Control of Distributed Parameter Systems

Mathematical control theory of applied partial differential
equations is built on linear and nonlinear functional analysis and
many existence theorems in control theory result from applications
of theorems in functional analysis.  Thus,
many  advanced control theory books  were
written  in an abstract setting,  using functional analysis and semigroup theory. This makes control theory
inaccessible to  students who do not have a background in
the abstract mathematical theory. This motivates
me to write this text for my control theory class in a way to
present control theory by concrete examples and try to minimize the use of the
abstract mathematical theory. Analysis and calculations are elementary with using calculus.

This book contains the essential knowledge, concepts, methods, and theories
on the fundamental problem of disturbance rejection and asymptotic tracking
in control
theory. It covers feedback control, feedforward control, robust control for the linear system of ordinary differential equation, the linear one-dimensional heat equation, and the linear one-dimensional  wave equation.  Applications discussed in this book include the blood glucose regulation, the temperature regulation,  and the asymptotic wave tracking.
Because I choose to focus on the two basic classes of partial differential equations,  many other applied partial differential equations, such as elastic equations, thermoelastic equations, viscoelastic equations, Schr\"{o}dinger's
equations, and Navier-Stokes equations, are not included. However, this book could
provide an appropriate starting point to look at those more complicated equations.

This book is constituted of four chapters. Elementary functional analysis is given in
Chapter 1,
linear systems of ordinary differential equations are presented in
Chapter 2, the one-dimensional heat equation is presented in Chapter 3, and the one-dimensional wave equation is presented in Chapter 4.

This book can be used as a textbook for an advanced undergraduate/first-year
graduate course on feedback control of differential equations. The prerequisites
for such a course would be undergraduate linear algebra and undergraduate
ordinary differential equations. I have used it for my feedback control class
for a number of semesters.

The RPL overview, control messages format and network topology are deeply studied in this paper. RPL uses link quality Expected Transmission count (ETX) as single metric for default route selection. The ETX ensures the transmission quality, but may cause load imbalance between nodes. The life time of networks possibly be shortened by individual heavily used nodes. In wireless sensor networks, the power consumption which is a sensitive parameter for those battery-powered nodes should be considered in the routing process. A new routing metric named Energy Aware Metric (EAM) is proposed by combining the energy consumption situation with link quality ETX. The use of EAM makes it possible to consider both link quality and residual energy of nodes as routing metric. By using EAM, the relative balance of energy consumption between nodes is achieved. Simulation results show that the EAM has worked in improving the energy consumption and prolonging the life time of networks.

In this paper, microwave element is designed
using the concept of split ring resonator. In this branch
line coupler is loaded with ring resonator structure to
reduce overall size than conventional element and to
have excellent stop band characteristics. The proposed
coupler has good performance than conventional
element and also hasharmonic suppression in addition.

ABSRACT: The aim of this chapter is to introduce the concept of regional boundary exponential reduced observability in connection with the sensors characterizations on a sub-region Γ of the considered domain boundary ∂Ω. More precisely, we explore the original results devoted to this concept in linear dynamical systems which is generated by a strongly continuous semi-group in Hilbert space H 1 (Ω). Thus, the existence of sufficient conditions is presented and examined for regional boundary exponential reduced estimator in parabolic infinite dimensional systems. Finally, we apply these results to the exchange systems with various strategic sensors.

The aim of this paper is to study the regional observer concept through the consideration of sensors. For a class of distributed diffusion systems, we propose an approach derived from the Luenberger observer type as introduced by Gressang and Lamont [1]. Furthermore, we show that the structures of sensors allow the existence of regional observer and we give a sufficient condition for each regional observer. We also show that, there exists a dynamical system for diffusion systems is not observer in the usual sense, but it may be regional observer.

The main objective of this paper is to study the problem of regional boundary asymptotic gradient full order observer (Γ * ‫-ܱܨܩܣ‬observer) in link with the structures of sensors and actuators. Important results have been obtained related to diverse kinds of measurements and controls, of domains and boundary conditions. It has been shown that the structures of measurements and feedback controls allow the existence of Γ * ‫-ܱܨܩܣ‬observer in order to achieve regional boundary gradient closed loop control system (ܴ‫-ܥܮܥܩܤ‬system). It has also been found that there is a dynamical system which does not represent the observer in the usual sense, but it could be interpreted as a regional observer.

In this paper, we deal with the linear in®nite dimensional systems in a Hilbert space where the dynamics of the system is governed by strongly continuous semi-groups. We study the concept of asymptotic (exponential) regional detectability in connection with the structures of sensors for a class of parabolic distributed parameter systems. For different sensors structures, we give the characterization of the asymptotic (exponential) regional detectability in order that asymptotic (exponential) regional observability be achieved. Furthermore, we apply these results to the regional observer for distributed parameter diffusion systems. #

The purpose of this paper is to characterize regional boundary gradient strategic sensors notions for different cases of regional boundary gradient observation to be achieved. Then, the characterizations based on how to a cross from internal gradient region to the boundary gradient region. Thus, the obtained results are applied in two dimensional linear infinite distributed systems in Hilbert space where the dynamics are governed by strongly continuous semi-group. Moreover, we give the relation between the regional gradient observability on a subregion í µí¼” of the spatial domain Ω with the regional boundary gradient observability on a subregion Г of the boundary í µí¼•Ω of Ω. Finally, sufficient conditions of regional boundary gradient strategic sensors notions are explored, analyzed and discussed in connection with the regional boundary gradient of exact (weak) observability, positive definite observability operator and rank conditions. Keywords: í µí¼” í µí°º-strategic sensors, Г í µí°º-strategic sensors, exactly Г í µí°º-observability, weakly Г í µí°º-observability

The paper is devoted to stability and stabilization of a class of evolution equations arising from mathematical modeling of hybrid mechanical systems with flexible parts. A sufficient condition is obtained for partial strong asymptotic stability of nonlinear, infinite-dimensional dynamic systems in Banach spaces. This result is applied to deriving a control law that stabilizes a part of the variables describing a rotating rigid body endowed with a number of elastic beams. Results of numerical simulations are presented.

The main purpose of this paper is to study and characterize the existing of general
asymptotic regional gradient observer which observe the current gradient state of the
original system in connection with gradient strategic sensors. Thus, we give an
approach based to Luenberger observer theory of linear distributed parameter systems
which is enabled to determinate asymptotically regional gradient estimator of current
gradient system state. More precisely, under which condition the notion of asymptotic
regional gradient observability can be achieved. Furthermore, we show that the
measurement structures allows the existence of general asymptotic regional gradient
observer and we give a sufficient condition for such asymptotic regional gradient
observer in general case. We also show that, there exists a dynamical system for the
considered system is not general asymptotic gradient observer in the usual sense, but it
may be general asymptotic regional gradient observer. Then, for this purpose we
present various results related to different types of sensor
boundary conditions in two dimensional distributed diffusion systems.

The aim of this paper is to introduce the concept of regional exponential observability in connection with the strategic sensors. Then, we give characterization of such sensors in order that regional exponential observability can be achieved. The obtained results are applied to two-dimensional systems, and various cases of sensors are considered. We also show that there exists a dynamical system for diffusion system which is not exponentially observable in the usual sense but it may be regionally exponentially observable.

A linear control system in Hilbert space is considered. An estimate of solutions of the system is carried out for the class of optimal control corresponding to a subsystem withfinite number of degrees of freedom. It is proved that the family of controllers considered solves the problem of approximate controllability for the infinite dimensional system.