# Control and Dynamic Systems

### Advances in Theory and Applications

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Control and Dynamic Systems: Advances in Theory and Applications reviews progress in the field of control and dynamic systems theory and applications, with emphasis on filtering and stochastic control in dynamic systems. Topics include linear and nonlinear filtering techniques; concepts and methods in stochastic control; and discrete-time optical stochastic observers. The theory of disturbance-accommodating controllers is also presented. Comprised of nine chapters, this volume begins with an overview of filtering and stochastic control in dynamic systems, followed by a discussion on linear and nonlinear filtering techniques. The reader is then introduced to concepts and methods in stochastic control, as well as the innovations process and its applications to sensitivity analysis and system identification. Subsequent chapters focus on the status of observer theory and its major results as applied to discrete-time linear systems; the properties of the class of discrete-time Riccati equations that arise in the filtering problem; and the theory of disturbance-accommodating controllers. The identification of noise characteristics in a Kalman filter and estimation of adaptive minimum variance in discrete-time linear systems round out the book. This monograph will be useful to practicing technologists and research workers interested in filtering and stochastic control in dynamic systems.

## Table of Contents

Contributors

Preface

Contents of Previous Volume

An Overview of Filtering and Stochastic Control in Dynamic Systems

I. General Stochastic Control Problem

A. Definition of the Basic Problem

B. Some Types of Control Policies

II. General Solution of the Optimal Stochastic Control Problem

A. A Recursive Solution of the Control Problem

B. Solving the Nonlinear Filtering Problem

C. Practical Considerations in Determining a Stochastic Control Policy

III. Linear Quadratic Systems and Extensions

A. The linear Recursive FilteringProblem

B. The Optimal,Closed-Loop, Stochastic Control Policy

IV. Summary of Proposed Algorithms

A. Nonlinear Filtering Algorithms

B. Stochastic Control Algorithms

References

Linear and Nonlinear Filtering Techniques

I. Introduction

II. Linear Filtering

III. System Modeling

IV. Suboptimal Filter Design and Sensitivity Analysis

V. Partitioned Filter Approach

VI. Data Prefiltering

VII. Square Root Techniques

VIII. Divergence of the Filter

IX. Nonlinear Filtering

X. Concluding Remarks

References

Concepts and Methods in Stochastic Control

I. Introduction

II. Classes of Stochastic Control Policies and Some Properties of the Control

A. Formulation of the Stochastic Control Problem

B. Classes of Stochastic Control Policies

C. The Dual Effect of the Control, Probing, and Caution

III. Optimal Stochastic Control

IV. The Optimal Control for a Class of Systems

A. The Certainty Equivalence Result and Its Connection with the Dual Effect

B. Discussion and Examples

V. A Stochastic Closed-Loop Control Method for Nonlinear Systems

A. Formulation of the Problem

B. The Algoritiun

C. Simulation Results

VI. A Stochastic Resource Allocation Problem

A. Formulation of the Problem

B. The Algorithm

C. Simulation Results

VII. Conclusions

Appendix A. Proof of the Connection Between the Certainty Equivalence and the Dual Effect

Appendix B. The Closed-Loop Optimization of the Cost-To-Go

References

The Innovations Process with Applications to Identifications

I. Introduction

A. Problem Definition

B. Solution Overview

II. Mathematical Specifications and Background

A. Objectives and Restrictions

B. Method of Approach

III. Sensitivity Analysis

A. The Error Model

B. Behavior of the Error Mean

C. Behavior of the Error Covariance

D. Error Correlation

E. Behavior of the Innovations

F. Summary

IV. System Identification

A. Introduction and Assumptions

B. Limiting Behavior

C. Some Necessary and Sufficient Conditions

D. Some Steady State Considerations

E. Variance and Correlation of Residuals

F. Stochastic Approximation

G. The Partial Derivatives

V. Simulation Results

A. The Boozer Example

B. The Ohap Example

C. Summary

References

Discrete-Time Optical Stochastic Observer

I. Introduction

II. Definition of Discrete Observer for StochasticSystems

III. Construction of a Reduced-Order Observer

IV. An Alternate Reduced-Order Observer Algorithm

V. limiting Cases of the Reduced-Order Observer Solution

A. Minimal-Order Observer

B. Kalman Filter

C. Some Perfect Measurements

VI. Computational Advantages of Reduced Order Observers

VII. An Optimal Continuous-Time Observer Solution

VIII. Concluding Remarks

References

Discrete Riccati Equations: Alternative Algorithms,Asymptotic Properties,and System Theory Interpretations

