MaSiMS-A-1

Modelling and Simulation of Mechatronic Systems

Prof.Ing.Petr Noskievi, CSc.

Department of Automatic Control and Instrumentation

VB Technical University of Ostrava


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.


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.


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.


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


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:


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:


!,

!

!,

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

Documents

MaSiMS-A-1