Linear and Non-Linear System Theory

✍ Scribed by T Thyagarajan, D Kalpana

• Publisher: CRC Press
• Year: 2020
• Tongue: English
• Leaves: 435
• Edition: 1
• Category: Library

✦ Synopsis

Linear and Non-Linear System Theory focuses on the basics of linear and non-linear systems, optimal control and optimal estimation with an objective to understand the basics of state space approach linear and non-linear systems and its analysis thereof. Divided into eight chapters, materials cover an introduction to the advanced topics in the field of linear and non-linear systems, optimal control and estimation supported by mathematical tools, detailed case studies and numerical and exercise problems. This book is aimed at senior undergraduate and graduate students in electrical, instrumentation, electronics, chemical, control engineering and other allied branches of engineering.

Features

• Covers both linear and non-linear system theory

• Explores state feedback control and state estimator concepts

• Discusses non-linear systems and phase plane analysis

• Includes non-linear system stability and bifurcation behaviour

• Elaborates optimal control and estimation

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Table of Contents
Preface
Acknowledgements
Authors
1 Introduction
Determinants
Second-Order Determinant
Third-Order Determinant
Minor
Cofactor
Properties of Determinants
Matrices
Definition
Order of a Matrix
Row Matrix
Column Matrix
Square Matrix
Null Matrix
Principle Diagonal
Diagonal Matrix
Unit Matrix or Identity Matrix
Scalar Matrix
Upper Triangular Matrix
Lower Triangular Matrix
Transpose of a Matrix
Symmetric Matrix
Skew Symmetric Matrix
Singular Matrix
Adjoint of a Matrix
Inverse of a Matrix
Equality of Matrix
Addition of Matrices
Subtraction of Matrices
Multiplication of Matrices
Conjugate of a Matrix
Hermitian Matrix
Skew Hermitian Matrix
Rank of a Matrix
Definition of Transfer Function
Limitations of Transfer Function Approach
Introduction to State Space Analysis
Input and Output Variables
State Model
Review of State Models
Non-Uniqueness of State Model
2 State Space Approach
Role of Eigen Values and Eigen Vectors
How to Find Eigen Vectors?
Free and Forced Responses
Properties of State Transition Matrix
Evaluation of State Transition Matrix
Minimal Realization
Minimal Realization Using Transfer Function Matrix
Non-Minimal Realization
Non-Minimal Realization Using Transfer Function Matrix
Balanced Realization
3 State Feedback Control and State Estimator
Concept of Controllability and Observability
Controllability
a) State Controllability
Condition for Complete State Controllability in the s-Plane
Output Controllability
Uncontrollable System
Stabilizability
Observability
Complete Observability
Condition for Complete Observability in the s-Plane
Detectability
Kalman’s Tests for Controllability and Observability
State Space Representation in Canonical Forms
Controllable Canonical Form
Observable Canonical Form
Diagonal Canonical Form
Jordan Canonical Form
State Feedback Control (Pole Placement Technique)
Determination of State Feedback Gain Matrix (K(f))
State Observers
Full-Order State Observers
Reduced-Order State Observers
Minimum-Order State Observers
Mathematical Model of an Observer
Determination of State Observer Gain Matrix (K(0))
4 Non-Linear Systems and Phase Plane Analysis
Characteristics of Non-Linear Systems
Jump Resonance
Types of Nonlinearities
Saturation
Deadzone
Backlash
Friction
Describing Function Fundamentals
Describing Function of Deadzone
Describing Function of Saturation Nonlinearity
Describing Function of Deadzone and Saturation
Describing Function of On–Off Controller with a Deadzone
Describing Function of Backlash Nonlinearity
Describing Function of Relay with Deadzone and Hysteresis
Phase Plane Analysis
Phase Plane
Phase Trajectory
Phase Portrait
Phase Trajectory of a Linear Second-Order Servo System
Advantages of Phase Plane Analysis
Disadvantages of Phase Plane Analysis
Singular Point
Behaviour of Trajectories in the Vicinity of Singular Point
Stable Focus Singular Point
Unstable Focus Singular Point
‘Vortex’ or ‘Centre’ Singular Point
Stable Node Singular Point
Unstable Node Singular Point
Unstable System with One Negative Real Root and One Positive Real Root
Phase Plane Analysis (Analytical Approach – Illustration)
Isocline Method [Illustration]
Procedure for Construction of Phase Trajectories Using Isocline Method
Construction of Phase Trajectories Using Delta Method (Graphical)
Limit Cycle [For Non-Linear Systems]
Limit Cycle [For Linear Systems]
5 Stability of Non-Linear Systems
Concept of Stability
Stability Conditions for a Linear Time-Invariant System
Equilibrium Point
Stability Analysis of Non-Linear System Using Describing Function Method
Assumptions
Stable and Unstable Limit Cycles
Case 1: Stable Limit Cycle
Case 2: Unstable Limit Cycle
