  We investigate quantum parameter estimation based on linear and Kerr-type
nonlinear controls in an open quantum system, and consider the dissipation rate
as an unknown parameter. We show that while the precision of parameter
estimation is improved, it usually introduces a significant deformation to the
system state. Moreover, we propose a multi-objective model to optimize the two
conflicting objectives: (1) maximizing the Fisher information, improving the
parameter estimation precision, and (2) minimizing the deformation of the
system state, which maintains its fidelity. Finally, simulations of a
simplified {\epsilon}-constrained model demonstrate the feasibility of the
Hamiltonian control in improving the precision of the quantum parameter
estimation.
