State space lecture

8/10/2019 State space lecture

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2AE2235-I : Aerospace Systems & Control Theory - State-Space models

Lecture Schedule

When Where Time Topic

Week 17 April 22 Aula CZ-B / CZ-C 10.45-12.30 Introduction, dynamicalsystems, open & closed loop

Week 18 April 29 Aula CZ-B / CZ-C 10.45-12.30 Transfer functions

Week 19 May 6 Aula CZ-B / CZ-C 10.45-12.30 State-space systems

Week 20 May 13 Aula CZ-B / CZ-C 10.45-12.30 Transient and steady-stateresponses

Week 21 No lectures

Week 22 May 27 Aula CZ-B / CZ-C 10.45-12.30 Controller tuning with rootlocus

Week 23 June 3 Aula CZ-B / CZ-C 10.45-12.30 Frequency response, BodeDiagrams

Week 24 June 10 Aula CZ-B / CZ-C 10.45-12.30 Stability


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Recap of previous lectures

Open-loop and closed-loop control

Open-loop control

Closed-loop control


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Firstprinciples


Nonlinear differential eq.

Controller

Model Type

p

hysicalprocess

Purpose

Simulation / prediction

Influence a process,modify behavior

Transfer function State-space model Analysis, control design

design

implementation

Data Linear differential eq.

linearization

Basis for control-orientedmodels


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From physical system to Transfer Function

Physical system

InverseLaplace

transform

Laplacetransform

( ) ( )0 1 0 1( ) ( )n mn mY s a a s a s U s b b s b s+ + + = + + +

Transfer function(output divided by input in Laplace domain):

0 1

0 1

( )

( )

m

m

n

n

b b s b sY s

U s a a s a s

+ + +=

+ + +

nthorder lineardifferential equation

0 1 0 1

n m

n mn m

dy d y du d ua y a a b u b b

dt dt dt dt + + + = + + +

modelling


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From physical system to Transfer Function

Physical system

Transfer function:

Laplacetransform

InverseLaplace

transform

( ) ( )wV

s s sl

=

( )

( )w

s V

s ls

=

nthorder lineardifferential equationmodelling

( ) ( )wV

t tl

=


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1. Determine system transfer function ().

2. Transform input signal to Laplace domain:

3. Multiply input with transfer function in Laplace domain:

4. Transform back to time domain using inverse Laplace transform:


Step plan to Calculate system responses

( ) ( ) ( )Y s U s H s=

{ }( ) ( )U s u t = L

{ } { }1 1( ) ( ) ( ) ( )y t Y s U s H s = =L L Use Laplacetransform table!

Use Laplacetransform table!


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From physical system to Block diagrams

Physical system

Transfer function:

Laplacetransform

InverseLaplace

transform

( ) 1

( )w

s V

s l s

=

nthorder linear

differential equationmodelling

( ) ( )wV

t t

l

=

System Block Diagram (Time domain)

System Block Diagram (Laplacedomain)

( )w

s V

l

1

s

( )s( )s s

( )w

t V

l

( )t( )t


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Block diagram simplification

2

2 2

/( )( )

( ) / /

Y

Y

r Y

K K V lY sH s

Y s s K V ls K K V l

= =+ +

Using the tools in Lecture 3, any block diagram can be simplified to:

( )YH s( )

rY s ( )Y s

V

l

( )sy

K+

-

( )r

Y sV

( )sY s ( )Y s( )w s

+-

( )r

sK

( )s

( )s s 1

s

1

s

In this case the equivalent transfer function()is given as:


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Worksheet 2: Write transfer functionV

ls

( )s( )w s

+-

( )r

sK

( )s

( )( ) ( ) ( )rV

s K s sls

=

( ) ( ) ( )r

V Vs K s K s

ls ls

=

( ) 1 ( )r

V Vs K K s

ls ls

+ =

/ ( )( )( )

( ) 1 / ( )r

K V lssH s

s K V ls

= =

+

( ) ( )( ) ( )

( )( )

( )

r

r

y s su s s

sH s

s

==

=


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What you should know by now:

1. Understand the concepts of input, output, control error,

disturbance.

2. Derive transfer functions from differential equations.

3. Calculate system responses to input signals.

4. Reading and drawing block diagrams.

5. Manipulate and simplify block diagrams.

6. Design a proportional controller (by trial and error).


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Lecture Outline

1. Introduction to state-space models

2. State-space from a transfer function

3. Transfer function from state-space

4. State-space from a block diagram

5. The space in state-space

6. Outlook & Summary


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1. Introduction to State-Space models

Introducing the State-space model

Is a formulation of a systems dynamics in the time domain.

