This thesis introduces, develops and applies methods for analysing nonlinear systems with the multiple challenges of time-varying, non-polynomial, uncertain or large-scale proper- ties. Both computational and analytic methods using Lyapunov functions are developed and the methods are applied to a range of examples. Generalised Absolute stability is introduced, which is a method of treating polynomial systems with non polynomial, uncertain or time-varying feedback. Analysis is completed with Sum of Squares programming, and this method extends both the applicability of sum of squares as well as existing absolute stability theory. Perturbation methods for invariant Sum of Squares and Semidefinite programs are introduced, which significantly improves scalability of computations and al- lows sum of squares programming to be used for large scale systems. Finally, invariance principles are introduced for nonlinear, time-varying systems. The concept of trajectories leaving sets uniformly in time is introduced, which allows a non-autonomous version of Barbashin-Krasovskii theorem.

A dissertation submitted to the Faculty Of Science in partial fulfilment of the requirements for the degree of Master of Science by research only in the School of Physics, University of the Witwatersrand, Johannesburg. July 11, 2017. / A discrete Q-switched laser that gives a side-lobed single pulse as a laser output was implemented; followed by studying energy extraction e ciencies and pulse characterisation. The aim was to help design a passively Q-switched laser that gives a smooth single pulse of optimum energy as a laser output. The smoothness feature in a single pulse is important in some applications such as range nding. The concepts are demonstrated both experimentally and numerically; the latter using Fox-Li approach to modeling resonator modes with the Fresnel's integral for the system under study. In the rst two chapters, fundamentals of how a laser works and the spatial mode development are studied. In chapter 1, the principles of a laser are discussed: absorption, spontaneous and stimulated emission. Also, di erent types of pumping sources and resonator con gurations that can be used are discussed. In chapter 2, the focus is on developing spatial modes of a laser. The fundamental and high order modes are discussed together with their propagation laws. Then a numerical method is used to nd the eigenmodes of an arbitrary resonator con guration. This numerical method is used to simulate propagation of a fundamental mode and the simulation results are compared to analytical propagation laws. Then, this numerical method is used to simulate a laser resonator. The eigenmode of the lowest loss in the resonator was found. In chapters 3 and 4, experimental work is done on a Q-switched laser where the focus is on the overall laser performance. In chapter 3, Q-switched laser output energies are studied for di erent combinations of Q-switch transmission values and output coupler re ectivities. In addition, the in uence of spatial modes on a Q-switched pulse shape and pulse width are studied, taking into account beam divergence. In chapter 4, conclusions and future work are presented. In future work, the knowledge of spatial mode in uence on pulse shape, pulse width and beam divergence from chapter 3 is exploited. Then particular resonator con guration that gives optimised output results (Q-switched laser output energy, beam divergence, pulse shape and pulse width) is chosen. On that particular resonator, di erent Q-switch transmission values are studied, but now looking only at beam divergence and pulse width. Also, some suggestions on further improving laser performance are given. / LG2018

Programme in Advanced Mathematics of Finance, University of the Witwatersrand, Johannesburg. / Peng introduced a typical ltration consistent nonlinear expectation, called a g-expectation in [40]. It satis es all properties of the classical mathematical expectation besides the linearity. Peng's conditional g-expectation is a solution to a backward stochastic di erential equation (BSDE) within the classical framework of It^o's calculus, with terminal condition given at some xed time T. In addition, this g-expectation is uniquely speci ed by a real function g satisfying certain properties. Many properties of the g-expectation, which will be presented, follow from the speci cation of this function. Martingales, super- and submartingales have been de ned in the nonlinear setting of g-expectations. Consequently, a nonlinear Doob-Meyer decomposition theorem was proved. Applications of g-expectations in the mathematical nancial world have also been of great interest. g-Expectations have been applied to the pricing of contingent claims in the nancial market, as well as to risk measures. Risk measures were introduced to quantify the riskiness of any nancial position. They also give an indication as to which positions carry an acceptable amount of risk and which positions do not. Coherent risk measures and convex risk measures will be examined. These risk measures were extended into a nonlinear setting using the g-expectation. In many cases due to intermediate cash ows, we want to work with a multi-period, dynamic risk measure. Conditional g-expectations were then used to extend dynamic risk measures into the nonlinear setting. The Choquet expectation, introduced by Gustave Choquet, is another nonlinear expectation. An interesting question which is examined, is whether there are incidences when the g-expectation and the Choquet expectation coincide.

