3.State-Space Representation of Systems.pdf

Warp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be usedWarp : Ply is preferred for warps

Weft : Singles can be used

1. 1
EEE 2221
Mathematical Models of Systems: State-Space
Representation
Nayeema Hasan
Lecturer
Dept. of EEE, KUET.
Reference Books:
‘Modern Control Engineering’-by Katsuhiko Ogata.

2. Complex Variable and Complex Function
2

3. Euler’s Theorem
3

4. Euler’s Theorem
4

5. Laplace Transformation
5

6. Laplace Transformation
6

7. Laplace Transform of Different Functions
7

8. Laplace Transform of Different Functions
8

9. Laplace Transform of Different Functions
9

10. Properties of Laplace Transform
10

11. Properties of Laplace Transform
11

12. Properties of Laplace Transform
12

13. Modeling in State-Space
13
STATE:Thestateofasystemisamathematicalstructurecontainingaset
ofnvariablesx
1(t),x
2(t),...,x
i(t),...,x
n(t),calledthestatevariables,
suchthattheinitialvaluesx
i(t
0)ofthissetandthesysteminputsu
j(t)are
sufficienttodescribeuniquelythesystem’sfutureresponseoft≥t
0.
Aminimumsetofstatevariablesisrequiredtorepresentthesystem
accurately.Theminputs,u
1(t),u
2(t),...,u
j(t),...,u
m(t),aredeterministic;
i.e.,theyhavespecificvaluesforallvaluesoftimet≥t
0.
Generallytheinitialstartingtimet
0istakentobezero.
StateVariables.Thestatevariablesofadynamicsystemarethevariables
makingupthesmallestsetofvariablesthatdeterminethestateofthe
dynamicsystem.Ifatleastnvariablesx,x
1,x
2,...,x
n,areneededto
completelydescribethebehaviorofadynamicsystem(sothatoncethe
inputisgivenfortztoandtheinitialstateatt=toisspecified,thefuture
stateofthesystemiscompletelydetermined),thensuchnvariablesarea
setofstatevariables.

14. Modeling in State-Space
14
STATEVECTOR:Thesetofstatevariablesx
i(t)representstheelements
orcomponentsofthen-dimensionalstatevectorx(t);thatis,
Theorderofthesystemcharacteristicequationisn,andthestateequation
representationofthesystemconsistsofnfirst-orderdifferentialequations.
Whenalltheinputsu
j(t)toagivensystemarespecifiedfort>t
0,the
resultingstatevectoruniquelydeterminesthesystembehaviorforanyt>
t
0.

15. Modeling in State-Space
15
STATEVECTOR:Thesetofstatevariablesx
i(t)representstheelements
orcomponentsofthen-dimensionalstatevectorx(t);thatis,
Theorderofthesystemcharacteristicequationisn,andthestateequation
representationofthesystemconsistsofnfirst-orderdifferentialequations.
Whenalltheinputsu
j(t)toagivensystemarespecifiedfort>t
0,the
resultingstatevectoruniquelydeterminesthesystembehaviorforanyt>
t
0.

16. Modeling in State-Space
16
STATESPACE:Statespaceisdefinedasthen-dimensionalspaceinwhich
thecomponentsofthestatevectorrepresentitscoordinateaxes.Then-
dimensionalspacewhosecoordinateaxesconsistofthex
laxis,x
2axis,...,
x,axis,wherex
l,x
2,...,x
n,arestatevariables;iscalledastatespace.Any
statecanberepresentedbyapointinthestatespace.
STATETRAJECTORY:Statetrajectoryisdefinedasthepathproducedin
thestatespacebythestatevectorx(t)asitchangeswiththepassageof
time.
Statespaceandstatetrajectoryinthetwo-dimensionalcasearereferredto
asthephaseplaneandphasetrajectory,respectively.

17. First Step: Selection of State Variables
17

18. First Step: Selection of State Variables
18

19. First Step: Selection of State Variables
19

20. State Variable Selection for Electric circuits
20
Example1:
Writedownthestate-spaceequationsfortheseriesresistor-inductor
circuitbelow.
.

