Manuel S.

Enverga University Foundation

Lucena City, Philippines
Granted Autonomous Status
CHED CEB Res. 076-2009

Control System and

Feedback
Part 2

Sherwin C. Lagrama.,RECE,MMEM
College Of Engineering and Technical Department
2nd Sem 2013-2014

COLLEGE OF ENGINEERING AND TECHNICAL DEPARTMENT

Telefax No. (042) 710-3151; e-mail:engg.dept_mseuf@yahoo.com.ph
Agenda or Summary Layout
SLIDE Discussion Item One – Review Laplace Transform Theorem
[3] Add a second line of text here

SLIDE Discussion Item Two – Review Cramer’s rule

[4] Add a second line of text here

SLIDE Discussion Item Three – Mechanical Translational System

[7-16] Add a second line of text here

SLIDE
Discussion Item Four – Electrical System
[17-23]
Add a second line of text here

SLIDE
Discussion Item Five – Operational Amplifier
[24-25]
Add a second line of text here

SLIDE Discussion Item Six – Mechanical rotational System

[26-29] Add a second line of text here

SLIDE
Discussion Item Six – Mechanical Rotational with reflected load
[30-31] system
Add a second line of text here
“Laplace Transform Review”
Laplace transform theorems
“Review cramer’s rule”

click

See sample equations

 “Modeling of Systems”
• The first step in control design is the • Outputs: These represent the variables
development of a mathematical model which the designer ultimately wants to
for the process or system under control and that can be measured by the
designer. For example, in a flight control
consideration. In the modeling of application an output may be the altitude of
systems, we assume a cause and the aircraft, and in automobile cruise control
effect relationship described by the the output is the speed of the vehicle.
simple input/output diagram below. An • System or Plant: This represents the
input is applied to a system, and the dynamics of a physical process which relate
system processes it to produce an the input and output signals. For example,
output. In general, a system has the in automobile cruise control, the output is
three basic components listed below the vehicle speed, the input is the supply of
gasoline, and the system itself is the
automobile

• Inputs: These represent the variables under

the designer's disposal. The designer
produces these signals directly and applies
them to the system under consideration. For
example, the voltage source to a motor and
the external torque input to a robotic
manipulator both represent inputs. Systems
may have single or multiple inputs.
Block diagram representation of a system.
 “Modeling of Systems”
• Systems we will be looking at. • n. A preliminary work or
 electrical systems construction that serves as a
 mechanical systems plan from which a final product
(translational and is to be made.
rotational) • n. A schematic description of a
 electromechanical system, theory, or
systems phenomenon that accounts for
its known or inferred properties
and may be used for further
 What is a Model? study of its characteristics.
Dictionary definition. Model. • n. A style or design of an item.
• n. One serving as an example
• n. A small object, usually built to be imitated or compared.
to scale, that represents in
detail another, often larger
object.
“Modeling of Systems-Mechanical”
 “Modeling of Systems-Mechanical”

Example: Mechanical system: Spring-mass-

damper system.
In this example we model the spring-mass-system
shown in Figure. The mass, m is subjected to an
external force f. Let's suppose that we are
interested in controlling the position of m. The way
Diagram of the mechanical system components. (b) Free
to control the position of the mass is by choosing f. body diagram of the mechanical system.
We first identify the input and output.
Input: external force, f, output: mass position, x. http://homepage.mac.com/sami_ashhab/courses/control/lectures/lecture_3/spring_mass.html
 “Modeling of Systems-”
• mass, M- is the property of an elements that store the kinetic energy of a
translational
f (t )  ma  mD 2 v f(t) m x(t)
d 2v dx
a 2 v
dt dt
• Elastance of stiffness-property of an elements that stores potential energy

f (t )  k ( x1  x) f(t)
x
x1

• B (damping)
a) viscous-represents the retarding force that has a linear relationship between
the applied force and velocity
f (t )  B (v 2  v1)
dx
but v f(t) x x1
dt
 dx 2 dx1 
f (t )  B     f (t )  BD ( x 2  x1)
 dt dt 
 “Modeling of Systems-Mechanical”
other reference

Force-velocity,
force-
displacement, and
impedance
translational
relationships for
springs, viscous
dampers, and
mass
 “Modeling of Systems-Mechanical”
other reference

Find the transfer function X(s)/F(s):

d 2x dx
Solutions M 2  f (t )  f v  f k x(t )
a. Mass, spring, and damper system; b. block dt dt
diagram Ms 2 X ( s)  F ( s)  Fv sX ( s )  Fk X ( s )
X ( s)  Ms 2  Fv s  Fk   F ( s )
X ( s) 1
 2
F ( s) Ms  Fv s  Fk

Figure 2.16
a. Free-body diagram of mass, spring, and damper
system;
b. transformed free-body diagram
 “Modeling of Systems-Mechanical”
We apply Newton's second law to obtain the differential equation of this mechanical
system. Using the free-body-diagram shown in Figure 3(b), we have
where, b is the damping coefficient and k is the spring stiffness (see Table 2.2,
page 35 in the text book). Equation (2) is the differential equation that describes the
dynamics of the spring-mass-damper system. Note that the input and output
appear in this equation.

