《自动控制原理》试卷及答案(英文10套)
AUTOMATIC CONTROL THEOREM (1)
⒈ Derive the transfer function and the differential equation of the electric network
⒉ Consider the system shown in Fig.2. Obtain the closed-loop transfer function )()(S R S C , )
()
(S R S E . （12%）
⒊ The characteristic equation is given 010)6(5)(123=++++=+K S K S S S GH . Discuss the distribution of the closed-loop poles. (16%)
① There are 3 roots on the LHP ② There are 2 roots on the LHP
② There are 1 roots on the LHP ④ There are no roots on the LHP . K=?
⒋ Consider a unity-feedback control system whose open-loop transfer function is
)
6.0(1
4.0)(++=
S S S S G . Obtain the response to a unit-step input. What is the rise time for
this system? What is the maximum overshoot? （10%）
Fig.1
5. Sketch the root-locus plot for the system )
1()(+=
S S K
S GH . ( The gain K is
assumed to be positive.)
① Determine the breakaway point and K value.
② Determine the value of K at which root loci cross the imaginary axis. ③ Discuss the stability. (12%)
6. The system block diagram is shown Fig.3. Suppose )2(t r +=, 1=n . Determine the value of K to ensure 1≤
e . (12%)
Fig.3
7. Consider the system with the following open-loop transfer function:
)
1)(1()(21++=
S T S T S K
S GH . ① Draw Nyquist diagrams. ② Determine the
stability of the system for two cases, ⑴ the gain K is small, ⑵ K is large. (12%)
8. Sketch the Bode diagram of the system shown in Fig.4. (14%)
⒈ 2
121211
21212)()()(C C S C C R R C S C C R S V S V ++++= ⒉
2
423241321121413211)()
(H G H G G G G G G G H G G G G G G G S R S C ++++++=
⒊ ① 0 ⒌①the breakaway point is –1 and –1/3; k=4/27 ② The imaginary axis S=±j; K=2③ ⒍5.75.3≤≤K ⒎ )154 .82)(181.34)(1481.3)(1316.0() 11.0(62.31)(+++++=S S S S S S GH AUTOMATIC CONTROL THEOREM (2) ⒈Derive the transfer function and the differential equation of the electric network ⒉ Consider the equation group shown in Equation.1. Draw block diagram and obtain the closed-loop transfer function ) () (S R S C . （16% ） Equation.1 ?? ? ? ???= -=-=--=)()()()()]()()([)()]()()()[()()()]()()[()()()(3435233612287111S X S G S C S G S G S C S X S X S X S G S X S G S X S C S G S G S G S R S G S X ⒊ Use Routh ’s criterion to determine the number of roots in the right-half S plane for the equation 0400600226283)(12345=+++++=+S S S S S S GH . Analyze stability.（12% ） ⒋ Determine the range of K value ,when )1(2t t r ++=, 5.0≤SS e . （12% ） Fig.1 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%） （18% ） ⒎ Determine the transfer function. Assume a minimum-phase transfer function.（10% ） ⒈ 1 )(1 )()(2122112 221112++++=S C R C R C R S C R C R S V S V ⒉ ) (1)() (8743215436324321G G G G G G G G G G G G G G G G S R S C -+++= 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 208<≤K ⒌ K=20 ⒍ ⒎ )154 .82)(181.34)(1481.3)(1316.0() 11.0(62.31)(+++++=S S S S S S GH 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 ⒊ Consider the system shown in Fig.2. Obtain the closed-loop transfer function )()(S R S C , )()(S R S E , ) () (S P S C . （12%） ⒋ The characteristic equation is given 02023)(123=+++=+S S S S GH . Discuss the distribution of the closed-loop poles. (16%) 5. Sketch the root-locus plot for the system ) 1()(+= S S K S GH . (The gain K is 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. 