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Modelling and Simulation of Mechatronic Systems
Prof.Ing.Petr Noskievi, CSc.
Department of Automatic Control and Instrumentation
VB Technical University of Ostrava
Modelling and Simulation of Mechatronic Systems
Noskievi, P.: Identifikace a modelovn. Montanex a.s., Ostrava, 1999. ISBN 80-7225-030-2
Close,Charles, M., Frederick,K.: Modeling and Analysis of Dynamic Systems. John Wiley & Sons, Inc. New York.
Modelling and Simulation of Mechatronic Systems
Mathematical modelling is an effective method for investigation of the properties of real objects. The realization of the mathematical models using computers the system simulation have become a very important part of the design process of the complex systems. Using the computer simulation we can do experiments with the mathematical model in the similar way like with the real system, but without risk of the crash states, without the real object, with lower costs.
The development of the computers and simulation software contributed to the wide use of the system simulation. This fact underlines the need of the new skills methods of creating the mathematical models mathematical modelling and system identification.
Modelling and Simulation of Mechatronic Systems
The approach for creating the mathematical model is called system identification and can be divided into two groups of methods: Analytical identification also called mathematical modelling is based on the use of physical laws. Experimental identification based on the evaluation of the data from the realized experiment with the real system.
Summary of the identification methods
Modelling
Transfer function
Experimental Identification
Mechanical System
Electrical system
Use of physical Laws Newtons law Krchhoff law Etc.
Parameterization of the transfer function
Bode plot measurement and evaluation
Parameterization of the transfer function in frequence domain
Transfer function Deterministic methods Stochastic methods
Bode plot
Other methods of parameterization
Bode plot computation from Transfer function
Stochastic model of the system
Numerical deconvolution
Cross-correlation function
Basic terms
Modelling is an experimental process in which the physical or abstract model is using the specific criterion defined to the real discovered object - the machine the modelled system. Modelling is one of the oldest methods of discovering the real world, which at the beginning used only the imitation of the of the phenomenon in the nature and it was later developed into the modelling using the principle of the geometric similarity.
Geometric similarity, physical model
Geometric similarity: the model has the same shape, keeps the shape similarity the created model can be touched, it is a physical model the physical model allows to realize experiments and study the properties of the original using the same physical processes (for example the airflow around the model of the car in the wind tunnel).
Car (real) Model of the car
Mathematical model
We can define also another model, abstract mathematical model of the original mathematical model. Mathematical model it is not possible to realize the experiments based on the same physical processes, it allows to investigate the processes of the original using their mathematical description solution of the mathematical models. Creation of the mathematical model has the following steps: definition of the discovered processes, definition of the observed symptoms definition of the system on the real object.
Car suspension
Experiment we can discover the degree of the movement caused by the force working on the body of the car. This experiment can be done directly on the car.
t
x
t
F,x
Car suspension mathematical model
It is possible to analyze the same phenomena using the mathematical model of the system.
Car suspension Mechanical model Mathematical model
)()()()(22
tFtkxdttdxb
dttxdm =++
Simulation model of the car suspension
)()()()(22
tFtkxdttdxb
dttxdm =++
Mathematical model differential equation
Simulation model MATLAB - Simulink
Output of the simulation course of the car suspension
position
The relation model original
Criterion used for the assigning of the model to the original: Similarity Analogy.
Similarity similarity between different systems in their structure, properties and behaviour. Physical similarity similarity between systems and processes from the same physical domain geometric similarity, similarity of the parameters and state variables. Mathematical similarity - similarity between the systems and processes with the same mathematical description (structure of the mathematical model). Analogy mathematical similarity between the systems from different domains and processes (analogous systems, analogous variables).
Cybernetic similarity
Cybernetic similarity - expresses the mathematical similarity in the input-output description of the behaviour of the system. We can imagine the system like the black box without any information of the inner structure and state variables. We have only information on the in-out system behaviour. Grey box this term is used if we have only limited information on the system structure. White box we have total information on the inner structure of the studied system. The experimental identification is based on the principle of the cybernetic similarity.
