Describing Function
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Control Theory
Control theory S. Simrock DESY, Hamburg, Germany Abstract In engineering and mathematics, control theory deals with the behaviour of dynamical systems. The desired output of a system is called the reference. When one or more output variables of a system need to follow a certain ref- erence over time, a controller manipulates the inputs to a system to obtain the desired effect on the output of the system. Rapid advances in digital system technology have radically altered the control design options. It has become routinely practicable to design very complicated digital controllers and to carry out the extensive calculations required for their design. These advances in im- plementation and design capability can be obtained at low cost because of the widespread availability of inexpensive and powerful digital processing plat- forms and high-speed analog IO devices. 1 Introduction The emphasis of this tutorial on control theory is on the design of digital controls to achieve good dy- namic response and small errors while using signals that are sampled in time and quantized in amplitude. Both transform (classical control) and state-space (modern control) methods are described and applied to illustrative examples. The transform methods emphasized are the root-locus method of Evans and fre- quency response. The state-space methods developed are the technique of pole assignment augmented by an estimator (observer) and optimal quadratic-loss control. The optimal control problems use the steady-state constant gain solution. Other topics covered are system identification and non-linear control. System identification is a general term to describe mathematical tools and algorithms that build dynamical models from measured data. -
Analyzing Oscillators Using Describing Functions
Analyzing Oscillators using Describing Functions Tianshi Wang Department of EECS, University of California, Berkeley, CA, USA Email: [email protected] Abstract In this manuscript, we discuss the use of describing functions as a systematic approach to the analysis and design of oscillators. Describing functions are traditionally used to study the stability of nonlinear control systems, and have been adapted for analyzing LC oscillators. We show that they can be applied to other categories of oscillators too, including relaxation and ring oscillators. With the help of several examples of oscillators from various physical domains, we illustrate the techniques involved, and also demonstrate the effectiveness and limitations of describing functions for oscillator analysis. I. Introduction Oscillators can be found in virtually every area of science and technology. They are widely used in radio communication [1], clock generation [2], and even logic computation in both Boolean [3–6] and non-Boolean [7–9] paradigms. The design and use of oscillators are not limited to electronic ones. Integrated MEMS oscillators are designed with an aim of replacing traditional quartz crystal ones as on-chip frequency references [10–12]; spin torque oscillators are studied for potential RF applications [13–15]; chemical reaction oscillators work as clocks in synthetic biology [16, 17]; lasers are optical oscillators with a wide range of applications. The growing importance of oscillators in circuits and systems calls for a systematic design methodology for oscillators which is independent from physical domains. However, such an approach is still lacking. While structured design methodologies are available for oscillators with negative-feedback amplifiers [18, 19], different types of oscillators are still analyzed with vastly different methodologies. -
2376. Analysis of Multiple Oscillations in a Sampling Reduced-Order Sliding Mode Control in Frequency Domain
2376. Analysis of multiple oscillations in a sampling reduced-order sliding mode control in frequency domain Yu Shen1, Xing He2 1School of Computer and Information Science, Southwest University, Chongqing 400715, China 2College of Electronics and Information Engineering, Southwest University, Chongqing 400715, China 2Corresponding author E-mail: [email protected], [email protected] Received 17 June 2016; received in revised form 30 October 2016; accepted 15 November 2016 DOI https://doi.org/10.21595/jve.2016.17299 Abstract. The phenomena of oscillations in sliding mode control system usually cause potential damage or danger in engineering applications. This paper focuses on the estimation and adjustment of oscillation parameters (frequency and amplitude) using an improved describing function method and the characterization of the influence of the reduced-order sliding surface coefficients on the performance of the nonlinear system. The model of series connection of the zero-order holder and switching function is established in frequency domain. The improved describing function method could intuitively estimate the number, stability and parameters of oscillations for the coupled nonlinear system by its amplitude-phase characteristics and pole placement. The stability, parameters and attraction zones of three oscillations in a fourth-order plant are analyzed and simulated. The analytical results show that the reduced-order sliding mode controller possesses complex and colorful nonlinear behaviors. Besides, a coefficient tuning method is presented to eliminate the undesired oscillation which may lead to poor precision or break the stability. The simulations demonstrate that appropriate tuning on sliding surface may greatly improve the efficiency and robustness of the sliding mode control system. -
