ch3 modeling

7/23/2019 Ch3 Modeling

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Lamar U.Lamar U.

Chapter 3Chapter 3

MODELING OFMODELING OF

DYNAMIC SYSTEMDYNAMIC SYSTEM


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Lamar U.Lamar U. Transfer Function

Ratio of Laplace transform of the output to theLaplace transform of input under the assumptionthat all I.C. = 0

Used to describe forced response of system

initially in equilibrium (I.C.=0) roperty of the system! same for any input"

#ummari$es all information about %&' (model)


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Lamar U.Lamar U. Transfer Function

ransfer function completely characteri$essystems performance.

he transfer function can be represented as a bloc*

dia+ram ,ith input- output- and the system transferfunction inside the bloc*.

he Laplace transform of the output (response) is

equal to the product of the transfer function and

the Laplace transform of the input-


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Lamar U.Lamar U. Impulse

.he notion of an impulse function ( or function) rather theoretical-ho,e/er- its role in the system analysis is in/aluable.

. Definition: he impulse function (function) is a real /alued function of

real ar+ument ,ith the follo,in+ properties

. Consider a rectan+ular pulse ,ith base c > 0 and hei+ht 12c sho,n in fi+ure.

. If c 0 then the pulse becomes to be /ery tall and /ery thin- ,hile its area

remains equal to 1.

. 3rom practical point of /ie,- for /ery small c > 0 such a pulse +i/es +ooda ro4imation of the im ulse function.


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Lamar U.Lamar U. Impulse Response Function


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Lamar U.Lamar U. BLOCK DIAGRAMBLOCK DIAGRAM

5loc* dia+ram pictorial representation of the functions performed bycomponents and flo, of si+nals.

5loc*

#ummin+ point

5ranch point

%penloop transfer function 52' = 67

3eed for,ard transfer function C2' = 6Closedloop transfer function C2R = 62(1867)

E(s)

B(s)


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Lamar U.Lamar U. Com!inin" !loc#s in casca$e


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Lamar U.Lamar U. %arallel u!s&stem


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Lamar U.Lamar U.Mo'in" a summin" point !e(in$ a !loc#


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Lamar U.Lamar U.Mo'in" a summin" point a(ea$ of a !loc#


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BLOCK DIAGRAM R)DUCTIO*BLOCK DIAGRAM R)DUCTIO*

1. he product of the 3 in the feedfor,ard direction remains the same.9. he product of the 3 around the loop remains the same.


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Lamar U.Lamar U. AUTOMATIC CO*TROLL)RAUTOMATIC CO*TROLL)R

:n automatic controllerautomatic controllercompares the actual /alue the plantoutput ,ith the reference input (desired /alue)- determines

the de/iation- and produces a control si+nal that ,ill reduce

the de/iation to $ero or to a small /alue. he manner in

,hich the automatic controller produces the control si+nal is

called the control action.


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Lamar U.Lamar U. AUTOMATIC CO*TROLL)RAUTOMATIC CO*TROLL)R

,oposition (onoff) control action

roportional control action

Inte+ral control action

&eri/ati/e control action

I control action

& control action

I& control action


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Lamar U.Lamar U. Close$+Loop &stem u!,ecte$ to a Distur!anceClose$+Loop &stem u!,ecte$ to a Distur!ance

,o inputs (the reference input and disturbance) in a linear system each input can be treated

independently of the other! and the outputs correspondin+ to each input alone can be

added to +i/e the complete output.

:d/anta+e of the closedloop system1. Consider 6171; 1 and G1G2H1; 1. In this case- C&2 & becomes almost $ero-

and the effect of the disturbance is suppressed.

9. If G1G2H1; 1then CR2Rbecomes independent of 61and G2and becomes

in/ersely proportional toH so that the /ariations of 61and G2do not affect the

CR2R. It can easily be seen that any closedloop system ,ith unity


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Lamar U.Lamar U. MOD)LI*G I* TAT) %AC)MOD)LI*G I* TAT) %AC)

Comple4 system in/ol/es multiple inputs and multiple outputsand may be time /aryin+. It can be analy$ed based on theconcept of state(modern control theory).

#tate < the smallest set of /ariables such that the *no,led+e ofthese /ariables and the input functions ,ill- ,ith the equations

describin+ the dynamics- pro/ide the future state and output ofthe system.

#tate ariables describe the future response of a system- +i/enthe present state (initial conditions)- the e4citation inputs- andthe e-uations describin+ the dynamics.


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Lamar U.Lamar U. A %RI*G+MA+DAM%)R T)MA %RI*G+MA+DAM%)R T)M

In this case the position and the 'elocit& of themass are sufficient to describe the state of the

system.

&efine a set of state /ariables as (x1, x2)- ,here

x1(t) = y(t); x2(t) = dy(t)/dt

'.%.>.

M dx2(t)/dt !x2(t) "x1(t) = #(t)

,o firstorder differential equationsdx1/dt = x2

dx2/dt = $(!/M) x2$ ("/M) x1 (1/M) #

#

%x

x

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22

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Lamar U.Lamar U. TAT) DIFF)R)*TIAL )/UATIO*TAT) DIFF)R)*TIAL )/UATIO*

he state of a system is described by a set of firstorder differential

equations ,ritten in terms of the state /ariables ?x1, x2,&, xn@.

In matri4 form state differential equation and output equation

0(ere1

2 = state /ector! & = output /ector! u = input /ector

A = system matri4! B = input matri4! C = output matri4! D = feedfor,ard matri4


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Lamar U.Lamar U.R:A#3'R 3UACI%A 3R%> #:' 'BU:I%AR:A#3'R 3UACI%A 3R%> #:' 'BU:I%A

6i/en the state /ariable model- obtain the transfer function of a

sin+leinput- sin+leoutput system.

here- y is the sin+le output and u is the sin+le input.

he Laplace transform of these equations are +i/en by- assume 2(0)=0

he transfer function 6(s) = D(s)2U(s) is


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Lamar U.Lamar U.#:' R'R'#'A:I%A %3 &DA:>IC #D#'>#:' R'R'#'A:I%A %3 &DA:>IC #D#'>

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Define state 'aria!les as tate e-uation

Or1

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Lamar U.Lamar U.

3rom ransfer 3unction to #tate #pace

< ?:- 5- C- &@ = tf9ss(num-den)

< '4. pEF

3rom #tate #pace to ransfer 3unction

< ?num- den@ = ss9tf(:- 5- C- &- iu)

:I%A %3R:A#3%R>:I%A %3

>:7'>:IC:L >%&'L# I7 >:L:5>:7'>:IC:L >%&'L# I7 >:L:5

ch3 modeling

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