Basic terms from the system theory
System is a set of the elements and linkages between them which has defined properties. Surrounding of the system is a set of the elements, which are not elements of the defined systems, but they have important relations to the system.
Structure of the system, relations
The structure of the system is the representation of the collection of the inner elements and their interaction represented by the links. The structure can be shown using different methods: Description Using graphical method drawing, block diagram The links can be inner (internal) and external. The inner links are between the system elements, the external ones are between the system and the environment. The system variable corresponds to each link. The inputs (excitations), Outputs (responses) and inner state variables.
P1
P2 P3 P4
P5
P6
System
Surrounding (Environment)
Separability of the system
System
Inputs Outputs
Environment
The system can be separated.
It is not possible to separate the system. The system has to be modelled with the surrounding.
Coordinate system of the car
roll
pitch
lateral motion vehicle longitudinal
motion
vehicle vertical motion
yaw
body
wheel
steering motion
rolling motion
wheel liftl
Structure of the dynamic system of the car
PORUCHY
IDI Subsystm:
Subsystm: Subsystm:
Subsystm:
podln pohyboten kol
pohyby karoserie
svisl pohyb kol
naklpn
sklpn
podln zrychlena zpodn
sly psobcna kolo
pn zrychlenbon pohybnaten, naklnn
Horizonzln dynamika
Podln dynamika Svisl dynamika
Pn dynamika
psoben vtru
brzdovpedl
plyn. pedl
rychl.st.
volant
rychlost odporv zatkch
zmny povrchuvlastnosti vozu
nerovnosti vozovky
Break pedal
Driver
Gas pedal
gear
Steering wheel
Quality of the road
Changes of the quality of the road wind
Disturbances
Subsystem: Horizontal Dynamics
Subsystem: Vertical Dynamics Vertical motion of the wheels
Subsystem: Longitudinal Dynamics Longitudinal motion Wheel rotation
Subsystem: Cross Dynamics lateral motion
Wheel forces
Longitudinal acceleration and
deceleration
Lateral acceleration
velocity resistance in curve
Steering System Driver Car
Driver Car
Goal of the trip
Road Traffic on the
road
Steering: Steering wheel, gearshift, break
pedal
Side wind Surface of the road Quality of the road
Car position
Car position, velocity, direction of the movement
Steering System Driver Car with subordinate control system
Driver Car
Control System
Road Trafic Road
Traffic on the road
Side wind Surface of the road Quality of the road
Steering: Steering wheel, gearshift, break
pedal
Car position
Goal of the
trip
Action
Selected state variables
Car position, velocity, direction of the movement
Subsystem of the rotating wheel
vF g
r
Fx
m
M
g
m relative mass of the car on one wheel g gravity acceleration v velocity of the car J momentum of inertia of the wheel angular velocity of the wheel M breaking momentum produced by the
break on the wheel Fx breaking force working on the contact surface FN normal force on the wheel r radius of the wheel
zx FF =mgFz =
,mgFx =z
x
FF
=
mv Fx! =
J rF Mx! =
u r=
Motion equation of the rotating wheel
Motion equation of the car
Circumferential speed of the wheel
FN
Friction coefficient
Slip
The friction coefficient between the surface and the wheel (tyre) depends on the slip .
brzd.moment
v
sou
inite
l
G
skluz
led
FzFx
r
00 20 40 60 80 100 %
0,2
0,4
0,6
0,8
1,2such asfalt
mokr
snh
asfalt =
=
v uv
uv
1 Break momentum
asphalt
dry asphalt
wet asphalt
snow
ice
slip
Fric
tion
coef
ficie
nt
Simulation model of the wheel braking
-1/mk
deceleration x v, u
r
u
1/J
Wheel inertia
s 1
Angular velocity w
s 1
Velocity v
r
Momentum of the force Fx Working on the wheel
mi=f(lambda)
f(u)
lambda =1-u/v
lambda
break momentum M(t)
s 1
breaking path
final momentum
Mux
Mux
mk*g
Fx Clock
Simulation model of the wheel braking in the programe MATLAB Simulink.
