Dr. Boris J. Lurie


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Classical Feedback Control with MATLAB

Boris J. Lurie and Paul J. Enright, Marcel Dekker, NY, 2000
(470 pp. with 543 figures)

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From the Preface:

Classical Feedback Control describes design and implementation of high-performance feedback controllers for engineering systems. The book emphasizes the frequency-domain approach which is widely used in practical engineering. It presents frequency-domain design methods for high-order SISO and MIMO, linear and nonlinear, analog and digital control systems.

Modern technology allows implementation of high-performance controllers at a very low cost. Conversely, several analysis tools which were previously considered an inherent part of control system courses limit the design to low-order (and therefore low-performance) compensators. Among these are the root-locus method, the detection of right-sided polynomial roots using the Routh-Hurwitz criterion, and manual calculations using the Laplace and Fourier transforms. These methods have been rendered obsolete by computers and are granted only a brief treatment in the book, making room for loop shaping, Bode integrals, structural simulation of complex systems, multiloop systems, and nonlinear controllers, all of which are essential for good design practice.

In the design philosophy adopted by Classical Feedback Control, Bode integral relations play a key role. The integrals are employed to estimate the available system performance and to determine the frequency responses which maximize the disturbance rejection and the feedback bandwidth. This ability to quickly estimate the attainable performance is critical for system-level trades in the design of complex engineering systems, of which the controller is one of many subsystems. Only at the final design stage and only for the finally selected option of the system configuration do the compensators need to be designed in detail, by approximation of the already found optimal frequency responses.

Nonlinear dynamic compensation is employed to provide global and process stability, and to improve transient responses. The nearly-optimal high-order compensators are then economically implemented using analog and digital technology.

The first six chapters support a one-semester course in linear control. The rest of the book considers the issues of complex system simulation, robustness, global stability, and nonlinear control.

It was the authors' intention to make Classical Feedback Control not only a textbook but also a reference for students as they become engineers, enabling them to design high-performance controllers and easing the transition from school to the competitive industrial environment. The methods described in this book were used by the authors and their colleagues as the major design tools for feedback loops of aerospace and telecommunication systems.

MATLAB ® from MathWorks, Inc. is the most popular control CAD software package. Inexpensive student versions of MATLAB with control toolbox and of the associated block diagram simulation package Simulink ® available at university bookstores are more than adequate for the examples in this book and even some professional work. No preliminary knowledge of MATLAB is assumed, and only a small subset of MATLAB commands is used, listed below in the order of their introduction: logspace, bode, conv, tf2zp, zp2tf, step, gtext, title, set, grid, hold on, hold off, rlocus, plot(x,y), inv, linspace, lp2lp, lp2bp, format, roots, poly, inv, bilinear, residue, ezplot, linmod, laplace, invlaplace, impulse. It is standard procedure to do the preliminary design in MATLAB/Simulink since it's quick to code up, and when it's complete, to transfer the code to C.

CONTENTS     (Chapter 4 and Appendices 11 and 14 are on-line)


To Instructors

Chapter 1  Feedback and Sensitivity
1.1 Feedback control system
1.2 Feedback: positive and negative
1.3 Large feedback
1.4 Loop gain and phase frequency responses
  1.4.1 Gain and phase responses
  1.4.2 Nyquist diagram
  1.4.3 Nichols chart
1.5 Disturbance rejection
1.6 Example of system analysis
1.7 Effect of feedback on the actuator nonlinearity
1.8 Sensitivity
1.9 Effect of finite plant parameter variations
1.10 Automatic volume control
1.11 Lead and PID compensators
1.12 Conclusion and a look ahead
1.13 Problems

Chapter 2  Feedforward, Multiloop, and MIMO Systems

2.1 Command feedforward
2.2 Prefilter and the feedback path equivalent
2.3 Error feedforward
2.4 Black's feedforward method
2.5 Multiloop feedback systems
2.6 Local, common, and nested loops
2.7 Crossed loops and main/vernier loops
2.8 Manipulations of block diagrams and calculations of transfer functions
2.9 MIMO feedback systems
2.10 Problems

