⒌Fig.3 shows a unity-feedback control system. By sketching the Nyquist diagram of the system, determine the maximum value of K consistent with stability, and check the result using Routh’s criterion. Sketch the root-locus for the system（20%）

R C KR S ( S 2 ? 4 S ? 5 ) E Fig.3

⒍Sketch root-locus diagram.（18% ） Im Im Im Re Re Re Im Im Im Re Re Re

⒎ Determine the transfer function. Assume a minimum-phase transfer function.（10% ）

L(dB) 0 20 0 –20 –40 -60

40

30

20

5

ω

0.1 ω1 ω2 ω3 ω4

⒈ ⒉

V2(S)1? 2V1(S)R1C1R2C2S?(R1C1?R2C2?R1C2)S?1G1G2G3G4C(S)? R(S)1?G2G3G6?G3G4G5?G1G2G3G4(G7?G8)

⒊ There are 4 roots in the left-half S plane, 2 roots on the imaginary axes, 0 root in the RSP. The system is unstable.

⒋ 8?K?20

⒌ K=20 ⒍

S31.62(?1)0.1⒎ GH(S)?

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

AUTOMATIC CONTROL THEOREM (3)

⒈List the major advantages and disadvantages of open-loop control systems. (12% )

⒉Derive the transfer function and the differential equation of the electric network shown in Fig.1.（16% ） C1

R1 R2 U2 U1 Fig.1 C2

⒊ Consider the system shown in Fig.2. Obtain the closed-loop transfer function

C(S)E(S)C(S), , . （12%） R(S)R(S)P(S)G5 E R P C G1 G2 G3 G4 H2 Fig.2 H1 H3

⒋ The characteristic equation is given 1?GH(S)?S3?3S2?2S?20?0. Discuss the distribution of the closed-loop poles. (16%)

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

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

④ Determine the breakaway point and K value.

⑤ Determine the value of K at which root loci cross the imaginary axis. ⑥ Discuss the stability. (14%)

6. The system block diagram is shown Fig.3. G1?4K, G2?. Suppose

S(S?3)S?2r?(2?t), n?1. Determine the value of K to ensure eSS?1. (15%)

N

R E C G1 G2

Fig.3

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

GH(S)?K. ① Draw Nyquist diagrams. ② Determine the

S(T1S?1)(T2S?1)stability of the system for two cases, ⑴ the gain K is small, ⑵ K is large. (15%)