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minimal realizations

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2: Minimal Realizations

a: Find a minimal realization of the following system:

$$ \begin{array}{l} \dot{x}(t)=\left[\begin{array}{cc} -1 & 1 \\ 0 & -2 \end{array}\right] x(t)+\left[\begin{array}{l} 1 \\ 0 \end{array}\right] u(t) \\ y(t)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x(t) \end{array} $$

b: Check if the following realization is minimal:

$$ \dot{x}(t)=\left[\begin{array}{cc} -1 & 1 \\ 0 & -2 \end{array}\right] x(t)+\left[\begin{array}{l} 0 \\ 1 \end{array}\right] u(t) $$

$$ y(t)=\left[\begin{array}{ll} 1 & 0 \end{array}\right] x(t) $$

ci Consider a single-input, single-output system given by:

$$ \begin{array}{l} \dot{x}(t)=\left[\begin{array}{cccc} -2 & 3 & 0 & 0 \\ 1 & -5 & 0 & 0 \\ 2 & 4 & 6 & 1 \\ 0 & 3 & 0 & 2 \end{array}\right] x(t)+B u(t) \\ y(t)=\left[\begin{array}{llll} 1 & -1 & 0 & 0 \end{array}\right] x(t) \end{array} $$

Is there a vector \(B\) that makes this s minimal realization?

d: For the system from part \(c\) find a vector \(B\) so that the minimal realization has state dimension 1.

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