## Problem 4.1

Determine whether the following continuous-time linear time-invariant system is fully controllable $$\dot{x} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -4 & -3 \\ \end{bmatrix} x+ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ -1 & 1 \\ \end{bmatrix} u$$

This system is fully controllable


## Problem 4.2

Determine the range of values a,b,c for the following continuous-time linear time-invariant systems to be fully controllable

$$\dot{x} = \begin{bmatrix} -2 & 0 & 0 \\ 0 & -2 & 0 \\ 0 & 0 & -2 \\ \end{bmatrix} x+ \begin{bmatrix} a & 1 \\ 2 & 4 \\ b & 1 \\ \end{bmatrix} u$$

$\displaystyle \left[\begin{matrix}a & 1 & - 2 a & -2 & 4 a & 4\\2 & 4 & -4 & -8 & 8 & 16\\b & 1 & - 2 b & -2 & 4 b & 4\end{matrix}\right]$

This system is not fully controllable


## Problem 4.３

Determine whether the following continuous-time linear time-invariant system is fully observable

$$\dot{x} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -2 & -4 & -3 \end{bmatrix} x, y= \begin{bmatrix} 1 & 0 & 4 \\ 2 & 0 & 8 \end{bmatrix} x$$

This system is fully controllable


## Problem 4.４

Determine the range of values a,b,c for the following continuous-time linear time-invariant system to be fully observable

$$\dot{x} = \begin{bmatrix} -2 & 0 & 0 \\ 1 & -2 & 0\\ 0 & 0 & -2 \end{bmatrix} x, y= \begin{bmatrix} 1 & a & b \\ 4 & 0 & 4 \end{bmatrix} x$$

Condition 1: a!=0 or b!=1

$\displaystyle \left[\begin{matrix}1 & a & b\\4 & 0 & 4\\a - 2 & - 2 a & - 2 b\\-8 & 0 & -8\end{matrix}\right]$

$$Q=\begin{bmatrix}1 & a & b\\4 & 0 & 4\\a - 2 & - 2 a & - 2 b\\-8 & 0 & -8\end{bmatrix}$$ So, when a is not equal to 0 and b is not equal to 0, this system is fully observable

Condition 2: a==0 and b==1

$\displaystyle \left[\begin{matrix}1 & 0 & 1\\-2 & 0 & -2\\4 & 0 & 4\end{matrix}\right]$

rank(Q) = 3, thus in this condition, this system is not fully observable

## Problem 4.５

Determine the range of values a,b,c for the following continuous-time linear time-invariant system to be fully controllable and observable

$$\dot{x} = \begin{bmatrix} -1 & 1 & a \\ 0 & -2 & 1 \\ 0 & 0 & -3 \end{bmatrix} x+ \begin{bmatrix} 0 \\ 0 \\ 1 \\ \end{bmatrix} u, y= \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} x$$

## Problem 4.６

Calculate the controllability and observability index of the following continuous-time linear time-invariant system

$$\dot{x} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 3 & -1 \end{bmatrix} x+ \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 0 & 0 \end{bmatrix} u, y= \begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} x$$

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