Nonlinear Time-Frequency Control Theory with Applications

Nonlinear control is an important subject drawing much attention. When a nonlinear system undergoes route-to-chaos, its response is naturally bounded in the time-domain while in the meantime becoming unstably broadband in the frequency-domain. Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. It is necessary to facilitate nonlinear control in both the time and frequency domains without obscuring or misinterpreting the true dynamics. The objective of the dissertation is to formulate a novel nonlinear control theory that addresses the fundamental characteristics inherent of all nonlinear systems undergoing route-to-chaos, one that requires no linearization or closed-form solution so that the genuine underlying features of the system being considered are preserved. The theory developed herein is able to identify the dynamic state of the system in real-time and restrain time-varying spectrum from becoming broadband. Applications of the theory are demonstrated using several engineering examples including the control of a non-stationary Duffing oscillator, a 1-DOF time-delayed milling model, a 2-DOF micro-milling system, unsynchronized chaotic circuits, and a friction-excited vibrating disk. Not subject to all the mathematical constraint conditions and assumptions upon which common nonlinear control theories are based and derived, the novel theory has its philosophical basis established in the simultaneous time-frequency control, on-line system identification, and feedforward adaptive control. It adopts multi-rate control, hence enabling control over nonstationary, nonlinear response with increasing bandwidth ? a physical condition oftentimes fails the contemporary control theories. The applicability of the theory to complex multi-input-multi-output (MIMO) systems without resorting to mathematical manipulation and extensive computation is demonstrated through the multi-variable control of a micro-milling system. The research is of a broad impact on the control of a wide range of nonlinear and chaotic systems. The implications of the nonlinear time-frequency control theory in cutting, micro-machining, communication security, and the mitigation of friction-induced vibrations are both significant and immediate

State switching in multi-stable systems: control and optimisation.

This thesis studies state-switching in multistable systems, so that they can switch from inefficient operating states to a more efficient one, in order to achieve performance enhancement in real-life engineering systems. Multistable systems have more than one stable state under a set of parameters and the process of switching from an undesired state to a desired state is achieved by the proposed PD-like controller. It exploits the difference of the displacement and velocity between the undesired and the desired stable conditions for feedback in state switching. Three test systems are used for investigating the performance of this PD-like controller, namely: the Duffing oscillator, which is a typical smooth multistable system; the non-smooth soft-impact oscillator; and the soft-impact oscillator with a drift. A randomised triangular subdivision algorithm is proposed to reconstruct the basins of attraction of the target multistable systems, in order to identify the desired state for switching. Due to the limited capacity of physical actuators, behaviours of the constrained PD-like controller are investigated using extensive simulation on the test systems. Moreover, optimisation of the controller (based on multiple performance objectives) can further improve system performance. Two performance objectives - maximum peak of control input and switching duration - are adopted in optimising the proposed PD-like controller. The first objective is minimised in order to avoid output limit and reduce energy consumption in the actuator, while the second objective is minimised in order to shorten the time required for state switching. These two performance objectives are considered independently in performance optimisation, using particle swarm optimisation (PSO). Since these two objectives are in conflict with each other, both objectives are minimised simultaneously in multiobjective optimisation of the performance of the PD-like controller using Non-Dominated Sorting Genetic Algorithms-II (NSGA-II). A trade-off in performance enhancement is achieved through selecting control parameters from the Pareto optimal set

The Effect of Time-Delay Feedback Controller on an Electrically Actuated Resonator

This paper presents a study of the effect of a time-delay feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated by DC and AC voltages. It is shown that negative time-delay feedback control gain can lead to an unstable system, even if AC voltage is relatively small compared to DC voltage. Perturbation method is utilized to present analytically the nonlinear dynamic characteristics of the MEMS capacitor. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved

Complex dynamical networks constructed with fully controllable nonlinear nanomechanical oscillators

Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network’s full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization

Nonlinear Time-Frequency Control Theory with Applications

Nonlinear control is an important subject drawing much attention. When a nonlinear system undergoes route-to-chaos, its response is naturally bounded in the time-domain while in the meantime becoming unstably broadband in the frequency-domain. Control scheme facilitated either in the time- or frequency-domain alone is insufficient in controlling route-to-chaos, where the corresponding response deteriorates in the time and frequency domains simultaneously. It is necessary to facilitate nonlinear control in both the time and frequency domains without obscuring or misinterpreting the true dynamics. The objective of the dissertation is to formulate a novel nonlinear control theory that addresses the fundamental characteristics inherent of all nonlinear systems undergoing route-to-chaos, one that requires no linearization or closed-form solution so that the genuine underlying features of the system being considered are preserved. The theory developed herein is able to identify the dynamic state of the system in real-time and restrain time-varying spectrum from becoming broadband. Applications of the theory are demonstrated using several engineering examples including the control of a non-stationary Duffing oscillator, a 1-DOF time-delayed milling model, a 2-DOF micro-milling system, unsynchronized chaotic circuits, and a friction-excited vibrating disk. Not subject to all the mathematical constraint conditions and assumptions upon which common nonlinear control theories are based and derived, the novel theory has its philosophical basis established in the simultaneous time-frequency control, on-line system identification, and feedforward adaptive control. It adopts multi-rate control, hence enabling control over nonstationary, nonlinear response with increasing bandwidth ? a physical condition oftentimes fails the contemporary control theories. The applicability of the theory to complex multi-input-multi-output (MIMO) systems without resorting to mathematical manipulation and extensive computation is demonstrated through the multi-variable control of a micro-milling system. The research is of a broad impact on the control of a wide range of nonlinear and chaotic systems. The implications of the nonlinear time-frequency control theory in cutting, micro-machining, communication security, and the mitigation of friction-induced vibrations are both significant and immediate

