A Mathematical Introduction to Control Theory

by Engelberg, Schlomo

• ISBN13: 9781860945700
• ISBN10: 1860945708
• Format: Hardcover
• Copyright: 2005-06-01
• Publisher: World Scientific Pub Co Inc

Summary

Striking a careful balance between mathematical rigor and engineering-oriented applications, this textbook aims to maximize the readers' understanding of both the mathematical and engineering aspects of control theory. The bedrock elements of classical control theory are comprehensively covered: the Routh-Hurwitz theorem and applications, Nyquist diagrams, Bode plots, root locus plots, the design of controllers (phase-lag, phase-lead, lag-lead, and PID), and three further advanced topics: non-linear control, modern control and discrete-time control.

Table of Contents

Preface	vii
1. Mathematical Preliminaries	1	(34)
1.1 An Introduction to the Laplace Transform	1	(1)
1.2 Properties of the Laplace Transform	2	(13)
1.3 Finding the Inverse Laplace Transform	15	(5)
1.3.1 Some Simple Inverse Transforms	16	(2)
1.3.2 The Quadratic Denominator	18	(2)
1.4 Integro-Differential Equations	20	(5)
1.5 An Introduction to Stability	25	(4)
1.5.1 Some Preliminary Manipulations	25	(1)
1.5.2 Stability	26	(2)
1.5.3 Why We Obsess about Stability	28	(1)
1.5.4 The Tacoma Narrows Bridge-a Brief Case History	29	(1)
1.6 MATLAB	29	(3)
1.6.1 Assignments	29	(2)
1.6.2 Commands	31	(1)
1.7 Exercises	32	(3)
2. Transfer Functions	35	(26)
2.1 Transfer Functions	35	(2)
2.2 The Frequency Response of a System	37	(3)
2.3 Bode Plots	40	(2)
2.4 The Time Response of Certain "Typical" Systems	42	(4)
2.4.1 First Order Systems	43	(1)
2.4.2 Second Order Systems	44	(2)
2.5 Three Important Devices and Their Transfer Functions	46	(5)
2.5.1 The Operational Amplifier (op amp)	46	(3)
2.5.2 The DC Motor	49	(1)
2.5.3 The "Simple Satellite"	50	(1)
2.6 Block Diagrams and How to Manipulate Them	51	(3)
2.7 A Final Example	54	(3)
2.8 Exercises	57	(4)
3. Feedback-An Introduction	61	(14)
3.1 Why Feedback-A First View	61	(1)
3.2 Sensitivity	62	(2)
3.3 More about Sensitivity	64	(1)
3.4 A Simple Example	65	(1)
3.5 System Behavior at DC	66	(4)
3.6 Noise Rejection	70	(1)
3.7 Exercises	71	(4)
4. The Routh-Hurwitz Criterion	75	(16)
4.1 Proof and Applications	75	(9)
4.2 A Design Example	84	(3)
4.3 Exercises	87	(4)
5. The Principle of the Argument and Its Consequences	91	(40)
5.1 More about Poles in the Right Half Plane	91	(1)
5.2 The Principle of the Argument	92	(1)
5.3 The Proof of the Principle of the Argument	93	(2)
5.4 How are Encirclements Measured?	95	(3)
5.5 First Applications to Control Theory	98	(2)
5.6 Systems with Low-Pass Open-Loop Transfer Functions	100	(6)
5.7 MATLAB and Nyquist Plots	106	(1)
5.8 The Nyquist Plot and Delays	107	(4)
5.9 Delays and the Routh-Hurwitz Criterion	111	(2)
