# Modern Digital Control Systems

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## Book Description

This work presents traditional methods and current techniques of incorporating the computer into closed-loop dynamic systems control, combining conventional transfer function design and state variable concepts. Digital Control Designer - an award-winning software program which permits the solution of highly complex problems - is available on the CRC Press Website at http://www.crcpress.com/product/isbn/9780824789145. This edition: supplies new coverage of the Ragazzini technique; describes digital filtering, including Butterworth prototype filters; and more.

## Table of Contents

**Preface to the Second Edition**

**Preface to the First Edition**

**Introduction to Digital Control **

The Basic Idea of System Control

The Computer as a Control Element

Single-Loop Digital Control System

Why Digital Control Instead of Analog?

Data Converters

The State of Microprocessor Technology

Example of a Microprocessor-Based Thermal Controller

Summary

References

Linear Difference Equations and the z-Transform

Introduction

Scalar Differences Equations

z-Transform of Simple Sequences

Useful Theorems Associated with the z-Transform

Inversion of the z-Transform

Method of Partial Fraction Expansion

Solving Linear Difference Equations with the z-Transform

z-Domain Transfer Function and Impulse Response Sequence

Relation Between z-Plane Pole Locations and the Nature of Temporal Sequences

Frequency Response of Discrete-Data Systems

Relationship Between s- and z-Domain Poles of Sampled Functions

Discrete-Time Convolution Theorem (Theorem 2.4)

Final Value Theorem (Theorem 2.5)

Backdoor Approach to the Sampling Theorem

Summary

Problems

References

Elementary Digital Control System Design Using Transform Techniques

Introduction

Analog-to-Digital and Digital-to-Analog Convertors

Continuous-Time Plant Driven by a Zero-Order Hold with Sampled Output

Implementation of Digital Control Strategies

Closed-Loop Characteristic Equation

Conventional Digital Control Design Techniques

Conventional Control System Design Specifications

Elementary z-Domain Design Considerations

Effect of Disturbances on the Closed-Loop System

Concept of the Dynamic Controller or Compensator

Summary

Problems

References

Advanced Digital Control System Design Techniques Employing the z-Transform

Introduction

General PID Direct Digital Control Algorithm

Ziegler-Nichols Tuning Procedure for PID Control

Direct Design Method of Ragazzini

Summary

Problems

References

Digital Filtering and Digital Compensator Design

Introduction

Conventional Design Techniques for Digital Compensators

Approximate Numerical Integration Techniques

Another Look at the Bilinear Transformation

Bilinear Transformation with Prewraping

Matched Pole-Zero Technique

Zero-Order-Hold Approximation

Impulse Invariant Method

Using Prototypes to Design Digital Filters

z-Plane Design of Digital Compensators

Summary

Problems

References

State-Variable Representation in Digital Control Systems

Introduction

Continuous-Time State-Variable Problem

Solution of the State Equation

Matrix Exponential Series Approach

Solution of the Discrete State Equation

Transfer Functions from State Equation

Controllability

Observability

State-Variable Representation of Discrete Single-Input/Single-Output Systems

State-Variable Representation of Composite Control Systems

Summary

Problems

References

Quantization and Error Effects

Introduction

Quantization Errors (Type 1 Errors)

Response of a Discrete Transfer Function to Quantization Errors

Bound on the Output Magnitude (Bertramâ€™s Bound)

Multiplication Errors (Type 2 Errors)

Finite-World-Length Representation of Digital Filter Coefficients (Type 3 Errors)

Root Sensitivity Analysis

Summary

Problems

References

State-Space Approach to Control System Design

Introduction

State-Variable Feedback and System Design

Feedback Control with Incomplete State Information

Open-Loop Estimator or Observer

Asymptotic Prediction Estimator or Observer

Current Estimator or Observer

Reduced-Order Estimator or Observer

Algorithm for Gain Calculations for Single-Input Systems

Regulation with Nonzero Reference Inputs

Reference Inputs for Systems with Prediction Observers

Summary

Problems

References

Linear Discrete-Time Optimal Control

Introduction

Discrete Linear Regulator Problem

Cost of Control

Reciprocal Eigenvalues and Reciprocal Root Locus

Steady-State Regulator Problem by Eigenvector Decomposition

Optimal control About Nonzero Set Points

Suboptimal Control Employing Estimated State Feedback

Summary

Problems

References

Discrete-Time Stochastic Systems

Introduction

Probability and Random Variables

Expectation Operator and Statistical Moments

Dependence, Independence, and Conditional Probabilities

Joint Gaussian Random Variables

Linear Combinations and Linear Transformations of Gaussian Random Variables

Scalar Discrete Random Sequences

Markov and Purely Random Sequences

Vector Random Sequence

Random Sequences in Discrete-Time Dynamic Systems

Stationary Solutions

Summary

Problems

References

State Estimation in the Presence of Noise

Introduction

Derivation of the Discrete-Time Vector Kalman Filter

Steady-State Kalman Filter Gains by Eigenvector Decomposition

Summary

Problems

References

Discrete-Time Stochastic Control Systems

Introduction

Optimal Control with Random Disturbances and Noiseless Measurements

Control of Randomly Distributed Systems with Noise-Contaminated Measurements

Average Behavior of the Controlled System

Steady-State Control System Dynamics

Summary

Problems

References

Introduction to System Identification

Introduction

Least-Squares Technique

Transfer Function Estimation Using Least Squares

Weighted Least Squares

Recursive Least Squares

Effects of Noise

Summary

Problems

References

Appendix A: Tables and Properties of z-Transforms

A.1 Proof of the Complex Inversion Integral for the z-Transform

Appendix B: Algebraic Eigenvalue-Eigenvector Problem

B.1 Introduction

B.2 Statement of the Problem

B.3 Application of the Eigenvalue Problem to Discrete-Time Systems

B.4 Controllability and Observability

B.5 Cayley-Hamilton Theorem

References

Appendix C: Proof of the Matrix Inversion Lemma

Appendix D: Digital Control Designer

Index

## Reviews

". . .ideal for any student of classical theory. "

---Proceedings of the Institution of Mechanical Engineers Vol 209