214 March 31, 214, Rev.2.1

4........................ 4........................ 5............................. 7............................... 7 1 8 1.1............................... 8 1.2....................... 9 2 15 2.1..................... 15 2.2............................ 17 2.3....................... 18 2.3.1................... 18 2.3.2...................... 19 2.3.3 t............... 19 2.3.4 δ(t)......................... 2 2.3.5 sin(ωt)....................... 21 2.3.6........................... 24 2.4.......................... 25 2.4.1................................ 25 2.4.2............................. 25 2.4.3............................. 27 2.4.4................... 28 2.4.5................... 28 3 3 3.1......................... 31 3.2.......................... 33 3.3................................ 34 3.3.1 - D(s)............ 34 3.3.2 - D(s)............ 37 1

3.4.................. 39 3.5.................... 4 3.5.1 v i (t) = u(t), v o () =... 4 3.5.2 v i (t) = t, v o () =...... 41 3.5.3 v i (t) = u(t), v o () = v... 42 4 43 4.1............................ 43 4.2....................... 45 4.3.................. 46 4.4......................... 48 4.4.1........................ 48 4.4.2............ 49 4.5...................... 51 5 52 5.1............................... 52 5.2................................ 54 5.3........................... 55 6 56 6.1 G(s).................... 56 6.2......................... 59 6.2.1..................... 59 6.2.2................ 61 6.3 s.......................... 63 7 65 7.1.................. 65 7.2.......................... 68 7.2.1............................... 68 7.2.2............................... 69 7.2.3........................ 7 7.2.4................. 7 8 73 8.1.............................. 73 8.2............................... 74 8.2.1............................... 74 8.2.2............................... 74 2

8.2.3............................ 75 8.2.4............................... 76 8.2.5............................ 77 8.2.6............................... 78 9 8 9.1.............................. 8 9.2.......................... 8 9.3.......................... 83 9.3.1........................... 84 9.3.2........................ 85 9.4........................... 89 1 9 1.1............................... 9 1.2.......................... 92 1.3............................... 93 11 96 11.1......................... 96 11.2........................... 97 11.3 s.......................... 97 11.4.............................. 98 11.5............................ 1 11 3

2 (1) (2) (3) 15 1. 2. 4

3. 4. 5. 6. 7. 8. 9. 1. 11. 12. 13. 14. 15. 2 (1) 5

(2) (3) 15 1. 2. 3. 4. 5. 6. 7. 8. 9. (1) 1. (2) 11. 12. 13. 14. PID 15. 2 6

I II,HP 7

1 1.1 17 1776 1788 1.1 1.1: 1. 2. 8

3. 4. 5. 6. 1877 1895 19 193 1932 1952 1948 1956 1.2 1.2 2 9

1.2 1.3 1.2: 2 r(t) x(t) 1.3: 1.4 x r r x y r x 1.4 1.5 1

1.4: 2 r(t) r x x(t) x 1.5: 1.3 1.5 11

1.6 1.6: y r y m w u m u d d m 1.6 1.6 12

1.7 e = r y C u = K <, y > r(e < ), y = +K >, y < r(e > ) (1.1) e < u > e > u < e y r u +K t -K t 1.7: P C [O] [C] [P ] [O] = [C] [P ] (1.2) 13

[P ] PID [C] [O] P C [P ] C 1.8 1.8: 14

2 x y y = ax + b 2.1 t t 2.1: 2.1 2.2 i i(t) = C dv dt (2.1) Ri(t) + v(t) = E (2.2) 15

2.2: RC dq(t) = v s (t) = E dt RC dv t(t) + v t = (2.3) dt dv t (t) = 1 dt (2.4) v t RC log v t (t) = 1 RC t + c (2.5) v t (t) = Ae 1 RC t (2.6) t v t (t) v(t) = v t (t) + v s (t) t = v = A = E v(t) = E + Ae 1 RC t (2.7) v(t) = v t (t) + v s (t) = E 1 e 1 RC t ) (2.8) i(t) = C dv dt = E 1 R e RC t (2.9) E E (E/s) (2.2) s V (s) RCsV (s) + V (s) = E s V (s) = E s RCs + 1) = E s RCE RCs + 1 16

V (s) v(t) L 1 ( 1 s ) = 1 L 1 ( 1 s+α ) = e α v(t) = L 1 ( E s ) E L 1 ( s + 1 ) = E(1 e 1 RC t ) (2.1) RC i(t) = C dv dt = E 1 R e RC t (2.11) 1 + 1 = 2 2.2 1 f(t) = 1 2 a + (a n cos nωt + b n sin nωt) (2.12) n=1 f(t) = C n e jnωt (2.13) T f(t) = 1 { f(t)e jnωt dt}e jnωt dω (2.14) 2π { } F(jw) = f(t)e jnωt dt (2.15) F(jω) f(t) 1 17

f(t) F(jw) f(t) γ(t) = e σt ( t) f(t)γ(t) = 1 { f(t)γ(t)e jnωt dt}e jnωt dω (2.16) 2π γ 1 (t) f(t) = 1 { f(t)e (σ+jnω)t dt}e (σ+jnω)t dω (2.17) 2π s = σ + jnω jdω = ds f(t) = 1 σ+ { f(t)e st dt}e st ds (2.18) 2πj σ f(t) =, t > F (s) = f(t)e st dt (2.19) f(t) F (s) = L[f(t)] F (s) 2.18 f(t) = 1 σ+ F (s)e st ds (2.2) 2πj σ 2.3 2.3.1 t c (2.19) f(t) = c t (2.21) L[f(t)] = ce st dt (2.22) = [ c s e st ] = ( c s ) = c1 s (2.23) 18

2.3.2 f(t) = ce pt t (2.24) L[f(t)] = ce pt e st dt = = [ c s p e (s p)t ] = ( c s p ) = ce (s p)t dt (2.25) c s p (2.26) p = f(t) = ce = c L[f(t)] = c s 2.23 2.3.3 t f(t) = t k e pt t (2.27) h(t)g (t)dx = h(t)g(t) h (t)g(t)dt (2.28) h(t) = t k 1 (s p) g(t) = e (s p)t L[f(t)] = (t k e pt )e st dt(= = [t k 1 (s p) e (s p)t ] = I k h(t)g(t) dt) = [h(t)g(t)] kt k 1 1 (s p) e (s p)t dt h(t) g(t)dt (2.29) [t k 1 (s p) e (s p)t ] = (2.27) k 1 I k 1 = I k = kt k 1 1 (s p) e (s p)t dt (2.3) (t k 1 e pt )e st dt = t k 1 e (s p)t dt (2.31) 19