5.10 Relative Stability	113	(5)
5.11 The Bode Plots	118	(1)
5.12 An (Approximate) Connection between Frequency Specifications and Time Specification	119	(3)
5.13 Some More Examples	122	(4)
5.14 Exercises	126	(5)
6. The Root Locus Diagram	131	(36)
6.1 The Root Locus-An Introduction	131	(2)
6.2 Rules for Plotting the Root Locus	133	(14)
6.2.1 The Symmetry of the Root Locus	133	(1)
6.2.2 Branches on the Real Axis	134	(1)
6.2.3 The Asymptotic Behavior of the Branches	135	(3)
6.2.4 Departure of Branches from the Real Axis	138	(5)
6.2.5 A "Conservation Law"	143	(1)
6.2.6 The Behavior of Branches as They Leave Finite Poles or Enter Finite Zeros	144	(1)
6.2.7 A Group of Poles and Zeros Near the Origin	145	(2)
6.3 Some (Semi-)Practical Examples	147	(14)
6.3.1 The Effect of Zeros in the Right Half-Plane	147	(1)
6.3.2 The Effect of Three Poles at the Origin	148	(2)
6.3.3 The Effect of Two Poles at the Origin	150	(1)
6.3.4 Variations on Our Theme	150	(3)
6.3.5 The Effect of a Delay on the Root Locus Plot	153	(3)
6.3.6 The Phase-lock Loop	156	(3)
6.3.7 Sounding a Cautionary Note-Pole-Zero Cancellation	159	(2)
6.4 More on the Behavior of the Roots of Q(s)/K + P(s) = 0	161	(2)
6.5 Exercises	163	(4)
7. Compensation	167	(36)
7.1 Compensation-An Introduction	167	(1)
7.2 The Attenuator	167	(1)
7.3 Phase-Lag Compensation	168	(7)
7.4 Phase-Lead Compensation	175	(5)
7.5 Lag-lead Compensation	180	(1)
7.6 The PID Controller	181	(7)
7.7 An Extended Example	188	(8)
7.7.1 The Attenuator	189	(1)
7.7.2 The Phase-Lag Compensator	189	(2)
7.7.3 The Phase-Lead Compensator	191	(2)
7.7.4 The Lag-Lead Compensator	193	(2)
7.7.5 The PD Controller	195	(1)
7.8 Exercises	196	(7)
8. Some Nonlinear Control Theory	203	(24)
8.1 Introduction	203	(1)
8.2 The Describing Function Technique	204	(12)
8.2.1 The Describing Function Concept	204	(3)
8.2.2 Predicting Limit Cycles	207	(1)
8.2.3 The Stability of Limit Cycles	208	(3)
8.2.4 More Examples	211	(3)
8.2.4.1 A Nonlinear Oscillator	211	(1)
8.2.4.2 A Comparator with a Dead Zone	212	(1)
8.2.4.3 A Simple Quantizer	213	(1)
8.2.5 Graphical Method	214	(2)
8.3 Tsypkin's Method	216	(5)
8.4 The Tsypkin Locus and the Describing Function Technique	221	(2)
8.5 Exercises	223	(4)
9. An Introduction to Modern Control	227	(24)
9.1 Introduction	227	(1)
9.2 The State Variables Formalism	227	(2)
9.3 Solving Matrix Differential Equations	229	(1)
9.4 The Significance of the Eigenvalues of the Matrix	230	(2)
9.5 Understanding Homogeneous Matrix Differential Equations	232	(1)
9.6 Understanding Inhomogeneous Equations	233	(1)
9.7 The Cayley-Hamilton Theorem	234	(1)
9.8 Controllability	235	(1)
9.9 Pole Placement	236	(1)
9.10 Observability	237	(1)
9.11 Examples	238	(7)
9.11.1 Pole Placement	238	(2)
9.11.2 Adding an Integrator	240	(1)
9.11.3 Modern Control Using MATLAB	241	(1)
