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1. 2. 3. 4. 5. 6.

1. 2. 3. 1-1 (inputs) (actuating signals) u (outputs) (controlled variables) y

MATLAB Simulink (multivariable systems)

()

() PID

1-11 - (1-1) r y e b G H

( 1-11 GH 1 + GH 1

1-13

1) (analog data) (discrete-data) 2) (modulated) (unmodulated) : :1) (linear) (nonlinear)2) (time-varying) (time-invariant): (position-control system) (velocity-control system)

r(t) e(t) () 1-16

1. 2. 3. MATLAB z

1. 2. s f (t) f (t) (2-3) s = + j (one-sided Laplace transform) (causal system) (physically realizable system) t = 0 t = 0 t = 0 t 0

f (t) f (t) f (t)

PS:c F(s) MATLAB (TFtool) 1. k 2.

3. (2-13) f (t) (2-14) f (i) (0) f (t) t i t = 0 4. (2-15) n (2-16)

5. (2-17) us (t T) T 6. (2-18) 7. (2-19) sF(s) j s sF(s) sF(s)

Findfind(2-22) 8. 9. t < 0 f1(t) = 0f2(t) = 0 (2-24) t (convolution)

P(s) Q(s) s G(s) s1 s2 sn (i = 12n)

Ks1

G(s) s = si r ploes(i 12n r ) G(s) (n r) Ks1Ks2Ks(n r) (n r) A1A r

jG(s) G(s) damping ratio) 1g (t) = ?

(2-55)

1. s2. Y(s)3. 4. y(t) y(0) = 1

y (t) (2-68) y(t) dy(t)/dt

SISO-- n (2-86) a0a1an1 b0b1bm u(t) y(t)

1. 2. 4. 5. s z z ( n > m) (strictly proper) n = m (proper) m > n (improper) S domaIn

(a) (b) 1. - 2. K

1. 2. 3. (a) (b) (c) (d) a)

r(t)R(s) () y(t)Y(s) () b(t)B(s) u(t)U(s) = e(t)E(s) H(s) = 1H(s) G(s)H(s) L(s) = G(s) M(s) Y(s)/R(s) = M(s)

- N

N (3-21) SFG

Ex. y2 = a12y1 y1 y2 a12 y1 y2 a12 SFG

1. 2. 3. 4. 5. yk yj yj yk 6. yk yk yj yj akj akj yk SFG () ()

y2 (path) (forward path) y1 y3 y1 y4 y1 y5 (d) (loop)

(d) (d) (d)

(path gain) (forward-path gain) (loop gain) y2 y4 y3 y2 a24a43a32 SFG SFG (nontouching) (d) SFG y2 y3 y2 y4 y4 (d)

SFG Mason rule SFG N K yin yout yin = yout = M = yin yout N = yin youtMk = yin yout k

Lmr = r (1 r K) m (m = ijk) = 1 () + ( ) () + k = k SFG 3. Y(s)/R(s)

1. R(s) Y(s)

(d) (3-34) 2. (3-35) 3. 1 = 1 (3-31) SFGmason rule y1 y5

1. 2. SFG 3. 4. M1 M3 1 = 3 = 1 5. M2 y3 y4 y4 y4 y4

6. y1 y5 y2 5. SFGmason-

SFG y7/y2 yin SFG yout y2 yout /y2 y2 y7 1. - 2.

(b) SFG Y(s)/R(s) = ?

1. (b) 2. 1. 2. SFG 3. SFG

1. x1(t) x2(t) (3-54) X1(s) = [X2(s)/s] + [x1(t0)/s] 2.

3-18 X1(s) = [X2(s)/s] + [x1(t0)/s] 1. 2. 3. 4. 5. 6.

B- - 1. n n i = 12n xi (t) i uj (t)j = 12p j wk(t) k k = 12v 2. y1(t)y2(t)yq(t) q j = 12q (dynamic equation)

3. - 1) 2) (5-3) 3) (5-5) 4) 4.

x (t0) u (t) w (t)(t t0) 1. 2. (t) n n x(0) t = 0 (t) t 0

3. (sI A) (t) eAt t 0 eAt At

1. (free response) 2. (t) t = 0 t 1. () 2.

(5-27) e At (5-30) 1(t) (5-24)

1. 2. x(0) t = 0 t0 x(t0) u(t) w(t) t 0 (5-41) t = t0 x(0)

3. t 0 u (t) = 1 t 0 (t) x(t)

1. A B 2. A 3. t 0

1. SFG (5-40) 2. t0 (5-40) 3. Xi (s)i = 12n 4.

1. n2.

