Sistem Kendali - Modeling of Dynamic Systems
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Transcript of Sistem Kendali - Modeling of Dynamic Systems
Lecture 2 Lecture 2 ––Modeling of Dynamic SystemsModeling of Dynamic Systems
Electrical Engineering DepartmentUniversity of Indonesia
Lecturer:Aries Subiantoro
The Control Design CycleThe Control Design Cycle
The Control Design CycleThe Control Design Cycle1. Establish control goals
2. Identify the variables to control
3. Write the specifications for the variables
4. Establish system configuration (sensors+
actuator+process+ controller hardware)
5. Obtain a model of the process+actuator+sensor
6. Determine controller parameters to be adjusted
7. Optimize the parameters and analyze the controlled system’s
performance
Performance specs met
Performance does not meet the specs
Summary: Given a model of the system to be controlled (process, sensors, actuators) and design goals, find a controller or determine that none exists
Modeling and SimulationModeling and Simulation
mModel types: ODE, PDE, state machines, hybrid
mModeling approaches:q Physics based (white box)
q Input-output models (black box)
m Linear systems
m Simulation
mModeling uncertainty
Dynamic ModelsDynamic Models
m Energy Domain:
m Electric Circuit
mMechanical Systems
m Electromechanical Systems
m Heat and Flow Systems
mTo make progress on the control system design problem, it is first necessary to gain an understanding of how the process operates. This understanding is typically expressed in the form of a mathematical model.
Dynamic ModelsDynamic Models
Dynamic ModelsDynamic Models
The power of a mathematical model lies in the fact that it can be simulated in hypothetical situations, be subject to states that would be dangerous in reality, and it can be used as a basis for synthesizing controllers.
Modelling of SystemsModelling of Systems
m Classical Control
m based on continuous time systems
m takes system differential equation and using Laplace Transforms models system as a transfer function
m Note that it is essential to be familiar with Laplace Transforms
Modelling of SystemsModelling of Systems
mModern Control
m usually based on discrete time systems
m transfer function approach (z transform)
m or state space (time domain ) approach
m This subject is primarily concerned with classical control.
Linear vs NonLinear vs Non--Linear ModellingLinear Modelling
m In this course we will assume we are dealing with Linear Time Invariant systems
q Linear
§ superposition holds
q Time Invariant
§ system dynamics as described by system differential equation does not change with time
Linear vs NonLinear vs Non--Linear ModellingLinear Modelling
l Note that with non-linear systems we can often linearise the system about a certain operating point and apply the theory we will develop in this course.
Laplace TransformsLaplace Transforms
mThe study of differential equations of the type described above is a rich and interesting subject. Of all the methods available for studying linear differential equations, one particularly useful tool is provided by Laplace Transforms.
Definition of the TransformDefinition of the Transform
lConsider a continuous time signal y(t); 0 ≤ t < ∞. The Laplace transform pair associated with y(t) is defined as
mA key result concerns the transform of the derivative of a function:
Laplace TransformsLaplace Transforms
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 2.1Laplace transform table
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 2.2Laplace transform theorems
Transfer FunctionsTransfer Functions
l We start from the differential equation relating system output to input , take the Laplace transform of the D.E. and rearrange to get the ratio of the L.T. of the output to the L.T. of the input.
l Note in so doing we are employing the derivative property of the L.T.
Transfer FunctionsTransfer Functions
l Recall derivative property of L.T.
( ) ( )
( ) ( ) ( )00)(
0)(
2
2
2
fsfsFstfdt
dL
fssFtfdt
dL
&−−=
−=
Linear vs NonLinear vs Non--Linear ModellingLinear Modelling
l Note that with non-linear systems we can often linearise the system about a certain operating point and apply the theory we will develop in this course.
Transfer FunctionsTransfer Functions
l We start from the differential equation relating system output to input , take the Laplace transform of the D.E. and rearrange to get the ratio of the L.T. of the output to the L.T. of the input.
l Note in so doing we are employing the derivative property of the L.T.
Transfer FunctionsTransfer Functions
l Recall derivative property of L.T.
( ) ( )
( ) ( ) ( )00)(
0)(
2
2
2
fsfsFstfdt
dL
fssFtfdt
dL
&−−=
−=
Transfer FunctionsTransfer Functions
l
( ) ( )( )
( ) ( )
{ }
( ) ( ) ( )s
sXssXsXs
sL
xxxxx
fssFstfdt
dL
n
k
kknn
n
n
352
33 that Noting
00,00,352
of Transform Laplace heconsider t examplean As
0)(
generalIn
2
1
1
=++→
=
===++
−=
∑=
−−
&&&&
Transfer FunctionsTransfer Functions
l In constructing transfer functions we make the assumption that the initial conditions are zero
l in effect this means their effects have long died out.
Transfer FunctionsTransfer Functions
l
)(...
)(...
