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Advantages of State Space

ApproachClassical

State space approach/ModernControl Approach

Transfer Function

Linear Time InvariantSystem, SISO

Laplace Transform,Frequency domain

Only Input-output

Description: Less DetailDescription on SystemDynamics

State Variable Approach

Linear Time Varying,Nonlinear, Time Invariant,MIMO

Time domain

Detailed description of

Internal behaviour inaddition to I-O properties

2

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CLASSICAL APPROACH (TRANSFER

FUNCTION) The classical approach or frequency domain

technique is based on converting a systems

differential equation to a transfer function. It relates a representation of the output to a

representation of the input.

It can be applied only to linear, time-invariant

systems.

It rapidly provides stability and transient responseinformation.

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The state-space approach (also referred to as the modern, ortime-domain, approach) is a unified method for modeling,analyzing, and designing a wide range of systems.

The state-space approach can be used to represent nonlinear

systems. Also, it can handle, conveniently, systems withnonzero initial conditions and time varying

The state-space approach is also attractive because of theavailability of numerous state-space software package for the

personal computer.

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MODERN APPROACH (STATE-

SPACE) It can be used to model and analyze nonlinear

(backlash, saturation), time-varying (missiles withvarying fuel levels), multi-input multi-outputsystems (i.e. an airplane) with nonzero initialconditions.

But it is not as intuitive as the classical approach.The designer has to engage in several calculationsbefore the physical interpretation of the model isapparent.

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STATE-SPACE REPRESENTATION Select a particular subset of all possible system

variables and call them state variables.

For an nth-order system, write n simultaneousfirst-order differential equations in terms of state

variables.

If we know the initial conditions of all state

variables at t0 and the system input for tt0, we cansolve the simultaneous differential equations forthe state variables for tt0.

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System variable: Any variable that responds to an input or initial conditions in

a system.

State variables: The smallest set of linearly independent systemVariables such that the values of the members of the set at time

along with known forcing function completely determine the value of all

system variables for all .

Sate vector: A vector whose elements are the state variables.

State space: The n-dimensional space whose axes are the state variables.

State equations: A set of n simultaneous, first-order differential equations withn variables, where the n variables to be solved are the state variables.

Output equation: The algebraic equation that expresses the output variables ofa system as linear combinations of the state variables and the inputs.

0t

0tt

Concept of State Variable

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State SpaceThe state variables are the smallest number of states that arerequired to describe the dynamic nature of the system, and itis not a necessary constraint that they are measurable.

The manner in which the state variables change as a functionof time may be thought of as a trajectory in n dimensionalspace, called the state-space.

Two-dimensional state-space is sometimes referred to as thephase-plane when one state is the derivative of the other.

8

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Dynamic System

9

Dynamic System must involve elements that memorize thevalues of the input for

Integrators in CT serve as memory devices

Outputs of integrators are considered as internal state variables

of the dynamic system

Number of state variables to completely define the dynamics ofthe system=number of integrators involved

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Representation of a system in state-space

10

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RLC Circuit

11

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12

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EXAMPLE

+ i(t)Lv(t)

R

Ldi

dtRi v t

v t Ri t

( )

( ) ( ) ( )

(state equation)

Output equation

L sI s i V s

I s R s s

i

s

i tR

e i e

RL

RL

R L t R L t

[ ( ) ( )] ( )

( )( )

( ) ( ) ( )( / ) ( / )

0

1 1 1 0

11 0

Assuming that v(t) is a unit step and knowing i(0), taking the LT of the stateequation

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Writing the equations in matrix-vector form

i

v0

i

v 0v(t)

c

RL

1L

1

C c

1L

Assuming the voltage across the resistor as the output

v (t) Ri(t) R 0i

vR c

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Example: Given the electric network, find a state-space representation.(Hint: state variables and , output )

Cv Li

)(/1

0

0/1

/1)/(1tv

Li

v

L

CRC

i

v

L

C

L

C

L

C

Ri

vRi 0/1

Ri

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Block Diagram of CT CS in SS

16

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The general form of state and output equations for a linear,time-invariant system can be written as

