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E-raamat: Feedback Control: Linear, Nonlinear and Robust Techniques and Design with Industrial Applications

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This book develops the understanding and skills needed to be able to tackle original control problems. The general approach to a given control problem is to try the simplest tentative solution first and, when this is insufficient, to explain why and use a more sophisticated alternative to remedy the deficiency and achieve satisfactory performance. This pattern of working gives readers a full understanding of different controllers and teaches them to make an informed choice between traditional controllers and more advanced modern alternatives in meeting the needs of a particular plant. Attention is focused on the time domain, covering model-based linear and nonlinear forms of control together with robust control based on sliding modes and the use of state observers such as disturbance estimation.

Feedback Control is self-contained, paying much attention to explanations of underlying concepts, with detailed mathematical derivations being employed where necessary. Ample use is made of diagrams to aid these conceptual explanations and the subject matter is enlivened by continual use of examples and problems derived from real control applications. Readers learning is further enhanced by experimenting with the fully-commented MATLAB®/Simulink® simulation environment made accessible at insert URL here to produce simulations relevant to all of the topics covered in the text. A solutions manual for use by instructors adopting the book can also be downloaded from insert URL here.

Feedback Control is suitable as a main textbook for graduate and final-year undergraduate courses containing control modules; knowledge of ordinary linear differential equations, Laplace transforms, transfer functions, poles and zeros, root locus and elementary frequency response analysis, and elementary feedback control is required. It is also a useful reference source on control design methods for engineers practicing in industry and for academic control researchers.

Arvustused

People with a basic background in feedback control will discover many aspects of traditional design methods not commonly considered in the literature, and those who need to tackle real-life control problems will enjoy the author's approach and the richness of the solutions studied. (Blas M. Vinagre, Mathematical Reviews, August, 2016)

This is the original book on feedback control of linear and nonlinear systems with some interesting applications. This book can be recommended to undergraduate and graduate students. (Tadeusz Kaczorek, zbMATH 1336.93002, 2016)

