Recent Advances in The Stabilization Problem for Low Dimensional Systems

Abstract We survey recent advances on the stabilization problem for two and three dimensional, single input, affine nonlinear systems. Among the new results given here is a theorem which states that a generic, single input, three dimensional, homogeneous polynomial system of a fixed odd degree p can be asymptotically stabilized by using homogeneous feedback of degree p .

[1]  S. Samelson,et al.  Asymptotic stabilization of three dimensional homogeneous quadratic systems , 1992, [1992] Proceedings of the 31st IEEE Conference on Decision and Control.

[2]  W. Boothby,et al.  Feedback stabilization of planar nonlinear systems. II , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[3]  A. Isidori,et al.  Asymptotic stabilization of minimum phase nonlinear systems , 1991 .

[4]  Henry Hermes,et al.  Asymptotic stabilization of planar systems , 1991 .

[5]  M. Kawski.  Homogeneous feedback laws in dimension three , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[6]  J. Tsinias.  Optimal controllers and output feedback stabilization , 1990 .

[7]  Eduardo Sontag,et al.  Remarks on continuous feedback , 1980, 1980 19th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[8]  M. Kawski.  Stabilization of nonlinear systems in the plane , 1989 .

[9]  Wolfgang Hahn,et al.  Stability of Motion , 1967 .

[10]  J. Coron.  A necessary condition for feedback stabilization , 1990 .

[11]  R. Marino.  On the largest feedback linearizable subsystem , 1986 .

[12]  Andrea Bacciotti,et al.  Local Stabilizability of Nonlinear Control Systems , 1991, Series on Advances in Mathematics for Applied Sciences.

[13]  J. Tsinias,et al.  Output feedback stabilization , 1990 .

[14]  Eduardo D. Sontag,et al.  FEEDBACK STABILIZATION OF NONLINEAR SYSTEMS , 1990 .

[15]  Nikola Samardzija,et al.  Stability properties of autonomous homogeneous polynomial differential systems , 1983 .

[16]  Vladimir I. Arnold,et al.  Dynamical Systems I , 1988 .

[17]  W. Dayawansa,et al.  Some sufficient conditions for the asymptotic stabilizability of three dimensional homogeneous polynomial systems , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[18]  J. Zabczyk.  Some comments on stabilizability , 1989 .

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

[20]  K. Deimling.  Nonlinear functional analysis , 1985 .

[21]  W. P. Dayawansa,et al.  Asymptotic stabilization of a class of smooth two-dimensional systems , 1990 .

[22]  J. Coron,et al.  Adding an integrator for the stabilization problem , 1991 .

[23]  Henry Hermes,et al.  Asymptotically stabilizing feedback controls and the nonlinear regulator problem , 1991 .

[24]  Christopher I. Byrnes,et al.  On the attitude stabilization of rigid spacecraft , 1991, Autom..

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

[26]  Clyde F. Martin,et al.  Asymptotic Stabilization of Low Dimensional Systems , 1991 .

[27]  Riccardo Marino,et al.  Feedback stabilization of planar nonlinear systems , 1989 .

[28]  R. Ho.  Algebraic Topology , 2022 .

[29]  W. Respondek,et al.  Feedback Equivalence of Planar Systems and Stabilizability , 1990 .

[30]  Harold Levine,et al.  Singularities of differentiable mappings , 1971 .

[31]  Eduardo Sontag,et al.  Further comments on the stabilizability of the angular velocity of a rigid body , 1989 .

[32]  P. Kokotovic,et al.  A positive real condition for global stabilization of nonlinear systems , 1989 .

[33]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[34]  J. Mather,et al.  Stability of C∞ mappings, I, the division theorem , 1968 .

[35]  W. Dayawansa,et al.  A remark on a theorem of Andreini, Bacciotti and Stefani , 1989 .

[36]  E. Abed,et al.  Local feedback stabilization and bifurcation control, II. Stationary bifurcation , 1987 .

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

[38]  P. Crouch,et al.  Stabilization of non-linear control systems: the role of Newton diagrams , 1989 .

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

[40]  H. Hermes.  Nilpotent approximations of control systems and distributions , 1986 .

[41]  W. P. Dayawansa,et al.  Asymptotic stabilization of two dimensional real analytic systems , 1989 .

[42]  B. d'Andrea-Novel,et al.  Lyapunov design of stabilizing controllers , 1989, Proceedings of the 28th IEEE Conference on Decision and Control,.

[43]  A. Krener.  Approximate linearization by state feedback and coordinate change , 1984 .

[44]  Dirk Aeyes,et al.  Comments on the stabilization of the angular velocity of a rigid body , 1988 .

[45]  H. Sussmann.  Subanalytic sets and feedback control , 1979 .

[46]  P. Moylan.  Implications of passivity in a class of nonlinear systems , 1974 .

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

[48]  E. Abed,et al.  Local feedback stabilization and bifurcation control, I. Hopf bifurcation , 1986 .

[49]  Herbert Forster,et al.  Über das Verhalten der Integralkurven einer gewöhnlichen Differentialgleichung erster Ordnung in der Umgebung eines singulären Punktes , 1938 .

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

[51]  J. Tsinias,et al.  Existence of control Lyapunov functions and application to state feedback stabilizabilty of nonlinear systems , 1991 .

[52]  A. Isidori,et al.  Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems , 1991 .

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