Digital laboratory report writing, assessment

1
A Joint Laboratory Report
Student’s Name
Institutional Affiliation
Date
Laboratory 3

2
Modelling of dynamic systems in Matlab and Simulink
Objective
To model dynamic control systems including vehicle cruise control, DC motor velocity and
position control using Matlab and Simulink
Method
Matlab Simulation software and Simulink were used to model the various control systems and
to invetigate their time and frequency response. The transfer function for each of the control
systems provided was first derived and the system’s analyzed in Matlab and Simulink.
Analysis
Question 1:
Derivation of the transfer function relating the speed of the vehicle with the applied force.
m∙ a=F1−F2
m∙ dv
dt =F1 −b ∙ v
F1=b ∙ v +m∙ dv
dt
F1(t)=bv (t )+ m∙ dv ( t)
dt
Converting into frequency domain by taking laplace transforms gives,
F ( s ) =bV ( s ) + msV (s )= ( b+ ms ) V ( s)
F ( s )
V (s )=b+ms∨V (s)
F ( s ) = 1
ms+b
This is a first order transfer function. With the given values for the mass and damping ratio
the equation becomes,
V (s )
F ( s ) = 1
2000 s +50
DC Motor Speed: System Modelling
Question 2
Derivation of the transfer function relating the angular velocity of the motor with the applied
voltage.
T e=Kt i ,e=K e ω
J dω(t)
dt +bω (t)=T e=Ki( t)
L di(t )
dt + Ri (t)=v (t)−Kω (t)
Converting the equations into frequency domain by taking the laplace transforms gives,
LsI ( s )+ RI ( s )=V ( s )−Kω ( s )

3
( Ls+ R ) I ( s ) =V ( s ) −Kω ( s ) (i)
Jsω ( s ) +bω ( s ) =KI (s)
( Js+b ) ω ( s )=Kt I ( s)
ω ( s )= KI (s)
Js +b
I ( s )=(Js +b) ω ( s )
K t
(ii)
Substituting the value of I ( s ) in equation (i) we have,
( Ls+ R ) ( Js+ b)ω ( s )
Kt
=V ( s ) −Kω ( s )
( Ls+ R ) ( Js+ b)ω ( s )
Kt
=V ( s ) −Kω ( s )
V ( s ) =Kω ( s ) + ( Ls+ R ) ( Js+ b)ω ( s )
Kt
= Kt Ke ω ( s ) + ( Ls+ R ) (Js +b)ω ( s )
Kt
V ( s )= [ ( Ls+ R ) ( Js+b )+ Kt Ke ] ω ( s )
Kt
V ( s )
ω ( s ) = [ ( Ls+ R ) ( Js+ b ) + Kt Ke ]
Kt
∨ ω ( s )
V ( s ) = Kt
[ ( Ls+ R ) ( Js+b )+ Kt Ke ]
DC Motor Position: System Modelling
Question 3
The relationship between the angular velocity ω and the angular position θ is,
ω= dθ(t )
dt , Te=K t i, e=Ke
dθ (t)
dt
J d
dt
d θ(t )
dt +b dθ(t )
dt =T e=Kt i(t )
J d2 θ(t )
dt2 +b dθ(t )
dt =T e=K t i(t)
L di(t )
dt + Ri (t)=v (t)−K e
dθ(t )
dt
Converting the equations into frequency domain by taking the laplace transforms gives,
LsI ( s ) + RI ( s ) =V ( s ) −K e sθ ( s )
( Ls+ R ) I ( s )=V ( s )−Ke sθ ( s )( i)
J s2 θ ( s ) +bsθ ( s )=Kt I ( s ) , ( J s2 +bs ) θ ( s )=Kt I (s)

4
θ ( s )= K t I (s)
J s2+ bs
I ( s )=(J s2+ bs)θ ( s )
Kt
(ii)
Substituting the value of I ( s ) in equation (i) we have,
( Ls+ R ) ( J s2 +bs) θ ( s )
K t
=V ( s ) −Ke sθ ( s )
( Ls+ R ) ( J s2 +bs) θ ( s )
K t
=V ( s ) −Ke sθ ( s )
V ( s )=K e sθ ( s )+ ( Ls+ R ) (J s2+ bs)θ ( s )
Kt
= Kt Ke sθ ( s ) + ( Ls+ R ) ( J s2+ bs)θ ( s )
Kt
V ( s )= [ ( Ls+ R ) (J s2+bs )+ Kt Ke s ] θ ( s )
Kt
V ( s )
θ ( s ) = [ ( Ls+ R ) (J s2+ bs)+ Kt Ke s ]
K t
∨ θ ( s )
V ( s ) = Kt
[ ( Ls+ R ) (J s2 +bs)+ Kt K e s ]
Important results
Question 1:
Figure 1: a) Step response obtained using tf function and b) Step response obtained using Matlab's Simulink
DC Motor Speed: System Modelling
Question 2