2 1+= S K G , )3(42+=S S G . Suppose )2(t r +=, 1=n . Determine the value of K to ensure 1≤SS e . (15%) 7. Consider the system with the following open-loop transfer function: ) 1)(1()(21++= S T S T S K S GH . ① Draw Nyquist diagrams. ② Determine the stability of the system for two cases, ⑴ the gain K is small, ⑵ K is large. (15%) ⒈ Solution: The advantages of open-loop control systems are as follows: ① Simple construction and ease of maintenance ② Less expensive than a corresponding closed-loop system ③ There is no stability problem ④ Convenient when output is hard to measure or economically not feasible. (For example, it would be quite expensive to provide a device to measure the quality of the output of a toaster.) The disadvantages of open-loop control systems are as follows: ① Disturbances and changes in calibration cause errors, and the output may be different from what is desired. ② To maintain the required quality in the output, recalibration is necessary from time to time. ⒉ 1)(1)()()(2122112 221122112221112+++++++=S C R C R C R S C R C R S C R C R S C R C R S U S U ⒊ 3 51343212321215143211)() (H G G H G G G G H G G H G G G G G G G G S R S C +++++= 3 5134321232121253121431)1()() (H G G H G G G G H G G H G G H G G H G G G G S P S C ++++-+= ⒋ R=2, L=1 ⒌ S:①the breakaway point is –1 and –1/3; k=4/27 ② The imaginary axis S=±j; K=2 ⒍5.75.3≤≤K AUTOMATIC CONTROL THEOREM (4) ⒈ Find the poles of the following )(s F : s e s F --= 11 )( (12%) ⒉Consider the system shown in Fig.1,where 6.0=ξ and 5=n ωrad/sec. Obtain the rise time r t , peak time p t , maximum overshoot P M , and settling time s t when the system is subjected to a unit-step input. （10%） ⒊ Consider the system shown in Fig.2. Obtain the closed-loop transfer function )()(S R S C , )()(S R S E , ) () (S P S C . （12%） ⒋ The characteristic equation is given 02023)(123=+++=+S S S S GH . Discuss the distribution of the closed-loop poles. (16%) 5. Sketch the root-locus plot for the system ) 1()(+= S S K S GH . (The gain K is assumed to be positive.) ⑦ Determine the breakaway point and K value. ⑧ Determine the value of K at which root loci cross the imaginary axis. ⑨ Discuss the stability. (12%) 6. The system block diagram is shown Fig.3. 2 1+= S K G , )3(42+=S S G . Suppose )2(t r +=, 1=n . Determine the value of K to ensure 1≤SS e . (12%) 7. Consider the system with the following open-loop transfer function: ) 1)(1()(21++= S T S T S K S GH . ① Draw Nyquist diagrams. ② Determine the stability of the system for two cases, ⑴ the gain K is small, ⑵ K is large. (12%) 8. Sketch the Bode diagram of the system shown in Fig.4. (14%) ⒈ Solution: The poles are found from 1=-s e or 1)sin (cos )(=-=-+-ωωσωσj e e j From this it follows that πωσn 2,0±== ),2,1,0( =n . Thus, the poles are located at πn j s 2±= ⒉Solution: rise time sec 55.0=r t , peak time sec 785.0=p t , maximum overshoot 095.0=P M , and settling time sec 33.1=s t for the %2 criterion, settling time sec 1=s t for the %5 criterion. ⒊ 3 51343212321215143211)() (H G G H G G G G H G G H G G G G G G G G S R S C +++++= 3 5134321232121253121431)1()() (H G G H G G G G H G G H G G H G G H G G G G S P S C ++++-+= ⒋R=2, L=1 5. S:①the breakaway point is –1 and –1/3; k=4/27 ② The imaginary axis S=±j; K=2 ⒍5.75.3≤≤K AUTOMATIC CONTROL THEOREM (5) ⒈ Consider the system shown in Fig.1. Obtain the closed-loop transfer function )()(S R S C , ) () (S R S E . （18%） ⒉ The characteristic equation is given 0483224123)(12345=+++++=+S S S S S S GH . Discuss the distribution of the closed-loop poles. (16%) ⒊ Sketch the root-locus plot for the system ) 15.0)(1()(++= S S S K S GH . (The gain K is assumed to be positive.) ① Determine the breakaway point and K value. ② Determine the value of K at which root loci cross the imaginary axis. ③ Discuss the stability. (18%) ⒋ The system block diagram is shown Fig.2. 