Nonlinear Systems and Controls

Preface
Contents
1 Fundamentals of Nonlinear Systems
	1.1 System Description and System Behavior
		1.1.1 Linear and Nonlinear Systems
		1.1.2 System Description and Nonlinear Control Loops
		1.1.3 Equilibrium Points of Nonlinear Systems
		1.1.4 Example: Satellite
		1.1.5 Equilibrium Points of Linear Systems
		1.1.6 Stability and Asymptotic Stability
		1.1.7 Exponential Stability of Equilibrium Points
		1.1.8 Instability of Equilibrium Points
		1.1.9 Stability in the Case of Variable Input Signals
		1.1.10 Limit Cycles
		1.1.11 Sliding Modes
		1.1.12 Chaos
		1.1.13 Discrete-Time Systems
	1.2 Solution of Nonlinear Differential Equations
		1.2.1 Existence of Solutions
		1.2.2 Numerical Solution and Euler Method
		1.2.3 Accuracy of the Numerical Solution
		1.2.4 The Modified Euler Method
		1.2.5 The Heun and Simpson Methods
		1.2.6 The Runge-Kutta Methods
		1.2.7 Adaptation of the Step Size
		1.2.8 The Adams-Bashforth Methods
		1.2.9 The Adams-Moulton Predictor-Corrector Method
		1.2.10 Stability of Numerical Integration Methods
		1.2.11 Stiff Systems and Their Solutions
	1.3 Exercises
2 Limit Cycles and Stability Criteria
	2.1 The Describing Function Method
		2.1.1 Idea behind the Method
		2.1.2 Illustrative Example
		2.1.3 Characteristic Curves and Their Describing Functions
		2.1.4 Stability Analysis of Limit Cycles
		2.1.5 Example: Power-Assisted Steering System
	2.2 Absolute Stability
		2.2.1 The Concept of Absolute Stability
		2.2.2 The Popov Criterion and Its Application
		2.2.3 The Aizerman and Kalman Conjectures
		2.2.4 Example: Controlling a Ship
		2.2.5 The Circle Criterion
		2.2.6 The Tsypkin Criterion for Discrete-Time Systems
	2.3 Lyapunov’s Stability Theory
		2.3.1 The Concept and the Direct Method
		2.3.2 Illustrative Example
		2.3.3 Quadratic Lyapunov Functions
		2.3.4 Example: Mutualism
		2.3.5 The Direct Method for Discrete-Time Systems
		2.3.6 The Indirect Method
		2.3.7 Determining Exponential Stability
		2.3.8 Example: Underwater Glider
		2.3.9 Catchment Regions
		2.3.10 LaSalle’s Invariance Principle
		2.3.11 Instability Criterion
	2.4 Passivity and Stability
		2.4.1 Passive Systems
		2.4.2 Stability of Passive Systems
		2.4.3 Passivity of Connected Systems
		2.4.4 Passivity of Linear Systems
		2.4.5 Example: Transporting System for Material Webs
		2.4.6 Positive Real Transfer Functions
		2.4.7 Equivalence of Positive Realness and Passivity
		2.4.8 Lossless Hamiltonian Systems
		2.4.9 Example: Self-Balancing Vehicle
		2.4.10 Dissipative Hamiltonian Systems
		2.4.11 Example: Separately Excited Direct-Current Machine
		2.4.12 Linear Hamiltonian Systems
	2.5 Exercises
3 Controllability and Flatness
	3.1 Controllability
		3.1.1 Definition of Controllability
		3.1.2 Global and Local Controllability
		3.1.3 Proving Controllability
		3.1.4 Example: Industrial Robot
		3.1.5 Small-Time Local Controllability of Driftless Systems
		3.1.6 Example: Motor Vehicle with Trailer
		3.1.7 Omnidirectional Controllability
		3.1.8 Example: Steam Generator
	3.2 Flatness
		3.2.1 Basic Concept and Definition of Flatness
		3.2.2 The Lie-Bäcklund Transformation
		3.2.3 Example: VTOL Aircraft
		3.2.4 Flatness and Controllability
		3.2.5 Flat Outputs of Linear Systems
		3.2.6 Verification of Flatness
	3.3 Nonlinear State Transformations
		3.3.1 Transformations and Transformed System Equations
		3.3.2 Illustrative Example
		3.3.3 Example: Park Transformation
		3.3.4 Determining the Transformation Rule
		3.3.5 Illustration Using Linear Systems
	3.4 Exercises
4 Nonlinear Control of Linear Systems
	4.1 Control with Anti-Windup
		4.1.1 The Windup Effect
		4.1.2 PID Controller with Anti-Windup Element
		4.1.3 Example: Direct-Current Motor
		4.1.4 A General Anti-Windup Method
		4.1.5 Dimensioning the General Anti-Windup Controller
		4.1.6 Stability
	4.2 Time-Optimal Control
		4.2.1 Fundamentals and Fel\'dbaum’s Theorem
		4.2.2 Computation of Time-Optimal Controls
		4.2.3 Example 1/s2
		4.2.4 Time-Optimal Control of Low-Order Systems
		4.2.5 Example: Submarine
		4.2.6 Time-Optimal Pilot Control
	4.3 Variable Structure Control Without Sliding Mode
		4.3.1 Fundamentals of Variable Structure Control
		4.3.2 Piecewise Linear Control
		4.3.3 Example: Ship-to-Shore Gantry Crane
