Systems of First Order ODEs
Linear Systems in State Space Format
Linearization better defined
Let be the known solution of a nonlinear nth order system defined by
Let be the unknown solution of a slightly different nth order
nonlinear ODE with slightly different initial condition
)(tz
),( ztfz =
0
0)( ztz =
)(tw
Linearization better defined(2)
Remembering that is known, call
Then we have
)())(,( tAtzt
w
g
ij
j
i=
)(tz
Linearization better defined(3)
If , a constant, then is an equilibrium point of the system
Suppose also that
0
)( ztz =
0
z
),( ztfz =
General Form for Linear Equations
In state space format, a linear system is represented as
+
=
−
)()(…)()(
)()()(
11,11211
tAtAtAtA
tutBxtA
dt
xd
nn
where
General Form for Linear Equations (2)
=
…
)(
)(
)(
2
1
tu
tu
tu
System Block Diagram
Representing Nth order ODE as
System of N First Order ODEs
Consider
Let
)()(…)( 1
1
1tgyta
dt
yd
ta
dt
yd
n
n
n
n
n
=+++ −
−
1
=
yx
Representing Nth order ODE as
System of N First Order ODEs (2)
Then
2
1
x
dt
dx
=
Representing Nth order ODE as
System of N First Order ODEs (3)
In matrix form:
+=
)()()(
tgtbxtA
dt
xd
=
1
0
…
0
0
)(tb
Homework Assignment 16
Prepare for short quiz #2 in next class
Read:
Work:
(Problems in text)