Feedback is used to control systems whose open-loop behaviour is uncertain. Over the last twenty years a mature theory of robust control has been developed for linear multivariable systems in continuous time. But most practical control systems have constraints such as saturation limits on the actuators, which render the closed-loop nonlinear. Most of the modern controllers are also implemented digitally using computers.The study of this research is divided in two directions: the stability analysis of discrete-time Lur’e systems and the synthesis of static discrete-time anti-windup schemes. With respect to stability analysis, the main contributions of this thesis are the derivations of new LMI-based stability criteria for the discrete-time Lur’e systems with monotonic, slope-restricted nonlinearities via the Lyapunov method. The criteria provide convex stability conditions via LMIs, which can be efficiently computed via convex optimization methods. They are also extended to the general case that includes the non-diagonal MIMO nonlinearities. The importance of extending them to the general case is that it can eventually be applied to the stability analysis of several optimization-based controllers such as an input-constrainedmodel predictive control (MPC), which is inherently discrete. With respect to synthesis, the contribution is the convex formulation of a static discrete-time anti-windup scheme via one of the Jury-Lee criteria (a discrete-time counterpart of Popov criterion), which was previously conjectured to be unachievable. The result is also in the form of LMI, and is extended to several existing static anti-windup schemes with open-loop stable plants.

In the present climate of deregulation and privatisation, theutilities are often separated into generation, transmission anddistribution companies so as to help promote economic efficiencyand encourage competition. Also, environmental concerns,right-of-way and cost problems have delayed the construction ofboth generation facilities and new transmission lines while thedemand for electric power has continued to grow, which must bemet by increased loading of available lines. A consequence isthat power system damping is often reduced which leads to a poordamping of electromechanical power oscillations and/or impairmentof transient stability. The aim of this thesis is to examine theability of Controllable Series Devices (CSDs), such as Unified Power Flow Controller (UPFC)   Controllable Series Capacitor (CSC)   Quadrature Boosting Transformer (QBT)   for improving transient stability and damping ofelectromechanical oscillations in a power system. For these devices, a general model is used in power systemanalysis. This model is referred to as injection model which isvalid for load flow and angle stability analysis. The model isalso helpful for understanding the impact of the CSDs on powersystem stability. A control strategy for damping of electromechanical poweroscillations is also derived based on Lyapunov theory. Lyapunovtheory deals with dynamical systems without input. For thisreason, it has traditionally been applied only to closed-loopcontrol systems, that is, systems for which the input has beeneliminated through the substitution of a predetermined feedbackcontrol. However, in this thesis, Lyapunov function candidatesare used in feedback design itself by making the Lyapunovderivative negative when choosing the control. This controlstrategy is called Control Lyapunov Function (CLF) for systemswith control input. / QC 20100609

<p>Noise-induced stabilization occurs when an unstable deterministic system is stabilized by the addition of white noise. Proving that this phenomenon occurs for a particular system is often manifested through the construction of a global Lyapunov function. However, the procedure for constructing a Lyapunov function is often quite ad hoc, involving much time and tedium. In this thesis, a systematic algorithm for the construction of a global Lyapunov function for planar systems is presented. The general methodology is to construct a sequence of local Lyapunov functions in different regions of the plane, where the regions are delineated by different behaviors of the deterministic dynamics. A priming region, where the deterministic drift is directed inward, is first identified where there is an obvious choice for a local Lyapunov function. This priming Lyapunov function is then propagated to the other regions through a series of Poisson equations. The local Lyapunov functions are lastly patched together to form one smooth global Lyapunov function.</p><p>The algorithm is applied to a model problem which displays finite time blow up in the deterministic setting in order to prove that the system exhibits noise-induced stabilization. Moreover, the Lyapunov function constructed is in fact what we define to be a super Lyapunov function. We prove that the existence of a super Lyapunov function, along with a minorization condition, implies that the corresponding system converges to a unique invariant probability measure at an exponential rate that is independent of the initial condition.</p> / Dissertation

In the first part of the thesis we present several interior point algorithms for solving certain positive definite programming problems. One of the algorithms is adapted for finding out whether there exists or not a positive definite matrix which is a real linear combination of some given symmetric matrices A1,A2, . . . ,Am. In the second part of the thesis we discuss stability of nonlinear dynamical systems. We search using algorithms described in the first part, for Lyapunov functions of a few forms. A suitable Lyapunov function implies the existence of a hyperellipsoidal attraction region for the dynamical system, thus guaranteeing stability.

