CONVERTING TO THE STATE SPACE
2
CONVERTING TO THE
STATE SPACE
Differential equations can be solved analytically
when the equations are linear and when the
nonlinearities are relatively simple. Otherwise, the
equations need to be solved numerically. To solve
the
equations
numerically,
the
differential
equations are converted to a standard format. The
standard format is a set of 1
st
-order nonlinear
differential equations, called
state
equations. The
equations are converting to a standard format to
produce a corresponding numerical procedure that
is standardized, as well.
1. Nonlinear State Equations
The single degree-of-freedom of the pendulum is
associated with the pendulum’s configuration. For
this reason, the linearization described in the
previous chapter is sometimes referred to as being
carried out in the
configuration space.
In contrast,
we’ll momentarily carry out the linearization in the
state space
. The pendulum’s state consists of the
angle
θ
(
t
) and its angular rate
Once
θ
(
t
) and
are prescribed as initial conditions, the future
“state” of the system can be predicted – which is
).
(
t
θ
&
)
(
t
θ
&
MAE 461: DYNAMICS AND CONTROLS

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CONVERTING TO THE STATE SPACE
why the term
state
is used. The pendulum’s state
variables are
(2 – 1)
)
(
)
(
)
(
)
(
2
1
t
t
x
t
t
x
θ
θ
&
=
=
From Eq. (1 – 3), the two state equations that
describe the motion of the pendulum are expressed
in terms of its state variables as
(2 – 2)
)
sin(
1
2
2
1
x
L
g
x
x
x
−
=
=
&
&
Equations (2 – 2) are two 1
st
-order differential
equations. The first of the two equations defines
x
2
(
t
) as the time derivative of
x
1
(
t
) and the second
of the two equations is the equation of motion
coming from Eq. (1 – 3). So, one 2
nd
-order
differential equation has been converted into two
1
st
-order differential equations.
The benefit of the state format is a matter of
standardization.
Just
about
any
differential
equation or system of differential equations, not
just
2
nd
-order
differential
equations,
can
be
converted into a system of 1
st
-order equations. In
fact, the state format is not only used for numerical
integration, but in all kinds of methods of analysis
and design. Control methods, filtering techniques,
estimation procedures, to name a few, have been
standardized for state equations. These methods
are called
state variable methods
.
Let’s now retrace our steps and re-develop the
material previously covered in Chapter 1 using this
new state-variable format.
MAE 461: DYNAMICS AND CONTROLS