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

2. Automatic Control Theory Exercises (36) 7 — 1,4

3. Automatic Control Theory （ Lecture 36 ） §7 Nonlinear Systems §7.1 Introduction §7.2 Phase Plane Method §7.3 Describing Function Method §7.4 Methods to Improve the Performance of Nonlinear Control System

4. Automatic Control Theory （ Lecture 36 ） §7 Nonlinear Systems §7.1 Introduction to Nonlinear Control System §7.2 Phase Plane Method

5. §7 Nonlinear Systems （ 1 ） §7.1 Introduction to Nonlinear Control System §7.1.1 Nonlinearities in Physics Systems Nonlinearity is the universal law in the universe There are a lot of kinds of nonlinear systems and responses. The linear model is the approximate description of practical systems under the specific conditions. §7.1.2 Typical Nonlinear Factors in Control Systems Saturation Dead Zone Clearance Relay characteristic

6. §7 Nonlinear Systems （ 2 ） §7.1.3 Characteristics of Nonlinear Control System (1) Does not satisfy Superposition principle—The linear theory does not apply. (2) Stability — Not only depends on the structure and parameters, but also the input and initial condition. The equilibriums may not be unique. (3) Self-Excited Oscillation — The unique motion of nonlinear systems (4) Complex in frequency response — Frequency hopping ， frequency division/double Frequency, chaos. §7.1.4 Methods to Analyze Nonlinear Control System (1) Linearization by Taylor’s Expansion (2) The research method for nonlinear system (3) Simulation method: Digital simulation, Hardware-in-loop simulation Phase Plane Describing function Popov method Feedback linearization Differential geometry method

7. §7 Nonlinear Systems （ 3 ） Analysis of nonlinear characteristics Saturation Dead zone relay characteristic Nonlinear Characteristics Equivalent K* Affection on the system Example Oscillation↓,s  ↓ Bounded tracking velocity Transistor Steady state error↑ Remove small gain disturbance Electromotor Restrain divergence Self-excited oscillation Switches

8. §7 Nonlinear Systems （ 4 ） Analysis of nonlinear characteristics Relay and its equivalent gain

9. §7.2 Phase Plane Method （ 1 ） §7.2.1 Phase Plane Phase Plane: Phase locus ： The track of the system variable and its derivative varing with time in the phase plane. Example 1 Unity feedback system (1) Phase plane and phase locus The phase plane, which can describe the state of system, is constructed by the system variable and its derivative ( )

10. §7.2 Phase Plane Method （ 2 ） (2) Features of phase locus For linear time-invariant system, the origin is the unique equilibrium point. The direction of movement When the phase locus intersects with x axis, it always passes through with an angle of 90° Singular point (Equilibrium point): Suppose the system equation is ： Points on the phase locus with uncertain slop upper half plane — moving to the right Clockwise movement under half plane — moving to the left

11. §7.2 Phase Plane Method （ 3 ） Example 2 Consider the system Sketch the phase locus for the system Solution: — Elliptic Equation

12. §7.2 Phase Plane Method （ 4 ） Location of poles (3) Phase locus of second order linear systems Singular point Phase locus center point stable focus stable node saddle point unstable focus unstable node Location of poles Singular point Phase locus

13. §7.2 Phase Plane Method （ 5 ） Example 3 Consider the system Obtain the equilibriums x e and determine the characteristic of phase locus around the equilibriums Solution. Let Unstable focus By linearization Characteristic equation Saddle point

14. §7.2 Phase Plane Method （ 6 ） Solution. Let When Characteristic equation Linearization Center Point Saddle Point Example 4 Consider the system. Obtain the equilibriums x e and determine the characteristic of phase locus around the equilibriums

15. §7.2 Phase Plane Method （ 7 ） Example 5 Consider the system. Analyze its free response. Solution. Characteristic equation Stable focus Saddle point Analyze a class of nonlinear systems by the phase locus of 2nd order systems. Singular point Poles Switch Line

16. §7.2 Phase Plane Method （ 8 ） Solution. Characteristic equation Center Point Singular point Poles Center Point Switch line — The boundary line to divide different linear area. Equilibrium line (Singular line) — Generated by the interaction between phase locus in different area. Example 6 Consider the system. Analyze its free response.

17. Summary 7.1 Introduction to Nonlinear Control Systems 7.1.1 Nonlinearities in Physics Systems 7.1.2 Typical Nonlinear Factors in Control Systems 7.1.3 Characteristics of Nonlinear Control System 7.1.4 Methods to Analyze Nonlinear Control System 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)

18. Automatic Control Theory Exercises (36) 7 — 1,4