Passivity Methods for the Stabilization of Closed Sets in Nonlinear Control Systems

In this thesis we study the stabilization of closed sets for passive nonlinear control systems, developing necessary and sufficient conditions under which a passivity-based feedback stabilizes a given goal set. The development of this result takes us to a journey through the so-called reduction problem: given two nested invariant sets G1 subset of G2, and assuming that G1 enjoys certain stability properties relative to G2, under what conditions does G1 enjoy the same stability properties with respect to the whole state space? We develop reduction principles for stability, asymptotic stability, and attractivity which areapplicable to arbitrary closed sets. When applied to the passivity-based set stabilization problem, the reduction theory suggests a new definition of detectability which is geometrically appealing and captures precisely the property that the control system must possess in order for the stabilization problem to be solvable.The reduction theory and set stabilization results developed in this thesis are used tosolve a distributed coordination problem for a group of unicycles, whereby the vehiclesare required to converge to a circular formation of desired radius, with a specific ordering and spacing on the circle.

[1]  C. Samson.  Control of chained systems application to path following and time-varying point-stabilization of mobile robots , 1995, IEEE Trans. Autom. Control..

[2]  V. Jurdjevic,et al.  Controllability and stability , 1978 .

[3]  Giuseppe Oriolo,et al.  Feedback control of a nonholonomic car-like robot , 1998 .

[4]  J. Willems.  Dissipative dynamical systems Part II: Linear systems with quadratic supply rates , 1972 .

[5]  L. Marconi,et al.  Output Stabilization via Nonlinear Luenberger Observers , 2006, SIAM J. Control. Optim..

[6]  Manfredi Maggiore,et al.  Path following using transverse feedback linearization: Application to a maglev positioning system , 2009, ACC.

[7]  Naomi Ehrich Leonard,et al.  Collective Motion of Self-Propelled Particles: Stabilizing Symmetric Formations on Closed Curves , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[8]  H. Khalil,et al.  A separation principle for the stabilization of a class of nonlinear systems , 1997 .

[9]  P. Olver.  Nonlinear Systems , 2013 .

[10]  Robert Sutton,et al.  Advances in Unmanned Marine Vehicles , 2006 .

[11]  Kimon P. Valavanis,et al.  Advances in Unmanned Aerial Vehicles: State of the Art and the Road to Autonomy , 2007 .

[12]  Roger Skjetne,et al.  Stabilization of Sets Parametrized by a Single Variable : Application to Ship Maneuvering , 2002 .

[13]  Dirk Aeyels,et al.  Stabilization of positive systems with first integrals , 2002, Autom..

[14]  Ludovic Rifford,et al.  Semiconcave Control-Lyapunov Functions and Stabilizing Feedbacks , 2002, SIAM J. Control. Optim..

[15]  Romeo Ortega,et al.  Passivity-based Control of Euler-Lagrange Systems , 1998 .

[16]  Brian D. O. Anderson,et al.  Network Analysis and Synthesis , 1973 .

[17]  Z. Artstein.  Stabilization with relaxed controls , 1983 .

[18]  Jean-Pierre Aubin,et al.  Viability theory , 1991 .

[19]  Bruce A. Francis,et al.  The internal model principle of control theory , 1976, Autom..

[20]  Anton S. Shiriaev,et al.  The notion of V-detectability and stabilization of invariant sets of nonlinear systems , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[21]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[22]  Manfredi Maggiore,et al.  On Local Transverse Feedback Linearization , 2008, SIAM J. Control. Optim..

[23]  Anton S. Shiriaev.  Stabilization of compact sets for passive affine nonlinear systems , 2000, Autom..

[24]  P. Encarnacao,et al.  Combined trajectory tracking and path following: an application to the coordinated control of autonomous marine craft , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[25]  C. Kellett,et al.  Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[26]  David J. Hill,et al.  Stability results for nonlinear feedback systems , 1977, Autom..

