An Introduction to System Modeling and Control E-Book

John Chiasson

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 729

Veröffentlichungsjahr: 2022

Beschreibung

A practical and straightforward exploration of the basic tools for the modeling, analysis, and design of control systems

In An Introduction to System Modeling and Control, Dr. Chiasson delivers an accessible and intuitive guide to understanding modeling and control for students in electrical, mechanical, and aerospace/aeronautical engineering. The book begins with an introduction to the need for control by describing how an aircraft flies complete with figures illustrating roll, pitch, and yaw control using its ailerons, elevators, and rudder, respectively. The book moves on to rigid body dynamics about a single axis (gears, cart rolling down an incline) and then to modeling DC motors, DC tachometers, and optical encoders. Using the transfer function representation of these dynamic models, PID controllers are introduced as an effective way to track step inputs and reject constant disturbances.

It is further shown how any transfer function model can be stabilized using output pole placement and on how two-degree of freedom controllers can be used to eliminate overshoot in step responses. Bode and Nyquist theory are then presented with an emphasis on how they give a quantitative insight into a control system's robustness and sensitivity. An Introduction to System Modeling and Control closes with chapters on modeling an inverted pendulum and a magnetic levitation system, trajectory tracking control using state feedback, and state estimation. In addition the book offers:

• A complete set of MATLAB/SIMULINK files for examples and problems included in the book.
• A set of lecture slides for each chapter.
• A solutions manual with recommended problems to assign.
• An analysis of the robustness and sensitivity of four different controller designs for an inverted pendulum (cart-pole).

Perfect for electrical, mechanical, and aerospace/aeronautical engineering students, An Introduction to System Modeling and Control will also be an invaluable addition to the libraries of practicing engineers.

Leseprobe

Table of Contents

Cover

Title Page

Copyright

Preface

About the Companion Website

1 Introduction

1.1 Aircraft

1.2 Quadrotors

1.3 Inverted Pendulum

1.4 Magnetic Levitation

1.5 General Control Problem

Notes

2 Laplace Transforms

2.1 Laplace Transform Properties

2.2 Partial Fraction Expansion

2.3 Poles and Zeros

2.4 Poles and Partial Fractions

Appendix: Exponential Function

Problems

Notes

3 Differential Equations and Stability

3.1 Differential Equations

3.2 Phasor Method of Solution

3.3 Final Value Theorem

3.4 Stable Transfer Functions

3.5 Routh–Hurwitz Stability Test

Problems

Notes

4 Mass–Spring–Damper Systems

4.1 Mechanical Work

4.2 Modeling Mass–Spring–Damper Systems

4.3 Simulation

Problems

5 Rigid Body Rotational Dynamics

5.1 Moment of Inertia

5.2 Newton's Law of Rotational Motion

5.3 Gears

5.4 Rolling Cylinder

Problems

Notes

6 The Physics of the DC Motor

6.1 Magnetic Force

6.2 Single‐Loop Motor

6.3 Faraday's Law

6.4 Dynamic Equations of the DC Motor

6.5 Optical Encoder Model

6.6 Tachometer for a DC Machine*

6.7 The Multiloop DC Motor*

Problems

Notes

7 Block Diagrams

7.1 Block Diagram for a DC Motor

7.2 Block Diagram Reduction

Problems

Note

8 System Responses

8.1 First‐Order System Response

8.2 Second‐Order System Response

8.3 Second‐Order Systems with Zeros

8.4 Third‐Order Systems

Appendix: Root Locus Matlab File

Problems

Note

9 Tracking and Disturbance Rejection

9.1 Servomechanism

9.2 Control of a DC Servo Motor

9.3 Theory of Tracking and Disturbance Rejection

9.4 Internal Model Principle

9.5 Design Example: PI‐D Control of Aircraft Pitch

9.6 Model Uncertainty and Feedback*

Problems

Notes

10 Pole Placement, 2 DOF Controllers, and Internal Stability

10.1 Output Pole Placement

10.2 Two Degrees of Freedom Controllers

10.3 Internal Stability

10.4 Design Example: 2 DOF Control of Aircraft Pitch

10.5 Design Example: Satellite with Solar Panels (Collocated Case)

Appendix: Output Pole Placement

Appendix: Multinomial Expansions

Appendix: Overshoot

Appendix: Unstable Pole‐Zero Cancellation

Appendix: Undershoot

Problems

Notes

11 Frequency Response Methods

11.1 Bode Diagrams

11.2 Nyquist Theory

11.3 Relative Stability: Gain and Phase Margins

11.4 Closed‐Loop Bandwidth

11.5 Lead and Lag Compensation

11.6 Double Integrator Control via Lead‐Lag Compensation

11.7 Inverted Pendulum with Output

Appendix: Bode and Nyquist Plots in Matlab

Problems

Notes

12 Root Locus

12.1 Angle Condition and Root Locus Rules

12.2 Asymptotes and Their Real Axis Intersection

12.3 Angles of Departure

12.4 Effect of Open‐Loop Poles on the Root Locus

12.5 Effect of Open‐Loop Zeros on the Root Locus

12.6 Breakaway Points and the Root Locus

12.7 Design Example: Satellite with Solar Panels (Noncollocated)

Problems

Note

13 Inverted Pendulum, Magnetic Levitation, and Cart on a Track

13.1 Inverted Pendulum

13.2 Linearization of Nonlinear Models

13.3 Magnetic Levitation

13.4 Cart on a Track System

Problems

Notes

14 State Variables

14.1 Statespace Form

14.2 Transfer Function to Statespace

14.3 Laplace Transform of the Statespace Equations

14.4 Fundamental Matrix

14.5 Solution of the Statespace Equation*

14.6 Discretization of a Statespace Model*

Problems

Note

15 State Feedback

15.1 Two Examples

15.2 General State Feedback Trajectory Tracking

15.3 Matrix Inverses and the Cayley–Hamilton Theorem

15.4 Stabilization and State Feedback

15.5 State Feedback and Disturbance Rejection

15.6 Similarity Transformations

15.7 Pole Placement

15.8 Asymptotic Tracking of Equilibrium Points

15.9 Tracking Step Inputs via State Feedback

15.10 Inverted Pendulum on an Inclined Track*

15.11 Feedback Linearization Control*

Appendix: Disturbance Rejection in the Statespace

Problems

Notes

16 State Estimators and Parameter Identification

16.1 State Estimators

16.2 State Feedback and State Estimation in the Laplace Domain*

16.3 Multi‐Output Observer Design for the Inverted Pendulum*

16.4 Properties of Matrix Transpose and Inverse

16.5 Duality*

16.6 Parameter Identification

Problems

Note

17 Robustness and Sensitivity of Feedback

17.1 Inverted Pendulum with Output

x

17.2 Inverted Pendulum with Output

17.3 Inverted Pendulum with State Feedback

17.4 Inverted Pendulum with an Integrator and State Feedback

17.5 Inverted Pendulum with State Feedback via State Estimation

Problems

Notes

References

Index

Wiley End User License Agreement

List of Illustrations

Preface

Figure 1 Control system in standard block diagram form.

