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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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4AE2235-I : Aerospace Systems & Control Theory - State-Space models
Firstprinciples
Recap of previous lectures
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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Recap of previous lectures
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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6AE2235-I : Aerospace Systems & Control Theory - State-Space models
Recap of previous lectures
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:
Recap of previous lectures
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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Recap of previous lectures
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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Recap of previous lectures
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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Recap of previous lectures
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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Recap of previous lectures
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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
Nise, Section 3.3, p123
( ) ( ) ( )
( ) ( ) ( )
x t A x t B u t
y t C x t D u t
= +
= +
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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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
=
= +
Nise, Section 3.5, p132
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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
An Aerospace Example
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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1. Introduction to State-Space models
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 =
An Aerospace Example
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1. Introduction to State-Space models
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
= +
An Aerospace Example
52 4
3 3 3
e
m Vm V mq q
m c m m c = + +
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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
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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1. Introduction to State-Space models
Extending state-space systems
Response of theaircraft model to a
block input,() = 1for0 < t < 5
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1. Introduction to State-Space models
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
Physical system modelling nthorder linear
differentialequation
Transfer function(Laplacedomain)
Laplacetransform In
verseLaplace
transform
State-spacemodel (time
domain)
Writeinmatrixform
Set of nfirst
order differentialequations
introduce state vector
Nise, Section 3.5, p133
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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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= +
Nise, Section 3.5, p134
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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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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2. Transfer function to State-Space
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
Physical system modelling nthorder linear
differentialequation
Transfer function(Laplacedomain)
Cramers rule
State-spacemodel (time
domain)
Writeinmatrixform
Set of nfirst
order differentialequations
introduce state vector
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51AE2235-I : Aerospace Systems & Control Theory - State-Space models
3. State-space to Transfer function
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:
Nise, Section 3.6, p139
T f f i
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3. State-space to Transfer function
Transforming state space models into transfer functions
Starting with the state equation:
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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3. State-space to Transfer function
Example (1/3): Aircraft state space model to TF
Starting with the state equation:
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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3. State-space to Transfer function
Example (2/3): Aircraft state space model to TF
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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3. State-space to Transfer function
Example (3/3): Aircraft state space model to TF
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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3. State-space to Transfer function
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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3. State-space to Transfer function
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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4. Block Diagram to State-Space
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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4. Block Diagram to State-Space
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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5. The Space in state-space
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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5. The Space in state-space
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
= +
5 The Space in state space
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5. The Space in state-space
Analysis on phase plane
5 The Space in state space
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5. The Space in state-space
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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6. Outlook & Summary
Relationship between model forms
Physical system modelling nthorder linear
differentialequation
Transfer function(Laplacedomain)
Laplacetransform In
verseLaplace
transform
Cramers rule
State-spacemodel (time
domain)
WriteinmatrixformW
riteassetof
equations
Set of nfirst
order differentialequations
introduce state vector
collapse state vector
BlockDiagram
6 Outlook & Summary
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6. Outlook & Summary
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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6. Outlook & Summary
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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6. Outlook & Summary
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
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