[27]  Mireille E. Broucke,et al.  Stabilization of infinitesimally rigid formations of multi-robot networks , 2008, 2008 47th IEEE Conference on Decision and Control.

[28]  L. Foulds,et al.  Graph Theory Applications , 1991 .

[29]  Reza Olfati-Saber,et al.  Consensus and Cooperation in Networked Multi-Agent Systems , 2007, Proceedings of the IEEE.

[30]  Alexander L. Fradkov.  Swinging control of nonlinear oscillations , 1996 .

[31]  P. Seibert,et al.  Relative stability and stability of closed sets , 1970 .

[32]  Olav Egeland,et al.  On global properties of passivity-based control of an inverted pendulum , 2000 .

[33]  Derek A. Paley.  Stabilization of Collective Motion in a Uniform and Constant Flow Field , 2008 .

[34]  P. Seibert,et al.  On the reduction to a subspace of stability properties of systems in metric spaces , 1995 .

[35]  Rogelio Lozano,et al.  Stabilization of the Furuta Pendulum Around Its Homoclinic Orbit , 2001 .

[36]  Ura Taro,et al.  Sur le courant exterieur a une region invariante; Theoreme de Bendixson , 1960 .

[37]  Luca Consolini,et al.  Path following for the PVTOL aircraft , 2010, Autom..

[38]  Franck Plestan,et al.  Stable walking of a 7-DOF biped robot , 2003, IEEE Trans. Robotics Autom..

[39]  G. P. Szegö,et al.  Stability theory of dynamical systems , 1970 .

[40]  P. Moylan,et al.  Dissipative Dynamical Systems: Basic Input-Output and State Properties , 1980 .

[41]  Andrew R. Teel,et al.  Weak Converse Lyapunov Theorems and Control-Lyapunov Functions , 2003, SIAM J. Control. Optim..

[42]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion: All-to-All Communication , 2007, IEEE Transactions on Automatic Control.

[43]  Karl Johan Åström,et al.  Adaptive Control , 1989, Embedded Digital Control with Microcontrollers.

[44]  J. V. Rauff,et al.  Multi-Agent Systems: An Introduction to Distributed Artificial Intelligence , 2005 .

[45]  Emilio Frazzoli,et al.  Decentralized Policies for Geometric Pattern Formation and Path Coverage , 2007 .

[46]  Andrea Garulli,et al.  Collective circular motion of multi-vehicle systems , 2008, Autom..

[47]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[48]  Rachid Outbib,et al.  Semidefinite lyapunov functions stability and stabilization , 1996, Math. Control. Signals Syst..

[49]  A. Teel,et al.  Global stabilizability and observability imply semi-global stabilizability by output feedback , 1994 .

[50]  Petros A. Ioannou,et al.  Linear Time-Varying Systems: Control and Adaptation , 1992 .

[51]  N. P. Bhatia,et al.  Dynamical Systems: Stability, Theory and Applications , 1967 .

[52]  D. Youla,et al.  Bounded Real Scattering Matrices and the Foundations of Linear Passive Network Theory , 1959 .

[53]  Mireille E. Broucke,et al.  Formations of vehicles in cyclic pursuit , 2004, IEEE Transactions on Automatic Control.

[54]  Ludovic Rifford,et al.  Existence of Lipschitz and Semiconcave Control-Lyapunov Functions , 2000, SIAM J. Control. Optim..

[55]  R. Freeman,et al.  Robust Nonlinear Control Design: State-Space and Lyapunov Techniques , 1996 .

[56]  Alberto Isidori,et al.  Nonlinear Control Systems II , 1999 .

[57]  Alexander L. Fradkov,et al.  Stabilization of invariant sets for nonlinear non-affine systems , 2000, Autom..

[58]  Eduardo Sontag.  Further facts about input to state stabilization , 1990 .

[59]  Eduardo Sontag.  A Lyapunov-Like Characterization of Asymptotic Controllability , 1983, SIAM Journal on Control and Optimization.