Figure 2 Position control of a DC motor.

Chapter 1

Figure 1.1 The four forces on an aircraft.

Figure 1.2 A lift force is due to the shape of the airfoil that results in t...

Figure 1.3 Airflow streamlines on a wing in a wind tunnel.

Figure 1.4 Venturi tube showing air flow through a bottleneck reducing the c...

Figure 1.5 Angle of attack.

Figure 1.6 The airfoils and the control surfaces on the aircraft.

Figure 1.7 Control surface used to pitch the airfoil up or down. With the co...

Figure 1.8 Using the elevators to pitch the aircraft up.

Figure 1.9 Using the ailerons to roll the aircraft.

Figure 1.10 Using the rudder to change the heading of the aircraft.

Figure 1.11 Pilot using a yoke, pedals, and throttle to control the aircraft...

Figure 1.12 Photo of a PARROT quadrotor in hover.

Figure 1.13 Conservation of angular momentum requires

Figure 1.14 Quadrotor with the inertial and body coordinate systems shown.

Figure 1.15 A hovering quadrotor. The front and back propellers spin counter...

Figure 1.16 The quadrotor executing a roll motion.

Figure 1.17 The quadrotor executing a pitch motion.

Figure 1.18 The quadrotor executing a yaw motion.

Figure 1.19 Inverted pendulum.

Figure 1.20 A simple magnetic levitation system.

Figure 1.21 A laboratory magnetic levitation system.

Chapter 2

Figure 2.1

and

Figure 2.2 Converting

to polar coordinate form.

Figure 2.3

.

Figure 2.4 Location of the poles and zero of

Figure 2.5 Location of the poles and zero of

Figure 2.6 The open left half‐plane is where

.

Figure 2.7 Graph of

.

Figure 2.8 Graph of

Chapter 3

Figure 3.1

is stable if and only if all of its roots are in

, that is, in...

Chapter 4

Figure 4.1 Force acting on a mass

.

Figure 4.2 (a) Mass–spring–damper system. (b) Damper cross section.

Figure 4.3 Vertical mass–spring–mass–damper system.

Figure 4.4 Equations of motion about the equilibrium point.

Figure 4.5 Two masses connected by a spring and a damper.

Figure 4.6 Mass–spring–damper system all connected in series.

Figure 4.7 Mass–spring–damper system with the position

as input.

Figure 4.8 SIMULINK block diagram for

.

Figure 4.9 SIMULINK block diagram for

.

Figure 4.10 Simulation of

Figure 4.11 Dialog box for the configuration parameters.

Figure 4.12 Dialog box for the

To File

block.

Figure 4.13 Vertical mass–spring system.

Figure 4.14 Vertical mass–spring–damper system.

Figure 4.15 Moving mass–spring–damper system.

Figure 4.16 Vertical mass–spring–damper system.

Figure 4.17 Vertical mass–spring–damper modeling a suspension system.

Figure 4.18 Horizontal mass–spring–damper system.

Figure 4.19 Vertical mass–spring–damper system.

Figure 4.20 SIMULINK simulation of a DC motor.

Figure 4.21 Dialog box for the

Saturation

block.

Figure 4.22 SIMULINK block diagram for

Figure 4.23 Dialog block for the

Sine Wave

source block for

Figure 4.24 Dialog block for

Sine Wave

source for

Chapter 5

Figure 5.1 Cylinder constrained to rotate about a fixed axis.

Figure 5.2 Cylinder is considered to be made up of small masses

Figure 5.3 Force

applied to the cylinder is resolved into a normal and tan...

Figure 5.4 Viscous friction torque.

Figure 5.5 Sign convention for torque.

Figure 5.6 Rack and pinion system.

is the force of the pinion tooth on the...

Figure 5.7 Rack and pinion system.

Figure 5.8 (a) Satellite with solar panels for power. (b) Lumped parameter m...

Figure 5.9 Two gear system.

Figure 5.10 Dynamic equations for a two gear system.

Figure 5.11 Illustration of load torque.

Figure 5.12 Rolling mill.

Figure 5.13

is the tension in the cable or rope.

Figure 5.14 Massless pulley.

Figure 5.15 (a) A cylinder moving at velocity

with no angular velocity abo...

Figure 5.16 Translational and rotational motion.

Figure 5.17 Cylinder rolling on a flat surface with no slip.

Figure 5.18 Cylinder rolling down an incline under the influence of gravity....

Figure 5.19 The interaction of the surfaces of the incline and cylinder mode...

Figure 5.20 Static friction.

Figure 5.21 Height of the cylinder axis above the horizontal.

Figure 5.22 Block of mass

sliding down a frictionless inclined plane.

Figure 5.23 Cylinder going up an incline using a motor.

Figure 5.24 The interaction of the surfaces of the incline and cylinder mode...

Figure 5.25 Uniform density rod rotating on a pivot at one end.

Figure 5.26 Rod of length

rotating about its center of mass.

Figure 5.27 Torsional and linear mass–spring–damper system.

Figure 5.28 Winding up a cable with a load on it.

Figure 5.29 Gear system for an elevator.

Figure 5.30 Two pulley system.

Figure 5.31 Cable wire unwinding due to gravity.

Figure 5.32 Rotational mass–spring–damper System.

Figure 5.33 Beam attached to a wall and suspended from the ceiling.

Figure 5.34 Cylinder pulled up an incline by a sliding mass.

Figure 5.35 Mass, cylinder, and pulley system.

Figure 5.36 Mass, cylinder, and pulley system.

Figure 5.37 Box sliding on an incline being pulled up by

Figure 5.38 Cylinder rolling on a flat surface.

Figure 5.39 Kinetic friction and a slipping cylinder.

Figure 5.40 Calculation of the vertical height of the axis of rotation.

Figure 5.41 Vehicle rolling down an incline.

Figure 5.42 Vertical heights of the vehicle's axels and center of mass.

