Establishing consensus is a key probleminmulti-agent systems (MASs). This thesis proposes a novel methodology based on convex optimization in the form of linear matrix inequalities (LMIs) for establishing consensus in linear and nonlinear MAS in the presence of model uncertainties, i.e., robust consensus. Firstly, this thesis investigates robust consensus for uncertain MAS with linear dynamics. Specifically, it is supposed that the system is described by a weighted adjacency matrix whose entries are generic polynomial functions of an uncertain vector constrained in a set described by generic polynomial inequalities. For continuous-time dynamics, necessary and sufficient conditions are proposed to ensure the robust first-order consensus and the robust second-order consensus, in both cases of positive and non-positive weighted adjacency matrices. For discrete-time dynamics, necessary and sufficient conditions are provided for robust consensus based on the existence of a Lyapunov function polynomially dependent on the uncertainty. In particular, an upper bound on the degree required for achieving necessity is provided. Furthermore, a necessary and sufficient condition is provided for robust consensus with single integrator and nonnegative weighted adjacency matrices based on the zeros of a polynomial. Lastly, it is shown how these conditions can be investigated through convex optimization by exploiting LMIs. Secondly, local and global consensus are considered in MAS with intrinsic nonlinear dynamics with respect to bounded solutions, like equilibrium points, periodic orbits, and chaotic orbits. For local consensus, a method is proposed based on the transformation of the original system into an uncertain polytopic system and on the use of homogeneous polynomial Lyapunov functions (HPLFs). For global consensus, another method is proposed based on the search for a suitable polynomial Lyapunov function (PLF). In addition, robust local consensus in MAS is considered with time-varying parametric uncertainties constrained in a polytope. Also, by using HPLFs, a new criteria is proposed where the original system is suitably approximated by an uncertain polytopic system. Tractable conditions are hence provided in terms of LMIs. Then, the polytopic consensus margin problem is proposed and investigated via generalized eigenvalue problems (GEVPs). Lastly, this thesis investigates robust consensus problem of polynomial nonlinear system affected by time-varying uncertainties on topology, i.e., structured uncertain parameters constrained in a bounded-rate polytope. Via partial contraction analysis, novel conditions, both for robust exponential consensus and for robust asymptotical consensus, are proposed by using parameter-dependent contraction matrices. In addition, for polynomial nonlinear system, this paper introduces a new class of contraction matrix, i.e., homogeneous parameter-dependent polynomial contraction matrix (HPD-PCM), by which tractable conditions of LMIs are provided via affine space parametrizations. Furthermore, the variant rate margin for robust asymptotical consensus is proposed and investigated via handling generalized eigenvalue problems (GEVPs). For each section, a set of representative numerical examples are presented to demonstrate the effectiveness of the proposed results. / published_or_final_version / Electrical and Electronic Engineering / Doctoral / Doctor of Philosophy

Thesis (Ph. D.)--University of Hong Kong, 2007. / Title proper from title frame. Also available in printed format.