(2.3) I k 1 I k = kt k 1 1 (s p) e (s p)t dt k = t k 1 e (s p)t dt = (s p) k s p I k 1 (2.32) k = f(t) = t e pt = e pt L[e pt ] = 1 s p ( 2.26)) I = 1 (2.33) s p k = 1 (2.32) I 1 = 1 s p I = 1 (s p) 2 (2.34) I 2 = 2 1 s p I 1 = 2 1 (s p) 3 (2.35) I 3 = 3 1 s p I 2 = 3 2 1 (s p) 4 (2.36) I k = k 1 s p I k 1 = k! (s p) k+1 (2.37) L[f(t)] = k! (s p) k+1 (2.38) 2.3.4 δ(t) T 1 f(t) = T, T > t > (2.39), otherwise 1 2.3 T δ L[f(t)] = T 1 T e st dt (2.4) = [ e st s T ] T = e s T ( e ) s T 2 = 1 e s T s T

1 e s T L[δ(t)] = lim T s T / T (2.41) se s T L[δ(t)] = lim T s = s s = 1 (2.42) 2.3: δ 2.3.5 sin(ωt) e jωt = cos(ωt) + j sin(ωt) (2.43) sin(ωt) cos(ωt) = ejωt e jωt 2j = ejωt + e jωt 2 sin(ωt) L 21

L[sin(ωt)] = sin(ωt)e st e jωt e jωt dt = e st dt (2.44) 2j = 1 2j [ (e (s jω)t e (s+jω)t )dt = 1 2j [ 1 s jω e (s jω)t ( 1 s + jω e (s+jω)t )] = 1 2j [ 1 s jω 1 = 1 2j 2jω s 2 + ω 2 = s + jω ] = 1 2j ω s 2 + ω 2 [(s + jω) (s jω) (s + jω)(s jω) ] cos(ωt) Im A A sin Re A cos 2.4: 22

2.3 s I = f(t)e st dt Re(s) > a Re(s) < a Re(s) > a I s F (s) = L(f(t)) f(t) Im F(s) I Re 2.5: 23

2.3.6 f(t) 2.1 f(t) F (s) F (s) f(t) 2.1: Laplace transform table Item No. f(t) F (s) 1 δ(t) 1 2 u(t) 3 tu(t) 4 t n u(t) 5 e at u(t) 6 sin(ωt)u(t) 7 cos(ωt)u(t) 1 s 1 s 2 n! s n+1 1 s + a ω s 2 + ω 2 s s 2 + ω 2 u(t) t t 24

2.4 2.4.1 L[f 1 (t)] = F 1 (s) (2.45) L[f 2 (t)] = F 2 (s) (2.46) L[f 1 (t) + f 2 (t)] = F 1 (s) + F 2 (s) (2.47) a L[af 1 (t)] = af 1 (s) (2.48) 2.4.2 F (s) = F (s) = L[f(t)] = g(t) = h(t) = f(t) f(t)e st dt (2.49) e st dt = e st s h(t)g(t) dt (= [h(t)g(t)] = [f(t)( e st s )] = [f(t)( e st s )] + 1 s df(t) ( e st dt L h(t) g(t)dt) s )dt df(t) dt e st dt (2.5) df(t) dt e st dt = L[ df(t) dt ] (2.51) 25

L[ df(t) dt ] F (s) = [f(t)( e st s )] + t = + 1 s L[df(t) dt ] = ( 1 s f(+)) + 1 s L[df(t) dt ] (2.52) L[ df(t) ] = sf (s) f(+) (2.53) dt f(t) F (s) = h(t) = e st g(t) = f(t)e st dt = L[f(t)] f(t)dt F (s) = = f(t)e st dt (2.54) h(t)g(t) dt (= [h(t)g(t)] = [e st f(t)dt] L F (s) = [e st f(t)dt] = [ f(+)dt + sl[ h(t) g(t)dt) f(t)dt]( s)e st dt + sl[ f(t)dt] (2.55) f(t)dt] L[ f(t)dt] = 1 s F (s) + 1 f(+)dt (2.56) s 26

2.4.3 u(t) t + f(t) = g(t)u(t) (2.57) T t + T H(s) = L[h(t)] = h(t) = g(t T )u(t T ) (2.58) t < T u(t T ) = t = t + T H(s) = = g(t T )u(t T )e st dt (2.59) g(t T )u(t T )e st dt (2.6) g(t )u(t )e s(t +T ) dt = e st g(t )u(t )e st dt = e st f(t )e st dt = e st F (s) T s e st 2.6: 27

2.4.4 f(t) F (s) f(t) f( ) f( ) = lim t f(t) = lim s sf (s) (2.61) f(t) F (s) f(t) f() f(+) = lim t f(t) = lim s sf (s) (2.62) 2.4.5 28

2.2: Laplace transform theorems Item No. Theorem Name 1 L[f(t)] = F (s) = f(t)e st dt 2 L[kf(t)] = kf (s) 3 L[f 1 (t) + f 2 (t)] = F 1 (s) + F 2 (s) 4 L[e at f(t)] = F (s + a) 5 L[f(t T )] = e st F (s) 6 L[f(at)] = 1 a F ( s a ) 7 L[ df ] = sf (s) f(+) dt 8 L[ dn dt n ] = sn F (s) n s n k f k 1 (+) k=1 t 9 L[ f(τ)dτ] = F (s) + 1 f(+)dt s s (n 1 f( ) = lim s sf (s) 11 f() = lim s sf (s) 29

3 n n n n d 2 u(t) dt 2 = a 1 du(t) dt + a 2 u(t) (3.1) u() = α du(t) t= = β u() = dt α u(1) = β (1) (2) (3) (4) (5) (6) (1) (4) 3

3.1 dx(t) dt = x(t) + t, x() = 1 (3.2) dx(t) dt = x(t) (3.3) dx(t) x(t) = dt (3.4) log x(t) = t + c (3.5) x t (x) = Ce t (3.6) t x t (t) x x s (t) = at + b (3.2) dx s (t) dt = x s (t) + t = (at + b) + t (3.7) (3.7) x s (t) dx s(t) = a b = a (1 a)t = dt a = 1 b = a(= 1) x s (t) = t 1 x(t) = x t (t) + x s (t) = Ce t + t 1 (3.8) x() = 1 C 1 = C + 1 C = 2 x(t) = x t (t) + x s (t) = 2e t + t 1 (3.9) 31

x(t) = C(t)e t (3.1) 1 dx(t) dt = dc(t) e t + C(t) de t dt dt = dc(t) e t C(t)e t dt = x(t) + t = [C(t)e t ] + t dc(t) e t = t dt dc(t) dt = te t (3.11) (3.1) C(t) = c + te t dt (3.12) x(t) = C(t)e t = (c + [te t e t ])e t = ce t + t 1 (3.13) x() = 1 x() = ce + 1 = c 1 = 1 c = 2 x(t) = 2e t + t 1 (3.14) 1 Cξ(t) C(t)ξ(t) 32

