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

2. Excecies(21) 5 —9, 10, 11, 12 (Use the Semilog coordinate paper for No. 9 and No. 12 ) Automatic Control Theory

3. （ Lecture 21) §5. Analysis and Design of Linear Systems in Frequency-Domain §5.1 Concept of Frequency-Response Characteristics §5.2 Amplitude-phase Frequency Characteristics §5.3 Bode Diagrams §5.4 Nyquist Stability Criterion §5.5 Stability Margins §5.6 System Analysis by Frequency Response Characteristics of Open-Loop Systems §5.7 Nichols Chart §5.8 System Analysis by Frequency Response Characteristics of Closed-Loop Systems §5.9 Control Systems Design by Frequency Response

4. Automatic Control Theory （ Lecture 21 ） §5.3 Bode Diagrams

5. Review Frequency characteristic of the typical link

6. §5.3.2 Bode Diagram For Open-loop Systems （ 1 ） §5.3.2 Bode Diagrams For Open-loop Systems

7. §5.3.2 Bode Diagram For Open-loop Systems （ 2 ） The steps to sketch Bode diagram for open-loop system (1) Rewrite the open-loop transfer function G(j  ) to the lowest “1” type. ⑵ Listing the turning frequency in turn. (4) Addition Plotting Example 1 0.2 First-Order Factor 0.5 Reciprocal First-Order Factor 1 Quadratic Factor Base Point Slope First-Order Factor -20 dB/Dec (Reciprocal) +20 dB/Dec  First-order Factor -20  Reciprocal First-Order Factor +20  Quadratic Factor -40 (3) Determine the base line The line on the left hand side of first turning frequency and its extension Quadratic Factor -40 dB/Dec (Reciprocal) +40 dB/Dec

8. §5.3.2 Bode Diagram For Open-loop Systems （ 3 ） ⑸ Correction ⑹ Check ① The turning frequencies of two first-order factors are close to each other ② When oscillation  (0.38, 0.8) ① The rightmost slope of L(  ) -20(n-m) dB/dec ② The no. of turning points =(no. of F.O.F.)+(no. R.F.O.F)+(no. of Q.F)+(no. of R.Q.F) ③  -90°(n-m) Base Point Slope  First-order Factor -20  Reciprocal First-Order Factor +20  Quadratic Factor -40

9. §5.3.2 Bode Diagram For Open-loop Systems （ 4 ） Base point Example 2. Sketch Bode diagram Solution. ① Standard form ② Turning frequencies ③ Base line ④ Plotting Slope ⑤ Check The rightmost slope of L(w) is -20(n-m) =0 The number of turning points = 3  tends to -90 º(n-m)=0º

10. §5.3.2 Bode Diagram For Open-loop Systems （ 5 ） Example 3 Sketch the Bode Diagram and the Nyquist Plot. Solution. ① Base line Point Slope ② ③ ④ Check The rightmost slope of L(  ) is -20(n-m) =-80dB/dec The number of turning points = 3  -90 o  n-m  =-360 o

11. §5.3.2 Bode Diagram For Open-loop Systems （ 6 ） Example 3 Sketch the Bode Diagram and the Nyquist Plot.

13. §5.3.2 Bode Diagram For Open-loop Systems （ 7 ） Example 4 Obtain G(s) from the Bode diagram. Solution. Solution Ⅱ Solution Ⅰ Solution Ⅲ Proof ：

14. §5.3.2 Bode Diagram For Open-loop Systems （ 8 ） Example 5 Obtain G(s) and sketch  and the Nyquist plot for given L(  ) of a minimum phase system. Solution ⑴ I II ⑵ Sketching  ⑶

15. §5.3.2 Bode Diagram For Open-loop Systems （ 9 ） Example 6 Obtain G(s) of a minimum phase system from  Solution. Remark:  does not depend on K.

16. §5.3.2 Bode Diagram For Open-loop Systems （ 10 ） Example 7 Consider the Bode diagram shown in the figure. Obtain G(s) Solution: From the Bode diagram, we have : Determine K:

17. §5.3.2 Bode Diagram For Open-loop Systems （ 11 ） ⑴ ⑵ ⑶ ⑷

18. §5.3.2 Bode Diagram For Open-loop Systems （ 12 ） Non-minimum phase system —There are open-loop zeros or poles in the right half s-plane ★ Non-minimum phase systems may not be unstable ★ Non-minimum phase systems may not have 0° root-locus For non-minimum phase systems, L(  ) could not determine an unique G(s) ★ For minimum phase systems, L(  ) determines an unique G(s) ★ The absolute value of phase angle variation of non-minimum phase system is usually greater than minimum phase system

19. Summary Base Point Slope The steps to sketch Bode diagram for open-loop system (1) Rewrite the open-loop transfer function G(s) to the lowest “1” type. ⑵ Listing the turning frequency in turn. (4) Addition Plotting First-Order Factor -20 dB/Dec (Reciprocal) +20 dB/Dec (3) Determine the base line The line on the left hand side of first turning frequency and its extension Quadratic Factor -40 dB/Dec (Reciprocal) +40 dB/Dec ⑸ Correction ⑹ Check ① The turning frequencies of two first-order factors are close to each other ② When oscillation  (0.38, 0.8) ① The rightmost slope of L(  ) -20(n-m) dB/dec ② The no. of turning points =(no. of F.O.F.)+(no. R.F.O.F)+(no. of Q.F)+(no. of R.Q.F) ③  -90°(n-m)

20. Excecies(21) 5 —9, 10, 11, 12 (Use the Semilog coordinate paper for No. 9 and No. 12 ) Automatic Control Theory

22. §5.3.2 Bode Diagram For Open-loop Systems （ 10 ） Example 7 The second-order closed-loop system is known (doesn't have closed-loop zeros). The closed-loop gain K   The delay of c s (t) phase angle is 90º when the input signal frequency f=5/π Hz (shown in figure). Determine the Φ(s). Solution According to the problem