1 Introduction 1(72)
1.1 Overview
1(1)
1.2 Notation and Nomenclature
1(3)
1.2.1 Scalars, Vectors and Matrices
1(1)
1.2.2 Subscripts and Superscripts
2(1)
1.2.3 Constants and Variables
2(1)
1.2.4 Nomenclature and Standard Symbols
3(1)
1.2.5 Variables and Their Laplace Transforms
4(1)
1.3 Review of Traditional PID Controllers and Their Variants
4(39)
1.3.1 Traditional Error-Actuated Controllers
4(15)
1.3.2 Zero-Less Versions of the Traditional Controllers
19(3)
1.3.3 Traditional Controller Selection Guidelines
22(3)
1.3.4 Measurement Noise Filtering for Derivative Term
25(7)
1.3.5 Anti-windup Loop for Integral Term
32(11)
1.4 Dominance in the Pole-Zero Distribution
43(26)
1.4.1 Background
43(1)
1.4.2 Modes of Linear Systems
43(6)
1.4.3 Dominance in Pole Distributions
49(4)
1.4.4 Dominance in Systems with Zeros
53(16)
1.5 The Steps of Control System Design
69(2)
1.6 The Flexibility of Digital Implementation
71(1)
Reference
71(2)
2 Plant Modelling 73(96)
2.1 Introduction
73(8)
2.1.1 Overview
73(1)
2.1.2 Dynamical and Non-Dynamical Systems
73(3)
2.1.3 Linearity and Nonlinearity
76(4)
2.1.4 Modelling Categories and Basic Forms of Model
80(1)
2.2 Physical Modelling
81(40)
2.2.1 Introduction
81(1)
2.2.2 Mechanical Modeling Principles
81(11)
2.2.3 Two Basic Mechanical Components
92(5)
2.2.4 Modelling for Vehicle Attitude and Position Control
97(7)
2.2.5 Electric Motors
104(5)
2.2.6 Vector-Controlled AC Motors as Control Actuators
109(8)
2.2.7 Fluid and Thermal Subsystems
117(4)
2.3 Identification of LTI Plants from Measurements
121(46)
2.3.1 Overview
121(1)
2.3.2 Plant Model Determination from Step Response
122(5)
2.3.3 Plant Model Determination from Frequency Response
127(24)
2.3.4 Recursive Parameter Estimation: An Introduction
151(16)
References
167(2)
3 Plant Model Manipulation and Analysis 169(126)
3.1 Introduction
169(1)
3.2 The State Space Model
170(24)
3.2.1 Introduction
170(1)
3.2.2 Forming a State-Space Model
171(1)
3.2.3 The General State-Space Model
172(1)
3.2.4 The General LTI State-Space Model
173(2)
3.2.5 Some Preliminary Control Theory
175(4)
3.2.6 Controllability Analysis of Continuous LTI Plant Models
179(4)
3.2.7 Observability Analysis of Continuous LTI Plant Models
183(4)
3.2.8 The State-Variable Block Diagram
187(3)
3.2.9 Transfer Function from Continuous LTI State Space Model
190(2)
3.2.10 Relative Degree
192(2)
3.3 State Representation
194(61)
3.3.1 Introduction
194(6)
3.3.2 LTI SISO State-Space Models from Transfer Functions
200(11)
3.3.3 Transformation Matrices Connecting Linear Models
211(6)
3.3.4 Modal Forms for Multivariable LTI Plants: Transformations
217(20)
3.3.5 SISO Controller and Observer Canonical Forms
237(8)
3.3.6 Multivariable Controller and Observer Canonical Forms
245(10)
3.4 Discrete LTI Plant Models
255(38)
3.4.1 Formation of the Discrete State Space Model
255(4)
3.4.2 State Space Model Derivation from Modal Basis Functions
259(15)
3.4.3 Plant z-Transfer Function Model
274(8)
3.4.4 Change of Sampling Period for z-Transfer Function Models
282(5)
3.4.5 Controllability Analysis of Discrete LTI Plant Models
287(3)
3.4.6 Analysis of Discrete LTI Plant Models
290(3)
References
293(2)
4 Traditional Controllers: Model Based Design 295(60)
4.1 Approach
295(2)
4.2 Pole Assignment
297(2)
4.3 Definition of Settling Time
299(1)
4.4 PID Controllers and Their Variants
300(25)
4.4.1 First Order Systems
300(7)
4.4.2 Second Order Systems
307(12)
4.4.3 Cascade Control Structure
319(6)
4.5 Systems of Third and Higher Order
325(12)
4.5.1 Attainable Closed Loop Dynamics
325(2)
4.5.2 The Laplace to Time Domain Inverse Scaling Law
327(2)
4.5.3 Step Responses with Coincident Closed Loop Poles
329(2)
4.5.4 Derivation of the Settling Time Formulae
331(2)
4.5.5 Settling Time Formula Error Determination and Correction
333(2)
4.5.6 Closed Loop Poles for Given Overshoot and Settling Time
335(2)
4.6 Performance Specifications in the Frequency Domain
337(17)
4.6.1 Background
337(1)
4.6.2 Closed Loop System Bandwidth
337(2)
4.6.3 Sensitivity and Robustness
339(8)
4.6.4 Stability Analysis in the Frequency Domain
347(7)
References
354(1)
5 Linear Controllers for LTI SISO Plants of Arbitrary Order: Model-Based Design 355(60)
5.1 Overview
355(1)
5.2 Linear Continuous State Feedback Control
356(42)
5.2.1 Introduction
356(1)
5.2.2 Linear State Feedback Control Law
357(1)
5.2.3 Matrix—Vector Formulation
358(2)
5.2.4 Closed-Loop Transfer Function
360(1)
5.2.5 Pole Assignment Using the Matrix—Vector Formulation
361(3)
5.2.6 Pole Assignment Using Mason's Formula
364(4)