3.2 L[u(x)] = U(s) dx dt = x + t (3.15) sx(s) x() = X(s) + 1 s 2 sx(s) + X(s) = x() + 1 s 2 (3.16) X(s) = = 1 s + 1 x() + 1 (s + 1)s 2 1 s + 1 x() + 1 s 1 2 s + 1 s + 1 (3.17) x() = 1 x(t) = L 1 1 [ s + 1 ]x() + L 1 [ 1 s ] 2 L 1 [ 1 s ] + 1 L 1 [ s + 1 ] = x()e t + t 1 + e t = e t + t 1 + e t = 2e t + t 1 (3.18) 3.1 3.1: 33

step1: s step2: step3: t step1 s step2 s step3 t 3.3 step3 3.3.1 - D(s) F (s) = N(s) D(s) = b s m + b 1 s m 1 + + b m (3.19) s n + a 1 s n 1 + + a n m n D(s) = (s s 1 )(s s 2 ) (s s n ) (3.2) s 1 s 1 = s n F (s) F (s) = K 1 + K 2 + K n (3.21) s s 1 s s 2 s s n (s s 1 ) N(s)(s s 1 ) D(s) = K 1 + K 2(s s 1 ) s s 2 + K n(s s 1 ) s s n (3.22) s = s 1 K 2,, K n K 1 N(s)(s s 1) N(s) = (s s 1 ) D(s) s=s1 (s s = K 1 )(s s 2 ) (s s n ) 1 (3.23) s=s1 34

F (s) = a 1 s + a (s + α)(s + β) = K 1 s + α + K 2 s + β (3.24) K 1 = (a 1s + a ) (s + α) = a 1α + a (s + α)(s + β) s= α α + β K 2 = (a 1s + a ) (s + β) = a 1β + a (s + α) (s + β) s= β β + α (3.25) K 1 f(t) = L 1 [F (s)] = L 1 [ s + α ] + L 1 [ s + β ] (3.26) K 2 = L 1 [F (s)] = a 1α + a α + β e αt + a 1β + a α β e βt F (s) = = a 1 s + a (s + σ) 2 + ω = a 1 s + a 2 (s + σ jω)(s + σ + jω) K 1 s + σ jω + K 2 s + σ + jω K 1 = K 2 = (a 1 s + a ) (s + σ jω) = a 1σ + ja 1 ω + a (s + σ jω)(s + σ + jω) s= σ+jω 2jω (a 1 s + a ) (s + σ + jω) = a 1σ ja 1 ω + a (s + σ jω) (s + σ + jω) s= σ jω 2jω = a 1σ + ja 1 ω a 2jω f(t) = L 1 [F (s)] = K 1 e ( σ+jω)t + K 2 e ( σ jω)t = e σt {K 1 e jωt + K 2 e jωt } = e σt {K 1 (cos ωt + j sin ωt) + K 2 (cos ωt j sin ωt)} = e σt {(K 1 + K 2 ) cos ωt + (K 1 K 2 )j sin ωt} 35

K 1 K 2 = a 1σ + a + ja 1 ω 2jω K 1 + K 2 = a 1σ + a + ja 1 ω + a 1σ a + ja 1 ω 2jω 2jω a 1σ a + ja 1 ω 2jω = 2a 1 σ + 2a 2jω = 2a 1 j ω 2j ω = a 1 = a 1σ + a jω (3.27) f(t) = e σt {(K 1 + K 2 ) cos ωt + (K 1 K 2 )j sin ωt} = e σt {a 1 cos ωt + a a 1 σ sin ωt} (3.28) ω K M K M x(t) x() = A dx dt () = B M d2 x(t) dt 2 + Kx(t) = (3.29) L{ d2 x(t) } = s 2 X(s) sx() d x() (3.3) dt 2 dt Ms 2 X(s) Msx() M d x() + KX(s) dt = Ms 2 X(s) MsA MB + KX(s) = X(s) = As + B s 2 + K M (Ms 2 + K)X(s) = MsA + MB (s 2 + K )X(s) = sa + B (3.31) M s = A s 2 + K M + B K M K M s 2 + K M (3.32) K M K y(t) =L 1 [X(s)] = A cos M t + B K sin M t (3.33) B = cos 36 K M t

3.3.2 - D(s) D(s) = F (s) s = s 1 k F (s) = N(s) D(s) = N(s) (s s 1 ) k (s s 2 ) (s s n ) (3.34) F (s) = N(s) D(s) = K 1k (s s 1 ) k + K 1 k 1 (s s 1 ) k 1 + + K 1 1 (s s 1 ) + K 2 (s s 2 ) + + K n (s s n ) (s s 1 ) k N(s) D(s) (s s 1) k = K 1k + K 1k 1 (s s 1 ) + + K 11 (s s 1 ) k 1 s = s 1 K 1k K 2 +(s s 1 ) k { (s s 2 ) + + K n (s s n ) } N(s) D(s) (s s 1) k = K 1k (3.35) s=s1 (3.35) s { } d N(s) ds D(s) (s s 1) k s s 1 K 1k 1 = K 1k 1 + K 1k 2 (s s 1 ) + + K 11 (s s 1 ) k 2 K 2 +k(s s 1 ) k 1 { (s s 2 ) + + K n (s s n ) } +(s s 1 ) k d ds { K 2 (s s 2 ) + + K n (s s n ) } { } d N(s) ds D(s) (s s 1) k = K 1k 1 (3.36) s=s1 37

1 s 2 (T s + 1) 1 s 2 (T s + 1) = K 1 s 2 + K 2 s + K 3 T s + 1 (3.37) s 2 K 1 = s 2 = 1 (T s + 1) s= K 2 = d s 2 T ds s 2 = (T s + 1) s= (T s + 1) 2 = T s= T s + 1 K 3 = s 2 = T 2 (T s + 1) s= 1 T 1 s 2 (T s + 1) = 1 s 2 T s + T 2 T s + 1 (3.38) 38

