Advances in Design and Control – serie

Linear matrix inequalities (LMIs) have recently emerged as useful tools for solving a number of control problems. This book provides an up-to-date account of the LMI method and covers topics such as recent LMI algorithms, analysis and synthesis issues, nonconvex problems, and applications. It also emphasizes applications of the method to areas other than control. The basic idea of the LMI method in control is to approximate a given control problem via an optimization problem with linear objective and so-called LMI constraints. The LMI method leads to an efficient numerical solution and is particularly suited to problems with uncertain data and multiple (possibly conflicting) specifications.Since the early 1990s, with the development of interior-point methods for solving LMI problems, the LMI approach has gained increased interest. One advantage of this technique is its ability to treat large classes of control problems via efficient numerical tools. This approach is widely applicable, not only in control but also in other areas where uncertainty arises. LMI techniques provide a common language for many engineering problems. Notions now popular in control, such as uncertainty and robustness, are being used in other areas through the use of LMIs. This technique is particularly attractive for industrial applications. It is well suited for the development of CAD tools that help engineers solve analysis and synthesis problems.

H-Infinity control originated from an effort to codify classical control methods, where one shapes frequency response functions to meet certain objectives. H-Infinity control underwent tremendous development in the 1980s and made considerable strides toward systematizing classical control. This book addresses the next major issue of how this extends to nonlinear systems.At the core of nonlinear control theory lie two partial differential equations (PDEs). One is a first-order evolution equation called the information state equation, which constitutes the dynamics of the controller. One can view this equation as a nonlinear dynamical system. Much of this volume is concerned with basic properties of this system, such as the nature of trajectories, stability, and, most important, how it leads to a general solution of the nonlinear H-Infinity control problem.The second PDE actually builds on a classical type of partial differential inequality (PDI) called a Bellman–Isaacs inequality. While the information state PDE determines the dynamics of the controller, the PDI determines the output of the controller. The authors explore the system theoretic significance of the PDI and present its gross structure. These equations are only a few years old and their study is an expanding area of research.This book also emphasizes the theory effecting computer solvability of the information state equation, which at the outset looks numerically intractable, but which surprisingly is in many cases tractable. For example, the theory shows that careful initialization has a major influence on computer solvability.The authors keep the book self-contained by using the appendices to help explain certain prerequisite material. The reader should have a basic knowledge of control theory, real analysis and differential equations, nonlinear operator theory, and nonlinear PDEs.

Flow control and optimization has been an important part of experimental flow science throughout the last century. As research in computational fluid dynamics (CFD) matured, CFD codes were routinely used for the simulation of fluid flows. Subsequently, mathematicians and engineers began examining the use of CFD algorithms and codes for optimization and control problems for fluid flows. The marriage of mature CFD methodologies with state-of-the-art optimization methods has become the center of activity in computational flow control and optimization.Perspectives in Flow Control and Optimization presents flow control and optimization as a subdiscipline of computational mathematics and computational engineering. It introduces the development and analysis of several approaches for solving flow control and optimization problems through the use of modern CFD and optimization methods. The author discusses many of the issues that arise in the practical implementation of algorithms for flow control and optimization, such as choices to be made and difficulties to overcome. He provides the reader with a clear idea of what types of flow control and optimization problems can be solved, how to develop effective algorithms for solving such problems, and potential problems to be aware of when implementing the algorithms.

The efficiency and reliability of manufactured products depend on, among other things, geometrical aspects; it is therefore not surprising that optimal shape design problems have attracted the interest of applied mathematicians and engineers. This self-contained, elementary introduction to the mathematical and computational aspects of sizing and shape optimization enables readers to gain a firm understanding of the theoretical and practical aspects so they may confidently enter this field.In contrast to existing texts on structural optimization, Introduction to Shape Optimization: Theory, Approximation, and Computation treats sizing and shape optimization in a comprehensive way, covering everything from mathematical theory (existence analysis, discretizations, and convergence analysis for discretized problems) through computational aspects (sensitivity analysis, numerical minimization methods) to industrial applications. Some of the applications included are contact stress minimization for elasto-plastic bodies, multidisciplinary optimization of an airfoil, and shape optimization of a dividing tube. By presenting sizing and shape optimization in an abstract way, the authors are able to use a unified approach in the mathematical analysis for a large class of optimization problems in various fields of physics.