published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy

近年来，随着机会约束规划被广泛应用以及凸分析和概率论的新进展，如何有效的处理机会约束成为一个炙手可热的研究方向。其中，一个成功的解决方法就是考虑其安全可解近似，也就是说将机会约束转化成一组方便处理的确定性约束，并且保持原机会约束在新的约束下成立。目前这样的方法主要应用于带有独立分布的数据扰动的机会约束规划，或者已知扰动的协方差矩阵的情况。同时，带有相关数据扰动的机会约束下的锥不等式广泛应用于供应链管理、金融、控制以及信号处理等学科，而现有的优化理论却极少涵盖。 / 在这篇论文中我们主要研究机会约束下的线性矩阵不等式，并假设扰动分布不必相互独立，其仅有的相关性信息只由一系列子扰动的独立关系结构提供。通过推导矩阵值随机变量的大偏差上界，我们得出这一类条件约束的安全可解近似。我们随后考虑了基于条件风险价值度量的机会约束规划问题, 以及带多项式扰动的机会约束优化问题。另外，通过构造相应的鲁棒对等式的不确定集合，我们把机会约束规划转换成鲁棒优化问题。由于这种近似可以表示为一组线性矩阵不等式，因而可以使用现成的优化软件方便地求解。最后，我们把该安全可解近似方法运用到一个控制理论问题，以及一个带风险价值约束的投资组合优化问题中。 / The wide applicability of chanceconstrained programming, together with advances in convex optimization and probability theory, has created a surge of interest in finding efficient methods for processing chance constraints in recent years. One of the successes is the development of so-called safe tractable approximations of chance-constrained programs, where a chance constraint is replaced by a deterministic and efficiently computable inner approximation. Currently, such an approach applies mainly to chance-constrained linear inequalities, in which the data perturbations are either independent or define a known covariance matrix. However, its applicability to the case of chanceconstrained conic inequalities with dependent perturbations--which arises in supply chain management, finance, control and signal processing applications--remains largely unexplored. / In this thesis, we consider the problem of processing chance-constrained affinely perturbed linear matrix inequalities, in which the perturbations are not necessarily independent, and the only information available about the dependence structure is a list of independence relations. Using large deviation bounds for matrix-valued random variables, we develop safe tractable approximations of those chance constraints. Extensions to the Matrix CVaR (Conditional Value-at-Risk) risk measure and general polynomials perturbations are also provided separately. Further more, we show that the chanceconstrained linear matrix inequalities optimization problem can be converted to a robust optimization problem by constructing the uncertainty set of the corresponding robust counterpart. A nice feature of our approximations is that they can be expressed as systems of linear matrix inequalities, thus allowing them to be solved easily and efficiently by off-the-shelf optimization solvers. We also provide a numerical illustration of our constructions through a problem in control theory and a portfolio VaR (Value-at-Risk) optimization problem. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Wang, Kuncheng. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 94-101). / 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 --- Motivations and Philosophy --- p.1 / Chapter 1.2 --- Background --- p.2 / Chapter 1.3 --- Literature Review --- p.4 / Chapter 1.4 --- Contribution --- p.7 / Chapter 2 --- Preliminaries --- p.10 / Chapter 2.1 --- Probabilistic Inequalities --- p.10 / Chapter 2.2 --- Exact Proper Fractional Covers --- p.12 / Chapter 2.2.1 --- Exact Proper Fractional Cover of Quadratic Perturbations --- p.15 / Chapter 3 --- Large Deviations of Sums of Dependent Random Matrices --- p.18 / Chapter 3.1 --- The Matrix Exponential Function and Its Properties --- p.18 / Chapter 3.2 --- Main Theorem --- p.19 / Chapter 4 --- From Large Deviations to ChanceConstrained LMIs --- p.26 / Chapter 4.1 --- General Results --- p.26 / Chapter 4.2 --- Application to ChanceConstrained Quadratically Perturbed Linear Matrix Inequalities --- p.30 / Chapter 4.3 --- Bounding the Matrix Moment Generating Functions --- p.31 / Chapter 4.4 --- Iterative Improvement of the Proposed Approximations --- p.42 / Chapter 5 --- Computational Studies --- p.49 / Chapter 5.1 --- Application to Control Problems --- p.49 / Chapter 5.2 --- Application to Value-at-Risk Portfolio Optimization --- p.57 / Chapter 6 --- ChanceConstrained LMIs with CVaR Risk Measure --- p.64 / Chapter 6.1 --- Matrix CVaR Risk Measure --- p.65 / Chapter 6.2 --- Some Useful Inequalities --- p.68 / Chapter 6.3 --- From Matrix CVaR to ChanceConstrained LMIs --- p.69 / Chapter 6.3.1 --- Bound π¹(A₀, · · · ,A[subscript m]) --- p.70 / Chapter 6.3.2 --- Bound π²(A₀, · · · ,A[subscript m]) --- p.71 / Chapter 6.3.3 --- Bound π³(A₀, · · · ,A[subscript m]) --- p.72 / Chapter 6.3.4 --- Convex Approximation of π[superscript i](A0, · · · ,Am) --- p.73 / Chapter 7 --- Extension to Polynomials Perturbations --- p.75 / Chapter 7.1 --- Decoupling Theory --- p.75 / Chapter 7.2 --- Safe Tractable Approximation by SecondOrder Cone Programming --- p.77 / Chapter 8 --- Construct Uncertainty Set for Chance Constraints --- p.81 / Chapter 8.1 --- Problem Statement --- p.82 / Chapter 8.2 --- Fractional Cover for Quartic Perturbations --- p.83 / Chapter 8.3 --- Probabilistic Guarantees --- p.85 / Chapter 8.3.1 --- Probabilistic Bound Based on Large Deviations --- p.85 / Chapter 8.4 --- The Value of Ω for Bounds --- p.88 / Chapter 8.5 --- Computational Study --- p.89 / Chapter 8.5.1 --- Independent Standard Normal Perturbations --- p.89 / Chapter 8.5.2 --- Independent Bounded Quadratic Perturbations --- p.91 / Chapter 9 --- Conclusion --- p.93 / Bibliography --- p.94

