Fractional System Toolbox for Matlab - mechatronics

BORDEAUX 1 UNIVERSITY - ENSEIRB Engineering School UMR 5131 CNRS
CONTROL engineering, CIM, SIGNAL&IMAGE processing
41st IEEE Conference on Decision and Control
Tutorial Workshop
"Fractional Calculus Applications in Automatic Control and Robotics"
December 9, 2002, Las Vegas, Nevada, USA
Fractional systems toolbox for Matlab: Matlab
applications in System Identification
Dr Olivier COIS, Dr Pierre MELCHIOR, Dr Patrick LANUSSE,
Frédéric DANCLA and Pr Alain Oustaloup
LAP - UMR 5131 CNRS
Université Bordeaux 1 - ENSEIRB
351 cours de la Libération - F33405 TALENCE Cedex - FRANCE
Tél. : +33 (0)5 56 84 66 07 - Fax : +33 (0)5 56 84 66 44
Email : melchior@lap.u-bordeaux.fr - URL : http://www.lap.u-bordeaux.fr

A new CAD toolbox: toolbox:
"The CRONE toolbox: toolbox
Fractional Systems Toolbox"
Toolbox
• Applications
• Concept
The original theoretical mathematical concepts developed in
our laboratory are based on fractional or non integer
differentiation.
CRONE control (French abbreviation for Non Integer Order
Robust Control) and System Identification by fractional model
are typical applications.
•Aims
•Facilitate transfer of concepts to :
- engineers, researchers in mathematics and engineering sciences,
- universities,
- industrials.
•Make our research results widely available and develop international relations
• Identification & Robust control (electromechanical, thermal, pneumatic,
hydraulic systems, …)
• Electronic (filter synthesis),
• Path planning : Path description (cutting table) & Path tracking
• Signal and image processing, ...

1. System Identification
Content
A new tool for mathematical models of physical systems
Principles of parameter estimation methods
Matlab Toolbox
Application example

A new tool for mathematical models
of physical systems
What is exactly a fractional derivative ?
⎛
⎜
⎝
n
d ⎞
⎟ f
dt⎠
() t
d n
⎛ ⎞
⎜ ⎟
⎝ dt ⎠
f
() t
Mathematical viewpoint
Time domain Operational domain
• Riemann Liouville definition
∆
=
Γ
( m − n)
• Complicated formula
1
m
⎛ d ⎞ ⎛ t
⎜ ⎟ ⎜
⎝ dt⎠
⎜∫0
−
⎝
Laplace transform
( )
( ) ( ) ⎟ f τ ⎞
dτ
−
1−
m n
t τ
• Global operator: takes into account
all the values of function f(t)
⎠
Magnitude (dB)
Phase (deg)
-50
s F()
s
n
• Expression simpler than
in the time domain
50
0
200
10 -1
0
-200
10 -1
n=-1.5
n=-1
n=-0.5
n=0
n=0.5
n=1
n=1.5
Bode diagrams
10 0
Frequency (rad/s)
10 0
Frequency (rad/s)
10 1
10 1

A new tool for mathematical models
of physical systems
Fractional differential equation
n
⎛ d ⎞
a0⎜
⎟
⎝ dt
⎠
a0
y
n
⎛ d ⎞
1⎜
⎟
⎝ dt
⎠
a1
n
⎛ d ⎞
L⎜
⎟
⎝ dt
⎠
aL
n
⎛ d ⎞
0⎜
⎟
⎝ dt
⎠
n
⎛ d ⎞
1⎜
⎟
⎝ dt
⎠
n
⎛ d ⎞
M ⎜ ⎟
⎝ dt
⎠
() t + a y()
t + ... + a y()
t = b u()
t + b u()
t + ... + b u()
t
Fractional state space representation
⎧
⎛ ⎞
⎪⎜
⎟
⎨⎝
dt
⎠
⎪
⎩y
d n
x
() t = A x()
t + B u()
t
() t = C x()
t + D u()
t
Mathematical viewpoint
b0
b1
•Modal decomposition
•Output analytical expression
•Stability properties
bM

A new tool for mathematical models
of physical systems
Thermal system studies
Semi-infinite medium
• Diffusive equation
→ x = 0
⎛ d ⎞
⎜ ⎟
⎝ dt
⎠
0.
5
→ x > 0 : series expansion
⎛
⎜
⎝
d
dt
⎞
⎟
⎠
0
. 5
T
T
∂T
∂t
1
λ ρ C
( 0,
t)
= φ()
t
1
λ ρ C
∑ ∞
n=
0
( x,
t)
=
c φ()
t
p
( x,
t)
= c ( x,
t)
n
∂
2
∂x
T
p
2
⎛ d ⎞
⎜ ⎟
⎝ dt
⎠
n
2
Physical viewpoint
φ(t)
0
T(x,t)
Semi-infinite medium
x