Provides a comprehensive and in-depth treatment of one of the most important control problems: the nonlinear output regulation problem. Contains up-to-date research results and algorithms and tools for approaching and solving the output regulation problem and other related problems, such as robust stabilization of nonlinear systems. The author also offers personal insights about solving the output regulation problem. Output regulation is a general mathematical formulation of many control problems encountered in daily life including cruise control of automobiles, landing and takeoff of aircraft, manipulation of robot arms, orbiting of satellites, and speed regulation of motors.Nonlinear Output Regulation provides a self-contained treatment starting with an introduction to the linear output regulation problem and a review of the fundamental nonlinear control theory. The author’s presentation strikes a balance between the theoretical foundation of the problem and the practical applications of the theory. The book is accompanied by many examples, including practical case studies with numerical simulations based on MATLAB /SIMULINK.In addition to the time-tested results on the output regulation problem obtained over the past two decades by the author and other leading researchers in this field, the book highlights a newly developed general framework that converts the robust output regulation problem for a given nonlinear system into a robust stabilization problem for an augmented system. This general framework is based on the construction of nonlinear internal models and offers greater flexibility to incorporate recent new stabilization techniques, thus setting the stage for systematically tackling robust output regulation with global stability. This general framework has been successfully applied to solve the output regulation problem with global stability for several classes of important nonlinear problems.

Based on the results of over 10 years of research and development by the authors, this book presents a broad cross section of dynamic programming (DP) techniques applied to the optimization of dynamical systems. The main goal of the research effort was to develop a robust path planning/trajectory optimization tool that did not require an initial guess. The goal was partially met with a combination of DP and homotopy algorithms. DP algorithms are presented here with a theoretical development, and their successful application to variety of practical engineering problems is emphasized. Applied Dynamic Programming for Optimization of Dynamical Systems presents applications of DP algorithms that are easily adapted to the reader’s own interests and problems. The book is organized in such a way that it is possible for readers to use DP algorithms before thoroughly comprehending the full theoretical development. A general architecture is introduced for DP algorithms emphasizing the solution to nonlinear problems. DP algorithm development is introduced gradually with illustrative examples that surround linear systems applications. Many examples and explicit design steps applied to case studies illustrate the ideas and principles behind DP algorithms.DP algorithms potentially address a wide class of applications composed of many different physical systems described by dynamical equations of motion that require optimized trajectories for effective maneuverability. The DP algorithms determine control inputs and corresponding state histories of dynamic systems for a specified time while minimizing a performance index. Constraints may be applied to the final states of the dynamic system or to the states and control inputs during the transient portion of the manoeuvre.

Organizes the analysis and design of iterative numerical methods from a control perspective. The authors discuss a variety of applications, including iterative methods for linear and nonlinear systems of equations, neural networks for linear and quadratic programming problems, support vector machines, integration and shooting methods for ordinary differential equations, matrix preconditioning, matrix stability, and polynomial zero finding.This book opens up a new field of interdisciplinary research that should lead to insights in the areas of both control and numerical analysis and shows that a wide range of applications can be approached from - and benefit from - a control perspective.

Designed to meet the needs of a wide audience without sacrificing mathematical depth and rigor, Adaptive Control Tutorial presents the design, analysis, and application of a wide variety of algorithms that can be used to manage dynamical systems with unknown parameters. Its tutorial-style presentation of the fundamental techniques and algorithms in adaptive control make it suitable as a textbook.Adaptive Control Tutorial is designed to serve the needs of three distinct groups of readers: engineers and students interested in learning how to design, simulate, and implement parameter estimators and adaptive control schemes without having to fully understand the analytical and technical proofs; graduate students who, in addition to attaining the aforementioned objectives, also want to understand the analysis of simple schemes and get an idea of the steps involved in more complex proofs; and advanced students and researchers who want to study and understand the details of long and technical proofs with an eye toward pursuing research in adaptive control or related topics. The authors achieve these multiple objectives by enriching the book with examples demonstrating the design procedures and basic analysis steps and by detailing their proofs in both an appendix and electronically available supplementary material; online examples are also available. A solution manual for instructors can be obtained by contacting SIAM or the authors.