Chapter 3  Frequency Response Methods

3.1 Conversion of time-domain requirements to frequency-domain
  3.1.1 Approximate relations
  3.1.2 Filters
3.2 Closed-loop transient response
3.3 Root locus
3.4 Nyquist stability criterion
3.5 Robustness and stability margins
3.6 Nyquist criterion for unstable plant
3.7 Successive loop closure stability criterion
3.8 Nyquist diagrams for loop transfer functions with poles at the origin
3.9 Bode integrals
  3.9.1 Minimum phase functions
  3.9.2 Integral of feedback
  3.9.3 Integral of resistance
  3.9.4 Integral of the imaginary part
  3.9.5 Gain integral over finite bandwidth
  3.9.6 Phase-gain relations
3.10 Phase calculations
3.11 From the Nyquist diagram to the Bode diagram
3.12 Non-minimum phase lag
3.13 Ladder networks and parallel connections of m.p. links
3.14 Problems

Chapter 4  Shaping the Loop Frequency Response

4.1 Optimality in the compensator design
4.2 Feedback maximization
  4.2.1 Structural design
  4.2.2 Bode step
  4.2.3 Example of a system having loop response with Bode step
  4.2.4 Reshaping the feedback response
  4.2.5 Bode cutoff
  4.2.6 Band-pass systems
  4.2.7 Nyquist-stable systems
4.3 Feedback bandwidth limitations
  4.3.1 Feedback bandwidth
  4.3.2 Sensor noise at the system output
  4.3.3 Sensor noise at the system input
  4.3.4 Non-minimum phase shift
  4.3.5 Plant tolerances
  4.3.6 Lightly damped flexible plants; collocated and non-collocated control
  4.3.7 Unstable plants
4.4 Coupling in MIMO systems
4.5 Shaping parallel channel responses
4.6 Problems

Chapter 5  Compensator Design

5.1 Accuracy of the loop shaping
5.2 Asymptotic Bode diagram
5.3 Approximation of constant slope gain response
5.4 Lead and lag links
5.5 Complex poles
5.6 Cascaded links
5.7 Parallel connection of links
5.8 Simulation of a PID controller
5.9 Analog and digital controllers
5.10 Digital compensator design
  5.10.1 Discrete trapezoid integrator
  5.10.2 Laplace and Tustin transforms
  5.10.3 Design sequence
  5.10.4 Block diagrams, equations, and computer code
  5.10.5 Compensator design example
  5.10.6 Aliasing and noise
  5.10.7 Transfer function for the fundamental
5.11 Command profiling
5.12 Problems

Chapter 6  Analog Controller Implementation

6.1 Active RC circuits
  6.1.1 Operational amplifier
  6.1.2 Integrator and differentiator
  6.1.3 Noninverting configuration
  6.1.4 Op-amp dynamic range, noise, and packaging
  6.1.5 Transfer functions with multiple poles and zeros
  6.1.6 Active RC filters
  6.1.7 Nonlinear links
6.2 Design and iterations in the element value domain
  6.2.1 Cauer and Foster RC two-poles
  6.2.2 RC-impedance chart
6.3 Analog compensator, analog or digitally controlled
6.4 Switched-capacitor filters
  6.4.1 Switched-capacitor circuits
  6.4.2 Example of compensator design
6.5 Miscellaneous hardware issues
  6.5.1 Ground
  6.5.2 Signal transmission
  6.5.3 Stability and testing issues
6.6 PID tunable controller
  6.6.1 PID compensator
  6.6.2 TID compensator
6.7 Tunable compensator with one variable parameter
  6.7.1 Bilinear transfer function
  6.7.2 Symmetrical regulator
  6.7.3 Hardware implementation
6.8 Loop response measurements
6.9 Problems

