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On Stability and Trajectory Boundedness in Mean-square Sense for ARMA Processes

Authors:

Email author" target="_blank">Han-Fu?Chen Email author

Institution:

(1) Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

Abstract:

Abstract For the multidimensional ARMA system A(z)y _k = C(z)w _k it is shown that stability (det A(z) ≠= 0, ∀ z : |z| ≤ 1) of A(z) is equivalent to the trajectory boundedness in the mean square sense (MSS)

${\mathop {\lim {\kern 1pt} {\kern 1pt} \sup }\limits_{n \to \infty } }{\kern 1pt} \frac{1} {n}{\sum\limits_{k = 1}^n {{\left\| {y_{k} } \right\|}^{2} < \infty {\kern 1pt} {\kern 1pt} {\text{a}}{\text{.s}}.,{\kern 1pt} } }$

which, as a rule, is a consequence of a successful stochastic adaptive control leading the closed-loop of an ARMAX system to a steady state ARMA system. In comparison with existing results the stability condition imposed on C(z) is no longer needed. The only structural requirement on the system is that det A(z) and det C(z) have no unstable common factor. Supported by the National Natural Science Foundation of China (No. 60074003) and by the Ministry of Science and Technology of China.

Keywords:

ARMA stability boundedness in MSS equivalence

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