An embedding approach for the design of state‐feedback tracking controllers for references with jumps

We study the problem of designing state‐feedback controllers to track time‐varying state trajectories that may exhibit jumps. Both plants and controllers considered are modeled as hybrid dynamical systems, which are systems with both continuous and discrete dynamics, given in terms of a flow set, a flow map, a jump set, and a jump map. Using recently developed tools for the study of stability in hybrid systems, we recast the tracking problem as the task of asymptotically stabilizing a set, the tracking set, and derive conditions for the design of state‐feedback tracking controllers with the property that the jump times of the plant coincide with those of the given reference trajectories. The resulting tracking controllers guarantee that solutions of the plant starting close to the reference trajectory stay close to it and that the difference between each solution of the controlled plant and the reference trajectory converges to zero asymptotically. Constructive conditions for tracking control design in terms of LMIs are proposed for a class of hybrid systems with linear maps and input‐triggered jumps. The results are illustrated by various examples. Copyright © 2013 John Wiley & Sons, Ltd.

[1]  Bastian Goldlücke,et al.  Variational Analysis , 2014, Computer Vision, A Reference Guide.

[2]  Anthony N. Michel,et al.  Relaxation of hypotheses in LaSalle-Krasovskii type invariance results , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[3]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[4]  Laura Menini,et al.  Robust Trajectory Tracking for a Class of Hybrid Systems: An Internal Model Principle Approach , 2012, IEEE Transactions on Automatic Control.

[5]  Anthony N. Michel,et al.  Relaxation of Hypotheses in LaSalle-Krasovskii-Type Invariance Results , 2011, SIAM J. Control. Optim..

[6]  Ricardo G. Sanfelice,et al.  Hybrid controllers for tracking of impulsive reference state trajectories: a hybrid exosystem approach , 2011, HSCC '11.

[7]  Bernard Brogliato,et al.  Trajectory Tracking Control of Multiconstraint Complementarity Lagrangian Systems , 2010, IEEE Transactions on Automatic Control.

[8]  Ricardo G. Sanfelice,et al.  Asymptotic Stability in Hybrid Systems via Nested Matrosov Functions , 2009, IEEE Transactions on Automatic Control.

[9]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[10]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[11]  Thomas Parisini,et al.  Global asymptotic stabilization of the attitude and the angular rates of an underactuated non-symmetric rigid body , 2008, Autom..

[12]  Boris M. Miller,et al.  Dynamical systems with active singularities: Input/state/output modeling and control , 2008, Autom..

[13]  Nathan van de Wouw,et al.  Uniform Convergence of Monotone Measure Differential Inclusions: with Application to the Control of Mechanical Systems with Unilateral Constraints , 2008, Int. J. Bifurc. Chaos.

[14]  M. Kanat Camlibel,et al.  Observer‐based control of linear complementarity systems , 2008, HSCC.

[15]  N. Wouw,et al.  Stability and Convergence of Mechanical Systems with Unilateral Constraints , 2008 .

[16]  Laura Menini,et al.  Trajectory tracking for a particle in elliptical billiards , 2008, Int. J. Control.

[17]  Ricardo G. Sanfelice,et al.  Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability , 2007, IEEE Transactions on Automatic Control.

[18]  Ricardo G. Sanfelice,et al.  A hybrid systems approach to trajectory tracking control for juggling systems , 2007, 2007 46th IEEE Conference on Decision and Control.

[19]  Boris M. Miller,et al.  Dynamical Systems With Active Singularities of Elastic Type: A Modeling and Controller Synthesis Framework , 2007, IEEE Transactions on Automatic Control.

[20]  M. Kanat Camlibel,et al.  Lyapunov Stability of Complementarity and Extended Systems , 2006, SIAM J. Optim..

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

[22]  Bernard Brogliato,et al.  Tracking Control of complementarity Lagrangian Systems , 2005, Int. J. Bifurc. Chaos.

[23]  Geraldo Nunes Silva,et al.  Lyapunov Stability of Measure Driven Impulsive Systems , 2004 .

[24]  Karl Henrik Johansson,et al.  Dynamical properties of hybrid automata , 2003, IEEE Trans. Autom. Control..

[25]  A. Tornambe,et al.  State estimation for a class of linear mechanical systems that become observable thanks to non-smooth impacts , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[26]  Bernard Brogliato,et al.  Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[27]  A. Tornambe,et al.  Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts , 2001, IEEE Trans. Autom. Control..

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

[29]  B. Brogliato,et al.  On tracking control of a class of complementary-slackness hybrid mechanical systems , 1998 .

[30]  B. Brogliato,et al.  On the control of finite-dimensional mechanical systems with unilateral constraints , 1997, IEEE Trans. Autom. Control..

[31]  B. Paden,et al.  Nonlinear inversion-based output tracking , 1996, IEEE Trans. Autom. Control..

[32]  A. Michel,et al.  Stability theory for hybrid dynamical systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

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

[35]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[36]  Theodosios Pavlidis,et al.  Stability of a Class of Discontinuous Dynamical Systems , 1966, Inf. Control..

[37]  Tosio Kato.  Perturbation theory for linear operators , 1966 .