A $k$-Winners-Take-All $(k\text{WTA})$ Network with Noise Characteristics Captured

Competition-based <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k-\mathbf{winners}-\mathbf{take}-\mathbf{all}\ \ (k \mathbf{WTA})$</tex> networks play a crucial role in multi-agent systems. However, existing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k \mathbf{WTA}$</tex> networks either neglect the impact of noise or only consider simple forms, such as constant noise. In practice, noises often exhibit time-varying and nonlinear characteristics, which can be modeled using nonlinear functions and approximated by high-order polynomials. Such noises pose significant challenges for current <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k \mathbf{WTA}$</tex> networks, limiting their practical applications. To address this, a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k \mathbf{WTA}$</tex>. network with noise characteristics captured <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(k \mathbf{WTA}-\mathbf{NCC})$</tex> is proposed in this article. Theoretical analyses demonstrate that the residual error of the proposed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k\mathbf{WTA}- \mathbf{NCC}$</tex> network converges to zero globally, while simulation results confirm its robustness against polynomial noises. Additionally, a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k \mathbf{WTA}$</tex> coordination model is constructed by integrating the proposed network with a consensus estimator to achieve multi-agent tracking tasks. Finally, simulations and physical experiments are conducted further to demonstrate the validity and practicality of the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> WTA coordination model.