Transcript 1.1 Why Nonlinear Control？

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
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Office Room: 工學大樓六樓
Telephone: Ext. 3221
E-mail: [email protected]
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,
Springer-Verlag, 1999.
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
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.
• 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.
• 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
x  Ax
with x being a vector of states and A being the
system matrix.
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
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 x  0 ).
Example 1.2: A first-order system
x  x  x
2
Its linearization around x(t )  0 is
x  x
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.
- Integrating equation dx/(－x + x2)=dt
t
x0e
x(t ) 
t
1  x0  x0e
-Tow equilibrium points, x = 0 and x = 1.
- Qualitative behavior strongly depends on its
initial condition.
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 x0 ＜ 1 converges.
- Motions starting with x0 > 1 diverges.
Properties of LTI Systems:
In the presence of an external input u(t), i. e.,
with
x  Ax  Bu
-Principle of superposition.
-Asymptotic stability implied BIBO stability
in the presence of u.
Stability of Nonlinear Systems May Depend
on Input Values:
A bilinear system
x  xu
u  1 , converges.
u  1 , diverges.
II. Limit Cycles
-Oscillations of fixed amplitude and fixed
period without external excitation.
Example 1.3: Van der Pol Equation
mx  2c( x 1) x  kx  0
2
where m, c and k are positive constants.
- 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.
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.
II. Limit Cycles (continued)
-Oscillations of fixed amplitude and fixed
period without external excitation.
Example 1.4:
x  ( x  x 1) x  x  0
2
2
•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.
-Topic of bifurcation theory: Quantitative
change of parameters leading to qualitative
change of system properties.
- Undamped Duffing equation
x  x  x  0
3
(the damped Duffing Equation is
x  cx   x   x  0
3
, which may represent a mass-damper-spring
system with a hardening spring).
- As  varies from + to -, one equilibrium
point splits into 3 points (xe   ,   ,0), as
shown in Figure 1.5(a).
  0 is a critical bifurcation value.
Figure 1.5: (a) a pitchfork bifurcation
(b) a Hopf bifurcation
•Common Nonlinear System Behaviors
IV. Chaos
-The system output is extremely sensitive to
initial conditions.
-Essential feature: the unpredictability of the
system output.
•Simple Nonlinear system
x  0.1x  x  6sin t
-Two almost identical initial conditions,
Namely x(0)  2, x(0)  3 , and
5
x(0)  2.01, x(0)  3.01.
- The two responses are radically different
after some time.
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