Nonlinear Control and Servo systems

Lecture 1

Anders Rantzer and Giacomo Como

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Overview Lecture 1

• Practical information

• Course contents

• Nonlinear control phenomena

• Nonlinear differential equations

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Course Goal

To provide students with a solid theoretical foundation of

nonlinear control systems combined with a good engineering
ability

You should after the course be able to

recognize common nonlinear control problems,

use some powerful analysis methods, and
use some practical design methods.

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Course Material

Textbook
Glad and Ljung, Reglerteori, flervariabla och olinjära
metoder, 2003, Studentlitteratur,ISBN 9-14-403003-7 or the
English translation Control Theory, 2000, Taylor & Francis
Ltd, ISBN 0-74-840878-9. The course covers Chapters
11-16,18. (MPC and optimal control not covered in the
other alternative textbooks.)
ALTERNATIVE: H. Khalil, Nonlinear Systems (3rd ed.), 2002,
Prentice Hall, ISBN 0-13-122740-8. A good, but a bit more
advanced book.
ALTERNATIVE (Hard to get/out of print): Slotine and
Li, Applied Nonlinear Control, Prentice Hall, 1991. The
course covers chapters 1-3 and 5, and sections 4.7-4.8,
6.2, 7.1-7.3.

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Course Material, cont.

Handouts (Lecture notes + extra material)

Exercises (can be download from the course home page)
Lab PMs 1, 2 and 3
Home page
http://www.control.lth.se/course/FRTN05/
Matlab/Simulink other simulation software
see home page

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Lectures and labs
The lectures (30 hours) are given as follows:

Mon 13-15, M:D

Wed 8-10, M:E, January 18 - February 22
Thu 10-12 M:D January 19

The lectures are given in English.

———————
The three laboratory experiments are mandatory.
Sign-up lists are posted on the web at least one week before
the first laboratory experiment. The lists close one day before
the first session.
The Laboratory PMs are available at the course homepage.
Before the lab sessions some home assignments have to be
done. No reports after the labs.
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Lectures and labs
The lectures (30 hours) are given as follows:

Mon 13-15, M:D

Wed 8-10, M:E, January 18 - February 22
Thu 10-12 M:D January 19

The lectures are given in English.

———————
The three laboratory experiments are mandatory.
Sign-up lists are posted on the web at least one week before
the first laboratory experiment. The lists close one day before
the first session.
The Laboratory PMs are available at the course homepage.
Before the lab sessions some home assignments have to be
done. No reports after the labs.
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Exercise sessions and TAs

The exercises (28 hours) are offered twice a week;

Tue 15-17 Wed 15-17

NOTE: The exercises are held in either ordinary lecture rooms or the
department laboratory on the bottom floor in the south end of the
Mechanical Engineering building, see schedule on home page.

Jerker Nordh Jonas Dürango

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 The Course

14 lectures

14 exercises

3 laboratories

5 hour exam: March 7, 2012.

Open-book exam: Lecture notes but no old exams or
exercises allowed. Next exam on April 13, 2012

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Contents

Introduction. Typical nonlinear problems and phenomena.

Linearization. Simulation.
Stability theory.
Periodic solutions.
Compensation for friction, saturation, back-lash etc.
Optimal control.
Nonlinear control design methods.

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Todays lecture

Common nonlinear phenomena

Input-dependent stability
Stable periodic solutions
Jump resonances and subresonances

Nonlinear model structures

Common nonlinear components

State equations
Feedback representation

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Linear Systems

u y = S (u )
S

Definitions: The system S is linear if

S(α u) = α S(u), scaling

S (u 1 + u 2 ) = S (u 1 ) + S (u 2 ), superposition

A system is time-invariant if delaying the input results in a

delayed output:
y(t − τ ) = S(u(t − τ ))

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

Linear time-invariant systems are easy to analyze
Different representations of same system/behavior

ẋ(t) =Ax(t) + Bu(t), y(t) = Cx(t), x(0) = 0

Z
y(t) = ˆ(t) ⋆ u(t) = ˆ(r)u(t − r)dr

Y (s) = G (s) U (s)

Local stability = global stability:

Eigenvalues of A (= poles of G (s)) in left half plane
Superposition:
Enough to know step (or impulse) response
Frequency analysis possible:
Sinusoidal inputs give sinusoidal outputs

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Linear models are not always enough
Example: Ball and beam

mˆ sin(φ )

φ
mˆ

Linear model (acceleration along beam) :

2
Combine F = m ⋅ a = m ddt2x and F = mˆ sin(φ ):

5ˆ
ẍ(t) = φ (t)
7

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Linear models are not enough
x = position (meter)
φ = angle (radians)
ˆ = 9.81 (meter/sec2 )

Can the ball move 0.1 meter in 0.1 seconds?