11
1
10
11
1
10
txbdt
dxb
dt
xdb
dt
xdb
tyadt
dya
dt
yda
dt
yda
mmm
m
m
m
nnn
n
n
n
++++=
++++
−−
−
−−
−
LTI Systemx(t) y(t)
System differential equation
Transfer FunctionsTransfer Functions
l
)()(...)()(
)()(...)()(
11
10
11
10
sXbssXbsXsbsXsb
sYassYasYsasYsa
mmmm
nnnn
++++=
++++
−−
−−
LTI Systemx(t) y(t)
Taking Laplace Transforms assuming zero initial conditions
Transfer FunctionsTransfer Functions
l
zeros system theasknown are 0 of Roots
poles system theasknown are 0)( of Roots
)(
)(
...
...)(
11
10
11
10
=
=
=++++
++++=
−−
−−
N(s)
sD
sD
sN
asasasa
bsbsbsbsG
nnnn
nnmm
LTI Systemx(t) y(t)
Hence transfer function G(s)=Y(s)/X(s)
Derivation of Transfer Derivation of Transfer Function Function -- ExampleExample
m Electric Circuit
mMechanical Systems
m Electromechanical Systems
m Flow Systems
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 2.3Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.3RLC network
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.4Block diagram of series RLC electrical network
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.11Inverting operationalamplifier circuit for Example 2.14
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 2.4Force-velocity, force-displacement, and impedance translational relationshipsfor springs, viscous dampers, and mass
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.15a. Mass, spring, and damper system; b. block diagram
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Table 2.5Torque-angular velocity, torque-angular displacement, and impedancerotational relationships for springs, viscous dampers, and inertia
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.22a. Physical system; b. schematic; c. block diagram
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.25Three-degrees-of-freedom rotationalsystem
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure P2.35a. Coupling ofpantograph andcatenary;b. simplifiedrepresentationshowing theactive-controlforce
© 1997 ASME.
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure P1.9High-speed rail system showing pantograph and catenary
© 1997, ASME.
Derivation of Transfer Function Derivation of Transfer Function --ExampleExample
l Attitude Control of a satellite
θ
Thrusters
Reference
Centre of
mass
Derivation of Transfer Function Derivation of Transfer Function --ExampleExample
l Attitude Control of a satellite
Reference θ
Tdt
dJ
T
J
=2
2
torque thruster theas & satellite theof
inertia ofmoment theas Defining
θ
Derivation of Transfer Function Derivation of Transfer Function --ExampleExample
l Attitude Control of a satellite
Reference θ2
2
1
)(
)()(
function transfer System
)()(
sidesboth of Transforms Laplace Taking
JssT
ssG
sTsJs
=Θ
=
→
=Θ
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.34NASA flightsimulatorrobot arm withelectromechanicalcontrol systemcomponents
© Debra Lex.
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.35DC motor:a. schematic12;b. block diagram
Control Systems Engineering, Fourth Edition by Norman S. NiseCopyright © 2004 by John Wiley & Sons. All rights reserved.
Figure 2.36Typical equivalentmechanical loading on a motor
Sistem Tangki TerhubungSistem Tangki TerhubungModel CEModel CE--105105
cQhhghhadt
dhA
hhghhaQdt
dhA
−−−=
−−−=
212112
2
2121111
1
2)(sign
2)(sign
Model Sistem Tangki TerhubungModel Sistem Tangki Terhubung
Block DiagramsBlock Diagrams
l Control system elements or sub-systems are represented by block diagrams
l Each block will contain the transfer function for that sub-system and possibly the name of the subsystem
Block DiagramsBlock Diagrams
l Signal flow denoted by arrows and a description
l summing blocks sum two or more signals with the a plus or minus sign at the arrowhead indicating if signal is added or subtracted
l branch points are points where signal goes concurrently to two or more points.