I. Introduction

II. Square Root Algorithms and the Riccati Equation

A. The Time-Invariant, Zero Terminal, Cost Problem

B. The General Time-Variable Problem

C. Square Root Algorithms 33

D. Structure Algorithms

E. Equivalent Optimization Problems

III. System Structure

A. Observability

B. Invertibility and Detectability

C. Matrix Characterizations

IV. Singular Riccati Equations

A. Asymptotic Properties

B. Reduced Order Riccati Equations

References

Theory of Disturbance-Accommodating Controllers

I. Introduction

II. The Waveform-Mode Description of Realistic Disturbances

III. The Waveform-Mode Characterization Versus the Statistical Characterization

IV. State Models for Disturbances with Waveform Structure

A. Some Examples of State Models for Common Disturbances

B. Waveform Description of Unfamiliar Disturbances

C. "Unfamiliar Disturbances" Arising from Modeling Errors in System Parameters

D. Waveform Description of State-Dependent Disturbances

E. Waveform Description with Linear State Models

F. Experimental Determination of Linear State Models for Disturbances

G. Noise Combined with Disturbances Having Waveform Structure

H. Disturbance Waveform Models Equations (39) and (40) Versus Coloring Filters for White Noise

V. Design of Disturbance-Accommodating Controllers for Stabilization, Regulation, and Servo Tracking Control Problems

A. The Class of Systems and Disturbances to Be Considered

B. Practical Constraints on the Structure of Disturbance-Accommodating Controllers

C. Description of the Stabilization, Regulation and Servo-Tracking Control Problems

D. Philosophies of Disturbance Accommodation in Control Problems

E. The Notion of State Constructors (Observers) for Signals with Waveform Structure

F. Design of Disturbance-Absorbing Controllers

G. Design of Disturbance-Minimization Controllers

H. Design of Disturbance-Utilization Controllers

I. Design of Multimode Disturbance-Accommodating Controllers

J. Transfer Function Interpretation of Disturbance-Accommodating Controllers

VI. Conclusions

References

Appendix

Identification of the Noise Characteristics in a Kaiman Filter

I. Introduction

A. Background

B. Outline

II. System Description

III. Moment System Formulation

A. Mean State Model

B. Mean Measurement Model

C. Mean System Statistics

D. Covariance State Model

E. Covariance Measurement Model

F. Covariance System Statistics

IV. Estimates of the Moments

A. Estimates of the Means

B. Estimates of the Covariance Parameters

C. Adaptive Estimates of Both Moments

D. Comparison of Adaptive Techniques

V. Correlated Moment System Measurement Noise

A. Nonwhite Moment System Measurement Noise

B. Weighted Least Squares Estimates for No System State Noise

C. Linear Minimum Variance Estimates for No System State Noise

D. Weighted Leasted Squares Estimates for the General Case

VI. Conclusions

References

Appendix A

Appendix B

Appendix C

Adaptive Minimum VarianceEstimationin Discrete-Time LinearSystems

I. Introduction

II. Adaptive Filter Algorithm

III. Hypothesis Test for Time Correlation of Residuals

IV. Summary of the Adaptive Algorithm

V. Convergence of Algorithm

VI. Example

A. Description of System

B. Determination of Adaptive Filter Parameters

C. Step Size Control

D. Practical Considerations

E. Results of ComputerSimulation

VII. Conclusions

Appendix

References

Subject Index

## Product details

- No. of pages: 648
- Language: English
- Copyright: © Academic Press 1976
- Published: March 28, 1976
- Imprint: Academic Press
- eBook ISBN: 9781483191249