Simulation results
Constant brake momentum
Car velocity v, wheel velocity u Breaking momentum
Breaking path Deceleration
Simulation results
Variable brake momentum Car velocity v, wheel velocity u Breaking momentum
Breaking path Deceleration
Simulation model of the wheel breaking with ABS
Model ABS
-1/mk
deceleration x
0.19 w lambda
final momentum On the wheel
v, u
r
u
s 1
tlak
1/J
inertia
s 1
Angular velocityt w
s 1
velocity v
valve ABS
Slip error
r
Momentumof the force Fx On the wheeel
mi=f(lambda)
f(u)
lambda =1-u/v
lambda
100 0.01s+1
Hydraulic system dynamics
6
break momentum
s 1
Breaking path
Mux
Mux
mk*g
Fx
The influence of the ABS is observable: Oscillation of the rotating
speed Oscillation of the slip The wheel is not blocked during the intensive breaking.
Simulation model of the wheel breaking with ABS simulation results
Car velocity v, wheel velocity u
Slip
Types and forms of the systems and their description
x =
xx
xn
1
2
! u =
uu
ur
1
2
!y =
yy
yl
1
2
!
x state vector, u input vector, y output vektor
y f u= ( )
Static systm description only using the static characteristic:
System variables
State model of the system
! ( , )( , )
x f x uy g x u=
=
x x( )0 0= Initial state vector
Dynamic system non linear, t-invariant
).,,(),,,(tt
uxgyuxfx
=
=!
Dynamic system non linear, t-variant
x x( )0 0=
Linear state models of the system
! ( ) ( ) ( ) ( )( ) ( ) ( ) ( )
x A x B uy C x D u= +
= +
t t t tt t t t
x x( )0 0=
! ( ) ( )( ) ( )
x Ax Buy Cx Du= +
= +
t tt t
! ( ) ( )( ) ( )
x Ax by c x= +
= +
t u tt du tT
x x( )0 0=
x x( )0 0=
Linear, t-variant dynamic system
Linern, t-invariant dynamic system
Linear, t-invariant single input single output dynamic system
Transfer Function
G sb s b s b
a s a s a s am
m
nn( ) =
+ + +
+ + + +
!!
1 0
22
1 0
G jb j b j b
a j a j a j am
m
nn( )( )
( ) ( ) ( )
=+ + +
+ + + +
!!
1 0
22
1 0
nn
mm
zazazbzbb
zUzYzG
+++
+++==
!!
11
1101
1)()()(
Transfer function
Transfer function in the frequency domain
Z-Transfer function
Creating of the state space model from differential equation Definition of the state variables
( ) ( ) ubyayayay nnn
0011
1 ... =++++
!
! , , , ...,x xi i= = +1 1 2 1 i n
x y1 =
!xd ydtnn
n=
( ) ( ) ubyayayay nnn
01
110 ... +=
!
( )
( )d ydt
xi
i i= = +1 1 2 1, , , ..., i n
! ...x a x a x a x a x b un n n= +0 1 1 2 2 3 1 0
The system is described by the ODE (Ordinary differential equation) order n:
The first state variable is equal to the output y:
The second and next variable is defined as a derivative of the previous:
The time derivative of the last state variable xn can be expressed from the given differential equation:
Initial condition
The initial conditions- initial values of the state variables x1(0), xn(0) are equal to the initial values of the output variables y(0). This can be obtained from the definition formulas for the state variable for t=0.
niyx ii ,...,2,1),0()0()1( ==
)0()0(
)0()0()0()0(
)1(
2
1
=
=
=
nn yx
yxyx
!
"
Definition of the state variables
!!
.
.!
.
.! . .
x xx x
x x
x a x a x a x a x b u
i i
n n n
1 2
2 3
1
0 1 1 2 2 3 1 0
=
=
=
=
=
=
=
= +
+
y x= 1
!x1!x2..!xi..!xn
!
"
###########
$
%
&&&&&&&&&&&
=
0 1 0 0 . . 00 0 1 0 . . 0. .. .. 1 . 0. 1 .. 1a0 a1 a2 a3 . . an1
!
