In this talk, we present results for the stabilization and observation of nonlinear systems over analog erasure channels. The stochastic nature of the problem requires us to consider notions of stability adapted from ergodic theory of random dynamical system. In particular, we employ almost sure and second moment exponential stability as the stability and performance metric, respectively, to study this problem. The main result shows that while there are no limitations for the almost sure stabilization, fundamental limitation arise for the second moment stabilization of nonlinear system over erasure channels. The limitation for second moment exponential stabilization is expressed in terms of probability of erasure and global instability, as captured by the positive Lyapunov exponents, of the open loop uncontrolled system. The dependence on the Lyapunov exponents highlight, for the first time, the important role played by the global non-equilibrium dynamics of the nonlinear systems in determining the performance limitation. These results generalize the existing results known in the context of linear time invariant system, where the Lyapunov exponents emerge as the natural generalization of eigenvalues from linear systems to nonlinear systems. Furthermore these results differ from the existing Bode-like fundamental limitation results for the stabilization of nonlinear system with no channel uncertainty, where the limitation is expressed in terms of the eigenvalues of the linearization.
Dr. Umesh Vaidya has obtained his Bachelor’s degree in Electrical Engineering from University of Mumbai, Master’s Degree in Systems and Control Engineering from Indian Institute of Technology, Bombay and PhD in Mechanical Engineering from University of California, Santa Barbara in 2004. From August 2004 to September 2005, he worked as a senior scientist at United Technologies Research Center (UTRC), East Hartford, CT. He is currently an Assistant Professor in the department of Electrical and Computer Engineering at Iowa State University, Ames, IA. His research interest is in the area of dynamical systems and control theory with particular emphasis on the application of ergodic theory methods to control problems.
