Reachable Set Estimation for Discrete-Time Systems with Interval Time-Varying Delays and Bounded Disturbances

The reachable set estimation problem for discrete-time systems with delay-range-dependent and bounded disturbances is investigated. A triple-summation term, the upper bound, and the lower bound of time-varying delay are introduced into the Lyapunov function. In this case, an improved delay-range-dependent criterion is established for the addressed problem by constructing the appropriate Lyapunov functional, which guarantees that the reachable set of discrete-time systems with time-varying delay and bounded peak inputs is contained in the ellipsoid. It is worth mentioning that the initial value of the system does not need to be zero. Then, the reachable set estimation problem for time-delay systems with polytopic uncertainties is investigated. The effectiveness and the reduced conservatism of the derived results are demonstrated by an illustrative example

Control Theory: On the Way to New Application Fields

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently, deep interactions are emerging with new application areas, such as systems biology, quantum control and information technology. In order to address the new challenges posed by the new application disciplines, a special focus of this workshop has been on the interaction between control theory and mathematical systems biology. To complement these more biology oriented focus, a series of lectures in this workshop was devoted to the control of networks of systems, fundamentals of nonlinear control systems, model reduction and identification, algorithmic aspects in control, as well as open problems in control

Feedback Systems: An Introduction for Scientists and Engineers

This book provides an introduction to the basic principles and tools for the design and analysis of feedback systems. It is intended to serve a diverse audience of scientists and engineers who are interested in understanding and utilizing feedback in physical, biological, information and social systems.We have attempted to keep the mathematical prerequisites to a minimum while being careful not to sacrifice rigor in the process. We have also attempted to make use of examples from a variety of disciplines, illustrating the generality of many of the tools while at the same time showing how they can be applied in specific application domains. A major goal of this book is to present a concise and insightful view of the current knowledge in feedback and control systems. The field of control started by teaching everything that was known at the time and, as new knowledge was acquired, additional courses were developed to cover new techniques. A consequence of this evolution is that introductory courses have remained the same for many years, and it is often necessary to take many individual courses in order to obtain a good perspective on the field. In developing this book, we have attempted to condense the current knowledge by emphasizing fundamental concepts. We believe that it is important to understand why feedback is useful, to know the language and basic mathematics of control and to grasp the key paradigms that have been developed over the past half century. It is also important to be able to solve simple feedback problems using back-of-the-envelope techniques, to recognize fundamental limitations and difficult control problems and to have a feel for available design methods. This book was originally developed for use in an experimental course at Caltech involving students from a wide set of backgrounds. The course was offered to undergraduates at the junior and senior levels in traditional engineering disciplines, as well as first- and second-year graduate students in engineering and science. This latter group included graduate students in biology, computer science and physics. Over the course of several years, the text has been classroom tested at Caltech and at Lund University, and the feedback from many students and colleagues has been incorporated to help improve the readability and accessibility of the material. Because of its intended audience, this book is organized in a slightly unusual fashion compared to many other books on feedback and control. In particular, we introduce a number of concepts in the text that are normally reserved for second-year courses on control and hence often not available to students who are not control systems majors. This has been done at the expense of certain traditional topics, which we felt that the astute student could learn independently and are often explored through the exercises. Examples of topics that we have included are nonlinear dynamics, Lyapunov stability analysis, the matrix exponential, reachability and observability, and fundamental limits of performance and robustness. Topics that we have deemphasized include root locus techniques, lead/lag compensation and detailed rules for generating Bode and Nyquist plots by hand. Several features of the book are designed to facilitate its dual function as a basic engineering text and as an introduction for researchers in natural, information and social sciences. The bulk of the material is intended to be used regardless of the audience and covers the core principles and tools in the analysis and design of feedback systems. Advanced sections, marked by the “dangerous bend” symbol shown here, contain material that requires a slightly more technical background, of the sort that would be expected of senior undergraduates in engineering. A few sections are marked by two dangerous bend symbols and are intended for readers with more specialized backgrounds, identified at the beginning of the section. To limit the length of the text, several standard results and extensions are given in the exercises, with appropriate hints toward their solutions. To further augment the printed material contained here, a companion web site has been developed and is available from the publisher’s web page: http://press.princeton.edu/titles/8701.html The web site contains a database of frequently asked questions, supplemental examples and exercises, and lecture material for courses based on this text. The material is organized by chapter and includes a summary of the major points in the text as well as links to external resources. The web site also contains the source code for many examples in the book, as well as utilities to implement the techniques described in the text. Most of the code was originally written using MATLAB M-files but was also tested with LabView MathScript to ensure compatibility with both packages. Many files can also be run using other scripting languages such as Octave, SciLab, SysQuake and Xmath. The first half of the book focuses