"
###########
$
%
&&&&&&&&&&&
x1x2..xi..xn
!
"
###########
$
%
&&&&&&&&&&&
+
00..0..b0
!
"
###########
$
%
&&&&&&&&&&&
u[ ]y
xx
x
x
i
n
=
1 0 0 0
1
2
. . .
.
.
.
.
Matrix formulation of the state model:
The set of the state equations can be written using the matrix form.
Definition of the state variables
If the derivatives of the input occur on the right side of the differential equation:
( ) ( ) ( ) ( ) ububububyayayay mmm
mn
nn
011
1011
1 ... ... ++++=++++
!!
We define an additional dynamic system with the state variable x and input only u:
( ) ( ) uxaxaxax nnn =++++ 01
11 ... !
!!
.
.
.!
. .
. .
.
.. . .
.
.
.
.
.
.
xx
x a a a
xx
x
u
n n n
1
2
0 1 1
1
2
0 1 0 00 0 1 0
01
00
1
=
+
x x= 1
Definition of the state variables
The output of the original system is a linear combination of the state variables x1, x2, ..., xn and of the coefficients on the right side of the differential equation describing the system. The state variables x1, x2, ..., xn represent the partial outputs of the system for the input signal u and his derivatives u,u etc.. The right side of the given differential equation is a linear combination of the input signal u and its derivatives. The state variables represent the outputs of the system for the inputs x1 u, x2 u, x3 u, etc.
y b x b x b x b xm m= + + + + +0 1 1 2 2 3 1 ...
!!
.
.!
.
.! . .
x xx x
x x
x a x a x a x a x u
i i
n n n
1 2
2 3
0 1 1 2 2 3 1
=
=
=
=
=
=
=
= +
y b x b x b x b xm m= + + + + +0 1 1 2 2 3 1. . ( ) ( ) ( )( ) ( ) ( ) ( )( )0 ..., ,0 ,0 ,0)0( x..., ,0)0( x,0)0( 11n21 === mn uuuyyyx !!
Definition of the state variables structure of the state model
!x1!x2..!xi..!xn
!
"
###########
$
%
&&&&&&&&&&&
=
0 1 0 0 . . 00 0 1 0 . . 0. .. .. 1 . 0. 1 .. 1a0 a1 a2 a3 . . an1
!
"
###########
$
%
&&&&&&&&&&&
x1x2..xi..xn
!
"
###########
$
%
&&&&&&&&&&&
+
00..0..1
!
"
#########
$
%
&&&&&&&&&
u[ ]y b b b
xx
x
x
mi
n
=
0 1
1
2
0 0. . .
.
.
.
.
y(t)
y(t)
x1=x x2
x10
a0
....
xn x3 xm+1
....
.... .... x2
x20
a1
b0 b1
a2
b2
am
bm
xm+10
an-1
xn0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )y y un m0 0 0 0 0 01 1, ! , ..., , , ! , ..., y u u
Block diagram of the state model of the dynamic system
Definition of the state variables calculation of the initial values of the state variables if the derivatives of the input u occur
The initial conditions for the defined state variables x1,xn must be calculated from the given initial conditions for y and u as follows:
Creation of the state space model from the transfer function
( )( )( )G sY sU s
= ( )G sb s b s s b
s a s s am
mm
m
nn
n=+ + +
+ + +
11
0
11
0
... + b ... + a
1
1
( )( )( ) ( )( )( ) ( )G ss n s ns p s p
=
1 2
1 2
s - n s - p
m
n
......