3.4 3.2 RC v o (t) 3.2: RC i(t) = C dv o dt (3.39) Ri(t) = v i (t) v o (t) (3.4) RC dv (t) + v o (t) = v i (t) (3.41) dt [ L RC dv ] o(t) + L[v o (t)] = L[v i (t)] (3.42) dt L[v o (t)] = V o (s) L[v i (t)] = V i (s) [ ] dvo (t) L = sv o (s) v o () (3.43) dt (RCsV o (s) RCv o ()) + V o (s) = V i (s) V o (s) = 1 RCs + 1 V i(s) + RC RCs + 1 v o() v o (t) [ ] [ ] v o (t) = L 1 [V o (t)] = L 1 1 RCs + 1 V i(s) + L 1 RC RCs + 1 v o() v i (t) v o (t) (3.44) 39

3.5 v i (t) v o () v o (t) 3.5.1 v i (t) = u(t), v o () = V i (s) = L[v i (t)] = 1 s [ ] [ ] v o (t) = L 1 [V o (t)] = L 1 1 RCs + 1 V i(s) + L 1 RC RCs + 1 v o() [ ] = L 1 1 1 RCs + 1 s [ ] v o (t) =L 1 1 (RCs + 1)s [ ] [ = L 1 K1 + L 1 s K 2 RCs + 1 ] (3.45) 1s K 1 = = 1 (RCs + 1) s s= K 2 = 1 (RCs + 1) = RC (3.46) (RCs + 1)s s= 1 RC [ ] [ ] v o (t) = L 1 K1 + L 1 K2 1 s RC s + 1/RC = K 1 + K 2 t RC e RC = 1 e t RC (3.47) 3.3: v i (t) = u(t), v o () = 4

3.5.2 v i (t) = t, v o () = V i (s) = L[v i (t)] = 1 s 2 [ ] v o (t) = L 1 [V o (t)] = L 1 1 1 RCs + 1 s 2 (3.48) [ ] v o (t) =L 1 1 (RCs + 1)s 2 [ ] = L 1 K1 s 2 [ ] [ + L 1 K2 + L 1 s K 3 RCs + 1 ] (3.49) 1 s 2 K 1 = (RCs + 1) s 2 = 1 s= K 2 = d 1 s 2 RC ds (RCs + 1) s 2 = s= (RCs + 1) 2 = RC s= K 3 = 1 (RCs + 1) (RCs + 1)s 2 = (RC) 2 (3.5) s= 1 RC [ ] v o (t) = L 1 K1 s 2 = K 1 t + K 2 + K 3 [ ] [ + L 1 K2 + L 1 s t RC e RC K 3 RCs + 1 = t RC + RCe t RC = t RC(1 e t RC ) (3.51) ] 3.4: v i (t) = t, v o () = 41

3.5.3 v i (t) = u(t), v o () = v V i (s) = L[v i (t)] = 1 s [ v o (t) = L 1 [V o (t)] = L 1 1 RCs + 1 ] [ 1 + L 1 s RC RCs + 1 v o() ] (3.52) [ ] L 1 RC RCs + 1 v o() = v o ()e t RC (3.53) [ ] [ v o (t) = L 1 K1 + L 1 K 2 s RCs + 1 = K 1 + K 2 t RC e RC ] + L 1 [ ] RC RCs + 1 v o() + v e t RC (3.54) K 1 = 1 (3.55) K 2 = RC v o (t) = 1 e t RC + v e t RC 3.3 v 3.5: v i (t) = u(t), v o () = v 42

4 1 4.1 4.1 1 43

4.1: x F = kx (4.1) m F ẍ F = mẍ (4.2) F ẋ F = dẋ (4.3) 4.2 x P 2 P 1 q q = 1 R h (P 2 P 1 ) (4.4) R h dt qdt A Adx qdt = Adx (4.5) 44

4.2: R h A 2 = d A dx dt = q = 1 R h (P 2 P 1 ) (4.6) F = (P 2 P 1 )A = R h A 2 dx dt (4.7) F = dẋ (4.8) 4.2 4.3 4.3 m 2 m 1 m 1 F F m 2 x 2 = x 2 = ẍ 2 = m 1 m 1 m 2 m 2 m 1 m 2 4.4 m 1 m 2 m 2 m 1 m 1 m 2 m 1 4.5 m 2 m 1 ẍ 1 + k 1 x 1 + d 1 ẋ 1 = F + k 2 (x 2 x 1 ) + d 3 (ẋ 2 ẋ 1 ) m 2 ẍ 2 + k 3 x 2 + d 2 ẋ 2 = k 2 (x 1 x 2 ) + d 3 (ẋ 1 ẋ 2 ) 45

4.3: 4.4: 4.3 4.6 RLC q = t di dt L + Ri + 1 C t idt idt = v i (4.9) d 2 q dt 2 L + R q + q C = v i (4.1) 4.7 46

4.5: - - 4.6: RLC 4.7 47

4.7: 4.4 4.4.1 4.8 l 2 m θ(t) = lmgsin(θ) + f(t) M = -lmgsinq M q M q M = -lmgq 4.8: θ sin(θ) θ l 2 m θ(t) = lmgθ + f(t) θ = 4.8 M = lmgsin(θ) M = lmgθ 48

f 1 (t) θ 1 (t) f 2 (t) θ 2 (t) l 2 m θ 1 (t) + lmgθ 1 (t) = f 1 (t) l 2 m θ 2 (t) + lmgθ 2 (t) = f 2 (t) l 2 m( θ 1 (t) + θ 1 (t)) + lmg(θ 1 (t) + θ(t)) = f 1 (t) + f 2 (t) 4.4.2 4.9 h q i q o q q 4.9: v = 2gh q o = k 2g h (4.11) g 49

P + 1 2 ρv2 + ρgz = const[pa] P ρ z v 2 ρgz = 1 2 ρv2 v = 2gz h h q = q o + q o = k 2g( h + h) (4.12) q( h) = q o + q o q( h) =k 2g( h + h) = k 2g h 1 + h h = q o 1 + h h (4.13) a q( h) = q(a) + q (a)( h a) + + q(n) (a) ( h a) n (4.14) n! ( q(a) = q 1 + ā ) 1 2 h q (a) = 1 2 a = q h ( 1 + ā ) 1 2 h q( h) = q + q h + (4.15) 2 h 2 R 1 R = q 2 h q( h) = q + 1 R h = q + q (4.16) S h + q t = S h + 1 R h t = q i t (4.17) S d h dt + 1 R h = q i (4.18) 4.9 h q o 5