Mathematical Models of system

1. Define the system of its components
2. Formulate the mathematical and list the necessary assumptions
3. Write DE describing the model
4. Solve the equation for the desired output variables
5. Examine the solution and assumptions
6. If necessary re-examine and analyze the system

Methods of Writing DE: the “D” operator

d d 2x dx 1
D   Dx  dt 
2
2
D x
dt dt dt dt
 “Modeling of Systems-Mechanical”
Example Mechanical System: (2 mass)-spring-damper system
This system contains two masses, a spring, and a damper. An external force is applied to the first mass and we
would like to control the position of the second mass. The external force, f indirectly affects the motion of the second
mass.
Solution: First identify the input and output of the
system
input: external force, f, output: displacement of the
second mass, z.
Next we find a set of differential equations describing this
system. Newton's second law is applied to each mass to
obtain these differential equations.

(a) Diagram of the mechanical system components. (b) Free body diagram of the system.

http://homepage.mac.com/sami_ashhab/courses/control/lectures/lecture_3/spring_mass.html
 “Modeling of Systems-Mechanical”
a)
Sample Problem:
Consider the mechanical system in figure below
a) Write DE that describe the system below
 F  ma
b) Find G(s)=Y(s)/F(s) from the applied forces d 2 y (t )
to the displacement ft  fb1  fb 2  f s  m
dt 2
 dy (t )   dy (t )  d 2 y (t )
ft  b1    b2    ky (t )  m
f(t) b1  dt   dt  dt 2
b)
d 2 y (t )  dy (t )   dy (t ) 
ft  m  b1    b2    ky (t )
M
 dt   dt 
2
y(t) dt
F ( s )  Ms 2Y ( s )  B1sY ( s )  B2 sY ( s )  KY (s )
b2 k F ( s )  Y ( s )  Ms 2  B1s  B2 s  K 
Y (s) 1

F ( s )  Ms 2  B1s  B2 s  K 
 “Modeling of Systems-Mechanical”
Sample Problem: X 2 (s)
Find the transfer function F ( s ) a. Forces on M1 due only to motion of M1
b. forces on M1 due only to motion of M2
c. all forces on M1

a. Forces on M2 due only to motion of M2;

b. forces on M2 due only to motion of M1;
c. all forces on M2
Note: check your signs for X1-X2 and
X2-X1. See the preceding sample on
W-Z. Ask your student w/c reference they want
to apply for the uniformity of silution.
 “Modeling of Systems-Mechanical”
Solutions:
@ M1
( K1  K 2 )  ( Fv1  Fv 2 ) s  M 1 s 2  X 1 ( s )  ( K 2  fv3 s ) X 2 ( s )  F ( s )
@M2
  Fv 3 s  K 2  X 1 ( s )  ( K 2  K 3 )  ( Fv 2  Fv 3 ) s  M 2 s 2  X 2 ( s )  0
cramer ' s rul ( PLS  RECOMPUTE  CLARITY  ANS )
( K1  K 2 )  ( FV 1  Fv 2 ) s  M 1 s 2 F (s)
( Fv 3 s  K 2 ) 0
X 2 ( s) 
( K1  K 2 )  ( Fv1  Fv 2 ) s  M 1 s 2  ( Fv 3 s  K 2 )
( Fv 3 s  K 2 ) ( K 2  K 3 )  ( Fv 3  Fv 2 ) s  M 2 s 2 
F ( s )( Fv 3 s  K 2 )
X 2 ( s) 
( K1  K 2 )  ( Fv1  Fv 2 ) s  M 1 s 2  ( K 2  K 3 )  ( Fv 3  Fv 2 ) s  M 2 s 2   ( Fv 3 s  K 2 )( K 2  f v 3 )
X 2 ( s) ( Fv 3 s  K 2 )

