# Solution to the State-Space Control System Problem ## System Description The state-space system is given by: \[ \begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \\ \dot{x}_4 \end{bmatrix} = \begin{bmatrix} & 1 & & \\ & & & \\ & & & 2 \\ & & & \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} + \begin{bmatrix} \\ 1 \\ \\ \end{bmatrix} u \] Let \( A \) and \( B \) be the system and input matrices, respectively. --- ## (a) **Controllability Analysis** To determine controllability, construct the controllability matrix: \[ \mathcal{C} = [B\ AB\ A^2B\ A^3B] \] Let's compute each term: ### \( B \): \[ B = \begin{bmatrix} \\ 1 \\ \\ \end{bmatrix} \] ### \( AB \): \[ AB = A \cdot B = \begin{bmatrix} & 1 & & \\ & & & \\ & & & 2 \\ & & & \end{bmatrix} \begin{bmatrix} \\ 1 \\ \\ \end{bmatrix} = \begin{bmatrix} 1 \\ \\ \\ \end{bmatrix} \] ### \( A^2B \): \[ A^2B = A \cdot AB = \begin{bmatrix} & 1 & & \\ & & & \\ & & & 2 \\ & & & \end{bmatrix} \begin{bmatrix} 1 \\ \\ \\ \end{bmatrix} = \begin{bmatrix} \\ \\ \\ \end{bmatrix} \] ### \( A^3B \): \[ A^3B = A \cdot A^2B = A \cdot \begin{bmatrix} \\ \\ \\ \end{bmatrix} = \begin{bmatrix} \\ \\ \\ \end{bmatrix} \] ### **Controllability Matrix** \[ \mathcal{C} = \left[ B\ AB\ A^2B\ A^3B \right] = \begin{bmatrix} & 1 & & \\ 1 & & & \\ & & & \\ & & & \end{bmatrix} \] #### **Rank of the Controllability Matrix** The rank is 2 (since only the first 2 rows/columns are nonzero and linearly independent). The system is **not controllable** (it is a 4-state system, but the controllability matrix has rank 2). --- ## (b) **Input Decoupling for Uncontrollable Modes** To decouple the input and obtain the uncontrollable modes: - The controllable states are associated with the first two states (\(x_1, x_2\)). - The uncontrollable modes are associated with the remaining states (\(x_3, x_4\)), which are not affected by the input. ### **Transformation to Decouple** Let \( x = Tz \), where \(T\) is a transformation matrix that puts the system into controllable and uncontrollable canonical form. One possible \(T\) is: \[ T = \begin{bmatrix} & 1 & & \\ 1 & & & \\ & & 1 & \\ & & & 1 \end{bmatrix} \] Rewriting the system in these new coordinates will separate the controllable and uncontrollable states. The uncontrollable subsystem is: \[ \dot{x}_3 = \\ \dot{x}_4 = 2x_3 \] This subsystem is unaffected by the input \(u\). --- ## **Final Answers** ### (a) **The system is not controllable.** - **Rank of controllability matrix = 2 < 4.** ### (b) **The uncontrollable modes are \(x_3, x_4\).** - **Input does not affect these modes.** - **They evolve independently:** \[ \dot{x}_3 = , \quad \dot{x}_4 = 2x_3 \] --- Let me know if you need the explicit decoupling transformation!