solution to homework 2

President University Erwin Sitompul SMI 3/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 3

System Modeling and Identification

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Solution to Homework 2Chapter 2 Examples of Dynamic Mathematical Models

1,s

2,s

0.6378 m0.3189 m

hh

1,0 2,0 0h h

1,0 2,00.1 m, 0.8 mh h


A suitable model for a large class of continuous theoretical processes is a set of ordinary differential equations of the form:

11 1 1 1

( ), ( ), , ( ), ( ), , ( ), ( ), , ( )n m s

dx tf t x t x t u t u t r t r t

dt

State EquationsChapter 2 General Process Models

22 1 1 1

( ), ( ), , ( ), ( ), , ( ), ( ), , ( )n m s


dt

1 1 1

( ), ( ), , ( ), ( ), , ( ), ( ), , ( )n

n n m s


dt

t : Time variable x1,...,xn : State variablesu1,...,um : Manipulated variablesr1,...,rs : Disturbance, nonmanipulable variablesf1,...,fn : Functions


A model of process measurement can be written as a set of algebraic equations:

1 1 1 1 1( ) , ( ), , ( ), ( ), , ( ), ( ), , ( )n m m mty t g t x t x t u t u t r t r t

Output EquationsChapter 2 General Process Models

2 2 1 1 1( ) , ( ), , ( ), ( ), , ( ), ( ), , ( )n m m mty t g t x t x t u t u t r t r t

1 1 1( ) , ( ), , ( ), ( ), , ( ), ( ), , ( )r r n m m mty t g t x t x t u t u t r t r t

t : Time variable x1,...,xn : State variablesu1,...,um : Manipulated variablesrm1,...,rmt : Disturbance, nonmanipulable variables at outputy1,...,yr : Measurable output variablesg1,...,gr : Functions


State Equations in Vector FormChapter 2 General Process Models

If the vectors of state variables x, manipulated variables u, disturbance variables r, and vectors of functions f are defined as:

1 1 1 1

, , ,

n m s n

x u r f

x u r f

x u r f

Then the set of state equations can be written compactly as:

( ), ( ), ( ), ( )

d tt t t t

dt

xf x u r


Output Equations in Vector FormChapter 2 General Process Models

If the vectors of output variables y, disturbance variables rm, and vectors of functions g are defined as:

1 1 1

, , m

m

r mt r

y r g

y r g

y r g

Then the set of algebraic output equations can be written compactly as:

( ) , ( ), ( ), ( )mt t t t ty g x u r


Chapter 2

Heat Exchanger in State Space Form

Tj

qTl

qT

Vρ Tcp

l jp p

p p p

V c q cdT AT T T

q c A dt q c A q c A

j l

( )p

p p

q c AdT A qT T T

dt V c V c V

If ,1 1 j 1 l, , x T u T r T

then 11 1 1 1, ,

dxf x u r

dt

1 1 1 1

( )p

p p

q c A A qx x u r

V c V c V

1y xState Space Equations

General Process Models


Chapter 2

Double-Pipe Heat Exchanger in State Space Form General Process Models

Processes with distributed parameters are usually approximated by a series of well-mixed lumped parameter processes.

This is also the case for the heat exchanger, as shown in the next figure, which is divided into n well-mixed heat exchangers.

The space variable is divided into n equal lengths within the interval [0, L].

After rearrangement, the mathematical model of the heat exchanger is of the form:

i ii o

i i ip p

dT Tq A AT T

dt A A c A c

1 2 2 o

dT Tc c T c T

dt

where 1

i

qc

A and 2

i p

Ac

A c

After rearrangement, the mathematical model of the heat exchanger is of the form:


Double-Pipe Heat Exchanger in State Space Form We introduce the state parameters

Chapter 2 General Process Models

1( ) ,L

x t T tn

2

2( ) ,

Lx t T t

n

( ) ,nx t T L t

1 ou T

1 (0, )r T t


Difference Quotient The derivation with respect to space, δT/δτ, will now be

approximated by using a difference quotient. The difference quotient itself is the equation that can be used to

approximately calculate the slope of a function at a certain point. There are three formations of difference quotient:

Chapter 2 General Process Models

1( ) ( )( ) i iF

ix x

f x f xf x

x x

1( ) ( )( ) i iB

ix x

f x f xf x

x x

1 1( ) ( ) ( )

2C i i

ix x

f x f x f x

x x

●Forward Difference

●Backward Difference

●Central Differenceix

( )if x

x

( )f x

1ix 1ix

1( )if x

1( )if x


Double-Pipe Heat Exchanger in State Space Form Chapter 2 General Process Models

Replacing δT/δτ with its corresponding difference will result a model that consists of a set of ordinary differential equations only:

1 1 11 2 1 2 1

( )dx x rc c x c u

dt L n

2 2 11 2 2 2 1

( )dx x xc c x c u

dt L n

1

1 2 2 1

( )n n nn

dx x xc c x c u

dt L n

1y x

State Space Equations


Chapter 2 Linearization

Linearization Linearization is a procedure to replace a nonlinear original model

with its linear approximation. Linearization is done around a constant operating point. It is assumed that the process variables change only very little and

their deviations from steady state remain small.

0x

0( )f x

Operating point

Linearization

x

( )f x

Nonlinear Model

Linear Model

Taylor series expansion



Linearization

( ) ( ), ( )t t tx f x u

( ) ( ), ( )t t ty g x u

The approximation model will be in the form of state space equations

An operating point x0(t) is chosen, and the input u0(t) is required to maintain this operating point.

