561 research outputs found
On stabilizability of nonlinearly parameterized discrete-time systems
published_or_final_versio
Global parameter identification and control of nonlinearly parameterized systems
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2002.Includes bibliographical references (leaves 109-114).Nonlinearly parameterized (NLP) systems are ubiquitous in nature and many fields of science and engineering. Despite the wide and diverse range of applications, there exist relatively few results in control systems literature which exploit the structure of the nonlinear parameterization. A vast majority of presently applicable global control design approaches to systems with NLP, make use of either feedback-linearization, or assume linear parameterization, and ignore the specific structure of the nonlinear parameterization. While this type of approach may guarantee stability, it introduced three major drawbacks. First, they produce no additional information about the nonlinear parameters. Second, they may require large control authority and actuator bandwidth, which makes them unsuitable for some applications. Third, they may simply result in unacceptably poor performance. All of these inadequacies are amplified further when parametric uncertainties are present. What is necessary is a systematic adaptive approach to identification and control of such systems that explicitly accommodates the presence of nonlinear parameters that may not be known precisely. This thesis presents results in both adaptive identification and control of NLP systems. An adaptive controller is presented for NLP systems with a triangular structure. The presence of the triangular structure together with nonlinear parameterization makes standard methods such as back-stepping, and variable structure control inapplicable. A concept of bounding functions is combined with min-max adaptation strategies and recursive error formulation to result in a globally stabilizing controller.(cont.) A large class of nonlinear systems including cascaded LNL (linear-nonlinear-linear) systems are shown to be controllable using this approach. In the context of parameter identification, results are derived for two classes of NLP systems. The first concerns systems with convex/concave parameterization, where min-max algorithms are essential for global stability. Stronger conditions of persistent excitation are shown to be necessary to overcome the presence of multiple equilibrium points which are introduced due to the stabilization aspects of the min-max algorithms. These conditions imply that the min-max estimator must periodically employ the local gradient information in order to guarantee parameter convergence. The second class of NLP systems considered in this concerns monotonically parameterized systems, of which neural networks are a specific example. It is shown that a simple algorithm based on local gradient information suffices for parameter identification. Conditions on the external input under which the parameter estimates converge to the desired set starting from arbitrary values are derived. The proof makes direct use of the monotonicity in the parameters, which in turn allows local gradients to be self-similar and therefore introduces a desirable invariance property. By suitably exploiting this invariance property and defining a sequence of distance metrics, global convergence is proved. Such a proof of global convergence is in contrast to most other existing results in the area of nonlinear parameterization, in general, and neural networks in particular.by Aleksandar M. KojiÄ.Ph.D
Implicit regularization and momentum algorithms in nonlinear adaptive control and prediction
Stable concurrent learning and control of dynamical systems is the subject of
adaptive control. Despite being an established field with many practical
applications and a rich theory, much of the development in adaptive control for
nonlinear systems revolves around a few key algorithms. By exploiting strong
connections between classical adaptive nonlinear control techniques and recent
progress in optimization and machine learning, we show that there exists
considerable untapped potential in algorithm development for both adaptive
nonlinear control and adaptive dynamics prediction. We first introduce
first-order adaptation laws inspired by natural gradient descent and mirror
descent. We prove that when there are multiple dynamics consistent with the
data, these non-Euclidean adaptation laws implicitly regularize the learned
model. Local geometry imposed during learning thus may be used to select
parameter vectors - out of the many that will achieve perfect tracking or
prediction - for desired properties such as sparsity. We apply this result to
regularized dynamics predictor and observer design, and as concrete examples
consider Hamiltonian systems, Lagrangian systems, and recurrent neural
networks. We subsequently develop a variational formalism based on the Bregman
Lagrangian to define adaptation laws with momentum applicable to linearly
parameterized systems and to nonlinearly parameterized systems satisfying
monotonicity or convexity requirements. We show that the Euler Lagrange
equations for the Bregman Lagrangian lead to natural gradient and mirror
descent-like adaptation laws with momentum, and we recover their first-order
analogues in the infinite friction limit. We illustrate our analyses with
simulations demonstrating our theoretical results.Comment: v6: cosmetic adjustments to figures 4, 5, and 6. v5: final version,
accepted for publication in Neural Computation. v4: significant updates,
revamped section on dynamics prediction and exploiting structure. v3: new
general theorems and extensions to dynamic prediction. 37 pages, 3 figures.
v2: significant updates; submission read
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