4.5 4.1 m g mẍ + kx = f + mg (4.19) f = x mg = kx x = x x x = x + x x ẍ = ẍ m ẍ + k( x + x ) = f + mg = f + kx m ẍ + k x = f (4.2) (mg) 4.1: 51

5 jω s 5.1 5.1 RC 5.1: RC i(t) = C dv(t) (5.1) dt Ri(t) = v i (t) v o (t) (5.2) [ ] [ ] v o (t) = L 1 [V o (s)] = L 1 1 RCs + 1 V i(s) + L 1 RC RCs + 1 v o() (5.3) 52

v i (t) = u(t), v o () = V i (s) = L[u(t)] = 1 s v o (t) = 1 e t RC (5.4) v i (t) = t, v o () = V i (s) = L[u(t)] = 1 s 2 v o (t) = t RC + RCe t RC = t RC(1 e t RC ) (5.5) v o(t) v i (t) (3.47) (3.51) (5.2) RCsV o (s) + V o (s) = V i (s) + RCv o () V o (s) = 1 RCs + 1 V i(s) + RC RCs + 1 v o() v o () = V o (s) V i (s) = 1 RCs + 1 (5.6) s R C V i (s) s L[ ] = G L[ ] (5.7) 53

(1) (2) (3) G(s) = V o(s) = 1 V i (1) (s) RCs+1 (2) R C (3) 1 2 G 1 (s) G 2 (s) G 1 (s)g 2 (s) 5.2 v o () RCsV o (s) + V o (s) = V i (s) + RCv o () (5.8) t = v o () RC RCs + 1 v o() 1 RCs + 1 RCv o() v o () V i (s) V i (s) + RCv o () 1 V o (s) = RCs + 1 (V i(s) + RCv o ()) (5.9) 2 1 2 54

5.3 (1) (2) (3) = L[ ] L[ ] 5.2 K x D M f 5.2: Mẍ(t) = Dẋ(t) Kx(t) + f (5.1) s 2 MX(s) + sdx(s) + KX(s) = F (s) (5.11) L(x(t)) = X(s), L(f(t)) = F (s) G(s) = X(s) F (s) = 1 Ms 2 + Ds + k (5.12) 4 4.3 55

6 s t convolution 2 f g f g x(t) y(t) G(s) Y (s) = G(s)X(s) Y (s) = G(s)U(s) s t g(t) u(t) 6.1 G(s) δ(t) 2 x(t) g(t) x(t) g(t) 1 fig:1-1a 2 x(t) g(t) g(t) L[δ(t)] = 1 G g(t) = L 1 [G(s)L[δ(t)]] = L 1 [G(s)] (6.1) G(s) g(t) G(s) g(t) x(t) y(t) x(t) 6.2 t 1 56

6.1: 6.2: τ = n t x(τ) t τ x(τ) tg(t τ) (6.2) y(t) y(t) = x(τ) tg(t τ) (6.3) n t y(t) = lim x(τ) tg(t τ) = t t n x(τ)g(t τ)dτ (6.4) 57

τ = t τ y(t) = = t t x(t τ )g(τ )d( τ ) (6.5) x(t τ )g(τ )dτ (6.6) [ ] g(t) = L 1 1 = e αt u(t) y(t) (t ) s + α y(t) = = t t = e αt 1 α x(τ)g(t τ)dτ t [ ] 1 t e α(t τ) dτ = e αt e ατ dτ = e αt α eατ [ e αt 1 ] = 1 [ ] 1 e αt α u(t) 1 y(t) s +α 6.3: 58

6.2 6.4 1 6.4: 6.2.1 6.4 6.5 g(t) 6.5: 59

g(t) f(t) g(t) = u(t) u(t 1) (6.7) f(t) = u(t) 3u(t 1) + 2u(t 2) (6.8) u(t 1) t = 1 1 G(s) = 1 s 1 s e s = 1 s [1 e s ] (6.9) F (s) = 1 s 3 s e s + 2 s e 2s = 1 s [1 3e s + 2e 2s ] (6.1) Y (s) = G(s)F (s) = 1 s [1 e s ] 1 s [1 3e s + 2e 2s ] y(t) = 1 s 2 [1 e s ][1 3e s + 2e 2s ] = 1 s 2 [1 4e s + 5e 2s 2e 3s ] (6.11) y(t) = L 1 [Y (s)] = t 4(t 1)u(t 1) + 5(t 2)u(t 2) 2(t 3)u(t 3) t t < 1 3t + 4 1 t < 2 = (6.12) 2t 6 2 t < 3 3 t 6.6: 6

6.2.2 t y(t) = x(τ)g(t τ)dτ (6.13) 6.5 f(τ) = τ < 1 τ < 1 2 1 τ < 2 2 τ (6.14) g(t ) = t < 1 t < 1 (6.15) 1 t g(t τ) 6.7 τ g(t τ) = t < τ 1 t 1 < τ t τ t 1 (6.16) t < t τ < t < τ 1 t τ t 1 t < 1 t 1 < τ t (6.16) 6.7 g(t ) τ t τ 6.7: g(t τ) t 6.8 61

t < 1 t y(t) = 1 1dτ = t (6.17) 1 t < 2 y(t) = t 1 1dτ + 1 t 1 1 1dτ + t 1 1 ( 2)dτ = + [t] 1 t 1 2 [t]t 1 = [1 (t 1)] 2[t 1] = 3t + 4 (6.18) 2 t < 3 y(t) = 1 2 + 1dτ + t 1 t 1 1 1 ( 2)dτ + ( 2)dτ t 2 1 dτ = + 2 [t] 2 t 1 + = 2[2 (t 1)] = 2t 6 (6.19) 3 t y(t) = 1 3 + 1dτ + t 1 2 1 1 dτ + ( 2)dτ + t 3 1 dτ = t 1 2 dτ (6.2) (6.12) y(t) = t t < 1 3t + 4 1 t < 2 2t 6 2 t < 3 3 t (6.21) 62

6.8: 6.3 s (6.4) y(t)e st dt = t = t + τ y(t)e st dt = = = = [ G(s) = X(s) = Y (s) = x(τ) ] x(τ)g(t τ)dτ [ [ x(τ) τ [ x(τ) x(τ)e sτ dτ 63 e st dt ] g(t τ)e st dt dτ g(t )e st dt ] e sτ dτ g(t )e st dt ] e sτ dτ g(τ)e sτ dτ x(τ)e sτ dτ y(τ)e sτ dτ g(t )e st dt

