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# Robust nonlinear receding horizon control with constraint tightening: off line approximation and application to networked control system

Pin, Gilberto

2009-04-20

Abstract

Nonlinear Receding Horizon (RH) control, also known as moving horizon control or nonlinear
Model Predictive Control (MPC), refers to a class of algorithms that make explicit use of a
nonlinear process model to optimize the plant behavior, by computing a sequence of future ma-
nipulated variable adjustments. Usually the optimal control sequence is obtained by minimizing
a multi-stage cost functional on the basis of open-loop predictions. The presence of uncertainty
in the model used for the optimization raises the question of robustness, i.e., the maintenance
of certain properties such as stability and performance in the presence of uncertainty.
The need for guaranteeing the closed-loop stability in presence of uncertainties motivates
the conception of robust nonlinear MPC, in which the perturbations are explicitly taken in
account in the design of the controller. When the nature of the uncertainty is know, and it is
assumed to be bounded in some compact set, the robust RH control can be determined, in a
natural way, by solving a min–max optimal control problem, that is, the performance objective
is optimized for the worst-case scenario. However, the use of min-max techniques is limited
by the high computational burden required to solve the optimization problem. In the case of
constrained system, a possibility to ensure the robust constraint satisfaction and the closed-loop
stability without resorting to min-max optimization consists in imposing restricted (tightened)
constraints on the the predicted trajectories during the optimization.
In this framework, an MPC scheme with constraint tightening for discrete-time nonlinear
systems affected by state-dependent and norm bounded uncertainties is proposed and discussed.
A novel method to tighten the constraints relying on the nominal state prediction is described,
leading to less conservative set contractions than in the existing approaches. Moreover, by
imposing a stabilizing state constraint at the end of the control horizon (in place of the usual
terminal one placed at the end of the prediction horizon), less stringent assumptions can be posed on the terminal region, while improving the robust stability properties of the MPC closed-loop
system.
The robust nonlinear MPC formulation with tightened constraints is then used to design off-
line approximate feedback laws able to guarantee the practical stability of the closed-loop system.
By using off-line approximations, the computational burden due to the on-line optimization is
removed, thus allowing for the application of the MPC to systems with fast dynamics. In this
framework, we will also address the problem of approximating possibly discontinuous feedback
functions, thus overcoming the limitation of existent approximation scheme which assume the
continuity of the RH control law (whereas this condition is not always verified in practice, due
to both nonlinearities and constraints).
Finally, the problem of stabilizing constrained systems with networked unreliable (and de-
layed) feedback and command channels is also considered. In order to satisfy the control ob-
jectives for this class of systems, also referenced to as Networked Control Systems (NCS’s), a
control scheme based on the combined use of constraint tightening MPC with a delay compen-
sation strategy will be proposed and analyzed.
The stability properties of all the aforementioned MPC schemes are characterized by using
the regional Input-to-State Stability (ISS) tool. The ISS approach allows to analyze the depen-
dence of state trajectories of nonlinear systems on the magnitude of inputs, which can represent
control variables or disturbances. Typically, in MPC the ISS property is characterized in terms
of Lyapunov functions, both for historical and practical reasons, since the optimal finite horizon
cost of the optimization problem can be easily used for this task. Note that, in order to study
the ISS property of MPC closed-loop systems, global results are in general not useful because,
due to the presence of state and input constraints, it is impossible to establish global bounds for
the multi-stage cost used as Lyapunov function. On the other hand local results do not allow to
analyze the properties of the predictive control law in terms of its region of attraction. There-
fore, regional ISS results have to employed for MPC controlled systems. Moreover, in the case of
NCS, the resulting control strategy yields to a time-varying closed-loop system, whose stability
properties can be analyzed using a novel regional ISS characterization in terms of time-varying
Lyapunov functions.

Insegnamento

Publisher

Università degli studi di Trieste

Languages

en

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