Lyapunov Design for Event-Triggered Exponential Stabilization

Control Lyapunov Functions (CLF) method gives a constructive tool for stabilization of nonlinear systems. To find a CLF, many methods have been proposed in the literature, e.g. backstepping for cascaded systems and sum of squares (SOS) programming for polynomial systems. Dealing with continuous-time systems, the CLF-based controller is also continuous-time, whereas practical implementation on a digital platform requires sampled-time control. In this paper, we show that if the continuous-time controller provides exponential stabilization, then an exponentially stabilizing event-triggered control strategy exists with the convergence rate arbitrarily close to the rate of the continuous-time system.

[1]  Keck Voon Ling,et al.  Inverse optimal adaptive control for attitude tracking of spacecraft , 2005, IEEE Trans. Autom. Control..

[2]  Koushil Sreenath,et al.  Rapidly Exponentially Stabilizing Control Lyapunov Functions and Hybrid Zero Dynamics , 2014, IEEE Transactions on Automatic Control.

[3]  C. C. Chien,et al.  Regulation layer controller design for automated highway systems , 1995 .

[4]  P. Olver.  Nonlinear Systems , 2013 .

[5]  Murat Arcak,et al.  Constructive nonlinear control: a historical perspective , 2001, Autom..

[6]  Dragan Nesic,et al.  A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models , 2004, IEEE Transactions on Automatic Control.

[7]  F. G. Shinskey,et al.  2.19 Nonlinear and Adaptive Control , 2008 .

[8]  Roberto Horowitz,et al.  SAFE PLATOONING IN AUTOMATED HIGHWAY SYSTEMS. PART II, VELOCITY TRACKING CONTROLLER , 1999 .

[9]  Kevin Fiedler,et al.  Robust Nonlinear Control Design State Space And Lyapunov Techniques , 2016 .

[10]  C. J. Himmelberg,et al.  On measurable relations , 1982 .

[11]  A. Anta,et al.  Self-triggered stabilization of homogeneous control systems , 2008, 2008 American Control Conference.

[12]  Paulo Tabuada,et al.  Correct-by-Construction Adaptive Cruise Control: Two Approaches , 2016, IEEE Transactions on Control Systems Technology.

[13]  W. P. M. H. Heemels,et al.  Event-Separation Properties of Event-Triggered Control Systems , 2014, IEEE Transactions on Automatic Control.

[14]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[15]  Paulo Tabuada,et al.  Control Barrier Function Based Quadratic Programs for Safety Critical Systems , 2016, IEEE Transactions on Automatic Control.

[16]  Manuel Mazo,et al.  System Architectures, Protocols and Algorithms for Aperiodic Wireless Control Systems , 2014, IEEE Transactions on Industrial Informatics.

[17]  Andrew R. Teel,et al.  Discrete-time asymptotic controllability implies smooth control-Lyapunov function , 2004, Syst. Control. Lett..

[18]  Kazuo Tanaka,et al.  An SOS-Based Control Lyapunov Function Design for Polynomial Fuzzy Control of Nonlinear Systems , 2017, IEEE Transactions on Fuzzy Systems.

[19]  J. Martínez,et al.  Lyapunov event-triggered control: a new event strategy based on the control , 2013 .

[20]  Yuandan Lin,et al.  A universal formula for stabilization with bounded controls , 1991 .

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

[22]  Bin Liu,et al.  Exponential stability via event-triggered impulsive control for continuous-time dynamical systems , 2014, Proceedings of the 33rd Chinese Control Conference.

[23]  Nicolas Marchand,et al.  A General Formula for Event-Based Stabilization of Nonlinear Systems , 2013, IEEE Transactions on Automatic Control.

[24]  Fabian R. Wirth,et al.  Control Lyapunov Functions and Zubov's Method , 2008, SIAM J. Control. Optim..

[25]  Mrdjan Jankovic,et al.  Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems , 2001, IEEE Trans. Autom. Control..

[26]  Bayu Jayawardhana,et al.  Stabilization with guaranteed safety using Control Lyapunov-Barrier Function , 2016, Autom..

[27]  Paulo Tabuada,et al.  Event-Triggered Real-Time Scheduling of Stabilizing Control Tasks , 2007, IEEE Transactions on Automatic Control.

[28]  Dragan Nesic,et al.  A framework for nonlinear sampled-data observer design via approximate discrete-time models and emulation , 2004, Autom..

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

[30]  Ricardo G. Sanfelice,et al.  On the Existence of Control Lyapunov Functions and State-Feedback Laws for Hybrid Systems , 2013, IEEE Transactions on Automatic Control.

[31]  HRISTOPHE,et al.  Stability of nonlinear systems by means of event-triggered sampling algorithms , 2018 .

[32]  Jeroen Ploeg,et al.  Event-Triggered Control for String-Stable Vehicle Platooning , 2017, IEEE Transactions on Intelligent Transportation Systems.

[33]  Nicolas Marchand,et al.  Stability of non-linear systems by means of event-triggered sampling algorithms , 2014, IMA J. Math. Control. Inf..

[34]  Jean-Baptiste Pomet,et al.  Esaim: Control, Optimisation and Calculus of Variations Control Lyapunov Functions for Homogeneous " Jurdjevic-quinn " Systems , 2022 .

[35]  Petros A. Ioannou,et al.  Robust platoon-stable controller design for autonomous intelligent vehicles , 1995 .

[36]  K. Åström,et al.  Comparison of Riemann and Lebesgue sampling for first order stochastic systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[37]  R. E. Kalman,et al.  Control System Analysis and Design Via the “Second Method” of Lyapunov: I—Continuous-Time Systems , 1960 .

[38]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[39]  W. P. M. H. Heemels,et al.  Output-Based and Decentralized Dynamic Event-Triggered Control With Guaranteed $\mathcal{L}_{p}$- Gain Performance and Zeno-Freeness , 2017, IEEE Transactions on Automatic Control.

[40]  P. Kokotovic,et al.  Inverse Optimality in Robust Stabilization , 1996 .

[41]  Paulo Tabuada,et al.  To Sample or not to Sample: Self-Triggered Control for Nonlinear Systems , 2008, IEEE Transactions on Automatic Control.