Networked control systems (NCSs) are feedback control systems with the feedback control loops closed via network. The origin of the term NCSs is from industrial systems where the plant and controller are often connected through networks. The applications of NCSs cover a wide range of industries, for example, manufactory automation, domestic robots, aircraft, automobiles and tele-operations. The research activities in NCSs are focused on the following three areas: control of networks, control over networks and multi-agent systems. Control of networks is mainly concerned with the problem of how to efficiently utilize the network resource by controlling and routing the network data flows. Control over networks is mainly concerned with the design of feedback control strategies of control systems in which signals are transmitted through unreliable communication links. Multi-agent systems deal with two problems: how the topology of the network connections between each component influences global control goals and how to design local control law describing the behavior of each individual to achieve the global control goal of the whole systems. The objective in this thesis is to deal with control over networks and multi-agent systems. The most challenging problem in the control over networks field is that the unreliable communication channels can degrade system performance greatly. The main unreliable properties of networks are delays and packet loss. In order to deal with this problem, a Lyapunov-based method has been used to design the sampled-data stabilization control strategy for a networked single system by choosing proper delay and packet loss dependent Lyapunov functional candidates. Linear matrix inequality techniques have been used to find the sufficient and necessary conditions for the controller design. Furthermore, the consensus formation control problem of multiple robotic vehicle systems has been investigated. The consensus-based design scheme has been applied to the formation control of multiple wheeled mobile-robot group with a virtual leader. A novel delay-dependent Lyapunov functional candidate has been constructed to investigate the convergence of the system states. The proposed control strategy is experimentally implemented for multiple wheeled mobile robots under neighbor-to-neighbor information exchange with group communication delays involved. In conclusion, through the simulation results and experimental validations, the proposed new Lyapunov-based control methods can effectively deal with the networked control systems discussed in this thesis.

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By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model is formulated and it leads to a system of multi-time scale stochastic differential equations. The classical Picard-Lindel\"{o}f successive approximations scheme is expended to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this generates to a problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations. To illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are exhibited. Without loss in generality, the modeling and analysis of three time-scale fractional stochastic differential equations is followed by the development of the numerical algorithm for multi-time scale dynamic equations. The development of numerical algorithm is based on the idea if numerical integration in the context of the notion of multi-time scale integration. The multi-time scale approach is applied to explore the study of higher order stochastic differential equations (HOSDE) is presented. This study utilizes the variation of constant parameter technique to develop a method for finding closed form solution processes of classes of HOSDE. Then then probability distribution of the solution processes in the context of the second order equations is investigated.