almost exclusively on state space control systems. We begin in Chapter 2 with a description of modeling of physical, biological and information systems using ordinary differential equations and difference equations. Chapter 3 presents a number of examples in some detail, primarily as a reference for problems that will be used throughout the text. Following this, Chapter 4 looks at the dynamic behavior of models, including definitions of stability and more complicated nonlinear behavior. We provide advanced sections in this chapter on Lyapunov stability analysis because we find that it is useful in a broad array of applications and is frequently a topic that is not introduced until later in one’s studies. The remaining three chapters of the first half of the book focus on linear systems, beginning with a description of input/output behavior in Chapter 5. In Chapter 6, we formally introduce feedback systems by demonstrating how state space control laws can be designed. This is followed in Chapter 7 by material on output feedback and estimators. Chapters 6 and 7 introduce the key concepts of reachability and observability, which give tremendous insight into the choice of actuators and sensors, whether for engineered or natural systems. The second half of the book presents material that is often considered to be from the field of “classical control.” This includes the transfer function, introduced in Chapter 8, which is a fundamental tool for understanding feedback systems. Using transfer functions, one can begin to analyze the stability of feedback systems using frequency domain analysis, including the ability to reason about the closed loop behavior of a system from its open loop characteristics. This is the subject of Chapter 9, which revolves around the Nyquist stability criterion. In Chapters 10 and 11, we again look at the design problem, focusing first on proportional-integral-derivative (PID) controllers and then on the more general process of loop shaping. PID control is by far the most common design technique in control systems and a useful tool for any student. The chapter on frequency domain design introduces many of the ideas of modern control theory, including the sensitivity function. In Chapter 12, we combine the results from the second half of the book to analyze some of the fundamental trade-offs between robustness and performance. This is also a key chapter illustrating the power of the techniques that have been developed and serving as an introduction for more advanced studies. The book is designed for use in a 10- to 15-week course in feedback systems that provides many of the key concepts needed in a variety of disciplines. For a 10-week course, Chapters 1–2, 4–6 and 8–11 can each be covered in a week’s time, with the omission of some topics from the final chapters. A more leisurely course, spread out over 14–15 weeks, could cover the entire book, with 2 weeks on modeling (Chapters 2 and 3) — particularly for students without much background in ordinary differential equations — and 2 weeks on robust performance (Chapter 12). The mathematical prerequisites for the book are modest and in keeping with our goal of providing an introduction that serves a broad audience. We assume familiarity with the basic tools of linear algebra, including matrices, vectors and eigenvalues. These are typically covered in a sophomore-level course on the subject, and the textbooks by Apostol [10], Arnold [13] and Strang [187] can serve as good references. Similarly, we assume basic knowledge of differential equations, including the concepts of homogeneous and particular solutions for linear ordinary differential equations in one variable. Apostol [10] and Boyce and DiPrima [42] cover this material well. Finally, we also make use of complex numbers and functions and, in some of the advanced sections, more detailed concepts in complex variables that are typically covered in a junior-level engineering or physics course in mathematical methods. Apostol [9] or Stewart [186] can be used for the basic material, with Ahlfors [6], Marsden and Hoffman [146] or Saff and Snider [172] being good references for the more advanced material. We have chosen not to include appendices summarizing these various topics since there are a number of good books available. One additional choice that we felt was important was the decision not to rely on a knowledge of Laplace transforms in the book. While their use is by far the most common approach to teaching feedback systems in engineering, many students in the natural and information sciences may lack the necessary mathematical background. Since Laplace transforms are not required in any essential way, we have included them only in an advanced section intended to tie things together for students with that background. Of course, we make tremendous use of transfer functions, which we introduce through the notion of response to exponential inputs, an approach we feel is more accessible to a broad array of scientists and engineers. For classes in which students have already had Laplace transforms, it should be quite natural to build on this background in the appropriate sections of the text. Acknowledgments: The authors would like to thank the many people who helped during the preparation of this book. The idea for writing this book came in part from a report on future directions in control [155] to which Stephen Boyd, Roger Brockett, John Doyle and Gunter Stein were major contributors. Kristi Morgansen and Hideo Mabuchi helped teach early versions of the course at Caltech on which much of the text is based, and Steve Waydo served as the head TA for the course taught at Caltech in 2003–2004 and provided numerous comments and corrections. Charlotta Johnsson and Anton Cervin taught from early versions of the manuscript in Lund in 2003–2007 and gave very useful feedback. Other colleagues and students who provided feedback and advice include Leif Andersson, John Carson, K. Mani Chandy, Michel Charpentier, Domitilla Del Vecchio, Kate Galloway, Per Hagander, Toivo Henningsson Perby, Joseph Hellerstein, George Hines, Tore Hägglund, Cole Lepine, Anders Rantzer, Anders Robertsson, Dawn Tilbury and Francisco Zabala. The reviewers for Princeton University Press and Tom Robbins at NI Press also provided valuable comments that significantly improved the organization, layout and focus of the book. Our editor, Vickie Kearn, was a great source of encouragement and help throughout the publishing process. Finally, we would like to thank Caltech, Lund University and the University of California at Santa Barbara for providing many resources, stimulating colleagues and students, and pleasant working environments that greatly aided in the writing of this book

