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