Chapter 7  Linear Links and System Simulation

7.1 Mathematical analogies
  7.1.1 Electro-mechanical analogies
  7.1.2 Electrical analogy to heat transfer
  7.1.3 Hydraulic systems
7.2 Junctions of unilateral links
  7.2.1 Structural design
  7.2.2 Junction variables
  7.2.3 Loading diagram
7.3 Effect of the plant and actuator impedances on the plant transfer function uncertainty
7.4 Effect of feedback on impedance (mobility)
  7.4.1 Large feedback with velocity and force sensors
  7.4.2 Blackman's formula
  7.4.3 Parallel feedback
  7.4.4 Series feedback
  7.4.5 Compound feedback
7.5 Effect of load impedance on feedback
7.6 Flowchart for the chain connection of bi-directional two-ports
  7.6.1 Chain connection of two-ports
  7.6.2 DC motors
  7.6.3 Motor output mobility
  7.6.4 Piezoelements
  7.6.5 Drivers, transformers, and gears
  7.6.6 Coulomb friction
7.7 Examples of system modeling
7.8 Flexible structures
  7.8.1 Impedance (mobility) of a lossless system
  7.8.2 Lossless distributed structures
  7.8.3 Collocated control
  7.8.4 Non-collocated control
7.9 Sensor noise
  7.9.1 Motion sensors Position and angle sensors Rate sensors Accelerometers Noise responses
  7.9.2 Effect of feedback on the signal-to-noise ratio
7.10 Mathematical analogies to the feedback system
  7.10.1 Feedback to parallel channel analogy
  7.10.2 Feedback to two-pole connection analogy
7.11 Linear time-variable systems
7.12 Problems

Chapter 8  Introduction to Alternative Methods of Controller Design

8.1 QFT
8.2 Root locus and pole placement methods
8.3 State-space methods and full-state feedback
8.4 LQR and LQG
8.5 H , µ-synthesis, and LMI

Chapter 9  Adaptive Systems

9.1 Benefits of adaptation to the plant parameter variations
9.2 Static and dynamic adaptation
9.3 Plant transfer function identification
9.4 Flexible and n.p. plants
9.5 Disturbance and noise rejection
9.6 Pilot signals and dithering systems
9.7 Adaptive filters

Chapter 10  Provision of Global Stability

10.1 Nonlinearities of the actuator, feedback path, and plant
10.2 Types of self-oscillation
10.3 Stability analysis of nonlinear systems
  10.3.1 Local linearization
  10.3.2 Global stability
10.4 Absolute stability
10.5 Popov criterion
  10.5.1 Analogy to passive two-poles' connection
  10.5.2 Different forms of the Popov criterion
10.6 Applications of the Popov criterion
  10.6.1 Low-pass system with maximum feedback
  10.6.2 Band-pass system with maximum feedback
10.7 Absolutely stable systems with nonlinear dynamic compensation
  10.7.1 Nonlinear dynamic compensator
  10.7.2 Reduction to equivalent system
  10.7.3 Design examples
10.8 Problems

Chapter 11  Describing Functions

11.1 Harmonic balance
  11.1.1 Harmonic balance analysis
  11.1.2 Harmonic balance accuracy
11.2 Describing functions
11.3 Describing functions for symmetrical piece-linear characteristics
  11.3.1 Exact expressions
  11.3.2 Approximate formulas
11.4 Hysteresis
11.5 Nonlinear links yielding phase advance for large amplitude signals
11.6 Two nonlinear links in the feedback loop
11.7 NDC with a single nonlinear nondynamic link
11.8 NDC with parallel channels
11.9 NDC made with local feedback
11.10 Negative hysteresis and Clegg Integrator
11.11 Nonlinear interaction between the local and common feedback loops
11.12 NDC in multiloop systems
11.13 Harmonics and intermodulation
  11.13.1 Harmonics
  11.13.2 Intermodulation
11.14 Verification of global stability
11.15 Problems