F ( s ) ( K1  K 2 )  ( Fv1  Fv 2 ) s  Ms 2  ( K 2  K 3 )  ( Fv 3  Fv 2 ) s  M 2 s 2   ( Fv 3 s  K 2 )( K 2  f v 3 )
“Modeling of Systems-Electrical”
The system below shows an electrical
circuit with a current source i, resistor R,
inductor L, and capacitor C. All of these
parts are connected in parallel. It is
required to regulate the capacitor
voltage V.
 “Modeling of Systems-Electrical”

Voltage-current, voltage-charge, and impedance relationships for

capacitors, resistors, and inductors (passive linear components)
“Modeling of Systems-Electrical”
first identify the input and output of the system
input: current, i, output: voltage, V.
Next we find a set of differential equation that describes this system. Kirchoff's current
law is applied. The sum of the three currents (R, L, and C) is equal to the overall source
current i.

RLC circuit (parallel connection).

“Modeling of Systems-Electrical”
Find the transfer function relating the capacitor, V C (s ),
to the input voltage, V (s ) .

Figure 2.3 Figure 2.5

RLC network Laplace-transformed network

Figure 2.4
Block diagram of series RLC
electrical network
 “Modeling of Systems-Electrical”
(a)Laplace transform
Given the circuit diagram shown below (show your solution)
(1)……’v1(s) =I(s) R1 + I(s) R2 + I(s)/sC
(a)Write the Differential Equation v1(t), v2(t)
(b)Write the Laplace Transform]
(c)Solve for Transfer Function
v 2( s)  1 
G( s)  v1( s )  I ( s )  R1  R 2 
v1( s)  sC 


v1( s )
I ( s) 
R1 R2 1
R1  R 2 
sC

i(t) V2(t) (2)’……v2(s) =I(s) R2 + I(s)/sC

V1(t)

C  1 
v 2( s )  I ( s )  R 2 
 sC 


 
(a) Write the differential equation  
v 2( s )   v1( s )  R2  1 
(1) v1 (t) =i (t) R1 + i(t)R2 + 1/c∫i(t)dt  1  sc 

(2) (2) v2 (t) =i(t) R2 + i(t)1/c∫i(t)dt  R1  R 2  
 sC 
 1 
 R2  
v 2( s )  v1( s )  sC

 R1  R 2  1 

 
sC 
“Modeling of Systems-Electrical”
v 2( s )
(c) Find the transfer function of G(s) =
v1( s )
1
R2 
v 2( s )
 sC  sCR2  1  sC 
v1( s ) 1   
R1  R 2   sC  R1sC  sCR 2  1 
sc
R2 1

 1 sc
sCR2  1
R1  R 2 
1 
sC R1sC  sCR 2  1

sCR 2  1 v2 (s) sCR2  1


v1 (s) sC  R1  R2   1
 sC
sCR 2  1
R1 
sC

sCR 2  1
 sC
R1sC  sCR 2  1
sC
 “Modeling of Systems-Electrical”
Find the following it  il  ir  ic
(a)Write the two equations using Laplace transform,
I(s), V(s) 1 t V dv
V ( s)
(b) Find the value of I ( s )
it  0 Vt dt  c  diffrential eqn.
L R dt
L
i(t) V( s ) V( s )
I(s)    sCV( s )  laplace
sL R
vt  vl  vz
V(t) R C Vs  LsI ( s )  Z p ( s ) I ( s )
where
1 1 1 R
  ; Z p(s) 
Z p(s) R 1 RCs  1
sC
Note: V(s)=total voltage
I(s)=total current V( s )
 Z( s) therefore
Z(s)=total resistance I(s)
V( s ) RLCs 2  Ls  R
Z(s)   sL  Z p ( s ) 
I(s) RCs  1
 “Modeling of Systems-Electrical”
Operational Amplifiers Figure 2.10
a. Operational amplifier;
b. schematic for an
inverting operational
amplifier;
c. inverting operational
amplifier configured for
transfer function realization.
Typically, the amplifier gain,
A, is omitted.
An operational amplifier has the
following characteristics
1. Differential inputs,
2. High input impedance,
(ideal)
3. Low output impedance,
(ideal)
4. High constant gain amplification,
(ideal)
 “Modeling of Systems-Electrical”
Consider the ckt below find the ff: a)
a) Write DE for Vi(t)
b) Write the laplace transform  1
t