Du+Cx=yBu+Ax=x

Where x is the nx1 state vector, u is rx1 input vector,yis the mx1output vector.Ais nxn state matrix, B is nxr, C is mxn and D ismxr matrices. For the previous example

0=D0R=C

0=B0

--=Av

i=x L

R

1L

1

L

R

c

C

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System Description

18

Consider a MIMO System with n integrators

r inputs

m outputs

Define n outputs of integrators as state variables

Dynamics ofthe System:

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The outputs

19

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Define

20

Linearizedabout

operatingpoints

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Time Varying/Invariant Systems

21

Time Varying System (f and g involve explicitly t)

Time Invariant System (f and g do not involve explicitly t)

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State & Output Equation:

Matrices/Vectors

22

A(t)= State MatrixB(t)= Input Matrix

C(t)=Output MatrixD(t)=Direct Transmission Matrix

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State-space Equations

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DC Servo Motor

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Selection of State Variables

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Simulation Diagrams/Block Dagrams

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Transfer Function from SS Equations

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Evaluation of Transfer Function Matrix

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Linearization of State Equations

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Linearisation about Nominal Trajectory

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Perturbations

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Ex Spin Stabilized Satellite

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Solution of State Equations

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State Transition Matrix

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Laplace Transform Method (STM

Computation)

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STM using MATLAB

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Total Response

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System Response Using MATLAB

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LTI Viewer

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Total Response using Symbolic Math

MATLAB

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SS Manipulations in MATLAB

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Natural Modes

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Natural Motions

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Eigenvalue Problem

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Diagonalisation

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Rapid Calculation of Modal Matrix

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Diagonalisation Repeated Eigen Values

Case

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Engineering/Scientific Theories

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g g/A model or framework for understandingA set of statements closed under certain rules of

inferenceValidated & tested (not mere conjectures) Summarizes (infinitely) many practical situations

Requires abstraction

Types Categorization (system of naming things) Summarizes past experiences Predicts future outcomes

Tool to design with State space theory

super theory (physics, chemistry, etc hence abstract) Helps understand engineering analysis and design techniques

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CT vs DT Discrete time state equations Continuous time state equations

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Discrete time State Equations

( 1) ( ( ), ( ), )

( ) ( ( ), ( ), )

X t F X t U t t

Y t G X t U t t t

( 1) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

X t A t X t B t U t

Y t C t X t D t U t t

( 1) ( ) ( )

( ) ( ) ( )

X t AX t BU t

Y t CX t DU t t

Possibly nonlinear, most general, hardest to analyze

Linear, possibly time-varying

Linear & time-invariant, easiest to analyze, provides mostconvenient design techniques

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Continuous time State Equations

( ) ( ( ), ( ), )

( ) ( ( ), ( ), )

X t F X t U t t

Y t G X t U t t t

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

X t A t X t B t U t

Y t C t X t D t U t t

( ) ( ) ( )

( ) ( ) ( )

X t AX t BU t

Y t CX t DU t t

Possibly nonlinear, most general, hardest to analyze

Linear, possibly time-varying

Linear & time-invariant, easiest to analyze, provides mostconvenient design techniques

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Solve LTI DT state equations Free response Forced response

Weighting sequence (Markov parameters)

External equivalence

Impulse response

Convolution

S l LTI CT t t ti

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Solve LTI CT state equations Scalar equation

Vector-matrix equation Matrix exponential

Existence, uniqueness, Lipschitz condition

Free response, forced response State transition matrix

Linearity

Complete response

Impulse response

convolution

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Discretization

STATE SPACE REPRESENTATION OF DYNAMIC

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STATE SPACE REPRESENTATION OF DYNAMIC

SYSTEMS

Consider the following n-th order differential equation in which the forcingfunction does not involve derivative terms.

d y

dta

d y

dta

dy

dta y b u

Y s

U s

b

s a s a s a

n

n

n

n n n

n n

n n

1

1

1 1 0

0

1

1

1

( )

( )Choosing the state variables as

x y, x y, x y, , xd y

dtx x , x x , , x x

x a x a x a x b u

1 2 3 n

n 1

n 1

1 2 2 3 n-1 n

n n 1 n 1 2 1 n 0

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x

x

x

x

x

x

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

a a a a a

x

x

x

x

x

x

1

2

3

n 2

n 1

n n n 1 n 2 2 1

1

2

3

n 2

n 1

n

0

0

0

0

0

b

u

y = [1 0 0 0 0]x

0

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EXAMPLE

Y s

U s

s

s s s

x

x

x

x

x

x

u

y

x

x

x

( )