A new tool for mathematical models
of physical systems
T
Other geometries : fractional differential equation
0.
5
1
0.
5M
0.
5 N
() t + a D T () t + a D T () t + ... + a D T () t = b φ()
t + b D φ()
t + ... + b D φ()
t
1
2
M
Physical viewpoint
Studies of various geometries (cylindrical, spherical …) show that fractional
differential equations, with differentiation orders being multiples of 0.5, permit better
approximations of analytical solutions than classical differential equations
The result can be extended to all the physical systems governed by
- the diffusive partial differential equation (electrochemical systems,
electromagnetic systems)
- the Telegraphist partial differential equation (long transmission cables…)
Preliminary conclusion: fractional differentiation is an efficient tool
for modeling some real physical systems
0
1
0.
5
N

1.5
1
0.5
0
y
n
⎛ d ⎞
1⎜
⎟
⎝ dt
⎠
a1
System Identification
Goal: establish a mathematical model capable of reproducing the system's
physical behavior as faithfully as possible from a series of observations
u ( t)
y(
t)
Real physical System
0 20 40 60 80
Parameter estimation method
n
⎛ d ⎞
L⎜
⎟
⎝ dt
⎠
aL
n
⎛ d ⎞
0⎜
⎟
⎝ dt
⎠
n
⎛ d ⎞
1⎜
⎟
⎝ dt
⎠
n
⎛ d ⎞
M ⎜ ⎟
⎝ dt
⎠
() t + a y()
t + ... + a y()
t = b u()
t + b u()
t + ... + b u()
t
b0
b1
3
2
1
0
-1
0 20 40 60 80
bM

System Identification:
parameter estimation methods
for fractional models
y
n
⎛ d ⎞
1⎜
⎟
⎝ dt
⎠
a1
linearity with respect to coefficients
non-linearity with respect to differentiation orders
2 types of estimation methods
Only coefficients are estimated
orders are fixed by the user
Equation error method
n
⎛ d ⎞
L⎜
⎟
⎝ dt
⎠
linear optimisation techniques
aL
Least Square, Instrumental Variable...
Fractional models
n
⎛ d ⎞
0⎜
⎟
⎝ dt
⎠
n
⎛ d ⎞
M ⎜ ⎟
⎝ dt
⎠
() t + a y()
t + ... + a y()
t = b u()
t + ... + b u()
t
b0
Both orders and coefficients
are estimated
Output error method
non-linear optimisation techniques
Gradient, Marquardt ...
bM

Matlab Toolbox

Application example
Application Field: Machining by turning
The efficency of a machining depends
on the heat flux through the tool
It cannot be measured directly
during machining
Tools Temperature, Flux ?
How to estimate the heat flux through the tool during machining

2 stages
Strategy
Heat surface
Thermocouple
T
Insert tool
Sensor
Tool holder
1. Before machining: Identify the thermal dynamic behavior between
- heat flux through the tool
- temperature close to the tip of the tool
2. During machining: Estimate the heat flux through the tool using:
- the identified thermal model
- temperature measurments close to the tip of the tool
q

T
Identification of the thermal dynamic behavior of the tool
Heat resistor
Model structure choice
thermocouple embeded in the tool
Heat surface
q
Thermocouple
T
Thermal system Fractional model
0.
5
1
0.
5M
Insert tool
Sensor
Tool holder
0.
5 N
() t + a D T () t + a D T () t + ... + a D T () t = b φ()
t + b D φ()
t + ... + b D φ()
t
1
2
M
0
1
0.
5
N

T
Power supply
1.5
1
0.5
Identification of the thermal dynamic behavior of the tool
0
0 20 40 60 80
Thermocouple
3
2
1
0
-1
0 20 40 60 80
Heat flux Temperature
0.
5
1
0.
5M
0.
5 N
() t + a D T () t + a D T () t + ... + a D T () t = b φ()
t + b D φ()
t + ... + b D φ()
t
1
2
Parameter estimation method
M
0
1
0.
5
N

Identification of the thermal dynamic behavior of the tool
Data acquisistion : estimation and validation data (Ts=0.1s)

Identification of the thermal dynamic behavior of the tool
Parameter estimation : 5 parameters
0.
5
1
() t + 10. 90 D T () t −1.
04 D T () t + 39.
81D
T () t = 6.
22φ()
t −1.
65D
() t
T φ
1.
5
0.
5

Identification du comportement thermique de l'outil
Model validation

Conclusion (Syst ( Syst. . Ident.) Ident.)
A further class of mathematical models, called fractional models, has been
developed and analyzed (stability theorem, output analytical expression…)
Various estimation methods of fractional models have been developed and
applied to the identification of real systems
A Matlab Toolbox has been developed
Fractional differentiation is shown to be an efficient tool for
modeling certain real systems, using few parameters
Application to the identification of electro-chemical systems (batteries)