非 線 性 控 制Nonlinear Control

林心宇長庚大學電機工程學系

2012 春

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

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, Spring

er-Verlag, 1999.

課程目標及背景需求 1. 介紹如何以 Phase Portrait 及 Lyapunov M

ethod 分析非線性系統穩定性。 2. 介紹 Feedback Linearization, Sliding Contr

ol 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

評 量 標 準 作業 (20%)

正式考試 2 次 ( 各 40%)

Chapter 1

Introduction

1.1 Why Nonlinear Control ？

A: To control nonlinear 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.

• Improvement of Existing Control Systems

• 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.

• 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.

• 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

1.2 Nonlinear System Behavior

•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.

•Linear Systems

Linear time-invariant (LTI) control systems, of the form

with x being a vector of states and A being the system matrix.

x Ax

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

•Common Nonlinear System Behaviors

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 ).

I. Multiple Equilibrium Points

0x

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.

2x x x

x x

( ) 0x t

- Integrating equation dx/( － x + x2)=dt

-Tow equilibrium points, x = 0 and x = 1.

- Qualitative behavior strongly depends on its

initial condition.

0

0 0

( )1

t

t

x ex t

x x e

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

Stability of Nonlinear Systems May Depend on Initial Conditions:

- Motions starting with ＜ 1 converges.

- Motions starting with > 1 diverges.

0x

0x

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.

x Ax Bu

Stability of Nonlinear Systems May Depend on Input Values:

A bilinear system

, converges.

, diverges.

x xu1

1

u

u

-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.

II. Limit Cycles

22 ( 1) 0mx c x x kx

- 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.

-Neither grow unboundedly nor decay to zero.

- Oscillate independent of initial conditions.

- Limit cycle (case for m=1, c=1 and k=1)

The trajectories starting from both outside and inside converge to this curve.

Figure 2.8:Phase portrait of the Van der Pol equation

-Oscillations of fixed amplitude and fixed

period without external excitation.

Example 1.4:

II. Limit Cycles (continued)

2 2( 1) 0x x x x x

-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.

•Common Nonlinear System Behaviors

III. Bifurcations

-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).

3 0x x x

3 0x cx x x

- As varies from + to -, one equilibrium point splits into 3 points ( ), as shown in Figure 1.5(a).

is a critical bifurcation value. , ,0ex

0

Figure 1.5: (a) a pitchfork bifurcation

(b) a Hopf bifurcation

-The system output is extremely sensitive to

initial conditions.

-Essential feature: the unpredictability of the

system output.

•Common Nonlinear System Behaviors

IV. Chaos

•Simple Nonlinear system

-Two almost identical initial conditions,

Namely , and

- The two responses are radically different after some time.

50.1 6sinx x x t

(0) 2, (0) 3x x

(0) 2.01, (0) 3.01.x x

Figure 1.6: Chaotic behavior of a nonlinear system

Outlines of this Course

I. Phase plane analysis

II. Lyapunov theory

III. Feedback linearization

IV. Sliding control

VI. Adaptive control