Homework 2: Interacting Tank-in-Series System

President University Erwin Sitompul SMI 3/1

Dr.-Ing. Erwin SitompulPresident University

Lecture 3System Modeling and Identification

http://zitompul.wordpress.com


Homework 2: Interacting Tank-in-Series SystemChapter 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 sdx t f t x t x t u t u t r t r tdt

State EquationsChapter 2 General Process Models

22 1 1 1

( ) , ( ), , ( ), ( ), , ( ), ( ), , ( )n m sdx t f t x t x t u t u t r t r tdt

1 1 1( ) , ( ), , ( ), ( ), , ( ), ( ), , ( )n

n n m sdx t f t x t x t u t u t r t r tdt

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 equatios can be written compactly as:

( ) , ( ), ( ), ( )d t t t t tdt

x f 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 equatios 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 Tq c A dt q c A q c A

j l

( )p

p p

q c AdT A qT T Tdt V c V c V

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

then 11 1 1 1, ,dx f x u r

dt

1 1 1 1

( )p

p p

q c A A qx x u rV c V c V

1y xState Space Equations

General Process Models


Chapter 2 Linearization

LinearizationLinearization 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 xf x f x x x x

n

A point near x0

Only the linear terms are used for the linearization



Taylor Expansion SeriesVector Case

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

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

00

00

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 00 0

( ) ( )( ) ( )

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

( ) ( )t tt t

t t t tt t t

t t

x x

u u

f x u f x ux x u

x u



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

00

( )( )

( ), ( ),

( ) tt

t tt

xu

f x uA

x

00

( )( )

( ), ( )

( ) tt

t tt

xu

f x uB

u

00

1 1

1

( )1( )

,n

n n

tnt

f fx x

f fx x

x

u

A

00

1 1

1

( )1( )

, m

n n

tmt

f fu u

f fu u

x

u

B

n : Number of statesm : Number of inputs



Taylor Expansion SeriesPerforming 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

00

00

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 00 0

( ) ( )( ) ( )

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

( ) ( )t tt t

t t t tt t t

t t

x x

u u

g x u g x uy x u

x u



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

00

( )( )

( ), ( ),

( ) tt

t tt

xu

g x uC

x

00

( )( )

( ), ( )( ) t

t

t tt

xu

g x uD

u

00

1 1

1

( )1( )

,n

r r

tnt

g gx x

g gx x

x

u

C

00

1 1

1

( )1( )

,m

r r

tmt

g uu u

g gu 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 ah ghA A

v1

qi

qoV 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 0i,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 12 hh

qq

a gh h qA h A

1i

0

1 2 12a gh h qA h A

i 1 2q ah ghA A

y h



Single Tank System

0 0i,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 ah 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 SystemThe Matlab-Simulink model of the linearized system is

shown below. All parameters take the previous values.

1i

0

1 2 12a gh h qA h A

y h



Single Tank SystemThe 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: Interacting Tank-in-Series System

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 digit of your Student ID.

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

Simulink file + scope.



Homework 3: Triangular-Prism-Shaped TankLinearize 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 2 digits of your Student ID.

For example: Bernard Andrew qi2= 0.3 liter/s, Sugianto qi2= 1.0 liter/s.

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

NEW v

qi1

qo

a

qi2

hmaxh

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Homework 2: Interacting Tank-in-Series System