On reduction of differential inclusions and Lyapunov stability

In this paper, locally Lipschitz regular functions are utilized to identify and remove infeasible directions from differential inclusions. The resulting reduced differential inclusion is point-wise smaller (in the sense of set containment) than the original differential inclusion. The reduced inclusion is utilized to develop a generalized notion of a time derivative for locally Lipschitz candidate Lyapunov functions. The developed generalized derivative yields less conservative statements of Lyapunov stability results and invariance-like results for differential inclusions. Illustrative examples are included to demonstrate the utility of the developed stability theorems.

[1]  W. Rudin.  Principles of mathematical analysis , 1964 .

[2]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[3]  Emilio Roxin,et al.  Stability in general control systems. , 1965 .

[4]  B. Paden,et al.  Lyapunov stability theory of nonsmooth systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[5]  J. Alvarez,et al.  An Invariance Principle for Discontinuous Dynamic Systems With Application to a Coulomb Friction Oscillator , 2000 .

[6]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[7]  Warren E. Dixon,et al.  LaSalle-Yoshizawa Corollaries for Nonsmooth Systems , 2013, IEEE Transactions on Automatic Control.

[8]  Shankar Sastry,et al.  A calculus for computing Filippov's differential inclusion with application to the variable structure control of robot manipulators , 1986, 1986 25th IEEE Conference on Decision and Control.

[9]  A. I. Subbotin,et al.  Game-Theoretical Control Problems , 1987 .

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

[11]  E. Ryan.  An Integral Invariance Principle for Differential Inclusions with Applications in Adaptive Control , 1998 .

[12]  Sanjay P. Bhat,et al.  Semistability, Finite-Time Stability, Differential Inclusions, and Discontinuous Dynamical Systems Having a Continuum of Equilibria , 2009, IEEE Transactions on Automatic Control.

[13]  Hartmut Logemann,et al.  Asymptotic Behaviour of Nonlinear Systems , 2004, Am. Math. Mon..

[14]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[15]  Andrea Bacciotti,et al.  An invariance principle for nonlinear switched systems , 2005, Syst. Control. Lett..

[16]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

[17]  E. P. Ryan,et al.  Discontinuous Feedback and Universal Adaptive Stabilization , 1990 .

[18]  Wilfrid Perruquetti,et al.  Finite time stability of differential inclusions , 2005, IMA J. Math. Control. Inf..

[19]  V. Matrosov.  On the stability of motion , 1962 .

[20]  J. Moreau,et al.  A chain rule involving vector functions of bounded variation , 1987 .

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

[22]  Francesca Maria Ceragioli,et al.  Discontinuous ordinary differential equations and stabilization , 2000 .

[23]  A. Bacciotti,et al.  Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions , 1999 .

[24]  Antonio Loría,et al.  A nested Matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems , 2005, IEEE Transactions on Automatic Control.

[25]  Brad Paden,et al.  Globally asymptotically stable ‘PD+’ controller for robot manipulators , 1988 .

[26]  Ying Tan,et al.  A refinement of Matrosov's theorem for differential inclusions , 2016, Autom..

[27]  A. Michel,et al.  Qualitative Theory of Dynamical Systems: The Role of Stability Preserving Mappings , 1997, Proceedings of the IEEE.