Order poles on s-domain

Poles are ordered on s-domain of the transfer function inputted form of α and β. G(s) is rewritten that it solve the following equation.
  G(s) = {the transfer function of inputted old α and β}×H(s)

• 1st order system
        p=
  
• 2nd order system
    
      p1,p2=±j
      p1,p2=exp(±j )

Order zeros on s-domain

The system of H(s) is setted zeros z, z1 and z2 of a given the following form. The matrix of α and β as transfer function G(s) is rewritten to add zeros. The G(s) is solved the following equation.
  G(s) = {the transfer function of inputted old α and β}×H(s)
If α and β was blank, G(s) = H(s).

• 1st order system
    H(s)= s - z    z=
  
• 2nd order system
    H(s)=(s - z₁ )(s - z₂ )
      z1,z2= ± j
      z1,z2=exp(±j )

Design from ζ and ω₀ on a 2nd order system

Poles are ordered on s-domain of the transfer function inputted form of α and β. G(s) is rewritten that it solve the following equation.
  G(s) = {the transfer function of inputted old α and β}×H(s)
If α and β was blank, G(s) = H(s).

2nd order system
  
•Natural angular frequency ω₀=[rad/s]
•Damping ratio ζ=

Design the coefficient of s², s¹, s⁰

G(s) is rewritten that it solve the following equation.
  G(s) = {the transfer function of inputted old α and β}×H(s)
If α and β was blank, G(s) = H(s).

the transfer function
  H(s)=β₁s²+β₂s+β₃   

---

  β₁=0  β₁=1  β₁=

---

  β₂=0  β₂=2ζω₀ β₂=
    •Use the above form to input of ω₀, ζ

---

  β₃=0  β₃=ω₀²  β₃=
    •Use the above form to input of ω₀

Convert frequency

→

ω₀(natural angular frequency) from the transfer function of inputted α and β is converted that ω₀ is multiplied by a given ω_k.

  ω_k=    

Convert gain

G(s) transfer function of inputted α and β is multiplied by a given K.
    G(s)←KG(s)

  K=