Seriel programming
( ) ( )G s G sii
n=
=1
( )G s
s ns p
s p
i
i
i
i
=
1
u....
yn
p - pn1
yi+1 yi
p - pip - ni ....
y2 y=y1
p - p1p - n1
Structure of the model using the decomposition of the polynomial
( )( )
Y sY s
s ns p
i
i
i
i+=
1
( )!x p x p n yi i i i i i= + +1y x yi i i= + +1
The system is described by the transfer function:
Creation of the state space model from the transfer function
!x p x yi i i i= + +1
y n x p x yi i i i i ixi
= + +!"# $#
yi+1
yi xi
xi0
pi
pi - ni
Structure of the partial model type 1
.
yi+1 yi xi xi
xi0
-pi
-ni
Structure of the partial model type 2
Parallel structure of the state model
( )G sAs p
ba
i
ii
nn
n=
+
=1
( ) ( )A s p G sis p
ii
= limThe partial transfer functions are calculated from the
original transfer function, constants Ai are given:
The state variables are defined x1, x2, ..., xn using the partial transfer functions:
( )( )
X sU s
As p
i i
i=
, i = 1, 2, ..., n
! ,x p x A ui i i i= + i = 1, 2, ..., n
Output equation: ( ) ( )Y s X s
baUsi
n
ni
n= +
=1
y x xba
unn
= + + +1 2 ... + xn
Parallel structure
!
. . . .
. . . .. . .. .. . .. . .
. . . . .
.
.
.
x
pp
p
p
x
AA
A
A
ui
n
i
n
=
+
1
2
1
2
0 00 0
0
[ ]y x ba un
n= + 1 1 1 1 1. . . .
Matrix form:
The obtained form is also called Jordan form of the state equations.
Parallel structure block diagram
.
u y
u x1 x1
x10
p1
A1
. xi xi
xi0
.
.
.
.
.
.
pi
Ai
. xn xn
xn0
An
pn
an bn
Structure of the state model using the parallel programing Jordan form
Parallel structure complex eigen values
p jp j
i
i
= +
= +
,.1
The pair of the complex eigen values A A Bj
A A Bji
i
= +
= +
,.1Complex coefficients of the partial
transfer functions:
( ) ( ) ( )G sA
s p j ji=
+
+1 ... +
A + Bjs- +
+A - Bj
s- ... +
As- p
n
n
!x x x Aui i i= + ++ 1 2
Buxxx iii 211 ++= ++ !
ii xy =
Parallel programming
!
.
.
.!!
.
.
.!
. . . . . . . .. . .. . .. . .. .. .. . .. . .. . .
. . . . . . . .
.
.
.
.
.
.
.
x
xx
x
p
p
x
xx
x
i
i
n n
i
i
n
1
1
1 1
1
0
0
+ +
=
+
+
A
AA
A
uii
n
1
1
.
.
.
.
.
.
[ ]y
x
xx
x
i
i
n
=
+1 1 0 1
1
1. . . . . .
.
.
.
.
.
.
Matrix form of the state model with complex eigen values: (all matrix elements are real numbers)
Series combination of the transfer funtions - example
Using the series combination of the transfer functions you have to create the state model:
( )( )G s s s s= + +
35 42
( )G ss s s
= +
+
1 11
34Solution:
u x3
s + 4 3
s + 1 1
x2 y = x1
s 1
( ) ( )
( ) ( )
( ) ( )
X ssX s
X ss
X s
X ss
U s
1 2
2 3
3
1
1134
=
=+
=+
State variables:
!!!
xx x xx u
1
2 2 3
3 3
=
= +
= +
x
- 4x
2
3
!!!
xxx
xxx
u1
2
3
1
2
3
0 1 00 1 10 0 4
003
=
+
Parallel combination of the transfer functions - example
( )( )
G ss ss s s
=+ +
+ +
2
25 66 5
The system is described using the transfer function:
You have to create the state model using the parallel combination of the transfer functions (parallel programming).
( )( ) ( )G s s s s
= ++
++
65
12 1
310 5Solution: Partial transfer functions:
( ) ( )
( )( )
( )
( )( )
( )sUs
sX
sUs
sX
sUs
sX
+
=
+
=
=
5103121
56
3
2
1State variables are defined as the outputs of the partial transfer functions.
( ) ( ) ( ) ( )Y s X s X s X s= + +1 2 3The Laplace transformation of the output defines the output equation:
Parallel programing
Solution in the time domain:
!
!
