This paper addresses the problem of drop-point constrained state modeling in the set-valued framework, where only drop-point information of state trajectories is available. Due to the lack of complete trajectory information, existing estimation methods with linear equality constraints struggle to achieve effective state prediction. This paper primarily investigates the integration of drop-point constraint information into the entire system state evolution process via convex optimization projection in the set-valued framework, aiming to reconstruct a linear dynamic system model. Subsequently, by employing multiple affine transformations of ellipsoids and designing a weight matrix, the smoothness of state trajectories is enhanced to better align with real motion patterns. Finally, through simulation experiments, we validate the significant advantages of the reconstructed system model over traditional unconstrained system models under the set-valued framework. Additionally, we demonstrate the impact of drop-point constraints on state trajectory evolution under different initial point conditions with the same drop point.

Multiple-source localization problem based on acoustic energy measurements is investigated by set-membership estimation theory. When the probability density function of measurement noise is unknown-but-bounded, multiple-source localization is a difficult problem since not only the acoustic energy measurement is a complicated nonlinear function of multiple sources, but also the multiple sources bring about a high-dimensional state estimation problem. The main contribution of this paper is as follows. Firstly, to deal with the nonlinear function, it is linearized by the first-order Taylor expansion with a remainder error. The point is that the bounding box of the remainder is derived in each iteration based on the convex bounding set of the state. Especially, when the state can be bounded in a cylinder, the remainder bound can be achieved analytically. Secondly, based on the separate property of the nonlinear observation function, an efficient estimation procedure is developed to deal with the high-dimensional state estimation problem by using an alternately optimization iterative algorithm. In the process of iteration, the remainder bound requires to be known on-line. Finally, a typical numerical example in multiple-source localization demonstrates the effectiveness of the set-membership localization algorithms. In particular, it shows that when the noise is non-Gaussian, the set-membership localization algorithm performs better than the maximum likelihood localization algorithm.