The subject of this thesis is the design and practical application of a model-based controller with variable structure control (VSC). Robot manipulators are highly non-linear systems, however they form a specific class in the non-linear group. Exact mathematical descriptions of the robot dynamics can be achieved and further, robot manipulators have specific useful properties that can be used for the design of advanced controllers. The inclusion of the inverse dynamic description of the robot manipulator as a feedforward term of the controller (model-based controller) is used to transform two non-linear systems i.e. the controller and the robot, into one linear system. The limitation of this technique arises from the accuracy of the inverse dynamic model. The linearisation only takes place if the model is known exactly. To deal with the uncertainties that arise in the model, a control methodology based on variable structure control is proposed. The design of the controller is based on a Lyapunov approach and engineering considerations of the robot. A candidate Lyapunov function of a pseudo-energy form is selected to start the controller design. The general form of the controller is selected to satisfy the negative definiteness of the Lyapunov function. The initial uncertainties between the actual robot dynamics and the model used in the controller are dealt with using a classical VSC regulator. The deficiencies of this approach are evident however because of the chattering phenomenum. The model uncertainties are examined from an engineering point of view and adjustable bounds are then devised for the VSC regulator, and simulations confirm a reduction in the chattering. Implementation on the SPRINTA robot reveals further limitations in the proposed methodology and the bound adjustment is enhanced to take into account the position of the robot and the tracking errors. Two controllers based on the same principle are then obtained and their performances are compared to a PID controller, for three types of trajectory. Tests reveal the superiority of the devised control methodology over the classic PID controller. The devised controller demonstrates that the inclusion of the robot dynamics and properties in the controller design with adequate engineering considerations lead to improved robot responses.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-19Bitstream added on 2014-06-13T18:06:55Z : No. of bitstreams: 1 pavan_jf_me_sjrp.pdf: 525723 bytes, checksum: 14295e01658745f42b4e6dd2b22c1791 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / No presente trabalho consideramos o sistema de equações diferenciais ordinároas x1 = afλ 1 (x1)+ bfµ 2 (x2) ˙ x2 = cfη 1 (x1)+ dfζ 2 (x2) (I) onde a,b,c e d são coeficientes constantes, λ, ,η e ζ são números racionais positivos numeradores e denominadores ímpares, as funções fi :(−h,h) → R, h> 0, são contínuas e satisfazem as condições fi(0)=0,i =1, 2e xifi(xi) > 0,para xi =0,i =1, 2. Associado ao sistema(I) consideramos a seguinte função V = α Z x1 0 fξ 1 (τ )dτ + Z x2 0 fθ 2 (τ )dτ, (II) onde ξ e θ são número racionais numeradores e denominadores ímpares. Nosso objetivo principal é encontar é encontrar sob quais condições dos parâmetros a,b,c,d e α> 0 a função V definidaem(II) é uma função de Liapunov estita para a solução nula dos sitema (I), o que leva a concluir a estabilidade assintótica da solução nula. / In this work we consider the system of ordinary differential equations x1 = afλ 1 (x1)+ bfµ 2 (x2) ˙ x2 = cfη 1 (x1)+ dfζ 2 (x2) (I) where a,b,c and d are constantco efficients, λ, ,η and ζ a repositive rational numbers with odd numerators and denominators ,and the functions fi :(−h,h) → R, h> 0,are continuous and satisfy the conditions fi(0)=0,i =1, 2and xifi(xi) > 0,for xi =0,i = 1, 2. Associated to the system(I) we consider the following function V = α Z x1 0 fξ 1 (τ )dτ + Z x2 0 fθ 2 (τ )dτ, (II) where ξ and θ are positive rational numbers with odd numerators and denominators and α is a positive constant. Our main goal is find under what conditions the parameters a,b,c,d and α> 0 the function V defined in(II) is a strict Liapunov function for the zero solution of the system (I), which leads us to conclude the asymptotic stability of zero solution.

Made available in DSpace on 2014-06-11T19:22:31Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-05-27Bitstream added on 2014-06-13T20:47:51Z : No. of bitstreams: 1 esteves_tt_me_ilha.pdf: 642482 bytes, checksum: 6d87652f7c370af770e00c245690a225 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho é realizada a análise da estabilidade de sistemas fuzzy Takagi-Sugeno (TS) contínuos no tempo, através de Funções de Lyapunov Fuzzy (FLF), Funções de Lyapunov Metzler (FLM) e de Funções de Lyapunov Fuzzy-Metzler (FLFM) introduzida nesta disser- tação. Novas propostas são feitas a partir destas análises, sendo apresentadas condições su- ficientes para a estabilidade assintótica destes sistemas no sentido de Lyapunov. As soluções obtidas são baseadas em desigualdades lineares matriciais (LMIs, do inglês Linear Matrix Ine- qualities) e dependem da solução de um conjunto de desigualdades de Lyapunov-Metzler, que podem ser de difícil solução. Então, foram apresentadas condições de estabilidade baseadas em uma subclasse de matrizes de Metzler que, quando factíveis, podem ser resolvidas através de LMIs com a necessidade de uma busca unidimensional. Foram propostos métodos que genera- lizam os já existentes na literatura, baseados em FLF, para a estabilidade assintótica dos sistemas fuzzy TS / This work addresses the stability analysis of Takagi-Sugeno (TS) fuzzy systems via Fuzzy Lyapunov Functions (FLF), Metzler Lyapunov Functions (MLF) and Fuzzy-Metzler Lyapunov Functions (FMLF) that was proposed in this dissertation. New proposals are made from these analyses, and sufficient conditions for asymptotic stability of these systems in the sense of Lyapunov are presented. The results obtained are based on LMIs (Linear Matrix Inequalities) and depend on the solutions of a set of Lyapunov-Metzler inequalities, that are usually difficult to solve. Then, conditions for stability based on a subclass of Metzler matrices that, when feasible, can be described by a set of LMIs with an unidimensional search, are presented. The proposed methods generalize the similar methods available in the literature, based on FLF, for the asymptotic stability of TS fuzzy systems