Further results on merging control Lyapunov functions for linear differential inclusions

For the robust stabilization of constrained linear differential inclusions, we consider non-homogeneous, smooth, composite control Lyapunov functions (CLFs) which belong to the class of “merging” CLFs. Previous work showed the equivalence between the possibility to always merge two given CLFs and the fact that these two share a common control law. In presence of state and control constraints, this latter property may hold only in a small domain of the state space. For such cases, we provide a novel constructive procedure to merge a CLF having a large controlled invariant set and a CLF with locally-optimal performance but with a smaller controlled invariant set. Our merging allows the explicit derivation of a Lyapunov-based, robustly stabilizing, continuous control law with the large controlled invariant set of the former CLF and the locally-optimal performance of the latter. The theoretical results are illustrated via a numerical example.

[1]  F. Blanchini.  Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions , 1994, IEEE Trans. Autom. Control..

[2]  Sergio Grammatico,et al.  Stabilizability of constrained uncertain linear systems via smooth control Lyapunov R-functions , 2011, IEEE Conference on Decision and Control and European Control Conference.

[3]  S. Miani,et al.  MAXIS-G: a software package for computing polyhedral invariant sets for constrained LPV systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[4]  Alexandre Megretski,et al.  Robust state feedback control of LTV systems: Nonlinear is better than linear , 1997 .

[5]  Per Hagander,et al.  A new design of constrained controllers for linear systems , 1982, 1982 21st IEEE Conference on Decision and Control.

[6]  Tingshu Hu,et al.  Control Systems with Actuator Saturation: Analysis and Design , 2001 .

[7]  Sergio Grammatico,et al.  Stabilization of constrained linear systems via smoothed truncated ellipsoids , 2011 .

[8]  Franco Blanchini,et al.  Nonquadratic Lyapunov functions for robust control , 1995, Autom..

[9]  Christophe Prieur.  Uniting Local and Global Controllers with Robustness to Vanishing Noise , 2001, Math. Control. Signals Syst..

[10]  Sergio Grammatico,et al.  Multivariable constrained process control via Lyapunov R-functions , 2012 .

[11]  Sergio Grammatico,et al.  A universal class of non-homogeneous control Lyapunov functions for linear differential inclusions , 2013, 2013 European Control Conference (ECC).

[12]  Sergio Grammatico,et al.  Control-Sharing and Merging Control Lyapunov Functions , 2014, IEEE Transactions on Automatic Control.

[13]  Christophe Prieur,et al.  Uniting two Control Lyapunov Functions for affine systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[14]  Franco Blanchini,et al.  Set invariance in control , 1999, Autom..

[15]  A. Hmamed,et al.  Regulator problem for linear continuous-time systems with nonsymmetrical constrained control , 1993, IEEE Trans. Autom. Control..

[16]  E. Gilbert,et al.  Computation of minimum-time feedback control laws for discrete-time systems with state-control constraints , 1987 .

[17]  F. Blanchini,et al.  Constrained stabilization via smooth Lyapunov functions , 1998 .

[18]  Alexandre Megretski,et al.  Robust state feedback control of LTV systems: Nonlinear is better than linear , 1997, 1997 European Control Conference (ECC).

[19]  Mircea Lazar,et al.  On infinity norms as Lyapunov functions: Alternative necessary and sufficient conditions , 2010, 49th IEEE Conference on Decision and Control (CDC).

[20]  Franco Blanchini,et al.  A new class of universal Lyapunov functions for the control of uncertain linear systems , 1999, IEEE Trans. Autom. Control..

[21]  Franco Blanchini,et al.  Any domain of attraction for a linear constrained system is a tracking domain of attraction , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[22]  P. R. Bélanger,et al.  Piecewise-linear LQ control for systems with input constraints , 1994, Autom..

[23]  Fred C. Schweppe,et al.  Uncertain dynamic systems , 1973 .

[24]  Abdellah Benzaouia,et al.  Piecewise linear constrained control for continuous-time systems , 1999, IEEE Trans. Autom. Control..

[25]  Sergio Grammatico,et al.  A new class of Lyapunov functions for the constrained stabilization of linear systems , 2012, Autom..

[26]  T. Hu,et al.  Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions , 2010, Autom..

[27]  Graziano Chesi,et al.  Rational Lyapunov functions for estimating and controlling the robust domain of attraction , 2013, Autom..

[28]  Per-olof Gutman,et al.  Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states , 1984, The 23rd IEEE Conference on Decision and Control.

[29]  Sergio Grammatico,et al.  Stabilizability of Linear Differential Inclusions via R-Functions , 2010 .

[30]  Franco Blanchini,et al.  Set-theoretic methods in control , 2007 .

[31]  Francis Clarke,et al.  Lyapunov functions and discontinuous stabilizing feedback , 2011, Annu. Rev. Control..

[32]  Thomas Thibodeau,et al.  Set invariance and performance analysis of linear systems via truncated ellipsoids , 2009, Autom..

[33]  Tingshu Hu,et al.  Composite quadratic Lyapunov functions for constrained control systems , 2003, IEEE Trans. Autom. Control..

[34]  E. Misawa,et al.  Semi-ellipsoidal controlled invariant sets for constrained linear systems , 2002 .

[35]  Mircea Lazar,et al.  On infinity norms as Lyapunov functions for continuous-time dynamical systems , 2011, IEEE Conference on Decision and Control and European Control Conference.

[36]  Randy A. Freeman,et al.  Robust Nonlinear Control Design , 1996 .

[37]  Sergio Grammatico,et al.  Stability analysis of dynamical systems via R-functions , 2009, 2009 European Control Conference (ECC).