Introduction to linear control systems

Cover
Introduction to Linear Control Systems
Copyright
Dedication
Preface
Acknowledgments
Part I: Foundations
1 Introduction
	1.1 Introduction
	1.2 Why control?
	1.3 History of control
	1.4 Why feedback?
	1.5 Magic of feedback
	1.6 Physical elements of a control system
	1.7 Abstract elements of a control system
	1.8 Design process
	1.9 Types of control systems
	1.10 Open-loop control
		1.10.1 Stability and performance
		1.10.2 Sensitivity and robustness
		1.10.3 Disturbance
		1.10.4 Reliability, economics, and linearity
	1.11 Closed-loop control
		1.11.1 Stability and performance
		1.11.2 Sensitivity and robustness
		1.11.3 Disturbance and noise
		1.11.4 Reliability, economics, and linearity
	1.12 The 2-DOF control structure
	1.13 The Smith predictor
	1.14 Internal model control structure
	1.15 Modern representation—Generalized model
	1.16 Status quo
		1.16.1 Overview
			1.16.1.1 Summary
			1.16.1.2 The forgotten
		1.16.2 Relation with other disciplines
		1.16.3 Challenges
		1.16.4 Outlook
	1.17 Summary
	1.18 Notes and further readings
	1.19 Worked-out problems
	1.20 Exercises
	References
	Further Reading
2 System representation
	2.1 Introduction
	2.2 System modeling
		2.2.1 State-space
			2.2.1.1 Linearization
			2.2.1.2 Number of inputs and outputs
		2.2.2 Frequency domain
			2.2.2.1 Finding the output
		2.2.3 Zero, pole, and minimality
	2.3 Basic examples of modeling
		2.3.1 Electrical system as the plant
		2.3.2 Mechanical system as the plant
		2.3.3 Liquid system as the plant
		2.3.4 Thermal system as the plant
		2.3.5 Hydraulic system as the plant
		2.3.6 Chemical system as the plant
		2.3.7 Structural system as the plant
		2.3.8 Biological system as the plant
		2.3.9 Economics system as the plant
		2.3.10 Ecological system as the plant
		2.3.11 Societal system as the plant
		2.3.12 Physics system as the plant
		2.3.13 Delay
			2.3.13.1 Exact modeling of delay
			2.3.13.2 Approximate modeling of delay
		2.3.14 The other constituents
			2.3.14.1 Sensors
			2.3.14.2 Amplifiers
	2.4 Block diagram
	2.5 Signal flow graph
		2.5.1 Basic terminology of graph theory
		2.5.2 Equivalence of BD and SFG methods
		2.5.3 Computing the transmittance of an SFG
	2.6 Summary
	2.7 Notes and further readings
	2.8 Worked-out problems
	2.9 Exercises
	References
3 Stability analysis
	3.1 Introduction
	3.2 Lyapunov and BIBO stability
	3.3 Stability tests
	3.4 Routh’s test
		3.4.1 Special cases
	3.5 Hurwitz’ test
	3.6 Lienard and Chipart test
	3.7 Relative stability
	3.8 D-stability
	3.9 Particular relation with control systems design
	3.10 The Kharitonov theory
	3.11 Internal stability
	3.12 Strong stabilization
	3.13 Stability of LTV Systems
	3.14 Summary
	3.15 Notes and further readings
	3.16 Worked-out problems
	3.17 Exercises
	References
4 Time response
	4.1 Introduction
	4.2 System type and system inputs
	4.3 Steady-state error
	4.4 First-order systems
		4.4.1 Impulse input
		4.4.2 Step, ramp, and parabolic inputs
	4.5 Second-order systems
		4.5.1 System representation
		4.5.2 Impulse response
		4.5.3 Step response
			4.5.3.1 Time response characteristics
		4.5.4 Ramp and parabola response
	4.6 Bandwidth of the system
		4.6.1 First-order systems
		4.6.2 Second-order systems
		4.6.3 Alternative derivation
		4.6.4 Higher-order systems
		4.6.5 Open-loop and closed-loop systems
	4.7 Higher-order systems
	4.8 Model reduction
	4.9 Effect of addition of pole and zero
	4.10 Performance region
	4.11 Inverse response
	4.12 Analysis of the actual system
		4.12.1 Sensor dynamics
		4.12.2 Delay dynamics
	4.13 Introduction to robust stabilization and performance
		4.13.1 Open-loop control
		4.13.2 Closed-loop control
			4.13.2.1 Disturbance and noise rejection and setpoint tracking
				Design for disturbance and noise rejection
				Design for sinusoidal reference tracking
