Feedback design using nonsmooth control Lyapunov functions: A numerical case study for the nonholonomic integrator

Theoretical results for the existence of (non-smooth) control Lyapunov functions (CLFs) for nonlinear systems asymptotically controllable to the origin or a closed set have been available since the late 1990s. Additionally, robust feedback stabilizers based on such CLFs have also been available though, to the best of our knowledge, these stabilizers have not been implemented. Here, we numerically investigate the properties of the closed loop solutions of the nonholonomic integrator using three control techniques based on the knowledge of two different nonsmooth CLFs. In order to make the paper self-contained, we review theoretical results on the existence of nonsmooth CLFs.

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

[2]  Eduardo Sontag.  A Lyapunov-Like Characterization of Asymptotic Controllability , 1983, SIAM Journal on Control and Optimization.

[3]  Hisakazu Nakamura,et al.  Discontinuous Control of Nonlinear Systems with convex input constraint via Locally Semiconcave Control Lyapunov Functions , 2014 .

[4]  Hisakazu Nakamura,et al.  Control of two-wheeled mobile robot via homogeneous semiconcave control Lyapunov function: Verification by experimental results , 2014 .

[5]  Yu. S. Ledyaev,et al.  Asymptotic controllability implies feedback stabilization , 1997, IEEE Trans. Autom. Control..

[6]  Hisakazu Nakamura.  Generalized Minimum Projection Method and Its Application to Two Wheeled Mobile Robot Control , 2015 .

[7]  Nicole Fruehauf.  Semiconcave Functions Hamilton Jacobi Equations And Optimal Control , 2016 .

[8]  C. Kellett,et al.  Uniform asymptotic controllability to a set implies locally Lipschitz control-Lyapunov function , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

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

[10]  Karl Worthmann,et al.  Model Predictive Control of Nonholonomic Mobile Robots Without Stabilizing Constraints and Costs , 2016, IEEE Transactions on Control Systems Technology.

[11]  Yu. S. Ledyaev,et al.  Asymptotic Stability and Smooth Lyapunov Functions , 1998 .

[12]  Ludovic Rifford,et al.  Existence of Lipschitz and Semiconcave Control-Lyapunov Functions , 2000, SIAM J. Control. Optim..

[13]  Karl Worthmann,et al.  Quadratic costs do not always work in MPC , 2017, Autom..

[14]  R. W. Brockett,et al.  Asymptotic stability and feedback stabilization , 1982 .

[15]  Mrdjan J. Jankovic,et al.  Constructive Nonlinear Control , 2011 .

[16]  Andrew R. Teel,et al.  Model predictive control: for want of a local control Lyapunov function, all is not lost , 2005, IEEE Transactions on Automatic Control.

[17]  D. Mayne.  Nonlinear and Adaptive Control Design [Book Review] , 1996, IEEE Transactions on Automatic Control.

[18]  Hyungbo Shim,et al.  Further results on robustness of (possibly discontinuous) sample and hold feedback , 2004, IEEE Transactions on Automatic Control.

[19]  Andrew R. Teel,et al.  Weak Converse Lyapunov Theorems and Control-Lyapunov Functions , 2003, SIAM J. Control. Optim..

[20]  Sergey V. Drakunov,et al.  Stabilization and tracking in the nonholonomic integrator via sliding modes , 1996 .