Figure 5.43 Vehicle rolling down an incline.

Chapter 6

Figure 6.1 Magnetic force law.

Figure 6.2 Only the component

of the magnetic field that is perpendicular ...

Figure 6.3 A linear DC motor.

Figure 6.4 Soft iron cylindrical core placed inside a hollowed out permanent...

Figure 6.5 A single‐loop motor.

Figure 6.6 Cylindrical coordinate system used in Figure 6.5.

Figure 6.7 (a) DC motor with a field winding. (b) Radial magnetic field stre...

Figure 6.8a

.

Figure 6.8b Rotor loop just prior to commutation where

.

Figure 6.8c The ends of the rotor loop are shorted when

Figure 6.8d Rotor loop just after commutation where

Figure 6.9 A magnet moving upwards produces a changing flux in the loop that...

Figure 6.10 (a) Positive direction of travel around a surface element with t...

Figure 6.11 Positive direction of travel around two joined surface elements....

Figure 6.12 Positive direction of travel around a surface boundary.

Figure 6.13 With

, the direction of positive travel is in the counterclockw...

Figure 6.14 Flux surface for the single loop motor.

Figure 6.15 Surface element vector for the flux surface of Figure 6.14.

Figure 6.16 The rotor flux

due to the external magnetic field vs.

.

Figure 6.17 Computation of the inductance of the rotor loop. The surface ele...

Figure 6.18 (a) Rotor loop. (b) Equivalent circuit.

Figure 6.19 Equivalent circuit of the armature electrical dynamics.

Figure 6.20 DC motor drawing and schematic.

Figure 6.21 Saturation model of an amplifier.

Figure 6.22 Schematic diagram of an optical encoder.

Figure 6.23 (a) Voltage waveforms for clockwise rotation. (b) Voltage wavefo...

Figure 6.24 Plot of

and the encoder output

.

Figure 6.25 DC tachometer (generator).

Figure 6.26 Single loop motor and tachometer.

Figure 6.27 Cutaway view of the DC tachometer.

Figure 6.28 A multiloop armature for a DC motor.

Figure 6.29 Commutator for the rotor in Figure 6.28.

Figure 6.30 Photo of the rotor of a DC motor (left) and its tachometer (righ...

Figure 6.31a (a). Rotor loops and commutator for four sets of rotor loops. B...

Figure 6.31b Rotor turned

with respect to Figure 6.31.

Figure 6.31c Rotor turned

with respect to Figure 6.31a.

Figure 6.32 Induced emf in a loop due to a moving magnet.

Figure 6.33 Induced emf in a loop due to a moving magnet.

Figure 6.34 A linear DC motor.

Figure 6.35 Linear DC machine with

,

.

Figure 6.36 Computing the torque produced by a DC motor.

Figure 6.37 Computing the flux with

.

Figure 6.38 Rotor loop where

Figure 6.39 Flux surface with the normal radially out.

Figure 6.40 Flux surface with the normal radially in.

Figure 6.41 DC motor SIMULINK block diagram.

Figure 6.42 SIMULINK block diagram for a DC motor and optical encoder.

Figure 6.43 Dialog box for the

Rounding Function

block.

Figure 6.44 Dialog box for the

Discrete Transfer Fcn

block.

Figure 6.45 Dialog box for the

Zero Order Hold

block.

Figure 6.46 DC motor with a field winding and gear system.

Figure 6.47 Rolling mill powered by a DC motor.

Chapter 7

Figure 7.1 Block diagram of a DC motor.

Figure 7.2 Block diagram of the DC motor with

.

Figure 7.3 Equivalent block diagram of the DC motor with the disturbance tor...

Figure 7.4 Equivalent block diagram of the DC motor with the disturbance tor...

Figure 7.5 Standard block diagram.

Figure 7.6 Block diagram of a DC motor.

Figure 7.7 Block diagram of simple proportional feedback for a DC motor.

Figure 7.8 Typical block diagram of a control system.

Figure 7.9 Equivalent block diagram of Figure 7.8.

is no longer available ...

Figure 7.10 Block diagram reduction example.

Figure 7.11 Equivalent block diagram to that of Figure 7.10.

Figure 7.12 Reduction of the block diagram of Figure 7.11

Figure 7.13

Figure 7.14 Block diagram reduction example.

Figure 7.15 Block diagram equivalent to Figure 7.14.

Figure 7.16 Block diagram equivalent to Figure 7.15.

Figure 7.17 Block diagram equivalent to Figure 7.16.

Figure 7.18 Block diagram reduction.

Figure 7.19 Block diagram equivalent to Figure 7.18

Figure 7.20 Block diagram equivalent to Figure 7.18.

Figure 7.21 DC motor with an inner current control loop.

Figure 7.22 Equivalent block diagram.

Figure 7.23 Simplified block diagram.

Figure 7.24 Reduced block diagram.

Figure 7.25 Current command model of a DC motor.

Figure 7.26 Block diagram reduction.

Figure 7.27 Block diagram reduction.

Figure 7.28 Block diagram reduction.

Figure 7.29 Block diagram equivalent to Figure 7.28.

Figure 7.30 Motor whose output shaft is flexible.

Figure 7.31 Block diagram of a motor whose output shaft is flexible.

Figure 7.32 Block diagram reduction.

Figure 7.33 PI current controller.

Figure 7.34 Simulation diagram for a third‐order transfer function.

Figure 7.35 Block diagram for a DC motor control system.

Figure 7.36

SIMULINK

block diagram for the DC motor.

Chapter 8

Figure 8.1 Block diagram of a DC motor.

Figure 8.2 Simplified block diagram of a DC motor.

Figure 8.3 Time constant form for a first‐order transfer function.

Figure 8.4 Step response of a first‐order system.

Figure 8.5 Position control of a DC motor.

Figure 8.6 Proportional feedback position control of a DC motor.

Figure 8.7 The closed‐loop poles

for

.

Figure 8.8 The locus of the closed‐loop poles

for

Figure 8.9 The unit step response

for

.

Figure 8.10 Rise time

, peak time

and settling time

.

Figure 8.11 Using the envelope of the step response to compute an approximat...

Figure 8.12 Simple proportional feedback control.

Figure 8.13 Sketch of closed‐loop poles as

is varied from

to

.

Figure 8.14 Choose

so that

Figure 8.15 Speed control system for a DC motor.

Figure 8.16 Block diagram of a PI speed control system for a DC motor.