Y (s) = X(s)G(s) (6.22) s L[x(t) g(t)] = X(s)G(s) (6.23) * s 6.9 s F 2 (s) = G 1 (s)f 1 (s) Y (s) = G 2 (s)f 2 (s) Y (s) = G 2 (s)g 1 (s)f 1 (s) 6.9: L[f 1 (t) f 2 (t)] = F 1 (s)f 2 (s) L[f 1 (t)f 2 (t)] = F 1 (s)f 2 (s) (6.24) F 1 (s)f 2 (s) s 64

7 7.1 (1) (2) (3) 7.1 7.2 Ms 2 X(s) = KX(s) DsX(s) + F (s) (7.1) 65

7.1: F (s) X(s) F (s) KX(s)+DsX(s) X(s) 1 Ms 2 2 7.3 7.3 f f D M K x D x M Kx M x 7.2: 7.3: 66

X(s) F (s) = 1 Ms 2 + Ds + K (7.2) F(s) X(s) 7.4: ( ) 1 7.5 R 1 R 2 vi i1 v 2 C 1 1 i C 2 vo 7.5: C 1 V 1 (s) V 1 (s) = V i (s) R 1 [I 1 (s) + I 2 (s)] V o (s) = V 1 (s) R 2 I 2 (s) V 1 (s) = I 1(s) sc 1 V o (s) = I 2(s) sc 2 V i (s) V o (s) I 2 = sc 2 V o (s) V o = V 1 (s) R 2 I 2 (s) I 1 (s)+i 2 (s) V 1 (s) = V i (s) R 1 [I 1 (s) + I 2 (s)] 7.6 67

V i (s) + V o (s) + R 1 sc 1 R 2 + I 1 + sc 2 ( s) I 2 2 7.6: 7.2 (a) (b) (a) (b) (a) (b) (c) (d) 7.2.1 G 1 G 2 G 1 (s)g 2 (s) 7.7: 68

7.2.2 G 1 G 2 U 1 (s) U 2 (s) Y 1 (s) Y 2 (s) Y 1 (s) = G 1 (s)u 1 (s) Y 2 (s) = G 2 (s)u 2 (s) U 1 (s) = U 2 (s) 7.8: Y (s) =Y 1 (s) + Y 2 (s) = [G 1 (s) + G 2 (s)]u(s) (7.3) ( 7.8 3 ) G 1 (s) = G 2 (s) = G(s) Y (s) =Y 1 (s) + Y 2 (s) = G(s)[U 1 (s) + U 2 (s)] (7.4) ( 7.8 1 ) 69

7.2.3 G(s) E(s) E(s) = U(s) H(s)Y (s) Y (s) = G(s)E(s) Y (s) U(s) = G(s) 1 + G(s)H(s) (7.5) H(s) = G(s) 7.9: 7.2.4 7.1 7.11 2 7

7.1: u + G ( ) G ( s) 2 1 s H ( s 1 ) y u u + G ' 1 ( s) G ( ) 2 s y G ( ) G ( s) 2 1 s H ( s 1 ) 1/ G 2( s ) y 7.11: ) + + G 1 ( s) + H 2 ( s) G 2 ( s) H 1 ( s) G ( s 3 ) H 3 ( s) 7.12: 71

7.13 7.13: Y (s) U(s) = G 1 G 2 G 3 1 + G 2 G 3 H 2 + G 1 G 2 H 1 + G 1 G 2 G 3 H 3 (7.6) 72

8 s 8.1 δ(t) u(t) t(t) G(s) g(t) = L 1 [G(s)1] (8.1) [ f(t) = L 1 G(s) 1 ] (8.2) s [ h(t) = L 1 G(s) 1 ] (8.3) s 2 g(t) f(t) h(t) s G(s) = N(s) D(s) = b s m + b 1 s m 1 + + b m e Ls s n + a 1 s n 1 + + a n = Ksp n p1 i=1(1 + T i s) n p2 i=1 1(s 2 + 2ω i ξ i s + ωi 2 ) s l j l1 j=1(1 + T j s) n l2 e Ls (8.4) j=1(s 2 + 2ω j ξ j s + ωj 2 ) { K, s, 1 s, 1 1 + T s, 1 } e Ls s 2 +2ωξs+ω 2 n 2 73

8.2 8.2.1 8.1 N A θ A N B θ B θ A θ B θ B = N A N B θ A (8.5) N A N B 8.1: 8.2.2 8.2 A h(t) V (t) = Ah(t) Ah(t) = t u(t)dt (8.6) H(s) U(s) = 1 As (8.7) 1 s 1 Cs 74

8.2: q +u h o +h U(s) 1 sa+ 1 R h H(s) A q o +y 8.3: 1 8.2.3 8.3 h q R h = 2h y = 1 h(t) R h dh dt A = u(t) 1 R h h(t) (8.8) sah(s) = U(s) 1 R h H(s) {sa + 1 R h }H(s) = U(s) q H(s) U(s) = 1 sa + 1 R h (8.9) s 75

8.4 x(t) y(t) y() = K(x(t) y(t)) = D dy dt G(s) = Y (s) X(s) = K K + Ds K x y D 8.4: 2 8.2.4 8.5 2 8.5: 1 v 2 (t) i 1 (t) v 2 (t) = M di 1 dt (8.1) 76

M Ls 8.6 f(t) = D dx dt oil D dx D dt f 8.6: 2 8.2.5 8.7 D M f K x F(s) 1 2 Ms + Ds+ K X(s) 8.7: 77

M d2 x(t) dt 2 + D dx(t) dt + Kx(t) = f(t) (8.11) X(s) F (s) = 1 Ms 2 + Ds + K (8.12) s 2 2 2 8.2.6 8.8 L v y(t) = {, t < L v u(t L), L t (8.13) v v Y (s U(s) = L e v s (8.14) l A v B 8.8: 78

A B 79

9 (1) (2) (3) (a) (b) (c)2 9.1 f(t) = L 1 [ K s 1 ] = K (9.1) g(t) = L 1 [ K s 1 s ] = Kt (9.2) h(t) = L 1 [ K s 1 s 2 ] = Kt2 2 (9.3) 9.2 L [ ] f(t) =L 1 K T s + 1 1 [ ] K/T = L 1 = K t s + 1/T T e T (9.4) 8

!"#$%&!'()*#%& +!,%& u(t) u(t) y(t)! t y(t) t y(t) t y(t)! t t t 9.1: [ g(t) =L 1 K T s + 1 1 ] s [ ] [ = L 1 K + L 1 s K = K K 1 = K K 1 s + 1/T ] (9.5) g(t) = K(1 e t T ) (9.6) [ h(t) =L 1 K T s + 1 1 ] = K [ ] t T (1 e tt ) s 2 (9.7) (3.47) (3.51) 9.2 t t 81