5.2.7 Pole Assignment for Plants with Significant Zeros
368(19)
5.2.8 State Feedback Controllers with Additional Integral Terms
387(11)
5.3 Polynomial Control
398(16)
5.3.1 Introduction
398(1)
5.3.2 Formulation of Polynomial Controller Structure
399(2)
5.3.3 Constraints on Controller Polynomial Degrees
401(2)
5.3.4 Determination of the Controller Parameters
403(4)
5.3.5 The Polynomial Integral Controller
407(7)
References
414(1)
6 Discrete Control of LTI SISO Plants 415(66)
6.1 Introduction
415(1)
6.2 Real-Time Operation of Digital Controllers
416(1)
6.3 Dynamics of Discrete Linear Systems
417(18)
6.3.1 Stability Analysis in the z-Plane
417(8)
6.3.2 Connection Between Dynamic Behaviour and the z-Plane Pole Locations
425(7)
6.3.3 The Effects of Zeros in the z-Plane
432(3)
6.4 Criterion for Applicability of Continuous LTI System Theory
435(4)
6.5 Discrete Control for Small Iteration Intervals
439(13)
6.5.1 Introduction
439(1)
6.5.2 Discrete Equations of the Basic Elements
439(4)
6.5.3 Discrete Controller Block Diagrams for Simulation
443(4)
6.5.4 Control Algorithms and Flow Charts
447(5)
6.6 Discrete Control with Unlimited Iteration Intervals
452(28)
6.6.1 Pole Placement Design with the Settling Time Formulae
452(4)
6.6.2 Pole Placement for Negligible Digital Processing Time
456(8)
6.6.3 Computational Delay Allowance
464(8)
6.6.4 Discrete Integral Polynomial Control
472(4)
6.6.5 Control of Plants Containing Pure Time Delays
476(4)
References
480(1)
7 Model Based Control of Nonlinear and Linear Plants 481(80)
7.1 Introduction
481(1)
7.2 Linearisation About an Operating Point
482(9)
7.2.1 Basic Principle
482(2)
7.2.2 Linear State-Space Model
484(7)
7.2.3 Limitation
491(1)
7.3 Feedback Linearising and Forced Dynamic Control
491(69)
7.3.1 Preliminaries
491(4)
7.3.2 Feedback Linearising Control of Plants with Full Relative Degree
495(14)
7.3.3 Feedback Linearising Control of Plants Less Than Full Relative Degree
509(13)
7.3.4 Forced Dynamic Control of Continuous LTI Plants
522(21)
7.3.5 FDC and FLC Using Discrete Plant Models
543(7)
7.3.6 Near-Time-Optimal Position Control Through FDC
550(10)
References
560(1)
8 State Estimation 561(64)
8.1 Introduction
561(1)
8.2 The Full State Continuous Observer for LTI SISO Plants
562(25)
8.2.1 Introduction
562(3)
8.2.2 The Separation Principle and the Transparency Property
565(2)
8.2.3 Design of the Real-Time Model Correction Loop
567(8)
8.2.4 Estimation of Disturbances
575(12)
8.3 The Full State Discrete Observer for LTI SISO Plants
587(5)
8.3.1 Introduction
587(1)
8.3.2 Observer Algorithm and Design Procedure
587(5)
8.4 The Full State Observer for Multivariable Plants
592(4)
8.4.1 Introduction
592(1)
8.4.2 Matrix—Vector Design Method for SISO LTI Plants
592(2)
8.4.3 Matrix—Vector Design Method for Multivariable LTI Plants
594(2)
8.5 The Noise Filtering Property of the Observer
596(17)
8.5.1 Background
596(1)
8.5.2 Lumped Plant Noise and Measurement Noise Sources
597(1)
8.5.3 State Estimation Error Variation with Observer Gains
598(1)
8.5.4 State Estimation Error Transfer Function Relationship
599(3)
8.5.5 Considering Noise Levels in Observer Design
602(11)
8.6 The Kalman Filter
613(11)
8.6.1 Introduction
613(1)
8.6.2 The Discrete Observer
614(1)
8.6.3 The Kalman Filter: State Difference and Error Equations
615(2)
8.6.4 Derivation of the Discrete Kalman Gain Algorithm
617(4)
8.6.5 The Steady-State Kalman Filter
621(1)
8.6.6 The Kalman—Bucy Filter
622(2)
References
624(1)
9 Switched and Saturating Control Techniques 625(80)
9.1 Introduction
625(3)
9.1.1 Switched Control
625(2)
9.1.2 Saturating Control
627(1)
9.2 Pulse Modulation for Use with Continuous Controllers
628(10)
9.2.1 Basic Concept
628(1)
9.2.2 Implementation
629(9)
9.3 Switched State Feedback Control: Basic Concepts
638(3)
9.4 Switching Function Sign Convention
641(1)
9.5 Boundary Layer for Saturating Control Systems
642(2)
9.6 Supporting Theory
644(7)
9.6.1 Background
644(1)
9.6.2 Optimal Control Through Pontryagin's Maximum Principle
644(7)
9.7 Feedback Control of First-Order Plants
651(15)
9.7.1 Time-Optimal Feedback Control: Analytical Method
651(3)
9.7.2 Time-Optimal Feedback Control: Graphical Approach
654(1)
9.7.3 Limit Cycling and Its Control
655(3)
9.7.4 Control with Time-Varying Reference Inputs
658(4)
9.7.5 Continuous Control with Saturation
662(4)
9.8 Feedback Control of Second-Order Plants
666(21)
9.8.1 Introduction
666(1)
9.8.2 State Trajectories and State Portraits
666(3)
9.8.3 Time-Optimal Feedback Control of the Double Integrator Plant
669(3)
9.8.4 Time-Optimal Control Law Synthesis Using State Portraits