Figure 8.17 Pole–zero plot of

with

.

Figure 8.18 Unit step responses for

(

).

Figure 8.19 Unit step response of a second‐order system with a zero in the o...

Figure 8.20 Pole–zero plot for

with

.

Figure 8.21 Unit step response of a second‐order system with a zero in the o...

Figure 8.22 Pole–zero plot for

Figure 8.23 Unit step responses of a third‐order system which has one real p...

Figure 8.24 Step response of a second‐order system.

Figure 8.25

and

.

Figure 8.26

SIMULINK

simulation using the output of an optical encoder to ca...

Figure 8.27

SIMULINK

setup for a DC motor and optical encoder.

Figure 8.28 Block diagram of an antenna pointing system.

Figure 8.29 System with a right half‐plane zero.

Figure 8.30

Figure 8.31 Proportional speed control of a DC motor.

Figure 8.32 Block diagram for proportional speed controller.

Figure 8.33 Block diagram for PI speed control of a DC motor.

Figure 8.34 Values

and

can result in left‐half or right‐half plane zeros...

Chapter 9

Figure 9.1 DC motor, power amplifier, gears. and encoder of a servo system....

Figure 9.2 Schematic diagram of a servomechanism.

Figure 9.3 Block diagram for the servo system.

Figure 9.4 Equivalent block diagram of the servo system with

Figure 9.5 Block diagram equivalent to Figure 9.4.

Figure 9.6 A simplified block diagram of the servo system.

Figure 9.7 (a) DC motor servo system. (b) Block diagram representation.

Figure 9.8 Block diagram of servo system in standard form.

Figure 9.9 Control system in standard block diagram form.

Figure 9.10 Equivalent block diagram to Figure 9.9.

Figure 9.11 Tracking a step input with

.

Figure 9.12 Tracking a ramp input with

.

Figure 9.13 Error with a ramp input and using a proportional controller.

Figure 9.14 An integral controller.

Figure 9.15 A proportional plus integral (PI) controller.

Figure 9.16 A proportional controller with a load acting on the system.

Figure 9.17 Examples of torque loads on DC motors.

Figure 9.18 PI controller for disturbance rejection.

Figure 9.19

to cancel out the effect of the load torque.

Figure 9.20 Shaded area is

and

Figure 9.21 A proportional plus integral plus derivative controller.

Figure 9.22 (a) An error signal. (b) An error signal with high‐frequency low...

Figure 9.23 Differentiation followed by low pass filtering.

Figure 9.24 PI‐D implementation of derivative feedback.

Figure 9.25 Equivalent block diagram of Figure 9.24.

Figure 9.26 Equivalent block diagram of Figure 9.25 with

Figure 9.27 Example of a PI‐D controller design.

Figure 9.28 Block diagram for tracking a type

input.

Figure 9.29 Block diagram for rejecting a type

disturbance.

Figure 9.30 Asymptotically rejecting a sinusoidal disturbance.

Figure 9.31 Using the elevators to pitch the aircraft up.

Figure 9.32 PI‐D unity feedback control system.

Figure 9.33 Output responses in radians for the P and PI controllers.

Figure 9.34 Output response in radians of the PI‐D controller with no actuat...

Figure 9.35 Elevator command

in degrees.

Figure 9.36 Pitch angle and ramp reference.

Figure 9.37 Elevator angle vs. time when using the ramp reference input.

Figure 9.38 Pitch response using the design model

and the truth model

Figure 9.39 Elevator commands

using the design model

and the truth model...

Figure 9.40 DC motor servo system.

Figure 9.41 Block diagram of the DC motor servo system. Source: Adapted from...

Figure 9.42 Simplified block diagram of a DC motor servo system.

Figure 9.43 Simple proportional feedback control of speed.

Figure 9.44 Open‐loop controller to speed up the response.

Figure 9.45 Open‐loop speed control.

Figure 9.46 Open‐loop speed control with uncertainty in

Figure 9.47 Proportional feedback speed control.

Figure 9.48 Operational amplifier feedback system.

Figure 9.49 Block diagram representation of feedback for an Op Amp.

Figure 9.50 Block diagram for a double integrator control system.

Figure 9.51 Tracking and disturbance rejection of an unstable system.

Figure 9.52 Pole‐placement and disturbance rejection.

Figure 9.53 Pole‐placement and disturbance rejection.

Figure 9.54 Pole‐placement and disturbance rejection.

Figure 9.55 Controller design for a non‐minimum phase system.

Figure 9.56 Disturbance rejection for a DC motor.

Figure 9.57 Geosynchronous satellite.

 rev/day.

Figure 9.58 Satellite control jets.

Figure 9.59 Block diagram for the satellite pointing control system.

Figure 9.60 Satellite pointing control system with a gyroscope added.

Figure 9.61 Block diagram reduction of the satellite control system with gyr...

Figure 9.62 Tracking of

stable

reference signals.

Figure 9.63 Rejection of a sinusoidal disturbance.

Figure 9.64 Asymptotically rejecting a sinusoidal disturbance.

Figure 9.65 Missile attitude control.

Figure 9.66 Missile control system.

Figure 9.67 Unity feedback control system.

Figure 9.68 Control system with both a lead and an integral controller.

Figure 9.69 Lead controller replaced by a PD controller to determine the gai...

Figure 9.70 Internal model principle.

Figure 9.71 Tracking a sinusoidal signal.

Figure 9.72 Using a tachometer to speed up the system response.

Figure 9.73 I‐PD unity feedback control system.

Chapter 10

Figure 10.1 Pole placement for a second‐order system.

Figure 10.2 Pole placement for a second‐order system.

Figure 10.3 Disturbance rejection for a second‐order system.

Figure 10.4 Asymptotic tracking of a sinusoidal input.

Figure 10.5 An unstable system with a right half‐plane zero.

Figure 10.6 Step response with all poles at

.

Figure 10.7 Block diagram of a DC motor.

Figure 10.8 Block diagram with the disturbance entering via the input.

Figure 10.9 Including the initial conditions in the block diagram of a syste...

Figure 10.10 Feedback control system with the initial conditions included.

Figure 10.11 Block diagram equivalent to Figure 10.10.

Figure 10.12 Inverted pendulum on a cart. The center of mass of the pendulum...

Figure 10.13 Closed‐loop controller for the inverted pendulum.

Figure 10.14 PI‐D controller.

Figure 10.15 Equivalent block diagram of Figure 10.14.

Figure 10.16 Step response of the system of Figure 10.14 with the three clos...

