Approximation, analysis and control of large-scale systems - Theory and Applications

This work presents some contributions to the fields of approximation, analysis and control of large-scale systems. Consequently the Thesis consists of three parts. The first part covers approximation topics and includes several contributions to the area of model reduction. Firstly, model reduction by moment matching for linear and nonlinear time-delay systems, including neutral differential time-delay systems with discrete-delays and distributed delays, is considered. Secondly, a theoretical framework and a collection of techniques to obtain reduced order models by moment matching from input/output data for linear (time-delay) systems and nonlinear (time-delay) systems is presented. The theory developed is then validated with the introduction and use of a low complexity algorithm for the fast estimation of the moments of the NETS-NYPS benchmark interconnected power system. Then, the model reduction problem is solved when the class of input signals generated by a linear exogenous system which does not have an implicit (differential) form is considered. The work regarding the topic of approximation is concluded with a chapter covering the problem of model reduction for linear singular systems. The second part of the Thesis, which concerns the area of analysis, consists of two very different contributions. The first proposes a new "discontinuous phasor transform" which allows to analyze in closed-form the steady-state behavior of discontinuous power electronic devices. The second presents in a unified framework a class of theorems inspired by the Krasovskii-LaSalle invariance principle for the study of "liminf" convergence properties of solutions of dynamical systems. Finally, in the last part of the Thesis the problem of finite-horizon optimal control with input constraints is studied and a methodology to compute approximate solutions of the resulting partial differential equation is proposed.Open Acces

Switched dynamical systems: Transition model, qualitative theory, and advanced control

Ph.DDOCTOR OF PHILOSOPH

Discrete Time Systems

Discrete-Time Systems comprehend an important and broad research field. The consolidation of digital-based computational means in the present, pushes a technological tool into the field with a tremendous impact in areas like Control, Signal Processing, Communications, System Modelling and related Applications. This book attempts to give a scope in the wide area of Discrete-Time Systems. Their contents are grouped conveniently in sections according to significant areas, namely Filtering, Fixed and Adaptive Control Systems, Stability Problems and Miscellaneous Applications. We think that the contribution of the book enlarges the field of the Discrete-Time Systems with signification in the present state-of-the-art. Despite the vertiginous advance in the field, we also believe that the topics described here allow us also to look through some main tendencies in the next years in the research area

Models of Delay Differential Equations

This book gathers a number of selected contributions aimed at providing a balanced picture of the main research lines in the realm of delay differential equations and their applications to mathematical modelling. The contributions have been carefully selected so that they cover interesting theoretical and practical analysis performed in the deterministic and the stochastic settings. The reader will find a complete overview of recent advances in ordinary and partial delay differential equations with applications in other multidisciplinary areas such as Finance, Epidemiology or Engineerin

Mathematical Modeling and Analysis of Epidemiological and Chemical Systems

This dissertation focuses on three interdisciplinary areas of applied mathematics, mathematical biology/epidemiology, economic epidemiology and mathematical physics, interconnected by the concepts and applications of dynamical systems.;In mathematical biology/epidemiology, a new deterministic SIS modeling framework for the dynamics of malaria transmission in which the malaria vector population is accounted for at each of its developmental stages is proposed. Rigorous qualitative and quantitative techniques are applied to acquire insights into the dynamics of the model and to identify and study two epidemiological threshold parameters reals* and R0 that characterize disease transmission and prevalence, and that can be used for disease control. It is shown that nontrivial disease-free and endemic equilibrium solutions, which can become unstable via a Hopf bifurcation exist. By incorporating vector demography; that is, by interpreting an aspect of the life cycle of the malaria vector, natural fluctuations known to exist in malaria prevalence are captured without recourse to external seasonal forcing and delays. Hence, an understanding of vector demography is necessary to explain the observed patterns in malaria prevalence. Additionally, the model exhibits a backward bifurcation. This implies that simply reducing R0 below unity may not be enough to eradicate the malaria disease. Since, only the female adult mosquitoes involved in disease transmission are identified and fully accounted for, the basic reproduction number (R0) for this model is smaller than that for previous SIS models for malaria. This, and the occurrence of both oscillatory dynamics and a backward bifurcation provide a novel and plausible framework for developing and implementing optimal malaria control strategies, especially those strategies that are associated with vector control.;In economic epidemiology, a deterministic and a stochastic model are used to investigate the effects of determinism, stochasticity, and safety nets on disease-driven poverty traps; that is, traps of low per capita income and high infectious disease prevalence. It is shown that economic development in deterministic models require significant external changes to the initial economic and health care conditions or a change in the parametric structure of the system. Therefore, poverty traps arising from deterministic models lead to more limited policy options. In contrast, there is always some probability that a population will escape or fall into a poverty trap in stochastic models. It is demonstrated that in stochastic models, a safety net can guarantee ultimate escape from the poverty trap, even when it is set within the basin of attraction of the poverty trap or when it is implemented only as an economic or health care intervention. It is also shown that the benefits of safety nets for populations that are close to the poverty trap equilibrium are highest for the stochastic model and lowest for the deterministic model. Based on the analysis of the stochastic model, the following optimal economic development and public health intervention questions are answered: (i) Is it preferable to provide health care, income/income generating resources, or both health care and income/income generating resources to enable populations to break cycles of poverty and disease; that is, escape from poverty traps? (ii) How long will it take a population that is caught in a poverty trap to attain economic development when the initial health and economic conditions are reinforced by safety nets?;In mathematical physics, an unusual form of multistability involving the coexistence of an infinite number of attractors that is exhibited by specially coupled chaotic systems is explored. It is shown that this behavior is associated with generalized synchronization and the emergence of a conserved quantity. The robustness of the phenomenon in relation to a mismatch of parameters of the coupled systems is studied, and it is shown that the special coupling scheme yields a new class of dynamical systems that manifests characteristics of dissipative and conservative systems

