Detectability, observability and Lyapunov-type theorems of linear discrete time-varying stochastic systems with multiplicative noise

Abstract

The objective of this paper is to study detectability, observability and related Lyapunov-type theorems of linear discrete-time time-varying stochastic systems with multiplicative noise. Some new concepts such as uniform detectability, K ∞ -exact detectability (resp. K W F T -exact detectability, K F T -exact detectability, K N -exact detectability) and K ∞ -exact observability (resp. K W F T -exact observability, K F T -exact observability, K N -exact observability) are introduced, respectively, and nice properties associated with uniform detectability, exact detectability and exact observability are also obtained. Moreover, some Lyapunov-type theorems associated with generalized Lyapunov equations and exponential stability in mean square sense are presented under uniform detectability, K N -exact observability and K N -exact detectability, respectively.

Adaptive tracking control for a class of random pure‐feedback nonlinear systems with Markovian switching

“…The Markovian process studied in this paper is a homogeneous irreducible stationary process, then how do we design the adaptive tracking controller for random nonlinear systems with more general Markovian switchings? Finally, it is also valuable to consider similar problems in discrete-time stochastic systems as done in the works of Lin and Zhang 45 and Zhang et al 46,47…”

Corrections to ‘Mixed control of time‐varying stochastic discrete‐time systems under uniform detectability’

“…Below, we give a correct version for the iterative formula φ l,k , and the proof can be found in [2].…”

“…Assume (2) holds for l = m > k + 1, then we only need to show that (2) still holds for l = m + 1. According to the formula (1), we get …”

Finite-time stability and stabilization of linear discrete time-varying stochastic systems

“…In this section, we mainly use the state transition matrix approach developed in [31,33] to study the finite-time stability and stabilization of the system (2).…”