Figure 10.17 Stable type 2 system with a step input.

Figure 10.18 Two degree of freedom controller to eliminate overshoot.

Figure 10.19 Step response of the system of Figure 10.18 with the closed‐loo...

Figure 10.20 Open‐loop system with a right half‐plane zero.

Figure 10.21 Two degree of freedom controller to eliminate overshoot.

Figure 10.22 Closed‐loop system with two right half‐plane zeros.

Figure 10.23 Reference filter for a system with 2 right half‐plane zeros.

Figure 10.24 Step response of the 2 DOF control system of Figure 10.23.

Figure 10.25 Two DOF controller.

Figure 10.26 Unity feedback control system.

Figure 10.27 Unstable pole–zero cancellation.

Figure 10.28 Unstable pole–zero cancellation.

Figure 10.29 Cancellation between the closed‐loop transfer function and the ...

Figure 10.30 Using the elevators to pitch the aircraft up.

Figure 10.31 2 DOF controller using pole placement.

Figure 10.32 (a) Pitch angle

. (b) Elevator command

.

Figure 10.33 Truth model. (a) Pitch angle

. (b) Elevator command

.

Figure 10.34 (a) Satellite with solar panels for power. (b) Lumped parameter...

Figure 10.35 Two DOF controller for the collocated case.

Figure 10.36 The responses

and

due to the reference input

.

Figure 10.37 General tracking and disturbance rejection problem.

Figure 10.38 Stable type 2 system with a step input.

Figure 10.39 As

is stable it follows that

. Thus for

the integral

e...

Figure 10.40 Standard unity feedback control system.

Figure 10.41 Step response of a stable system with a right half‐plane zero....

Figure 10.42 Relation between

and the settling time

.

Figure 10.43 Standard unity feedback control system.

Figure 10.44 Standard unity feedback control system.

Figure 10.45 Pole placement for an unstable system.

Figure 10.46 Tracking and disturbance rejection with pole placement.

Figure 10.47 Unity feedback control system.

Figure 10.48 Reference input filter for a 2 DOF controller.

Figure 10.49 Control system with a sinusoidal disturbance.

Figure 10.50 Elimination of overshoot with a 2 DOF controller.

Figure 10.51 Elimination of overshoot.

Figure 10.52 Elimination of overshoot for a system with a RHP zero.

Figure 10.53 Tracking a step input while rejecting a sinusoidal disturbance....

Figure 10.54 Tracking a step input while rejecting a sinusoidal disturbance....

Figure 10.55 Pole placement for a double integrator system.

Figure 10.56 Pole placement with a strictly proper

.

Figure 10.57 Disturbance rejection with a strictly proper

.

Figure 10.58 Stable pole–zero cancellation.

Figure 10.59 Stable pole‐zero cancellation.

Figure 10.60 Unbounded references and disturbances.

Figure 10.61 Pole placement for an unstable non‐minimum phase system.

Figure 10.62 Reference input filter to eliminate overshoot.

Figure 10.63 Tracking and disturbance rejection of an unstable non‐minimum p...

Figure 10.64 Reference input filter to eliminate overshoot.

Figure 10.65 Control of a mass–spring–damper system.

Figure 10.66 Control of a mass–spring–damper system.

Figure 10.67

SIMULINK

block diagram for the inverted pendulum.

Figure 10.68 Unity feedback control system.

Figure 10.69 Two DOF controller for the non‐collocated case.

Figure 10.70 Two DOF controller for the non‐collocated case.

Figure 10.71 Two DOF control system for a DC motor with a flexible shaft.

Figure 10.72 Cascade of a lead, a PI, and a notch controller.

Figure 10.73 Pole placement controller.

Figure 10.74 Two DOF controller.

Chapter 11

Figure 11.1. Bode magnitude plot of

where

Figure 11.2. Asymptotic magnitude plot of

where

Figure 11.3. Bode phase diagram of

where

Figure 11.4. Magnitude and phase plots of

with

Figure 11.5. Bode magnitude and phase plots of

with

Figure 11.6. Bode diagram of

Figure 11.7. Bode plot of

and

Figure 11.8. Bode plot of

Figure 11.9. Bode magnitude plot

Figure 11.10.

vs.

Figure 11.11.

vs.

Figure 11.12.

vs.

Figure 11.13.

vs.

.

Figure 11.14.

vs.

Figure 11.15. Bode diagram of

Figure 11.16.

vs.

for

.

Figure 11.17. (a)

vs.

(b)

vs.

Figure 11.18.

.

Figure 11.19.

vs.

with

Figure 11.20.

vs.

Figure 11.21.

vs.

.

Figure 11.22.

.

Figure 11.23.

.

Figure 11.24.

.

Figure 11.25.

.

Figure 11.26.

.

Figure 11.27. Nyquist polar plot of

Figure 11.28. Nyquist polar plot of

The plot of

(blue) is asymptotic to ...

Figure 11.29. Examples of Nyquist contours.

Figure 11.30. Nyquist polar plot of

Figure 11.31. Bode diagram of

Figure 11.32. Nyquist polar plot of

Figure 11.33. Bode diagram of

Figure 11.34. Proportional feedback control system.

Figure 11.35. Nyquist plot of

Figure 11.36. Nyquist plot of

for

Figure 11.37. Nyquist plot of

for

Figure 11.38. Nyquist plot of

for

or

Figure 11.39. Nyquist plot of

for

or

Figure 11.40. Nyquist plot of

Figure 11.41. Bode diagram of

Figure 11.42. Nyquist plot of

Figure 11.43. Nyquist plot of

Figure 11.44. Bode diagram of

Figure 11.45. Nyquist plot of

Figure 11.46. Nyquist plot of

Figure 11.47. Bode plot of

Figure 11.48. Unity feedback controller.

Figure 11.49.

Figure 11.50.

and

Figure 11.51. Nyquist plot of

Figure 11.52. Bode diagram of

Figure 11.53.

Figure 11.54. Bode diagram of

Figure 11.55. Bode diagram of

Figure 11.56. The vector (complex number) from

to

is

Figure 11.57.

vs.

Figure 11.58. Closed‐loop control system.

Figure 11.59. Standard unity feedback control configuration.

Figure 11.60. How increasing the gain can decrease the phase margin.

Figure 11.61. Bode diagram of a lag compensator

Figure 11.62.

and

for lag compensation.

Figure 11.63. Lead compensator

Figure 11.64.

and

for lead compensation.

Figure 11.65. Unity feedback control structure.

