Performance and robustness analysis of stochastic jump linear systems using Wasserstein metric

Abstract

This paper focuses on the performance and the robustness analysis of stochastic jump linear systems. The state trajectory under stochastic jump process becomes random variables, which brings forth the probability distributions in the system state. Therefore, we need to adopt a proper metric to measure the system performance with respect to stochastic switching. In this perspective, Wasserstein metric that assesses the distance between probability density functions is applied to provide the performance and the robustness analysis. Both the transient and steady-state performance of the systems with given initial state uncertainties can be measured in this framework. Also, we prove that the convergence of this metric implies the mean square stability. Overall, this study provides a unifying framework for the performance and the robustness analysis of general stochastic jump linear systems, but not necessarily Markovian jump process that is commonly used for stochastic switching. The practical usefulness and efficiency of the proposed method are verified through numerical examples.

A switched dynamical system framework for analysis of massively parallel asynchronous numerical algorithms

“…Lemma 5.1: Consider an i.i.d. jump linear system given by (13) with the switching probability Π = {π 1 , π 2 , . .…”

Stability analysis of large-scale distributed networked control systems with random communication delays: A switched system approach

“…It is computationally intractable to deal with 2 2×99 numbers of matrices to analyze system stability. However, in contrast, the reduce mode model (7) has total N i=1 qn i (n i −1) = 98 × (2 3×2 ) + 2 × (2 2×1 ) = 6280 modes. Furthermore, the proposed method fully maximizes its own advantage to reduce the mode numbers by considering the symmetric property between agents, which cannot be implemented on the full state model.…”

“…Without any relaxation or conservatism, theorem 4.1 proved the necessary and sufficient condition for stability, which is equivalent to (6) for the mean square stability of the entire system. Compared to the total number of modes of full state model (5), which is q N (N −1) , the reduced mode model (7) has total N i=1 qn i (n i −1) modes. Consequently, the growth of mode numbers in full state model is exponential with respect to N 2 , whereas that in reduced mode model is linear with regard to N .…”

“…Proof. If the Markov transition probability matrix for the system in (7) has the uncertainty denoted by P i =P i + ∆P i , then the term ρ (P i ⊗ I)…”

“…instability. Therefore, it has led various researches to analyze the NCSs with communication delays [1], [2], [3], [4], [5], [6], [7], [8]. In particular, [6] constructed a switched system structure for the analysis of NCS by including actuators, sensors, and the plant as a single system.…”