Chapter 12  Process Instability

12.1 Process instability
12.2 Absolute stability of the output process
12.3 Jump-resonance
12.4 Subharmonics
  12.4.1 Odd subharmonics
  12.4.2 Even subharmonics
12.5 Nonlinear dynamic compensation
12.6 Problems

Chapter 13  Multi-window Compensators

13.1 Composite nonlinear controllers
13.2 Multi-window control
13.3 Switching between hot and between cold controllers
13.4 Windup, and anti-windup controllers
13.5 Selection order
13.6 Acquisition and tracking
13.7 Time-optimal control
13.8 Examples
13.9 Problems


Appendix 1  Feedback control, elementary treatment

A1.1 Introduction
A1.2 Feedback control, elementary treatment
  A1.2.1 Feedback block diagram
  A1.2.2 Feedback control
  A1.2.3 Links
A1.3 Why control cannot be perfect
  A1.3.1 Dynamic links
  A1.3.2 Control accuracy limitations
A1.4 NDC with parallel channels
  A1.4.1 Self-oscillation
  A1.4.2 Loop frequency response
  A1.4.3 Control system design using frequency responses
  A1.4.4 Some algebra
  A1.4.5 Disturbance rejection
  A1.4.6 Conclusion
A1.5 New words

Appendix 2  Frequency responses

A2.1 Frequency responses
A2.2 Complex transfer function
A2.3 Laplace transform and the s-plane
A2.4 Laplace transfer function
A2.5 Poles and zeros of transfer functions
A2.6 Pole-zero cancellation, dominant poles and zeros
A2.7 Time-responses
A2.8 Problems

Appendix 3  Causal systems, passive systems, and positive real functions

Appendix 4  Derivation of Bode integrals

A4.1 Integral of the real part
A4.2 Integral of the imaginary part
A4.3 General relation

Appendix 5  Program for phase calculation

Appendix 6  Generic single-loop feedback system

Appendix 7  Effect of feedback on mobility, derivation

Appendix 8  Dependence of a function on a parameter

Appendix 9  Balanced bridge feedback

Appendix 10  Phase-gain relation for describing functions

Appendix 11  Discussions

A11.1 Compensator implementation
A11.2 Feedback: positive and negative
A11.3 Tracking systems
A11.4 Elements (links) of the feedback system
A11.5 Plant transfer function uncertainty
A11.6 The Nyquist stability criterion
A11.7 Actuator's output impedance
A11.8 Integral of feedback
A11.9 Bode integrals
A11.10 The Bode phase-gain relation
A11.11 What limits the feedback?
A11.12 Feedback maximization
A11.13 Feedback maximization in multi-loop systems
A11.14 Nonminimum phase function
A11.15 Feedback control design procedure
A11.16 Global stability and absolute stability
A11.17 Describing function and nonlinear dynamic compensation
A11.18 Multi-loop system
A11.19 MIMO system
A11.20 Bode's book

Appendix 12  Design sequence

Appendix 13  Examples

A13.1 Industrial furnace temperature control
A13.2 Scanning mirror of a mapping spectrometer
A13.3 Rocket booster nutation control
A13.4 Telecommunication repeater with an NDC
A13.5 Multi-loop pointing system with a resonant plant
A13.6 Voltage regulator with a main, vernier and local loops
A13.7 Telecommunication repeater
A13.8 Distributed regulators
A13.9 Saturn V flight control system
A13.10 PLL computer clock with duty cycle adjustment
A13.11 Attitude control of solar panels
A13.12 Conceptual design of an antenna attitude control
A13.13 Pathlength control of an optical delay line
A13.14 MIMO motor control system with responses with Bode step
A13.15 Mechanical snake control

Appendix 14  Bode Step toolbox





Downloads: Scripts from all chapters and Appendices 1 - 13, and the Bode Step Toolbox.