4. investigate the observability of the system
\\begin{aligned}
\mathbf{x} &= \mathbf{a}\mathbf{x} + \mathbf{b}\mathbf{u} \\
\mathbf{y} &= \mathbf{c}\mathbf{x}
\end{aligned}\
where
(a) $\mathbf{a} = \begin{bmatrix} -1 & 2 \\ 3 & -4 \end{bmatrix}$, $\mathbf{c} = \begin{bmatrix} 1 & 1 \end{bmatrix}$;
(b) $\mathbf{a} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -6 & -11 & -6 \end{bmatrix}$, $\mathbf{c} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}$.

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Accepted Answer

# Explanation:

🆕 New Concept Discovered: Observability Matrix
Determining if internal states can be reconstructed from outputs.

## Step 1: Understand Observability

A linear time-invariant system described by:
$$ \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} $$
$$ \mathbf{y} = \mathbf{C}\mathbf{x} $$
is completely observable if and only if the observability matrix $\mathcal{O}$ has full rank. For an $n \times n$ state matrix $\mathbf{A}$, the observability matrix is defined as:
$$ \mathcal{O} = \begin{bmatrix} \mathbf{C} \ \mathbf{C}\mathbf{A} \ \mathbf{C}\mathbf{A}^2 \ \vdots \ \mathbf{C}\mathbf{A}^{n-1} \end{bmatrix} $$
The system is observable if $\text{rank}(\mathcal{O}) = n$.

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## Step 2: Investigate System (a)

For system (a), we have:
$$ \mathbf{A} = \begin{bmatrix} -1 & 2 \ 3 & -4 \end{bmatrix}, \quad \mathbf{C} = \begin{bmatrix} 1 & 1 \end{bmatrix} $$
Here, the dimension of the state vector is $n = 2$. Therefore, the observability matrix is:
$$ \mathcal{O} = \begin{bmatrix} \mathbf{C} \ \mathbf{C}\mathbf{A} \end{bmatrix} $$

First, calculate $\mathbf{C}\mathbf{A}$:
$$ \mathbf{C}\mathbf{A} = \begin{bmatrix} 1 & 1 \end{bmatrix} \begin{bmatrix} -1 & 2 \ 3 & -4 \end{bmatrix} = \begin{bmatrix} (1)(-1) + (1)(3) & (1)(2) + (1)(-4) \end{bmatrix} = \begin{bmatrix} 2 & -2 \end{bmatrix} $$

Construct the observability matrix $\mathcal{O}$:
$$ \mathcal{O} = \begin{bmatrix} 1 & 1 \ 2 & -2 \end{bmatrix} $$

Calculate the determinant of $\mathcal{O}$ to check its rank:
$$ \det(\mathcal{O}) = (1)(-2) - (1)(2) = -2 - 2 = -4 $$

Since $\det(\mathcal{O})
eq 0$, the matrix $\mathcal{O}$ has full rank ($\text{rank}(\mathcal{O}) = 2 = n$). Thus, system (a) is completely observable.

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## Step 3: Investigate System (b)

For system (b), we have:
$$ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ -6 & -11 & -6 \end{bmatrix}, \quad \mathbf{C} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} $$
Here, the dimension of the state vector is $n = 3$. Therefore, the observability matrix is:
$$ \mathcal{O} = \begin{bmatrix} \mathbf{C} \ \mathbf{C}\mathbf{A} \ \mathbf{C}\mathbf{A}^2 \end{bmatrix} $$

First, calculate $\mathbf{C}\mathbf{A}$:
$$ \mathbf{C}\mathbf{A} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ -6 & -11 & -6 \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} $$

Next, calculate $\mathbf{C}\mathbf{A}^2$:
$$ \mathbf{C}\mathbf{A}^2 = (\mathbf{C}\mathbf{A})\mathbf{A} = \begin{bmatrix} 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 & 0 \ 0 & 0 & 1 \ -6 & -11 & -6 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} $$

Construct the observability matrix $\mathcal{O}$:
$$ \mathcal{O} = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} $$

This is the $3 \times 3$ identity matrix $\mathbf{I}_3$. Its determinant is:
$$ \det(\mathcal{O}) = 1 $$

Since $\det(\mathcal{O})
eq 0$, the matrix $\mathcal{O}$ has full rank ($\text{rank}(\mathcal{O}) = 3 = n$). Thus, system (b) is completely observable.

# Answer:

* **System (a):** Completely observable (the observability matrix $\mathcal{O}$ has full rank of 2).
* **System (b):** Completely observable (the observability matrix $\mathcal{O}$ has full rank of 3).

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QUESTION IMAGE

Question

Explanation:

Step 1: Understand Observability

Step 2: Investigate System (a)

Step 3: Investigate System (b)

Answer: