Energy- and Entropy-Based Stabilization for Lossless Dynamical Systems via Hybrid Controllers

A novel class of dynamic, energy-based hybrid controllers is proposed as a means for achieving enhanced energy dissipation in lossless dynamical systems. These dynamic controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that the closed-loop system is consistent with basic thermodynamic principles. In particular, the existence of an entropy function for the closed-loop system is established that satisfies a hybrid Clausius-type inequality. Special cases of energy-based and entropy-based hybrid controllers involving state-dependent switching are described.

[1]  Wassim M. Haddad,et al.  Nonlinear impulsive dynamical systems. I. Stability and dissipativity , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[2]  A. J. Schaft,et al.  Stabilization of Hamiltonian systems , 1986 .

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

[4]  Y. Kishimoto,et al.  Thermodynamic Modeling of Interconnected Systems: Conservative Coupling , 1993, 1993 American Control Conference.

[5]  Dennis S. Bernstein,et al.  Resetting Virtual Absorbers for Vibration Control , 1997, Proceedings of the 1997 American Control Conference (Cat. No.97CH36041).

[6]  Qing Hui,et al.  Thermodynamic modeling, energy equipartition, and nonconservation of entropy for discrete-time dynamical systems , 2005, Proceedings of the 2005, American Control Conference, 2005..

[7]  Drumi D. Bainov,et al.  Impulsive Differential Equations: Asymptotic Properties of the Solutions , 1995, Series on Advances in Mathematics for Applied Sciences.

[8]  D. Baĭnov,et al.  Systems with impulse effect : stability, theory, and applications , 1989 .

[9]  Kaining Wang,et al.  Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings , 1997, Proceedings of the IEEE.

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

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

[12]  David J. Hill,et al.  On output feedback stabilization of Euler-Lagrange systems with nondissipative forces , 1996 .

[13]  Jan C. Willems,et al.  Dissipative Dynamical Systems , 2007, Eur. J. Control.

[14]  Arjan van der Schaft,et al.  Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems , 2002, Autom..

[15]  Orhan Beker,et al.  Fundamental properties of reset control systems , 2004, Autom..

[16]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

[17]  S. Bhat,et al.  An invariance principle for nonlinear hybrid and impulsive dynamical systems , 2003 .

[18]  Dennis S. Bernstein,et al.  Thermodynamic modelling of interconnected systems, Part II: dissipative coupling , 1995 .

[19]  V. Lakshmikantham,et al.  Theory of Impulsive Differential Equations , 1989, Series in Modern Applied Mathematics.

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

[21]  Xuping Xu,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control - s[Book review; W. M. Haddad; V. S. Chellaboina, and S. G. Nersesov] , 2007, IEEE Transactions on Automatic Control.

[22]  B. Brogliato.  Nonsmooth Mechanics: Models, Dynamics and Control , 1999 .

[23]  W. Haddad,et al.  Optimal disturbance rejection control for nonlinear impulsive dynamical systems , 2005 .

[24]  B. Dundas,et al.  DIFFERENTIAL TOPOLOGY , 2002 .

[25]  Dennis S. Bernstein,et al.  Resetting Virtual Absorbers for Vibration Control , 2000 .

[26]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..

[27]  Wassim M. Haddad,et al.  Non-linear impulsive dynamical systems. Part I: Stability and dissipativity , 2001 .

[28]  A. Samoilenko,et al.  Impulsive differential equations , 1995 .

[29]  Bernhard Maschke,et al.  Dissipative Systems Analysis and Control , 2000 .

[30]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[31]  Franck Plestan,et al.  Asymptotically stable walking for biped robots: analysis via systems with impulse effects , 2001, IEEE Trans. Autom. Control..

[32]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[33]  Tao Yang,et al.  In: Impulsive control theory , 2001 .

[34]  Wassim M. Haddad,et al.  Thermodynamic modeling, energy equipartition, and nonconservation of entropy for discrete-time dynamical systems , 2005 .