Application of Model Order Reduction Techniques in PID controller Design

Table of Contents

1 INTRODUCTION

1.1 LOWER ORDER MODEL FORMULATION

1.2 NEED FOR LOWER ORDER MODEL FORMULATION

1.3 SURVEY OF LOWER ORDER MODEL FORMULATION SCHEMES

1.4 SCOPE OF THE THESIS WORK

1.5 OUTLINE OF THE THESIS

2 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION AND DESIGN OF PID CONTROLLER FOR SINGLE INPUT SINGLE OUTPUT LINEAR TIME INVARIANT CONTINUOUS SYSTEMS

2.1 INTRODUCTION

2.2 PROBLEM DEFINITION

2.3 INITIAL LOWER ORDER APPROXIMANTS FOR CONTINUOUS SYSTEM USING ADJUNCT POLYNOMIAL APPROACH

2.3.1 Methodology

2.3.2 Choice of Second Order Model

2.3.3 Proof of Adjunct Polynomial Approach

2.4 PARTICLE SWARM OPTIMIZATION

2.4.1 General Particle Swarm Optimization

2.4.2 Modified Particle Swarm Optimization Algorithm

2.5 PROPOSED ALGORITHM FOR LOWER ORDER MODEL FORMULATION OF SINGLE INPUT SINGLE OUTPUT CONTINUOUS SYSTEMS

2.6 ILLUSTRATIONS

2.7 SIGNIFICANTS OF LOWER ORDER FORMULATED MODELS IN PID CONTROLLER DESIGN

2.7.1 Aspects of PID Controller

2.7.2 Issues of PID Controller Design

2.8 PROPOSED ALGORITHM FOR DESIGN OF CONTINUOUS PID CONTROLLER USING FORMULATED LOWER ORDER MODELS

2.9 ILLUSTRATIONS

2.10 SUMMARY

3 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION AND DESIGN OF PID CONTROLLER FOR SINGLE INPUT SINGLE OUTPUT LINEAR TIME INVARIANT DISCRETE SYSTEMS

3.1 INTRODUCTION

3.2 PROBLEM DEFINITION

3.3 INITIAL LOWER ORDER APPROXIMANTS FOR DISCRETE SYSTEM USING ADJUNCT POLYNOMIAL APPROACH

3.4 PROPOSED ALGORITHM FOR LOWER ORDER MODEL FORMULATION OF SINGLE INPUT SINGLE OUTPUT DISCRETE SYSTEMS

3.5 ILLUSTRATIONS

3.6 DISCRETE PID CONTROLLER DESIGN

3.7 PROPOSED ALGORITHM FOR DESIGN OF DISCRETE PID CONTROLLER USING FORMULATED LOWER ORDER MODELS

3.8 ILLUSTRATIONS

3.9 SUMMARY

4 DESIGN OF STATE FEEDBACK CONTROLLER AND STATE SPACE OBSERVER FOR LINEAR TIME INVARIANT SYSTEMS USING LOWER ORDER FORMULATED MODELS

