Dynamics of Controlled Mechanical Systems With Delayed Feedback

by Hu, H. Y.; Wang, Z. H.; Zaihua, W.

• ISBN13: 9783540437338
• ISBN10: 3540437339
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Springer Verlag

Summary

The time delays in controllers and actuators can either deteriorate or improve the dynamic performance of a controlled mechanical system. Thus, it is desirable to gain an insight into the effect of time delays on the dynamics of a practical system in its design phase. This monograph represents the recent advances in system modeling, analysis of stability, robust stability and bifurcation by using some new mathematical tools such as generalized Sturm criterion and Dixon's resultant elimination of polynomials. The theoretical results are demonstrated through a number of examples of active vehicle chassis, structure control, as well as the control of chaos of mechanical systems.

Table of Contents

Modeling of Delayed Dynamic Systems	1	(26)
Mathematical Models	1	(8)
Dynamic Systems with Delayed Feedback Control	1	(4)
Dynamic Systems with Operator's Retardation	5	(4)
Experimental Modeling	9	(18)
Identification of Short Time Delays in Linear Systems	10	(4)
Identification of Arbitrary Time Delays in Nonlinear Systems	14	(7)
Discussions on Identifiability of Time Delays	21	(6)
Fundamentals of Delay Differential Equations	27	(32)
Initial Value Problems	27	(10)
Existence and Uniqueness of Solution	28	(5)
Solution of Linear Delay Differential Equations	33	(4)
Stability in the Sense of Lyapunov	37	(17)
The Lyapunov Methods	38	(4)
Method of Characteristic Function	42	(5)
Stability Criteria	47	(7)
Important Features of Delay Differential Equations	54	(5)
Stability Analysis of Linear Delay Systems	59	(56)
Delay-independent Stability of Single-degree-of-freedom Systems	60	(10)
Stability Criteria	61	(5)
Stability Criteria in Terms of Feedback Gains	66	(4)
The Generalized Sturm Criterion for Polynomials	70	(6)
Classical Sturm Criterion	70	(2)
Discrimination Sequence	72	(2)
Modified Sign Table	74	(1)
Generalized Sturm Criterion	75	(1)
Delay-independent Stability of High Dimensional Systems	76	(10)
Stability of Single-degree-of-freedom Systems with Finite Time Delays	86	(5)
Systems with Equal Time Delays	86	(4)
Systems with Unequal Time Delays	90	(1)
Stability Switches of High Dimensional Systems	91	(11)
Systems with a Single Time Delay	92	(6)
Systems with Commensurate Time Delays	98	(4)
Stability Analysis of an Active Chassis	102	(13)
A quarter Car Model of Suspension with a Delayed Sky-hook Damper	102	(7)
Four-wheel-steering Vehicle with a Time Delay in Drive's Response	109	(6)
Robust Stability of Linear Delay Systems	115	(36)
Robust Stability of a One-parameter Family of Quasi-polynomials	116	(8)
Non-convexity of the Set of Hurwitz Stable Quasi-polynomials	117	(3)
Sufficient and Necessary Conditions for Interval Stability	120	(4)
Edge Theorem for a Polytopic Family of Quasi-polynomials	124	(6)
Problem Formulation	125	(2)
Edge Theorem	127	(1)
Sufficient and Necessary Conditions	127	(3)
Dixon's Resultant Elimination	130	(10)
Dixon's Resultant Elimination	130	(6)
Robust D-stability of One-parameter Family of Polynomials	136	(4)
Robust Stability of Systems with Uncertain Commensurate Delays	140	(11)
Problem Formulation	141	(2)
Stability of Vertex Quasi-polynomials	143	(2)
Stability of Edge Quasi-polynomials	145	(2)
A sufficient and Necessary Condition	147	(1)
An Illustrative Example	147	(4)
Effects of a Short Time Delay on System Dynamics	151	(38)
Stability Estimation of High Dimensional Systems	151	(17)
Distribution of Eigenvalues Subject to a Short Time Delay	152	(3)
Estimation of Eigenvalues	155	(3)
Illustrative Examples	158	(9)
A Relation of Orthogonality of Mode Shapes	167	(1)
Stability Test Based on the Pade Approximation	168	(11)
Test of Stability	168	(8)
Test of Interval Stability	176	(3)
Dynamics of Simplified Systems via the Taylor Expansion	179	(10)
Linear Systems with Delayed State Feedback	180	(2)
Nonlinear Systems with Delayed Velocity Feedback	182	(7)
Dimensional Reduction of Nonlinear Delay Systems	189	(24)
Decomposition of State Space of Linear Delay Systems	190	(8)
Spectrum of a Linear Operator	192	(2)
Decomposition of State Space	194	(4)
Dimensional Reduction for Stiff-soft Systems	198	(7)
A quarter Car Model as a Singularly Perturbed System	199	(1)
Center Manifold Reduction in Critical Cases	200	(2)
Reduction for Singularly Perturbed Differential Equations	202	(3)
Stability Analysis of an Active Suspension	205	(8)
Center Manifold Reduction	206	(1)
Computation of the Approximated Center Manifold	207	(3)
Stability Analysis	210	(3)
Periodic Motions of Nonlinear Delay Systems	213	(54)
The Hopf Bifurcation of Autonomous Systems	213	(9)
Theory of the Hopf Bifurcations	214	(3)
Decomposition of Bifurcating Solution	217	(2)
Bifurcating Solutions in Normal Form	219	(3)
Computation of Bifurcating Periodic Solutions	222	(12)
Method of the Fredholm Alternative	222	(5)
Stability of Bifurcating Periodic Solutions	227	(3)
Perturbation Method	230	(4)
Periodic Motions of a Duffing Oscillator with Delayed Feedback	234	(14)
Stability Switches of Equilibrium	235	(2)
Periodic Motion Determined by Method of Fredholm Alternative	237	(6)
Periodic Motion Determined by Method of Multiple Scales	243	(5)
Periodic Motions of a Forced Duffing Oscillator with Delayed Feedback	248	(11)
Primary Resonance	249	(5)
1/3 Subharmonic Resonance	254	(5)
Shooting Scheme for Locating Periodic Motions	259	(8)
Basic Concepts and Computation Scheme	259	(3)
Case Studies	262	(5)
Delayed Control of Dynamic Systems	267	(20)
Delayed Linear Feedback for Linear Systems	267	(5)
Delayed Linear Feedback and Artificial Damping	267	(2)
Delayed Resonator: A Tunable Vibration Absorber	269	(3)
Stabilization to Critically Stable Nonlinear Systems	272	(10)
Statement of Problem	274	(1)
Analysis on Stabilization	275	(3)
Case Studies	278	(2)
Discussions on Approximate Integrals	280	(2)
Controlling Chaotic Motion	282	(5)
Basic Idea	283	(1)
Choice of Feedback Gain	284	(3)
References	287	(6)
Index	293