Terahertz (THz) frequencies comprise the portion of the electromagnetic spectrum more energetic than microwaves, but less energetic than infrared light. The THz band presents many opportunities for condensed matter physics and optics engineering. From the physics perspective, advances in the generation and detection of THz radiation have opened the door for spectroscopic studies of a range of solid-state phenomena that manifest at THz frequencies. From an engineering perspective, THz frequencies are an under-used spectral region, ripe for the development of new devices. In both cases, the challenge for researchers is to overcome a lack of sources, detectors, and optics for THz light, termed the THz Gap. Metamaterials (MMs), composite structures with engineered index of refraction, n, and impedance, Z, provide one path towards realizing THz optics. MMs are an ideal platform for the design of local EM field distributions, and far-field optical properties. This is especially true at THz frequencies, where fabrication of inclusions is easily accomplished with photolithography. Historically, MM designs have been based around static configurations of resonant inclusions that work only in a narrow frequency band, limiting applications. Broadband and tunable MMs are needed to overcome this limit. This dissertation focuses on creating tunable and controllable MM structures through the manipulation of electromagnetic interactions between MM inclusions. We introduce three novel MM systems. Each system is studied computationally with CST-Studio, and experimentally via THz spectroscopy. First, we look at the tunable transmission spectrum of two coupled split ring resonators (SRRs) with different resonant frequencies. We show that introducing a lateral displacement between the two component resonators lowers the electromagnetic coupling between the SRRs, activating a new resonance. Second, we study an SRR array, coupled to a non-resonant closed ring array. We show that lowering the interaction strength through lateral displacement changes the MM oscillator strength by ~ 40% and electric field enhancement by a factor of 4. Finally, we show that interactions between a superconducting SRR array and a conducting ground plane result in a temperature and field strength dependent MM absorption. The peak absorption changes by ~ 40% with increasing electric field and by ~ 66% with increasing temperature.

For decades, the increasing applications in engineering field, such as space exploration, ocean sampling, aerial vehicle formation flight and so on, have made the cooperative control problem of multi-agent systems a research focus. In this thesis, we will further study the consensus problem, one of the basic cooperative control problems, for two classes of nonlinear multi-agent systems: the Euler-Lagrange systems and the rigid spacecraft attitude control systems. / An Euler-Lagrange system refers to a system whose motion equation is derived by the Euler-Lagrange equation. Euler-Lagrange systems can describe many practical systems, such as robot manipulators and AC motors. The rigid spacecraft attitude control system is usually embedded in the overall control system for a rigid spacecraft to perform attitude maneuver, such as spin stabilization or target acquisition. The consensus problems for these two classes of systems have many applications, like robot manipulators coordination and spacecraft formation flying. Being one of the basic cooperative control problems, consensus problem lays the foundation for some typical cooperative control problems, such as rendezvous, flocking and formation control. So far, two kinds of consensus problems have been investigated, i.e., the leader-following consensus problem and the leaderless consensus problem. For a multi-agent system, each agent will be viewed as a subsystem of the overall multi-agent system. The leader-following consensus problem aims to design a control law such that the state and/or output of each subsystem will asymptotically track a prescribed trajectory, which is usually generated by another subsystem called the leader system, while the leaderless consensus problem requires that the control law drive the state and/or output of each subsystem to a common trajectory. / The leader-following consensus problems for both the Euler-Lagrange systems and the rigid spacecraft attitude control systems present certain technical difficulties. On one hand, the controller is constrained by the communication network, which describes the information flow among all subsystems. On the other hand, both the Euler-Lagrange system and the rigid spacecraft attitude control system are nonlinear systems with complex kinematic or dynamic equations. To overcome these difficulties, we have employed the distributed observer approach, which consists of two steps: first, given a leader system, a distributed observer is designed for each follower subsystem