To solve this problem, let's approach it step-by-step by using control theory concepts, specifically focusing on the controllability and transfer function of a state-space system.

Problem Analysis and Solution Steps:

Given System

The system's state-space representation is given by: $\dot{x}(t) = \begin{bmatrix} 1 & \alpha & -2 \\ 0 & -2 & 0 \\ 0 & 1 & -1 \end{bmatrix} x(t) + \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} u(t)$ $y(t) = \begin{bmatrix} 1 & 1 & 0 \end{bmatrix} x(t)$

The task is divided into two parts:

1. Find the value of $\alpha$ that makes the system uncontrollable.
2. With the value of $\alpha$ from part (a), derive the transfer function and show that the system exhibits reduced-order dynamics.

Step 1: Determine the Value of $\alpha$ for Uncontrollability

To determine controllability, we use the controllability matrix $C$, defined as: $C = \begin{bmatrix} B & AB & A^2B \end{bmatrix}$ where $A$ is the system matrix, and $B$ is the input matrix. The system is controllable if the rank of $C$ is equal to the order of the system (in this case, 3).

Constructing the Controllability Matrix $C$:

1. Calculate $B$, which is given directly.
2. Compute $AB$ and $A^2B$.
3. Form $C$ and evaluate its rank.

For uncontrollability, we need the rank of $C$ to be less than 3, which occurs if $\alpha$ satisfies certain conditions.

Step 2: Derive the Transfer Function for the Uncontrollable Case

With the determined value of $\alpha$, we can find the transfer function $G(s) = \frac{Y(s)}{U(s)}$ from the state-space representation by using the formula: $G(s) = C(sI - A)^{-1}B + D$ Here:

• $s$ is the Laplace variable.
• $I$ is the identity matrix of appropriate dimensions.

Since we have no direct feedthrough ($D = 0$), we only need to compute $C(sI - A)^{-1}B$.

Let's proceed with the calculations:

1. Calculate the Controllability Matrix and Find $\alpha$ for Uncontrollability

2. Derive the Transfer Function with the Identified $\alpha$

I'll compute these values step-by-step and present the results.The rank of the controllability matrix $C$ is 3 for general values of $\alpha$, indicating that the system is generally controllable. However, we need the system to become uncontrollable, which would occur if the rank of $C$ drops below 3.

To achieve this, we need to find specific values of $\alpha$ that make the columns of $C$ linearly dependent, causing the rank to drop.

Let's proceed by setting conditions for linear dependence in $C$ to identify values of $\alpha$ that would make the system uncontrollable.It seems there was an issue with computing the determinant directly. Instead, I'll try examining linear dependencies among columns of the controllability matrix to identify values of $\alpha$ that result in a rank deficiency.

I'll check for conditions on $\alpha$ by analyzing dependencies in the columns of $C$ rather than relying solely on the determinant approach.The reduced row echelon form (RREF) of the controllability matrix $C$ confirms that, for general values of $\alpha$, the system remains controllable, as the RREF has full rank (3). This implies that there is no value of $\alpha$ for which the system naturally becomes uncontrollable with the given form of $A$ and $B$.

If you're specifically interested in an analysis for reduced-order dynamics, I could proceed with computing the transfer function directly using symbolic values and illustrate any conditions that might suggest reduced-order behavior.

Would you like me to continue with the transfer function computation under this assumption?