This self-contained, practical, entry-level text integrates the basic principles of applied mathematics, applied probability, and computational science for a clear presentation of stochastic processes and control for jump-diffusions in continuous time. The author covers the important problem of controlling these systems and, through the use of a jump calculus construction, discusses the strong role of discontinuous and nonsmooth properties versus random properties in stochastic systems. The book emphasizes modeling and problem solving and presents sample applications in financial engineering and biomedical modeling. Computational and analytic exercises and examples are included throughout. While classical applied mathematics is used in most of the chapters to set up systematic derivations and essential proofs, the final chapter bridges the gap between the applied and the abstract worlds to give readers an understanding of the more abstract literature on jump-diffusions.An additional 160 pages of online appendices are available on a Web page that supplements the book.

Discusses analysis and design techniques for linear feedback control systems using MATLAB® software. By reducing the mathematics, increasing MATLAB working examples, and inserting short scripts and plots within the text, the authors have created a resource suitable for almost any type of user. For beginners, the book provides an efficient entrance into the field; for readers who have already had a first course in control, the text helps bridge the gap between control theory and the use of MATLAB for control systems; for practicing engineers, it serves as a handy reference. The book begins with a summary of the properties of linear systems and addresses modeling and model reduction issues. In the subsequent chapters on analysis, the authors introduce time domain, complex plane, and frequency domain techniques. Their coverage of design includes discussions on model-based controller designs, PID controllers, and robust control designs. A unique aspect of the book is its inclusion of a chapter on fractional-order controllers, which are useful in control engineering practice.Especially useful for self-study, Linear Feedback Control uses the MATLAB companion package CtrlLAB, which allows readers to learn quickly while simply clicking through options with a mouse. Each chapter ends with a set of problems to help readers strengthen their understanding of the material. All example scripts within the book, as well as the CtrlLAB package, are freely downloadable from MATLAB Central.

Lagrange multiplier theory provides a tool for the analysis of a general class of nonlinear variational problems and is the basis for developing efficient and powerful iterative methods for solving these problems. This comprehensive monograph analyzes Lagrange multiplier theory and shows its impact on the development of numerical algorithms for problems posed in a function space setting. The book is motivated by the idea that a full treatment of a variational problem in function spaces would not be complete without a discussion of in?nite-dimensional analysis, proper discretization, and the relationship between the two.The authors develop and analyze efficient algorithms for constrained optimization and convex optimization problems based on the augumented Lagrangian concept and cover such topics as sensitivity analysis, convex optimization, second order methods, and shape sensitivity calculus. General theory is applied to challenging problems in optimal control of partial differential equations, image analysis, mechanical contact and friction problems, and American options for the Black–Scholes model.

This concise and highly-usable textbook presents an introduction to backstepping, an elegant new approach to boundary control of partial differential equations (PDEs). Backstepping provides mathematical tools for constructing coordinate transformations and boundary feedback laws for converting complex and unstable PDE systems into elementary, stable, and physically intuitive ""target PDE systems"" that are familiar to engineers and physicists. Readers will be introduced to constructive control synthesis and Lyapunov stability analysis for distributed parameter systems.The text’s broad coverage includes parabolic PDEs; hyperbolic PDEs of first and second order; fluid, thermal, and structural systems; delay systems; PDEs with third and fourth derivatives in space (including variants of linearized Ginzburg–Landau, Schrodinger, Kuramoto–Sivashinsky, KdV, beam, and Navier–Stokes equations); real-valued as well as complex-valued PDEs; stabilization as well as motion planning and trajectory tracking for PDEs; and elements of adaptive control for PDEs and control of nonlinear PDEs. Boundary Control of PDEs is appropriate for courses in control theory and includes homework exercises and a solutions manual that is available from the authors upon request. The results are explicit and the style is accessible; students are not expected to have a background beyond that of a typical engineering or physics graduate. Even an instructor who is not an expert on control of PDEs will find it possible to teach effectively from this book. At the same time, an expert researcher in PDEs looking for novel technical challenges will find many topics of interest, particularly in control synthesis for unstable PDEs, nonlinear PDEs, and PDEs with unknown coefficients.

Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational, accuracy, and storage capabilities, model reduction (system approximation) is often necessary.Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.

Unmanned aerial vehicles (UAVs) are increasingly used in military missions because they have the advantages of not placing human life at risk and of lowering operation costs via decreased vehicle weight. These benefits can be fully realized only if UAVs work cooperatively in groups with an efficient exchange of information. This book provides an authoritative reference on cooperative decision and control of UAVs and the means available to solve problems involving them.The contributors present the information in a manner that abstracts the challenges from the concrete problems, making it possible to leverage the solution methods over a broader range of applications. The first chapter offers representative scenarios to describe the problem and its challenges. The second chapter closely examines these challenges by providing an overview of the algorithms that could be used for cooperative control of UAV teams. Later chapters offer methods for performing multiple tasks on multiple targets and assigning multiple tasks to multiple UAVs in one step. Other topics addressed are the application of mixed integer linear programming and genetic algorithms in situations with strict time constraints, the cooperation of UAVs when there are communication delays, and effectiveness measures derived for operations in uncertain environments. The book has two appendices. The first describes the operation of the MultiUAV2 simulation software used to test the cooperation control algorithms, while the second details the UAV path planning problem and Dubins’ optimal trajectories. A supplementary website offers a MultiUAV2 software manual and relevant code.

The performance of a process – for example, how an aircraft consumes fuel – can be enhanced when the most effective controls and operating points for the process are determined. This holds true for many physical, economic, biomedical, manufacturing, and engineering processes whose behavior can often be influenced by altering certain parameters or controls to optimize some desired property or output.Primer on Optimal Control Theory:Provides a rigorous introduction to analyzing these processes and finding the best modes of control and operation for them.Makes optimal control theory accessible to a large class of engineers and scientists who are not mathematicians but have a basic mathematical background and need to understand the sophisticated material associated with optimal control theory.Presents the important concepts of weak and strong control variations leading to local necessary conditions, as well as global sufficiency of Hamilton–Jacobi–Bellman theory.Gives the second variation for local optimality where the associated Riccati equation is derived from the transition matrix of the Hamiltonian system, ideas that lead naturally to the development of H2 and H? synthesis algorithms.

Presents a comprehensive overview of the recently developed L1 adaptive control theory, including detailed proofs of the main results. The key feature of the L1 adaptive control theory is the decoupling of adaptation from robustness. The architectures of L1 adaptive control theory have guaranteed transient performance and robustness in the presence of fast adaptation, without enforcing persistent excitation, applying gain-scheduling, or resorting to high-gain feedback. It covers detailed proofs of the main results and also presents the flight test results that have used this theory and contains results not yet published in technical journals and conference proceedings.The material is organized into six chapters and concludes with an appendix that summarizes the mathematical results used to support the proofs. Software is available on a supplementary Web page.

This considerably enriched new edition provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is the shape or the structure of a geometric object. Shapes and Geometries presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field. A series of generic examples has been added to the introduction and special emphasis has been put on the construction of important metrics.Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new version of the book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization. This area of research is very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, shape and geometric identification, image processing, and design of endoprotheses in interventional cardiology.

Uncertainty and risk are integral to engineering because real systems have inherent ambiguities that arise naturally or due to our inability to model complex physics. The authors discuss probability theory, stochastic processes, estimation, and stochastic control strategies and show how probability can be used to model uncertainty in control and estimation problems. The material is practical and rich in research opportunities.The authors provide a comprehensive treatment of stochastic systems from the foundations of probability to stochastic optimal control. The book covers discrete- and continuous-time stochastic dynamic systems leading to the derivation of the Kalman filter, its properties, and its relation to the frequency domain Wiener filter as well as the dynamic programming derivation of the linear quadratic Gaussian (LQG) and the linear exponential Gaussian (LEG) controllers and their relation to H2 and H-inf controllers and system robustness.Stochastic Processes, Estimation, and Control is divided into three related sections. First, the authors present the concepts of probability theory, random variables, and stochastic processes, which lead to the topics of expectation, conditional expectation, and discrete-time estimation and the Kalman filter. After establishing this foundation, stochastic calculus and continuous-time estimation are introduced. Finally, dynamic programming for both discrete-time and continuous-time systems leads to the solution of optimal stochastic control problems, resulting in controllers with significant practical application.