It is an alternativeto the transfer function system

representation.

It is very well suited for use with CACSD packages such as Matlab. Many systems have more than one output and more than one

input (= Multiple Input, Multiple Output or MIMO systems). State-

space models are naturally suited for MIMO systems.

Modern control theory is based on MIMO state-space models.


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The route from transfer function to state space

Physical system modelling nthorder linear

differentialequation

Transfer function(Laplacedomain)

Laplacetransform In

verseLaplac

e

transform

State-spacemodel (time

domain)

Writeinmatrixform

Set of nfirst

order differentialequations

introduce state vector

Nise, Section 3.5


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State variables: the smallest set of variables

which combines all necessary knowledge of

the system at =0such that behaviour of

the system can be determined for 0. State vector : an -dimensional vector

containing all state variables.

State space: -dimensional space whos axesare the state variables.

State equations: A set of simultaneous 1storder differential equations with variableswhich are the state variables.

The elements of a State-Space model

V

h Vx

h

=

State space for system with state, and state vector = [].

Nise, Section 3.3, p123


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Watch the drone demo

For yourself make a list of possible variables which could

describe the stateof the drone during flight.

Take 3 minutes discuss your variables.

State vector of a drone


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Some words on the state : The state contains all information needed

to determine future system behaviour

without reference to the derivatives of

input and output variables.

The state is often determined from

physical considerations (related to energy

storage in the system).

The dimension of the state vector is theorder of the system.

( )

( )( )

( )

t

M tx t n

h t

=


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Introducing the State-space model

The state-space form of a linear time invariant (LTI) system is:

(): the state vector: the state matrix : the input matrix

: the output matrix : the feedthrough matrix

with:

state equation

output equation


( ) ( ) ( )

( ) ( ) ( )

x t A x t B u t

y t C x t D u t

= +

= +


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Block diagrams of state-space models

( )x t( )x t

D

CB

A

( )u t ( )y t

( ) ( ) ( )

( ) ( ) ( )

x t A x t B u t

y t C x t D u t

= +

= +


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Relation to transfer functions

( ) ( ) ( )

( ) ( ) ( )

x t A x t B u t

y t C x t D u t

= +

= +

System

( )( )

( )

Y sH s

U s=

Laplacedomain

Timedomain


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Example: Converting an ODE to state-space form

We want to convert the 2ndorder ODE:

Step 1: define the state variables, and state vector:

2y y y u+ + =

1

2 1

x y

x x

=

= 1

2

x yx

x y

= =

Step 2: reduce the ODE to a set of ODEs in terms of the state variables:

1 2

2 2 12

x x

x x x u

=

= +



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Example: Converting an ODE to state-space form

Step 3: Write in state space form with matrices A,B,C,D:

[ ]

0 1 0

1 2 1

1 0

x x u

y x

= +

=

so the state-space matrices are:

[ ]

0 1 0,1 2 1

1 0 , 0

A B

C D

= =

= =

1

2

1

2

x

x x

xx

x

=

=


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An Aerospace Example

Pitch dynamics of aircraft in state-space form

Symbol Description

pitch rate

angle of attack

velocity =constant!

elevator deflection

Aerodynamic force Aerodynamic moment

Aerodynamic chord

V

q

e

Z

M


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Vertical EOM:

V

q

e

Z

M

1 2 3 4 0ec c

z z z q zV V

+ + + =

2 3 4 5 0ec c

m m q m q mV V

+ + =

Step 1: define the state vector and input vector

Pitch EOM:

, ex uq

= =


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Step 2:

Vertical EOM reformulation:

V

q

e

Z

M

1 2 3 4 0ec c

z z z q zV V

+ + + =

2 3 4 5 0ec cm m q m q mV V

+ + =

2 4 5

3

e

V cq m m q m

m c V

= + +

Pitch EOM reformulation:

32 4

1 1 1

e

zz V z Vq

z c z z c =



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Step 3:

We now have:

V

q

e

Z

M

Resulting in the state-space model:

32 4

1 1 1

e

zz V z Vq

z c z z c =

3 42

11 1

52 4

3 3 3

e

z z Vz V

z cz c z

m Vq m V m q

m c m m c

= +


52 4

3 3 3

e

m Vm V mq q

m c m m c = + +


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Extending state-space systems By adding states to the state equation

By specifying more outputs in the output equation

These states are the integral of (combinations of) variables present inthe original state equation. For example, using climb speed one can

add a state variable for altitude.