Block DiagramsBlock Diagrams
l Example - Closed Loop Control System
+
s
1
5
10
+s
1
1
+s
-
controller plant
sensor
Y(s)U(s)E(s)
Summing block
Input signal
Error signal
Output signal
sensor
Block DiagramsBlock Diagrams
l Cascading Blocks
)(2 sG)(1 sG
U(s) X(s)Y(s)
)( and
)(between function transfer equivalent
)(
)(
)(
)(
)(
)()()(
)(
)()(,
)(
)()(
21
21
ty
tu
sU
sY
sX
sY
sU
sXsGsG
sX
sYsG
sU
sXsG
=
=
⋅=⋅→
==
Block DiagramsBlock Diagrams
l Cascading Blocks
)(2 sG)(1 sG
U(s) X(s)Y(s)
Can be replaced by:
)()()( 21 sGsGsG ⋅=U(s) Y(s)
Closed Loop Transfer FunctionClosed Loop Transfer Function
l
+- G(s)
H(s)B(s)
Y(s)E(s)U(s)
)()()(
)()()(
)()()(
sYsHsU
sBsUsE
sEsGsY
−=
−=
=
Closed Loop Transfer FunctionClosed Loop Transfer Function
l +- G(s)
H(s)B(s)
Y(s)E(s)U(s)
[ ]
)()()(1
)(
)(
)(
)()()()()(
get we gEliminatin
sGsHsG
sG
sU
sY
sYsHsUsGsY
E(s)
equiv=+
=→
−=
Closed Loop Transfer FunctionClosed Loop Transfer Function
l ++ G(s)
H(s)B(s)
Y(s)E(s)U(s)
)()(1
)()(
feedback positive of case in the that Note
sHsG
sGsGequiv
−=
Positive feedback
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
Y(s)U(s)G2(s)++
Disturbance D(s)
To analyse this we use superposition
1. Consider set point to be zero and compute output
2. Consider disturbance to be zero and compute output
3. Add both outputs in 1 And 2 together
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
YD(s)U(s)=0
G2(s)++
Disturbance D(s)
1. Consider set-pint U(s) to be zero and compute output
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
YD(s)
G2(s)++
Disturbance D(s)
)()()(1
)(
)(
)(
0h output wit of L.T. be )(let
21
2
sHsGsG
sG
sD
sY
U(s)sY
D
D
+=
=
U(s)=0
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
YU(s)E(s)
G2(s)++
Disturbance D(s)=0
U(s)
2. Consider disturbance to be zero and compute output
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
YU(s)E(s)
G2(s)++
Disturbance D(s)=0
U(s)
)()()(1
)()(
)(
)(
21
21
sHsGsG
sGsG
sU
sYU
+=
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
Y(s)E(s)
G2(s)++
Disturbance D(s)
U(s)
3. Add both outputs together
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
l
+- G1(s)
H(s)B(s)
Y(s)E(s)
G2(s)++
Disturbance D(s)
U(s)
[ ])()()()()()(1
)(
)()()(
2
21
1 sDsUsGsHsGsG
sG
sYsYsY UD
++
=
+=
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbancel
[ ]
0)(
)(
1)()()(&
1)()( If
)()()()()()(1
)()(
21
1
1
21
1
→
>>
>>→
++
=
sD
sY
sHsGsG
sHsG
sDsUsGsHsGsG
sGsY
D
Effect of disturbance is minimised
- one advantage of a closed loop system
Closed Loop System subjected to Closed Loop System subjected to a Disturbancea Disturbance
ll
)(),( oft independen is )(
)(then
1 )()()( if
increases )()()( as )(
1
)(
)(
21
21
21
sGsGsU
sY
sHsGsG
sHsGsGsHsU
sY
U
U
>>→
→
i.e. independent of small variations in G1(s),G2(s)
Another advantage of a closed loop system
Block Diagram AlgebraBlock Diagram Algebra
l Often control systems can be quite complex
l To adequately model and predict their behaviour it is often desirable to reduce system down to a simple closed loop transfer function
l Next lecture we will look at techniques for doing this
State Space ModellingState Space Modelling
l Time domain approach
l express system as a series of first order differential equations
l assemble this set of first order equations into a matrix-vector equation
l very useful in higher order systems
l will be covered in detail in EEB511
State Space Modelling State Space Modelling -- ExampleExample
l Spring Mass system with damping constant D
Equilibrium position
x
kxxDxM −=+ &&&Mass = M
Spring constant = k
State Space Modelling State Space Modelling -- ExampleExample
l Spring Mass system with damping constant D
Equilibrium position
x
xtxxtx &== )(,)( states Define 21
Mass = M
Spring constant = k
State Space Modelling State Space Modelling -- ExampleExample
l
−−=
−−==
2
1
2
1
12221
10
formtor matrix vecIn
,
sD.E.'order first twobecomes D.E.order 2nd
x
x
M
D
M
kx
x
xM
kx
M
Dxxx
&
&
&&
State Space ModellingState Space Modelling
l In general state space model is of the form
matrices sizedely appropriat are
toroutput vec
signalsinput of vector &
vectorstate
DC,B,A,
y
DuCxy
EquationOutput
u
xBuAxx
Equation State
=
+=
=
=+= where&
SensorsSensors
l Dependent on application
l Usually present in feedback path of closed loop system
l In time constant of sensor is very small compared with system time constants then sensor may be represented by a simple time constant
SensorsSensors
q Can also be a source of noise
q Effect of noise can be amplified by any differentiation blocks in loop
§ i.e.transfer function blocks of the form Ks
MATLABMATLAB
q Widely used in the control field
q many control designs are developed in MATLAB before converting to C or assembly code.
§ Automatic conversion software exists
q available as a student edition and on the EESE network
HomeworksHomeworks
q Nise chapter 1: 2, 3, 5, 17(a)
q Nise chapter 2: 17, 25, 29, 37, 42
Next LectureNext Lecture
l Block diagram algebra, transient response of LTI systems - 1st, 2nd, & higher order systems
l time domain performance measures
l significance of pole locations