Figure 11.66. Bode diagram of

Figure 11.67. Bode diagram for

Figure 11.68. Nyquist contour for

Figure 11.69. Polar plot for

Figure 11.70. Bode diagram for

Figure 11.71. Nyquist contour for

Figure 11.72. Polar plot of

Figure 11.73. Closed‐loop controller for the inverted pendulum.

Figure 11.74. (a) Nyquist contour. (b) Nyquist plot.

Figure 11.75.

?

Figure 11.76.

?

Figure 11.77.

Figure 11.78.

Figure 11.79.

Figure 11.80.

Figure 11.81. Asymptotic Bode magnitude plot.

Figure 11.82. Nyquist contour.

Figure 11.83. Nyquist contour.

Figure 11.84. Proportional feedback control system.

Figure 11.85.

Figure 11.86. Proportional feedback control system.

Figure 11.87.

and

.

Figure 11.88. Simple proportional feedback control system.

Figure 11.89.

Figure 11.90. Gain and phase margin.

Figure 11.91. Bode diagram of an open‐loop system.

Figure 11.92. Unity feedback control configuration.

Figure 11.93. Pole placement for a double integrator system.

Figure 11.94.

Figure 11.95. System with right half‐plane poles and zeros.

Figure 11.96. Bode diagram for

Figure 11.97. Nyquist contour and Nyquist plot for

.

Figure 11.98. Nyquist contour for

Figure 11.99. Polar plot of

Figure 11.100. Two DOF control system for a DC motor with a flexible shaft....

Figure 11.101. Nyquist contour for

Figure 11.102. Polar plot for

Figure 11.103. PID controller with the notch filter

.

Chapter 12

Figure 12.1. System block diagram.

Figure 12.2. A plot of the roots of

for

Figure 12.3. (a) The vectors “

” and “

”. (b) The angles

and

.

Figure 12.4. System block diagram.

Figure 12.5. Real axis root locus.

Figure 12.6.

at

Figure 12.7.

axis intercepts at

Figure 12.8. Root locus plot of

Figure 12.9. System block diagram.

Figure 12.10. Asymptotes and their intercept on the real axis.

Figure 12.11. Root locus plot with the three asymptotes showing their inters...

Figure 12.12. System block diagram.

Figure 12.13. Real axis root locus.

Figure 12.14.

at

Figure 12.15. Root locus plot of

Figure 12.16. Closed‐loop system block diagram.

Figure 12.17.

on the real axis.

Figure 12.18. Real axis root loci.

Figure 12.19.

axis intercept points.

Figure 12.20. Asymptotes and their real axis intercept.

Figure 12.21. Angle of departure from the open‐loop pole at

Figure 12.22. The angles of departure from a complex conjugate pair of open‐...

Figure 12.23. System block diagram.

Figure 12.24.

on the real axis.

Figure 12.25. Real axis root locus.

Figure 12.26. Angle of departure from the open‐loop pole at

Figure 12.27. Root locus plot of

Figure 12.28. Proportional control of

Figure 12.29. Real axis root locus.

Figure 12.30. Breakaway point,

axis intercepts, and asymptotes.

Figure 12.31. Angle of departure from

Figure 12.32. Root locus plot of

Figure 12.33. Effect of open‐loop poles on the root locus.

Figure 12.34. Effect of open‐loop zeros on the root locus.

Figure 12.35. Effect of the breakaway point on the root locus.

Figure 12.36. (a) Satellite with solar panels for power. (b) Lumped paramete...

Figure 12.37. Controller for

Figure 12.38. Root locus of

Figure 12.39. Zoomed in root locus of

with

Figure 12.40. Root locus of

with

Figure 12.41. Simulation of the responses

and

along with the reference i...

Figure 12.42. Root locus of

Figure 12.43. Root locus of

Figure 12.44. Root locus of

Figure 12.45. Control system for a magnetically levitated steel ball.

Figure 12.46. Control system for a magnetically levitated steel ball.

Figure 12.47. Pole placement controller for a magnetic levitation system.

Figure 12.48. Root locus for a PI controller.

Figure 12.49. Root locus with a lag controller.

Figure 12.50. Root locus for

Figure 12.51. Root locus for

Figure 12.52. Root locus for

Figure 12.53. Root locus for

Figure 12.54. Root locus of

Figure 12.55. Control system for a magnetically levitated steel ball.

Figure 12.56. Pole placement controller for the magnetic levitation system....

Chapter 13

Figure 13.1. (a) Inverted pendulum. (b) Free body diagram. (c)

and

.

Figure 13.2. Control structure to feedback

and

separately.

Figure 13.3. Block diagram equivalent to Figure 13.2.

Figure 13.4. Reference input filter added for output tracking of steps and r...

Figure 13.5. Block diagram used to compute

Figure 13.6. Schematic diagram of a magnetically levitated steel ball.

Figure 13.7. Current command amplifier for the magnetic levitation system.

Figure 13.8. Block diagram of the linear model of the magnetic levitation sy...

Figure 13.9. Cart on a track with one end of the track raised an angle

.

Figure 13.10. DC motor schematic.

and

Figure 13.11. Block diagram of the cart on a track.

Figure 13.12. Equivalent block diagram for the cart and track system.

Figure 13.13. Equivalent transfer function model of the cart and track syste...

Figure 13.14. Control of the inverted pendulum using nested loops.

Figure 13.15. Inside the

Linear Statespace Model

of Figure 13.14.

Figure 13.16. Nested loop control system with nonlinear statespace model of ...

Figure 13.17. Inside the

Pendulum on Cart

block.

Figure 13.18. A

zero_order_hold

bock followed by a

gain

block added ...

Figure 13.19. A

gain

block followed by a

floor

block added to model ...

Figure 13.20. Controller for the inverted pendulum with three nested loops....

Figure 13.21. Output feedback controller for the magnetic levitation system....

Figure 13.22. Steel ball resting on a pedestal to start.

Figure 13.23.

SIMULINK

block diagram for the closed‐loop controller.

Figure 13.24.

SIMULINK

model for the open‐loop magnetic levitation system.

Figure 13.25. Controller for the third‐order magnetic levitation model.

Figure 13.26. Simulation of the transfer function control system model.

Figure 13.27.

SIMULINK

block diagram for the open loop cart model.

Figure 13.28. Inside the

Linear Cart

block of Figure 13.27.

Figure 13.29. Voltage input to the cart motor.

Figure 13.30. Reference input for the cart position.