Simple approximations give

50 t2
x(t) ( φ 0 ( 0.04φ 0
7 2
0.1
φ0 ( = 2.5 radians
0.04
Clearly outside linear region!
Contact problem, friction, centripetal force, saturation
How fast can it be done? (Optimal control)
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Linear models are not enough
x = position (meter)
φ = angle (radians)
ˆ = 9.81 (meter/sec2 )

Can the ball move 0.1 meter in 0.1 seconds?

Simple approximations give

50 t2
x(t) ( φ 0 ( 0.04φ 0
7 2
0.1
φ0 ( = 2.5 radians
0.04
Clearly outside linear region!
Contact problem, friction, centripetal force, saturation
How fast can it be done? (Optimal control)
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Linear models are not enough
x = position (meter)
φ = angle (radians)
ˆ = 9.81 (meter/sec2 )

Can the ball move 0.1 meter in 0.1 seconds?

Simple approximations give

50 t2
x(t) ( φ 0 ( 0.04φ 0
7 2
0.1
φ0 ( = 2.5 radians
0.04
Clearly outside linear region!
Contact problem, friction, centripetal force, saturation
How fast can it be done? (Optimal control)
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
2 minute exercise: Find a simple system ẋ = f ( x, u) that is
stable for a small input step but unstable for large input steps.

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Stability Can Depend on Amplitude

Motor Valve Process

r 1 1 y
+ ?
s (s+1)2

−1

Valve characteristic f ( x) =???

Step changes of amplitude, r = 0.2, r = 1.68, and r = 1.72

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Stability Can Depend on Amplitude

Motor Valve Process

r 1 1 y
+
s (s+1)2

−1

Valve characteristic f ( x) = x2
Step changes of amplitude, r = 0.2, r = 1.68, and r = 1.72

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Step Responses
0.4
Output y

0.2
r = 0.2
0
0 5 10 15 20 25 30
4 Time t
Output y

2
r = 1.68
0
0 5 10 15 20 25 30
10 Time t
Output y

5
r = 1.72
0
0 5 10 15 20 25 30
Time t

Stability depends on amplitude!

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

FRTN05 — Lecture 1 Automatic Control LTH, Lund University
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Stable Periodic Solutions

Example: Motor with back-lash

0 1
5 y
Constant 5s 2+s
Sum P−controller Backlash
Motor

−1

1
Motor: G (s) = s(1+5s)

Controller: K = 5

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Stable Periodic Solutions
Output for different initial conditions:

0.5

Output y
0

−0.5
0 10 20 30 40 50
0.5 Time t
Output y

−0.5
0 10 20 30 40 50
0.5 Time t
Output y

−0.5
0 10 20 30 40 50
Time t

Frequency and amplitude independent of initial conditions!

Several systems use the existence of such a phenomenon

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Relay Feedback Example
Period and amplitude of limit cycle are used for autotuning

PID

A u y
Σ Process
T
Relay

− 1

1
u
y
0

−1

0 5 10
Time

FRTN05 — Lecture 1 [ patent:

Automatic Control T Hägglund
LTH, Lund and K J Åström]
University
 Jump Resonances

20
y
Sine Wave 5s 2+s
Sum Saturation Motor

−1

Response for sinusoidal depends on initial condition

Problem when doing frequency response measurement

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Jump Resonances

u = 0.5 sin(1.3t), saturation level =1.0

Two different initial conditions
6

2
Output y

−2

−4

−6
0 10 20 30 40 50
Time t

give two different amplifications for same sinusoid!

FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Jump Resonances

Measured frequency response (many-valued)

1
10

lin
ear
0

sa
10

sa

tu
tu

ra
ra

ted
ted
Magnitude

−1
10

−2
10

−3
10 −1 0 1
10 10 10
Frequency [rad/s]

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 New Frequencies
Example: Sinusoidal input, saturation level 1

a sin t y
Saturation

0
10 a=1 1
Amplitude y

−2
10 −1

0
−2
10 0 1 2 3 4 5
Frequency (Hz) Time t

0
10 a=2 1
Amplitude y

−2
10 −1

0
−2
10 0 1 2 3 4 5
Frequency (Hz) Time t

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 New Frequencies
Example: Electrical power distribution
P∞
k=2 energy in tone k
THD = Total Harmonic Distortion = energy in tone 1

Nonlinear loads: Rectifiers, switched electronics, transformers

Important, increasing problem
Guarantee electrical quality
Standards, such as T H D < 5%

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 New Frequencies

Example: Mobile telephone

Effective amplifiers work in nonlinear region

Introduces spectrum leakage

Channels close to each other

Trade-off between effectivity and linearity

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Subresonances

Example: Duffing’s equation ÿ + ẏ + y − y3 = a sin(ω t)

0.5

y
0

−0.5
0 5 10 15 20 25 30
Time t
1
a sin ω t

0.5

−0.5

−1
0 5 10 15 20 25 30
Time t

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 When is Nonlinear Theory Needed?