	4.14 Summary
	4.15 Notes and further readings
	4.16 Worked-out problems
	4.17 Exercises
	References
5 Root locus
	5.1 Introduction
	5.2 The root locus method
	5.3 The root contour
	5.4 Finding the value of gain from the root locus
	5.5 Controller design implications
		5.5.1 Difficult systems
			5.5.1.1 System without NMP zeros
			5.5.1.2 Systems with NMP zeros
			5.5.1.3 Examples of systems without NMP zeros
			5.5.1.4 Examples of system with NMP zeros
		5.5.2 Simple systems
	5.6 Summary
	5.7 Notes and further readings
	5.8 Worked-out problems
	5.9 Exercises
	References
Part II: Frequency domain analysis & synthesis
6 Nyquist plot
	6.1 Introduction
	6.2 Nyquist plot
		6.2.1 Principle of argument
		6.2.2 Nyquist stability criterion
		6.2.3 Drawing of the Nyquist plot
		6.2.4 The high- and low-frequency ends of the plot
		6.2.5 Cusp points of the plot
		6.2.6 How to handle the proportional gain/uncertain parameter
		6.2.7 The case of j-axis zeros and poles
		6.2.8 Relation with root locus
	6.3 Gain, phase, and delay margins
		6.3.1 The GM concept
			6.3.1.1 Definition of GM in the Nyquist plot context
		6.3.2 The PM and DM concepts
		6.3.3 Stability in terms of the GM and PM signs
		6.3.4 The high sensitivity region
	6.4 Summary
	6.5 Notes and further readings
	6.6 Worked-out problems
	6.7 Exercises
	References
7 Bode diagram
	7.1 Introduction
	7.2 Bode diagram
		7.2.1 Logarithm
		7.2.2 Decibel
		7.2.3 Log magnitude
		7.2.4 The magnitude diagram
		7.2.5 Octave and decade
		7.2.6 Some useful figures to remember
		7.2.7 Relation between the transfer function and its constituting components
			7.2.7.1 Gain K
			7.2.7.2 Zeros at origin (jω)+m
			7.2.7.3 Poles at origin (jω)−m
			7.2.7.4 Real zeros not at origin (1+jωT)+m
			7.2.7.5 Real poles not at origin (1+jωT)−m
			7.2.7.6 Error in Lm
			7.2.7.7 Error in φ
			7.2.7.8 Double zeros [1+2ζωnjω+1ωn2(jω)2]+m
			7.2.7.9 Double poles [1+2ζωnjω+1ωn2(jω)2]−m
		7.2.8 How to draw the Bode diagram with hand
	7.3 Bode diagram and the steady-state error
	7.4 Minimum phase and nonminimum phase systems
		7.4.1 NMP zero with positive gain: z−s, z%3e0
		7.4.2 NMP pole with positive gain: 1/(p−s), p%3e0
		7.4.3 NMP zero with negative gain: −(z−s)=s−z, z%3e0
		7.4.4 NMP pole with negative gain: −1/(p−s)=1/(s−p), p%3e0
		7.4.5 Determination of NMP systems from the Bode diagram
	7.5 Gain, phase, and delay margins
	7.6 Stability in the Bode diagram context
	7.7 The high sensitivity region
	7.8 Relation with Nyquist plot and root locus
	7.9 Standard second-order systems
	7.10 Bandwidth
	7.11 Summary
	7.12 Notes and further readings
	7.13 Worked-out problems
	7.14 Exercises
	References
8 Krohn-Manger-Nichols chart
	8.1 Introduction
	8.2 S-Circles
	8.3 M-Circles
	8.4 N-circles
	8.5 M- and N-Contours
	8.6 KMN chart
	8.7 System features: GM, PM, DM, BW, stability
		8.7.1 Gain, phase, and delay margins
		8.7.2 Stability
		8.7.3 Bandwidth
	8.8 The high sensitivity region
	8.9 Relation with Bode diagram, Nyquist plot, and root locus
	8.10 Summary
	8.11 Notes and further readings
	8.12 Worked-out problems
	8.13 Exercises
	References
9 Frequency domain synthesis and design
	9.1 Introduction
	9.2 Basic controllers: proportional, lead, lag, and lead-lag
	9.3 Controller simplifications: PI, PD, and PID
	9.4 Controller structures in the Nyquist plot context
	9.5 Effect of the controllers on the root locus
	9.6 Design procedure
	9.7 Specialized design and tuning rules of PID controllers
		9.7.1 Heuristic rules
		9.7.2 Analytical rules
			9.7.2.1 Pole placement method
			9.7.2.2 Direct synthesis
			9.7.2.3 Skogestad tuning rules
		9.7.3 Optimization-based rules
	9.8 Internal model control
	9.9 The Smith predictor
	9.10 Implementation with operational amplifiers
		9.10.1 Proportional control—P-term
		9.10.2 Integral control—I-term
		9.10.3 Proportional–integral—PI-term
