Question

Given are the following eigen-values of two adaptive systems. System 1: λ1 =3,λ 2 =1.5,λ3 =2, System 2:λ1 =0.04,λ2=0.01,λ3 =0.04,λ4 =0.08.

Which system may potentially converge faster (with properly chosen step size)? Justify your answer, don't guess.

Answer 1

System 2, with its smaller eigenvalues, may potentially converge faster than System 1 when proper step sizes are selected, because a larger step size can be used to maintain a stable convergence rate.

Step-by-step explanation:

In the context of adaptive systems, the eigenvalues given for the two systems indicate how quickly they may converge to a solution with a properly chosen step size. The convergence rate of an adaptive system is generally faster when the absolute values of the eigenvalues are smaller because the product of the step size (f) and the eigenvalue (λ) needs to be a constant. A larger eigenvalue requires a smaller step size to maintain the product as a constant, which can slow down the convergence.

System 1 has eigenvalues λ1 = 3, λ2 = 1.5, λ3 = 2. System 2 has eigenvalues λ1 = 0.04, λ2 = 0.01, λ3 = 0.04, λ4 = 0.08. Given that the eigenvalues for System 2 are much smaller than those of System 1, if the step sizes are chosen appropriately, System 2 may potentially converge faster since it can handle a larger step size, hence reaching equilibrium quicker without stability issues. This is predicated on the systems being at least critically damped or underdamped as an overdamped system would move slowly to equilibrium regardless of eigenvalues.