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6 1 Introduction
description of different NMPC algorithms as well as related background information
on dynamic programming in Chap. 3.
A distinctive feature of this book is that both schemes with stabilizing terminal
constraints as well as schemes without such constraints are considered and treated in
a uniform way. This “uniform way” consists of interpreting both classes of schemes
as relaxed versions of infinite horizon optimal control. To this end, Chap. 4 first de-
velops the theory of infinite horizon optimal control and shows by means of dynamic
programming and Lyapunov function arguments that infinite horizon optimal feed-
back laws are actually asymptotically stabilizing feedback laws. The main building
block of our subsequent analysis is the development of a relaxed dynamic program-
ming framework in Sect. 4.3. Roughly speaking, Theorems 4.11 and 4.14 in this
section extract the main structural properties of the infinite horizon optimal control
problem, which ensure
• asymptotic or practical asymptotic stability of the closed loop,
• admissibility, i.e., maintaining the imposed state constraints,
• a guaranteed bound on the infinite horizon performance of the closed loop,
• applicability to NMPC schemes with and without stabilizing terminal constraints.
The application of these theorems does not necessarily require that the feedback
law to be analyzed is close to an infinite horizon optimal feedback law in some
quantitative sense. Rather, it requires that the two feedback laws share certain prop-
erties which are sufficient in order to conclude asymptotic or practical asymptotic
stability and admissibility for the closed loop. While our approach allows for inves-
tigating the infinite horizon performance of the closed loop for most schemes under
consideration—which we regard as an important feature of the approach in this
book—we would like to emphasize that near optimal infinite horizon performance
is not needed for ensuring stability and admissibility.
The results from Sect. 4.3 are then used in the subsequent Chaps. 5 and 6 in
order to analyze stability, admissibility and infinite horizon performance properties
for NMPC schemes with and without stabilizing terminal constraints, respectively.
Here, the results for NMPC schemes with stabilizing terminal constraints in Chap. 5
can by now be considered as classical and thus mainly summarize what can be
found in the literature, although some results—like, e.g., Theorems 5.21 and 5.22—
generalize known results. In contrast to this, the results for NMPC schemes without
stabilizing terminal constraints in Chap. 6 were mainly developed by ourselves and
coauthors and have not been presented before in this way.
While most of the results in this book are formulated and proved in a mathemat-
ically rigorous way, Chap. 7 deviates from this practice and presents a couple of
variants and extensions of the basic NMPC schemes considered before in a more
survey like manner. Here, proofs are occasionally only sketched with appropriate
references to the literature.
In Chap. 8 we return to the more rigorous style and discuss feasibility and robust-
ness issues. In particular, in Sects. 8.1–8.3 we present feasibility results for NMPC
schemes without stabilizing terminal constraints and without imposing viability as-
sumptions on the state constraints which are, to the best of our knowledge, either