		9.10.4 Proportional–derivative—PD-term
		9.10.5 Nonideal/actual derivative—D-term
		9.10.6 Series proportional-integral-derivative—Series PID
		9.10.7 Lead
		9.10.8 Lag
		9.10.9 Lead or lag
		9.10.10 Lead-lag
	9.11 Summary
	9.12 Notes and further readings
	9.13 Worked-out problems
	9.14 Exercises
	References
Part III: Advanced Issues
10 Fundamental limitations
	10.1 Introduction
	10.2 Relation between time and frequency domain specifications
	10.3 The ideal transfer function
	10.4 Controller design via the TS method
	10.5 Interpolation conditions
	10.6 Integral and Poisson integral constraints
	10.7 Constraints implied by poles and zeros
		10.7.1 Implications of open-loop integrators
		10.7.2 MP and NMP poles and zeros
		10.7.3 Imaginary-axis poles and zeros
	10.8 Actuator and sensor limitations
		10.8.1 Maximal actuator movement
		10.8.2 Minimal actuator movement
		10.8.3 Sensor precision
		10.8.4 Sensor speed
	10.9 Delay
	10.10 Eigenstructure assignment by output feedback
		10.10.1 Regulation
		10.10.2 Tracking
	10.11 Noninteractive performance
	10.12 Minimal closed-loop pole sensitivity
	10.13 Robust stabilization
		10.13.1 Structured perturbations
		10.13.2 Unstructured perturbations
	10.14 Special results for positive systems
	10.15 Generic design procedure
	10.16 Summary
	10.17 Notes and further readings
	10.18 Worked-out problems
	10.19 Exercises
	References
Appendices A–G
Appendix A Laplace transform and differential equations
	A.1 Introduction
	A.2 Basic properties and pairs
		A.2.1 Inverse Laplace transform
		A.2.2 Table of some Laplace transform pairs
	A.3 Differentiation and integration in time domain and frequency domain
		A.3.1 Fourier transform of the Heaviside function
		A.3.2 Differentiation formula in time domain
		A.3.3 Integration formula in time domain
		A.3.4 Frequency domain formulae
		A.3.5 Some consequences
	A.4 Existence and uniqueness of solutions to differential equations
	References
Appendix B Introduction to dynamics
	B.1 Introduction
		B.1.1 Electrical systems
		B.1.2 Mechanical systems
		B.1.3 Chemical systems
	B.2 Equivalent systems
	B.3 Worked-out problems
	References
Appendix C Introduction to MATLAB®
	C.1 Introduction
	C.2 MATLAB®
		C.2.1 How to write an M.file
			C.2.1.1 Script file
			C.2.1.2 Function file
		C.2.2 MATLAB® functions by category—control system toolbox
			C.2.2.1 LTI models
			C.2.2.2 Model characteristics
			C.2.2.3 Model conversions
			C.2.2.4 Model order reduction
			C.2.2.5 State-space realizations
			C.2.2.6 Model dynamics
			C.2.2.7 Model interconnections
			C.2.2.8 Time responses
			C.2.2.9 Time delays
			C.2.2.10 Frequency response
			C.2.2.11 Pole placement
			C.2.2.12 LQG design
			C.2.2.13 Equation solvers
			C.2.2.14 Graphical user interfaces for control system analysis and design
	C.3 Simulink
	C.4 Worked-out problems
	References
Appendix D Treatise on stability concepts and tests
	D.1 Introduction
	D.2 A survey on stability concepts and tools
		D.2.1 Deterministic systems
		D.2.2 Stochastic systems
		D.2.3 Miscellaneous
	D.3 Lipschitz stability
	D.4 Lagrange, Poisson, and Lyapunov stability
	D.5 Finite-time and fixed-time stability
		D.5.1 Fixed-time decentralized stability of large-scale systems
			D.5.1.1 Large-scale system description
	D.6 Summary
	References
Appendix E Treatise on the Routh’s stability test
	E.1 Introduction
	E.2 Applications of the Routh’s array
	E.3 The case of imaginary-axis zeros
	References
Appendix F Genetic algorithm: A global optimization technique
	F.1 Introduction
	F.2 Convex optimization
	F.3 Nonconvex optimization
	F.4 Convexification
	F.5 Genetic algorithms
	References
Appendix G Sample exams
	G.1 Sample Midterm Exam (ME) (4h, closed book/notes)
	G.2 Sample Endterm Exam (EE) (4h, closed book/notes)
Index
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