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.



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


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


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.


l Recall derivative property of L.T.

( ) ( )

( ) ( ) ( )00)(

0)(

2

2

2

fsfsFstfdt

dL

fssFtfdt

dL

&−−=

−=


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.


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.


l Recall derivative property of L.T.

( ) ( )

( ) ( ) ( )00)(

0)(

2

2

2

fsfsFstfdt

dL

fssFtfdt

dL

&−−=

−=


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

=++→

=

===++

−=

∑=

−−

&&&&


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.


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


l

)()(...)()(

)()(...)()(

11

10

11

10

sXbssXbsXsbsXsb

sYassYasYsasYsa

mmmm

nnnn

++++=

++++

−−

−−

LTI Systemx(t) y(t)

Taking Laplace Transforms assuming zero initial conditions


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


Table 2.3Voltage-current, voltage-charge, and impedance relationships for capacitors, resistors, and inductors


Figure 2.3RLC network


Figure 2.4Block diagram of series RLC electrical network


Figure 2.11Inverting operationalamplifier circuit for Example 2.14


Table 2.4Force-velocity, force-displacement, and impedance translational relationshipsfor springs, viscous dampers, and mass


Figure 2.15a. Mass, spring, and damper system; b. block diagram


Table 2.5Torque-angular velocity, torque-angular displacement, and impedancerotational relationships for springs, viscous dampers, and inertia


Figure 2.22a. Physical system; b. schematic; c. block diagram


Figure 2.25Three-degrees-of-freedom rotationalsystem


Figure P2.35a. Coupling ofpantograph andcatenary;b. simplifiedrepresentationshowing theactive-controlforce

© 1997 ASME.


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



Reference θ

Tdt

dJ

T

J

=2

2

torque thruster theas & satellite theof

inertia ofmoment theas Defining

θ



Reference θ2

2

1

)(

)()(

function transfer System

)()(

sidesboth of Transforms Laplace Taking

JssT

ssG

sTsJs

=Θ

=

→

=Θ


Figure 2.34NASA flightsimulatorrobot arm withelectromechanicalcontrol systemcomponents

© Debra Lex.


Figure 2.35DC motor:a. schematic12;b. block diagram


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


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.


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


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

=

=

⋅=⋅→

==


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

−=

−=

=


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=+

=→

−=


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


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


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


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


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

+=


l

+- G1(s)

H(s)B(s)

Y(s)E(s)

G2(s)++

Disturbance D(s)

U(s)

3. Add both outputs together


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


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


l Spring Mass system with damping constant D

Equilibrium position

x

xtxxtx &== )(,)( states Define 21

Mass = M

Spring constant = k


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

sistem kendali - modeling of dynamic systems

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