Example: extend the state equation for the aircraft with attitude

and altitude .

The new state vector becomes: [ ]T

x q h =


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Extending state-space systemsAttitude is found by integrating pitch

rate, so: =

The difference between (wherethe nose is pointing) and (wherethe plane is going) is the angle of

attack: =

For small angles , the climb speedcan be approximated by: = ( )

V

q

e


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Extending state-space systemsV

q

e

3 42

11 1

52 4

3 3 3

0 0

0 0

0 1 0 0 0

0 0 0

e

z z Vz V

z cz c z

m Vq m V m q

m c m m c

h h

V V

= +

Putting everything together:


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Extending state-space systems

Response of theaircraft model to a

block input,() = 1for0 < t < 5


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The state-space vector choice is not unique!

You might have in [deg/s] or in [rad/s]. The state might be based on and , but also on and , or

and or any other combination.

If you ask Matlab or Python to convert a transfer function, itpicks a state that is numerically convenient.

So dont worry about the e-lectures/exam and getting a different

variation of your state-space system. The system accepts all valid

state-space systems!


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2. Transfer function to State-Space

Converting transfer functions to state-space models




Laplacetransform In

verseLaplace

transform


domain)

Writeinmatrixform

Set of nfirst





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From transfer function to state-space; Controllercanonical form(only one of many options).

0 1

0 1

( )m

m

n

n

b b s b sH s

a a s a s

+ + +=

+ + +

Here


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0 1

n

n n

dx d xa x a a u

dt dt

+ + + =

()

()leads to a differential equation for the input:

()

()leads to a differential equation for the output:

0 1

m

m m

dx d xy b x b b

dt dt = + + +

0 1( ) ( )( )n

nU s X s a a s a s= + + +

0 1( ) ( )( )m

mY s X s b b s b s= + + +

1L

1L

Step 2: transform transfer functions()

()and

()

()to the time domain


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With 1=, 2= , , =1 1we get:

1 2

2 3

0 111 2

1nn n

n n n n

x x

x x

a aax x x x u

a a a a

=

=

= +

Step 3A: introduce state variables into input differential equation

1

0 1 1 1

n n

n nn n

dx d x d x

a x a a a udt dt dt

+ + + + = 0 1 1 2 1n n n na x a x a x a x u+ + + + =

Writing out the all the 1storder differential equations leads to:


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With 1=, 2= , , =1 1and weget:

Step 3B: introduce state variables into output differential equation

1

0 1 1 1

m m

m mm m

dx d x d xy b x b b b

dt dt dt

= + + + + 0 1 1 2 m my b x b x b x= + + +


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Step 4A: Write state (input) equation in matrix form:

1 1

2 2

3 3

1 10 3 11 2

00 1 0 0 0

00 0 1 0 00

0 0 0 0 10

1n nn

n n n n nn nn

x x

x x

x xu

x xa a aa a

a a a a ax x a

= +

State equation:

x Ax Bu= +



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Step 4B: Write output equation in matrix form:

[ ]

1

2

3

0 1 1

1

0 [0]m n m

n

n

xx

xy b b b u

x

x

= +

Output equation:

y Cx Du= +


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Example (1/4): convert to state space

0 1b b s+

20 1 2

1

a a s a s+ +

( )U s ( )X s ( )Y s

Step 2: Transform equations from Laplace domain to time domain:

2

0 1 2 2

dx d x

u a x a adt dt = + +and

0 1

dxy b x b

dt= +

2

0 1 2( ) ( )( )U s X s a a s a s= + +

0 1( ) ( )( )Y s X s b b s= +

1L

1L

0 1

2

0 1 2( )

b b sH s a a s a s

+

= + +

Step 1: Split TF into 2 parts:


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Example (2/4):

2

0 1 2 2

dx d xu a x a a

dt dt = + +

0 1b b s+20 1 2

1

a a s a s+ +

( )U s ( )X s ( )Y s

Step 3A: Introduce state variables in state (input) equation:2

0 1

2

2 2 2

1a ad x dxx u

dt a a dt a= +

With 1=, 2= =1,we get:

1 2

0 12 1 2

2 2 2

1

x x

a ax x x u

a a a

=

= +


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Example (3/4):

0 1b b s+20 1 2

1

a a s a s+ +

( )U s ( )X s ( )Y s

Step 3B: Introduce state variables in output equation:

With 1=, 2= ,we get:

0 1 1 2y b x b x= + [ ] 10 12

xy b bx

=

0 1

dxy b x b

dt= +


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Example (4/4):

0 1b b s+20 1 2

1

a a s a s+ +

( )U s ( )X s ( )Y s

Step 4A: write state equation in matrix form:

1 2

0 12 1 2

2 2 2

1

x x

a ax x x u

a a a

=

= +

1 1

0 1

2 2

22 2

0 1 0

1x x

x ua ax x

aa a

= = +

0 1 1 2y b x b x= + [ ] 1

0 1

2

xy b b

x

=

Step 4B: write output equation in matrix form:


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What to do if ?

0 1

0 1

( )m

m

n

n

b b s b sH s

a a s a s

+ + +=

+ + +

2

2

2( )

2 1

sH s

s s=

+ +

2

0 1 2 2

dx d xu a x a a

dt dt = + +

2

2 2

d xy b

dt=

0 12 1 2

2 2 2

1a ax x x u

a a a= +

2y x= problem: 2is not a state!!!

For example:

input eq.:

output eq.:


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How to fix the case of ?2

2

2( )

2 1

sH s

s s=

+ +

subtract 2

2from numerator:

2 2

2 2

2 2 1( ) 2 2

2 1 2 1

s s sH s

s s s s

+ += +

+ + + +

2 2

2

2

2 2 4 22

2 14 2

22 1

s s s

s s

s

s s

= +

+ +

= ++ +

we have again


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How to fix the case of ?

2

4 2 ( )( ) 2

2 1 ( )

s Y sH s

s s U s

= + =

+ +

2

4 2( ) ( ) 2 ( )

2 1

sY s U s U s

s s

= +

+ +

write in terms of the input:

2

1

2 1s s+ +

( )X s( )U s4 2s

( )Y s

2( )U s

+

+

Block diagram:


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How to fix the case of ?

2

4 2( ) ( ) 2 ( )

2 1

sY s U s U s

s s

= +

+ +2

1

2 1s s+ +

( )X s( )U s4 2s

( )Y s

2( )U s

+

+

Split into state equation and output equation:

( )

2

1( ) ( )

2 1

( ) 4 2 ( ) 2 ( )

X s U ss s

Y s s X s U s

=+ +

= +

Inverse Laplace transform:

( ) 2 ( ) ( ) ( )

( ) 4 ( ) 2 ( ) 2 ( )

x t x t x t u t

y t x t x t u t

+ + =

= +

2 ( ) 2 ( ) ( ) ( )s X s sX s X s U s+ + =

( ) 4 ( ) 2 ( ) 2 ( )Y s sX s X s U s= +


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How to fix the case of ?( ) 2 ( ) ( ) ( )

( ) 4 ( ) 2 ( ) 2 ( )

x t x t x t u t

y t x t x t u t

+ + =

= +

with 1=, 2=1,we get:

2 2 1

2 1

( ) 2 ( ) ( ) ( )

( ) 4 ( ) 2 ( ) 2 ( )

x t x t x t u t

y t x t x t u t

= += +

So finally, in state-space form:

[ ]1 1 12 2 2

0 1 0 ( ), 2 4 21 2 1

x x xu t y ux x x = + = +

In general, if , the feed forward matrix 0!

Note: 0!


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Worksheet

2

2

2 4

( ) 2 1

s s

H s s s

+

= + +

Convert the following transfer function to a state-space system.

The input of the state-space system is the input of thetransfer function.

The output should contain the output of the transferfunction.


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3. State-space to Transfer function

Transforming state-space models to transfer functions




Cramers rule


domain)

Writeinmatrixform

Set of nfirst




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Transforming state space models to transfer functions

Starting with the state equation:

( ) ( ) ( )x t A x t B u t= +

with the Laplace differentiation theorem we get:

( ) ( ) ( )sX s AX s BU s= +

( ) ( ) ( )sX s AX s BU s =

rearranging we get:

( ) ( ) ( )sI A X s BU s =

resulting in:


T f f i


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Transforming state space models into transfer functions


divide the states (output) by the input

now left-multiply with 1resulting in a system of Transfer

functions:

( ) ( ) ( )sI A X s BU s =

( ) ( )

( )

X ssI A B

U s =

( ) 1( )

( )

X ssI A B

U s

=

3 S T f f i


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Example (1/3): Aircraft state space model to TF