!
x u
x x u
x x u
1
2 2
3 3
5612
53
10
=
=
= +
y x x x= + +1 2 3
!!!
xxx
xxx
u1
2
3
1
2
3
0 0 00 1 00 0 3
5612310
=
+
[ ]yxxx
=
1 1 11
2
3
The matrix form:
Creating of the state model from more differential equations
2221
1122111
54,22,15,02312uyyy
uuyyyyy=
+=++++
!
!!!!!
2212
1122111
51
51
54,2
122,1
121
125,0
61
41
121
uyyy
uuyyyyy
=
++=
!
!!!!!The highest derivatives are written on the left side of each equation:
( ) ( ) ( )
+++=t
duyyyyuuyyyy0
121202101101101 121
125,0
121
61
101
41
!
The system is described using two equations:
!,
x y y u1 1 2 1112
0 512
112
= +
1020101010 61
101
41 yyuyx !++=
12111 61
101
41 xyuyy ++=!
After substitution we obtain:
Creating of the state model from more differential equations
y y y u y x dt
1 10 1 1 2 10
14
110
16
= + +
After integration we obtain:
!x y u y x2 1 1 2 114
110
16
= + +
x y20 10=y x1 2=
y y y y u dt
2 20 1 2 20
2 45
15
15
=
,
From the second given differential equation we obtain:
!,
x y y u3 1 2 22 45
15
15
= x y30 20=
y x2 3=
Output equation for y1:
Output equation for y2:
Creating of the state model from more differential equations
!,
!
!,
x y y u
x y u y x
x y y u
1 1 2 1
2 1 1 2 1
3 1 2 2
112
0 512
112
14
110
16
2 45
15
15
= +
= + +
=
!,
!
!,
x x x u
x x x u
x x x u
1 2 3 1
2 1 2 3 1
3 2 3 2
112
0 512
112
14
16
110
2 45
15
15
= +
= +
=
x
!!!
,
,
xxx
xxx
uu
1
2
3
1
2
3
1
2
0112
0 512
114
16
02 45
15
15
0110
0
015
=
+
State equations:
Matrix form:
State variables:
yy
xxx
1
2
1
2
3
0 1 00 0 1
=
x y y u y
x yx y
10 10 0 0 20
20 10
30 20
14
110
16
= + +
=
=
Initial conditions:
Block diagram:
x1
x10
5 -1
24 -1
12 -1
1
u2
u1
x2
x20
4 -1
25 12
x3
x30
5 -1
6 -1
10 1
5 -1
Structure of the created state model.
Creation of the state model from the non-linear differential equation
( ) ( )( ) tu, ,y ..., ,y ,y , 1-n!!!yfy n = ( )( )
( ) ( ) ( )
y t y
y t y
y t yn n
0 0
0 0
10 0
1
=
=
=
,
,...
,
( )!x f x= , u, tState variable form : ( )y g x= , u, t
Description of the system:
Initial values:
( ),...,,
1
2
1
=
=
=
nn yx
yxyx!
Definition of the state variables:
....,,
1
32
21
nn xx
xxxx
=
=
=
!
!
!Definition of the state equations:
( )! , , ,x f xn = 1 x ..., x u, t2 n
Transfer of the non-linear differential equation into the system of differential equations first order
Final form of the state model state equations:
( )
! ,! ,! ,
.
.
.
! , , ,
x xx xx x
x f xn
1 2
2 3
3 4
1
=
=
=
= x ..., x u , t2 n
y x= 1Output equation:
( )( )
( )
x t y
x t y
x t ynn
1 0 0
2 0 0
0 01
=
=
=
,
,...
.
Initial values:
( )f x xi i1 1, x ..., x u, t i = 1, 2, ..., n -12 n, , ,= +
( ) ( )f x f xn 1 1, x ..., x u, t , x x , ..., x u, t2 n 2 3 n, , , ,=
Description using the vector functions f (first n-1 equations have the shown form, the last equation has another form which corresponds to the right side of the given differential equation).
Simulation programme MATLAB - Simulink
Simulation programme AMESim