Finding parameters of exponentially decaying sinusoids (Matrix Pencil Method)
In his book, Numerical Methods that Work, Forman S. Acton writes, page 253:


  For it is  well known that an exponential equation of this type [$y=Ae^{-at}+Be^{-bt}$] in which all four parameters are to be fitted is extremely ill conditionned. That is, there are many combinations of $(a,b,A,B)$ that will fit most exact data quite well indeed (will you believe four significant figures? and when experimental noise is thrown into into the pot, the entire operation becomes hopeless. But those with Faith in Science...


A four exponential sum seems quite ill-posed to me, yet there have been recent advances using sparsity assumptions or more involved optimization algorithms. Recent examples can be found for instance in Parameter estimation of monomial-exponential sums or Nonlinear Approximation by Sums of Exponentials and Translates, or citations thereof (I have not had the opportunity to test them by myself.