Abstract
This thesis considers the stabilization and the robust stabilization of certain
classes of hybrid systems using model predictive control. Hybrid systems
represent a broad class of dynamical systems in which discrete behavior
(usually described by a finite state machine) and continuous behavior (usually
described by differential or difference equations) interact. Examples of
hybrid dynamics can be found in many application domains and disciplines,
such as embedded systems, process control, automated traffic-management
systems, electrical circuits, mechanical and bio-mechanical systems, biological
and bio-medical systems and economics. These systems are inherently
nonlinear, discontinuous and multi-modal. As such, methodologies for stability
analysis and (robust) stabilizing controller synthesis developed for linear
or continuous nonlinear systems do not apply. This motivates the need for
a new controller design methodology that is able to cope with discontinuous
and multi-modal system dynamics, especially considering its wide practical
applicability.
Model predictive control (MPC) (also referred to as receding horizon control)
is a control strategy that offers attractive solutions, already successfully
implemented in industry, for the regulation of constrained linear or nonlinear
systems. In this thesis, the MPC controller design methodology will
be employed for the regulation of constrained hybrid systems. One of the
reasons for the success of MPC algorithms is their ability to handle hard constraints
on states/outputs and inputs. Stability and robustness are probably
the most studied properties of MPC controllers, as they are indispensable to
practical implementation. A complete theory on (robust) stability of MPC
has been developed for linear and continuous nonlinear systems. However,
these results do not carry over to hybrid systems easily. These challenges
will be taken up in this thesis.
As a starting point, in Chapter 2 of this thesis we build a theoretical
framework on stability and input-to-state stability that allows for discontinuous
and nonlinear system dynamics. These results act as the theoretical
foundation of the thesis, enabling us to establish stability and robust stability
results for hybrid systems in closed-loop with various model predictive
control schemes.
The (nominal) stability problem of hybrid systems in closed-loop with
MPC controllers is solved in its full generality in Chapter 3. The focus is
on a particular class of hybrid systems, namely piecewise affine (PWA) systems.
This class of hybrid systems is very appealing as it provides a simple
mathematical description on one hand, and a very high modeling power on
the other hand. For particular choices of MPC cost functions and constrained
PWA systems as prediction models, novel algorithms for computing a
terminal cost and a local state-feedback controller that satisfy the developed
stabilization conditions are presented. Algorithms for calculating low
complexity piecewise polyhedral invariant sets for PWA systems are also developed.
These positively invariant sets are either polyhedral, or consist of
a union of a number of polyhedra that is equal to the number of affine subsystems
of the PWA system. This is a significant reduction in complexity,
compared to piecewise polyhedral invariant sets for PWA systems obtained
via other existing algorithms. Hence, besides the study of the fundamental
property of stability, the aim is to create control algorithms of low complexity
to enable their on-line implementation.
Before addressing the robust stabilization of PWA systems using MPC
in Chapter 5, two interesting examples are presented in Chapter 4. These
examples feature two discontinuous PWA systems that both admit a discontinuous
piecewise quadratic Lyapunov function and are exponentially stable.
However, one of the PWA systems is non-robust to arbitrarily small perturbations,
while the other one is globally input-to-state stable (ISS) with
respect to disturbance inputs. This indicates that one should be careful
in inferring robustness from nominal stability. Moreover, for the example
that is robust, the input-to-state stability property cannot be proven via
a continuous piecewise quadratic (PWQ) Lyapunov function. However, as
ISS can be established via a discontinuous PWQ Lyapunov function, the
conservatism of continuous PWQ Lyapunov functions is shown in this setting.
Therefore, this thesis provides a theoretical framework that can be
used to establish robustness in terms of ISS of discontinuous PWA systems
via discontinuous ISS Lyapunov functions. The sufficient conditions for ISS
of PWA systems are formulated as linear matrix inequalities, which can be
solved efficiently via semi-definite programming. These sufficient conditions
also serve as a tool for establishing robustness of nominally stable hybrid
MPC controllers a posteriori, after the MPC control law has been calculated
explicitly as a PWA state-feedback. Furthermore, we also present a
technique based on linear matrix inequalities for synthesizing input-to-state
stabilizing state-feedback controllers for PWA systems.
In Chapter 5, the problem of robust stabilization of PWA systems using
MPC is considered. Previous solutions to this problem rely without exceptions
on the assumption that the PWA system dynamics is a continuous
function of the state. Clearly, this requirement is quite restrictive and artificial,
as a continuous PWA system is in fact a Lipschitz continuous system.
In Chapter 5 we present an input-to-state stabilizing MPC scheme for PWA
systems based on tightened constraints that allows for discontinuous system
dynamics and discontinuous MPC value functions. The advantage of this
new approach, besides being the first robust stabilizing MPC scheme applicable
to discontinuous PWA systems, is that the resulting MPC optimization
problem can still be formulated as mixed integer linear programming problem,
which is a standard optimization problem in hybrid MPC.
A min-max approach to the robust stabilization of perturbed nonlinear
systems using MPC is presented in Chapter 6. Min-max MPC, although
computationally more demanding, can provide feedback to the disturbance,
resulting in better performance when the controlled system is affected by
perturbations. We show that only input-to-state practical stability can be
ensured in general for perturbed nonlinear systems in closed-loop with minmax
MPC schemes. However, new sufficient conditions that guarantee inputto-
state stability of the min-max MPC closed-loop system are derived, via a
dual-mode approach. These conditions are formulated in terms of properties
that the terminal cost and a local state-feedback controller must satisfy.
New techniques for calculating the terminal cost and the local controller for
perturbed linear and PWA systems are also presented in Chapter 6.
The final part of the thesis focuses on the design of robustly stabilizing,
but computationally friendly, sub-optimal MPC algorithms for perturbed
nonlinear systems and hybrid systems. This goal is achieved via new, simpler
stabilizing constraints, that can be implemented as a finite number of linear
inequalities. These algorithms are attractive for real-life implementation,
when solvers usually provide a sub-optimal control action, rather than a
globally optimal one. The potential for practical applications is illustrated
via a case study on the control of DC-DC converters. Preliminary realtime
computational results are encouraging, as the MPC control action is
always computed within the allowed sampling interval, which is well below
one millisecond for the considered Buck-Boost DC-DC converter.
In conclusion, this thesis contains a complete framework on the synthesis
of model predictive controllers for hybrid systems that guarantees stable
and robust closed-loop systems. The latter properties are indispensable for
any application of these control algorithms in practice. In the set-ups of the
MPC algorithms, a clear focus was also on keeping the on-line computational
burden low via simpler stabilizing constraints. The example on the control of
DC-DC converters showed that the application to (very) fast systems comes
within reach. This opens up a completely new range of applications, next
to the traditional process control for typically slow systems. Therefore, the
developed theory represents a fertile ground for future practical applications
and it opens many roads for future research in model predictive control and
stability of hybrid systems as well.
Original language | English |
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Qualification | Doctor of Philosophy |
Awarding Institution |
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Supervisors/Advisors |
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Award date | 7 Sept 2006 |
Place of Publication | Eindhoven |
Publisher | |
Print ISBNs | 90-386-1823-9 |
DOIs | |
Publication status | Published - 2006 |