21. State Variable Selection for Electric circuits
21
Example2:
Writedownthestate-spaceequationsfortheseriesRLCcircuitbelow.
Fig a: The Series RLC Circuit

22. State Equations for Electric circuits
22
Fig b: The redrawn Series RLC Circuit;
With node b as the reference node

23. State Equations and Matrix Notation
23
Thestateequationsofasystemareasetofnfirst-orderdifferential
equations,wherenisthenumberofindependentstates.Thestate
equationsrepresentedbyaboveequationsareexpressedinmatrix
notation.FortheaboveseriesRLCcircuitfromthestateequations,we
getthematrixformasfollows:

24. State Equations in Matrix Notation
24
Where,
Here, x = n×1 column vector. A = n×n plant co-efficient matrix.
x = n×1 state vector. B = n×1 control matrix.
And input u = [u] is a one dimensional control vector

25. Output Equation in Matrix Notation
25
Iftheoutputquantityy(t)fortheRLCcircuitofFig.aisthevoltage
acrossthecapacitorv
C,then

26. State Equation for MIMO system
26
Theaboveequationsareforasingle-inputsingle-output(SISO)system.
Foramultiple-inputmultiple-output(MIMO)system,withminputs
andloutputs,theseequationsbecome
Where,

27. State Equations for Electric circuits
27
Example3:ObtainthestateequationforthecircuitinFig.C,wherei
2
isconsideredtobetheoutputofthissystem.theseriesRLCcircuit
below.
Fig C
Solution:
The assignedstate
variablesarex
1=i
1,x
2=i
2,
andx
3=v
C.Thus,twoloop
andonenodeequations
arewrittenasfollows:.

28. State Equations for Electric circuits
28
Thethreestatevariablesareindependent,andthesystemstateand
outputequationsare:

29. State Equations for Electric circuits
29
Example4:ObtainthestateequationsforthecircuitofFig.2.7.The
outputisthevoltagev1.Theinputorcontrolvariableisacurrent
sourcei(t).
Fig D

30. State Equations for Electric circuits
30
Solution:Theassignedstatevariablesarei1,i2,i3,v1,andv2.Three
loopequationsandtwonodeequationsarewritten:
WritingtheloopequationthroughL1,
L2,andL3,andthenintegrating
(multiplyingby1/D),gives
whereKisafunctionoftheinitial
conditions.Thisequationrevealsthat
oneinductorcurrentisdependentupon
theothertwoinductorcurrents.Thus,
thiscircuithasonlyfourindependent
physicalstatevariables,twoinductor
currentsandtwocapacitorvoltages.

31. State Equations for Electric circuits
31
Thefourindependentstatevariablesaredesignatedasx1=v1,x2=v2,
x3=i1,andx4=i2,andthecontrolvariableisu=i.

32. State Equations for Mechanical Systems
32
Example01:Considerthemechanicalsystemshown
inFigure3-16.Weassumethatthesystemislinear.
Theexternalforceu(t)istheinputtothesystem,
andthedisplacementy(t)ofthemassistheoutput.
Thedisplacementy(t)ismeasuredfromthe
equilibriumpositionintheabsenceoftheexternal
force.Thissystemisasingle-input-single-output
system.
Solution:Fromthediagram,thesystemequationis:
Thissystemisofsecondorder.Thismeansthatthe
systeminvolvestwointegrators.Letusdefinestate
variablesas:

33. State Equations for Mechanical Systems
33
Thenweobtain:
Or,
Theoutputequationis:

34. State Equations for Mechanical Systems
34
Invector-matrixform,thestateequationis:

35. State-Space Modeling of Servomotor
35
Figure:CircuitDiagramofaDCmotor

36. State-Space Modeling of Servomotor
36

37. State-Space Modeling of Servomotor
37
Figure: Inertia and friction as motor load

38. State-Space Modeling of Servomotor
38

39. State-Space Modeling of Servomotor
39
Thestate-spaceequationscanbederivedasfollows:

40. State-Space Modeling of Servomotor
40
Finally,thestate-spaceequationsinmatrixformcanbewrittenasfollows:

41. 41