607 research outputs found
Convex Incremental Dissipativity Analysis of Nonlinear Systems
Efficiently computable stability and performance analysis of nonlinear
systems becomes increasingly more important in practical applications. An
important notion connecting stability and performance is dissipativity.
However, this property is usually only valid around an equilibrium point of the
nonlinear system and usually involves cumbersome computations to find a valid
storage function. Analyzing stability using the trajectories of the nonlinear
system, i.e. incremental stability analysis, has shown to solve the first
issue. However, how stability and performance characterizations of nonlinear
systems in the incremental framework are linked to dissipativity, and how
general performance characterization beyond the -gain concept
can be understood in the incremental framework is largely unknown. By
systematically establishing the missing links, this paper presents a matrix
inequality based convex dissipativity analysis with the use of quadratic
storage and supply functions, for a rather general class of systems with smooth
nonlinearities. The proposed dissipativity analysis links the notions of
differential, incremental and general dissipativity by a chain of implications.
We show that through differential dissipativity, we give guarantees on
incremental and general dissipativity of the nonlinear system. Using the
results from the aforementioned chain of implications, incremental extensions
for the analysis of -gain, the generalized -norm,
-gain and passivity of a nonlinear system are presented.
Moreover, we give a convex computation method to solve the obtained conditions.
The effectiveness of the analysis tools are demonstrated by means of an
academic example
Diffeomorphic Transformations for Time Series Analysis: An Efficient Approach to Nonlinear Warping
The proliferation and ubiquity of temporal data across many disciplines has
sparked interest for similarity, classification and clustering methods
specifically designed to handle time series data. A core issue when dealing
with time series is determining their pairwise similarity, i.e., the degree to
which a given time series resembles another. Traditional distance measures such
as the Euclidean are not well-suited due to the time-dependent nature of the
data. Elastic metrics such as dynamic time warping (DTW) offer a promising
approach, but are limited by their computational complexity,
non-differentiability and sensitivity to noise and outliers. This thesis
proposes novel elastic alignment methods that use parametric \& diffeomorphic
warping transformations as a means of overcoming the shortcomings of DTW-based
metrics. The proposed method is differentiable \& invertible, well-suited for
deep learning architectures, robust to noise and outliers, computationally
efficient, and is expressive and flexible enough to capture complex patterns.
Furthermore, a closed-form solution was developed for the gradient of these
diffeomorphic transformations, which allows an efficient search in the
parameter space, leading to better solutions at convergence. Leveraging the
benefits of these closed-form diffeomorphic transformations, this thesis
proposes a suite of advancements that include: (a) an enhanced temporal
transformer network for time series alignment and averaging, (b) a
deep-learning based time series classification model to simultaneously align
and classify signals with high accuracy, (c) an incremental time series
clustering algorithm that is warping-invariant, scalable and can operate under
limited computational and time resources, and finally, (d) a normalizing flow
model that enhances the flexibility of affine transformations in coupling and
autoregressive layers.Comment: PhD Thesis, defended at the University of Navarra on July 17, 2023.
277 pages, 8 chapters, 1 appendi
Virtual Control Contraction Metrics: Convex Nonlinear Feedback Design via Behavioral Embedding
This paper proposes a novel approach to nonlinear state-feedback control
design that has three main advantages: (i) it ensures exponential stability and
-gain performance with respect to a user-defined set of
reference trajectories, and (ii) it provides constructive conditions based on
convex optimization and a path-integral-based control realization, and (iii) it
is less restrictive than previous similar approaches. In the proposed approach,
first a virtual representation of the nonlinear dynamics is constructed for
which a behavioral (parameter-varying) embedding is generated. Then, by
introducing a virtual control contraction metric, a convex control synthesis
formulation is derived. Finally, a control realization with a virtual reference
generator is computed, which is guaranteed to achieve exponential stability and
-gain performance for all trajectories of the targeted
reference behavior. Connections with the linear-parameter-varying (LPV) theory
are also explored showing that the proposed methodology is a generalization of
LPV state-feedback control in two aspects. First, it is a unified
generalization of the two distinct categories of LPV control approaches: global
and local methods. Second, it provides rigorous stability and performance
guarantees when applied to the true nonlinear system, while such properties are
not guaranteed for tracking control using LPV approaches
Recommended from our members
Computation and Learning in High Dimensions (hybrid meeting)
The most challenging problems in science often involve the learning and
accurate computation of high dimensional functions.