11 12 11

21 22 21

e

a a b

a a bq q

= +

with the Laplace differentiation theorem we have:

11 12 11

21 22 21

( ) ( )( )

( ) ( ) e

a a bs s ss

a a bsq s q s

= +

after rearranging we get:

11 12 11

21 22 21

( )( )

( ) e

s a a bss

a s a bq s

=

3 S T f f i


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Divide the states (outputs) by the inputs:

Left multiply with the inverse of the matrix results in asystem of 2 TFs (()/(), and ()/()):

11 12 11

21 22 21

( )

( )

( )( )

e

e

s

ss a a b

a s a bq ss

=

1

11 12 11

21 22 21

( )

( ) ( )

( ) ( )

( )

e

e

e

q

e

s

h s s s a a b

a s a bh s q s

s

= =

3 S T f f i


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We can solve this using for example Cramers rule:

11 12

21 22

11 12

21 22

11 11

21 21

11 12

21 22

( )( )( )

( )( )( )

e

e

e

q

e

b a

b s ash s s a as

a s a

s a b

a bq sh s s a as

a s a

= =

= =

Cramers rule:replace the i-th columnof det(sI-A) in thenumerator with thevector b.

3 St t t T f f ti


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Eigenvalues and Poles

Eigenvalues of the state matrix = poles of the transfer function:The denominator of the transfer function is calculated from:

Thus, the roots are at = 0. This is the solution to theeigenvalue problem:

( )D s sI A=

n n nAv v=

with being non-zero eigenvectors, and the correspondingeigenvalues.

3 St t t T f f ti


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Here are some guidelines

Remember that ()=() + (. You must be able tocalculate the derivative of an element in the state vector, using

the input vector and the state vector itself. Any time you encounter an integrator in a block diagram, the

output of that integrator is an excellent candidate.

If you encounter a transfer function in a block diagram, convert

the transfer function to state-space, with one of the availablemethods, and add the states to the state vector.

4 Bl k Di t St t S


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4. Block Diagram to State-Space

Converting a Block Diagram to State-Space form

Converting roll-attitude controller into a state space system

4 Bl k Di t St t S


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Worksheet: taxiing aircraft

Convert this block diagram into a state-space system.

Use as outputs , and .

4 Bl k Di t St t S


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Worksheet: taxiing aircraft

Convert this block diagram into a state-space system.

Use as outputs , and .

5 Th S i t t


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5. The Space in state-space

The state (vector) of an n-th order system can be seenas spanning an n-dimensional space.

Example system:

2

2 2

( )( )( ) 2

Y sH sU s s s

= =

+ +

V

h Vx

h

=

State space for system with state, and state vector = [].

with = 2, = 0.1, state variables 1=,2=:

1 2

2 2

2 1 22

x x

x x x u

=

= +

5 Th S i t t


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State-space response to = 1:

2

2 2( )

2H s

s s

=

+ +

time [s]

1, 2

1 1 2

2

2 2

0 1

2

x xu

x x

= +

5 The Space in state space


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State-space response in the 1, 2plane

2

2 2( )

2

H s

s s

=+ +

1

2

1 1 2

2

2 2

0 1

2

x xu

x x

= +



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Analysis on phase plane



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Analysis on phase plane The previous plot is made with experimental data on car

following (keeping a constant distance).

The states are time headway and relative speed

(approximately proportional to distance and velocity). The plots allowed us to identify for which combinations of time

headway and relative speed people release the gas pedal.

6 Outlook & Summary


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Relationship between model forms




Laplacetransform In

verseLaplace

transform

Cramers rule


domain)

WriteinmatrixformW

riteassetof

equations

Set of nfirst



collapse state vector

BlockDiagram

6 Outlook & Summary


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Today you learned:

1. That state-space models are time domain models.

2. How to represent a system in state-space form starting from

linearized equations of motion.

3. How to transform a state-space into a transfer function, and vice

versa.4. How construct a state-space model from a block diagram.

5. How to interpret phase plane analysis

6 Outlook & Summary


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Study Guide

From Control Systems Engineering (6thedition) Chapter 3.1, 3.3, 3.4 (focus on mechanical examples), 3.5, 3.6.

E-lecture (Lecture 6) Creating state-space models with Matlab and Python.

6 Outlook & Summary


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What will we do in the next lecture?

Focus on analysis of system dynamics

Nise 4.1 to 4.4, 4.8, 6.1 and 7.1 to 7.3

State space lecture

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