L1 State Space Analysis

8/12/2019 L1 State Space Analysis

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DIGITAL CONTROL

SUBMITTE D BY:

SWATI NEGI

SEMINAR


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1. State Variable

Response

2.State Transition

Matrix

3.Controllability 4.Observability

State SpaceAnalysis


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= A X + B U ( 1 )

Y = C X + D U

W H E R E X I S T H E N X 1 S TAT E V E C T O R ,

U I S T H E I S A S C A L A R I N P U T , A I S N X N C O N S T A N T M A T R I X A N D BI S T H E N X 1 C O N S T A N T V EC T O R

STATE -EQUATION FORM:


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STATE-VARIABLE RESPONSE OF LINEAR SYSTEMS

Matrix exponential power series is given by:

eAt = I+At+A2t2/2! +Aktk/K!+. (3)

Homogenous state equation is given by

= AxThe solution of above equation is given by

X(t)= eAt x(0) (4)

From the equation no.4 it is observed that the initialstate x(0) at t=0 is driven to a state x(t) at time t.

This transition in state is carried out by the matrixexponential eAt .Therefore eAt is known as State

Transition Matrix


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PROPERTIES OF STATE TRANSITION MATRIX

(0) = I, which simply states that the stateresponse at time t = 0 is identical to the initialconditions.(t) = 1(t).

(t1)(t2) = (t1+ t2)

If A is a diagonal matrix then eAtis also diagonal,

and each element on thediagonal is an exponential in the correspondingdiagonal element of the Amatrix, that is eaiit.


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EVALUATION OF STATE TRANSITION MATRIX

Evaluation using Inverse Laplace Transforms

eAt= L-1[sI-A]-1

Evaluation using Similarity Transformation

eAt =PetP-1


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THE FORCED STATE RESPONSE OF LINEAR SYSTEMS

Non Homogenous equation is given by

(t) = ax(t) +bu(t)

State transition Equation is given by

X(t)= eAtx(0)+ eA(t-) bu()dT

0


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SOLUTION OF STATE DIFFERENCE EQUATION

We will find the solution of the state equation

X(k+1)=Fx(k)+gu(k)

Recursion procedure is simple and convenient for

digital computations.By repeative procedure we obtain the solution of state

equation as

1

0

)1(k )()0()(k

i

ik iguFXFkX


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STATE TRANSITION MATRIX

We can also write the solution of the homogenousstate equation x(k+1)=Fx(k)

as x(k)=Fkx(0)

Fkis called state transition matrix. Evaluation using inverse z-transforms

Fk=z-1[(zI-F)-1z]

Evaluation using Similarity TransformationFk = PekP-1


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CONTROLLABILITY

Controllability: A control system is said to becompletely state controllable if it is possible totransfer the system from any arbitrary initial state to

any desired state in a finite time period. That is, acontrol system is controllable if every state variablecan be controlled in a finite time period by someunconstrained control signal. If any state variable is

independent of the control signal, then it isimpossible to control this state variable and thereforethe system is uncontrollable.


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CONTROLLABILITY FOR A DISCRETE-TIMECONTROL SYSTEM

Consider the discrete-time system defined by

. If given the initial state X(0), can we find acontrol signal to drive the system to a desiredstate X(k)? In other words, is this systemcontrollable.

Using the definition just given, we shall nowderive the condition for complete statecontrollability.

)()(

)()(1)X(k

kCXky

kgukFX


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Controllability For A Discrete-time Control System

Since the solution of the above equation is

)1()1()0()0(

)()0()(

21k

1

0

)1(k

kguguFguFXF

iguFXFkX

kk

k

i

ik


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Controllability For a Discrete-time Control System

That is

)0(

)2(

)1(

][

)0()2()1(

)1()1()0()0()(

12

1

21k

u

ku

ku

gFgFFgg

guFkFgukgu

kguguFguFXFkX

k

k

kk


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Controllability for a discrete-time control system

As g is an k1matrix, we find that each term ofthe matrix is ank1 matrix or column vector. That is thematrix

is an kk matrix. If itsdeterminant is not zero, we can get the

inverse of this matrix. Then we have

]g[ 12 gFgfFg k

)]0()([][

)0(

)2(

)1(

k112 XFkXgFgFFgg

u

ku

ku

k

][ 12 gFgFFgg k


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Controllability for a discrete-time control system

That means that if we chose the input as theabove, we can transfer the system from anyinitial stateX(0)to any arbitrary stateX(k)inat most ksampling periods.

