La linearizzazione di un sistema dinamico non lineare stazionario ne approssima, mediante lo sviluppo di Taylor del primo ordine, il comportamento a un modello dinamico lineare, detto sistema dinamico linearizzato .
Linearizzazione di sistemi a tempo continuo
modifica
Il sistema dinamico linearizzato può essere espresso in funzione delle perturbazioni
δ
x
(
t
)
{\displaystyle \delta x(t)}
δ
u
(
t
)
{\displaystyle \delta u(t)}
δ
y
(
t
)
{\displaystyle \delta y(t)}
movimento nominale di riferimento (
x
~
(
t
)
{\displaystyle {\tilde {x}}(t)}
u
~
(
t
)
{\displaystyle {\tilde {u}}(t)}
y
~
(
t
)
{\displaystyle {\tilde {y}}(t)}
{
x
˙
(
t
)
=
f
(
x
(
t
)
,
u
(
t
)
)
y
(
t
)
=
g
(
x
(
t
)
,
u
(
t
)
)
x
(
t
=
0
)
=
x
0
⇒
{
δ
x
˙
(
t
)
=
A
(
t
)
δ
x
(
t
)
+
B
(
t
)
δ
u
(
t
)
,
x
(
t
)
=
x
~
(
t
)
+
δ
x
(
t
)
δ
y
(
t
)
=
C
(
t
)
δ
x
(
t
)
+
D
(
t
)
δ
u
(
t
)
,
y
(
t
)
=
y
~
(
t
)
+
δ
y
(
t
)
δ
x
(
t
=
0
)
=
x
(
t
=
0
)
−
x
~
0
,
u
(
t
)
=
u
~
(
t
)
+
δ
u
(
t
)
{\displaystyle {\begin{cases}{\dot {x}}(t)=f\left(x(t),u(t)\right)\\y(t)=g\left(x(t),u(t)\right)\\x\left(t=0\right)=x_{0}\end{cases}}\Rightarrow {\begin{cases}{\dot {\delta x}}(t)=A(t)\delta x(t)+B(t)\delta u(t),&x(t)={\tilde {x}}(t)+\delta x(t)\\\delta y(t)=C(t)\delta x(t)+D(t)\delta u(t),&y(t)={\tilde {y}}(t)+\delta y(t)\\\delta x\left(t=0\right)=x\left(t=0\right)-{\tilde {x}}_{0},&u(t)={\tilde {u}}(t)+\delta u(t)\end{cases}}}
dove le matrici
A
(
t
)
{\displaystyle A(t)}
B
(
t
)
{\displaystyle B(t)}
C
(
t
)
{\displaystyle C(t)}
D
(
t
)
{\displaystyle D(t)}
f
(
x
,
u
)
{\displaystyle f\left(x,u\right)}
A
(
t
)
{\displaystyle A(t)}
C
(
t
)
{\displaystyle C(t)}
f
{\displaystyle f}
x
{\displaystyle x}
A
(
t
)
=
∂
f
(
x
,
u
)
∂
x
|
x
=
x
~
,
u
=
u
~
=
[
∂
f
1
∂
x
1
⋯
∂
f
1
∂
x
n
⋮
⋱
⋮
∂
f
n
∂
x
1
⋯
∂
f
n
∂
x
n
]
|
x
=
x
~
,
u
=
u
~
∈
R
n
×
n
{\displaystyle A(t)={\left.{\frac {\partial f\left(x,u\right)}{\partial x}}\right|}_{x={\tilde {x}},u={\tilde {u}}}={\left.{\begin{bmatrix}{\frac {\partial f_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial f_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{n}}{\partial x_{1}}}&\cdots &{\frac {\partial f_{n}}{\partial x_{n}}}\end{bmatrix}}\right|}_{x={\tilde {x}},u={\tilde {u}}}\in {\mathbb {R} }^{n\times n}}
C
(
t
)
=
∂
g
(
x
,
u
)
∂
x
|
x
=
x
~
,
u
=
u
~
=
[
∂
g
1
∂
x
1
⋯
∂
g
1
∂
x
n
⋮
⋱
⋮
∂
g
q
∂
x
1
⋯
∂
g
q
∂
x
n
]
|
x
=
x
~
,
u
=
u
~
∈
R
q
×
n
{\displaystyle C(t)={\left.{\frac {\partial g\left(x,u\right)}{\partial x}}\right|}_{x={\tilde {x}},u={\tilde {u}}}={\left.{\begin{bmatrix}{\frac {\partial g_{1}}{\partial x_{1}}}&\cdots &{\frac {\partial g_{1}}{\partial x_{n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial g_{q}}{\partial x_{1}}}&\cdots &{\frac {\partial g_{q}}{\partial x_{n}}}\end{bmatrix}}\right|}_{x={\tilde {x}},u={\tilde {u}}}\in {\mathbb {R} }^{q\times n}}
B
(
t
)
{\displaystyle B(t)}
D
(
t
)
{\displaystyle D(t)}
f
{\displaystyle f}
u
{\displaystyle u}
B
(
t
)
=
∂
f
(
x
,
u
)
∂
u
|
x
=
x
~
,
u
=
u
~
=
[
∂
f
1
∂
u
1
⋯
∂
f
1
∂
u
p
⋮
⋱
⋮
∂
f
n
∂
u
1
⋯
∂
f
n
