω) = 1

( jω + 1)3 =

(1− jω)3

(1 + ω2)3 =

(1− 3ω2) + j(−3ω + ω3)

(1 + ω2)3

ω =√

3, 0 ω =√

3

G( jω)

G( jω) = (1− 3 ∗ 3)

(1 + 3)3 = −8

64 = −0.125

−a,a = 0.125

N = P − Z

s = −1

G(s) P = 0

Z = 0 N = 0 s = −1

s =−Ka, a = 0.125 Ka = 1

Ka = 1 K = 1a

= 8

K = 8 18.06 dB

K = 8



18.1 dB 20 log(K ) = 18.1 dB K = 8.035

℄

K = 8.0011

Range of P roportional Gain K

∈[0, 8]



kx F = kxx + kθθ

θ, ω α

θ

Iα = F L− bvω − mgLsin(θ)

2

sin(θ) θ

Iα = F L− bvω − mgLθ

2

mω2L

2 = T − mgcos(θ)



2kx = Tsin(θ)

θ ω

X =

θ

ω

X ′

=

0 1−mgL2I

− bvI

X +

0LI

F

θ

F = kθθ

kθ 0

ω

X ′

=

0 1−mgL2I

− bvI

X +

0LI

∗

kθ 0

X

X ′

= 0 1

−mgL

2I + kθL

I −bv

I X

A =

0 1

−mgL2I

+ kθLI − bv

I

s2 + sbv

I +

mg L

2I − kθL

I = 0



b2vI 2 ≥ 4(

mgL

2I − kθL

I )

kθ

kθ ≤mg

2 − b2v

4IL

kθ = mg

2 − b2v

4IL

kθ > mg

2 − b2v

2IL

F = kxx + kθθ x

θ

x θ



G1(s) = (1−s)(1+s)(1+ s

2) G1(s) =

(1−s)0.5s2+1,5s+1

℄ ℄ s

G2(s) = (1−s)(1− s

2)

(1+s)(1+ s2)(1+ s

3) G2(s) = 3s2−9s+6

s3+6s2+11s+6

℄ ℄



G1(s)

G2(s)

limt→0f (0) = lims→∞sF (s)

H (s)s

H (s)s

limt→00f ′(0) = lims→∞sH (s)

∞

∞

(1−s)0.5s2+1,5s+1

∞

∞

3s2−9s+6s3+6s2+11s+6

H (s)s

H (s)s

srU (s)

U (s) = H (s)

s

sr−1U (s)

∞srH (s)



H (s) = K

(s− zi)(s− zi∗)

(s− zj)

(s + zk)

(s− pm)(s− pm∗)

(s + pn)

K (s−zi)(s−zi∗)

(s−zj)

(s+zk)

(s− pm)(s− pm∗)(s+ pn)

S = K ( |zi|2)(−1)b(

zj)(

zk)( | pm|2)(

pn)

∞sr

H (s)

D = lims−>∞sr.K

(s− zi)(s− zi∗)

(s− zj)

(s + zk)

(s− pm)(s− pm∗)

(s + pn)

∞

K 2( |zi|2)(−1)b(

zj)(

zk)( | pm|2)(

pn)

< 0

control engineering project submission

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