6 CONCLUSIONS
A novel framework for closed-form Nonlinear Model
Predictive Control (NMPC) for continuous state space
and a ﬁnite set of control inputs has been presented
that directly incorporates the noise inﬂuence in the
corresponding optimal control problem. By using the
proposed state prediction methods, which are based
on transition density approximation by Gaussian mix-
ture densities and complexity reduction techniques,
the otherwise not analytically solvable state predic-
tion of nonlinear noise affected systems can be per-
formed in an efﬁcient closed-form manner. Another
very important aspect of NMPC is the modeling of the
cost function. The proposed methods also use Gaus-
sian mixtures, which leads to a level of ﬂexibility far
beyond the traditional representations. By employing
the same representation for both the predicted proba-
bility density functions and the cost functions, NMPC
is solvable in closed-form for nonlinear systems with
consideration of noise inﬂuences. The effectiveness
of the presented framework and the importance of
the consideration of noise in the controller have been
shown in simulations of a two-wheeled differential-
drive robot following a speciﬁed trajectory.
Future research is intended to address various top-
ics. One is the optimization of the value function ap-
proximation by abandoning a ﬁxed grid in order to in-
crease performance and accuracy. An additional im-
portant task will be the consideration of stability as-
pects, especially in cases of approximated value func-
tions. This can, e.g. be tackled by the use of bounding
techniques for the approximation error (Lincoln and
Rantzer, 2006). Another interesting extension will be
the incorporation of effects of inhomogeneous noise,
i.e., noise with state and/or input dependent noise lev-
els. Together with the incorporation of nonlinear ﬁl-
tering techniques this is expected to increase the con-
trol quality even more.
Besides the addition of new features to the frame-
work, also the extension to new application ﬁelds is
intended. Of special interest is the extension of Model
Predicted Control to the related emerging ﬁeld of
Model Predictive Sensor Scheduling (He and Chong,
2004), which is of special importance, e.g. in sensor-
actuator-networks.
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A CLOSED-FORM MODEL PREDICTIVE CONTROL FRAMEWORK FOR NONLINEAR NOISE-CORRUPTED
SYSTEMS
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