[60]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[61]  Lionel Lapierre,et al.  Nonsingular path following control of a unicycle in the presence of parametric modelling uncertainties , 2006 .

[62]  Bong Wie,et al.  Space Vehicle Dynamics and Control , 1998 .

[63]  Aristotle Arapostathis,et al.  Remarks on smooth feedback stabilization of nonlinear systems , 1988 .

[64]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[65]  Eduardo Sontag.  A universal construction of Artstein's theorem on nonlinear stabilization , 1989 .

[66]  A. Garulli,et al.  Collective circular motion of multi-vehicle systems with sensory limitations , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[67]  J. Hauser,et al.  Feedback linearization of transverse dynamics for periodic orbits , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[68]  Romeo Ortega,et al.  Putting energy back in control , 2001 .

[69]  P. Moylan,et al.  Connections between finite-gain and asymptotic stability , 1980 .

[70]  Naomi Ehrich Leonard,et al.  Stabilization of Planar Collective Motion With Limited Communication , 2008, IEEE Transactions on Automatic Control.

[71]  S. R. Simanca,et al.  On Circulant Matrices , 2012 .

[72]  Alexander L. Fradkov,et al.  Stabilization of invariant sets for nonlinear systems with applications to control of oscillations , 2001 .

[73]  Derek A. Paley,et al.  Stabilization of Collective Motion in a Time-Invariant Flowfield , 2009 .

[74]  Mireille E. Broucke,et al.  Pursuit formations of unicycles , 2006, Autom..

[75]  Debasish Ghose,et al.  Generalization of nonlinear cyclic pursuit , 2007, Autom..

[76]  Katsuhisa Furuta,et al.  Swinging up a pendulum by energy control , 1996, Autom..

[77]  George Birkhoff.  Physical aspects of dynamical systems , 1927 .

[78]  Wei Ding,et al.  Formations on two-layer pursuit systems , 2009, 2009 IEEE International Conference on Robotics and Automation.

[79]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[80]  Yannick Morel,et al.  Applied Nonlinear Control of Unmanned Vehicles with Uncertain Dynamics , 2009 .

[81]  Antonio Bicchi,et al.  Closed loop steering of unicycle like vehicles via Lyapunov techniques , 1995, IEEE Robotics Autom. Mag..

[82]  Manfredi Maggiore,et al.  Necessary and sufficient graphical conditions for formation control of unicycles , 2005, IEEE Transactions on Automatic Control.

[83]  Alex Meystel.  Autonomous Mobile Robots - Vehicles with Cognitive Control , 1991, Series in Automation.

[84]  Daniel E. Koditschek,et al.  Hybrid zero dynamics of planar biped walkers , 2003, IEEE Trans. Autom. Control..

[85]  Manfredi Maggiore,et al.  Output stabilization and maneuver regulation: A geometric approach , 2006, Syst. Control. Lett..

[86]  Alexander L. Fradkov,et al.  Nonlinear and Adaptive Control of Complex Systems , 1999 .

[87]  Eduardo D. Sontag,et al.  Continuous control-Lyapunov functions for asymptotically controllable time-varying systems , 1999 .

[88]  M. Vidyasagar.  Decomposition techniques for large-scale systems with nonadditive interactions: Stability and stabilizability , 1980 .

[89]  Naomi Ehrich Leonard,et al.  Stabilization of symmetric formations to motion around convex loops , 2008, Syst. Control. Lett..

[90]  Alexander L. Fradkov,et al.  Introduction to Control of Oscillations and Chaos , 1998 .

[91]  E. Davison.  The robust control of a servomechanism problem for linear time-invariant multivariable systems , 1976 .

[92]  Yu. S. Ledyaev,et al.  Asymptotic controllability implies feedback stabilization , 1997, IEEE Trans. Autom. Control..

[93]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[94]  B. Francis.  The linear multivariable regulator problem , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.