to estimate the state of the leader system and recover the reference signal; second, by making use of the estimated reference signal, a certainty equivalence controller will be synthesized to control the overall multi-agent system. Whether or not the distributed observer approach will work depends on two crucial issues: the existence of the distributed observer for a given leader system and the stability of the closed-loop system subject to the certainty equivalence controller. / In the first part of this thesis, we have considered both the leader-following and leader-less consensus problems for multiple uncertain Euler-Lagrange systems under a switching communication network. The main results are summarized as follows: / 1. The leader-following consensus problem for multiple uncertain Euler-Lagrange systems under switching communication network was studied. Since the kinematic equation of the Euler-Lagrange system is linear, the leader system is linear. Given this leader system, we have established the distributed observers for all follower subsystems under a jointly connected communication network. To show the stability of the closed-loop system, we resorted to a generalized Barbalat’s lemma to deal with the discontinuity resulted from the switching of the communication network. / 2. Inspired by the idea of the distributed observer approach, we have employed an auxiliary system for each subsystem to further solve the leaderless consensus problem for multiple uncertain Euler-Lagrange systems under jointly connected communication network. When the communication network is undirected, we pointed out that the final common trajectory will be uniquely decided by the initial values and will not be affected by the communication network. / The second part of this thesis addresses the leader-following attitude consensus problem for multiple rigid spacecraft systems. Owing to the nonlinear kinematic equation of the rigid spacecraft attitude control system, the leader system is also nonlinear and we have established a nonlinear distributed observer for each follower subsystem to estimate the state of the leader system. Three scenarios were considered and are summarized as follows: / 1. In the first scenario, the controller uses the control feedback of both the attitude and angular velocity. We posed a technical lemma to show that the couplings between subsystems do not change the asymptotic behavior of each follower subsystem and hence the stability of the closed-loop system can be guaranteed. / 2. In the second scenario, only the attitude is available for control feedback. In the absence of the angular velocity, it is more difficult to prove the stability of the closed loop system. In order to do so, we have enhanced the performance of the distributed observers by making the convergent speed exponential. / 3. In the third scenario, we further solved the attitude consensus problem for multiple rigid spacecraft systems subject to unknown system parameters and external disturbances. For each follower subsystem, an adaptive law is used to estimate the unknown parameters and a dynamic compensator is attached to eliminate the influence of the external disturbances. / 近十年来，随着在空间探索，海洋监测，以及飞行器编队飞行等工程领域内的逐步应用，多智能体系统的协作式控制问题受到了广泛的关注。在本论文中，我们将进一步采用基于分布式观测器的方法来研究两类非线性多智能体系统，即Euler-Lagrange系统与刚性航天器姿态控制系统的一类基本协作式控制问题-趋同问题。 / Euler-Lagrange系统指代一类系统，其动态方程可由Euler-Lagrange方程导出。Euler-Lagrange系统可以描述许多实际系统，例如机器人手臂与交流电机。航天器姿态控制系统通常被嵌入航天器的全局控制系统中以调整航天器姿态，例如自旋镇定与目标跟踪。这两类系统的趋同问题有诸多应用，例如机器人手臂协作与航天器编队飞行。作为一类基本的协作式控制问题，趋同问题是一些典型的协作式控制问题的基础，例如蜂拥，聚类与编队。目前，趋同问题主要分为两类：有领导者的趋同问题与无领导者的趋同问题。在一个多智能体系统中，每一个智能体都作为全局系统的一个子系统。有领导者的趋同问题的控制目标是设计控制器使得所有子系统的状态和/或输出渐近地跟踪一个指定的轨迹，该轨迹通常由另一个被称作领导者的子系统产生，而无领导者的趋同问题则要求设计控制器使所有子系统的状态和/或输出渐近地趋于一个共同的轨迹。 / 目前，Euler-Lagrange系统与刚性航天器姿态控制系统的趋同问题都面临着技术上的挑战。一方面，控制器的设计受限于用于描述信息流动的通信网络。另一方面，Euler-Lagrange系统与刚性性航天器姿态控制系统都是非线性系统，且具有复杂的运动学方程或动力学方程。为了克服这些困难，我们采用了基于分布式观测器的方法。该方法包含两步：首先，为每一个跟随者设计一个分布式观测器以估测领导者的状态；其次，根据观测器的估测信号设计全局控制器。该方法能否成功取决于两个关键因素：一，对某个给定的领导者系统，分布式观测器是否存在；二，根据估测信号设计的确定性等价控制器是否能保证闭环系统的稳定性。 / 在本论文的第一部分中， 我们将分别考虑在通信网络联合连通条件下Euler-Lagrange 系统有领导者的和无领导者的趋同问题。主要结论总结如下 / 1. 我们将研究Euler-Lagrange系统的有领导者的趋同问题。因Euler-Lagrange系统的运动学方程是线性方程，其领导者系统也为线性系统。在联合联通的通信网络下，我们为每一个跟随者系统设计了线性的分布式观测器，并引用了一个广义Barbalat引理来证明闭环系统的稳定性。 / 2. 受分布式观测器方法的启发，我们将借由一个辅助系统来研究通信网络联合连通条件下Euler-Lagrange系统无领导者的趋同问题，并进一步指出，如果通信网络是无向的，那么系统的稳态将完全取决于系统的初态而与通信网络无关。 / 在本论文的第二部分中，我们将探讨刚性航天器系统的姿态趋同问题。由于刚性航天器系统的姿态运动学方程是非线性方程，其领导者系统也是一个非线性系统。我们为每一个跟随者系统设计了非线性的分布式观测器，并考虑了如下三类情况。 / 1. 在第一类情况下，控制器可同时采用角位置与角速度反馈。为确保子系统间的耦合不影响闭环系统稳定性，我们确立了一个引理用以说明此类耦合并不改变系统的稳态特性。 / 2. 在第二类情况下，控制器只能采用角位置反馈。这使得证明闭环系统的稳定性变得困难。为此，分布式观测器的收敛速度被提高至指数收敛。 / 3. 在第三类情况下，刚性航天器系统包含未知系统参数且受到外部干扰。我们采用了自适应控制技术以及动态补偿技术用以估计未知参数及抵消外部干扰。 / Cai, He. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 116-127). / Abstracts also in Chinese. / Title from PDF title page (viewed on 05, October, 2016). / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only.

