María Tomás-Rodríguez

Linear approximations to nonlinear dynamical systems with applications to stability and spectral theory

There are many approaches to the study of nonlinear dynamical systems, including local linearizations in phase space (Perko, 1991), global linear representations involving the Lie series solution (Banks & Iddir, 1992; Banks, 1992; Banks et al., 1996), Lie algebraic methods (Banks, 2001) and global results based on topological indices (McCaffrey & Banks, 2002; Perko, 1991). Linear systems, on the other hand are very well understood and there is, of course, a vast literature on the subject (see, for example, Banks, 1986). The simplicity of linear mathematics relative to nonlinear theory is evident and forms the basis of much of classical mathematics and physics. It is therefore attractive to try to attack nonlinear problems by linear methods, which are not local in their applicability. In this paper we study a recently introduced approach to nonlinear dynamical systems based on a representation of the system as the limit of a sequence of linear, time-varying approximations which converge in the space of continuous functions to the solution of the nonlinear system, under a very mild local Lipschitz condition. This approach has already been used in optimal control theory (Banks & Dinesh, 2000), in the theory of nonlinear delay systems (Banks, 2002) and in the theory of chaos (Banks & McCaffrey, 1998). In these papers, however, only a local (in time) proof of convergence was given—here we shall give a global proof for dynamical systems which do not have finite escape time. We shall also study two further applications of the method: to the study of stability of nonlinear systems and the definition of a spectral theory for nonlinear systems. In the former case, we shall define a generalized Lyapunov function, the existence of which is equivalent to the stability of the system. In the second case we first study the spectral theory of a linear, time-varying system as the perturbation of a time-invariant system and then use the approximation scheme to extend it to nonlinear systems.

Linear, Time-varying Approximations to Nonlinear Dynamical Systems

to Nonlinear Systems.- Linear Approximations to Nonlinear Dynamical Systems.- The Structure and Stability of Linear, Time-varying Systems.- General Spectral Theory of Nonlinear Systems.- Spectral Assignment in Linear, Time-varying Systems.- Optimal Control.- Sliding Mode Control for Nonlinear Systems.- Fixed Point Theory and Induction.- Nonlinear Partial Differential Equations.- Lie Algebraic Methods.- Global Analysis on Manifolds.- Summary, Conclusions and Prospects for Development.

Suppression of Burst Oscillations in Racing Motorcycles

Burst oscillations occurring at high speed, and under firm acceleration, can be suppressed with a mechanical steering compensator. Burst instabilities in the subject racing motorcycle are the result of interactions between the wobble and weave modes under firm-acceleration at high speed. Under accelerating conditions, the wobble-mode frequency (of the subject motorcycle) decreases, while the weave mode frequency increases so that destabilizing interactions can occur. The design analysis is based on a time-separation principle, which assumes that bursting occurs on time scales over which speed variations can be neglected. Even under braking and acceleration conditions linear time-invariant models corresponding to constant-speed operation can be utilized in the design process. The influences of braking and acceleration are modeled using d’Alembert-type inertial forces that are applied at the mass centers of each of the model’s constituent bodies. The resulting steering compensator is a simple mechanical network that comprises a conventional steering damper in series with a linear spring. In control theoretic terms, this network is a mechanical lag compensator. A robust control framework was used to optimize the compensator design because it is necessary to address the inevitable uncertainties in the motorcycle model, as well as the nonlinearities that influence the machine’s local behavior as the vehicle ranges over its operating envelope.

Vibration reduction for vision systems on board unmanned aerial vehicles using a neuro-fuzzy controller

In this paper, an intelligent control approach based on neuro-fuzzy systems performance is presented, with the objective of counteracting the vibrations that affect the low-cost vision platform onboard an unmanned aerial system of rotating nature. A scaled dynamical model of a helicopter is used to simulate vibrations on its fuselage. The impact of these vibrations on the low-cost vision system will be assessed and an intelligent control approach will be derived in order to reduce its detrimental influence. Different trials that consider a neuro-fuzzy approach as a fundamental part of an intelligent semi-active control strategy have been carried out. Satisfactory results have been achieved compared to those obtained by means of vibration reduction passive techniques.