An Introduction to Nonlinearity in Control Systems
Derek Atherton An Introduction to Nonlinearity in Control Systems 2 Download free eBooks at bookboon.com An Introduction to Nonlinearity in Control Systems © 2014 Derek Atherton & bookboon.com ISBN 978-87-7681-790-9 3 Download free eBooks at bookboon.com An Introduction to Nonlinearity in Control Systems Contents Contents Preface 10 1 Introduction 12 1.1 What is nonlinearity? 12 1.2 Forms of nonlinearity 13 1.3 Structure and Behaviour 16 1.4 Overview of contents 17 1.5 References 18 2 The Phase Plane Method 19 2.1 Introduction 19 2.2 Basic Principles 19 2.2.1 The Linear Case 20 2.2.2 The Nonlinear Case 23 2.2.3 An Example 24 2.3 The Phase Plane for Systems with Linear Segmented Nonlinearities 26 2.3.1 Example 1 – Nonlinear Output Derivative Feedback 27 2.3.2 Example 2 – Relay Position Control 28 2.3.3 Example 3 – Position Control with Torque Saturation 34 �e Graduate Programme I joined MITAS because for Engineers and Geoscientists I wanted real responsibili� www.discovermitas.comMaersk.com/Mitas �e Graduate Programme I joined MITAS because for Engineers and Geoscientists I wanted real responsibili� Maersk.com/Mitas Month 16 I wwasas a construction Month 16 supervisorI wwasas in a construction the North Sea supervisor in advising and the North Sea Real work helpinghe foremen advising and IInternationalnternationaal opportunities ��reeree wworkoro placements solves Real work problems helpinghe foremen IInternationalnternationaal opportunities ��reeree wworkoro placements solves problems 4 Click on the ad to read more Download free eBooks at bookboon.com An Introduction to Nonlinearity in Control Systems Contents 2. -
EL2620 Nonlinear Control Exercises and Homework
EL2620 Nonlinear Control Exercises and Homework Henning Schmidt, Karl Henrik Johansson, Krister Jacobsson, Bo Wahlberg, Per Hägg, Elling W. Jacobsen September 2012 Automatic Control KTH, Stockholm, Sweden Contents Preface 5 1 Homeworks 7 1.1 ReportRequirements .................................. ..... 7 1.1.1 Writing your report using Microsoft Word R ....................... 7 1.1.2 Writing your report using LATEX............................. 8 1.1.3 Workingroupsoftwo ................................. 9 1.1.4 Grading.......................................... 9 1.1.5 References ...................................... 11 1.2 Homework1 ......................................... 12 1.3 Homework2 ......................................... 13 1.4 Homework3 ......................................... 15 2 Exercises 19 2.1 Nonlinear Models and Simulation . 19 2.2 Computer exercise: Simulation . 24 2.3 Linearization and Phase-Plane Analysis . ........ 26 2.4 Lyapunov Stability . 30 2.5 Input-Output Stability . 34 2.6 Describing Function Analysis . ...... 43 2.7 Anti-windup......................................... 51 2.8 Friction, Backlash and Quantization . ...... 55 2.9 High-gain and Sliding mode . 58 2.10 Computer Exercises: Sliding Mode, IMC and Gain scheduling . .......... 63 2.11 Gain scheduling and Lyapunov based methods . .......... 66 2.12 Feedback linearization . ...... 71 2.13 Optimal Control . 74 2.14Fuzzycontrol ..................................... ...... 80 3 Solutions 81 3.1 Solutions to Nonlinear Models and Simulation . 81 3.2 Solutions to Computer exercise: Simulation . 85 3.3 Solutions to Linearization and Phase-Plane Analysis . ........ 86 3.4 Solutions to Lyapunov Stability . 90 3.5 Solutions to Input-Output Stability . 94 3.6 Solutions to Describing Function Analysis . 102 3.7 Solutions to Anti-windup . 108 3.8 Solutions to Friction, Backlash and Quantization . 110 3.9 Solutions to High-gain and Sliding mode . 114 3.10 Solutions to Computer Exercises: Sliding Mode, IMC and Gain scheduling ......... -
CHATTERING ANALYSIS of the SYSTEM with HIGHER ORDER SLIDING MODE CONTROL THESIS Presented in Partial Fulfillment of the Requirem
CHATTERING ANALYSIS OF THE SYSTEM WITH HIGHER ORDER SLIDING MODE CONTROL THESIS Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Abdalla Swikir Graduate Program in Electrical and Computer Engineering The Ohio State University 2015 Master's Examination Committee: Prof. Vadim I. Utkin, Advisor Prof. Lisa Fiorentini Copyright by Abdalla Swikir 2015 Abstract Sliding mode control methodology (SMC) is considered as an efficient technique in several aspects for control systems. Nevertheless, applying first-order sliding mode control method in real-life has what so-called chattering phenomenon. Chattering is a high frequency movement that makes the state trajectories quickly oscillating around the sliding surface. This phenomenon may lead to degrade the system effectiveness, or even worst it may lead to fast damage of mechanical parts of the system. Presently, the major solutions to this drawback are asymptotic observers and high-order sliding mode control. The higher order sliding mode control is an expansion of the conventional sliding mode, and it can cancel the imperfection of the conventional sliding mode methodology and sustain its advantages. Since the second-order sliding mode controller, for example, super twisting algorithm, has unsophisticated construction and requires a lesser amount of acquaintance, it is the most extensively technique that used in the higher order sliding mode methodology. However, in several cases, the chattering amplitude result from conventional method is less than the one result from super twisting algorithm. In this thesis, the describing function method, most suitable technique to analyze nonlinear systems, is used to make comparison between the two methods, the conventional and the super twisting sliding mode control.