by Zheng Yu-kun. / Thesis (Ph.D.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Preface --- p.1 / Chapter Chapter 1. --- Gauge Transformation and the Higher Order Korteweg-de Vries Equations --- p.6 / Chapter 1. --- Higher order KdV equations --- p.6 / Chapter 2. --- η2-dependent higher order mKdV equations --- p.9 / Chapter 3. --- η2-dependent Miura transformation and Backlund transformation --- p.13 / Chapter 4. --- Gauge transformation of the wave function --- p.15 / Chapter 5. --- Backlund transformation for the η2 -dependent higher order mKdV equation --- p.24 / Chapter 6. --- Applications --- p.25 / Chapter 7. --- References --- p.30 / Chapter Chapter 2. --- Solutions of a Nonisospectral and Variable Coefficient Korteweg-de Vries Equation --- p.31 / Chapter 1. --- Introduction --- p.31 / Chapter 2. --- Nonisospectral variable coefficient KdV-type equations --- p.32 / Chapter 3. --- Invariance of LP under the Crum transformation --- p.34 / Chapter 4. --- Backlund transformation for the h-t-KdV equation --- p.35 / Chapter 5. --- Solutions --- p.39 / Chapter 6. --- References --- p.43 / Chapter Chapter 3. --- Nonisospectral Variable Coefficient Higher Order Korteweg-de Vries Equations --- p.45 / Chapter 1. --- Introduction --- p.45 / Chapter 2. --- Nonisospectral t-ho-KdV equations --- p.47 / Chapter 3. --- Nonisospectral η2 dependent variable coefficient higher order modified KdV equation --- p.50 / Chapter 4. --- Backlund transformation and gauge transformation --- p.57 / Chapter 5. --- Example. Solutions of second order ni-t-KdV equation and its corresponding ni-t-η2-mKdV equation --- p.61 / Chapter 6. --- References --- p.66 / Chapter Chapter 4. --- Gauge and Backlund Transformations for the Variable Coefficient Higher-Order Modified Korteweg-de Vries Equation --- p.67 / Chapter 1. --- Introduction --- p.67 / Chapter 2. --- The t-ho-mKdV equation --- p.68 / Chapter 3. --- Some results about the t-ho-KdV equation --- p.74 / Chapter 4. --- A Backlund transformation for the t-ho-mKdV equation --- p.76 / Chapter 5. --- Gauge transformat ion and the Backlund transformation --- p.78 / Chapter 6. --- References --- p.85 / Chapter Chapter 5. --- Gauge and Backlund Transformat ions for the Generalized Sine-Gordon Equation and Its η Dependent Modified Equation --- p.86 / Chapter 1. --- Introduction --- p.86 / Chapter 2. --- Generalized Sine-Gordon equation --- p.87 / Chapter 3. --- Backlund transformation for the GSGE --- p.92 / Chapter 4. --- Gauge transformations for AKNS systems --- p.98 / Chapter 5. --- η dependent modified GSGE and its Backlund transformation --- p.102 / Chapter 6. --- Summary and example --- p.105 / Chapter 7. --- References --- p.110 / Chapter Chapter 6. --- Backlund Transformation for the Nonisospectral and Variable Coefficient Nonlinear Schrodinger Equation --- p.111 / Chapter 1. --- Introduction --- p.111 / Chapter 2. --- A generalized NLSE --- p.112 / Chapter 3. --- Γ Riccati equation system --- p.114 / Chapter 4. --- Invariance of the Γ-system --- p.116 / Chapter 5. --- Lax pair corresponding to the GNLSE --- p.119 / Chapter 6. --- BT´ةs for the Γ evolution equation and the GNLSE --- p.121 / Chapter 7. --- References --- p.126 / Chapter Chapter 7. --- Backlund Transformations for the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation and Its λ-Modified Equation --- p.127 / Chapter 1. --- Introduction --- p.127 / Chapter 2. --- The CDGSKE and the λ-mCDGSKE --- p.128 / Chapter 3. --- The general solution for the scattering problem of the CDGSKE --- p.130 / Chapter 4. --- The BT for the λ-mCDGSKE --- p.135 / Chapter 5. --- The BT for the CDGSKE --- p.136 / Chapter 6. --- References --- p.139 / Summary --- p.140

Zhu Minghui. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves 144-149). / Abstracts in English and Chinese. / Abstract --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Small Gain Theorem --- p.1 / Chapter 1.2 --- Stabilization for Feedforward Systems --- p.2 / Chapter 1.3 --- Output Regulation for Feedforward Systems --- p.4 / Chapter 1.4 --- Organization and Contributions --- p.5 / Chapter 2 --- Input-to-State Stability for Nonlinear Systems --- p.7 / Chapter 3 --- Small Gain Theorem with Restrictions for Uncertain Time-varying Non- linear Systems --- p.13 / Chapter 3.1 --- Input-to-State Stability Small Gain Theorem with Restrictions for Uncer- tain Nonlinear Time-varying Systems --- p.14 / Chapter 3.1.1 --- Nonlinear Time Invariant Systems Case --- p.14 / Chapter 3.1.2 --- Uncertain Time-varying Nonlinear Systems Case --- p.16 / Chapter 3.1.3 --- Remarks and Corollaries --- p.28 / Chapter 3.2 --- Semi-Uniform Input-to-State