9.11.4 A System that is not Observable	242	(2)
9.11.5 A System that is neither Observable nor Controllable	244	(1)
9.12 Converting Transfer Functions to State Equations	245	(1)
9.13 Some Technical Results about Series of Matrices	246	(2)
9.14 Exercises	248	(3)
10. Control of Hybrid Systems	251	(44)
10.1 Introduction	251	(1)
10.2 The Definition of the Z-Transform	251	(1)
10.3 Some Examples	252	(1)
10.4 Properties of the Z-Transform	253	(4)
10.5 Sampled-data Systems	257	(1)
10.6 The Sample-and-Hold Element	258	(2)
10.7 The Delta Function and its Laplace Transform	260	(1)
10.8 The Ideal Sampler	261	(1)
10.9 The Zero-Order Hold	261	(1)
10.10 Calculating the Pulse Transfer Function	262	(4)
10.11 Using MATLAB to Perform the Calculations	266	(2)
10.12 The Transfer Function of a Discrete-Time System	268	(1)
10.13 Adding a Digital Compensator	269	(2)
10.14 Stability of Discrete-Time Systems	271	(2)
10.15 A Condition for Stability	273	(3)
10.16 The Frequency Response	276	(2)
10.17 A Bit about Aliasing	278	(1)
10.18 The Behavior of the System in the Steady-State	278	(1)
10.19 The Bilinear Transform	279	(5)
10.20 The Behavior of the Bilinear Transform as T -> 0	284	(1)
10.21 Digital Compensators	285	(3)
10.22 When Is There No Pulse Transfer Function?	288	(1)
10.23 An Introduction to the Modified Z-Transform	289	(2)
10.24 Exercises	291	(4)
11. Answers to Selected Exercises	295	(50)
11.1 Chapter 1	295	(3)
11.1.1 Problem 1	295	(1)
11.1.2 Problem 3	296	(1)
11.1.3 Problem 5	297	(1)
11.1.4 Problem 7	298	(1)
11.2 Chapter 2	298	(5)
11.2.1 Problem 1	298	(1)
11.2.2 Problem 3	299	(1)
11.2.3 Problem 5	300	(1)
11.2.4 Problem 7	301	(2)
11.3 Chapter 3	303	(2)
11.3.1 Problem 1	303	(1)
11.3.2 Problem 3	304	(1)
11.3.3 Problem 5	304	(1)
11.3.4 Problem 7	305	(1)
11.4 Chapter 4	305	(5)
11.4.1 Problem 1	305	(1)
11.4.2 Problem 3	306	(1)
11.4.3 Problem 5	307	(1)
11.4.4 Problem 7	307	(2)
11.4.5 Problem 9	309	(1)
11.5 Chapter 5	310	(6)
11.5.1 Problem 1	310	(1)
11.5.2 Problem 3	311	(1)
11.5.3 Problem 5	311	(1)
11.5.4 Problem 7	312	(2)
11.5.5 Problem 9	314	(1)
11.5.6 Problem 11	315	(1)
11.6 Chapter 6	316	(6)
11.6.1 Problem 1	316	(1)
11.6.2 Problem 3	316	(2)
11.6.3 Problem 5	318	(1)
11.6.4 Problem 7	319	(1)
11.6.5 Problem 9	320	(2)
11.7 Chapter 7	322	(10)
11.7.1 Problem 1	322	(2)
11.7.2 Problem 3	324	(2)
11.7.3 Problem 5	326	(1)
11.7.4 Problem 7	327	(3)
11.7.5 Problem 9	330	(2)
11.8 Chapter 8	332	(5)
11.8.1 Problem 1	332	(3)
11.8.2 Problem 3	335	(1)
11.8.3 Problem 5	336	(1)
11.8.4 Problem 7	337	(1)
11.9 Chapter 9	337	(2)
11.9.1 Problem 6	337	(1)
11.9.2 Problem 7	338	(1)
11.10 Chapter 10	339	(6)
11.10.1 Problem 4	339	(1)
11.10.2 Problem 10	339	(1)
11.10.3 Problem 13	340	(2)
11.10.4 Problem 16	342	(1)
11.10.5 Problem 17	343	(1)
11.10.6 Problem 19	343	(2)
Bibliography	345	(2)
Index	347