(Solution) - Consider the single input single output system described by x t Ax t -(2025 Original AI-Free Solution)

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Consider the single-input, single-output system described by
x(t) = Ax(t) + Bu(t)
y(t) = Cx(t)
where

Assume that the input is a linear combination of the states, that is,
u(t) = -Kx(t) + r(t),
where r(t) is the reference input. Determine K = [K1 K2] so that the closed-loop system
x(t) = [A - BK]x(t) + Br(t)
y(t) = Cx(t)
possesses closed-loop eigenvalues at r1 and r2. If r1 = ? + jw is a complex number, then r2 = ? - jw is its complex conjugate.