Stability Small Gain Theorem with Restric- tions for Uncertain Nonlinear Time-varying Systems --- p.38 / Chapter 3.3 --- Asymptotic Small Gain Theorem with Restrictions for Uncertain Nonlinear Time-varying Systems --- p.44 / Chapter 3.4 --- Input-to-State Stability Small Gain Theorem with Restrictions for Uncer- tain Time-varying Systems of Functional Differential Equations --- p.49 / Chapter 4 --- A Remark on Various Small Gain Conditions --- p.52 / Chapter 4.1 --- Introduction --- p.52 / Chapter 4.2 --- Preliminary --- p.53 / Chapter 4.3 --- The Sufficient and Necessary Condition for Input-to-State Stability of Time-varying Systems --- p.56 / Chapter 4.3.1 --- ISS-Lyapunov functions for Time-varying Systems --- p.56 / Chapter 4.3.2 --- A Remark on Input-to-State Stability for Time-varying Systems --- p.61 / Chapter 4.4 --- Comparison of Various Small Gain Theorems --- p.63 / Chapter 4.4.1 --- Comparison of Theorem 4.1 and Theorem 4.2 --- p.63 / Chapter 4.4.2 --- "Comparison of Theorem 4.1 and Theorem 4.3, Theorem 4.2 and Theorem 4.3" --- p.68 / Chapter 4.5 --- Two Small Gain Theorems for Time-varying Systems --- p.70 / Chapter 4.6 --- Conclusion --- p.73 / Chapter 5 --- Semi-global Robust Stabilization for A Class of Feedforward Systems --- p.74 / Chapter 5.1 --- Introduction --- p.74 / Chapter 5.2 --- Main result --- p.76 / Chapter 5.3 --- Conclusion --- p.91 / Chapter 6 --- Global Robust Stabilization for A Class of Feedforward Systems --- p.93 / Chapter 6.1 --- Main Result --- p.93 / Chapter 6.2 --- Conclusion --- p.104 / Chapter 7 --- Global Robust Stabilization and Output Regulation for A Class of Feedforward Systems --- p.105 / Chapter 7.1 --- Introduction --- p.105 / Chapter 7.2 --- Preliminary --- p.107 / Chapter 7.3 --- Global Robust Stabilization via Partial State Feedback --- p.108 / Chapter 7.3.1 --- RAG with restrictions --- p.110 / Chapter 7.3.2 --- Fulfillment of the restrictions --- p.114 / Chapter 7.3.3 --- Small gain conditions --- p.117 / Chapter 7.3.4 --- Uniform Global Asymptotic Stability of Closed Loop System . . . . --- p.118 / Chapter 7.4 --- Global Robust Output Regulation --- p.118 / Chapter 7.5 --- Conclusion --- p.134 / Chapter 8 --- Conclusion --- p.136 / Chapter A --- Appendix --- p.138 / List of Figures --- p.143 / Bibliography --- p.144 / Biography --- p.150

Firman So. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1999. / Includes bibliographical references (leaves [159]-160). / Text in English; abstracts in English and Chinese. / Firman So. / Abstract --- p.i / Acknowledgement --- p.iii / Contents --- p.iv / List of Figures --- p.vii / List of Tables --- p.xii / Chapter Chapter 1. --- Introduction --- p.1 / Chapter Chapter 2. --- Derivation of the Fokker-Planck Equation --- p.4 / Chapter 2.1 --- Brownian Motion --- p.4 / Chapter 2.2 --- Non-Linear Langevin Equation --- p.7 / Chapter 2.3 --- Conditional Probability Density --- p.9 / Chapter 2.4 --- Kramers-Moyal Expansion --- p.11 / Chapter 2.5 --- Fokker-Planck Equation --- p.13 / Chapter Chapter 3. --- Method & Solution of the One-variable Fokker-Planck Equation with Time-Independent Coefficients --- p.15 / Chapter 3.1 --- Stationary Solution --- p.16 / Chapter 3.2 --- Ornstein-Ulhenbeck Process: An Exactly Solvable Fokker-Planck Equation --- p.17 / Chapter 3.3 --- Eigenfunction Expansion --- p.19 / Chapter 3.4 --- Ornstein-Ulhenbeck process by Eigenfunction Expansion --- p.29 / Chapter 3.5 --- Eigenfunctions and Eigenvalues of Inverted Potentials --- p.30 / Chapter 3.6 --- Kramers' Escape Rate --- p.32 / Chapter Chapter 4. --- Diffusion in Potential Wells --- p.36 / Chapter 4.1 --- Symmetric Double-Well Potential --- p.36 / Chapter 4.2 --- Asymmetric Bistable Potential --- p.61 / Chapter Chapter 5. --- Stochastic Resonance --- p.100 / Chapter 5.1 --- Introduction --- p.100 / Chapter 5.2 --- Probability Density........................... --- p.101 / Chapter 5.3 --- Power Spectrum of the Autocorrelation Function of x --- p.113 / Chapter 5.4 --- Stochastic Resonance --- p.120 / Chapter Chapter 6. --- Colored Noise --- p.124 / Chapter 6.1 --- Introduction --- p.124 / Chapter 6.2 --- Approximation Schemes for the Colored Noise Problem --- p.125 / Chapter 6.3 --- Stationary Probability Density of the Colored Noise Driven Bistable System --- p.132 / Chapter 6.4 --- Escape Rate in the Presence of Colored Noise --- p.140 / Chapter Chapter 7. --- Conclusion --- p.146 / Appendix A --- p.149 / Chapter A.1 --- State-Dependent Diagonalization Method --- p.149 / Chapter A.2 --- Infinite-Square-Well Basis Diagnalization --- p.153 / Chapter A.3 --- Solving the Fokker-Planck equation --- p.156 / References