Assessing plant design with regards to MPC performance using a novel multi-model prediction method

Model Predictive Control (MPC) is nowadays ubiquitous in the chemical industry and offers significant advantages over standard feedback controllers. Notwithstanding, projects of new plants are still being carried out without assessing how key design decisions, e.g., selection of production route, plant layout and equipment, will affect future MPC performance. The problem addressed in this Thesis is comparing the economic benefits available for different flowsheets through the use of MPC, and thus determining if certain design choices favour or hinder expected profitability. The Economic MPC Optimisation (EMOP) index is presented to measure how disturbances and restrictions affect the MPC’s ability to deliver better control and optimisation. To the author’s knowledge, the EMOP index is the first integrated design and control methodology to address the problem of zone constrained MPC with economic optimisation capabilities (today's standard in the chemical industry). This approach assumes the availability of a set of linear state-space models valid within the desired control zone, which is defined by the upper and lower bounds of each controlled and manipulated variable. Process economics provides the basis for the analysis. The index needs to be minimised in order to find the most profitable steady state within the zone constraints towards which the MPC is expected to direct the process. An analysis of the effects of disturbances on the index illustrates how they may reduce profitability by restricting the ability of an MPC to reach dynamic equilibrium near process constraints, which in turn increases product quality giveaway and costs. Hence the index monetises the required control effort. Since linear models were used to predict the dynamic behaviour of chemical processes, which often exhibit significant nonlinearity, this Thesis also includes a new multi-model prediction method. This new method, called Simultaneous Multi-Linear Prediction (SMLP), presents a more accurate output prediction than the use of single linear models, keeping at the same time much of their numerical advantages and their relative ease of obtainment. Comparing the SMLP to existing multi-model approaches, the main novelty is that it is built by defining and updating multiple states simultaneously, thus eliminating the need for partitioning the state-input space into regions and associating with each region a different state update equation. Each state’s contribution to the overall output is obtained according to the relative distance between their identification point, i.e., the set of operating conditions at which an approximation of the nonlinear model is obtained, and the current operating point, in addition to a set of parameters obtained through regression analysis. Additionally, the SMLP is built upon data obtained from step response models that can be obtained by commercial, black-box dynamic simulators. These state-of-the-art simulators are the industry’s standard for designing large-scale plants, the focus of this Thesis. Building an SMLP system yields an approximation of the nonlinear model, whose full set of equations is not of the user’s knowledge. The resulting system can be used for predictive control schemes or integrated process design and control. Applying the SMLP to optimisation problems with linear restrictions results in convex problems that are easy to solve. The issue of model uncertainty was also addressed for the EMOP index and SMLP systems. Due to the impact of uncertainty, the index may be defined as a numeric interval instead of a single number, within which the true value lies. A case of study consisting of four alternative designs for a realistically sized crude oil atmospheric distillation plant is provided in order to demonstrate the joint use and applicability of both the EMOP index and the SMLP. In addition, a comparison between the EMOP index and a competing methodology is presented that is based on a case study consisting of the activated sludge process of a wastewater treatment plant

