Lecture#4 State Space Model

7/27/2019 Lecture#4 State Space Model

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Prof. Wahied GHARIEB

EE455: Applied Control

Lecture# 4

State Space Model


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2 EE455: Applied Control

Advantages of State Space Approach

Classical Approach Modern Control Approach

Transfer Function

Linear Time Invariant

System, SISO Laplace Transform,

Frequency domain

Only Input-output

Description: Less DetailDescription on SystemDynamics

State Variable Approach

Linear Time Varying,

Nonlinear, TimeInvariant, MIMO

Time domain

Detailed description of

Internal behavior in

addition to I-O

properties


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State Space Representation

Select a particular subset of all possible independent

system variables and call them state variables.

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

variables.

If we know the initial conditions of all state variablesat t0 and the system input for tt0, we can solve the

simultaneous differential equations for the state

variables for tt0.


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Concept of State Variable

System variable: Any variable that responds to an input or initialconditions in a system.

State variables: The smallest set of linearly independent system

Variables.

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

with n variables, where the n variables to be solved are the state variables.

Output equation: The algebraic equation that expresses the output

variables of a system as linear combinations of the state variables and the

inputs.


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Dynamic System must involve elements that memorize

the values of the input

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 thedynamics of the system=number of integrators involved

Concept of State Variable


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State Space Model

Input equation

Output equation


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State Space Model

Most linear systems encountered are time-invariant: A, B,

C, D are constant, i.e., dont depend on t

Example: DC motor with constant coefficients

when u(t) and y(t) are scalar, system is called single-

input, single-output (SISO)

when input & output signal dimensions are vectors,MIMO

Example: Aircraft , Electrical power station


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Mass-Spring-damper system

Therefore, we define variable x1 andx2.

)()()()(

2

2

trtkydt

tdyb

dt

tydM

yx 1yx 2

Dynamic equation of thesystem:


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If the measured output of the system

is position, then we have:

In matrix form:

General State-Space

Model:

Mass-Spring-damper system


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Electrical System


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RLC Circuit


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RLC Circuit


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RLC Circuit

Example: Given the electric network, find a state-spacerepresentation. (Hint: state variables Vc and iL , output iR )

)(/1

0

0/1

/1)/(1

tvLi

v

L

CRC

i

v

L

C

L

C

L

C

Ri

vRi 0/1


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Block Diagram of State Space model

Time Invariant System

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

C(t)=Output Matrix

D(t)=Direct Transmission Matrix


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cx

cx

cx

3

2

1

1

3213

32

21

2492624

xy

rxxxx

xx

xx

Transfer Function to State Space


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Decomposed Transfer Function


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Decomposed Transfer Function


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State Space to Transfer Function

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)Substituting 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


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Eigen Values

Roots of Characteristics

Equation

Example


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Diagonalization of nxn matrix


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Another approach using the eigenvectors:For distinct eigenvalues solve the equation

Where i is the eigenvalues index

iii PPA

]P......P[Pmatrixnnsformatiolinear traThe n21


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Invariance of Eigenvalues


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Free Response of State Space Model

)()( tAxtxdtd

By taking Laplace transform

)0()()0()()(

)()0()(

1 xsxAsIsx

sAxxsxs

)()(

)0()()(

nxnmatrixtransitionstatet

xttx

By taking inverse Laplace transform


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(t)compute,4-3-

10

A

1)()( AsIs

1

4-3-

10

s0

0)(

ss

1

4s3

1-)(

ss


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s3-

14

3)4(

1)( s

sss

(s))(

(s))(

2221

1211

s

s

3

5.0

1

5.1

)1)(3(

4)(

11

ssss

ss

3

5.0

1

5.0

)1)(3(

1)(12

sssss


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35.1

15.1

)1)(3(3)(21

ssss

s

3

5.1

1

5.0

)1)(3()(22

ssss

ss

)e1.55.0()e1.55.1(

)e0.5-5.0()e0.5-5.1()(

3t-3t-

-3t-3t

tt

tt

ee

eet

)0()()( xttx


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Forced Response of state space model

)()()( tButAxtxdtd

By taking Laplace transform

)()()0()()(

)()()0()()(

)()()0()(

11

sBusxssx

sBuAsIxAsIsx

sBusAxxsxs

t

t

dButxttx

dtBuxttx

0

0

)()()0()()(

)()()0()()(

By taking inverse

Laplace transform


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Forced Response of state space model

1

0

(0)x

(0)x,

1

0B,

4-3-

10

2

1A

Free response

tt

tt

eetx

eetx

32

3

1

5.15.0)(

5.05.0)(

Forced response

t

t

tt

t

t

tt

edeetx

edeetx

3

0

)(3)(

2

3

0

)(3)(

1

]5.15.0[)(

)1(3

1]5.05.0[)(

Complete response = free response + forced response

Lecture#4 State Space Model

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