本文研究了多变量系统的非线性输出调节问题，并处理了机电和机械控制系统中的两个典型控制问题。 / 输出调节问题是控制中的核心问题之一。它同时处理轨迹跟踪和干扰抑制问题。参考输入和干扰是通过一定的动态系统产生，称为外部系统。因此输出调节问题和其它轨迹跟踪问题的区别在于它允许跟踪一类参考信号，而不需要精确的知道参考信号的值。而且，输出调节问题可应用于真实世界的很多控制问题，包括能源系统，交通和系统生物学。 / 众所周知，处理输出调节问题的基本框架包含两个步骤。首先把被控对象的鲁棒输 出调节问题转化成由被控对象和叫做内模的动态补偿器组成的增广系统的鲁棒镇定问题，然后鲁棒镇定增广系统。问题的关键在于如何设计一个合适的内模。这个内模不仅可以产生所有的稳态信息，而且使得增广系统可镇定。 / 虽然在过去的二十几年里关于非线性输出调节问题的研究取得了重大进展，大部分的结果只处理了单输入单输出的非线性系统。多变量非线性系统非线性输出调节问题有关的结果甚少。两大难点阻碍了多变量非线性系统非线性输出调节问题的进一步发展。一方面，由于被控对象的复杂结构，很难构造出一个合适的内模。这个内模既可以产生必需的稳态信息，又可以保证增广系统的镇定问题可解。另一方面，由于内模的引入，增广系统变成一个更加复杂的多输入多输出的非线性系统，其中既包含了动态不确定性，也包含了时变的静态不确定性。这个增广系统具有一定特殊的结构，并且从未出现在现有的文献中。增广系统的镇定问题很有挑战性。此外，关于实际系统非线性输出调节问题的研究甚少。因为输出调节方法对于未知的对象参数具有一定的鲁棒性，而且允许外部系统含有未知参数，通过输出调节方法我们可能得出更好的结果。基于以上的动机，我们将考虑具有特定结构的多变量系统的全局鲁棒输出调节问题，并解决两个典型控制问题，包括永磁同步电机的速度跟踪控制和球形倒立摆的轨迹跟踪控制。本文的贡献总结如下。 / 1. 本文考虑了一类多变量非线性系统的全周鲁棒输出调节问题。通过把内模附加到被控对象， 增广系统具有」定特殊的结构。它的镇定问题尚未有人处理过。我们把增广系统的镇定问题分解为多个单输入系统的镇定问题。然后利用changingsupply function 技术，我们通过法代的方法求解多个单输入控制问题，从而解决了这一镇定问题。理论结果可以解决面装式永磁同步电机的速度跟踪和负载干扰抑制问题。与现有的结果比较，我们的设计允许所有的电机参数未知。 / 2. 本文考虑了未知外部系统作用下的面装式永磁同步电机的速度跟踪和负载干扰抑制问题。由于外部系统未知，传统的鲁棒控制方法不能处理这一问题。而且，外部系统中的未知参数使得增广系统的镇定问题更具挑战性。因此我们需要把鲁棒控制和自适应控制相结合处理这一问题。利用changing supply function 技术和动态坐标变换技术相结合，我们用迭代的方法求解了两个单输入控制问题，进而解决了这一问题。 / 3. 本文考虑了用内模设计的方法处理永磁同步电机的速度跟踪和负载干扰抑制问题。当dq 轴电感相等时，这一问题己经得到了广泛的研究。然而dq 轴电感不等时，由于转速方程中的复杂非线性，这一问题的研究甚少。利用输出调节方法，电机参数可以具有更好的鲁棒性。由于增广系统不具有任何己知的特殊结构，我们研究出一种具体的工具来处理镇定问题。具体的说，我们研究出一种适用于non-ISS (输入到状态稳定)系统的广义changing supply function 技术。通过把这个技术与一个特殊的非线性内模相结合，这一问题得到了很好的解决。 / 4. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法基于神经网络近似调节器方程的解。求解调节器方程近似解的问题被转化成一个参数优化问题，进而有助于计算机实现的方便。而且，与现有结果中的基于三阶多项式的控制比较，对于参考输入信号的不同幅值的情况，这一方法的稳态跟踪误差保持了更加平稳的性能。 / 5. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法是基于神经网络的增强型设计，并采用输出反馈控制。与我们之前工作中的神经网络设计比较，我们给出了一种新的方法用于近似前馈控制，而不是调节器方程的解。求解前馈控制近似解的问题被转化成一个参数优化问题。我们采用Hooke-J eeeves 方法和Powell 方法解决了参数优化问题。当所有的状态可用的时候，我们的设计对于被控对象参数的变化具有一定的鲁棒性。 / In this thesis, we investigate nonlinear output regulation problem of multivariable systems and their applications to two typical control problems in electromechanical and mechanical control systems. / The output regulation problem is one of the central problems in control. It deals with trajectory tracking and disturbance rejection problem simultaneously. Both the reference inputs and disturbances are generated by some dynamical system called exosystem. Thus it is in contrast with some other traj ectory tracking problems in that it allows tracking of a family of reference signals which does not have to be known exactly. Moreover, it can be applied to many control problems in real world including energy systems, transportation and system biology. / It is well known that the general framework for tackling the output regulation problem consists of two steps. The first step is to convert the robust output regulation problem for the given plant into a robust stabilization problem for the so-called augmented system composed of the given plant and a dynamic compensator called internal model, and the second step aims to robustly stabilize the augmented system. The key issue lies in how to design a suitable internal model which can generate all the steady-state information and render the augmented system to be stabilizable. / Although the research on the nonlinear output regulation has made significant progress for over two decades, most of the results only deals with single-input, single-output nonlinear systems. There are few results on nonlinear output regulation of multivariable nonlinear systems. Two challenges hinder the further development of nonlinear output regulation of multivariable nonlinear systems. On the one hand, due to the complicated structure of the system, it is difficult to construct a suitable internal model which can generate the necessary steady-state information while ensuring the solvability of stabilization problem of the augmented system. On the other hand, due to the introduction of the internal model, the augmented system becomes a more complicated multi-input, multioutput nonlinear system containing both dynamic uncertainty and time-varying static uncertainty. Such augmented system takes certain special structure which has never been encountered before. The stabilization problem of such augmented system is very challenging. Moreover, t he research on the nonlinear output regulation of practical systems is very rare. Because output regulation approach offers certain robust property to unknown plant parameters and also allows unknown parameters in exosystem, better results may be achieved by using output regulation approach. Therefore we are motivated to consider global robust output regulation