672(4)
9.8.5 Continuous Control with Saturation
676(11)
9.8.6 Limit Cycling Control
687(1)
9.9 Feedback Control of Third and Higher-Order Plants
687(17)
9.9.1 Overview
687(1)
9.9.2 Time-Optimal Control of the Triple Integrator Plant
687(9)
9.9.3 Posicast Control of Fourth-Order Plants with Oscillatory Modes
696(8)
References
704(1)
10 Sliding Mode Control and Its Relatives 705(88)
10.1 Introduction
705(8)
10.1.1 Purpose and Origin
705(1)
10.1.2 Basic Principle
705(2)
10.1.3 Implementation for Robustness
707(6)
10.2 Control of SISO Second-Order Plants of Full Relative Degree
713(17)
10.2.1 The Plant Model
713(1)
10.2.2 Phase Portraits
714(2)
10.2.3 Sliding Motion
716(2)
10.2.4 The Equivalent Control
718(1)
10.2.5 Control Chatter
718(1)
10.2.6 Conditions for the Existence of Sliding Motion
719(1)
10.2.7 Reaching the Sliding Condition
720(3)
10.2.8 Closed-Loop Dynamics in the Sliding Mode
723(1)
10.2.9 Control with Time-Varying Disturbances and Reference Inputs
723(2)
10.2.10 Rate-Limiting Switching Boundary for Zero Overshoot
725(1)
10.2.11 Sub-Time-Optimal Control
726(4)
10.3 Control of SISO Plants of Arbitrary Order
730(12)
10.3.1 Control of Plants Having Full Relative Degree
730(8)
10.3.2 Control of Plants Less Than Full Relative Degree
738(4)
10.4 Methods for Elimination of Control Chatter
742(24)
10.4.1 The Boundary Layer Method
742(4)
10.4.2 The Control Smoothing Integrator Method
746(10)
10.4.3 Higher-Order Sliding Mode Control
756(10)
10.5 Controllers with Robust Pole Placement
766(7)
10.5.1 Introduction
766(1)
10.5.2 Output Derivative State Feedback Controller
767(3)
10.5.3 Dynamic Controllers with Robust Pole Placement
770(3)
10.6 Multivariable Sliding Mode Control: An Introduction
773(19)
10.6.1 Overview
773(1)
10.6.2 Simple Approach with Minimum Plant Information
774(4)
10.6.3 Discrete Sliding Mode Control
778(14)
References
792(1)
11 Motion Control 793(54)
11.1 Introduction
793(1)
11.2 Controlled Mechanisms
793(12)
11.2.1 The General-Purpose Jointed-Arm Robot
793(1)
11.2.2 General Model
794(4)
11.2.3 Feedback Linearising Control Law
798(1)
11.2.4 Simplified Model for Mechanisms with Geared Actuators
799(6)
11.3 Dynamic Lag Pre-compensation
805(21)
11.3.1 Definition of Dynamic Lag
805(1)
11.3.2 Derivative Feedforward Pre-compensation
806(16)
11.3.3 Implementation with Feedback Linearising Control
822(4)
11.4 Optimal Control for Minimising Frictional Energy Loss
826(20)
11.4.1 Motivation
826(1)
11.4.2 Formulation of Optimal Control
827(3)
11.4.3 Minimum Frictional Energy State Feedback Control Law
830(2)
11.4.4 Higher-Order Mechanisms
832(1)
11.4.5 Near-Optimal Control Using a Reference Input Generator
832(14)
References
846(1)
Erratum E1
Tables 847(6)
Laplace Transforms and z-Transfer Functions
847(3)
Characteristic Polynomial Coefficients of the Settling Time Formulae
850(3)
Appendices 853(150)
A2 Appendix to Chap. 2
853(47)
A2.1 Kinematics of Vehicle Attitude Control
853(14)
A2.2 Plant Model Determination from Frequency Response
867(14)
A2.3 A Case Study of Plant Modelling Undertaken in Industry: Modelling for a Throttle Valve Servomechanism
881(18)
References
899(1)
A4 Appendix to Chap. 4
900(20)
A4.1 Application of Mason's Formula Using Block Diagrams
900(8)
A4.2 Traditional Controller Zero Cancellation by Pole Assignment
908(7)
A4.3 Partial Pole Assignment for Traditional Controllers
915(4)
Reference
919(1)
A5 Appendix to Chap. 5
920(20)
A5.1 Computer Aided Pole Assignment
920(7)
A5.2 Linear Characteristic Polynomial Interpolation
927(9)
A5.3 Routh Stability Criterion
936(4)
A8 Appendix to Chap. 8
940(14)
A8.1 An Approach for State Estimation for Nonlinear Plants
940(14)
A8.1.2 Observer Based on Linearised Plant Model
941(1)
A8.1.3 Output Derivative Based State Estimator
941(13)
A9 Appendix to Chap. 9
954(17)
A9.1 Limit Cycling Control for Second Order Plants
954(16)
References
970(1)
A10 Appendix to Chap. 10
971(14)
A10.1 Observer Based Robust Control
971(13)
References
984(1)
A11 Appendix to Chap. 11
985(17)
A11.1 Path Planning and Reference Input Trajectory Generation
985(17)
Reference
1002(1)
Index 1003
My career in control engineering spans forty one years, commencing with control system design and research in the aerospace industry with Marconi Space Systems (Senior Engineer from 1973 to 1980; Principal Engineer from 1980 to 1984) followed by my appointment with the North East London Polytechnic, now the University of East London (Reader in Control Engineering from 1984 to 1994; Professor of Control Engineering from 1994 to the present time)

