Averaging and control of nonlinear systems

This dissertation investigates three principal areas regarding the dynamics and control of nonlinear systems: averaging theory, controllability of mechanical systems, and control of underactuated nonlinear systems. The most effective stabilizing controllers for underactuated nonlinear systems are time-periodic, which leads to the study of averaging theory for understanding the nonlinear effect generated by resonant oscillatory inputs. The research on averaging theory generalizes averaging theory to arbitrary order by synthesizing series expansion methods for nonlinear time-varying vector fields and their flows with nonlinear Floquet theory. It is shown that classical averaging theory is the application of perturbation methods in conjunction with nonlinear Floquet theory. Many known properties and consequences of averaging theory are placed within a single framework. The generalized averaging theory is merged with controllability analysis of underactuated nonlinear systems to derive exponentially stabilizing controllers. Although small-time local controllability (STLC) is easily demonstrated for driftless systems via the Lie algebra rank condition, STLC for systems with drift is more complicated. Furthermore, there exists a variety of techniques and canonical forms for determining STLC. This thesis exploits notions of geometric homogeneity to show that STLC results for a large class of mechanical systems with drift can be recovered by considering a class of nonlinear dynamical systems satisfying certain homogeneity conditions. These theorems generalize the controllability results for simple mechanical control systems found in Lewis and Murray [85]. Most nonlinear controllability results for classes of mechanical systems may be obtained using these methods. The stabilizing controllers derived using the generalized averaging theory and STLC analysis can be used to stabilize both systems with and without drift. Furthermore, they result in a set of tunable gains and oscillatory parameters for modification and improvement of the feedback strategy. The procedure can not only derive known controllers from the literature, but can also be used to improve them. Examples demonstrate the diversity of controllers constructed using the generalized averaging theory. This dissertation concludes with a chapter devoted to biomimetic and biomechanical locomotive control systems that have been stabilized using the generalized averaging theory and the controller construction procedure. The locomotive control systems roll, wriggle, swim, and walk, demonstrating the universal nature of the control strategy proposed.

[1]  N. Steenrod.  The Topology of Fibre Bundles. (PMS-14) , 1951 .

[2]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[3]  N. Fine,et al.  On the Group of Homeomorphisms of an Arc , 1955 .

[4]  Kuo-Tsai Chen,et al.  Integration of Paths, Geometric Invariants and a Generalized Baker- Hausdorff Formula , 1957 .

[5]  J. Gillis,et al.  Asymptotic Methods in the Theory of Non‐Linear Oscillations , 1963 .

[6]  K. Nomizu,et al.  Foundations of Differential Geometry , 1963 .

[7]  S. Smale.  Differentiable dynamical systems , 1967 .

[8]  J. Palis.  On Morse-Smale dynamical systems , 1969 .

[9]  Lawrence M. Perko,et al.  Higher order averaging and related methods for perturbed periodic and quasi-periodic systems , 1969 .

[10]  D. Epstein,et al.  The simplicity of certain groups of homeomorphisms , 1970 .

[11]  J. Hale,et al.  Ordinary Differential Equations , 2019, Fundamentals of Numerical Mathematics for Physicists and Engineers.

[12]  William P. Thurston,et al.  Foliations and groups of diffeomorphisms , 1974 .

[13]  John N. Mather.  Commutators of diffeomorphisms , 1974 .

[14]  Jacob Palis,et al.  Vector fields generate few diffeomorphisms , 1974 .

[15]  W. Boothby.  An introduction to differentiable manifolds and Riemannian geometry , 1975 .

[16]  Robert G. Littlejohn,et al.  A guiding center Hamiltonian: A new approach , 1979 .

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

[18]  Andrei A. Agrachev,et al.  Chronological algebras and nonstationary vector fields , 1981 .

[19]  Peter E. Crouch,et al.  Geometric structures in systems theory , 1981 .

[20]  Leopold Alexander Pars,et al.  A Treatise on Analytical Dynamics , 1981 .

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

[22]  Robert G. Littlejohn,et al.  Variational principles of guiding centre motion , 1983, Journal of Plasma Physics.

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

[24]  R. Littlejohn.  GEOMETRY AND GUIDING CENTER MOTION , 1983 .

[25]  M. Crampin.  Tangent bundle geometry Lagrangian dynamics , 1983 .

[26]  Particle channeling in crystals and the method of averaging , 1984 .

[27]  EMBEDDING OF A HOMEOMORPHISM IN A FLOW AND ASYMPTOTIC EMBEDDING , 1985 .

