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.

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.

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

(i) A pure small gain approach is proposed to handle a disturbance attenuation problem for a class of feedforward systems subject to both dynamic uncertainty and disturbance. Two versions of small gain theorem with restrictions are employed to establish the global attractiveness and local stability of the closed-loop system at the origin, respectively. Unlike Lyapunov's linearization method and asymptotic small gain theorem combined approach, the proposed approach does not require the stabilizability assumption of the Jacobian linearization of the system at the origin. / (i) We first identify structural properties of the plant so that an internal model candidate exists. Then, by looking for a suitable internal model and performing appropriate transformations on the augmented system, we succeed in converting the global robust output regulation problem for a class of feedforward systems into a global robust stabilization problem for a class of feedforward systems subject to both time-varying static and dynamic uncertainties. As a result, the global robust stabilization result obtained in the first part of this thesis is used to solve the global robust output regulation problem for a class of feedforward systems. / (ii) A small gain based bottom-up recursive design is developed to solve a global robust stabilization problem for a class of feedforward systems subject to both time-varying static and dynamic uncertainties. Unlike most existing results, our design does not require the bottom dynamics at each recursion be locally exponentially stable. / (ii) We apply the result of the global robust output regulation problem to solve two trajectory tracking problems for a chain of integrators with uncertain parameters and the Vertical Take-Off and Landing (VTOL) aircraft, respectively. In contrast with the existing designs, for the chain of integrators, our design is low gain and does not need to know the reference trajectory exactly, and for the VTOL aircraft, our design is a complete low gain design and thus is more cost effective. / (iii) The small gain based bottom-up recursive design is further extended to deal with a global robust stabilization problem for a class of feedforward systems which are approximated at the origin by a nonlinear chain of integrators and perturbed by some type of input unmodeled dynamics. Even in the special case where the input unmodeled dynamics is not present, our result is new in the sense that our approach can handle some cases that cannot be handled by any existing approaches. / (iii) We propose a Lyapunov approach to a special case of the output regulation problem, the input disturbance suppression problem for a class of feedforward systems. When the exosystem is known, we solve the problem via dynamic output feedback control. When the exosystem is unknown, we solve the problem via adaptive dynamic state feedback control and we also give the conditions under which an estimated parameter vector can converge to the true parameter vector. / It is now well known from the general framework for tackling the output regulation problem that the robust output regulation problem can be approached in two steps. In the first step, the problem is converted into a robust stabilization problem of a so-called augmented system which consists of the original plant and a suitably defined dynamic system called an internal model candidate, and in the second step, the robust stabilization problem of the augmented system is further pursued. The success of the first step depends on whether or not an internal model candidate exists. Even though the first step succeeds, the success of the second step is by no means guaranteed due to at least two obstacles. First, the stabilizability of the augmented system is dictated not only by the given plant but also by the particular internal model candidate employed. Second, the stabilization problem of the augmented system is much more challenging than that of the original plant with the exogenous signal set to 0, because the structure of the augmented system may be much more complex than that of the original plant. Perhaps, it is because of these difficulties, so far almost all papers on semi-global or global robust output regulation problem are focused on the lower triangular systems, feedback linearizable systems and output feedback systems. The second part of this thesis aims to study the global robust output regulation problem of feedforward systems. The major results are summarized as follows. / The stabilization problem of feedforward systems has absorbed a lot of attention during the past fifteen years. More recently, the stabilization problem of feedforward systems subject to input unmodeled dynamics is studied. Nevertheless, the more realistic case where the system is subject to both time-varying static and dynamic uncertainties has not been adequately investigated. The first part of this thesis focuses on the global robust stabilization problem for various classes of feedforward systems containing both time-varying static and dynamic uncertainties. The major results are summarized as follows. / This thesis contains two parts. The first part studies the global robust stabilization problem of feedforward systems and the second part further addresses the global robust output regulation problem of the same class of nonlinear systems. / Chen, Tianshi. / Adviser: Jie Huang. / Source: Dissertation Abstracts International, Volume: 70-09, Section: B, page: . / Thesis submitted in: December 2008. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (leaves 136-143). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis (Ph. D.)--University of Texas at Austin, 2003. / Vita. Includes bibliographical references. Available also from UMI Company.

Not available / text

No description available.

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: bicgstab, cgs, gmres, NITSOL, Newtons Method, nonlinear systems. Includes bibliographical references (p. 53).

M.Sc. / Roels [1] showed that on a two dimensional symplectic manifold, an arbitrary vector field can be locally decomposed into the sum of a gradient vector field and a Hamilton vector field. The Roels decomposition was extended to be applicable to compact even dimensional manifolds by Mendes and Duarte [2]. Some of the limitations of local decomposition are overcome by incorporating modern work on Hodge decomposition. This leads to a theorem which, in some cases, allows an arbitrary vector field on an even m-dimensional non-compact manifold to be decomposed into one gradient vector field and up to m-1 Hamiltonian vector fields. The method of decomposition is condensed into an algorithm which can be implemented using computer algebra. This decomposition is then applied to chaotic vector fields on non-compact manifolds [3]. This extended Roels decomposition is also compared to Helmholz decomposition in R 3 . The thesis shows how Legendre polynomials can be used to simplify the Helmholz decomposition in non-trivial cases. Finally, integral preserving iterators for both autonomous and non-autonomous first integrals are discussed [4]. The Hamilton vector fields which result from Roels' decomposition have their Hamiltonians as first integrals.