Ackermann’s Formula for Undamped Oscillator
(a) Use Ackermann’s formula to solve for the gains for the undamped oscillator of Example 7.15. (b) Verify the calculations with MATLAB for $ω_0 = 1.$

Question

Ackermann’s Formula for Undamped Oscillator
(a) Use Ackermann’s formula to solve for the gains for the undamped oscillator of Example 7.15. (b) Verify the calculations with MATLAB for [latex] ω_0 = 1.[/latex]

Accepted Answer

(a) The desired characteristic equation is [latex] α_c(s) = (s+2ω_0)^2. [/latex] Therefore, the desired characteristic polynomial coefficients,

[latex] α_1 = 4ω_0, \ \ \ \ \ \ \ \ α_2 = 4ω^2_0, [/latex]

are substituted into Eq. (7.93) and the result is

[latex] α_c( \pmb F) = \pmb F^n + α_1 \pmb F^{n−1} + α_2 \pmb F^{n−2} +···+ α_n \pmb I, [/latex] (7.93)

[latex] \alpha _c ( \pmb F) = \begin{bmatrix} -w^2_0 & 0 \ 0 & -w^2_0 \end{bmatrix} + 4w_0 \begin{bmatrix} 0 & 1 \ -w^2_0 & 0 \end{bmatrix} + 4w^2_0 \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, [/latex] (7.94a)

[latex] \ \ \ \ \ = \begin{bmatrix} 3w^2_0 & 4w_0 \ -4w^3_0 & 3w^2_0 \end{bmatrix} . [/latex] (7.94b)

The controllability matrix is

[latex] C = \begin{bmatrix} \pmb G & \pmb {FG} \end{bmatrix} = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} , [/latex]

which yields

[latex] C^{-1} = \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} . [/latex] (7.95)

Finally, we substitute Eqs. (7.95) and (7.94a) into Eq. (7.91) to get

[latex] \pmb K = \begin{bmatrix} 0 & ··· & 0 & 1 \end{bmatrix} C^{−1}α_c(\pmb F), [/latex] (7.91)

[latex] \pmb K = \begin{bmatrix} K_1 & K_2 \end{bmatrix} \ \ \ \ \ = \begin{bmatrix} 0 & 1 \end{bmatrix} \begin{bmatrix} 0 & 1 \ 1 & 0 \end{bmatrix} \begin{bmatrix} 3w^2_0 & 4w_0 \ -4w^3_0 & 3w^2_0 \end{bmatrix} . [/latex]

Therefore

[latex] \pmb K = \begin{bmatrix} 3w^2_0 & 4w_0 \end{bmatrix} , [/latex]

which is the same result we obtained previously.

(b) The MATLAB statements

Question 7.EX.16: Ackermann’s Formula for Undamped Oscillator (a) Use Ackerman...

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