A milestone in my career was my responsibility for the attitude control system design for the European Space Agencies X-ray astronomy satellite, Exosat. The performance specification was stringent, demanding 1 arc second peak pointing errors about all three control axes using on-off cold gas thrusters, time optimal large angle slewing and extreme robustness against disturbance torques approaching 90% of the available control torques during orbit change thruster firing.  Despite Exosat having substantially rigid body dynamics, this led me to originate non-standard nonlinear bang-off-bang state feedback control laws.  The result was Europe's first spacecraft attitude control system with software implemented control laws.  Following the success of this work, I instigated and carried out research programmes in Marconi Space Systems for the European Space Agency and DERA Space Department (Government research laboratory) aimed at extending my new pedigree of attitude control system to cater for spacecraft containing significant vibration modes due to flexible appendages such as long solar panels. Some of this work formed the basis of my PhD.  My personal research in spacecraft attitude control has continued during my time with the University of East London, having attracted several partners in the Institute of Control Sciences of the Russian Academy of Sciences, carrying out joint research, and this led to my election as an Academician of the Academy of Nonlinear Sciences.

During my time with the University of EastLondon, my research has developed into the more general areas of a) robust control (based on sliding mode control rather than H-infinity), which seeks to achieve a specified performance of an automatic control system in the presence of plant modelling uncertainties and unknown external disturbances, and b) model based nonlinear control.  Although this research encompasses two main application areas, i.e., spacecraft and electric drives, the control techniques emerging are of general application.  Notable is forced dynamic control, a model based control technique, which, through nonlinear state feedback, yields a specified closed-loop performance that may or may not be nonlinear and takes external disturbances into account.

