A $k$WTA Network Based on Linear Programming Under Polynomial Noises With Applications

Some existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>-winners-take-all (<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA) networks are constructed based on the quadratic programming (QP), which have a high complexity in problem modeling. Others are designed based on linear programming (LP) but cannot effectively handle dynamic inputs and external interferences. This article employs the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA problem based on the LP form, which offers a simpler framework for the network construction than those networks based on QP. Moreover, external interferences encountered during the implementation of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA operation are often characterized as complex noises which can be fitted by polynomials. To address these issues, a polynomial noise-tolerant <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA (PNT-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA) network based on the LP is constructed that possesses the capability to suppress complex noises. Theoretical analyses establish the robustness and convergence of the proposed network, while numerical simulations demonstrate its superior performance compared to existing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA networks. Finally, multi-robot target-tracking experiments confirm the PNT-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol{k}$</tex-math></inline-formula>WTA network’s practical effectiveness.