Automatic Control Theory School of Automation NWPU Teaching Group of Automatic Control Theory

Automatic Control Theory （ Lecture 40 ） Overall Review

Automatic Control Theory 各章概念融会贯通 解题方法灵活运用

Overall Review （ 1 ） Example 1 Consider an unity feedback minimum phase system. The open-loop Bode Diagram is shown in the figure. Solution 1 Obtain G(s), K and w n. 2 Determine the value of K for  =0.707

Overall Review （ 2 ） Solution. Sketch the root-locus of system Solution. 3.Sketch the root-locus for K=0→∞ and determine the location of closed-loop poles when K= Obtain the specifications (t p,  t s ) when K = Obtain e ss when r(t)=1(t) and t respectively (K =0.5) Solution.

Overall Review （ 3 ） Solution 7. Obtain the phase margin  and gain margin h 8. Obtain the closed-loop frequency indices (  r, M r,  b ) Solution 6. Sketch the corresponding Bode Diagram.

Overall Review （ 4 ） Solution 9. Obtain the steady state of the output c s (t), when

Overall Review （ 5 ） Solution 10 Determine the impact the tachometer feedback when t=0→∞. Sketch the root-locus.  ↑  →  e ss ↑ The system is stable ，  ↑  →  ↑  →  % ↓

Overall Review （ 6 ） Solution 11. The value of K is increased to improve the accuracy when r(t)=t. The corresponding L o (w) is shown in the figure. Determine which kind of cascade compensation method can improve the phase margin g with w 0 and K unchanged. Sketch the compensation diagram and discuss the impact of the compensation. Lower band: Middle band: Use lag-lead correction (The steps are shown in the diagram). Higher band: K remains unchanged to satisfy the requirement on e ss ;  c unchanged,  increased. Dynamic performance improved Higher frequency band raised, anti-high-frequency noise  Note: If two of L 0 (  ),L c (  ),L(  ) are known, another one is determined

Overall Review （ 7 ） Solution. (1) Without ZOH 12 When the system is sampled at the error signal with T=1, determine the stable range of K with/without the ZOH. Obtain the steady-state error e(∞) subject to a ramp function.

Overall Review （ 8 ） Solution (2) With ZOH

Overall Review （ 9 ） Solution (1) Sketch G(j  ). 13. Add a relay with hysteresis loop in the forward path.The -1/N(A) curve is given. (1)Determine whether there is SSO or there must be SSO. (2) Obtain the SSO parameters (A, w) when M=h=K=1 (3) Determine the impact of K and delay factor on (A, w). There must be SSO (2) Real part Imaginary part (3)

Overall Review （ 10 ） Example 2 Consider the system shown in the figure. Determine its stability. II. Stability analysis Solution I Routh criterion Solution Stable range of the parameters

Overall Review （ 11 ） Solution II Root-locus method Stable range: Sketch root-locus ① Root-locus on the real axis ③ Angle of departure ④ Crossing-point with the imaginary axis ② Asymptotes Example 2 Consider the system shown in the figure. Determine the its stability.

Overall Review （ 12 ） Solution III Nyquist criterion Solution Let

Overall Review （ 13 ） Solution IV Logarithm criterion Solution Sketch Bode diagram

Overall Review （ 14 ） Example 3 Consider the system shown in the figure. Analyze the impact of K 1, K 2, and  on the performance. III. Performance analysis Method I Time domain analysis Solution --

Overall Review （ 15 ） Solution （ -- ） Example 3 Consider the system shown in the figure. Analyze the impact of K 1, K 2, and  on the performance.

Overall Review （ 16 ） Method II Root-locus method Solution (--)

Overall Review （ 17 ） Method III Frequency domain method Solution Lower band Oscillation  Lower band not changed Higher band ↑, anti-noise ↓ not changed Bending frequency  Lower band B.F.  Higher band↑, Anti-high-frequency noise ↓

Automatic Control Theory 祝同学们 考出好成绩！ Good Luck!