That is the system is completely controllable ifand only if the inverse of controllability

matrixis available. Or the rank

of the controllability matrix is k.

][ 12 gFgFFggW kC


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Example for controllability

Example 1: Considering a system defined by

is this system controllable?

First, write the state equation for the abovesystem

16.0

2

)(

)()(

2

zz

z

zU

zYzG

)()(

)()(1)X(k

kCXky

kgukFX


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Examples for controllability

That is

Next, find the controllability matrix

)(

)(1)2()(

)(1

0

)(

)(

1-16.0

10

)1(

)1(

2

1

2

1

2

1

kx

kxky

kukx

kx

kx

kx

][]g[12

FgggFgFFg k


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OBSERVABILITY

Observability: A control system is said to beobservable if every initial state X(0) can bedetermined from the observation of Y(k)over a finitenumber of sampling periods. That is, a controlsystem is observable if every initial state isdetermined from the observation of Y(k) and inputin a finite time period. If any state is not determined,then the system is unobservable.


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Observability

u

y=

x Ax B

Cx

o A system is completely observable if and only if there existsa finite time T such that the initial state x(0) can bedetermined from the observation history y(t) given thecontrol u(t).

nn

nn

rank[ ]o nPrank[ ]o nP 0o P

1Observability Matrix [ ]n ToP C CA CA

nn


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Observability For a Discrete-time Control System

Consider the discrete-time system defined by

We assume that u(k) is a constant. If given theinput signal u(k) and the output y(k), can wetrack back to the initial state X(0)? Or, is thissystem observable?

)()(

)()(1)X(k

kCXky

kgukFX


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Since the solution of the above equation is

Andy(k)is

)()0()(1

0

)1(k

k

i

ik iguFXFkX

1

0

)1(k )()0()()(k

i

ik iguCFXCFkCXky


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Let k=0, 1, 2n-1, we have

2

0

)2(1-n

2

)()0()1()1(

)1()0()0()2()2(

)0()0()1()1(

)0()0(

n

i

in

iguCFXCFnCXny

CguCFguXCFCXy

CguCFXCXy

CXy


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Rewrite all the above equations in matrixequation, we have

)0(

)2(

)1(

CC0

C00

000

)0(

)1(

)1(

)0(

2-n1 u

nu

nu

gFg

gX

CF

CF

C

ny

y

y

n


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As C is an 1nmatrix, we find that each term ofthe following matrix is an 1nmatrix or row

vector. That is the matrix is an nnmatrix. If

its determinant is not zero, we can get theinverse of the observability matrix.

1n

O

CF

CF

C

W


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Then we have

)0(

)2(

)1(

CC0

C00

000

)1(

)1(

)0(

)0()0(

2-n

1

1

1

1

1

1

1

u

nu

nu

gFg

g

CF

CF

C

ny

y

y

CF

CF

C

X

CF

CF

C

CF

CF

C

X

nn

nn


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This means that if we knew the input and theoutput in n sampling periods, we can trackback to the system initial state.

That is the system is completely observable if andonly if the inverse of observability matrix isavailable. Or the rank of the observabilitymatrix is n.


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Example for observability

0 1 2

0 1 0 0

0 0 1 0 u

1

y= 1 0 0 0 u

a a a

x x

x

0 1 2

2

0 1 0

0 0 1 ,

1 0 0

0 1 0 ,

0 0 1

a a a

A

C

CA

CA

11 0 0

[ ] 0 1 0

0 0 1

n T

o

P C CA CA1

oP

Observablerank[ ]o nP


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Controllability And Observability For Continuous System

For a continuous system

We can draw the similar conclusions as thediscrete system

)(

)()()(

tCXy

tButAXtX

dt

d


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Controllability of Continuous System

A system is completely controllable if and only ifthe inverse of controllability matrix

is available. Or the rank

of the controllability matrix is n.

][ 12 BABAABB n


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Controllability And Observability Of a Two-stateSystem

Example

2 0 1u

1 1 1

y= 1 1

x x

x

1 2 1 2, ,

1 2 1 21 1

1 1 , 1 1 ,1 1

c

T

o

B AB P B AB

C CA P C CA

Rank =2=n hence,

0c o P P

Not Controllable and Not Observable


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THANK

YOU

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L1 State Space Analysis