Procedure for Investigating Stability of a Given Non-Linear System
Lyapunov’s Stability Criterion
Procedure to Comment on the Stability of Non-Linear System Using Lyapunov Stability Criterion
Krasovskii Method
Procedure to Deduce the Stability of Non-Linear System Using Krasovskii’s
Method
Variable Gradient Method
Popov’s Stability Criterion
Description of the Linear Part G(s)
Description of the Non-Linear Part Φ(.)
Popov’s Criterion
Procedure
Circle Criterion
Circle Criterion
Inference
6 Bifurcation Behaviour of Non-Linear Systems
Bifurcation Theory
Bifurcation Analysis
Bifurcation in One Dimension
Common Types of Bifurcation
Saddle Node Bifurcation
Transcritical Bifurcation
Pitchfork Bifurcation
Bifurcation in Two Dimension
Supercritical Hopf Bifurcation
Subcritical Hopf Bifurcation
Lorentz Equation
Solutions for Lorentz Equations
Stability Analysis of Lorentz Equations
Case 1: For Critical Point, P[sub(1)] (0,0,0)
Case 2: For Critical Point, P[sub(2)] ( β(γ −1), β(γ −1), (γ −1))
Chaos Theory
Definitions in the Study of Chaos Theory
Characteristics of Chaos
Application of Chaos Theory
Chaos in Chemical Process
Chaos in PI-Controlled CSTR
7 Optimal Control
Introduction
Classical Control versus Optimal Control
Classical Control
Optimal Control
Objective
Case 1: Linear System
Case 2: Non-Linear System
Function
Single Variable
Two Independent Variables
Maxima and Minima for Functions of Two Variables
Maximum Value
Minimum Value
Extreme Values
Necessary Conditions
Sufficient Conditions
Properties of Relative Maxima and Minima
Definition of Stationary Values
Single Variable
Two Variables
Conditions for Maximum or Minimum Values
Sufficient Conditions of Maximum and Minimum Values
Procedural Steps for Solving Maxima and Minima of Function ƒ(x)
Functional
Increment of a Function
Increment of a Functional
Differential of a Functional
Variation of a Functional
Sufficient Condition for a Minimum
Performance Measures in Optimal Control
Minimum Time Criterion
Minimum Energy Criterion
Minimum Fuel Criterion
State Regulator Criterion
Output Regulator Criterion
Servo or Tracking Criterion
Time-Varying Optimal Control
Continuous Time-Varying Optimal Regulator
LQR Optimal Regulator Design Using Hamiltonian–Jacobi Equation
Objective
Case 1: Finite Horizon
Assumptions
Case 2: Infinite Horizon
Assumptions
Procedural Steps for Hamilton–Jacobi Method for Solving LQR Optimal Control Problem
LQR Optimal Regulator Design Using Pontryagin’s Principle
Objective
Case 1: Finite Horizon
Case 2: Infinite Horizon
Assumptions
Procedural Steps for Pontryagin’s Principle Approach for Solving LQR Optimal Control Problem
LQR Steady-State Optimal Regulator
Assumptions
Objective
Symmetric Property of Matrix Riccati Equation
Numerical Solution of Matrix Riccati Equation
Direct Integration Method
Advantages of Direct Integration Method
Disadvantages of Direct Integration Method
Negative Exponential Method
Advantages of Negative Exponential Method
Disadvantages of Negative Exponential Method
Lyapunov Method
Advantages of Lyapunov Method
Disadvantages of Lyapunov Method
Discrete LQR Optimal Regulator Design
Objective
Case 1: Finite Horizon
Assumptions
Case 2: Infinite Horizon
Procedural Steps for Solving Discrete LQR Optimal Control Problem
Linear Quadratic Tracking (LQT) Problem
Continuous Linear Quadratic Tracking (LQT) Controller Design
Objective
Case 1: Finite Horizon
Assumptions
Case 2: Infinite Horizon
Procedural Steps for Solving LQT Optimal Control Problem
Discrete LQT Optimal Regulator Design
Objective
Case 1: Finite Horizon
Assumptions
Procedural Steps for Solving Discrete LQT Optimal Control Problem
Linear Quadratic Gaussian (LQG) Control
Continuous Time Linear Quadratic Gaussian (LQG) Control
Objective
Assumptions
Case 1: Finite Horizon
LQR Design
LQE Design
Case 2: Infinite Horizon
Assumptions
Procedural Steps for Solving LQG Optimal Control Problem
8 Optimal Estimation
Statistical Tools
Mean (X)
Standard Deviation (σ)
2 Variance (σ[sup(2)])
Covariance
Mean, Variance and Covariance and Cross-Covariance of Stochastic Process
Case 1: Sample Random Variable (x)
Case 2: Random Vector (x)
Case 3: Continuous-Time Vector Stochastic Process
Case 4: Discrete-Time Vector Stochastic Process
Gaussian Noise
Kalman Filter
Advantages of Kalman Filter
Disadvantages of Kalman Filter
Applications of Kalman Filter
Continuous-Time Kalman Filter Algorithm
Steps Involved in the Continuous-Time Kalman Filter Design
Flowchart for Continuous-Time Kalman Filter Design
State Estimation in Linear Time-Invariant Systems Using Kalman Filter
Discrete-Time Kalman Filter Design for a Multi-Dimension Model
Assumptions Used in the Design of Kalman Filter
Steps Involved in the Kalman Filter Algorithm for a Multi-Dimensional Model
Flowchart for Discrete-Time Kalman Filter Design (for Multi-Dimensional System)
Extended Kalman Filter (EKF)
Steps Involved in the Design of Discrete-Time EKF
Bibliography
Index