( )

3 2

1

2

3

1

2

3

1

2

3

14 56 160

0 1 0

0 0 1

160 56 14

0

1

14

1 0 0

a e pace epresen a on ot ith f i f ti

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system with forcing function

involves DerivativesConsider the following n-th order differential equation inwhich the forcing function involves derivative terms.

d ydt

a d ydt

a dydt

a y

bd u

dt

bd u

dt

bdu

dt

b u

Y s

U s

b s b s b s b

s a s a s a

n

n

n

n n n

n

n

n

n n n

n n

n n

n n

n n

1

1

1 1

0 1

1

1 1

0 1

1

1

1

1

1

( )

( )

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Define the following n variables as a set of n state variables

uxudt

du

dt

ud

dt

ud

dt

yd

x

uxuuuyxuxuuyx

uyx

nnnnn

n

n

n

n

n

n 11122

2

11

1

0

222103

11102

01

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.

.

.

. . . .

. . . .

. . . .

.

.

.

.

.

.

x

x

x

x a a a a

x

x

x

x

n

n n n n

n

n

n

1

2

1

1 2 1

1

2

1

1

2

1

0 1 0 0

0 0 1 0

0 0 0 1

n

n

u

y

x

x

x

u

1 0 0

1

2

0

.

.

.

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CONTROLLABLE CANONICAL FORMd y

dt

ad y

dt

ady

dt

a y

bd u

dtb

d u

dtb

du

dtb u

Y s

U s

b s b s b s b

s a s a s a

n

n

n

n n n

n

n

n

n n n

n n

n n

n n

n n

1

1

1 1

0 1

1

1 1

0 1

1

1

1

1

1

( )

( )

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Y s

U sb

b a b s b a b b a b

s a s a s a

Y s b U s Y s

Y sb a b s b a b b a b

s a s a s a

n

n n n n

n n

n n

nn n n n

n n

n n

( )

( )

( ) ( ) ( )

( ) ( ) ( )

( )( ) ( ) ( )

0

1 1 0

1

1 1 0 0

1

1

1

0

1 1 01

1 1 0 0

1

1

1

U s

Y s

b a b s b a b b a b

U s

s a s a s aQ s

n

n n n n

n n

n n

( )

( )

( ) ( ) ( )

( )( )

1 1 0

1

1 1 0 0

1

1

1

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s Q s a s Q s a sQ s a Q s U s

Y s b a b s Q s b a b sQ s

b a b Q s

n nn n

n

n n

n n

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

11

1

1 1 0

1

1 1 0

0

Defining state variables as follows:

X s Q s

X s sQ s

X s s Q s

X s s Q s

n

n

n

n

1

2

1

2

1

( ) ( )

( ) ( )

( ) ( )

( ) ( )

sX s X s

sX s X s

sX s X sn n

1 2

2 3

1

( ) ( )

( ) ( )

( ) ( )

x x

x x

x xn n

1 2

2 3

1

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sX s a X s a X s a X s U s

x a x a x a x u

Y s b U s b a b s Q s b a b sQ s

b a b Q s

b U s b a b X s

n n n n

n n n n

n

n n

n n

n

( ) ( ) ( ) ( ) ( )

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

( ) ( )

( ) ( ) ( )

1 1 2 1

1 1 2 1

0 1 1 0

1

1 1 0

0

0 1 1 0

+

=

( ) ( )

( ) ( )

( ) ( ) ( )

b a b X s

b a b X s

y b a b x b a b x b a b x b u

n n

n n

n n n n n

1 1 0 2

0 1

0 1 1 1 0 2 1 1 0 0

+

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ub

x

x

x

babbabbaby

u

xx

x

x

aaaaxx

x

x

n

nnnn

n

n

nnnn

n

0

2

1

0110110

1

2

1

121

1

2

1

.

.

.

10

.

.

.0

0

.

.

.

1000

....

....

....0100

0010

.

.

