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obvious, and the function that maps the state at time t to the state at time
t + T can be written explicitly. Generally, it will not be the case, neither
for other models in the following nor for most applications. Therefore, one
has to obtain a numerical approximation with a differential equation solver
(Runge-Kutta algorithm for instance [Demailly 1991]).
To control the mobile, we consider a scalar additive control on the speed,
denoted by u .
For instance, in our example, the second time derivative is easily obtained,
x 1
x 2
=
.
d 2
d t 2
x 1
x 2
From that expression, one can derive a second-order Taylor approximation of
the state evolution
x 1
x 2
( t + T )= x 1
x 2
( t )+ T d
d t
x 1
x 2
( t )+ T 2
2
x 1
x 2
( t )+ 0
u ( t )
.
d 2
d t 2
Thus, the following dynamical discrete-time controlled system is obtained
as
x 1 ( k +1)
x 2 ( k +1)
= f x 1 ( k )
x 2 ( k )
= x 1 ( k )+ Tx 2 ( k )
,
T 2 x 1 ( k )
T 2 x 2 ( k )+ u ( k )
x 2 ( k )
Tx 1 ( k )
such that the trajectories of that system are a close approximation of the
sampled trajectories of the continuous-time dynamical system.
4.1.4 Example: The Inverted Pendulum
We consider now the nonlinear dynamical system called inverted pendulum
because its unstable equilibrium is considered as the Reference State. The
device diagram is represented on Fig. 4.2.
The differential equation of the controlled system is
d 2 θ
d t 2
k d θ
= g sin ( θ )
d t + u.
Its continuous-time state representation is
x 1
x 2
0
u
x 2
g sin x 1
d
d t
=
+
.
kx 2
Notice that the state space is not really a vector state, since the angle θ is
only defined up to a multiple of 2 π . Actually the physical problem makes sense
only if the angle is constrained within a given viability domain. The differential
equation solver is not detailed. Simulations that are used to illustrate the
present chapter are performed using Matlab TM software.
 
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