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
Consider the difference
Now
)(tz
)(tw
zwx −=
Linearization better defined(2)
Remembering that is known, call
Then we have
Calling
we have a linear system:
)(tz
0
0
0
0)( xzwtx =−=
Linearization better defined(3)
If , a constant, then is an equilibrium point of the system
Suppose also that
is not a function of time. Then is constant and
is time-invariant.
0
)( ztz =
0
z
A
General Form for Linear Equations
In state space format, a linear system is represented as
−
)()(…)()(
11,11211
tAtAtAtA
nn
where
General Form for Linear Equations (2)
=
−
−
)()(…)()(
)(
)(
…
)(
11,11211
1
2
tBtBtBtB
tu
tu
tu
mm
m
m
System Block Diagram
Representing Nth order ODE as
System of N First Order ODEs
Consider
Let
dt
dt
2
1
=
=
dt
dy
x
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:
0…100
0…010
0
0
Homework Assignment 16
Prepare for short quiz #2 in next class
Read: