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

2. Automatic Control Theory Exercises (38) 7 — 9, 10

3. Review 7.2 Phase Plane Method 7.2.1 Phase Plane (1) Phase plane and phase locus (2) Features of phase locus (The direction of movement, Singular point, Singular line, Switched line ) (3) Phase locus of the second order linear system ( Analyze the free response a class of nonlinear systems) 7.2.2 Isoclines Method for Plotting Phase Locus 7.2.3 Phase Plain Analysis of Nonlinear Control Systems

4. Automatic Control Theory （ Lecture 38 ） §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

5. Automatic Control Theory （ Lecture 38 ） §7 Nonlinear Systems §7.3 Describing Functions Method

6. §7.3 Describing Functions Method (1) §7.3.1 Describing Functions (1) The Fourier series expansion for periodic function y(t)

7. §7.3 Describing Functions Method (2) (2) Definition of describing functions The fundamental wave of the output: Input ： Definition of describing functions N(A) The describing functions of ideal relays

8. §7.3 Describing Functions Method (2) Describing functions for relays

9. §7.3.2 Describing Functions of Typical Nonlinear Factors (1) Example The describing function of relays.

10. §7.3.2 Describing Functions of Typical Nonlinear Factors Describing functions for relays Ideal relays Relays with dead zone Relays with hysteresis Loop Generally, the describing function N(A) is a function of A, and does not depend on  If the nonlinear function is monotropic/non-monotropic, the N(A) is a real/complex function. 非线性环节为单 / 非单值函数时,N(A) 是实 / 复数, 虚部为 / 不为 0.

11. §7.3.2 Describing Functions of Typical Nonlinear Factors (1) 1. Stipulations ① N(A) and G(j  ) are in cascade. ② The input and the output of N(A) are symmetrical about the origin. y(x)=-y(-x) ③ G(j  ) is a low-pass filter. 2 Stability Analysis Example 1 The negative reciprocal describing functions of ideal relays 3 not encircling encircling, crossing with stable unstable may be SSO

12. §7.3.2 Describing Functions of Typical Nonlinear Factors (2) Example 2 The negative reciprocal describing function of relays with hysteresis loop 3

13. §7.3.2 Describing Functions of Typical Nonlinear Factors (3) 4 Self-sustained oscillation Crossing in Crossing out Tangential to Non-SSO point Neither stable nor unstable SSO Point

14. §7.3.2 Describing Functions of Typical Nonlinear Factors (4) Self-sustained oscillation

15. Summary (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

16. Automatic Control Theory Exercises (38) 7 — 9, 10