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

[29]  A. Isidori.  Nonlinear Control Systems: An Introduction , 1986 .

[30]  Eduardo D. Sontag,et al.  Time-optimal control of manipulators , 1986, Proceedings. 1986 IEEE International Conference on Robotics and Automation.

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

[32]  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.

[33]  Existence and stability of periodic motion under higher order averaging , 1986 .

[34]  Semyon M. Meerkov,et al.  Vibrational control of nonlinear systems: Vibrational controllability and transient behavior , 1986 .

[35]  Marc H. Raibert,et al.  Legged Robots That Balance , 1986, IEEE Expert.

[36]  S. Omohundro,et al.  Geometric Perturbation Theory In Physics , 1986 .

[37]  Jaroslav Kurweil.  Limit processes in oridinary differential equations , 1987 .

[38]  Jaroslav Kurzweil,et al.  Limit processes in ordinary differential equations , 1987 .

[39]  H. Sussmann.  A general theorem on local controllability , 1987 .

[40]  Conditions for a diffeomorghism to be embedded in aCr flow , 1988 .

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

[42]  M. Kawski.  Control variations with an increasing number of switchings , 1988 .

[43]  A. Jamiołkowski.  Applications of Lie groups to differential equations , 1989 .

[44]  S. Wiggins.  Introduction to Applied Nonlinear Dynamical Systems and Chaos , 1989 .

[45]  R. Bianchini,et al.  Graded approximations and controllability along a trajectory , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[46]  J. Palis,et al.  Rigidity of centralizers of diffeomorphisms , 1989 .

[47]  Improved Nth order averaging theory for periodic systems , 1990 .

[48]  Jack K. Hale,et al.  Averaging in infinite dimensions , 1990 .

[49]  M. Kawski.  Families of Dilations and Asymptotic Stability , 1991 .

[50]  Higher‐order averaging for nonperiodic systems , 1991 .

[51]  Gerardo Lafferriere,et al.  Motion planning for controllable systems without drift , 1991, Proceedings. 1991 IEEE International Conference on Robotics and Automation.

[52]  H. Sussmann,et al.  Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[53]  Henry Hermes,et al.  Nilpotent and High-Order Approximations of Vector Field Systems , 1991, SIAM Rev..

[54]  M. Krastanov.  A Necessary Condition for Small Time Local Controllability , 1998 .

[55]  M. Fliess,et al.  On Differentially Flat Nonlinear Systems , 1992 .

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

[57]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

[58]  O. J. Sordalen,et al.  Exponential stabilization of mobile robots with nonholonomic constraints , 1992 .

[59]  Jean-Baptiste Pomet,et al.  Explicit Design of Time Varying Stabilizing Feedback Laws for a Class of COntrollable Systems without Drift , 1992 .

[60]  A. Bloch,et al.  Control and stabilization of nonholonomic dynamic systems , 1992 .

[61]  Gerardo Lafferriere,et al.  A Differential Geometric Approach to Motion Planning , 1993 .

[62]  P. Michor,et al.  Natural operations in differential geometry , 1993 .

[63]  H. Sussmann,et al.  Lie Bracket Extensions and Averaging: The Single-Bracket Case , 1993 .

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

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

[66]  Jean-Paul Laumond,et al.  Singularities and Topological Aspects in Nonholonomic Motion Planning , 1993 .

[67]  S. Sastry,et al.  Nonholonomic motion planning: steering using sinusoids , 1993, IEEE Trans. Autom. Control..

[68]  Gianna Stefani,et al.  Controllability along a trajectory: a variational approach , 1993 .

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

[70]  Mahmut Reyhanoglu,et al.  Motion Planning for Nonholonomic Dynamic Systems , 1993 .

[71]  Joel W. Burdick,et al.  Nonholonomic mechanics and locomotion: the snakeboard example , 1994, Proceedings of the 1994 IEEE International Conference on Robotics and Automation.

[72]  J. Marsden,et al.  Introduction to mechanics and symmetry , 1994 .

[73]  Naomi Ehrich Leonard,et al.  High-order averaging on Lie groups and control of an autonomous underwater vehicle , 1994, Proceedings of 1994 American Control Conference - ACC '94.

[74]  Andrew D. Lewis,et al.  Aspects of Geometric Mechanics and Control of Mechanical Systems , 1995 .

[75]  J. Ostrowski.  The mechanics and control of undulatory robotic locomotion , 1995 .

