The analysis and synthesis of higher order models are complicated and are not desirable on economic and computational considerations. To circumvent the difficulties, lower order model formulation techniques are utilized to find a lower dimensional approximant for the original higher order model. The obtained lower order model preserves the characteristics of the original higher order model.
Firstly, the linear time invariant single input single output continuous systems are considered to investigate the efficiency of the proposed lower order model formulation approach. For this, the given linear time invariant higher order system represented in the form of transfer function is adopted to get adjunct lower order transfer function and its coefficients are tuned suitably with the help of modified particle swarm optimization along with transient and steady state gain adjustments. The lower order model is formed on an error based criterion. Moreover, the formulated second order models are used to design the continuous PID controllers.
Secondly, the single input single output linear time invariant discrete systems are dealt for model order formulation with the help of proposed approach. Discrete PID controllers are designed by employing the proposed formulated lower order model and it retains the desired performance specifications. The lower order models minimize the computational complexities for the process of output stabilization compared with higher order models. The proposed approach is direct and simple in approach for linear time invariant discrete systems.
Thirdly, certain procedures are proposed for designing the state feedback controller and state space observer of linear time invariant continuous and discrete systems. Further, the lower order model formulation approach for single input single output systems is extended to multi input multi output linear time invariant continuous and discrete systems. The analysis of the discrete system is carried out directly without applying any linear or bilinear transformations, which reduces computational complexities. This approach guarantees an absolutely stable lower order model if the considered higher order system is stable in nature. The proposed methodology extracts a second order model which has a better approximation compared to models obtained due to other methods.
Algorithms are also presented for all the contributions provided in the thesis with illustrations and results.
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
1.1 LOWER ORDER MODEL FORMULATION
Lower order model formulation is a mathematical computational process to find a lower dimensional approximation for the original higher order model. Lower order models are very much useful in designing controllers and compensators which in turn are used for stabilization of output response in a given control system. In general, a given LTIS can be represented in the form of open loop transfer function having numerator polynomial and denominator polynomial in s-variable. The degree of numerator polynomial should be less than or equal to the degree of denominator polynomial. During the course of designing a controller or a compensator, if the original higher order system is employed, the computations involved in estimating the parameters of controller or
compensator employing any procedure in time domain or frequency domain will be more in general.
To evade this situation, a lower order model of the given system is formulated which will almost maintain the characteristics of original system; this is done with the help of an error criterion (performance index or integral square error). After design, the controller or compensator is attached with the original system to observe the stabilization process; it should be noted that the stabilization involves minimizing of oscillations in the system output for a given input. Sometimes the periodic response is made into an aperiodic response.
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.
- Citation du texte
- Dr G. Sugumaran (Auteur), 2012, Application of Model Order Reduction Techniques in PID controller Design, Munich, GRIN Verlag, https://www.grin.com/document/1164249
