非 線 性 控 制 Nonlinear Control

1. 非 線 性 控 制Nonlinear Control 林心宇 長庚大學電機工程學系 2012春

2. 教 師 資 料 • 教師：林心宇 • Office Room: 工學大樓六樓 • Telephone: Ext. 3221 • E-mail: shinylin@mail.cgu.edu.tw • Office Hour: 2:00 – 4:00 pm, Friday

3. 教 科 書 • Textbook: • Jean-Jacques E. Slotine and Weiping Li, Applied Nonlinear Control, Pearson Education Taiwan Ltd., 1991. • Reference: • Alberto Isidori, Nonlinear Control Systems, Springer-Verlag, 1999.

4. 課程目標及背景需求 • 1.介紹如何以Phase Portrait及Lyapunov Method分析非線性系統穩定性。 • 2.介紹Feedback Linearization, Sliding Control及Adaptive Control等方法設計非線性系統的控制器。 • 背景需求 • Linear System Theory • Elementary Differential Equations

5. 評 量 標 準 • 作業 (20%) • 正式考試 2 次 (各40%)

6. Chapter 1 Introduction

7. 1.1 Why Nonlinear Control？A: To control nonlinear systems.

8. Improvement of Existing Control Systems - Linear control methods rely on the key assumption of small range operation for the linear model to be valid. - Nonlinear controllers may handle the nonlinearities in large range operation directly, because the controller is designed for handling the nonlinear system directly.

9. Analysis of hard nonlinearities • Linear control assumes the system model is linearizable. • Hard nonlinearities: nonlinearities whose discontinuous nature does not allow linear approximation. • Coulomb friction, saturation, dead-zones, backlash, and hysteresis.

10. Dealing with Model Uncertainties • In designing linear controllers, we assume that the parameters of the system model are reasonably well known. • In real world, control problems involve uncertainties in the model parameters. • The model uncertainties can be tolerated in nonlinear control, because the uncertainty is taken into account in the controller design.

11. Design Simplicity • Good nonlinear controller designs may be simpler and more intuitive than their linear counterparts. • This result comes from the fact that nonlinear controller designs are often deeply rooted in the physics of the plants. • Example: pendulum

12. 1.2 Nonlinear System Behavior

13. Nonlinearities • Inherent (natural) : Coulomb friction between contacting surfaces. • Intentional (artificial): adaptive control laws. • Continuous • Discontinuous: Hard nonlinearities (backlash, hysteresis) cannot be locally approximated by linear function.

14. Linear Systems Linear time-invariant (LTI) control systems, of the form with x being a vector of states and A being the system matrix.

15. Properties of LTI systems • Unique equilibrium point if A is nonsingular • Stable if all eigenvalues of A have negative real parts, regardless of initial conditions • General solution can be solved analytically

16. Common Nonlinear System Behaviors I. Multiple Equilibrium Points Nonlinear systems frequently have more than one equilibrium point (an equilibrium point is a point where the system can stay forever without moving, i.e. a point where ).

17. Example 1.2: A first-order system Its linearization around is with solution x(t) = x(0)e－t: general solution can be solved analytically. • Unique equilibrium point at x = 0. • Stable regardless of initial condition.

18. - Integrating equation dx/(－x + x2)=dt • Tow equilibrium points, x = 0 and x = 1. • Qualitative behavior strongly depends on its initial condition.

19. Figure 3.1: Responses of the linearized system (a) and the nonlinear system (b)

20. Stability of Nonlinear Systems May Depend on Initial Conditions: • Motions starting with ＜1 converges. • Motions starting with > 1 diverges.

21. Properties of LTI Systems: In the presence of an external input u(t), i. e., with -Principle of superposition. -Asymptotic stability implied BIBO stability in the presence of u.

22. Stability of Nonlinear Systems May Depend on Input Values: A bilinear system , converges. , diverges.

23. II. Limit Cycles • Oscillations of fixed amplitude and fixed period without external excitation. Example 1.3: Van der Pol Equation where m, c and k are positive constants.

24. A mass-spring-damper system with a position-dependent damping coefficient 2c (x2-1) • For large x, 2c (x2-1)>0 : the damper removes energy from the system - convergent tendency. • For small x, 2c (x2-1)<0 : the damper adds energy to the system - divergent tendency.

25. Neither grow unboundedly nor decay to zero. - Oscillate independent of initial conditions.

26. Figure 2.8:Phase portrait of the Van der Pol equation - Limit cycle (case for m=1, c=1 and k=1) The trajectories starting from both outside and inside converge to this curve.

27. II. Limit Cycles (continued) • Oscillations of fixed amplitude and fixed period without external excitation. Example 1.4:

28. Common Nonlinear System Behaviors III. Bifurcations -As parameters changed, the stability of the equilibrium point can change. -critical or bifurcation values : Values of the parameters at which the qualitative nature of the system’s motion changes.

29. Topic of bifurcation theory: Quantitative change of parameters leading to qualitative change of system properties. - Undamped Duffing equation (the damped Duffing Equation is , which may represent a mass-damper-spring system with a hardening spring).

30. - As  varies from + to -, one equilibrium point splits into 3 points ( ), as shown in Figure 1.5(a). is a critical bifurcation value.

31. Figure 1.5: (a) a pitchfork bifurcation (b) a Hopf bifurcation

32. Common Nonlinear System Behaviors IV. Chaos • The system output is extremely sensitive to initial conditions. • Essential feature: the unpredictability of the system output.

33. Simple Nonlinear system • Two almost identical initial conditions, Namely , and - The two responses are radically different after some time.

34. Figure 1.6: Chaotic behavior of a nonlinear system

35. Outlines of this Course • Phase plane analysis • II. Lyapunov theory • III. Feedback linearization • IV. Sliding control • VI. Adaptive control