Synthesis of Feedback Systems

Synthesis of Feedback Systems presents the feedback theory which exists in various feedback problems. This book provides techniques for the analysis and solution of these problems.

The text begins with an introduction to feedback theory and exposition of problems of plant identification, representation, and analysis. Subsequent chapters are devoted to the application of the feedback point of view to any system; the principal useful properties of feedback; the feedback control system synthesis techniques; and the class of two degree-of-freedom feedback configurations and synthesis procedures appropriate for such configurations. The final chapter considers how to translate specifications from their typical original formulation, to the language appropriate for detailed design.

The book is intended for engineers and graduate students of engineering design.

Preface

Chapter 1 Introduction to Feedback Theory

1.1 Formulation of the Feedback Problem

1.2 Classification of Feedback Problems

1.3 Coverage of the Book

1.4 The Problem of Plant Identification and Representation

1.5 Canonical Representation of a Plant

1.6 Signal-Flow Graph Representation and Analysis

1.7 Terminal Representation of Systems

1.8 More Detailed System Representations

1.9 Electric Circuit Models of Mechanical Systems

1.10 Combined Translational-Rotational Systems

1.11 Conditions under Which a Passive Linear Electrical Analog Is Possible

1.12 Electrical Analogs of the Gyroscope

1.13 Electric Models of Transducers

Chapter 2 Foundations of Linear Feedback Theory

2.1 Introduction

2.2 The Fundamental Feedback Equation and Signal-Flow Graph

2.3 The Representation of Two-Terminal Elements by Controlled Sources

2.4 The Subjective Nature of Feedback

2.5 Return Difference: The Bilinear Theorem and Its Exceptions

2.6 Null Return Difference

2.7 Active Impedances

2.8 Invariance of the Numerator of the Return Difference for All References

2.9 The Loaded Transistor Terminal Functions

2.10 Analysis of Transistor Feedback Circuits

2.11 Use of Two Controlled Sources

2.12 The Fundamental Feedback Matrix Equation

Chapter 3 Introduction to the Properties of Feedback

3.1 Introduction

3.2 Reduction in the Sensitivity of a System to Parameter Variation

3.3 Design Examples

3.4 Cost of Feedback

3.5 Sensitivity as a Function of Frequency

3.6 Sensitivity Function When Leakage Transmission Is Not Zero

3.7 Magnitude and Phase Sensitivities

3.8 Systems with Multiple Inputs

3.9 Sensitivity to Parameter Variation and Noise in Feedback Return Path

3.10 The Effect of Feedback in Nonlinear Systems

3.11 Classification of Reasons for Using Feedback

3.12 Effect of Feedback on System Response

Chapter 4 The Stability Problem in Feedback Systems with Plant Parameter Variations

4.1 Introduction to Root Locus

4.2 Rules for Constructing Root Loci

4.3 Sensitivity of Roots to Variation in k

4.4 Root Sensitivity in Terms of Open-Loop Poles and Zeros

4.5 Sensitivity of Roots of 1 + kB(s) — 0 to Variations in the Zeros and Poles of B(s)

4.6 Application of Root Sensitivity to Predistortion for Incidental Dissipation in Filters

4.7 Application of Root Sensitivity to Limit the Drift in Feedback System Poles

4.8 The Stability Problem from the Root Locus Point of View

4.9 The Nyquist Stability Criterion

4.10 Application of the Nyquist Criterion to Open-Loop Stable Feedback Systems; Bode Plots

4.11 Positive Feedback

4.12 Loop Shaping for Stability, with Parameter Variations — Single-Order Systems

4.13 Extension to Higher Order Systems

Chapter 5 Design of Feedback Control Systems with Single-Degree-of-Freedom Configuration