[76]  Richard M. Murray,et al.  Nonlinear Control of Mechanical Systems: A Lagrangian Perspective , 1995 .

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

[78]  O. J. Sørdalen,et al.  Exponential stabilization of nonholonomic chained systems , 1995, IEEE Trans. Autom. Control..

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

[80]  Richard M. Murray,et al.  Non-holonomic control systems: from steering to stabilization with sinusoids , 1995 .

[81]  Richard M. Murray,et al.  Geometric phases and robotic locomotion , 1995, J. Field Robotics.

[82]  M. Dickinson.  UNSTEADY MECHANISMS OF FORCE GENERATION IN AQUATIC AND AERIAL LOCOMOTION , 1996 .

[83]  Yoshihiko Nakamura,et al.  Nonlinear control of a nonholonomic free joint manipulator with the averaging method , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[84]  Henry Hermes.  Large-Time Local Controllability Via Homogeneous Approximations , 1996 .

[85]  I. Kolmanovsky,et al.  Controllability and motion planning for noncatastatic nonholonomic control systems , 1996 .

[87]  Olav Egeland,et al.  Feedback Control of a Nonholonomic Underwater Vehicle With a Constant Desired Configuration , 1996, Int. J. Robotics Res..

[88]  P. Krishnaprasad,et al.  Nonholonomic mechanical systems with symmetry , 1996 .

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

[90]  J. Burdick,et al.  Controllability Tests for Mechanical Systems withSymmetries and ConstraintsJim , 1997 .

[91]  Fundamental theorems of averaging for functional differential equations , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[92]  Héctor J. Sussmann,et al.  Noncommutative Power Series and Formal Lie-algebraic Techniques in Nonlinear Control Theory , 1997 .

[93]  R. Murray,et al.  Exponential stabilization of driftless nonlinear control systems using homogeneous feedback , 1997, IEEE Trans. Autom. Control..

[94]  J. Marsden.  Lectures on Mechanics Second Edition , 1997 .

[95]  Perinkulam S. Krishnaprasad,et al.  On approximate inversion and feedback stabilization for systems on matrix Lie groups , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

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

[97]  Matthias Kawski,et al.  Nonlinear Control and Combinatorics of Words , 1997 .

[98]  R. M’Closkey,et al.  Time-varying homogeneous feedback: Design tools for the exponential stabilization of systems with drift , 1998 .

[99]  Andrew Y. T. Leung,et al.  HIGHER ORDER NORMAL FORM AND PERIOD AVERAGING , 1998 .

[100]  S. Kelly.  The mechanics and control of robotic locomotion with applications to aquatic vehicles , 1998 .

[101]  Joel W. Burdick,et al.  Local Motion Planning for Nonholonomic Control Systems Evolving on Principal Bundles , 1998 .

[102]  V. Arnold,et al.  Topological methods in hydrodynamics , 1998 .

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

[104]  A. Gomez,et al.  Flows and diffeomorphisms , 1998 .

[105]  J. Goodwine.  Control of stratified systems with robotic applications , 1998 .

[106]  F. Bullo,et al.  On modeling and locomotion of hybrid mechanical systems with impacts , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[107]  Kazuyuki Yagasaki,et al.  HIGHER-ORDER AVERAGING FOR PERIODICALLY FORCED WEAKLY NONLINEAR SYSTEMS , 1999 .

[108]  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..

[109]  C. Samson,et al.  Robust point-stabilization of nonlinear affine control systems , 1999 .

[110]  F. Bullo.  Nonlinear control of mechanical systems : a Riemannian geometry approach , 1999 .

[111]  M. Kawski.  Controllability via chronological calculus , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

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

[113]  C. SIAMJ.,et al.  AVERAGING RESULTS AND THE STUDY OF UNIFORM ASYMPTOTIC STABILITY OF HOMOGENEOUS DIFFERENTIAL EQUATIONS THAT ARE NOT FAST TIME-VARYING∗ , 1999 .

[114]  Richard M. Murray,et al.  Configuration Controllability of Simple Mechanical Control Systems , 1997, SIAM Rev..

[115]  Joel W. Burdick,et al.  Propulsion and control of deformable bodies in an ideal fluid , 1999, Proceedings 1999 IEEE International Conference on Robotics and Automation (Cat. No.99CH36288C).

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

[117]  Arjan van der Schaft,et al.  Dynamics and control of a class of underactuated mechanical systems , 1999, IEEE Trans. Autom. Control..

