Finite‐Time Stability and Stabilization of Linear Itô Stochastic Systems with State and Control‐Dependent Noise

Abstract

In this paper, finite‐time stability and stabilization problems for a class of linear stochastic systems are studied. First, a new concept of finite‐time stochastic stability is defined for linear stochastic systems. Then, based on matrix inequalities, some sufficient conditions under which the stochastic systems are finite‐time stochastically stable are given. Subsequently, the finite‐time stochastic stabilization is studied and some sufficient conditions for the existence of a state feedback controller and a dynamic output feedback controller are presented by using a matrix inequality approach. An algorithm is given for solving the matrix inequalities arising from finite‐time stochastic stability (stabilization). Finally, two examples are employed to illustrate the results.

New conditions for the finite-time stability of stochastic linear time-varying systems

“…From this last inequality it follows that condition (10) in Lemma 2 is equivalent to (14), which proves the theorem.…”

“…guarantees the satisfaction of (14). Therefore we can state the following theorem, which is a sufficient, computationally tractable condition for SFTS of the SLTV system (4).…”

“…Now we give the definition of SFTS for SLTV systems, as proposed in [14] (by setting to zero the lower bound in order to maintain consistency with the deterministic definition of FTS [1], Chapter 1).…”

“…While stochastic linear quadratic optimal control and Kalman filtering theory mostly deal with additive white noise [5], the research on stochastic FTS (SFTS) has mainly focussed on the Itô and/or Markovian (when switching systems are dealt with) modeling of system uncertainty (see, among others, [14], [7]). …”

The Mixed Robust /FTS Control Problem Analysis and State Feedback Control

“…IO-FTS is a more practical concept, with respect to the classical Lyapunov IO-stability, useful for studying the behavior of the system within a finite interval, and therefore it finds application whenever it is desired that the output variables do not exceed a given threshold during the transients; moreover they are independent properties, since IO Lyapunov stability does not imply IO-FTS and vice versa. It is worth noting that the definition of IO-FTS given in [2] is consistent with the definition of (state) FTS [3,9,10,16,24,34,35,38,39], where the state of a zero-input system, rather than the input and the output, is involved.…”