∂
u
p
]
|
x
=
x
~
,
u
=
u
~
∈
R
n
×
p
{\displaystyle B(t)={\left.{\frac {\partial f\left(x,u\right)}{\partial u}}\right|}_{x={\tilde {x}},u={\tilde {u}}}={\left.{\begin{bmatrix}{\frac {\partial f_{1}}{\partial u_{1}}}&\cdots &{\frac {\partial f_{1}}{\partial u_{p}}}\\\vdots &\ddots &\vdots \\{\frac {\partial f_{n}}{\partial u_{1}}}&\cdots &{\frac {\partial f_{n}}{\partial u_{p}}}\end{bmatrix}}\right|}_{x={\tilde {x}},u={\tilde {u}}}\in {\mathbb {R} }^{n\times p}}
D
(
t
)
=
∂
g
(
x
,
u
)
∂
u
|
x
=
x
~
,
u
=
u
~
=
[
∂
g
1
∂
u
1
⋯
∂
g
1
∂
u
p
⋮
⋱
⋮
∂
g
q
∂
u
1
⋯
∂
g
q
∂
u
p
]
|
x
=
x
~
,
u
=
u
~
∈
R
q
×
p
{\displaystyle D(t)={\left.{\frac {\partial g\left(x,u\right)}{\partial u}}\right|}_{x={\tilde {x}},u={\tilde {u}}}={\left.{\begin{bmatrix}{\frac {\partial g_{1}}{\partial u_{1}}}&\cdots &{\frac {\partial g_{1}}{\partial u_{p}}}\\\vdots &\ddots &\vdots \\{\frac {\partial g_{q}}{\partial u_{1}}}&\cdots &{\frac {\partial g_{q}}{\partial u_{p}}}\end{bmatrix}}\right|}_{x={\tilde {x}},u={\tilde {u}}}\in {\mathbb {R} }^{q\times p}}
Se il movimento nominale è un punto di equilibrio
(
x
¯
,
u
¯
)
{\displaystyle \left({\bar {x}},{\bar {u}}\right)}
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
D
{\displaystyle D}
{
δ
x
˙
(
t
)
=
A
δ
x
(
t
)
+
B
δ
u
(
t
)
δ
y
(
t
)
=
C
δ
x
(
t
)
+
D
δ
u
(
t
)
{\displaystyle {\begin{cases}{\dot {\delta x}}(t)=A\delta x(t)+B\delta u(t)\\\delta y(t)=C\delta x(t)+D\delta u(t)\end{cases}}}
Linearizzazione di sistemi a tempo discreto
modifica
Il sistema dinamico linearizzato può essere espresso in funzione delle perturbazioni
δ
x
(
k
)
{\displaystyle \delta x(k)}
δ
u
(
k
)
{\displaystyle \delta u(k)}
δ
y
(
k
)
{\displaystyle \delta y(k)}
movimento nominale (
x
~
(
k
)
{\displaystyle {\tilde {x}}(k)}
u
~
(
k
)
{\displaystyle {\tilde {u}}(k)}
y
~
(
k
)
{\displaystyle {\tilde {y}}(k)}
{
x
(
k
+
1
)
=
f
(
x
(
k
)
,
u
(
k
)
)
y
(
k
)
=
g
(
x
(
k
)
,
u
(
k
)
)
x
(
k
=
0
)
=
x
0
⇒
{
δ
x
(
k
+
1
)
=
A
(
k
)
δ
x
(
k
)
+
B
(
k
)
δ
u
(
k
)
,
x
(
k
)
=
x
~
(
k
)
+
δ
x
(
k
)
δ
y
(
k
)
=
C
(
k
)
δ
x
(
k
)
+
D
(
k
)
δ
u
(
k
)
,
y
(
k
)
=
y
~
(
k
)
+
δ
y
(
k
)
δ
x
(
k
=
0
)
=
x
(
k
=
0
)
−
x
~
0
,
u
(
k
)
=
u
~
(
k
)
+
δ
u
(
k
)
{\displaystyle {\begin{cases}x\left(k+1\right)=f\left(x(k),u(k)\right)\\y(k)=g\left(x(k),u(k)\right)\\x\left(k=0\right)=x_{0}\end{cases}}\Rightarrow {\begin{cases}\delta x\left(k+1\right)=A(k)\delta x(k)+B(k)\delta u(k),&x(k)={\tilde {x}}(k)+\delta x(k)\\\delta y(k)=C(k)\delta x(k)+D(k)\delta u(k),&y(k)={\tilde {y}}(k)+\delta y(k)\\\delta x\left(k=0\right)=x\left(k=0\right)-{\tilde {x}}_{0},&u(k)={\tilde {u}}(k)+\delta u(k)\end{cases}}}
Se il movimento nominale è un punto di equilibrio
(
x
¯
,
u
¯
)
{\displaystyle \left({\bar {x}},{\bar {u}}\right)}
A
{\displaystyle A}
B
{\displaystyle B}
C
{\displaystyle C}
D
{\displaystyle D}
{
δ
x
(
k
+
1
)
=
A
δ
x
(
k
)
+
B
δ
u
(
k
)
δ
y
(
k
)
=
C
δ
x
(
k
)
+
D
δ
u
(
k
)
{\displaystyle {\begin{cases}\delta x\left(k+1\right)=A\delta x(k)+B\delta u(k)\\\delta y(k)=C\delta x(k)+D\delta u(k)\end{cases}}}
