Stationary Set Analysis for PD Controlled Mechanical Systems

This brief presents a comprehensive method to analyze the stationary set of the high-dimensional mechanical systems with proportional-derivative (PD) control. The system behavior is characterized by Filippov's differential inclusion, in which the discontinuous disturbance mainly caused by static friction is modeled by a convex set-valued map with respect to the velocity. The stationary set of the closed-loop system is obtained by calculating the static solution in the sense of Filippov. The stationary set is also proved to be asymptotically stable, while the asymptotic stability cannot be obtained for proportional-integral-derivative (PID) controlled systems. Finally, the experiments on a three-axis rotation table are carried out to illustrate our analysis results.

[1]  A. Bacciotti,et al.  Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions , 1999 .

[2]  E. A. Misawa,et al.  Sliding mode compensation of dry friction , 1996, Proceeding of the 1996 IEEE International Conference on Control Applications IEEE International Conference on Control Applications held together with IEEE International Symposium on Intelligent Contro.

[3]  J. Alvarez,et al.  An Invariance Principle for Discontinuous Dynamic Systems With Application to a Coulomb Friction Oscillator , 2000 .

[4]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[5]  Mark W. Spong,et al.  Adaptive motion control of rigid robots: a tutorial , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[6]  Bernard Friedland,et al.  On the Modeling and Simulation of Friction , 1990, 1990 American Control Conference.

[7]  Maarten Steinbuch,et al.  Friction induced hunting limit cycles: A comparison between the LuGre and switch friction model , 2003, Autom..

[8]  N. Matsui,et al.  Disturbance observer-based nonlinear friction compensation in table drive system , 1998, AMC'98 - Coimbra. 1998 5th International Workshop on Advanced Motion Control. Proceedings (Cat. No.98TH8354).

[9]  Francesca Maria Ceragioli,et al.  Discontinuous ordinary differential equations and stabilization , 2000 .

[10]  Clark J. Radcliffe,et al.  Robust nonlinear stick-slip friction compensation , 1991 .

[11]  Brian Armstrong,et al.  PID control in the presence of static friction: A comparison of algebraic and describing function analysis , 1996, Autom..

[12]  V V Filippov.  ON THE THEORY OF THE CAUCHY PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION WITH DISCONTINUOUS RIGHT-HAND SIDE , 1995 .

[13]  Bernard Friedland,et al.  On adaptive friction compensation , 1992 .

[14]  H. Kaminaga,et al.  Robust Nonlinear Control of Parametric Uncertain Systems With Unknown Friction and Its Application to a Pneumatic Control Valve , 2000 .

[15]  Yury Orlov,et al.  Switched chattering control vs. backlash/friction phenomena in electrical servo-motors , 2003 .

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

[17]  Bernard Friedland,et al.  On the Modeling and Simulation of Friction , 1990, 1990 American Control Conference.

[18]  Brad Paden,et al.  Globally asymptotically stable ‘PD+’ controller for robot manipulators , 1988 .

[19]  David E. Stewart,et al.  Rigid-Body Dynamics with Friction and Impact , 2000, SIAM Rev..

[20]  Nathan van de Wouw,et al.  Friction compensation in a controlled one-link robot using a reduced-order observer , 2004 .

[21]  K. Astrom,et al.  Friction generated limit cycles , 1996, Proceeding of the 1996 IEEE International Conference on Control Applications IEEE International Conference on Control Applications held together with IEEE International Symposium on Intelligent Contro.

[22]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[23]  Yury Orlov,et al.  Global position regulation of friction manipulators via switched chattering control , 2003 .

[24]  R. Ortega.  Passivity-based control of Euler-Lagrange systems : mechanical, electrical and electromechanical applications , 1998 .

[25]  Suguru Arimoto,et al.  A New Feedback Method for Dynamic Control of Manipulators , 1981 .

[26]  Carlos Canudas de Wit,et al.  A new model for control of systems with friction , 1995, IEEE Trans. Autom. Control..

[27]  Kostas J. Kyriakopoulos,et al.  Backstepping for nonsmooth systems , 2003, Autom..

[28]  Bernard Friedland,et al.  On adaptive friction compensation , 1991, [1991] Proceedings of the 30th IEEE Conference on Decision and Control.

[29]  P. Dupont.  Avoiding stick-slip through PD control , 1994, IEEE Trans. Autom. Control..

[30]  T.H. Lee,et al.  Adaptive friction compensation of servo mechanisms , 1999, Proceedings of the 1999 IEEE International Conference on Control Applications (Cat. No.99CH36328).

[31]  B. Paden,et al.  Lyapunov stability theory of nonsmooth systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[32]  Suguru Arimoto,et al.  Is a local linear PD feedback control law effective for trajectory tracking of robot motion? , 1988, Proceedings. 1988 IEEE International Conference on Robotics and Automation.

[33]  Patrizio Tomei,et al.  Adaptive PD controller for robot manipulators , 1991, IEEE Trans. Robotics Autom..

[34]  Jean-Jacques E. Slotine,et al.  Adaptive manipulator control: A case study , 1988 .