AUTOMATIC CONTROL THEOREM (1)

¢± Derive the transfer function and the differential equation of the electric network shown in Fig.1. (12% ) R1 C1 C2 V1(S) V2(S) Fig.1 R2

¢² Consider the system shown in Fig.2. Obtain the closed-loop transfer function

C(S)E(S), . £¨12%£© R(S)R(S)G4 R E C G1 G2 G3 H2 H1 Fig.2 ¢³ The characteristic equation is given 1?GH(S)?S3?5S2?(6?K)S?10K?0. Discuss the distribution of the closed-loop poles. (16%)

¢Ù There are 3 roots on the LHP ¢Ú There are 2 roots on the LHP

¢Ú There are 1 roots on the LHP ¢Ü There are no roots on the LHP . K=?

¢´ Consider a unity-feedback control system whose open-loop transfer function is

G(S)?0.4S?1 . Obtain the response to a unit-step input. What is the rise time for

S(S?0.6)this system? What is the maximum overshoot? £¨10%£©

5. Sketch the root-locus plot for the system GH(S)?K. ( The gain K is

S(S?1)assumed to be positive.)

¢Ù Determine the breakaway point and K value.

¢Ú Determine the value of K at which root loci cross the imaginary axis. ¢Û Discuss the stability. (12%)

6. The system block diagram is shown Fig.3. Suppose r?(2?t), n?1. Determine the value of K to ensure eSS?1. (12%)

N

C E R 4K S(S?3)S?2

Fig.3

7. Consider the system with the following open-loop transfer function:

GH(S)?K. ¢Ù Draw Nyquist diagrams. ¢Ú Determine the

S(T1S?1)(T2S?1)stability of the system for two cases, ¢Å the gain K is small, ¢Æ K is large. (12%)

8. Sketch the Bode diagram of the system shown in Fig.4. (14%)

R(S) C(S) (S?2)S?2

S3(S?5)(S?10)

Fig.4

¢± ¢²

V2(S)R2C1C2S?C1? V1(S)(R1?R2)C1C2S?C1?C2G1G2G3?G1G4C(S)? R(S)1?G1G2H1?G1G2G3?G1G4?G2G3H2?G4H2

¢³ ¢Ù 0

¢µ¢Ùthe breakaway point is ¨C1 and ¨C1/3; k=4/27 ¢Ú The imaginary axis S=¡Àj; K=2¢Û

¢¶3.5?K?7.5

S31.62(?1)0.1¢· GH(S)?

SSSS(?1)(?1)(?1)(?1)0.3163.48134.8182.54

AUTOMATIC CONTROL THEOREM (2)

¢±Derive the transfer function and the differential equation of the electric network shown in Fig.1. (12% ) R1 R1 C2 V1(S) C1 V2(S) Fig.1

¢² Consider the equation group shown in Equation.1. Draw block diagram and obtain the closed-loop transfer function

C(S). £¨16% £© R(S)?X1(S)?G1(S)R(S)?G1(S)[G7(S)?G8(S)]C(S)?X2(S)?G2(S)[X1(S)?G6(S)X3(S)]?Equation.1 ?

X(S)?[X(S)?C(S)G(S)]G(S)3253??C(S)?G4(S)X3(S)?

¢³ Use Routh¡¯s criterion to determine the number of roots in the right-half S plane for the equation 1?GH(S)?S5?3S4?28S3?226S2?600S?400?0. Analyze stability.£¨12% £©

¢´ Determine the range of K value ,when r?(1?t?t2), eSS?0.5. £¨12% £©

E C R 3S2?4S?K 2S(S?2)

Fig.2