3. (phase-variable canonical form, PVCF)4. -

5.

1. (5-80) 2.

3. - x(t) 3 1 u(t) 4. 1. nth-order system: The state variable must be chosen such that they will eliminate the derivatives of u in the state equation. 2. n state variables:

where are determined from With the present choice of state variables, we obtain

In terms of vector-matrix equations, Equation (3-36) and the output equation can be written as and

1. x(t) = n 1 y(t) = q 1 u(t) = p 1 w(t) = v 1 2. x (0) = 0 (5-89)

Gu(s) w(t) = 0 u(t) y(t) q p Gw(s) u(t) = 0 w(t) y(t) q v 5-5

1. 2. s 3.

:

: Determine of A

8.( A A

A 1. A 2. 12n A A A 3. ii = 12n A A' 4. A ii = 12n 1/ii = 12n A1 A

ii = 12n A pi A i (5-38) A

1. A 2. 1 = 1 2 = 1 3.

1=1 p1 2 = 1(5-126) eigenvalue1. A A (5-120) 2. A n q(< n) q i i i = 12q 3. j m (m n q) m

11. A A

1. A 1 = 22 = 3 = 1 2. 1 = 2 (5-128) p11 = 2 p21 = 1 p31 = 2 (5-131) 3. 2 = 1 (5-129) (5-133) (5-134) 3 = 1 (5-129)

1. - (SISO) : 2. 3.

4. (5-140) t 1.

2. (CCF) 1. (5-137) (5-138) 2. A 3.

(

(OCF) 1. 2. (5-164) 3. (5-141) (5-142)

4. M V (observability matrix) V 1 OCF

1. M (5-160)

2. 3. OCF 4. OCF (DCF) 1. 2. A

12n A n (DCF) 3. DCF T pii = 12n i 4. n n

A CCF DCF (Vandermonde) DCF

1. 1 = 12 = 2 3 = 3

2. A CCF DCF (5-183) 3. A DCF JORDEN (JCF) 1. A 2. JCF

1. A 2. A 3. 1 4. 1 5. A n n A r (r < n) 6. r 7. 1 n r 3. 4. T JCF

ADCF

1. A 211 2. DCF DCF

1. 2. n SISO U(s) Y(s) CCF 1. s 2. X(s) 3. 4.

5. 5-6 CCF 1) x1(t)x2(t)xn(t) 2) u(t) 3)

State equation:Output equation: OCF 1. 2. s n

3. CCF 4.

1. 2. 5-10 3. u(t) SFG

4. u(t) y(t) 5-10 1. 2. 3.

1. 2. DCF JCF 3. 4. (5-217) 5. 6.

DCF JCF

1.

2.

(controllability) (observability) (Kalman) 1. 2. 5-14(a)

3. K 4. (a) (b) 5. K (ABK) 6. 7. K (ABK)

1. 2. (unconstrained) u (t) (completely controllable) 3. 4. u(t) x1(t) x2(t) u(t) (tf t0) x2(t0) x2(t) x2(tf)

5-16 1. x(t) n 1 u (t) r 1 y(t) p 1 ABC D 2. (tf t0) 0 u(t)x(t0) x(tf) x(t) t = t0 x(t0) n nr n [ AB ] S n

S n n SS' SS' S n r = 1 - (SISO) A B CCF CCF [ AB ] A DCF JCF B [ AB ] Ex. JCF A B B A B b31 0b32 0b41 0 b42 0

S = [B AB]

1. )2. S

1. S 2. A 1 = 22 = 2 3 = 1 1

1. 2. 3. 5-17 x2 y(t) y(t) x1(t) x1(t) = y(t) x2 y(t) 1.

2. u(t) tf t0 t0 t < tf u(t) ABC D t0 t < tf y(t) x(t0) x(t0) A C n np n C 1 n V n n V (SISO) ( r = 1 p = 1) A C OCF OCF [ AC ]

A DCF JCF C [ AC ] A C 5-21 5-17

1. [ AC ] 2. A DCF C x2(t) --- - Ex. SISO

1. A x1 (C O) x2 (C UO) x3 (UC O) x4 (UC UO) 2. DCF 3. (5-240) - -

CCF OCF [A] CCF: 1. CCF CCF [ AB ] 2. CCF [ AC ] [B] OCF: 1. OCF OCF [ AC ] 2. (5-247) OCF [ AB ]

3~5 ..()...,.,.*

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