Huygens' synchronization of dynamical systems : beyond pendulum clocks

Synchronization is one of the most deeply rooted and pervasive behaviours in nature. It extends from human beings to unconscious entities. Some familiar examples include the fascinating motion of schools of fish, the simultaneous flashing of fireflies, a couple dancing in synchrony with the rhythm of the music, the synchronous firing of neurons and pacemaker cells, and the synchronized motion of pendulum clocks. In a first glimpse to these examples, the existence of selfsynchronization in nature may seem almost miraculous. However, the main "secret" behind this phenomenon is that there exists a communication channel, called coupling, such that the entities/systems can influence each other. This coupling can be, for instance, in the form of a physical interconnection or a certain chemical process. Although synchronization is a ubiquitous phenomenon among coupled oscillatory systems, its onset is not always obvious. Consequently, the following questions arise: How exactly do coupled oscillators synchronize themselves, and under what conditions? In some cases, obtaining answers for these questions is extremely challenging. Consider for instance, the famous example of Christiaan Huygens of two pendulum clocks exhibiting anti-phase or in-phase synchronized motion. Huygens did observe that there is a "medium" responsible for the synchronized motion, namely the bar to which the pendula are attached. However, despite the remarkably correct observation of Huygens, even today a complete rigorous mathematical explanation of this phenomenon, using proper models for pendula and flexible coupling bar, is still missing. The purpose of this thesis is to further pursue the nature of the synchronized motion occurring in coupled oscillators. The first part, addresses the problem of natural synchronization of arbitrary self-sustained oscillators with Huygens coupling. This means that in the analysis, the original setup of Huygens’ clocks is slightly modified in the sense that each pendulum clock is replaced by an arbitrary second order nonlinear oscillator and instead of the flexible wooden bar (called here Huygens’ coupling), a rigid bar of one degree of freedom is considered. Each oscillator is provided with a control input in order to guarantee steady-state oscillations. This requirement of having a control input to sustain the oscillations can be linked to Huygens’ pendulum clocks, where each pendulum is equipped with an escapement mechanism, which provides an impulsive force to the pendulum in order to keep the clocks running. Then, it is shown that the synchronized motion in the oscillators is independent of the kind of controller used to maintain the oscillations. Rather, the coupling bar, i.e. Huygens’ coupling is considered as the key element in the occurrence of synchronization. In particular, it is shown that the mass of the coupling bar determines the eventual synchronized behaviour in the oscillators, namely in-phase and anti-phase synchronization. The Poincaré method is used in order to determine the existence and stability of these synchronous motions. This is feasible since in the system there exists a natural small parameter, namely the coupling strength, which value is determined by the mass of the coupling bar. Next, the synchronization problem is addressed from a control point of view. First, the synchronization problem of two chaotic oscillators with Huygens’ coupling is discussed. It is shown that by driving the coupling bar with an external periodic excitation, it is possible to trigger the onset of chaos in the oscillators. The mass of the coupling bar is considered as the bifurcation parameter. When the oscillators are in a chaotic state, the synchronization phenomenon will not occur naturally. Consequently, it is demonstrated that by using a master-slave configuration or a mutual synchronization scheme, it is possible to achieve (controlled) synchronization. Secondly, the effect of time delay in the synchronized motion of oscillators with Huygens’ coupling is investigated. In this case, the wooden bar, is replaced by a representative dynamical system. This dynamical system generates a suitable control input for the oscillators such that in closed loop the system resembles a pair of oscillators with Huygens’ coupling. Under this approach, the oscillators do not need to be at the same location and moreover, the dynamical system generating the control input should be implemented separately, using for instance a computer. Consequently, the possibility of having communication time-delays (either in the oscillators or in the applied control input) comes into play. Then, the onset of in-phase and anti-phase synchronization in the coupled/controlled oscillators is studied as a function of the coupling strength and the time delay. In addition to computer simulations, the (natural and controlled) synchronized motion of the oscillators is validated by means of experiments. These experiments are performed in an experimental platform consisting of an elastically supported (controllable) rigid bar (in Huygens’ example the wooden bar) and two (controllable) mass-spring-damper oscillators (the pendulum clocks in Huygens’ case). A key feature of this platform is that its dynamical behaviour can be adjusted. This is possible due to the fact that the oscillators and the coupling bar can be actuated independently, then by using feedback, specific desirable oscillators’ dynamics are enforced and likewise the behaviour of the coupling bar is modified. This feature is exploited in order to experimentally study synchronous behaviour in a wide variety of dynamical systems. Another question considered in this thesis is related to the modeling of the real Huygens experiment. The models used in the first part of this thesis and the ones reported in the literature are simplifications of the real model: the coupling bar has been considered as a rigid body of one degree of freedom. However, in the real Huygens experiment, the bar to which the clocks are attached is indeed an infinite dimensional system for which a rigorous study of the in-phase and antiphase synchronized motion of the two pendula is, as far as is known, still never addressed in the literature. The second part of the thesis focuses on this. A Finite Element Modelling technique is used in order to derive a model consisting of a (finite) set of ordinary differential equations. Numerical results illustrating all the possible stationary solutions of the "true" infinite dimensional Huygens problem are included. In summary, the results contained in the thesis in fact reveal that the synchronized motion observed by Huygens extends beyond pendulum clocks