Sliding Mode Control for Nonlinear Systems: An Iterative Approach.

In this paper an alternative method of sliding mode control for nonlinear systems is presented. A time-varying approach to nonlinear sliding mode control is studied based on an iteration technique that approaches a nonlinear system by a sequence of LTV equations. In this paper, the convergence of this method is shown and applied to a practical example

Discrete time linear optimal multi-periodic repetitive control: a benchmark tracking solution

The use of optimal control tracking methodologies for the development of feedback control schemes for arbitrary discrete-time linear systems is considered for the special case when the demand signal is multi-periodic, i.e. can be expressed as a finite sum of periodic signals of different but known frequencies. By embedding the reference signal into plant dynamics, the problem is reduced to a classical linear quadratic control problem which yields an explicit formula for a globally stabilizing tracking controller. Asymptotic perfect tracking is guaranteed by the presence of the familiar internal model of the demand signal dynamics within the control structure. Transfer function analysis of the controller indicates that it consists of an inner state feedback loop plus an error actuated forward path control element containing the internal model.

PARAMETRIC APPROACH TO OPTIMAL NONLINEAR CONTROL PROBLEM USING ORTHOGONAL EXPANSIONS

Abstract This article presents a parametric approach to the optimal nonlinear control problem. This contribution is based on reducing the nonlinear optimal control problem to a sequence of linear time-varying ones and approaching the desired optimal control by a sequence of linear algebraic equations. This approach provides a solution which can be obtained in a simpler way than the usual one. Alternative choices of orthonormal basis for the parametric approach are discussed.

Limit sets and switching strategies in parameter-optimal iterative learning control

This paper characterizes the existence and form of the possible limit error signals in typical parameter-optimal iterative learning control. The set of limit errors has attracting and repelling components and the behaviour of the algorithm in the vicinity of these sets can be associated with the undesirable properties of apparent (but in fact temporary) convergence or permanent slow convergence properties in practice. The avoidance of these behaviours in practice is investigated using novel switching strategies. Deterministic strategies are analysed to prove the feasibility of the concept by proving that each of a number of such strategies is guaranteed to produce global convergence of errors to zero independent of the details of plant dynamics. For practical applications a random switching strategy is proposed to replace these approaches and shown, by example, to produce substantial potential improvements when compared with the non-switching case. The work described in this paper is covered by pending patent applications in the UK and elsewhere.

An Iterative Approach to Eigenvalue Assignment for Nonlinear Systems.

In this paper, the authors present a method of eigenvalue assignment for nonlinear systems. A time-varying approach to nonlinear exponential stability via eigenvalue placement is studied based on an iteration technique that approaches a nonlinear system by a sequence of LTV equations. The convergent behavior of this method is shown and applied to a practical example to illustrate these ideas

Influence of road camber on motorcycle stability

This paper studies the influence of road camber on the stability of single-track road vehicles. Road camber changes the magnitude and direction of the tire force and moment vectors relative to the wheels, as well as the combined-force limit one might obtain from the road tires. Camber-induced changes in the tire force and moment systems have knock-on consequences for the vehicles stability. In order to study camber-induced stability trends for a range of machine speeds and roll angles, we study the machine dynamics as the vehicle travels over the surface of a right circular cone. Conical road surfaces allow the machine to operate at a constant steady-state speed, a constant roll angle and a constant road camber angle. The results show that at low speed both the weave- and wobble-mode stability is at a maximum when the machine is perpendicular to the road surface. This trend is reversed at high speed, since the weave- and wobble-mode damping is minimized by running conditions in which the wheels are orthogonal to the road. As a result, positive camber, which is often introduced by road builders to aid drainage and enhance the friction limit of four-wheeled vehicle tires, might be detrimental to the stability of two-wheeled machines.