In other words, choose the i th transfer function as following transfer function of high order delay system,

　

　

 and consider the time constant Ti as design parameter at that time.

　

[4] Let φi(A) be defined as

　

and compute gain K and basic optimal gain

　

　

where,

　

[5] Calculate σ, a lower bound of σi by following method and choose the value of each σi (in σi > σ).

　

(a) By computing maximum eigenvalues of symmetry matrix KB + (KB)^T, the range of σ is given as

　

　

(b) Choose a σ(σ∈Σa) and compute the symmetry positive definite matrix E as

　

(c) Compute eigenvalues of following matrix F and discriminate whether they are stable or not.

　

F = AK + GH; G ≡ BE^-1/2, H ≡ E^-1/2KAK

　

 From the results, if F is stable, go to the next step. But if F is unstable, return to (b) and increase the value of σ.

　

(d) Compute eigenvalues of following Hamiltonian matrix,

　

　

and discriminate whether they are lie on the imaginary axis or not. From the results, if they are not lie on the imaginary axis, diminish the value of σ. But if any one of they lies on the imaginary axis, increase the value of σ and return to (b).

　

 Perform the above process over again and take σ as σ in case the renewed value of σ is smaller than the defined value. On the other hand, dichotomy is used to renew the value of σ in (c) and (d).