Samenvatting
Performance analysis is an important aspect in the design of dynamic (control) systems. Without
a proper analysis of the behavior of a system, it is impossible to guarantee that a certain
design satisfies the system’s requirements. For linear time-invariant systems, accurate performance
analyses are relatively easy to make and as a result also many linear (controller) design
methods have appeared in the past. For nonlinear systems, on the other hand, such accurate performance analyses and controller design methods are in general not available. A main reason
hereof is that nonlinear systems, as opposed to linear time-invariant systems, can have multiple
steady-state solutions. Due to the coexistence of multiple steady-state solutions, it is much
harder to define an accurate performance index. Some nonlinear systems, i.e. the so-called convergent nonlinear systems, however, are characterized by a unique steady-state solution. This
steady-state solution may depend on the system’s input signals (e.g. reference signals), but is
independent of the initial conditions of the system. In the past, the notion of convergent systems
has already been proven to be very useful in the performance analysis of nonlinear systems with
inputs.
In this thesis, new results in the field of performance analysis of nonlinear systems with inputs
are presented, based on the notion of convergent systems. One part of the thesis is concerned
with the question "how to analyse the performance for a convergent system?" Since the behavior
of a convergent system is independent of the initial conditions (after some transient time),
simulation can be used to find the unique steady-state solution that corresponds to a certain input
signal, but this can be very time-consuming. In this thesis, a computationally more efficient
approach is presented to estimate the steady-state performance of harmonically forced Lur’e
systems, in terms of nonlinear frequency response functions (nFRFs). This approach is based
on the method of harmonic linearization. It provides both a linear approximation of the nFRF
and an upper bound on the error between this linear approximation and the true nFRF. It is
shown in several examples that the approximation of the nFRF is accurate, and that it provides
more detailed information on the considered system than the often used ‘L2 gain’ performance
index. An additional observation that is made, is that the method of harmonic linearization can
sometimes be ‘misleading’ for Lur’e systems with a saturation-like nonlinearity: for the case
that the harmonic balance equation has a unique solution, it is shown that the corresponding
nonlinear system can have multiple distinct steady-state solutions.
Another part of the thesis is concerned with the question "under what conditions is a system
with inputs guaranteed to be convergent?" In particular two types of systems were investigated:
switched linear systems and Lur’e systems with a saturation nonlinearity and marginally stable
linear part.
For the switched linear systems, it is assumed that the dynamics of all the separate linear modes
are given. For this setting, it was investigated if it is possible to find a switching rule (which
defines when to switch between the available modes) such that the closed-loop system is convergent.
Both a state-based, an observer-based, and a time-based switching rule are presented
that guarantee a convergent system, assuming some conditions on the linear dynamics are met.
The second type of systems that are discussed, are Lur’e systems with a saturation nonlinearity
and marginally stable linear part. For this type of systems, the goal was to find sufficient
conditions under which the closed-loop system is convergent. Because of the marginally stable
linear part, however, a quadratically convergent system cannot be obtained. Instead, sufficient
conditions are discussed that guarantee uniform convergency of the system. The obtained theory
is shown to be also applicable to a class of anti-windup systems with a marginally stable
plant. For this class of systems, the results of the convergency-based performance analysis
are compared with the analysis results of existing anti-windup methods. It is shown that the
convergency-based performance analysis can in some cases provide more detailed information
on the steady-state behavior of the system.
The results of uniform convergency for anti-windup systems are shown to be also applicable in
the field of production and inventory control of discrete-event manufacturing systems. Since a
manufacturing machine has a certain production capacity and cannot produce at a negative rate,
it can be seen as an integrator plant (input: production rate, output: amount of finished products)
preceded by a saturation function. For this marginally stable plant, a controller was constructed
in such a way that the closed-loop system is uniformly convergent. The controller was also
implemented in the discrete-event domain and the results from discrete-event simulations were
compared with those of continuous-time simulations. Similarly, the controller was also applied
for the production and inventory control of a line of four manufacturing machines. For both the
single machine and the line of four machines, the resulting controlled discrete-event systems
are shown to have the desired tracking properties.
Besides these theoretical and numerical results, also experimental results are presented in this
thesis. By means of an electromechanical construction, several experimental results were obtained,
and used to validate the theoretical results for both the switched linear systems and the
anti-windup systems.
Originele taal-2 | Engels |
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Kwalificatie | Doctor in de Filosofie |
Toekennende instantie |
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Begeleider(s)/adviseur |
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Datum van toekenning | 19 aug. 2008 |
Plaats van publicatie | Eindhoven |
Uitgever | |
Gedrukte ISBN's | 978-90-386-1344-4 |
DOI's | |
Status | Gepubliceerd - 2008 |