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Die vorliegende Arbeit beschäftigt sich mit den finiten matriziellen Momentenproblemen von Stieltjes-Typ und beschreibt unter Verwendung der Methode der Fundamentalen Matrixungleichungen die Lösungsmenge durch gebrochen lineare Transformationen.

Networked control systems (NCSs) are known as distributed control systems (DCSs) which are based on traditional feedback control systems but closed via a real-time communication channel. In an NCS, the control and feedback signals are exchanged among the system’s components in the form of information packages through the communication channel. The research of NCSs is important from the application perspective due to the significant advantages over the traditional point-to-point control. However, the insertion of the communication links would also bring challenges and constraints such as the network-induced delays, the missing packets, and the inter symbol interference (ISI) into the system design. In order to tackle these issues and move a step further toward industry applications, two important design problems are investigated in the control areas: Tracking Control (Chapter 2–Chapter 5) and Signal Estimation (Chapter 6–Chapter8). With the fact that more than 90% of control loops in industry are controlled by proportional-integral-derivative (PID) controllers, the first work in this thesis aims to propose the design algorithm on PID controllers for NCSs. Such a design will not require the change or update of the existing industrial hardware, and it will enjoy the advantages of the NCSs. The second motivation is that, due to the network-induced constraints, there is no any existing work on tuning the PID gains for a general NCS with a state-space model. In Chapter 2, the PID tracking control for multi-variable NCSs subject to time-varying delays and packet dropouts is exploited. The H_infty control is employed to attenuate the load disturbance and the measurement noise. In Chapter 3, the probabilistic delay model is used to design the delay-scheduling tracking controllers for NCSs. The tracking control strategy consists of two parts: (1) the feedforward control can enhance the transient response, and (2) the feedback control is the digital PID control. In order to compensate for the delays on both communication links, the predictive control scheme is adopted. To make full use of the delay information, it is better to use the Markov chain to model the network-induced delays and the missing packets. A common assumption on the Markov chain model in the literature is that the probability transition matrix is precisely known. However, the assumption may not hold any more when the delay is time-varying in a large set and the statistics information on the delays is inadequate. In Chapter 4, it is assumed that the transition matrices are with partially unknown elements. An observer-based robust energy-to-peak tracking controller is designed for the NCSs. In Chapter 5, the step tracking control problem for the nonlinear NCSs is in- vestigated. The nonlinear plant is represented by Takagi-Sugeno (T-S) fuzzy linear model. The control strategy is a modified PI control. With an augmentation technique, the tracking controller design problem is converted into an H_infty optimization problem. The controller parameters can be obtained by solving non-iterative linear matrix inequality conditions. The state estimation problem for networked systems is explored in Chapter 6. At the sensor node, the phenomenon of multiple intermittent measurements is considered for a harsh sensing environment. It is assumed that the network-induced delay is time- varying within a bounded interval. To deal with the delayed external input and the non-delayed external input, a weighted H_infty performance is defined. A Lyapunov- based method is employed to deal with the estimator design problem. When the delay is not large, the system with delayed state can be transformed into delay-free systems. By using the probabilistic delay model and the augmentation, the H_infty filter design algorithm is proposed for networked systems in Chapter 7. Considering the phenomenon of ISI, the signals transmitted over the communication link would distort, that is, the output of the communication link is not the same with the input to the communication link. If the phenomenon occurs in the NCSs, it is desired to reconstruct the signal. In Chapter 8, a robust equalizer design algorithm is proposed to reconstruct the input signal, being robust against the measurement noise and the parameter variations. Finally, the conclusions of the dissertation are summarized and future research topics are presented. / Graduate

