Equivalent Stability Notions, Lyapunov Inequality, and Its Application in Discrete-Time Linear Systems With Stochastic Dynamics Determined by an i.i.d. Process

Abstract

This paper is concerned with stability analysis and synthesis for discrete-time linear systems with stochastic dynamics. Equivalence is first proved for three stability notions under some key assumptions on the randomness behind the systems. In particular, we use the assumption that the stochastic process determining the system dynamics is independent and identically distributed (i.i.d.) with respect to the discrete time. Then, a Lyapunov inequality condition is derived for stability in a necessary and sufficient sense. Although our Lyapunov inequality will involve decision variables contained in the expectation operation, an idea is provided to solve it as a standard linear matrix inequality; the idea also plays an important role in state feedback synthesis based on the Lyapunov inequality. Motivating numerical examples are further discussed as an application of our approach.

Stability Optimization of Positive Semi-Markov Jump Linear Systems via Convex Optimization

“…Let us allow the following box constraint 0 ≤ α ≤ᾱ (10) on the amount of doses. Under this scenario, we consider the following optimal intervention problem: In this numerical example, we assume that the cost for antibiotics is linear with their dose amount, i.e., we let c (α ) = r α for a constant r > 0 for all .…”

“…Several important issues on Markov jump linear systems have been addressed in the literature including controllability and stabilizability [5], robust optimal control [6], sampled-data control [7], and game theory [8]. Furthermore, the class of systems includes the basic class of stochastic dynamical systems with independent and identically distributed parameters [9], [10]. Despite the aforementioned advances, modelling by a Markov jump linear system suffers from the limitation that the sojourn time of the systems must follow an exponential distribution.…”

Linearization of expectation-based inequality conditions in control for discrete-time linear systems represented with random polytopes

“…The research team involved in the present paper, on the other hand, has studied control of systems with dynamics determined by an i.i.d. stochastic process (Hosoe et al, 2018;Hosoe & Hagiwara, 2019), which is the time sequence of random vectors that are independent and identically distributed (i.i.d.). One might consider that our system class is included in that for Markov jump systems, since i.i.d.…”

“…In particular, to take account of possibilities of the presence of modeling errors even in such stochastic systems, we assume uncertainties in the random coefficient matrices that are represented with random polytopes (Hug, 2013), as in Hosoe et al (2018). In Hosoe & Hagiwara (2019), a Lyapunov inequality condition characterizing nominal stability was proposed and proved to be necessary and sufficient. However, the inequality involves decision variables contained in the expectation operation, which we call an expectation-based inequality.…”

“…Section 2 briefly describes the discrete-time linear systems represented with random polytopes. Section 3 shows two key lemmas, and extends the Lyapunov inequality in Hosoe & Hagiwara (2019) toward robust stability by using the lemmas. In particular, the obtained robust stability condition is reformulated as a standard LMI.…”