.

BLOCK DIAGRAM

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BLOCK DIAGRAM

u +

b0

+ + +

+ + + +

+y

++

++

++

+

b1-a1b0 b2-a2b0 bn-1-an-1 b0 bn-an b0

a1 a2an-1 an

xn xn-1x2 x1

OBSERVABLE CANONICAL FORM

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OBSERVABLE CANONICAL FORM

s Y s b U s s a Y s bU s

s a Y s b U s a Y s b U s

Y s b U s bU s a Y s

b U s a Y s b U s a Y s

X s

n n

n n n n

s

s n n s n n

n s

n n

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] ( ) ( )

( ) ( ) [ ( ) ( )]

[ ( ) ( )] [ ( ) ( )]

( ) [

0

1

1 1

1 1

0

1

1 1

11 1

1

1

0

1

bU s a Y s X s

X s b U s a Y s X s

X s b U s a Y s X s

X s b U s a Y s

Y s b U s X s

n

n s n

s n n

s n n

n

1 1 1

1

1

2 2 2

2

1

1 1 1

1

1

0

( ) ( ) ( )]

( ) [ ( ) ( ) ( )]

( ) [ ( ) ( ) ( )]

( ) [ ( ) ( )]

( ) ( ) ( )

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.

.

.

. . .

. . . .

. . . .

.

.

.

.

x

x

x

x

a

a

a

a

x

x

x

x

b a b

b a

n

n

n

n

n

n

n n

n n

1

2

1

1

2

1

1

2

1

0

1

0 0 0

1 0 0

0 0

0 0 1

1 0

1 1 0

1

2

00 0 1

b

b a b

u

y

x

x

b u

.

.

.

.

.

.

.

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BLOCK DIAGRAM

b0

a1an-1an

+

+

+

+

+

+

+

y

u

bn-anb0 bn-1-an-1b0 b1-a1b0

x1x2 xn-1 xn

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DIAGONAL CANONICAL FORM

Consider that the denominator polynomial involves onlydistinct roots. Then,

Y s

U s

b s b s b s b

s p s p s p

bc

s p

c

s p

c

s p

n n

n n

n

n

n

( )

( ) ( )( ) ( )

0 1

1

1

1 2

0

1

1

2

2

where, ci, i=1,2, , n are the residues corresponding pi

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Y s b U sc

s pU s

c

s pU s

c

s p

X ss p

U s sX s p X s U s

X ss p

U s sX s p X s U s

X ss p

U s sX s p X s U s

n

n

n

n

n n n

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0

1

1

2

2

1 1 1 1 1

2

2

2 2 2

1

1

1

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( ) ( ) ( ) ( ) ( )

x p x u

x p x u

x p x u

Y s b U s c X s c X s c X s

y c x c x c x b u

n n n

n n

n n

1 1 1

2 2 2

0 1 1 2 2

1 1 2 2 0

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.

.

.

.

.

.

.

.

.

.

.

.

.

.

x

x

x

p

p

p

x

x

x

u

y c c c

x

x

x

b u

n

n

n

n

n

1

2

1

2

1

2

1 2

1

2

0

0

0

1

1

1

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BLOCK DIAGRAM

yu

x1

x2

xn

c1

b0

c2

cn

++

++

+

1

1s p

1

2s p

1

s pn

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JORDAN CANONICAL FORMConsider the case where the denominator polynomial involves multiple roots.Suppose that the pis are different from one another, except that the first three areequal.

Y s

U s

b s b s b s b

s p s p s p s p

Y s

U sb

c

s p

c

s p

c

s p

c

s p

c

s p

n n

n n

n

n

n

( )

( ) ( ) ( )( ) ( )

( )

( ) ( ) ( )

0 1

1

1

1

3

4 5

0

1

1

3

2

1

2

3

1

4

4

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Y s b U s cs p

U s cs p

U s

c

s pU s

c

s pU s

c

s pU s

X sc

s p

U s

X sc

s pU s

X s

X s s p

X sc

s pU s

X s

X

n

n

( ) ( )( )

( )( )

( )

( ) ( ) ( )

( )

( )

( )

( )( )

( )( )

( )

( ) ( )( )