Modelling and Adaptive Control; Proceedings of an IIASA Conference, Sopron, Hungary, July 1986

One of the main purposes of the workshop on Modelling and Adaptive Control at Sopron, Hungary, was to give an overview of both traditional and recent approaches to the twin theories of modelling and control which ultimately must incorporate some degree of uncertainty. The broad spectrum of processes for which solutions of some of these problems were proposed was itself a testament to the vitality of research on these fundamental issues. In particular, these proceedings contain new methods for the modelling and control of discrete event systems, linear systems, nonlinear dynamics and stochastic processes

Dynamical Systems

Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...

From Underactuation to Quasi‐Full Actuation: A Unifying Control Framework for Rigid and Elastic Joint Robot

The quest for animal-like performance in robots has driven the integration of elastic elements in their drive trains, sparking a revolution in robot design. Elastic robots can store and release potential energy, providing distinct advantages over traditional robots, such as enhanced safety in human-robot interaction, resilience to mechanical shocks, improved energy efficiency in cyclic tasks, and dynamic motion capabilities. Exploiting their full potential, however, necessitates novel control methods. This thesis advances the field of nonlinear control for underactuated systems and utilizes the results to push the boundaries of motion and interaction performance of elastic robots. Through real-life experiments and applications, the proposed controllers demonstrate that compliant robots hold promise as groundbreaking robotic technology. To achieve these objectives, we first derive a simultaneous phase space and input transformation that enables a specific class of underactuated Lagrangian systems to be treated as if fully actuated. These systems can be represented as the interconnection of actuated and underactuated subsystems, with the kinetic energy of each subsystem depending only on its own velocity. Elastic robots are typical representatives. We refer to the transformed system as quasi-fully actuated due to weak constraints on the new inputs. Fundamental aspects of the transforming equations are 1) the same Lagrangian function characterizes both the original and transformed systems, 2) the transformed system establishes a passive mapping between inputs and outputs, and 3) the solutions of both systems are in a one-to-one correspondence, describing the same physical reality. This correspondence allows us to study and control the behavior of the quasi-fully actuated system instead of the underactuated one. Thus, this approach unifies the control design for rigid and elastic joint robots, enabling the direct application of control results inherited from the fully-actuated case while ensuring closed-loop system stability and passivity. Unlike existing methods, the quasi-full actuation concept does not rely on inner control loops or the neglect and cancellation of dynamics. Notably, as joint stiffness values approach infinity, the control equivalent of a rigid robot is recovered. Building upon the quasi-full actuation concept, we extend energy-based control schemes such as energy shaping and damping injection, Euler-Lagrange controllers, and impedance control. Moreover, we introduce Elastic Structure Preserving (ESP) control, a passivity-based control scheme designed for robots with elastic or viscoelastic joints, guided by the principle of ``do as little as possible''. The underlying hope is that reducing the system shaping, i.e., having a closed-loop dynamics match in some way the robot's intrinsic structure, will award high performance with little control effort. By minimizing the system shaping, we obtain low-gain designs, which are favorable concerning robustness and facilitate the emergence of natural motions. A comparison with state-of-the-art controllers highlights the minimalistic nature of ESP control. Additionally, we present a synthesis method, based on purely geometric arguments, for achieving time-optimal rest-to-rest motions of an elastic joint with bounded input. Finally, we showcase the remarkable performance and robustness of the proposed ESP controllers on DLR David, an anthropomorphic robot implemented with variable impedance actuators. Experimental evidence reveals that ESP designs enable safe and compliant interaction with the environment and rigid-robot-level accuracy in free motion. Additionally, we introduce a control framework that allows DLR David to perform commercially relevant tasks, such as pick and place, teleoperation, hammer drilling into a concrete block, and unloading a dishwasher. The successful execution of these tasks provides compelling evidence that compliant robots have a promising future in commercial applications