5.1 Comparison of Feedback Amplifiers with Feedback Control Systems

5.2 Distinction between the Feedback Problem, the Filter Problem, and the Control Problem

5.3 Introduction to Single-Degree-of-Freedom Feedback Control Design

5.4 Classical Control System Specifications — the Error Constants

5.5 Design for Simultaneous Achievement of Error Constant and Phase Margin by Means of Lag Compensation

5.6 Use of Lead Compensation to Achieve Specified Error Constant and Phase Margin

5.7 Loop Shaping for Simultaneous Achievement of Error Constant, Crossover Frequency, and Stability Margins

5.8 Optimization of Loop Transmission Function

5.9 Design Example

5.10 Correlation between System Frequency Response and Time Response

5.11 Relation between the Loop Transmission L and the System Transfer Function T

5.12 Synthesis from Pole-Zero Specifications of T(s)

5.13 Realization of Any Combination of Kv, Bandwidth, and Overshoot with a Pair of Poles and One Zero

5.14 Increase of Velocity Constant by Lag Compensation

5.15 Comparison of Lead and Lag Compensation Having the Same Kv, Bandwidth, and Overshoot

5.16 Determination of the Loop Transmission L(s); Validity of Canceling Plant Dynamics

5.17 Relative Merits of the Open-Loop Frequency Response Method and the T(s) Pole-Zero Method

5.18 The Price That Is Paid for a Dominant Type T(s)

5.19 T(s) Pole-Zero Method for More Complicated Dominant Pole-Zero Patterns

5.20 Root Locus Method

5.21 Design of High-Order System

5.22 Inadequacies of the Single-Degree-of-Freedom Configuration

Chapter 6 Design of Feedback Control Systems for Independent Control of Transmission and Sensitivity Functions

6.1 Configuration with Two Degrees of Freedom

6.2 Root Locus Synthesis to Control System Sensitivity to Variations in Plant Gain Factor

6.3 Root Locus s-Plane Synthesis for General Plant Parameter Variations

6.4 Location of the Far-Off Poles of L

6.5 Sensitivity of the Dominant Zeros of T(s)

6.6 Feasibility of Root Locus Sensitivity Design in High-Order Systems

6.7 Philosophy of the Frequency Response Approach to the Sensitivity Problem

6.8 Realization of Sensitivity Specifications — Frequency Response Method

6.9 Cost of Feedback, and Comparison of Two-Degree-of-Freedom Structures

6.10 The Problem of the Far-Off Poles

6.11 Design for Multiple Inputs

6.12 Design for Disturbance Attenuation Accompanied by Plant Parameter Variation

6.13 Analytical Specification of the Sensitivity Function

6.14 Achievable Benefits of Feedback in Two-Degree-of-Freedom Structure

Chapter 7 Fundamental Properties and Limitations of the Loop Transmission Function

