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

2. Automatic Control Theory Exercises (39) 7 — 11, 12, 14, 18 7 —16, 17 （ Optional ）

3. Review (1) (1) Stipulation (2) Stability analysis (3) SSO 1. Describing function 2. System analysis by describing functions ① N(A) and G(j  ) are in cascade. ② The input and the output of N(A) are symmetry about the origin. y(x)=-y(-x) ③ G(j  ) is a low-pass filter. not circling circling, crossing with unstable stable May be SSO Crossing in Crossing out Tangential to Non-SSO point Neither stable nor unstable SSO Point

4. Review (2) 4 Self-sustained oscillation Crossing in Crossing out Tangential to Non-SSO point Neither stable nor unstable SSO Point

5. Automatic Control Theory （ Lecture 39 ） §7 Nonlinear Systems §7.1 Introduction to Nonlinear Control System §7.2 Phase Plane Method §7.3 Describing Functions Method §7.4 Methods to Improve Nonlinear Control System Performances

6. Automatic Control Theory （ Lecture 39 ） §7 Nonlinear Systems §7.3 Describing Functions Method §7.4 Methods to Improve Nonlinear Control System Performances

7. §7.3.3 Describing Function （ 5 ） 4 Analysis of SSO Necessary condition of SSO: Example 1 Determine the stability of the system (M=1) and obtain the parameters of SSO. Solution. From the plot, there is SSO By the SSO condition: Thus: comparing the real/imaginary part

8. Analysis: Obtain the required SSI by adjusting K and  Solution By and comparing the magnitude and phase Example 2 Consider the system shown in the figure. Determine the value of K and  to obtain a periodic signal with 。 §7.3.3 Describing Function （ 6 ）

9. Example 3 Consider the system shown in the figure with (1)When the system is in SSO, choose the value of K for Determine the value of K, SSO parameters (A, w) and the magnitude of the output A c. (2) Analyze how (A, w) changes with K. Solution (1) (2) From diagram §7.3.3 Describing Function （ 7 ）

10. Example 4 Consider the system shown in the figure, where (1)When, if the system is in SSO? Obtain the range of K for SSO and the parameters of SSO when K=2 (2) Determine the system stability when G 3 (s)=s Solution. Transform the block diagram into the standard form Solution II By the characteristic equation Solution I By BD tran. §7.3.3 Describing Function （ 8 ）

11. By SSO condition: Solution (1) when G 3 (s)=1 imaginary part real part §7.3.3 Describing Function （ 9 ）

12. (2) When G 3 (s)= s System is stable Thus: §7.3.3 Describing Function （ 10 ） (1)When G 3 (s)=1, if the system is in SSO? Obtain the range of K for SSO and the parameters of SSO when K=2 (2) Determine the system stability when G 3 (s)=s

13. Example 5 Consider the nonlinear system shown in the figure. Determine if there is SSO and the stable range of the initial value A. Solution. By block diagram transformation we have G*(s) §7.3.3 Describing Function （ 11 ）

14. G*(j  ) from the stable area to the unstable area – not a SSO point From the plot, we know that the stable range of A is Let §7.3.3 Describing Function （ 12 ）

15. Solution. By block diagram transformation, we have Example 6 Consider the nonlinear system shown in the figure, where Analyze if there is SSO. If there is, obtain the magnitude and frequency of the output signal c(t). By SSO condition Comparing the real/imaginary part §7.3.3 Describing Function （ 13 ）

16. There is SSO with §7.3.3 Describing Function （ 14 ）

17. §7.4.1 Adjustments of Linear Part Example 2 Reducing the impact of nonlinear characteristics by local feedback Example 1 Change the parameters of linear part §7.4.2 Nonlinear Alteration Example 4 Improvement of backlash characteristics Example 3 Saturation +Dead zone §7.4.3 Utilities of Nonlinearities Example 5 Add in nonlinear factor for special purpose Example 6 Add in dead zone into tachometer feedback §7.4.1 Alteration and utility of nonlinear systems （ 15 ）

18. Summary 2. Analyze nonlinear systems by describing function method 1. Describing function Analysis Calculation (1) Stipulation (2) Stability analysis (3) SSO ① N(A) and G(j  ) are in cascade. ② The input and the output of N(A) are symmetry about the origin. y(x)=-y(-x) ③ G(j  ) is a low-pass filter. not circling circling, crossing with unstable stable May be SSO Crossing in Crossing out Tangential to Non-SSO point Neither stable nor unstable SSO Point

19. Automatic Control Theory Exercises (39) 7 — 11,12,14,18 7 —16,17 （ optional ）