One of my contributions to linear control system design is a pair of formulae for the settling time of control systems with linear closed loop dynamics of arbitrarily high order and coincident closed loop poles, one for the 5% criterion and the other for the 2%criterion. This greatly simplifies and renders more effective the design method of pole assignment.  I have found this useful for state observer design as well as the design of various controllers.  I have been including the 5% formula in my lectures for several years and some of my European partners are now including it in their control systems teaching.  I therefore intend to include this in my book.

It was during the continuation of the spacecraft attitude control work at UEL that sliding modes entailing rapid switching on and off of the gas-jets were found to occur under certain conditions.  While this effect was undesirable regarding control valve wear and inefficiency, its potential for achieving closed loop performance independent of the dynamic parameters and external disturbances with continuous momentum exchange actuators was recognised.  This also triggered research into sliding mode control for other applications, the most significant being a) an advance in the state-of-the-art vector control methods for a.c. electrical drives (produced on my EU INCO-COPERNICUS project 960169) and b) a novel motion control system for loudspeakers undertaken under a DTI Smart Award.

More recently, I originated two new robust control techniques catering for plant model order uncertainty as well as external disturbances and parametric modelling errors.  One is Hyper-Sliding Mode Control, an extension of sliding mode control in which the loop closure using output derivatives beyond the usual rank minus one limit creates additional state variables taking part in the sliding mode that may not be plant state variables.  The other is Observer Based Robust Control using my plant model mismatch equivalent input premise in which the problem of controlling an unknown plant is converted to that of controlling a known real time model of an observer by state feedback with pole assignment.

My most recent research is in the direction of reducing the carbon footprint by means of a new, relatively simple and practicable motion controller for electric drives that minimises the wastage of energy due to friction in the driven mechanism, this being predicted to save terawatts of electrical energy consumption throughout industry if employed on a large scale.

I feel that the inclusion of the aforementioned new control techniques in my book as well as the standard ones will add significant value and stimulate the reader's interest and creativity.

During my time with the University of East London I have gained considerable experience of teaching, through creating and delivering three modules in control engineering: Control System Design (final year), Control Applications (final year) for the BEng (Hons) in Electrical and Electronic Engineering and Computer Control for the MSc in Computer Systems Engineering.  In addition, I have taught Control System Design and Control of Electric Drives to fourth year MSc students in Warsaw University of Technology, Poland, Wroclaw University of Technology, Poland, Silesian University of Technology, Gliwice, Poland and the University of Zilina, Slovakia.  These lectures have always been very well received, students often commenting on the enhanced understanding of control and interest in the subject that they have gained.  These fourth year studies are the most advanced taken on the 'continental' MSc courses, the fifth year being largely devoted to a substantial project.  I have experience supervising these projects (in the field of control of electric drives) as well as control projects in the final year of the BEng (Hons) in Electrical and Electronic Engineering and the MSc in Computer Systems Engineering at the University of East London.

I have continually taken advantage of student feedback to improve my handouts, which are in the form of books.  Over the years, many students have made positive comments about these handouts and my teaching style, having often encouraged me to writea book.  I intend to use the chapters of these handouts as the starting point in the development of my book.