Stabilizing Error-Correction Codes for Control Over Erasure Channels

Abstract

In this article, we propose <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(k,k^{\prime })$</tex-math></inline-formula> stabilizing codes, which are a type of delayless error-correction codes that are useful for control over networks with erasures. For each input symbol, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> output symbols are generated by the stabilizing code. Receiving at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k^{\prime }$</tex-math></inline-formula> of these outputs guarantees stability. Thus, both the system to be stabilized and the channel are taken into account in the design of the erasure codes. Receiving more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k^{\prime }$</tex-math></inline-formula> outputs further improves the performance of the system. In the case of i.i.d. erasures, we further demonstrate that the erasure code can be constructed such that stability is achieved if on average at least <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k^{\prime }$</tex-math></inline-formula> output symbols are received. Our focus is on linear and time-invariant systems, and we construct codes based on independent encodings and multiple descriptions. Stability is assessed via Markov jump linear system theory. The theoretical efficiency and performance of the codes are assessed, and their practical performances are demonstrated in a simulation study. There is a significant gain over other delayless codes such as repetition codes.