Introduction to Linear Control Systems

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 internal model control structure
       1.14 The Smith predictor
       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

    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 modelling of delay
                2.3.13.2 Approximate modelling of delay
            2.3.14 The other constituents
                2.3.14.1 Sensors
                2.3.14.2 Amplifiers
       2.4 Block diagrams
       2.5 Signal flow graphs
            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
       Reference

    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-Chipart’s test
       3.7 Relative stability
       3.8 D-stability
       3.9 Particular relation with control system 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
       4.5 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 system
            4.6.2 Second-order system
            4.6.3 Alternative derivation
            4.6.4 Higher-order systems
            4.6.5 Open-loop and closed-loop systems
       4.7 Higher-order system
       4.8 Model reduction
       4.9 Effect of addition of a pole and zero
       4.10 Performance region
       4.11 Inverse response
       4.12 Analysis and synthesis 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
       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
       5.3 The root contour
       5.4 Finding the value of the gain from the root locus
       5.5 Controller design implications
            5.5.1 Difficult systems
                5.5.1.1 Systems 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 systems 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 and 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
            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
            6.3.2 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
            7.2.8 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
       7.5 Gain, phase, and delay margins
       7.6 Stability in the Bode diagram context
       7.7 The high sensitivity region
       7.8 Relation with Nyquist and root locus
       7.9 Standard second-order system
       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 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 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 constraints
       10.3 Ideal transfer functions
       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 integrators
            10.7.2 MP and NMP open-loop 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 Non-interactive performance by output feedback
       10.12 Minimal closed-loop pole sensitivity by output feedback
       10.13 Robust stabilization by output feedback
            10.13.1 Structured perturbations
            10.13.2 Unstructured perturbations
       10.14 Special results on 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

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 Derivation 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

    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

    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

    D.Treatise on Stability Concepts & Tests
       D.1 Introduction
       D.2 A survey on stability concepts and tests
            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.6 Summary
       References

    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

    F. Genetic algorithm: A global optimization technique
       F.1 Introduction
       F2. Non-convex optimization
       F.3 Convex optimization
       F.4 Convexification
       F.5 Genetic algorithms
       References

    G. Sample Exams
       G.1 Sample midterm exam
       G.2 Sample endterm exam