Constrained adaptive filtering has developed rapidly in recent years. The constrained least lncosh (CLL) algorithm has good performance in heavy-tailed impulsive noise environments. However, it does not consider the situation where the input of the adaptive filter is corrupted by noise. To solve this problem, this work proposes a constrained least total lncosh algorithm (CLTL) based on the error-in-variables (EIV) model by employing the total least-squares framework and the lncosh loss function. In addition, to enhance the behavior of CLTL for estimating sparse systems or to save energy consumption in beamforming, an improved CLTL algorithm is proposed by using the constraint of the l1-norm of the weight vector. Simulation results are given to demonstrate the superiority of CLTL and its sparsity-induced version.