Development of Robust Control Strategies for Autonomous Underwater Vehicles

The resources of the energy and chemical balance in the ocean sustain mankind in many ways. Therefore, ocean exploration is an essential task that is accomplished by deploying Underwater Vehicles. An Underwater Vehicle with autonomy feature for its navigation and control is called Autonomous Underwater Vehicle (AUV). Among the task handled by an AUV, accurately positioning itself at a desired position with respect to the reference objects is called set-point control. Similarly, tracking of the reference trajectory is also another important task. Battery recharging of AUV, positioning with respect to underwater structure, cable, seabed, tracking of reference trajectory with desired accuracy and speed to avoid collision with the guiding vehicle in the last phase of docking are some significant applications where an AUV needs to perform the above tasks. Parametric uncertainties in AUV dynamics and actuator torque limitation necessitate to design robust control algorithms to achieve motion control objectives in the face of uncertainties. Sliding Mode Controller (SMC), H / μ synthesis, model based PID group controllers are some of the robust controllers which have been applied to AUV. But SMC suffers from less efficient tuning of its switching gains due to model parameters and noisy estimated acceleration states appearing in its control law. In addition, demand of high control effort due to high frequency chattering is another drawback of SMC. Furthermore, real-time implementation of H / μ synthesis controller based on its stability study is restricted due to use of linearly approximated dynamic model of an AUV, which hinders achieving robustness. Moreover, model based PID group controllers suffer from implementation complexities and exhibit poor transient and steady-state performances under parametric uncertainties. On the other hand model free Linear PID (LPID) has inherent problem of narrow convergence region, i.e.it can not ensure convergence of large initial error to zero. Additionally, it suffers from integrator-wind-up and subsequent saturation of actuator during the occurrence of large initial error. But LPID controller has inherent capability to cope up with the uncertainties. In view of addressing the above said problem, this work proposes wind-up free Nonlinear PID with Bounded Integral (BI) and Bounded Derivative (BD) for set-point control and combination of continuous SMC with Nonlinear PID with BI and BD namely SM-N-PID with BI and BD for trajectory tracking. Nonlinear functions are used for all P,I and D controllers (for both of set-point and tracking control) in addition to use of nonlinear tan hyperbolic function in SMC(for tracking only) such that torque demand from the controller can be kept within a limit. A direct Lyapunov analysis is pursued to prove stable motion of AUV. The efficacies of the proposed controllers are compared with other two controllers namely PD and N-PID without BI and BD for set-point control and PD plus Feedforward Compensation (FC) and SM-NPID without BI and BD for tracking control. Multiple AUVs cooperatively performing a mission offers several advantages over a single AUV in a non-cooperative manner; such as reliability and increased work efficiency, etc. Bandwidth limitation in acoustic medium possess challenges in designing cooperative motion control algorithm for multiple AUVs owing to the necessity of communication of sensors and actuator signals among AUVs. In literature, undirected graph based approach is used for control design under communication constraints and thus it is not suitable for large number of AUVs participating in a cooperative motion plan. Formation control is a popular cooperative motion control paradigm. This thesis models the formation as a minimally persistent directed graph and proposes control schemes for maintaining the distance constraints during the course of motion of entire formation. For formation control each AUV uses Sliding Mode Nonlinear PID controller with Bounded Integrator and Bounded Derivative. Direct Lyapunov stability analysis in the framework of input-to-state stability ensures the stable motion of formation while maintaining the desired distance constraints among the AUVs

Life Cycle Evaluation under Uncertain Environmental Policies Using a Ship-Centric Markov Decision Process Framework.

A novel design evaluation framework is offered to improve early stage design decisions relating to environmental policy change and similar non-technical disturbances. The goal of this research is to overcome the traditional treatment of policy as a static, external constraint and to address in early stage design the potential disruptions to performance posed by regulatory policy change. While a designer’s primary purpose is not to affect policy, it is the responsibility of the designer to be cognizant of how policy can change, of how to assess the implications of a policy change, and of how to deliver performance despite change. This research addresses a present need for a rigorous means to keep strategic pace with policy evolution. Use of a Markov Decision Process (MDP) framework serves as a unifying foundation for incorporating temporal activities into early stage design considerations. The framework employs probabilistic methods via a state-based structure to holistically address policy uncertainty. Presented research enables exploration of the performance of a design solution through time in the face of environmental instabilities and identifies decisions necessary to negotiate path dependencies. The outcome of this research is an advanced framework for addressing life cycle management needs that arise due to policy change, as judged from a life cycle cost perspective. Original metrics for evaluating decision paths provide insight into how the timing, location, and confluence of disturbances impact design decisions. Development of the metrics is driven by a desire to communicate the design-specific characteristics of a strategic response to policy change. Quantifying the amount and type of uncertainty present, changeability afforded, and life cycle changes exercised offer points of comparison among individual design solutions. The knowledge gained from path-centric measurements enables an enhanced ability to characterize design lock-in. Principles and metrics borne out of the design evaluation framework are validated through two ship design examples related to ballast water treatment and carbon emissions.PHDNaval Architecture & Marine EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/96130/1/ndniese_1.pd