In steady state, there will be no state change at the operating point, or x0(t) = 0

0 0 0( ) ( ), ( )t t t x f x u 0

0 0 0( ) ( ), ( )t t ty g x u



Taylor Expansion Series

0x

0( )f x

0x x x

0( ) ( )f x x f x

Scalar Case

0( ) ( )f x f x x

20 0 00

( ) ( ) ( )( ) ( ) ( ) ( )

1! 2! !

nnf x f x f x

f x f x x x xn

A point near x0

Only the linear terms are used for the linearization



Taylor Expansion Series Vector Case

( ) ( ), ( ) ,t t tx f x u0( ) ( ) ( )t t t x x x

0 0 0( ) ( ) ( ) ( ), ( ) ( )t t t t t t x x f x x u u

0

0

0

0

0 0 0( )( )

( )( )

( ), ( )( ) ( ) ( ), ( ) ( )

( )

( ), ( ) ( )

( )

tt

tt

t tt t t t t

t

t tt

t

xu

xu

f x ux x f x u x

x

f x uu

u

where

0 0

0 0

( ) ( )( ) ( )

( ), ( ) ( ), ( )( ) ( ) ( )

( ) ( )t tt t

t t t tt t t

t t

x xu u

f x u f x ux x u

x u



Taylor Expansion Series ( ) ( ) ( )t t t x A x B u

0

0

( )( )

( ), ( ),

( ) tt

t t

t

xu

f x uA

x

0

0

( )( )

( ), ( )

( ) tt

t t

t

xu

f x uB

u

0

0

1 1

1

( )1( )

,n

n n

tnt

f f

x x

f f

x x

x

u

A

0

0

1 1

1

( )1( )

, m

n n

tmt

f f

u u

f f

u u

x

u

B

n : Number of statesm : Number of inputs



Taylor Expansion Series Performing the same procedure for the output equations,

( ) ( ), ( )t t ty g x u

00 0( ) ( ) ( ) ( ), ( ) ( )t t t t t t y y g x x u u

0

0

0

0

00 0( )( )

( )( )

( ), ( )( ) ( ) ( ), ( ) ( )

( )

( ), ( ) ( )

( )

tt

tt

t tt t t t t

t

t tt

t

xu

xu

g x uy y g x u x

x

g x uu

u

0 0

0 0

( ) ( )( ) ( )

( ), ( ) ( ), ( )( ) ( ) ( )

( ) ( )t tt t

t t t tt t t

t t

x xu u

g x u g x uy x u

x u



Taylor Expansion Series( ) ( ) ( )t t t y C x D u

0

0

( )( )

( ), ( ),

( ) tt

t t

t

xu

g x uC

x

0

0

( )( )

( ), ( )

( ) tt

t t

t

xu

g x uD

u

0

0

1 1

1

( )1( )

,n

r r

tnt

g g

x x

g g

x x

x

u

C

0

0

1 1

1

( )1( )

,m

r r

tmt

g u

u u

g g

u u

x

u

D

r : Number of outputs



Taylor Expansion Series

0( ) ( ) ( )t t t x x x

0( ) ( ) ( )t t t y y y

( ) ( ) ( )t t t x A x B u

( ) ( ) ( )t t t y C x D u

( ) ( ), ( ) ,t t tx f x u

( ) ( ), ( )t t ty g x u

Nonlinear Model Linear Model



Single Tank System

i 1 2q a

h ghA A

v1

qi

qo

V h

The model of the system is already derived as:

The relationship between h and h in the above equation is nonlinear.

i( , )f h q

An operating point for the linearization is chosen, (h0,qi,0).

y h i( , )g h q



Single Tank System

0 0

i,0 i,0

i ii

i

, ,

h hq q

f h q f h qh h q

h q

The linearization around (h0,qi,0) for the state equation can be calculated as:

00i,0i,0

1i

1 2 1

2 hhqq

a gh h q

A h A

1i

0

1 2 1

2

a gh h q

A h A

i 1 2q a

h ghA A

y h



Single Tank System

0 0

i,0 i,0

i ii

i

, ,

h hq q

g h q g h qy h q

h q

y h

The linearization for the ouput equation is:

i 1 2q a

h ghA A

y h

0h h h

0y y y h

Note that the input of the linearized model is now Δqi.

To obtain the actual value of state and output, the following equation must be enacted:

i i i,0q q q



Single Tank System The Matlab-Simulink model of the linearized system is shown

below. All parameters take the previous values.

1i

0

1 2 1

2

a gh h q

A h A

y h

i3.5418 4h h q

y h



Single Tank System The simulation results

: Original model: Linearized model

i i,0

5 liters sq q

0

0.3185 mh h



Single Tank System If the input qi deviates from the operating point, the linearized

model will deliver inaccurate output.


i i,0

6 liters sq q



Single Tank System If the input qi deviates from the operating point, the linearized

model will deliver inaccurate output.


i i,0

7 liters sq q



Homework 3

v1

qi

h1 h2

v2

q1

a1 a2

Linearize the the interacting tank-in-series system for the operating point resulted by the parameter values as given in Homework 2. For qi, use the last two digits of your Student ID.

For example: 08 qi= 8 liters/s. Submit the mdl-file and the screenshots of the Matlab-Simulink file

+ scope.



Homework 3 (New)Linearize the the triangular-prism-shaped tank for the operating point resulted by the parameter values as given in Homework 2 (New). For qi2, use the last two digits of your Student ID.

For example: 03 qi2= 0.3 liter/s, 17 qi2= 1.7 liter/s.

Submit the mdl-file and the screenshots of the Matlab-Simulink file + scope.

NEWv

qi1

qo

a

qi2

hmax

h

solution to homework 2

Documents

set of state equations

state space form chapter

state equationschapter

state variablesu1

state parameterschapter

general process models

general process modelsif

vectors of state variables