Stability Analysis of Large Scale Networks of Autonomous Work Systems with Delays

Abstract

This paper considers the problem of stability analysis for a class of production networks of autonomous work systems with delays in the capacity changes. The system under consideration does not share information between work systems and the work systems adjust capacity with the objective of maintaining a desired amount of local work in progress (WIP). Attention is focused to derive explicit sufficient delay-dependent stability conditions for the network using properties of matrix norm. Finally, numerical results are provided to demonstrate the proposed approach.

Appendix

$\left\| . \right\|$ is any matrix norm which satisfies the following conditions:

(I) $\left\| {A \ge 0} \right\|$, and $\left\| {A = 0} \right\|$ if and only if A = 0,

(II) for each c ∈ R, $\left\| {cA} \right\| = \left| c \right|\left\| A \right\| ,$

(III) $\left\| {A + B} \right\| \le \left\| A \right\| + \left\| B \right\| ,$

(IV) $\left\| {A B} \right\| \le \left\| A \right\| \cdot \left\| B \right\|$ for all (m × m) matrices A, B.

In addition, matrix norm should be concordant with the vector norm $\left\| . \right\|_{*}$, that is,

$$ \left\| {Ax} \right\|_{ * } \le \left\| A \right\|.\left\| x \right\|_{ * } $$

for all $x \in \Re^{m}$ and any (m × m) matrix A. For real (m × m) matrix A, we define, as usual, $\left\| A \right\|_{1} {\text{ = max}} _{1 \le j \le m} \sum _{i = 1}^{m} \left| {a_{ij} } \right|$ and $\left\| A \right\|_{\infty } = \max _{1 \le i \le m} \sum _{j = 1}^{m} \left| {a_{ij} } \right|$