Uniform Detectability of Linear Time Varying Systems With Exponential Dichotomy

Exponential dichotomies play a central role in stability theory for dynamical systems. They allow to split the state space into two subspaces, where all trajectories in one subspace decay whereas all trajectories in the other subspace grow, uniformly and exponentially. This letter studies uniform detectability and observability notions for linear time varying systems, which admit an exponential dichotomy. The main contributions are necessary and sufficient detectability conditions for this class of systems.

[1]  Qinghua Zhang,et al.  Observability conservation by output feedback and observability Gramian bounds , 2015, Autom..

[2]  Luca Dieci,et al.  Detecting exponential dichotomy on the real line: SVD and QR algorithms , 2011 .

[3]  Martin Horn,et al.  Detectability Analysis and Observer Design for Linear Time Varying Systems , 2020, IEEE Control Systems Letters.

[4]  Luca Dieci,et al.  Exponential dichotomy on the real line: SVD and QR methods , 2010 .

[5]  S. Shankar Sastry,et al.  The robustness of controllability and observability of linear time-varying systems , 1982 .

[6]  P. Khargonekar,et al.  Normalized coprime factorizations for linear time-varying systems , 1992 .

[7]  K. Maciej Przyłuski.  On asymptotic stability of linear time-varying infinite-dimensional systems , 1985 .

[8]  Luca Dieci,et al.  Lyapunov and Sacker–Sell Spectral Intervals , 2007 .

[9]  Brian D. O. Anderson,et al.  Stabilizability of linear time-varying systems , 2013, Syst. Control. Lett..

[10]  W. A. Coppel,et al.  Dichotomies and reducibility , 1967 .

[11]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[12]  Sergiy Zhuk,et al.  A detectability criterion and data assimilation for nonlinear differential equations , 2018, Nonlinearity.

[13]  Heng-Ming Tai,et al.  Equivalent characterizations of detectability and stabilizability for a class of linear time-varying systems , 1987 .

[14]  M. G. Kreïn.  Stability of solutions of differential equations in Banach space , 2007 .

[15]  R E Kalman,et al.  CANONICAL STRUCTURE OF LINEAR DYNAMICAL SYSTEMS. , 1962, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Christian Pötzsche,et al.  Nonautonomous Dynamical Systems , 2010 .

[17]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[18]  Li︠u︡dmila I︠A︡kovlevna Adrianova.  Introduction to linear systems of differential equations , 1995 .

[19]  Karthik S. Gurumoorthy,et al.  Degenerate Kalman Filter Error Covariances and Their Convergence onto the Unstable Subspace , 2016, SIAM/ASA J. Uncertain. Quantification.

[20]  Achim Ilchmann,et al.  Stabilizability of systems with exponential dichotomy , 1987 .

[21]  C. Langenhop.  On the stabilization of linear systems , 1964 .