1. **Problem statement:** Given a discrete-time system described by difference equations, we need to determine the state-space model in controlled canonical form and find the output for a given input. 2. **Controlled canonical form:** For a system described by a difference equation of order $n$, the controlled canonical form state-space model is given by: $$\mathbf{x}[k+1] = \mathbf{A} \mathbf{x}[k] + \mathbf{B} u[k]$$ $$y[k] = \mathbf{C} \mathbf{x}[k] + D u[k]$$ where $\mathbf{A}$ is an $n \times n$ companion matrix, $\mathbf{B}$ is a column vector, $\mathbf{C}$ is a row vector, and $D$ is a scalar. 3. **Forming the state model:** - Identify the coefficients of the difference equation. - Construct matrix $\mathbf{A}$ with the negative coefficients in the last row and ones on the superdiagonal. - Vector $\mathbf{B}$ has zeros except a 1 in the last position. - Vector $\mathbf{C}$ contains the output coefficients. - Scalar $D$ is the direct feedthrough term. 4. **Finding the output:** - Use the state-space equations to compute $\mathbf{x}[k]$ iteratively for the given input $u[k]$. - Compute output $y[k]$ using $y[k] = \mathbf{C} \mathbf{x}[k] + D u[k]$. Since the exact difference equation and input are not provided, this is the general method to find the controlled canonical form and output. **Final answer:** The state-space model in controlled canonical form is constructed as above, and the output is found by iterating the state equations with the given input.