Thanks Given a linear time-invariant systemx₁ (t)*₂ (t)(a) Check for controllability of the system.==−2x₁(t) + 4x2(t) + 2u(t),4x₁(t) — 2x₂(t) + 2u(t),(b) If the system is uncontrollable, separate the controllable and uncontrol-lable states.(c) Draw the state variable diagram showing clearly the controllable and un-controllable states.

Given LTI system isx1.t=-2x1t+4x2t+2ut,x2.t=4x1t-2x2t+2ut,Now the state matrix will…

Answered: Given a linear time-invariant system…

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Given a linear time-invariant system
x₁ (t)
*₂ (t)
(a) Check for controllability of the system.
=
=
−2x₁(t) + 4x2(t) + 2u(t),
4x₁(t) — 2x₂(t) + 2u(t),
(b) If the system is uncontrollable, separate the controllable and uncontrol-
lable states.
(c) Draw the state variable diagram showing clearly the controllable and un-
controllable states.

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Step 1

$Given LTI system is \dot{x_{1}} (t) = - 2 x_{1} (t) + 4 x_{2} (t) + 2 u (t), \dot{x_{2}} (t) = 4 x_{1} (t) - 2 x_{2} (t) + 2 u (t), Now the state matrix will be [\begin{matrix} \dot{x_{1}} (t) \\ \dot{x_{2}} (t) \end{matrix}] = [\begin{matrix} - 2 & 4 \\ 4 & - 2 \end{matrix}] [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] + [\begin{matrix} 2 \\ 2 \end{matrix}] u (t) Here A = [\begin{matrix} - 2 & 4 \\ 4 & - 2 \end{matrix}], B = [\begin{matrix} 2 \\ 2 \end{matrix}], AB = [\begin{matrix} - 2 & 4 \\ 4 & - 2 \end{matrix}] [\begin{matrix} 2 \\ 2 \end{matrix}] = [\begin{matrix} 4 \\ 4 \end{matrix}]$

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