Vi (t )  i (t ) R1 , Vo (t )   i (t ) R2   i (t )dt 
c) Transfer function Vo(t)/Vi(t)
 c0 
b)
 I (s) 
Vi ( s )  I ( s ) R1 , Vo ( s )    I ( s ) R2 
 sC 
c)
 I (s)   1 
 I ( s ) R  I ( s )  2
R  
sC 
2
Vo (t )
   sC 
Vi (t ) I ( s ) R1 I ( s ) R1
R2 sC  1
Vo (t ) 1  R sC  1  R2 sC  1
  sC   2   
Vi (t ) R1 R1  sC  sCR1
Torque-angular velocity, torque-angular displacement, and impedance
rotational relationships for springs,
viscous dampers, and inertia
Torque-angular velocity, torque-angular displacement, and impedance
rotational relationships for springs,
viscous dampers, and inertia
 Torque-angular velocity, torque-angular displacement, and impedance
rotational relationships for springs,
viscous dampers, and inertia
 2 (s)
Find the transfer function T (s)

a. Torques on J1 due only to the motion of J1

b. torques on J1 due only to the motion of J2
c. final free-body diagram for J1

a. Physical system; b.
schematic; c. block
diagram

T ( s)  J1s 21 ( s )  D1s1 ( s )  K1 ( s )  K 2 ( s )  0

J1s 21 ( s )  D1s1 ( s )  K1 ( s )  K 2 ( s )  T ( s )
 1  D1s  K 1 (s)  K2 (s)  T (s)  @ J1
J s 2
 Torque-angular velocity, torque-angular displacement, and impedance
rotational relationships for springs,
viscous dampers, and inertia

a. Torques on J2 due only to the motion of J2;

b. torques on J2 due only to the motion of J1
c. final free-body diagram for J2

J 2 s 2 2 ( s)  D1s 2 ( s)  K 2 ( s )  K1 ( s )  0
 K1 ( s)   J 2 s 2  D1s  K  2 ( s )  0  @ J 2

Using Cramer’s rule

( J1s 2  D1s  K ) T ( s )
K 0 0  T ( s)( K )
 2 ( s)  
( J1s 2  D1s  K ) K ( J1s 2  D1s  K )( J 2 s 2  D1s  K )  ( K )( K ) 
K ( J 2 s 2  D1s  K )
 2 ( s) K K
 
T ( s) ( J1s 2  D1s  K )( J 2 s 2  D1s  K )  K 2 J1 J 2 s 4  ( J1D 1  J 2 D1 ) s 3  ( J1K  D12  J1K )s 2  (2 D1K )s
Torque-angular velocity, torque-angular displacement, and impedance
rotational relationships for springs,
viscous dampers, and inertia
 2 (s)
Find the transfer function T1 ( s )

a. Rotational mechanical system with gears;

b. system after reflection of torques and impedances to the output
shaft;
c. block diagram
T  J11 " D11 ' T1
T2  J 2 2 " D2 2 ' K 2 2
 N   N1 
but T1  T2  1  and  2  1  
 N2   N2 
 N   N 
T  J11 " D11 " T2  1   J11 " D11 "  1   J 2 2 " D2 2 ' K 2 2 
 2
N  N2 
For larger
N2
but 1   2 size
 N1 
N2 N2   N1 
T  J1    2 " D1    2 '    J 2 2 " D2 2 ' K 2 2 
 1 
N  N1   N2 
 N2   N2 N2   N1  
  T  J1    2 " D1    2 '    J 2 2 " D2 2 ' K 2 2  
 1 
N  1 
N  N1   N2  
2 2
N  N2 N2
T  2   J1    2 " D1    2 ' J 2 2 " D2 2 ' K 2 2
 1
N  1 
N  N1 
2 2
 N2  N2 N2
 T (s)    J 1 s  2 ( s)    D1s 2 ( s )  J 2 s  2 ( s )  D2 s 2 ( s )  K 2 2 ( s )
2 2

 1
N  1 
N  1 
N

 N2   N2
2
N2
2

 T ( s )    J 1s    D1s  J 2 s  D2 s  K 2   2 ( s )
2 2

 N1  
 1 
N  N1  

 N2   N2 
   
 2 (s)  N1   N1 
 
T ( s)  N  2
N2
2
  N  2
  N  2 
  J 1s    D1s  J 2 s  D2 s  K 2    J 1  J 2  s    D1  D2  s  K 2
2 2 2 2 2 2


 1 
N  N1  
 
 1 
N 
 
 1 
N 

let
 N  2 
Je   2
 J 1J2 

 1 
N 

 N  2 
De   2
 D1  D2 

 1 
N 

Ke  K 2
Therefore
 N2 
 
 2 (s)  N1 

T (s) Je  De  Ke
Thank you
Instructional materials by
Engr. SCLagrama.,RECE,MMEM