Figure 13.31. Sinusoidal tracking of cart position by the inverted pendulum....

Chapter 14

Figure 14.1. (a) Spring–mass–damper system. (b) Damper cross section.

Figure 14.2 DC motor and its schematic diagram.

Figure 14.3. Simulation block diagram for

Figure 14.4. Simulation diagram for

Figure 14.5. Simulation diagram

Figure 14.6 Simulation diagram for

Figure 14.7. First step of a block diagram reduction of Figure 14.6.

Figure 14.8. Second step of a block diagram reduction of Figure 14.6.

Figure 14.9. Block diagram reduction of Figure 14.6.

Figure 14.10 Simulation diagram of

Figure 14.11 Unity feedback control system.

Figure 14.12 Simulation diagram for the controller given Figure 14.11.

Figure 14.13 Unity feedback control system.

Figure 14.14. Mass–spring–damper system.

Figure 14.15. Output feedback controller for the inverted pendulum.

Chapter 15

Figure 15.1. Inverted pendulum on a cart.

Figure 15.2. Cart on track system.

Figure 15.3. Transfer function model of the cart on a track system.

Figure 15.4. Reference speed and position profiles.

Figure 15.5. Acceleration reference and jerk reference.

Figure 15.6. State feedback trajectory tracking controller.

Figure 15.7. General setup for state feedback trajectory tracking.

Figure 15.8. Equivalent setup for state feedback trajectory tracking.

Figure 15.9. Trajectory tracking and disturbance rejection by state feedback...

Figure 15.10. Equivalent state feedback setups for tracking step inputs.

Figure 15.11. Stabilizing the inverted pendulum using state feedback.

Figure 15.12. This diagram is equivalent to Figure 15.11.

Figure 15.13. Transfer function representation of Figure 15.12.

Figure 15.14. Use integrator feedback to asymptotically track step inputs.

Figure 15.15. Statespace tracking of step inputs.

Figure 15.16. Step response

with the input applied at

second.

Figure 15.17. The cart and pendulum as it goes from

to

Figure 15.18. Cart acceleration.

Figure 15.19. Response of the pendulum angle

to a step input in position....

Figure 15.20. Inverted pendulum with the track raised at one end.

Figure 15.21. Control structure to reject the gravity force disturbance

on...

Figure 15.22. Block diagram for feedback linearization.

Figure 15.23. Block diagram for a state feedback trajectory tracking control...

Figure 15.24. Current command amplifier for the magnetic levitation system....

Figure 15.25. Statespace controller for the magnetic levitation system.

Figure 15.26.

SIMULINK

diagram for cart on the track state feedback controll...

Figure 15.27.

SIMULINK

block diagram for state feedback control of the inver...

Figure 15.28. Feedback architecture for tracking a sinusoidal reference.

Chapter 16

Figure 16.1 Combined state feedback trajectory tracking controller and speed...

Figure 16.2 Combined state feedback trajectory tracking controller and speed...

Figure 16.3 Simulation of the system.

Figure 16.4 State estimator.

Figure 16.5 Trajectory tracking with a state estimator.

Figure 16.6 Equivalent setup to Figure 16.5.

Figure 16.7 Transfer function representation of Figure 16.6.

Figure 16.8 Block diagram equivalent to Figure 16.7.

Figure 16.9 Block diagram equivalent to Figure 16.8.

Figure 16.10 The

SIMULINK

state‐space

block.

Figure 16.11 Dialog box for

State‐Space

block.

Figure 16.12

SIMULINK

block diagram for collecting data.

Figure 16.13 Dialog box for the chirp signal. The

Initial Frequency

and the

Figure 16.14 Dialog box for the

zero‐order hold

block.

Figure 16.15 Dialog box for the

To Workspace

block.

Figure 16.16

SIMULINK

block diagram for collecting data.

Figure 16.17 Dialog box for the

discrete filter

block.

Figure 16.18 Dialog box for the differentiation filter.

Chapter 17

Figure 17.1 Output pole placement controller.

Figure 17.2 (a) Nyquist contour. (b) Nyquist plot.

Figure 17.3 Nyquist plot of

Figure 17.4

For

is close to

and

is close to

Figure 17.5 Bode diagram of

Figure 17.6 Bode diagram of

Figure 17.7 Close up of the

QUANSER

cart used to carry the pendulum bar [34]...

Figure 17.8 Bode diagram of

Figure 17.9 Bode diagram of

Figure 17.10 Plot of

where

Figure 17.11 Plot of

for

. The horizontal axis is

(not

).

Figure 17.12 Plot of

for the inverted pendulum with the closed‐loop poles ...

Figure 17.13 Output pole placement controller.

Figure 17.14 (a) Nyquist contour. (b) Nyquist plot.

Figure 17.15

vs.

Figure 17.16 (a) State feedback controller. (b) Equivalent transfer function...

Figure 17.17 Nyquist contour and plot for

Figure 17.18

Figure 17.19 Statespace tracking of step inputs.

Figure 17.20 Equivalent block diagram of Figure 17.19.

Figure 17.21 Plot of

vs.

.

Figure 17.22 State feedback control of the inverted pendulum using a state e...

Figure 17.23 Block diagram equivalent to Figure 17.22.

Figure 17.24 Block diagram equivalent to Figure 17.23.

Figure 17.25 Pole placement controller using cart position for feedback.

Figure 17.26

SIMULINK

simulation to control the inverted pendulum using cart...

Figure 17.27 Inside the

Linear Statespace Model

block of Figure 17.26.

Figure 17.28 Real axis root locus for

Figure 17.29 Pole–zero plot of

along with a presumed right half‐plane pole...