problem of multivariable systems with certain structures and solve two typical control problems including speed tracking control of permanent magnet synchronous motor and trajectory tracking control of spherical invert d pendulum. The main contributions of the thesis are summarized as follows. / 1. The global robust output regulation problem for a class of multivariable nonlinear systems is considered. By attaching the internal model to the given plant, the augmented system takes certain special structure and the stabilization problem of such a system has never been handled. We decompose the stabilization problem of the augmented system into the stabilization problem of several single-input systems. Then we solve the problem by solving several single-input control problems via a recursive approach utilizing the changing supply function technique. The theoretical result is applied to the speed tracking control and load torque disturbance rejection problem of a surface-mounted PM synchronous motor. Compared with existing results, our design allows all the motor parameters to be uncertain. / 2. A speed tracking and load torque disturbance rejection problem of surface-mounted PM synchronous motor subject to an unknown exosystem is considered. Because the exosystem is unknown, traditional robust control approach cannot be used to tackle this problem. Moreover, the unknown parameters in the exosystem make the stabilization problem of augmented system even more challenging. Therefore we need to incorporate robust control with adaptive control to t ackle the problem. We solve the problem by solving two single-input control problems via a recursive approach utilizing the changing supply function and dynamic coordinate transformation techniques. / 3. A speed tracking and load torque disturbance rejection problem of PM synchronous motor by internal model design is considered. When dq axes inductances are equal, the problem has been extensively studied. However, when dq axes inductances are not equal, such problem has received little attention for the complicated nonlinearity in speed equation. By using output regulation approach, a better robust property with respect to motor parameters can be achieved. As the augmented system does not take any known special form, we develop a specific tool to deal with the stabilization problem. In particular, a generalized changing supply function technique applicable to non-ISS (input-to-state stable) systems is developed. This technique, in conjunction with a particular nonlinear internal model leads to an effective solution to the problem. / 4. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method is based on a neural network approximation of the solution of the regulator equations. The problem of seeking approximate solution of the regulator equations has been converted into a parameter optimization problem which has lent itself to a convenient implementation in computer. Moreover, compared with a third order polynomial based control in existing results, the steady-state tracking errors of the method maintains a more uniform performance over different amplitudes of the reference input signals. / 5. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method has been based on the neural network enhanced design with output feedb ack control. Different from the neural network design in our previous work, we present a novel method to approximate the feedforward control instead of the solution of regulator equations. The problem of seeking approximate solutions of feedforward control is converted into a parameter optimization problem. We solve the parameter optimization problem by both Hooke-Jeeves method and Powell method. When all the states are available, our design also offers certain robustness to plant parameter variations. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ping, Zhaowu. / "October 2011." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 160-170). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract --- p.i / Acknowledgement --- p.VI / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Literature Overview --- p.1 / Chapter 1.2 --- Contribution of Thesis --- p.7 / Chapter 1.3 --- Organization of Thesis --- p.9 / Chapter 2 --- Background and Preliminaries --- p.10 / Chapter 2.1 --- Fundamentals of Nonlinear Systems --- p.10 / Chapter 2.1.1 --- Lyapunov Stability --- p.11 / Chapter 2.1.2 --- Input-to-state Stability --- p.13 / Chapter 2.2 --- Framework of Robust Output Regulation --- p.15 / Chapter 2.3 --- Stabilization of Nonlinear Systems --- p.19 / Chapter 2.3.1 --- Technical Lemmas for ISS Systems --- p.20 / Chapter 2.3.2 --- Technical Lemmas for Non-ISS Systems --- p.23 / Chapter 2.4 --- Feedforward Control of Nonlinear Output Regulation --- p.28 / Chapter 2.4.1 --- Polynomial Based Approximation Method --- p.31 / Chapter 2.4.2 --- Neural Network Based Approximation Method --- p.32 / Chapter 3 --- Global Robust Output Regulation for a Class of Multivariable Systems --- p.39 / Chapter 3.1 --- Introduction --- p.40 / Chapter 3.2 --- Preliminaries --- p.42 / Chapter 3.3 --- Main Results --- p.48 / Chapter 3.4 --- Application to Speed Control of Surface-Mounted PM Synchronous Motor --- p.56 / Chapter 3.5 --- Conclusion --- p.71 / Chapter 4 --- Speed Tracking Control of Surface-Mounted PM Synchronous Motor Subject to an Unknown Exosystem --- p.72 / Chapter 4.1 --- Problem Formulation and Preliminaries --- p.73 / Chapter 4.2 --- Solvability of The Motor Control Problem --- p.79 / Chapter 4.3 --- Evaluation of The Control Law --- p.87 / Chapter 4.4 --- Conclusion --- p.99 / Chapter 5 --- Speed Tracking Control of PM Synchronous Motor by Internal Model Design --- p.100 / Chapter 5.1 --- Problem Formulation and Preliminaries --- p.101 / Chapter 5.2 --- Solvability of the Motor Control Problem --- p.107 / Chapter 5.3 --- Evaluation of the Control Law --- p.110 / Chapter 5.4 --- Conclusion --- p.119 / Chapter 6 --- Approximate Output Regulation of Spherical Inverted Pendulum by Neural Network Control --- p.120 / Chapter 6.1 --- Introduction --- p.121 / Chapter 6.2 --- Neural Network Based Approximate Solution of the Regulator Equations --- p.124 / Chapter 6.3 --- Approximate Control Law --- p.131 / Chapter 6.4 --- Conclusion --- p.135 / Chapter 7 --- Neural Network Enhanced Output Regulation of Spherical Inverted Pendulum --- p.136 / Chapter 7.1 --- Neural Network Based Output Feedback Control Law --- p.137 / Chapter 7.1.1 --- Hooke-J eeves Method --- p.139 / Chapter 7.1.2 --- Powell Method --- p.143 / Chapter 7.2 --- Robustness Analysis --- p.149 / Chapter 7.2.1 --- Hooke-Jeeves Method --- p.149 / Chapter 7.2.2 --- Powell Method --- p.153 / Chapter 7.3 --- Conclusion --- p.156 / Chapter 8 --- Conclusions --- p.157 / Bibliography --- p.160 / Biography --- p.171

Until this present work, the acoustics of waveguides has been divided into two broadly distinct fields---linear acoustics in ducts of complex geometry such as those with curvature or varying width, and nonlinear acoustics restricted to simple geometry ducts without curvature or flare. This PhD unites these distinct branches to give a complete mathematical description of weakly nonlinear wave propagation in a general shaped duct in both two and three dimensions. Such ducts have important applications---the clearest example is that of brass instruments, where it has been demonstrated that nonlinear wave steepening gives rise to the characteristic 'brassy' sounds of, for example, the trombone. As the ducts of these instruments have a very complicated geometry involving curvature, torsion and varying width, the goal of the PhD is to address what effect, if any, such changes in duct geometry have on the acoustic properties of such instruments. Other potential applications include the study of acoustics in curved aircraft engine intakes and even the nonlinear sound propagation through the trunk of an elephant. The first results chapter is focused on the exposition of the method used for the remainder of the paper, with the introduction of a new ``nonlinear admittance term'' as well as the associated algebra for it. An elegant notation for the nonlinear algebra is also developed, greatly simplifying the equations. The method is applied to one and two dimensional ducts and some analytical results are derived relating the work to previously published results. Numerical results are also presented and compared to other sources. The concept of nonlinear reflectance is also introduced---illustrating the effect of wave amplitude on the amount of energy reflected in a duct. The next results chapter builds on this work extending it to three dimensions. Numerical results are presented for three characteristic ducts---a curved duct, a horn and a helical duct, being one of the first works to study acoustics in helical pipes for both linear and nonlinear sound propagation. The final results chapter, utilising all of the previous work, addresses the problem of an open ended duct of finite length with nonlinear effects included. Results are compared with the linear results from the Wiener-Hopf method and new results are presented illustrating the effect of geometry and nonlinearity on the resonances of finite length waveguides culminating in the study of the resonances of a trombone.