9.2: 9.2 Y (s) = t = u() 1 K U(s) (9.8) T s + 1 sy (s) = 1 { Y (s) + KU(s)} (9.9) T dy(t) t= = K dt T (9.1) (9.6) y( ) = lim t {K(1 e t T )} = K (9.11) t = K T T K t = 82

τ T s[sec] (R) (C) RC (τ) (τ) R(Ω) C(F ) 1 v(t) = E(1 e t RC ) τ = RC v(τ) = E(1 e τ RC ) = E(1 e 1 ).632E.632 9.3 G(s) = K s 2 + as + b = ω 2 n s 2 + 2ω n ζs + ω 2 n (9.12) s 2 + 2ω n ζs + ωn 2 = (s s 1 )(s s 2 ) (9.13) s 1 s 2 (1)ζ = 1 s 1 = s 2 s 2 + 2ω n ζs + ωn 2 = (s + ω n ) 2 (9.14) (2)ζ > 1 s = ζω n ± ζ 2 ω 2 n ω 2 n = ζω n ± ω n ζ 2 1 (9.15) ζ 2 1 > s 1 s 2 s 2 + 2ω n ζs + ω 2 n = (s s 1 )(s s 2 ) (9.16) (3)ζ < 1 ζ 2 1 < s = ζω n ± ω n j s 1 s 2 1 ζ 2 (9.17) 83

9.3.1 2 g(t) s 1, s 2 [ g(t) =L 1 ωn 2 ] 1 s 2 + 2ω n ζs + ωn 2 [ ] [ ] = L 1 K1 + L 1 K2 s s 1 s s 2 (9.18) (1)ζ = 1 G(s) = ωn 2 (s + ω n ) = ωn1! 2 (9.19) 2 (s + ω n ) (1+1) (t k = 1 ) g(t) = ω 2 nt 1 e ωnt (9.2) (2)ζ > 1 s 1 = ζω n + ω n ζ 2 1 s 2 = ζω n ω n ζ 2 1 (9.21) s 1 s 2 = 2ω n ζ2 1 (9.18) K 1 = K 2 = [ ] g(t) = L 1 K1 s s 1 = = = [ ωn 2 (s s ] 1 ) (s s = ω2 n ω n = 1 )(s s 2 ) s=s 1 s 1 s 2 2 ζ 2 1 [ ωn 2 (s s ] 2 ) (s s 1 ) (s s = ω2 n = ω n 2 ) s=s 2 s 1 s 2 2 ζ 2 1 [ ] + L 1 K2 s s 2 ω n 2 ζ 2 1 e( ζωn+ωn ζ 2 1)t ζ ζ ω 2 1t n eωn e ωn 2 1t ζ2 1 e ζωnt 2 ω n 2 ζ 2 1)t ζ 2 1 e( ζωn ωn ω n ζ2 1 e ζω nt sinh(ω n ζ 2 1t) (9.22) 84

x 2 + y 2 = 1 x = sin θ y = cos θ x 2 y 2 = 1 cosh θ = eθ + e θ sinh θ = eθ e θ 2 2 (3)ζ < 1 s 1 = ζω n + ω n j 1 ζ 2 s 2 = ζω n ω n j 1 ζ 2 (9.23) s 1 s 2 = 2jω n 1 ζ2 (9.18) [ ωn 2 (s K 1 = s ] 1 ) (s s = ω2 n ω n = 1 )(s s 2 ) s=s 1 s 1 s 2 2j 1 ζ 2 [ ωn 2 ] (s s 2 ) K 2 = (s s 1 ) (s s = ω2 n ω n = 2 ) s=s 2 s 1 s 2 2j 1 ζ 2 [ ] g(t) = L 1 K1 s s 1 = = = [ ] + L 1 K2 s s 2 ω n 2j 1 ζ 2 e( ζω n+jω n ω n 1 ζ 2 e ζωnt ejω n ζ = 1 ζ 2 )t ω n 2j 1 ζ 2 e( ζω n jω n 1 ζ 2 )t 1 ζ 2t e jω n 1 ζ 2 t 2j ω n 1 ζ 2 e ζω nt sin(ω n 1 ζ 2 t) (9.24) g(t) = ω n sin ω n t (9.25) 9.3.2 2 f(t) [ 1 h(t) = L 1 s (s 2 + 2ω n ζs + ωn) 2 [ ] [ ] [ ] = L 1 K + L 1 K1 + L 1 K2 s s + s 1 s + s 2 ω 2 n ] (9.26) 85

9.3: (1)ζ = 1 h(t) = L 1 [ 1 s ωn 2 ] (s 2 + 2ω n ζs + ωn) 2 = ω 2 n s(s + ω n ) 2 (9.27) (9.26) K = K 1 = K 2 = [ ωn s 2 ] = 1 s(s + ω n ) 2 s= [ ω 2 n (s + ω n ) 2 ] s (s + ω n ) 2 = ω n s= ω [ n d ωn 2 (s ω n ) 2 ] ds s (s ω n ) 2 = 1 s= ω n [ ] [ ] [ ] h(t) = L 1 K + L 1 K 1 + L 1 K 2 s (s + s n ) 2 (s + s n ) = 1 ω n te ωnt e ωnt = 1 (ω n t + 1)e ωnt (9.28) 86

(2)ζ > 1 h(t) = 1 e ζω nt ζ2 1 sinh( ζ 2 1ω n t + tan 1 ζ2 1 ) (9.29) ζ (3)ζ < 1 (9.26) K = K 1 = K 2 = [ ωn s 2 ] s(s 2 + 2ζω n s + ωn) 2 [ ωn 2 (s s ] 1 ) s (s s 1 )(s s 2 ) [ ωn 2 (s s ] 2 ) s(s s 1 ) (s s 2 ) s= s=s 1 = s=s 2 = = 1 ω 2 n s 1 (s 1 s 2 ) ω 2 n s 2 (s 2 s 1 ) s 1 = ζω n + ω n j 1 ζ 2 s 2 = ζω n ω n j 1 ζ 2 (9.3) s 1 s 2 = 2jω n 1 ζ2 K 1 = ω 2 n s 1 (s 1 s 2 ) = K 2 = ζ j 1 ζ 2 2j 1 ζ 2 ωn 2 ( ζω n + ω n j 1 ζ 2 )(2jω n 1 ζ2 ) = ζ j 1 ζ 2 2j 1 ζ 2 [ ] [ ] h(t) = L 1 K + L 1 K1 s s + s 1 = K + K 1 e s1t + K 2 e s2t [ (ζ + j = 1 e ζωnt 2j 1 ζ 2 = 1 e ζω nt 1 ζ 2 [ ζ sin( [ ] + L 1 K2 s + s 2 1 ζ 2 )e j 1 ζ 2 ω nt + ( ζ + j 1 ζ 2 ω n t) + = 1 e ζω nt 1 ζ sin( 1 ζ 2 ω 1 ζ 2 n t + tan 1 2 ) ζ ] 1 ζ 2 cos( 1 ζ 2 ω n t) ] 1 ζ 2 )e j 1 ζ 2 ω nt 87