Guide

Cover

Table of Contents

Title Page

Copyright

Preface

About the Companion Website

Begin Reading

References

Index

Wiley End User License Agreement

Pages

iii

iv

ix

x

xi

xii

xiii

xiv

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

246

247

248

249

250

251

252

253

254

255

256

257

258

259

260

261

262

263

264

265

266

267

268

269

270

271

272

273

274

275

276

277

278

279

280

281

282

283

284

285

286

287

288

289

290

291

292

293

294

295

296

297

298

299

300

301

302

303

304

305

306

307

308

309

310

311

312

313

314

315

316

317

318

319

320

321

322

323

324

325

326

327

328

329

330

331

332

333

334

335

336

337

338

339

340

341

342

343

344

345

346

347

348

349

350

351

352

353

354

355

356

357

358

359

361

362

363

364

365

366

367

368

369

370

371

372

373

374

375

376

377

378

379

380

381

382

383

384

385

386

387

388

389

390

391

392

393

394

395

396

397

398

399

400

401

402

403

404

405

406

407

408

409

410

411

412

413

414

415

416

417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

439

440

441

442

443

444

445

447

448

449

450

451

452

453

454

455

456

457

458

459

460

461

462

463

464

465

466

467

468

469

470

471

472

473

474

475

476

477

478

479

480

481

482

483

484

485

486

487

488

489

490

491

492

493

494

495

496

497

498

499

500

501

502

503

504

505

506

507

508

509

510

511

512

513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

528

529

530

531

532

533

534

535

536

537

538

539

540

541

542

543

544

545

546

547

548

549

550

551

552

553

554

555

556

557

558

559

560

561

562

563

564

565

566

567

569

570

571

572

573

574

575

576

577

578

579

580

581

582

583

584

585

586

587

588

589

590

591

592

593

594

595

596

597

598

599

600

601

602

603

604

605

606

607

608

609

610

611

612

613

614

615

616

617

618

619

620

621

622

623

624

625

626

627

628

629

630

631

632

633

634

635

636

637

638

639

640

641

643

644

645

646

647

648

649

650

651

652

653

654

655

656

657

An Introduction to System Modeling and Control

John Chiasson

Boise State UniversityBoise, IdahoUnited States

This edition first published 2022© 2022 John Wiley & Sons, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of John Chiasson to be identified as the author of this work has been asserted in accordance with law.

Registered OfficeJohn Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA

Editorial Office111 River Street, Hoboken, NJ 07030, USA

For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.

Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats.

Limit of Liability/Disclaimer of WarrantyThe contents of this work are intended to further general scientific research, understanding, and discussion only and are not intended and should not be relied upon as recommending or promoting scientific method, diagnosis, or treatment by physicians for any particular patient. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of medicines, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each medicine, equipment, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging‐in‐Publication Data

Names: Chiasson, John, author.Title: An introduction to system modeling and control / John Chiasson.Description: First edition. | Hoboken, NJ : John Wiley & Sons, Inc., 2022.  | Includes bibliographical references and index.Identifiers: LCCN 2021030670 (print) | LCCN 2021030671 (ebook) | ISBN  9781119842897 (cloth) | ISBN 9781119842903 (adobe pdf) | ISBN  9781119842910 (epub)Subjects: LCSH: Systems engineering–Mathematics. | Mathematical models.Classification: LCC TA168 .C55 2022 (print) | LCC TA168 (ebook) | DDC  620.001/171–dc23LC record available at https://lccn.loc.gov/2021030670LC ebook record available at https://lccn.loc.gov/2021030671

Cover Design: WileyCover Image: © By Walter Barie

Preface

Quite honestly, I never thought I would write a book for a first controls course. So, what happened? Well, in my teaching career I have taught out of the textbooks by Ogata [1], Kuo [2], Franklin et al. [3], and Phillips and Harbor [4] with lecture notes taken from Qiu and Zhou [5] and Goodwin et al. [6]. I did find many great ideas in these texts that I do use.1 However, I was also disappointed that the modeling was not done using first principles of physics, but rather the model was simply stated without a derivation. To address this I made up my own lecture notes on rigid body dynamics (Chapter 05), DC motors (Chapter 06), and the inverted pendulum and magnetic levitation systems (Chapter 13). I realize that my emphasis on detailed modeling may be a “bug” to some (rather than a “feature”) as students find it difficult and it takes away from doing control “stuff”. However, as a colleague of mine put it, students develop important insight when they understand where the dynamic models come from and any linearization, approximation, or simplification used to obtain them.

It seemed to me early in my career that teaching the first controls course seemed to be more about its techniques, i.e., manipulating block diagrams, drawing root locus, Bode and Nyquist plots, doing the Routh–Hurwitz test, etc. Yet I think the course should be about making some physical system do what you want it to do such as having a robot arm rotate despite the weight of the object in its end effector, or ensure a magnetic bearing maintains an air gap despite various loads on it, keeping a pendulum rod pointing straight up, etc. I recall a colleague commenting on a lecture he gave in which he referred to the standard unity feedback controller block diagram (Figure 1) and told the class that the controller was to be designed so that A student then simply asked why not just get rid of the blocks and set It seems that in teaching the first controls course we end up manipulating block diagrams so much and so easily that students get lost in the abstraction not understanding what they represent.

Figure 1 Control system in standard block diagram form.

Figures such as Figure 2 on the next page are included to help the students remember what a block diagrams represents.

Spending the time to do detailed derivations of a few models helps the student to understand and remember what the transfer function models represent. A similar confusion arises in the modeling of disturbances. When a disturbance is shown on the block diagram model, it is typically placed as input to the physical system. For example, one might have a load torque on a motor, but in the block diagram this disturbance is modeled coming into the motor input, which is a voltage (Figure 1). I explain how this load torque is modeled as an equivalent voltage disturbance which has the same effect on the position/speed of the rotor as the actual load torque. This sort of understanding seems to be lost in the standard manipulations of converting a differential equation model into a block diagram model. Of course, some good laboratory work can really help to clarify these ideas as well.

Figure 2 Position control of a DC motor.

Chapter 01 presents a qualitative description of the operation of an aircraft, a quadrotor, an inverted pendulum, and a magnetically levitated steel ball to motivate the need for modeling and control.

Chapter 02 is a standard presentation of the Laplace transform theory with an emphasis on partial fractions as a way to connect the time domain to the Laplace domain.

Chapter 03 on differential equations introduces stability by giving special attention to the final value theorem (FVT). This is important as it is used over and over again to determine asymptotic tracking and/or disturbance rejection of step inputs by showing the error via the FVT . This chapter also explains how to check a differential equation for stability using the Routh–Hurwitz test.

Chapters – are modeling chapters. Chapter 04 develops mass–spring–damper systems and uses them to introduce simulation using SIMULINK. Chapter 05 presents rigid body dynamics applied to gears and rolling motion. Chapter 06 on DC motors uses the first principles of physics to develop the equations that model a DC motor and explains how both an optical encoder and a tachometer work.

Chapter 07 on block diagrams is pretty standard. It is emphasized that a block diagram is simply a graphical representation of the relationships between the various (Laplace transformed) variables of a physical system. It is shown how to rearrange and simplify them using block diagram reduction which provides a straightforward and simple way to manipulate all the block diagrams considered in the textbook.

Chapter 08