High-dimensionality is a typical feature for a multitude of problems
in various areas of science.
The so-called curse of dimensionality typically negates the use of
traditional numerical techniques for the solution of
high-dimensional problems. Instead, novel theoretical and
computational approaches need to be developed to make them tractable
and to capture fine resolutions and relevant features. Paradoxically,
increasing computational power may even serve to heighten this demand,
since the wealth of new computational data itself becomes a major
obstruction. Extracting essential information from complex
problem-inherent structures and developing rigorous models to quantify
the quality of information in a high-dimensional setting pose
challenging tasks from both theoretical and numerical perspective.
This has led to the emergence of several new computational methodologies,
accounting for the fact that by now well understood methods drawing on
spatial localization and mesh-refinement are in their original form no longer viable.
Common to these approaches is the nonlinearity of the solution method.
For certain problem classes, these methods have
drastically advanced the frontiers of computability.
The most visible of these new methods is deep learning. Although the use of deep neural
networks has been extremely successful in certain
application areas, their mathematical understanding is far from complete.
This workshop proposed to deepen the understanding of
the underlying mathematical concepts that drive this new evolution of
computational methods and to promote the exchange of ideas emerging in various
disciplines about how to treat multiscale and high-dimensional problems
Stability-preserving model reduction for linear and nonlinear systems arising in analog circuit applications
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 221-229).Despite the increasing presence of RF and analog components in personal wireless electronics, such as mobile communication devices, the automated design and optimization of such systems is still an extremely challenging task. This is primarily due to the presence of both parasitic elements and highly nonlinear elements, which makes simulation computationally expensive and slow. The ability to generate parameterized reduced order models of analog systems could serve as a first step toward the automatic and accurate characterization of geometrically complex components and subcircuits, eventually enabling their synthesis and optimization. This thesis presents techniques for reduced order modeling of linear and nonlinear systems arising in analog applications. Emphasis is placed on developing techniques capable of preserving important system properties, such as stability, and parameter dependence in the reduced models. The first technique is a projection-based model reduction approach for linear systems aimed at generating stable and passive models from large linear systems described by indefinite, and possibly even mildly unstable, matrices. For such systems, existing techniques are either prohibitively computationally expensive or incapable of guaranteeing stability and passivity. By forcing the reduced model to be described by definite matrices, we are able to derive a pair of stability constraints that are linear in terms of projection matrices.(cont.) These constraints can be used to formulate a semidefinite optimization problem whose solution is an optimal stabilizing projection framework. The second technique is a projection-based model reduction approach for highly nonlinear systems that is based on the trajectory piecewise linear (TPWL) method. Enforcing stability in nonlinear reduced models is an extremely difficult task that is typically ignored in most existing techniques. Our approach utilizes a new nonlinear projection in order to ensure stability in each of the local models used to describe the nonlinear reduced model. The TPWL approach is also extended to handle parameterized models, and a sensitivity-based training system is presented that allows us to efficiently select inputs and parameter values for training. Lastly, we present a system identification approach to model reduction for both linear and nonlinear systems. This approach utilizes given time-domain data, such as input/output samples generated from transient simulation, in order to identify a compact stable model that best fits the given data. Our procedure is based on minimization of a quantity referred to as the 'robust equation error', which, provided the model is incrementally stable, serves as up upper bound for a measure of the accuracy of the identified model termed 'linearized output error'. Minimization of this bound, subject to an incremental stability constraint, can be cast as a semidefinite optimization problem.by Bradley Neil Bond.Ph.D
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