Feedback linearization
We develop in this section the feedback linearization first presented in.
Consider the following output function
(1.15)
Differentiating (1.15), we obtain
(1.16)
(1.17)
(1.18)
(1.19)
From (1.9) and (1.10), we get
(1.20)
Introducing (1.20) in (1.19) yields
(1.21)
Thus,
the system has a relative degree of four with respect to the output
in the region 
, i.e. when 
.
We can define a controller
,
so the closed-loop system is given by
(1.22)
where
is a stable poly-nominal. Therefore,
.
Finally, from (1.16) and (1.17), it follows that 
.
The above shows that the reaction wheel pendulum is feedback
linearizable in the region 
,
i.e. when the pendulum angle
is above the horizontal.
Energy-based control design
Define the following Lyapunov function candidate
(1.23)
where
and 
are strictly positive constants. The time derivative of 
is
given by
We propose a controller such that
(1.24)
which leads to
(1.25)
Equations
(1.23) and (1.25) imply that
is a non-increasing function,
converges to a constant and 
.
This implies that the energy 
remains bounded as well as 
and 
. Therefore the closed-loop state vector
is bounded and we can thus apply LaSalle's invariance principle.
In
order to apply LaSalle ' s theorem, we are required to define a
compact (closed and bounded) set 
with the property that every solution of the system 
that starts in
remains in
for all future time. Therefore, the solutions of the closed-loop
system
remain inside a compact set
that is defined by the initial value of
.
Let 
be the set of all points in
such that 
.
Let 
be the largest invariant set in
.
LaSalle "s theorem ensures that every solution starting in
approaches
as 
.
Let us now compute the largest invariant set
in
.
In
the set
,
and from (1.25) it follows that 
in
.
Note that
also at the stable equilibrium point
.
Recall that the pendulum's energy is
at the stable equilibrium point. A way to avoid this undesired
convergence point is to constrain the initial conditions. Indeed, if
the initial state is such that
(1.26)
then,
in view of (7.23) and given that
,
the energy
will never reach the value 
,
which characterizes the stable equilibrium point
.
The inequality (1.26) mainly imposes upper bounds on 
and
.
Since
(see (1.12)) and
in the invariant set, then 
is constant. From (1.23), it follows that
(1.27)
for
some constant
.
Then, from (1.24), we get 
.
We
will consider two cases: a) 
and b) 
,
for some constant 
.
Case a: . From (1.24) , we have
(1.28)
Introducing (1.28) in (1.11), with , it then follows that
(1.29)
Since
is a constant, equation (1.29) defines a particular trajectory called
a homoclinic orbit of the pendulum in the 
phase plane, which is a two-dimensional subspace of the
four-dimensional state space of the complete system. It means that
only when 
.
The pendulum moves clockwise or counter-clockwise until it reaches
the equilibrium point 
.
Then, from (1.28), it follows that 
also.
Case
b:
.
Since 
is constant, then
is also constant. Thus from (1.27),
is also constant. From (1.7) - (1.10) and since
,
we obtain
(1.30)
Since
is constant, we have 
and from (7.30), we conclude that 
.
Note that the case when 
has been excluded by imposing the constraint (1.26). Suppose now that
.
From (1.24), since
and
,
it follows that
(1.31)
However,
since 
,
the energy (1.11) becomes
(1.32)
Therefore
equations (1.31) and (1.32) lead to a contradiction, which proves
that the assumption 
is false. We finally conclude that
.
Moreover, when
,
and 
,
is also zero, which contradicts the assumption 
.
Finally, the largest invariant set is given by the homoclinic orbit (1.29) together with the kinematic constraint (1.28). LaSalle's invariance principle guarantees that the system trajectories asymptotically converge to this invariant set, provided that the initial conditions are such that (1.26) is satisfied.