4.1 INTRODUCTION

4.2 ASPECTS OF STATE FEEDBACK CONTROLLER

4.2.1 Design of State Feedback Controller

4.3 PROPOSED PROCEDURE FOR CONTINUOUS STATE FEEDBACK CONTROLLER DESIGN

4.4 PROPOSED PROCEDURE FOR DISCRETE STATE FEEDBACK CONTROLLER DESIGN

4.5 ASPECTS OF STATE SPACE OBSERVER

4.5.1 Design of State Space Observer

4.6 PROPOSED PROCEDURE FOR CONTINUOUS STATE SPACE OBSERVER DESIGN

4.7 PROPOSED PROCEDURE FOR DISCRETE STATE SPACE OBSERVER DESIGN

4.8 SUMMARY

5 DESIGN OF SUB-OPTIMAL CONTROL FOR LINEAR TIME INVARIANT SYSTEMS USING LOWER ORDER FORMULATED MODELS

5.1 INTRODUCTION

5.2 PROBLEM DEFINITION

5.3 PROPOSED PROCEDURE FOR SUB-OPTIMAL CONTROL DESIGN USING FORMULATED LOWER ORDER MODEL

5.4 ILLUSTRATIONS

5.5 SUMMARY

6 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION OF MULTI INPUT MULTI OUTPUT LINEAR TIME INVARIANT SYSTEMS

6.1 INTRODUCTION

6.2 PROBLEM DEFINITION

6.3 PROPOSED ALGORITHM FOR LOWER ORDER MODEL FORMULATION OF MULTI INPUT MULTI OUTPUT CONTINUOUS SYSTEMS

6.4 ILLUSTRATION

6.5 PROPOSED ALGORITHM FOR LOWER ORDER MODEL FORMULATION OF MULTI INPUT MULTI OUTPUT DISCRETE SYSTEMS

6.6 ILLUSTRATION

6.7 SUMMARY

7 CONCLUSION AND FUTURE SCOPE

7.1 CONCLUSION

7.2 SUGGESTION FOR FUTURE SCOPE

APPENDIX 1 PROGRAM FOR LOWER ORDER MODEL FORMULATION OF LTICS

APPENDIX 2 PROGRAM FOR LOWER ORDER MODEL FORMULATION OF LTIDS

APPENDIX 3 PROGRAM FOR DESIGN OF STATE FEEDBACK CONTROLLER AND STATE SPACE OBSERVER FOR LTIS

APPENDIX 4 PROGRAM FOR DESIGN OF OPTIMAL AND SUB-OPTIMAL CONTROL FOR LTIS

APPENDIX 5 PROGRAM FOR LOWER ORDER MODEL FORMULATION OF MIMO LTIS

Objectives and Research Scope

The primary objective of this thesis is to develop an algebraic approach for the formulation of lower order models of linear time invariant systems to mitigate computational complexities associated with higher order systems. The research investigates techniques to maintain the essential characteristics of higher order models in lower order approximations, enabling efficient design of PID controllers, state feedback controllers, and state space observers across continuous and discrete systems.

• Formulation of lower order models for Linear Time Invariant Continuous (LTICS) and Discrete (LTIDS) systems.
• Development of modified Particle Swarm Optimization (MPSO) techniques to enhance model order reduction.
• Design and validation of continuous and discrete PID controllers using formulated lower order models.
• Implementation of state feedback controllers and state space observers based on reduced order models.
• Extension of methodologies to Multi-Input Multi-Output (MIMO) systems for both continuous and discrete domains.

Excerpt from the Book

Summary of Chapters

1 INTRODUCTION: This chapter provides an overview of control system modeling and discusses the necessity of lower order model formulation to manage computational complexities.

2 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION AND DESIGN OF PID CONTROLLER FOR SINGLE INPUT SINGLE OUTPUT LINEAR TIME INVARIANT CONTINUOUS SYSTEMS: This chapter presents the adjunct polynomial approach and Modified Particle Swarm Optimization (MPSO) for reducing continuous system models and designing associated PID controllers.

3 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION AND DESIGN OF PID CONTROLLER FOR SINGLE INPUT SINGLE OUTPUT LINEAR TIME INVARIANT DISCRETE SYSTEMS: This chapter extends the model formulation and PID controller design strategies specifically to discrete systems.

4 DESIGN OF STATE FEEDBACK CONTROLLER AND STATE SPACE OBSERVER FOR LINEAR TIME INVARIANT SYSTEMS USING LOWER ORDER FORMULATED MODELS: This chapter details procedures for designing state feedback controllers and observers using the previously formulated lower order models.

5 DESIGN OF SUB-OPTIMAL CONTROL FOR LINEAR TIME INVARIANT SYSTEMS USING LOWER ORDER FORMULATED MODELS: This chapter evaluates the reliability of lower order models by comparing optimal and sub-optimal control cost functions.

6 PROPOSED APPROACH FOR LOWER ORDER MODEL FORMULATION OF MULTI INPUT MULTI OUTPUT LINEAR TIME INVARIANT SYSTEMS: This chapter introduces techniques for extending the lower order formulation approach to multivariable (MIMO) continuous and discrete systems.

7 CONCLUSION AND FUTURE SCOPE: This chapter summarizes the research findings and provides potential directions for future study.

Keywords

Lower order model, Linear Time Invariant System, Model reduction, Particle Swarm Optimization, PID controller, State feedback controller, State space observer, MIMO, Continuous system, Discrete system, Integral square error, Control engineering, Lyapunov equation, Pole placement, Optimization

Frequently Asked Questions

What is the core focus of this thesis?

The thesis focuses on developing algebraic techniques to formulate lower order models for higher order linear time invariant systems, aiming to reduce computational complexity while preserving system characteristics.

Which systems are covered in the scope of this work?

The work covers Single-Input Single-Output (SISO) and Multi-Input Multi-Output (MIMO) linear time invariant systems in both continuous and discrete time domains.

What is the primary goal of the proposed methods?

The primary goal is to provide a simplified model that captures the essential dynamic features of a complex higher-order system, which can then be utilized for controller and observer design.

Which scientific methodology is central to the thesis?

The thesis utilizes an adjunct polynomial approach combined with a Modified Particle Swarm Optimization (MPSO) algorithm to achieve stable and accurate model order reduction.

What topics are addressed in the main sections?

The main sections cover model reduction strategies, PID controller design, state feedback controller and state space observer design, sub-optimal control implementation, and extension to MIMO systems.

What are the primary characteristics of the models developed?

The models are characterized by their ability to minimize integral square error, maintain transient and steady-state gain ratios, and guarantee stability for stable higher-order systems.

How is the reliability of the lower order models verified?

Reliability is verified through comparative studies using Integral Square Error (ISE) metrics and by comparing the cost functions of optimal control of higher-order systems against sub-optimal control of the formulated lower-order models.

Does the thesis provide practical implementations?

Yes, the thesis includes extensive numerical illustrations and provides sample program code in the appendices for implementing the proposed methodologies.