This work details the development of a stability and control module for implementation into a Multidisciplinary Design Optimization (MDO) framework for the conceptual design of conventional and advanced aircraft. A novel approach, called the Variance Constrained Flying Qualities (VCFQ) approach, is developed to include closed-loop dynamic performance metrics in the design optimization process. The VCFQ approach overcomes the limitations of previous methods in the literature, which only functioned for fully decoupled systems with single inputs to the system. Translation of the modal parameter based flying qualities requirements into state variance upper bounds allows for multiple-input control laws which can guarantee upper bounds on closed-loop performance metrics of the aircraft states and actuators to be rapidly synthesized. A linear matrix inequality (LMI) problem formulation provides a general and scalable numerical technique for computing the feedback control laws using convex optimization tools. The VCFQ approach is exercised in a design optimization study of a relaxed static stability transonic transport aircraft, wherein the empennage assembly is optimized subject to both static constraints and closed-loop dynamic constraints. Under the relaxed static stability assumption, application of the VCFQ approach resulted in a 36% reduction in horizontal tail area and a 32% reduction in vertical tail area as compared to the baseline configuration, which netted a weight savings of approximately 5,200 lbs., a 12% reduction in cruise trimmed drag, and a static margin which was marginally stable or unstable throughout the flight envelope. State variance based dynamic performance constraints offer the ability to analyze large, highly coupled systems, and the linear matrix inequality problem formulation can be extended to include higher-order closed-loop design objectives within the MDO. Recommendations for further development and extensions of this approach are presented at the end. / This material is based on research sponsored by Air Force Research Laboratory under agreement number FA8650-09-2-3938. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of Air Force Research Laboratory or the U.S. Government. / Ph. D.

This report describes a method to control the density distribution of a large number of autonomous agents. The approach is based on the fact that there are a large number of agents in the system, and hence the time evolution of the probabilistic density distribution of agents can be described as a Markov chain. The main contribution of this paper is the synthesis of a Markov matrix which will guide the multi-agent system density to a desired steady-state density distribution, in a probabilistic sense, while satisfying some motion and safety constraints. Also, an adaptive density control method based on real time density feedback is introduced to synthesize a time-varying Markov ma- trix, which leads to better convergence to the desired density distribution. Finally, a decentralized density computation method is described. This method guarantees that all agents will have a best, and common, density estimate in a finite, with an explicit bound, number of communication updates. / text

Les contrôleurs à réinitialisation sont une classe de systèmes hybrides dont la valeur de tout ou partie des états peut être instantannément modifiée sous certaines conditions algébriques. Cette interaction entre dynamique temps-continu et temps-discret de ces contrôleurs permet souvent de dépasser les limites des contrôleurs temps-continu. Dans cette thèse, nous proposons des conditions constructives (sous forme d’Inégalités Matricielles Linéaires) pour analyser la stabilité et les performances de boucle de commande incluant un contrôleur à réinitialisation. En particulier, nous prenons en compte la présence de saturation en amplitude des actionneurs du système. Ces non-linéarités sont souvent source d'une dégradation des performances voir d’instabilité. Les résultats proposés permettent d’estimer le domaine de stabilité et un niveau de performance pour ces systèmes, en s’appuyant sur des fonctions de Lyapunov quadratiques ou quadratiques par morceaux. Au delà de l'aspect analyse, nous exposons deux approches pour améliorer la région de stabilité (nouvelle loi de réinitialisation et stratégie « anti-windup »). / Hybrid controllers are flexible tools for achieving system stabilization and/or performance improvement tasks. More particularly, hybrid controllers enrich the spectrum of achievable trade-offs. Indeed, the interaction of continuous- and discrete-time dynamics in a hybrid controller leads to rich dynamical behavior and phenomena not encountered in purely continuous-time system. Reset control systems are a class of hybrid controllers whose states are reset depending on an algebraic condition. In this thesis, we propose constructive conditions (Linear Matrix Inequalities) to analyze stability and performance level of a closed-loop system including a reset element. More particularly, we consider a magnitude saturation which could be the source of undesirable effects on these performances, including instability. Proposed results estimate the stability domain and a performance level of such a system, by using Lyapunov-like approaches. Constructive algorithms are obtained by exploiting properties of quadratic - or piecewise quadratic - Lyapunov functions. Beyond analysis results, we propose design methods to obtain a stability domain as large as possible. Design methods are based on both continuous-time approaches (anti-windup compensator) and hybrid-time approaches (design of adapted reset rules).