State Space Models, Linearization, Transfer
Function
From Lecture 1
State Space Models
n
+
−
−
+ . . . +
=
+
−
−
+ . . . +
the superposition principle
=
=⇒ =
=
=⇒ =
=
·
+
·
=⇒ =
·
+
·
State Space Models
n
+
−
−
+ . . . +
=
+
−
−
+ . . . +
the superposition principle
=
=⇒ =
=
=⇒ =
=
·
+
·
=⇒ =
·
+
·
State Space Models
+
−
−
+ . . . +
=
+
−
−
+ . . . +
alternative
coupled dierential equations, each or order one
˙
=
(
,
, ...
, )
˙
=
(
,
, ...
, )
...
˙
=
(
,
, ...
, )
= (
,
, ...
, )
states ()
State Space Models
+
−
−
+ . . . +
=
+
−
−
+ . . . +
alternative
coupled dierential equations, each or order one
˙
=
(
,
, ...
, )
˙
=
(
,
, ...
, )
...
˙
=
(
,
, ...
, )
= (
,
, ...
, )
states ()
State Space Models
+
−
−
+ . . . +
=
+
−
−
+ . . . +
alternative
coupled dierential equations, each or order one
˙
=
(
,
, ...
, )
˙
=
(
,
, ...
, )
...
˙
=
(
,
, ...
, )
= (
,
, ...
, )
states ()
State Space Models
+
−
−
+ . . . +
=
+
−
−
+ . . . +
alternative
Linear
˙
=
+ ... +
+
˙
=
+ ... +
+
...
˙
=
+ ... +
+
=
+
+ ... +
+
˙
˙
˙
=
+
=
...
+
Only states () and inputs () are allowed
State Space Models
+
−
−
+ . . . +
=
+
−
−
+ . . . +
alternative
Linear
˙
=
+ ... +
+
˙
=
+ ... +
+
...
˙
=
+ ... +
+
=
+
+ ... +
+
˙
˙
˙
=
+
=
...
+
Only states () and inputs () are allowed
State Space Models
˙
= +
= (+)
∈ R
∈ R ∈ R
Example
Example
¨
=
=
˙
=
Dynamical Systems
Linearization - Why?
Linearization - How?
˙
= (, ), = (, )
stationary point (
,
)
˙
= ⇔ (
,
) =
Taylor series expansions
(
,
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(
,
) =
= (
,
)
Introduce ∆ = −
∆ = −
∆ = −
˙
∆ =
˙
−
˙
= (, ) ≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
∆ = (, ) −
≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
Linearization - How?
˙
= (, ), = (, )
stationary point (
,
)
˙
= ⇔ (
,
) =
Taylor series expansions
(
,
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(
,
) =
= (
,
)
Introduce ∆ = −
∆ = −
∆ = −
˙
∆ =
˙
−
˙
= (, ) ≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
∆ = (, ) −
≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
Linearization - How?
˙
= (, ), = (, )
stationary point (
,
)
˙
= ⇔ (
,
) =
Taylor series expansions
(
,
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(
,
) =
= (
,
)
Introduce ∆ = −
∆ = −
∆ = −
˙
∆ =
˙
−
˙
= (, ) ≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
∆ = (, ) −
≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
Linearization - How?
˙
= (, ), = (, )
stationary point (
,
)
˙
= ⇔ (
,
) =
Taylor series expansions
(
,
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(
,
) =
= (
,
)
Introduce ∆ = −
∆ = −
∆ = −
˙
∆ =
˙
−
˙
= (, ) ≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
∆ = (, ) −
≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
Linearization - How?
˙
= (, ), = (, )
stationary point (
,
)
˙
= ⇔ (
,
) =
Taylor series expansions
(
,
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(, ) ≈ (
,
) +
∂
∂
(
,
)( −
) +
∂
∂
(
,
)( −
)
(
,
) =
= (
,
)
Introduce ∆ = −
∆ = −
∆ = −
˙
∆ =
˙
−
˙
= (, ) ≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
∆ = (, ) −
≈
∂
∂
(
,
)∆ +
∂
∂
(
,
)∆ = ∆ + ∆
Example - Linearization
Example
˙
=
˙
= −
+
+
√
+
=
+
=
˙
=
=
(
,
, )
˙
= −
+
+
√
+ =
(
,
, )
=
+
= (
,
, )
=
˙
=
˙
=
0 =
0 = −
+
+
√
3 +
=
+ 3
2
=⇒ (x
10
, x
20
, u
0
) = (−, , )
= (
,
,
) =
˙
=
=
(
,
, )
˙
= −
+
+
√
+ =
(
,
, )
=
+
= (
,
, )
=⇒ (x
10
, x
20
, u
0
) = (−, , )
= (
,
,
) =
(−, , )
∂
∂
= ,
∂
∂
= ,
∂
∂
= ,
∂
∂
= +
+ ,
∂
∂
= −
,
∂
∂
=
√
+
,
∂
∂
=
,
∂
∂
= ,
∂
∂
= ,
˙
=
=
(
,
, )
˙
= −
+
+
√
+ =
(
,
, )
=
+
= (
,
, )
=⇒ (x
10
, x
20
, u
0
) = (−, , )
= (
,
,
) =
(−, , )
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
=
,
∂
∂
|{
,
}
= −,
∂
∂
|{
,
}
= ,
∂
∂
|{
,
}
= ,
˙
=
=
(
,
, )
˙
= −
+
+
√
+ =
(
,
, )
=
+
= (
,
, )
=⇒ (x
10
, x
20
, u
0
) = (−, , )
= (
,
,
) =
(−, , )
(, )
∂
|{
,
}
= =
"
#
(, )
∂
|{
,
}
= =
"
#
(, )
∂
|{
,
}
= =
h
−
i
(, )
∂
|{
,
}
= =
h
i
˙
=
=
(
,
, )
˙
= −
+
+
√
+ =
(
,
, )
=
+
= (
,
, )
=⇒ (x
10
, x
20
, u
0
) = (−, , )
= (
,
,
) =
∆
=
−
, ∆
=
−
∆ = −
∆ = −
"
∆
∆
#
=
"
#"
∆
∆
#
+
"
#
∆ =
h
−
i
"
∆
∆
#
+
h
i
Laplace Transformation
() L(())()
L(())() = () =
Z
∞
−
()d
L
d()
d
() = () − ()
Initial values helps to calculate what happens in transient phase!
() =
0
() = ··· =
−
() =
L
d
()
d
() =
()
L
Z
(τ)
τ
() =
() ()
See Collection of Formulae for a table of Laplace transformations.
Example - Transfer Function
Example
¨
+
˙
+
=
˙
+
.
(
+
+
)() = (
+
)()
() =
()
z }| {
+
+
+
() = ()()
()
Transfer Function
() ()
() = ()()
()
() =
()
()
() ()
From State Space to Transfer Function
˙
= +
= +
() = ( − )
−
+
() () = det( − )
From Transfer Function to State Space
Example
() =
+
+ −
Three Ways to Describe a Dynamical System
¨
+
˙
+
=
˙
+
˙
= +
= +
() = ()() =
()
()
()
() = ( − )
−
+
L
d
()
d
() =
()
=
=
˙
Block Diagram - Transfer Function
() =
()()
Block Diagram - Components
+
−
, , , ()
() () ()
Determine Transfer Function From Block Diagram
+
−
=
, =
, = −
=
+
Example - Transfer Functions
Example
+
+
