Let's carefully analyze given circuit and the state model by step. # Circuit Description- The circuit: - resistors \( R) in series - Twoitors \( C) connected in parallel with the resistors. - Input voltage \( u(t) \) applied at the left. - Output voltage \( y(t) \) at the right. - Grounded at the bottom. --- # Step 1: Define State Variables The capacitors store energy, so the voltages across the capacitors are natural choices for state variables: \[ x_1(t) = v_{C_1}(t), \quad x_2(t) = v_{C_2}(t) \] --- # Step 2: Write KCL at each node ### Node at the left of \( C \): - The current through \( R \) from input to node: \( \frac{u(t) - v_{C_1}(t)}{R} \) - The current through \( C_1 \): \( C \frac{dv_{C_1}}{dt} \) Applying KCL: \[ \frac{u(t) - v_{C_1}}{R} = C \frac{dv_{C_1}}{dt} \] Rearranged: \[ C \frac{dv_{C_1}}{dt} = \frac{u(t) - v_{C_1}}{R} \] Similarly, for the node between \( C_1 \) and \( C_2 \): - The current through \( R \) from \( v_{C_1} \) to \( v_{C_2} \): \( \frac{v_{C_1} - v_{C_2}}{R} \) - The current through \( C_2 \): \( C \frac{dv_{C_2}}{dt} \) Applying KCL: \[ \frac{v_{C_1} - v_{C_2}}{R} = C \frac{dv_{C_2}}{dt} \] --- # Step 3: Write the state equations Expressed as differential equations: \[ \boxed{ \begin{aligned} C \frac{dx_1}{dt} &= \frac{u(t) - x_1}{R} \\ C \frac{dx_2}{dt} &= \frac{x_1 - x_2}{R} \end{aligned} } \] which simplifies to: \[ \boxed{ \begin{aligned} \frac{dx_1}{dt} &= - \frac{1}{RC} x_1 + \frac{1}{RC} u(t) \\ \frac{dx_2}{dt} &= \frac{1}{RC} x_1 - \frac{1}{RC} x_2 \end{aligned} } \] --- # Step 4: Output Equation The output voltage \( y(t) \) is the voltage across the second capacitor: \[ y(t) = v_{C_2} = x_2(t) \] --- # **Final State-Space Model** \[ \boxed{ \begin{aligned} \dot{\mathbf{x}}(t) &= A \mathbf{x}(t) + B u(t) \\ y(t) &= C \mathbf{x}(t) \end{aligned} } \] where \[ A = \begin{bmatrix} -\frac{1}{RC} & \\ \frac{1}{RC} & -\frac{1}{RC} \end{bmatrix}, \quad B = \begin{bmatrix} \frac{1}{RC} \\ \end{bmatrix}, \quad C = \begin{bmatrix} & 1 \end{bmatrix} \] --- Let me know if you'd like me to clarify any step or proceed further!