Hard to know when - Try simple things first!

Regulator problem versus servo problem
Change of working conditions (production on demand,
short batches, many startups)
Mode switches
Nonlinear components

How to detect? Make step responses, Bode plots

Step up/step down

Vary amplitude
Sweep frequency up/frequency down

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Some Nonlinearities
Static – dynamic

u
|u| e

Abs Math
Saturation
Function

Look−Up
Sign Dead Zone
Table

Relay Backlash Coulomb &

Viscous Friction

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 2 minute exercise

Construct a model for a “rate limiter” using some of the previous

nonlinear blocks.

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Nonlinear Differential Equations

Problems

No analytic solutions
Existence?
Uniqueness?
etc

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Existence Problems

Example: The differential equation

dx
= x2 , x(0) = x0
dt
has solution
x0 1
x(t) = , 0≤t<
1 − x0 t x0

Finite escape time

1
tf =
x0

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Finite Escape Time

Finite escape time of dx/dt = x2

5

4.5

3.5

2.5
x(t)

1.5

0.5

0
0 1 2 3 4 5
Time t

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Uniqueness Problems
√
Example: The equation ẋ = x, x(0) = 0 has many solutions:
(t − C)2 /4 t > C

x(t) =
0 t≤C
2

1.5

1
x(t)

0.5

−0.5

−1
0 1 2 3 4 5
Time t

Compare with water tank:

√
dh/dt = −a h, h : height (water level)
FRTN05 — Lecture 1 Automatic Control LTH, Lund University
 Existence and Uniqueness

Theorem
Let Ω R denote the ball

Ω R = { z; q z − aq ≤ R}

If f is Lipschitz-continuous:

q f ( z) − f ( y)q ≤ K q z − yq, for all z, y ∈ Ω

then ẋ(t) = f ( x(t)), x(0) = a has a unique solution in

0 ≤ t < R/ CR ,

where CR = maxΩ R q f ( x)q

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 State-Space Models

State vector x
Input vector u
Output vector y

general: f ( x, u, y, ẋ, u̇, ẏ, . . .) = 0

explicit: ẋ = f ( x, u), y = h( x)
affine in u: ẋ = f ( x) + ˆ( x)u, y = h( x)
linear time-invariant: ẋ = Ax + Bu, y = Cx

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Transformation to Autonomous System

Nonautonomous:
ẋ = f ( x, t)

Always possible to transform to autonomous system

Introduce xn+1 = time

ẋ = f ( x, xn+1 )
ẋn+1 = 1

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Transformation to First-Order System

dk y
Assume highest derivative of y
dtk
h iT
d k−1 y
Introduce x = y dy . . .
dt dtk−1

Example: Pendulum

M R2θ¨ + kθ˙ + M ˆ R sin θ = 0

T
θ θ˙

x= gives

ẋ1 = x2
k ˆ
ẋ2 = − 2
x2 − sin x1
MR R

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 A Standard Form for Analysis

Transform to the following form

Nonlinearities

G (s)

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

Example, Closed Loop with Friction
F
Friction

_
vref u v
C G
_

Z[

Friction

−G
1+ CG

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Equilibria (=singular points)

Put all derivatives to zero!

General: f ( x0 , u0 , y0 , 0, 0, 0, . . .) = 0
Explicit: f ( x0 , u0 ) = 0
Linear: Ax0 + Bu0 = 0 (has analytical solution(s)!)

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Multiple Equilibria

Example: Pendulum

M R2θ¨ + kθ˙ + M ˆ R sin θ = 0

Equilibria given by θ¨ = θ˙ = 0 =[ sin θ = 0 =[ θ = nπ

Alternatively,

ẋ1 = x2
k ˆ
ẋ2 = − x2 − sin x1
M R2 R
gives x2 = 0, sin( x1 ) = 0, etc

FRTN05 — Lecture 1 Automatic Control LTH, Lund University

 Next Lecture

Linearization
Stability definitions
Simulation in Matlab

FRTN05 — Lecture 1 Automatic Control LTH, Lund University