Analysis and design of oscillatory control systems

This paper presents analysis and design results for control systems subject to oscillatory inputs, i.e., inputs of large amplitude and high frequency. The key analysis results are a series expansion characterizing the averaged system and various Lie-algebraic conditions that guarantee the series can be summed. Various example systems provide insight into the results. With regards to design, we recover and extend a variety of point stabilization and trajectory tracking results using oscillatory controls. We present novel developments on stabilization of systems with positive trace and on tracking for second order underactuated systems.

[1]  H. Sussmann.  New Differential Geometric Methods in Nonholonomic Path Finding , 1992 .

[2]  R. Gamkrelidze,et al.  THE EXPONENTIAL REPRESENTATION OF FLOWS AND THE CHRONOLOGICAL CALCULUS , 1979 .

[3]  John Baillieul,et al.  Energy methods for stability of bilinear systems with oscillatory inputs , 1995 .

[4]  H. Sussmann,et al.  Continuous Dependence with Respect to the Input of Trajectories of Control-Affine Systems , 1999 .

[5]  Francesco Bullo,et al.  Averaging and Vibrational Control of Mechanical Systems , 2002, SIAM J. Control. Optim..

[6]  Herschel Rabitz,et al.  Coherent Control of Quantum Dynamics: The Dream Is Alive , 1993, Science.

[7]  Naomi Ehrich Leonard,et al.  Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups , 2000, IEEE Trans. Autom. Control..

[8]  R. Abraham,et al.  Manifolds, Tensor Analysis, and Applications , 1983 .

[9]  M. R. Spiegel.  Mathematical handbook of formulas and tables , 1968 .

[10]  Gene H. Golub,et al.  Matrix computations , 1983 .

[11]  Ilya Kolmanovsky,et al.  Developments in nonholonomic control problems , 1995 .

[12]  J. Hale.  Oscillations in Nonlinear Systems , 1963 .

[13]  R. Brockett.  Control Theory and Singular Riemannian Geometry , 1982 .

[14]  Mark Levi,et al.  Geometry and physics of averaging with applications , 1999 .

[15]  Pascal Morin,et al.  Design of Homogeneous Time-Varying Stabilizing Control Laws for Driftless Controllable Systems Via Oscillatory Approximation of Lie Brackets in Closed Loop , 1999, SIAM J. Control. Optim..

[16]  Benjamin Shapiro,et al.  High-frequency nonlinear vibrational control , 1997, IEEE Trans. Autom. Control..

[17]  Jaroslav Kurzweil,et al.  Iterated Lie brackets in limit processes in ordinary differential equations , 1988 .

[18]  Joseph Bentsman,et al.  Vibrational control of a class of nonlinear systems by nonlinear multiplicative vibrations , 1986, 1986 25th IEEE Conference on Decision and Control.

[19]  Joel W. Burdick,et al.  The Geometric Mechanics of Undulatory Robotic Locomotion , 1998, Int. J. Robotics Res..

[20]  G. Paramonov.  Coherent control of linear and nonlinear multiphoton excitation of molecular vibrations , 1993 .

[21]  F. Verhulst,et al.  Averaging Methods in Nonlinear Dynamical Systems , 1985 .

[22]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[23]  L. Dai,et al.  Non-holonomic Kinematics and the Role of Elliptic Functions in Constructive Controllability , 1993 .

[24]  Roger W. Brockett.  On the rectification of vibratory motion , 1989 .

[25]  Jean-Michel Coron,et al.  Global asymptotic stabilization for controllable systems without drift , 1992, Math. Control. Signals Syst..

[26]  M. Golubitsky,et al.  Symmetry in locomotor central pattern generators and animal gaits , 1999, Nature.

[27]  A. Schaft,et al.  Variational and Hamiltonian Control Systems , 1987 .

[28]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[29]  Jean-Baptiste Pomet.  Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift , 1992 .

[30]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[31]  A. Isidori.  Nonlinear Control Systems , 1985 .

[32]  A. D. Lewis,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997 .

[33]  P. Krishnaprasad,et al.  Oscillations, SE(2)-snakes and motion control: A study of the Roller Racer , 2001 .

[34]  S. Weiner,et al.  Control and Design Principles in Biological Mineralization , 1992 .

[35]  Natasha Loder,et al.  Journal under attack over controversial paper on GM food , 1999, Nature.

[36]  J. M. Noworolski,et al.  Generalized averaging method for power conversion circuits , 1990, 21st Annual IEEE Conference on Power Electronics Specialists.

[37]  Wensheng Liu,et al.  An Approximation Algorithm for Nonholonomic Systems , 1997 .

[38]  Anthony M. Bloch,et al.  Nonlinear Dynamical Control Systems (H. Nijmeijer and A. J. van der Schaft) , 1991, SIAM Review.

[39]  M. Fliess,et al.  Flatness and defect of non-linear systems: introductory theory and examples , 1995 .

[40]  Andrey V. Sarychev.  Lie- and chronologico-algebraic tools for studying stability of time-varying systems , 2001, Syst. Control. Lett..

[41]  John Baillieul,et al.  Stable average motions of mechanical systems subject to periodic forcing , 1993 .

[42]  Mark Levi,et al.  Geometry of Kapitsa's potentials , 1998 .

[43]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[44]  Wensheng Liu,et al.  Averaging Theorems for Highly Oscillatory Differential Equations and Iterated Lie Brackets , 1997 .

[45]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[46]  R. Brockett,et al.  Nonholonomic control based on approximate inversion , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[47]  P. L. Kapitsa,et al.  Dynamical Stability of a Pendulum when its Point of Suspension Vibrates , 1965 .

[48]  広瀬 茂男,et al.  Biologically inspired robots : snake-like locomotors and manipulators , 1993 .

[49]  Sofiane Meradji,et al.  Vibrational control of crystal growth from liquid phase , 1997 .

[50]  H. Wilf.  generatingfunctionology: Third Edition , 1990 .

[51]  A. Seifert,et al.  Oscillatory Control of Separation at High Reynolds Numbers , 1999 .

[52]  J. Baillieul,et al.  Scale dependence in the oscillatory control of micromechanisms , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[53]  S. Shankar Sastry,et al.  Nonlinear control design for slightly non-minimum phase systems: Application to V/STOL aircraft , 1992, Autom..

[54]  Semyon M. Meerkov,et al.  Vibrational control of nonlinear systems: Vibrational stabilizability , 1986 .