[118]  Continuous dependence of trajectories with respect to the input , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[119]  Richard M. Murray,et al.  Tracking for fully actuated mechanical systems: a geometric framework , 1999, Autom..

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

[121]  F. Golse,et al.  A mathematical theory of planar particle channeling in crystals , 2000 .

[122]  Francesco Bullo,et al.  On the homogeneity of the affine connection model for mechanical control systems , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[123]  Mahmut Reyhanoglu,et al.  Discontinuous Feedback Control of a Special Class of Underactuated Mechanical Systems , 2000 .

[124]  Andrew D. Lewis,et al.  Simple mechanical control systems with constraints , 2000, IEEE Trans. Autom. Control..

[125]  C. Samson,et al.  A characterization of the Lie algebra rank condition by transverse periodic functions , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

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

[127]  Jonathan E. Clark,et al.  Stride Period Adaptation for a Biomimetic Running Hexapod , 2001, ISRR.

[128]  Joel W. Burdick,et al.  Nonlinear control methods for planar carangiform robot fish locomotion , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[129]  Kristi A. Morgansen,et al.  Controllability and trajectory tracking for classes of cascade-form second order nonholonomic systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[130]  Francesco Bullo,et al.  Series Expansions for the Evolution of Mechanical Control Systems , 2001, SIAM J. Control. Optim..

[131]  Pascal Morin,et al.  Practical stabilization of driftless homogeneous systems based on the use of transverse periodic functions , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[132]  Richard M. Murray,et al.  Modeling and control of thin film morphology using unsteady processing parameters: Problem formulation and initial results , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

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

[134]  Jerrold E. Marsden,et al.  Geometric mechanics, Lagrangian reduction, and nonholonomic systems , 2001 .

[135]  F. Bullo,et al.  On nonlinear controllability and series expansions for Lagrangian systems with dissipative forces , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[136]  Alberto Isidori,et al.  Nonlinear control in the Year 2000 , 2001 .

[137]  Kazuyuki Ito,et al.  Local accessibility and stabilization of an underactuated crawling robot with changing constraints , 2001, Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164).

[138]  Jerrold E. Marsden,et al.  Lagrangian Reduction by Stages , 2001 .

[139]  Joel W. Burdick,et al.  Controllability of kinematic control systems on stratified configuration spaces , 2001, IEEE Trans. Autom. Control..

[140]  James P. Ostrowski,et al.  Experiments in closed-loop control for an underwater eel-like robot , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[141]  H. Zhang,et al.  Simple Mechanical Control Systems with Constraints and Symmetry , 2002, SIAM J. Control. Optim..

[142]  T. Y. Wu.  On theoretical modeling of aquatic and aerial animal locomotion , 2002 .

[143]  P.A. Vela,et al.  Second order averaging methods for oscillatory control of underactuated mechanical systems , 2002, Proceedings of the 2002 American Control Conference (IEEE Cat. No.CH37301).

[144]  Dirk Aeyels,et al.  Exponential Stability of Slowly Time-Varying Nonlinear Systems , 2002, Math. Control. Signals Syst..

[145]  Extension of Floquet's theory to nonlinear periodic differential systems and embedding diffeomorphisms in differential flows , 2002 .

[146]  Dirk Aeyels,et al.  Exponential Stability of Nonlinear Time-Varying Differential Equations and Partial Averaging , 2002, Math. Control. Signals Syst..

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

[148]  Joel W. Burdick,et al.  Underwater locomotion from oscillatory shape deformations , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[149]  Joel W. Burdick,et al.  Motion planning for kinematic stratified systems with application to quasi-static legged locomotion and finger gaiting , 2002, IEEE Trans. Robotics Autom..

[150]  Joel W. Burdick,et al.  Trajectory stabilization for a planar carangiform robot fish , 2002, Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292).

[151]  Jorge Cortes,et al.  Motion Control Algorithms for Simple Mechanical Systems with Symmetry , 2003 .

[152]  Rosa-Maria Bianchini,et al.  Needle Variations that Cannot be Summed , 2003, SIAM J. Control. Optim..

[153]  Joel W. Burdick,et al.  Fluid locomotion and trajectory planning for shape-changing robots , 2003 .

[154]  Tamer Basar,et al.  On nonlinear controllability of homogeneous systems linear in control , 2003, IEEE Trans. Autom. Control..

[155]  Sonia Martínez,et al.  Analysis and design of oscillatory control systems , 2003, IEEE Trans. Autom. Control..

[156]  V. A. Steklov.  Local Invariants of Smooth Control Systems , 2004 .