01

1

32

1

2

3

1

4

4

1

1

1

3

2

2

1

2

1

2 1

3

3

1

2

1

+

3 1

44

4

1

( )

( ) ( )

( ) ( )

s s p

X sc

s pU s

X sc

s pU sn

n

n

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sX s p X s X s x p x x

sX s p X s X s x p x x

sX s p X s U s x p x u

sX s p X s U s x p x u

sX s p X s U s xn n n

1 1 1 2 1 1 1 2

2 1 2 3 2 1 2 3

3 1 3 3 1 3

4 4 4 4 4 4

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n n n

n n

n n

p x u

Y s b U s c X s c X s c X s c X s

y b u c x c x c x c x

( ) ( ) ( ) ( ) ( ) ( )0 1 1 2 2 3 3

0 1 1 2 2 3 3

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BLOCK DIAGRAMb0

c3

c2

c1

c4

cn

x3x2 x1

x4

xn

.

.

....

1

1s p

1

1s p

1

4s p

1

1s p

1

s pn

u y

+

+

+

++

++

+

+

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Example: Express in CCF/OCF/DCF

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Example: Express in CCF/OCF/DCF

107

CCF

OCFDCF

Converting a Transfer Function to State Space

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g p

A set of state variables is called phase variables, where eachsubsequent state variable is defined to be the derivative of theprevious state variable.Consider the differential equation

Choosing the state variables

ubyadt

dya

dt

yda

dt

ydn

n

nn

n

0011

1

1

ubxaxaxadt

ydx

dt

ydx

xdt

ydxdt

ydx

xdt

ydx

dt

dyx

xdt

dyxyx

nnn

n

nn

n

n 0121101

1

43

3

32

2

3

32

2

22

211

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u

bx

x

x

x

x

aaaaaax

x

x

x

x

n

n

nn

n

0

1

3

2

1

143210

1

3

2

1

0

0

0

0

100000

001000

000100

000010

n

n

xx

x

x

x

y

1

3

2

1

00001

Converting a transfer function with constant term in numerator

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g

Step 1. Find the associated differential equation.

Step 2. Select the state variables.

Choosing the state variables as successive derivatives.

24269

24

)(

)(

23

ssssR

sC

rcccc

sRsCsss

2424269

)(24)()24269( 23

cx

cx

cx

3

2

1

1

3213

32

21

2492624

xy

rxxxx

xx

xx

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Figure 3.10a.Transfer function;

b. equivalent block diagram showing phase-variables.

Note: y(t) = c(t)

Converting a transfer function with polynomial in numeratorStep1. Decomposing a transfer function.

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Step 2. Converting the transfer function with constant term in numerator.

Step 3. Inverse Laplace transform.

01

2

2

3

3

1 1

)(

)(

asasasasR

sX

)()()()( 101

2

2 sXbsbsbsCsY

101

12

1

2

2)( xbdt

dxb

dt

xdbty

01

2

2

3

3

1 1

)(

)(

asasasasR

sX

)()()()( 1012

2 sXbsbsbsCsY

322110)( xbxbxbty

Transfer function;

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equivalent block

diagram.

decomposedtransfer function

Converting from state space to a transfer function

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DuCxy

BuAxx

Given the state and output equations

Take the Laplace transform assuming zero initial conditions:

(1)

(2)

Solving for in Eq. (1),

or(3)

Substitutin Eq. (3) into Eq. (2) yields

The transfer function is

)()()(

)()()(

sss

ssss

DUCXY

BUAXX

)(sX

)()()( sss BUXAI

)()()( 1 sss BUAIX

)()()()( 1 ssss DUBUAICY

)(])([ 1 ss UDBAIC

DBAICU

Y 1)(

)(

)(s

s

s

Ex. Find the transfer from the state-space representation

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u

0

0

10

321

100

010

xx

x001y

321

10

01

321100

010

0000

00

)(s

s

s

ss

s

s AI

Solution:

123

12

31

1323

)det()()( 23

2

2

2

1

sss

sss

sss

sss

ssadjs

AIAIAI

123

)23(10

)(

)(23

2

sss

ss

s

s

U

Y

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116

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117

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118

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Output Eq. modifies to

119

Invariance of Eigenvalues

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