A Novelty in Blahut-Arimoto Type Algorithms: Optimal Control Over Noisy Communication Channels

• Published in: IEEE transactions on vehicular technology Vol. 69; no. 6; pp. 6348 - 6358
• Main Author: Zamanipour, Makan
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> {Alternating\;\;optimisation}</tex-math> </inline-formula>; <inline-formula xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"> <tex-math notation="LaTeX"> {detectability-and-stabilisability}</tex-math> </inline-formula>; Algorithms; branch-and-bound; Channel capacity; Communication channels; Complexity theory; Computer simulation; control-theory; distortion function; Formulas (mathematics); information-theory; joint-source-channel-coding; Linear programming; Mathematical analysis; multi-objective optimisation; Multiple objective analysis; Noise measurement; Optimal control; Optimization; Parameter uncertainty; Perturbation; rate-distortion-theory; Robustness (mathematics); synchronisability; Uncertain systems
• ISSN: 0018-9545; 1939-9359
• DOI: 10.1109/TVT.2020.2986788

A probability-theoretic problem under information constraints for the concept of optimal control over a noisy-memoryless channel is considered. For our Observer-Controller block, i.e., the lossy joint-source-channel-coding (JSCC) scheme, after providing the relative mathematical expressions, we propose a Blahut-Arimoto -type algorithm <inline-formula><tex-math notation="LaTeX">-</tex-math></inline-formula> which is, to the best of our knowledge, for the first time. The algorithm efficiently finds the probability-mass-functions (PMFs) required for <inline-formula><tex-math notation="LaTeX"> \mathop {{\rm min}} _{\mathscr {P}(i), i \in \lbrace \mathscr {Y}, \hat{\mathscr {S}}, \mathscr {X},{\mathscr {S}},\hat{\mathscr {X}}\rbrace } {\rm \; } \phi _1 \mathscr {I}(\mathscr {Y};\hat{\mathscr {S}}|\mathscr {X,S})-\phi _2 \mathscr {I}(\mathscr {Y};\hat{\mathscr {X}}|\mathscr {X,S})</tex-math></inline-formula>. This problem is an <inline-formula><tex-math notation="LaTeX">NP-</tex-math></inline-formula>hard and non-convex multi-objective optimisation (MOO) one, were the objective functions are respectively the distortion function <inline-formula><tex-math notation="LaTeX">dim (Null (\mathscr {I}(\hat{\mathscr {S}};{\mathscr {S}})) \rightarrow \infty</tex-math></inline-formula> and the memoryless-channel capacity <inline-formula><tex-math notation="LaTeX">dim (Null (\mathscr {I}(\mathscr {X};\hat{\mathscr {X}})) \rightarrow 0</tex-math></inline-formula>. Our novel algorithm applies an Alternating optimisation method. Subsequently, a robust version of the algorithm is discussed with regard to the perturbed PMFs <inline-formula><tex-math notation="LaTeX">-</tex-math></inline-formula> parameter uncertainties in general. The aforementioned robustness is actualised by exploiting the simultaneous-perturbation-stochastic-approximation (SPSA). The principles of detectability-and-stabilisability as well as synchronisability are explored, in addition to providing the simulations - by which the efficiency of our work is shown. We also calculate the total complexity of our proposed algorithms respectively as <inline-formula><tex-math notation="LaTeX">\mathscr {O} (\mathscr {T}\mathscr {K}\mathscr {M}_0(\mathscr {K} \log \mathscr {K}))</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">\mathscr {O} (\mathscr {T}\mathscr {K}\mathscr {M}_0(\mathscr {K} \log \mathscr {K}+0.33 \mathscr {K}))</tex-math></inline-formula>. Our methodology is generic which can be applied to other fields of studies which are optimisation-driven.