	4.4 Saturation Controllers
		4.4.1 Basics and Stability
		4.4.2 Design in Multiple Steps
		4.4.3 Example: Helicopter
	4.5 Exercises
5 Nonlinear Control of Nonlinear Systems
	5.1 Gain-Scheduling Control
		5.1.1 Mode of Operation and Design
		5.1.2 Illustrative Example
		5.1.3 Example: Solar Power Plant
	5.2 Input-Output Linearization
		5.2.1 Basic Concept and Nonlinear Controller Canonical Form
		5.2.2 Nonlinear Controller and Linear Control Loop
		5.2.3 Example: Magnetic Bearing
		5.2.4 Plants with Internal Dynamics
		5.2.5 Design Procedure
		5.2.6 Example: Lunar Module
		5.2.7 Input-Output Linearization of General SISO Systems
		5.2.8 Relative Degree and Internal Dynamics of Linear Systems
		5.2.9 Control Law for the Linear Case
		5.2.10 Stability of Internal and Zero Dynamics
		5.2.11 Input-Output Linearization of MIMO Systems
		5.2.12 MIMO Control Loops in State-Space Representation
		5.2.13 Example: Combustion Engine
	5.3 Full-State Linearization
		5.3.1 Full-State Linearization of SISO Systems
		5.3.2 Example: Drilling Rig
		5.3.3 Full-State Linearization of MIMO Systems
		5.3.4 Flatness of Full-State Linearizable Systems
		5.3.5 Example: Rocket
	5.4 Feedforward and Feedback Control of Flat Systems
		5.4.1 Fundamentals
		5.4.2 Feedforward Controls Using Fictitious Flat Outputs
		5.4.3 Flatness-Based Feedforward Control of Linear Systems
		5.4.4 Example: Propulsion-Based Aircraft Control
		5.4.5 Flatness-Based Feedback Control of Nonlinear Systems
		5.4.6 Example: Pneumatic Motor
		5.4.7 Flat Inputs and Their Design
		5.4.8 Flat Inputs of Linear Systems
		5.4.9 Example: Economic Market Model
	5.5 Control Lyapunov Functions
		5.5.1 Fundamentals
		5.5.2 Control Lyapunov Functions for Linear Systems
		5.5.3 Control Lyapunov Functions for Control-Affine Systems
		5.5.4 Illustrative Example
		5.5.5 Example: Power Plant with Grid Feed-In
	5.6 The Backstepping Method
		5.6.1 Fundamentals
		5.6.2 Recursive Scheme for the Controller Design
		5.6.3 Illustrative Examples
		5.6.4 Example: Fluid System with Chaotic Behavior
	5.7 Exercises
6 Nonlinear Control of Linear and Nonlinear Systems
	6.1 Model-Based Predictive Control
		6.1.1 Basics and Functionality
		6.1.2 Linear Model Predictive Control without Constraints
		6.1.3 LMPC with Constraints
		6.1.4 Example: Drainage System
		6.1.5 Nonlinear Model Predictive Control
		6.1.6 Example: Evaporation Plant
	6.2 Variable Structure Control with Sliding Mode
		6.2.1 Basics and Characteristics
		6.2.2 Design for Linear Plants
		6.2.3 Dynamics in the Sliding Mode
		6.2.4 Verification of Robustness
		6.2.5 Example: DC-to-DC Converter
		6.2.6 Design for Nonlinear Plants
		6.2.7 Example: Optical Switch
	6.3 Passivity-Based Control
		6.3.1 Control of Passive Systems Using Static Controllers
		6.3.2 Example: Damping of Seismic Building Vibrations
		6.3.3 Passivation of Non-Passive Linear Systems
		6.3.4 Passivation of Non-Passive Control-Affine Systems
		6.3.5 Passivity-Based Control with IDA
		6.3.6 Example: Paper Machine
	6.4 Fuzzy Control
		6.4.1 Introduction
		6.4.2 Fuzzification
		6.4.3 Inference
		6.4.4 Defuzzification
		6.4.5 Fuzzy Systems and Fuzzy Controllers
		6.4.6 Example: Distance Control for Automobiles
	6.5 Exercises
7 Observers for Nonlinear Systems
	7.1 Observability of Nonlinear Systems
		7.1.1 Definition of Observability
		7.1.2 Observability of Autonomous Systems
		7.1.3 Example: Synchronous Generator
		7.1.4 Observability of General Nonlinear Systems
		7.1.5 Nonlinear Observability Canonical Form
		7.1.6 Observability of Control-Affine Systems
	7.2 Canonical Forms and the Canonical Form Observer
	7.3 Luenberger Observers for Nonlinear Control Loops
	7.4 Observer Design Using Linearization
		7.4.1 Basics and Design
		7.4.2 Control Loop with Observer
		7.4.3 Example: Bioreactor
	7.5 The Extended Kalman Filter
		7.5.1 Kalman Filter for Linear Systems
		7.5.2 The EKF for Nonlinear Systems
		7.5.3 Example: Jet Engine
	7.6 High-Gain Observer
		7.6.1 Concept and Design
		7.6.2 High-Gain Observers in General Form
		7.6.3 Example: Chemical Reactor
		7.6.4 The Case of Control-Affine Systems
	7.7 Exercises
8 Solutions to the Exercises
A Appendix
	A.1 Proof of the General Circle Criterion
	A.2 Parameters of the Container Crane Control
B List of Symbols
References
Index