Introduction to Control of Oscillations and Chaos

by Fradkov, A. L.; Pogromsky, Alexander Yu

• ISBN13: 9789810230692
• ISBN10: 9810230699
• Format: Hardcover
• Copyright: 1999-02-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Table of Contents

Preface	v
Notations and definitions	1	(4)
Chapter 1 Introduction	5	(24)
1.1 What is control?	5	(13)
1.1.1 Models of the controlled plants	6	(2)
1.1.2 Control goals	8	(2)
1.1.3 "Naive" control	10	(2)
1.1.4 Feedback	12	(1)
1.1.5 Uncertainty	13	(2)
1.1.6 Nonlinearity	15	(3)
1.2 What is chaos?	18	(3)
1.3 What use is it?	21	(8)
1.3.1 Mechanics and mechanical engineering	21	(1)
1.3.2 Electrical engineering and telecommunications	22	(2)
1.3.3 Chemistry and chemical engineering	24	(2)
1.3.4 Biology, biochemistry and medicine	26	(1)
1.3.5 Economics and finance	27	(2)
Chapter 2 The mathematics of nonlinear control	29	(92)
2.1 Mathematical models of controlled systems	29	(8)
2.2 Stability and boundedness	37	(24)
2.2.1 State-space stability	37	(6)
2.2.2 Stability theorems and Lyapunov functions	43	(10)
2.2.3 Absolute stability	53	(8)
2.3 Feedback linearization and normal forms	61	(6)
2.4 Feedback stabilization and passivity	67	(20)
2.4.1 Introductory comments	67	(2)
2.4.2 Passivity and dissipativity	69	(9)
2.4.3 Passification as a control design problem	78	(5)
2.4.4 Input-to-state stability	83	(4)
2.5 Speed gradient algorithms	87	(25)
2.5.1 Goal-oriented formulation of the control problem	87	(3)
2.5.2 Design of Speed Gradient Algorithms	90	(4)
2.5.3 Properties of the speed gradient algorithms	94	(16)
2.5.4 Identifying properties of SG algorithms	110	(2)
2.6 Robustness of speed gradient algorithms with respect to disturbances	112	(4)
2.7 Gradient control of discrete-time systems	116	(5)
Chapter 3 The mathematics of oscillations and chaos	121	(42)
3.1 What is oscillation?	121	(11)
3.1.1 General concepts	121	(4)
3.1.2 Oscillations in dynamical systems	125	(7)
3.2 Stability of oscillations	132	(17)
3.2.1 Convergence and synchronization	132	(7)
3.2.2 Lyapunov stability, Lyapunov exponents, Bol exponents	139	(5)
3.2.3 Computation of the Bol exponents	144	(1)
3.2.4 Orbital stability	145	(4)
3.3 Poincare maps	149	(9)
3.3.1 Definition and properties of the Poincare map	149	(2)
3.3.2 Controlled Poincare maps	151	(5)
3.3.3 Controlled closing lemma	156	(2)
3.4 What is chaos? (continued)	158	(5)
Chapter 4 Methods of nonlinear and adaptive control of oscillations	163	(110)
4.1 Adaptive control problem statement	163	(6)
4.2 Direct and identification approaches to adaptive control design	169	(4)
4.3 Adaptive systems with reference models	173	(13)
4.3.1 Problem statement	173	(1)
4.3.2 State feedback	174	(8)
4.3.3 Output feedback	182	(4)
4.4 Controlled synchronization of dynamical systems	186	(3)
4.5 Decomposition based synchronization	189	(4)
4.6 Passivity based synchronization	193	(25)
4.6.1 Semipassivity and L-dissipativity	193	(3)
4.6.2 Synchronization of two linearly coupled systems	196	(7)
4.6.3 Synchronization of several systems with multiple interconnections	203	(4)
4.6.4 Adaptive synchronization	207	(3)
4.6.5 Adaptive synchronization of uncertain semipassive systems	210	(5)
4.6.6 Adaptive synchronization of hyper-minimum-phase systems	215	(3)
4.7 Adaptive suppression of forced oscillations	218	(14)
4.8 Control of cascaded systems. Relaxation of the matching condition	232	(15)
4.8.1 Integrator backstepping	232	(6)
4.8.2 Adaptive control of unmatched systems	238	(9)
4.9 Speed Gradient control of Hamiltonian systems	247	(13)
4.9.1 Control of energy	247	(5)
4.9.2 The swinging (small control) property	252	(1)
4.9.3 Control of first integrals	253	(4)
4.9.4 Control of generalized Hamiltonian systems	257	(3)
4.10 Discrete adaptive control via linearization of Poincare map	260	(9)
4.10.1 Background and motivation	260	(1)
4.10.2 Linearization of the controlled Poincare map	261	(8)
4.11 Control of bifurcations	269	(4)
Chapter 5 Control of oscillatory and chaotic systems	273	(50)
5.1 Control of pendulums	273	(13)
5.1.1 Swinging a simple pendulum	273	(7)
5.1.2 Pendulum with a controlled suspension point	280	(6)
5.2 Stabilization of the equilibrium.point of the thermal convection loop model	286	(9)
5.3 Adaptive synchronization of two forced Duffing's systems	295	(9)
5.4 Adaptive synchronization of Chua's circuits	304	(5)
5.5 Gradient control of the Henon system	309	(5)
5.5.1 Stabilizing the unstable equilibrium of the Henon system	310	(2)
5.5.2 Synchronizing two identical Henon systems	312	(1)
5.5.3 Adaptive model reference control of the Henon system	313	(1)
5.6 Control of periodic and chaotic oscillations in the brussellator model	314	(9)
Chapter 6 Applications	323	(36)
6.1 How to tow a car out of a ditch	323	(4)
6.2 Synchronization of generators based on tunnel diodes	327	(6)
6.3 Stabilization of swings in power systems	333	(9)
6.4 Adaptive control of the thin film growth from a multicomponent gas	342	(4)
6.5 Control of oscillatory behavior of populations	346	(5)
6.6 Control of a nonlinear business-cycle model	351	(8)
Chapter 7 Conclusions: What is the message of the book?	359	(4)
Exercises	363	(4)
Bibliography	367	(22)
Index	389