1111+= S T K G , 1 22 2+=S T K G . ①Suppose 0=r , 1=n . Determine the value of SS e . ②Suppose 1=r , 1=n . Determine the value of SS e . (14%) ⒌ Sketch the Bode diagram for the following transfer function. ) 1()(Ts s K s GH += , 7=K , 087.0=T . (10%) ⒍ A system with the open-loop transfer function ) 1()(2 += TS s K S GH is inherently unstable. This system can be stabilized by adding derivative control. Sketch the polar plots for the open-loop transfer function with and without derivative control. (14%) ⒎ Draw the block diagram and determine the transfer function. (10%) ⒈ ? =3 21)()(G G G S R S C ⒉R=0, L=3,I=2 ⒋①2121K K K e ss +-= ②2 12 11K K K e ss +-= ⒎1 1)()(12+=RCs s U s U AUTOMATIC CONTROL THEOREM (6) ⒈ Consider the system shown in Fig.1. Obtain the closed-loop transfer function )()(S R S C , ) () (S R S E . （18%） ⒉The characteristic equation is given 012012212010525)(12345=+++++=+S S S S S S GH . Discuss the distribution of the closed-loop poles. (12%) ⒊ Sketch the root-locus plot for the system ) 3() 1()(-+= S S S K S GH . (The gain K is assumed to be positive.) ① Determine the breakaway point and K value. ② Determine the value of K at which root loci cross the imaginary axis. ③ Discuss the stability. (15%) ⒋ The system block diagram is shown Fig.2. S G 1 1= , )125.0(102+=S S G . Suppose t r +=1, 1.0=n . Determine the value of SS e . (12%) ⒌ Calculate the transfer function for the following Bode diagram of the minimum phase. (15%) ⒍ For the system show as follows, ) 5(4 )(+= s s s G ,1)(=s H , (16%) ① Determine the system output )(t c to a unit step, ramp input. ② Determine the coefficient P K , V K and the steady state error to t t r 2)(=. ⒎ Plot the Bode diagram of the system described by the open-loop transfer function elements ) 5.01() 1(10)(s s s s G ++=, 1)(=s H . (12%) w ⒈ 3 2221212321221122211)1()() (H H G H H G G H H G G H G H G H G G G S R S C +-++-+-+= ⒉R=0, L=5 ⒌) 16 11() 14)(1)(110(05.0)(2 s s s s s s G ++++= ⒍t t e e t c 431341)(--+-= t t e e t t c 412 1 3445)(---+-= ∞=P K , 8.0=V K , 5.2=ss e AUTOMATIC CONTROL THEOREM (7) ⒈ Consider the system shown in Fig.1. Obtain the closed-loop transfer function )()(S R S C , ) () (S R S E . （16%） ⒉ The characteristic equation is given 01087444)(123456=+-- +-+=+S S S S S S S GH . Discuss the distribution of the closed-loop poles. (10%) ⒊ Sketch the root-locus plot for the system 3 ) 1()(S S K S GH += . (The gain K is assumed to be positive.) ① Determine the breakaway point and K value. ② Determine the value of K at which root loci cross the imaginary axis. ③ Discuss the stability. (15%) ⒋ Show that the steady-state error in the response to ramp inputs can be made zero, if the closed-loop transfer function is given by: n n n n n n a s a s a s a s a s R s C +++++=---1111)() ( ;1)(=s H （12%） ⒌ Calculate the transfer function for the following Bode diagram of the minimum phase. （15%） w ⒍ Sketch the Nyquist diagram (Polar plot) for the system described by the open-loop transfer function ) 12.0(1 1.0)(++= s s s S GH , and find the frequency and phase such that magnitude is unity. （16%） ⒎ The stability of a closed-loop system with the following open-loop transfer function ) 1() 1()(12 2++= s T s s T K S GH depends on the relative magnitudes of 1T and 2T . Draw Nyquist diagram and determine the stability of the system. （16%） ( 00021>>>T T K ) ⒈ 3 213221132112)() (G G G G G G G G G G G G S R S C ++-++= ⒉R=2, I=2,L=2 ⒌)1 () 1( )(321 22++= ωωωs s s s G ⒍o s rad 5.95/986.0-=Φ=ω