7.1 Introduction

7.2 Mathematical Background

7.3 Resistance Integral Theorem and the Equality of Positive and Negative Feedback Areas

7.4 Real or Imaginary Part Sufficiency

7.5 Relation between Loop Transmission Lag Angle and Optimum Loop Transmission Function

7.6 Specification of F(s) from Its Real and Imaginary Parts in Different Frequency Ranges

7.7 The Ideal Bode Characteristic

7.8 A Different Kind of Optimum L(jω)

7.9 Minimum Phase Functions

7.10 Conditionally Stable Systems

7.11 Maximum Rate at Which | L(jω) | May Be Decreased for Conditionally Stable Systems

7.12 Loop Transmissions for Systems with Time Delay (Unconditional Stability)

7.13 Conditionally Stable Systems with Pure Time Delay

7.14 Systems with Unstable Loop Transmissions

7.15 Systems with Combined Positive and Negative Feedback; Zero-Sensitivity Systems

7.16 Summary

Chapter 8 Advanced Topics in Linear Feedback Control Theory

8.1 Introduction

8.2 Design of Multiple-Loop Systems for Disturbance Attenuation (Cascade Plants)

8.3 Multiple-Loop Design for Noncascade Plants

8.4 Comparison of Single-Loop and Multiple-Loop Systems for Insensitivity to Parameter Variation

8.5 Design for Parameter Variations in the Two-Loop System

8.6 Extension to More Than Two Loops

8.7 Effect of Finite N1

8.8 Multiple-Loop Systems with Parallel Plants

8.9 Application of Parallel Plants in Design (to Conditionally Stable Systems and Nonminimum Phase Plants)

8.10 Synthetic Multiple-Loop Feedback Systems

8.11 Control of Effect of Parameter Variations by Shaping of Root Loci

8.12 Design of L(s) with a Single Zero

8.13 Design of L(s) with Two Zeros

8.14 Control of Effect of Parameter Variations by the Frequency Response Method

8.15 The Rate of Parameter Variations

8.16 Abrupt Parameter Variations

8.17 Design Example — Time-Varying Plant

8.18 Sensitivity of the Transient Response

8.19 Application of Linear Techniques to Nonlinear Plants

8.20 Practical Design Techniques for Feedback Control Systems with Nonlinear Plants

8.21 Fundamental Limitations in Adaptive Capabilities of Linear Time-Invariant Feedback Systems

8.22 The Role of Plant Identification in Adaptive Systems

Chapter 9 Problems in the Specification of System Functions

9.1 Introduction

9.2 The Physical Capacity of the Plant

9.3 Limitations of Design Based on Step or Ramp Response

9.4 S-Plane Design Based on the Typical Input Signal

9.5 Effect of Plant Parameter and Input Signal Variations

9.6 Specification of Disturbance Attenuation

9.7 Inputs Bounded in Slope and/or Magnitude

9.8 Choice of T(s) from the Statistical Properties of the Input

9.9 Energy Density Spectra and Correlation Functions

9.10 Optimum but Unrealizable Response Function

9.11 Optimum Realizable Response Function

9.12 Derivation of the Realizable Optimum Filter Function

9.13 Optimum Filter When Message and Noise Signals Are Related

9.14 Conditions under Which the Optimum Linear Time-Varying Filter Degenerates into a Time-Invariant Filter

9.15 Applications to Joint Filter and Feedback Problems

9.16 Optimization with Nonminimum Phase Plant

9.17 Optimization under Constraints

9.18 The Optimum Filter Problem in Sampled-Data Systems

9.19 Optimum Filter for Minimizing the Continuous Squared Error in Sampled-Data Systems

Chapter 10 Synthesis of Linear, Multivariable Feedback Control Systems

10.1 Introduction

10.2 Clarification of Design Objectives and System Constraints

10.3 Role of System Configuration

10.4 Principal Steps in the Design of Multivariable Systems

10.5 The System Transfer Matrix

10.6 Effect of Parameter Variations on the System Transmission Matrix

10.7 Comparison of Diagonal and Antidiagonal Loop Transmission Matrices

10.8 Loop Shaping for Stability with Plant Parameter Variation

10.9 Design Example with a Diagonal L

10.10 Example — Stability Problem with an Antidiagonal L

10.11 Design Examples

10.12 Noninteraction

10.13 Loop Shaping for Stability — The General Second-Order Case

10.14 The General Loop Shaping Problem for Higher Order Multivariable Systems

10.15 Design for Rejection of Disturbances

10.16 The Feedback Problem for Plants with More Outputs than Inputs

10.17 Discussion of the Feedback Mechanism in Multivariable Systems

10.18 Multivariable Plants with Internal Variables Available for Feedback

Chapter 11 Sampled-Data Feedback Control Systems

11.1 Definition of Sampled-Data Systems

11.2 The Importance of Sampled-Data Systems

11.3 Effect of Sampling on Achievable System Accuracy

11.4 z Transforms

11.5 z Transforms and the Pulse Transfer Function

11.6 The Fundamental Feedback Equation for Sampled-Data Systems

11.7 Intersample Behavior and Inverse z Transformation

11.8 Elementary Design of Sampled-Data Feedback Control Systems

11.9 Design from Selected Form of L(s)

11.10 Elementary Design by the Frequency Response Method

11.11 Loop Transmission Function in the w Plane

11.12 Use of Pulsed Networks as Compensators

11.13 The Digital Controller

11.14 Sensitivity in the Single-Degree-of-Freedom Configuration

11.15 Sensitivity Limitations in the Two-Degree-of-Freedom Structure

11.16 Limitations on Achievable Sensitivity Reduction — Use of Multirate Sampling

11.17 Design for Attenuation of Disturbances

Chapter 12 Approximation Methods in Feedback System Design

12.1 Introduction

12.2 Time Domain Synthesis — The Method of Moments

12.3 Comments on the Method of Moments

12.4 The z-Transform Method

12.5 Guillemin's Fourier Series Method

12.6 Expansion in Orthogonalized Exponentials

12.7 Rational Function Approximation of Frequency Response Data

12.8 Numerical Convolution

12.9 Discussion of Technique and Application to Inverse Convolution

12.10 Numerical Solution of Differential Equations

12.11 Approximate Inverse Transformation

12.12 Inverse Transformation with Unfactored Transforms

12.13 Inverse Transformation — A Graphical Method

12.14 Numerical Fourier Analysis

Appendix - Active RC Realization of Transfer Functions with Complex Poles

Index