Differential-algebraic equations are the most natural way to mathematically model many complex systems in science and engineering. Once the model is derived, it is important to optimize the design parameters and control it in the most robust and efficient way to maximize performance. This book presents the latest theory and numerical methods for the optimal control of differential-algebraic equations. Readers will find the following features presented in a readable fashion so the results are accessible to the widest audience: the most recent theory, written by leading experts from a number of academic and nonacademic areas and departments, several state-of-the-art numerical methods, and real-world applications.

This book is devoted to the theory and applications of second-order necessary and sufficient optimality conditions in the calculus of variations and optimal control. The authors develop theory for a control problem with ordinary differential equations subject to boundary conditions of equality and inequality type and for mixed state-control constraints of equality type. The book is distinctive in that necessary and sufficient conditions are given in the form of no-gap conditions; the theory covers broken extremals where the control has finitely many points of discontinuity; and a number of numerical examples in various application areas are fully solved.

The authors have developed a methodology for control of nonlinear systems in the presence of long delays, with large and rapid variation in the actuation or sensing path, or in the presence of long delays affecting the internal state of a system. In addition to control synthesis, they introduce tools to quantify the performance and the robustness properties of the designs provided in the book.The book is based on the concept of predictor feedback and infinite-dimensional backstepping transformation for linear systems and the authors guide the reader from the basic ideas of the concept — with constant delays only on the input — all the way through to nonlinear systems with state-dependent delays on the input as well as on system states.Several examples are presented that help the reader digest some of the intricacies in the methodology.Readers will find the book useful because the authors provide elegant and systematic treatments of long-standing problems in delay systems, such as systems with state-dependent delays that arise in many applications. In addition, the authors give all control designs by explicit formulae, making the book especially useful for engineers who have faced delay-related challenges and are concerned with actual implementations and they accompany all control designs with Lyapunov-based analysis for establishing stability and performance guarantees.

This book introduces key concepts for systematically controlling engineering systems that possess interacting phenomena occurring at widely different speeds. The aim is to present the reader with control techniques that extend the benefits of model reduction of singular perturbation theory to a larger class of nonlinear dynamical systems. New results and relevant background are presented through insightful examples that cover a wide range of applications from different branches of engineering. This book is unique because it:Presents a new perspective on existing control methods and thus broadens their application to a larger class of nonlinear dynamical systems.Discusses general rather than problem-specific developments to certain applications or disciplines in order to provide control engineers with useful analytical tools. Addresses new control problems using singular perturbation methods, including closed-form results for control of nonminimum.

Time delays are important components of many systems in, for instance, engineering, physics, economics, and the life sciences, because the transfer of material, energy, and information is usually not instantaneous. Time delays may appear as computation and communication lags, they model transport phenomena and heredity, and they arise as feedback delays in control loops. This monograph addresses the problem of stability analysis, stabilization, and robust fixed-order control of dynamical systems subject to delays, including both retarded- and neutral-type systems. Within the eigenvalue-based framework, an overall solution is given to the stability analysis, stabilization, and robust control design problem, using both analytical methods and numerical algorithms and applicable to a broad class of linear time-delay systems.In this revised edition, the authors:Make the leap from stabilization to the design of robust and optimal controllers and from retarded-type to neutral-type delay systems, thus enlarging the scope of the book within control.Include new, state-of-the-art material on numerical methods and algorithms to broaden the book’s focus and to reach additional research communities, in particular numerical linear algebra and numerical optimization.Increase the number and range of applications to better illustrate the effectiveness and generality of their approach.In this revised edition, the authors make the leap from stabilization to the design of robust and optimal controllers and from retarded-type to neutral-type delay systems, thus enlarging the scope of the book within control; include new, state-of-the-art material on numerical methods and algorithms to broaden the book’s focus and to reach additional research communities, in particular numerical linear algebra and numerical optimization; and increase the number and range of applications to better illustrate the effectiveness and generality of their approach.

