Given
a
nonlinear function
as
X(t)
Vector
where x(t) is the
uxl
State
ust) is the
px1
input vector
F(X(t) u(t) is the
,
State Vector
Uxl
.
F(X (t) u(t)
=
For linearizing [Ct) F(xct) ucts) with respect to a nominal operating
=
,
point (Xo. Uo)
Assume X(t)
Then
Xi(t)
:
(XI (t) X2(t)
=
Xn(t)]
. . .
.
Fi(X(t) X2(t)
=
-
-
Xu(t)
-
,
* (t) Fz (X (t)
i
=
X2(t)
.
STEP 1
U Ct)
,
.
Xn(t) U.It)
...
,
Halt)
·
-
--Up(t))
Define the state
Uz(t)---Up(t))
.
.
variable
i
Xn(t) Fr ( X(t)
Xact)--- XuIt)
=
Define
.
U . St)
,
.
Nalt)
...
Up(te)
:
Xol
E Hol
=>
I
.
.
Xoz
Noz
·
,
F (Xo1 Xoz
Xo
Nos
· -
is nominal operating points
Hop
-------
You , U01 Noz
:
Hop)
-
.
.
# (X01 XO2
Xon
-------
Xon , Upl
...
.
=
STEP 2
0
Define the nominal
Uoz- · Hops = 0
.
.
operating points
i
Fn(X0 Xo2
· ·
.
Yon, U0l Koz
-
--
.
Hop)
=
8
limerizing the nonlinear system the Taylor expansion method is commonly used.
Taylor expansion approximates a nonlinear function as a polynomial series around
the equilibrium point expressed as follows
When
,
:
,
f(x)
=
f"(X)(X- Xo)
f(x0) + f(x)(X Xo) +
-
+
influence of high-order terms on the function
value becomes negligible. Therefore in the linearization process only the zeroth-order
and first-order terms are retained while the second-order and
high order
In this expansion , as
decreases ,
X-Xo
the
,
,
,
terms
are
omitted
X
STEP 3
.
Tyalor Series
=
+
.
So :
Fi(Xo1 Xoz- Xon Hol Hoz
,
.
...
.
Uo) +
Ex(X, Xoi)
-
+
...
Gn(x(Xn Xon) y x(u, 401) + (Up Hop)
+
-
-
+
...
-
-
i
xn Fn (X0 Xo2 Xon Ur Mon-- Kop) + x(X1 Xoilt--=
+
-
-
,
,
.
x(Xn Xon) E (4) 401)
-
+
-
+..
-
+
G(x(Up Hop)
-
that * equal X1 Xo1 Xz Xo2-- Xn Yon U1 Hol
=
=
=
=
-
.
Assume(X1 X1
=
o Ul = U1
-
,
OX2
:
-Ur ,, 8U2
Because of OX1
·
Xo1
,
X2-Xon
Uz-Hon ... Up
=
0X2 .. . X
"
.
X =x
+
* xn
...
.
OU1
.
=
:
Uz Coz
=
,
Xn-Yon
Based
on
Up Hop
:
Up-Hop
Onz. - UDU
- -Xn + +... 044
1
&
STEP 4
.
Define the
i
o
...
error
Xi = X1 +...X + 1 +...
the state space function we get below
be written
,
the state space matrix can
as :
I I
·
i
Final