ζ = h(t) = 1 cos ω n t (9.31) 1 9.4 K 9.4: 2 88

9.4 1 ζ ζ > 1 ζ < 1 ζ = 1 1 ζ > 1 2 ζ = 1 3 ζ < 1 ζ ω 2 s 1, s 2 = σ ± jω = ζω n ± jω n 1 ζ 2 (9.32) s 1, s 2 ω n Im s 1 +j - Re s 2 -j 9.5: 2 89

1 G(s) H(s) 7 H 1 (s) H 2 (s) H 3 (s) Y (s) U(s) = G 1 G 2 G 3 1 + G 2 G 3 H 2 + G 1 G 2 H 1 + G 1 G 2 G 3 H 3 (1.1) G(s) H(s) 8 s 1.1 1.1 Y (s) U(s) = G(s) 1 + G(s)H(s) (1.2) U(s) Y (s) U(s) Y (s) = K 1 s s 1 + K 2 s s 2 + + 9 K n s s n + K+ s s i + K s s i (1.3)

s 1 s n 1 + G(s)H(s) = (1.2) y(t) y(t) = K 1 e s 1t + K 2 e s 2t + + K n e s nt + K + e s i + K e s i (1.4) s i = σ i + jγ i y i (t) s i K + e s it = K + e (σ i+jγ i )t K e s it = K e (σ i jγ i )t (1.5) (1.6) y(t) [ ] [ ] K g(t) = L 1 + K + L 1 s s i s s i 1 = e (σ i+jω i )t 1 e (σ i jω i )t 2jω i 2jω i = 1 ω i e σ it ejω it e jω it 2j = 1 ω i e σ it sin(ω i t) (1.7) 1.1: (1)s 1 s n (1.7) σ i < t (2) s s = 91

(3)s 1 s n (1.7) σ i = g(t) = (1/ω i ) sin(ω i t) t (4)s 1 s n (1.7) σ i > ω i = ω i σ i > σ i = σ i < 1.2 1.2 jω > 1 + G(s)H(s) = H(s) 1.2: 92

1.3 G(s) = N(s) D(s) (1.8) (D(s) = ) (N(s) = ) U(s) Y (s) G(s) = K(s z 1)(s z 2 ) (s z m ) (s p 1 )(s p 2 ) (s p n ) (1.9) U(s) = 1 Y (s) s K 1 Y (s) = K s + K 1 + K n (1.1) s p 1 s p n y(t) = K + K 1 e p 1t + K n e pnt (1.11) K = [Y (s)s] s= = K( z 1)( z 2 ) ( z m ) ( p 1 )( p 2 ) ( p n ) (1.12) K 1 = [Y (s)(s p 1 )] s=p1 = K(p 1 z 1 )(p 1 z 2 ) (p 1 z m ) (1.13) p 1 (p 1 p 2 ) (p 1 p n ) K 2 K 3 (1) (K i p i K i ) (2) p i z i K i (3) (4) 93

t = 1.3 p 1 I R 1. t z1 p1 I R p/z 1. t I p/z R p 1 z 1 1. t 1.3: ( 1 1 1 Y (s) = p 1 z 1 s + z 1 s + p 1 1 s = K s + K 1 s + p 1 (1.14) K = p 1 [ (s + z 1) s z 1 (s + p 1 ) s ] s= = p 1 z 1 = 1 z 1 p 1 (1.15) K 1 = p 1 [ s + z 1 s + p 1 z 1 s + p ] s= p1 = p 1 p 1 + z 1 = p 1 z 1 1 s z 1 p 1 z 1 (1.16) y(t) = 1 + p 1 z 1 e p 1t z 1 (1.17) t = y() = p 1 z 1 z > p ( ) y() 1 p > z ( ) y() 1 94

1.4 2 3 1 I 1. I 1. p2 p1 R p 2 R 1. t p 1 t I I 1. p 2 p 1 z 1 R p 2 p 1 R I 1. t p 3 I 1. t p 2 z1 p 2 p1 R p 1 R t p 3 t 1.4: ( 3 1.3 1.4 95

11 11.1 f(t) =, t > F (s) = f(t)e st dt (11.1) f(t) F (s) = L[f(t)] t t s s t step1 s step2 step3 t step1 s step2 s s step3 t 96

11.2 L[ ] = G L[ ] (11.2) (1) (2) (3) m x F, X mẍ + kx = f (11.3) ms 2 X + kx = F (11.4) X F = 1 ms 2 + k (11.5) mg mg 11.3 s y(t) = t x(τ)g(t τ)dτ (11.6) 97

s Y (s) = X(s)G(s) (11.7) s 11.1 s F 2 (s) = G 1 (s)f 1 (s) Y (s) = G 2 (s)f 2 (s) Y (s) = G 2 (s)g 1 (s)f 1 (s) 11.1: L[f 1 (t) + f 2 (t)] = F 1 (s) + F 2 (s) L[f 1 (t) f 2 (t)] = F 1 (s)f 2 (s) L[f 1 (t)f 2 (t)] = F 1 (s)f 2 (s) (11.8) F 1 (s)f 2 (s) s 11.4 P 11.2 C 1 C 2 11.2 E E = C 2 R Y Y = C 1 P E 98

11.2: Y = C 1 P (C 2 R Y ) (1 + C 1 P )Y = C 1 C 2 RP R Y R = C 2C 1 P 1 + C 1 P (11.9) s 99

11.5 ( ) ( )L ( ) ( ) ( ) ( ) ( ) ( ) 1

Rev.. August 3 29 TEX Rev.1. March 3, 212 PPT Rev.1.5 March 25, 213 212 Rev.2.1 March 31 214 213 1. 2. 3. 4. 2-39-1 http://ictrl.cs.kumamoto-u.ac.jp/ 11