Many things around us have properties that depend on their shape - for example, the drag characteristics of a rigid body in a flow. This self-contained overview of differential geometry explains how to differentiate a function (in the calculus sense) with respect to a “shape variable”. This approach, which is useful for understanding mathematical models containing geometric partial differential equations (PDEs), allows readers to obtain formulas for geometric quantities (such as curvature) that are clearer than those usually offered in differential geometry texts.Readers will learn how to compute sensitivities with respect to geometry by developing basic calculus tools on surfaces and combining them with the calculus of variations. Several applications that utilize shape derivatives and many illustrations that help build intuition are included.

This is the first comprehensive book on the AIMD algorithm, the most widely used method for allocating a limited resource among competing agents without centralized control. The authors offer a new approach that is based on positive switched linear systems. It is used to develop most of the main results found in the book, and fundamental results on stochastic switched nonnegative and consensus systems are derived to obtain these results.The original and best known application of the algorithm is in the context of congestion control and resource allocation on the Internet, and readers will find details of several variants of the algorithm in order of increasing complexity, including deterministic, random, linear, and nonlinear versions. In each case, stability and convergence results are derived based on unifying principles. Basic and fundamental properties of the algorithm are described, examples are used to illustrate the richness of the resulting dynamical systems, and applications are provided to show how the algorithm can be used in the context of smart cities, intelligent transportation systems, and the smart grid.

Engineering systems are highly distributed collective systems - decisions, information, and objectives are distributed throughout - that have humans in the loop, and thus decisions may be influenced by socioeconomic factors. Engineering systems emphasize the potential of control and games beyond traditional applications. Game theory can be used to design incentives to obtain socially desirable behaviors on the part of the players, for example, a change in the consumption patterns on the part of the “prosumers” (producers-consumers) or better redistribution of traffic.This unique book addresses the foundations of game theory, with an emphasis on the physical intuition behind the concepts, an analysis of design techniques, and a discussion of new trends in the study of cooperation and competition in large complex distributed systems.

For over a quarter of a century, high-gain observers have been used extensively in the design of output feedback control of nonlinear systems. This book presents a clear, unified treatment of the theory of high-gain observers and their use in feedback control. Also provided is a discussion of the separation principle for nonlinear systems; this differs from other separation results in the literature in that recovery of stability as well as performance of state feedback controllers is given.The author provides a detailed discussion of applications of high-gain observers to adaptive control and regulation problems and recent results on the extended high-gain observers. In addition, the author addresses two challenges that face the implementation of high-gain observers: high dimension and measurement noise. Low-power observers are presented for high-dimensional systems. The effect of measurement noise is characterized and techniques to reduce that effect are presented. The book ends with discussion of digital implementation of the observers.Readers will find:comprehensive coverage of the main results on high-gain observersrigorous, self-contained proofs of all results; andnumerous examples that illustrate and provide motivation for the results.

This book presents new computational tools for the H∞ control of distributed parameter systems in which transfer functions are considered as input-output descriptions for the plants to be controlled. The emphasis is on the computation of the controller parameters and reliable implementation. The authors present recent studies showing that the simplified skew Toeplitz method is applicable to a wide class of systems, supply detailed examples from systems with time delays and various engineering applications, and discuss reliable implementation of the controller, complemented by a software based on MATLAB®.Frequency Domain Techniques for H∞ Control of Distributed Parameter Systems is intended for advanced undergraduate and early graduate students interested in robust control of distributed parameter systems—time delay systems—as well as researchers and engineers working in related fields. It can be used in the following courses: Introduction to Robust Control with Applications to Distributed Parameter Systems and Introduction to Robust Control with Applications to Time Delay Systems.

A resurgence of interest in network synthesis in the last decade, motivated in part by the introduction of the inerter, has led to the need for a better understanding of the most economical way to realize a given passive impedance. This monograph outlines the main contributions to the field of passive network synthesis and presents new research into the enumerative approach and the classification of networks of restricted complexity. Passive Network Synthesis: An Approach to Classification serves as both an ideal introduction to the topic and a definitive treatment of the Ladenheim catalogue.In particular, the authors provide a new analysis and classification of the Ladenheim catalogue, building on recent work, to obtain an improved understanding of the structure and realization power of the class within the biquadratic positive-real functions.