Let's break down the solution step-by-step for the **H∞ optimal controller design** for the given system, using **Riccati equations** and the **Bisection algorithm**. Here’s how you can approach it: --- ### **Step 1: Nominal Plant Model** Given: \[ G(s) = \frac{6s^3 + 14.75s^2 + 55.5s + 72}{s^4 + 21s^3 + 58s^2 + 104s + 96} \] --- ### **Step 2: Weighting Function** Performance weighting: \[ W_p(s) = \frac{250000}{s + 10} \] This will penalize the sensitivity function at low frequencies. --- ### **Step 3: State-Space Representation** Convert \(G(s)\) and \(W_p(s)\) to state-space forms: Let’s write general state-space equations: \[ \dot{x}_G = A_G x_G + B_G u \] \[ y = C_G x_G + D_G u \] and for \(W_p(s)\): \[ \dot{x}_W = A_W x_W + B_W e \] \[ z = C_W x_W + D_W e \] --- ### **Step 4: Generalized Plant Construction** Construct the **generalized plant** \(P(s)\) for H∞ synthesis: - **Inputs:** \( [r; w] \) (reference, disturbance) - **Outputs:** \( [z; y] \) (performance output, measured output) - **Control input:** \( u \) Augment the plant with the weighting function as follows: \[ \begin{bmatrix} \dot{x}_G \\ \dot{x}_W \end{bmatrix} = \begin{bmatrix} A_G & 0 \\ 0 & A_W \end{bmatrix} \begin{bmatrix} x_G \\ x_W \end{bmatrix} + \begin{bmatrix} B_G & 0 \\ 0 & B_W \end{bmatrix} \begin{bmatrix} u \\ e \end{bmatrix} \] \[ z = C_W x_W + D_W e \] \[ y = C_G x_G + D_G u \] --- ### **Step 5: Formulate H∞ Control Problem** The H∞ control problem is to find \(F(s)\) such that the closed-loop transfer function from \(w\) to \(z\) has \(H_\infty\) norm less than \(\gamma\). --- ### **Step 6: Solve Riccati Equations** Let the state-space of the generalized plant be: \[ \begin{align*} \dot{x} &= A x + B_1 w + B_2 u \\ z &= C_1 x + D_{11} w + D_{12} u \\ y &= C_2 x + D_{21} w + D_{22} u \end{align*} \] The H∞ controller synthesis involves solving two coupled algebraic Riccati equations (AREs): **Control ARE:** \[ A X + X A^T + X ( \gamma^{-2} B_1 B_1^T - B_2 B_2^T ) X + C_1^T C_1 = 0 \] **Filter ARE:** \[ A^T Y + Y A + Y ( \gamma^{-2} C_1^T C_1 - C_2^T C_2 ) Y + B_1 B_1^T = 0 \] --- ### **Step 7: Bisection Algorithm** 1. **Initialize** \(\gamma_{low}\) and \(\gamma_{high}\) (bounds for \(\gamma\)). 2. **Repeat:** - Set \(\gamma = (\gamma_{low} + \gamma_{high})/2\). - Try to solve the Riccati equations with this \(\gamma\). - If solution exists and closed-loop is stable, set \(\gamma_{high} = \gamma\). - Else, set \(\gamma_{low} = \gamma\). 3. **Iterate** until desired precision (\(\gamma_{high} - \gamma_{low} < \epsilon\)). --- ### **Step 8: Synthesize Controller** Once the optimal \(\gamma^*\) is found, extract the controller \(F(s)\) from the Riccati solutions. The controller will have the form: \[ F(s): \begin{cases} \dot{x}_K = A_K x_K + B_K y \\ u = C_K x_K + D_K y \end{cases} \] --- ### **Step 9: Final Answer (Summary)** #### **Summary Steps** - Convert \(G(s)\) and \(W_p(s)\) to state-space - Build generalized plant \(P(s)\) - Formulate H∞ problem and solve Riccati equations - Use bisection to find minimal \(\gamma\) for which solution exists - Extract optimal dynamic controller \(F(s)\) --- ### **Note** The actual numeric solution (controller matrices) would require MATLAB or Python with Control System Toolbox (e.g., `hinfsyn`) or manual implementation of Riccati solver and bisection. If you want, I can provide MATLAB code for this design. --- ## **Final Answer** To find the dynamic H∞ optimal controller: **1.** Convert \(G(s)\) and \(W_p(s)\) to state-space form. **2.** Construct the generalized plant \(P(s)\) for the closed-loop system. **3.** Formulate and solve the two coupled Riccati equations for a given \(\gamma\). **4.** Use the bisection algorithm to minimize \(\gamma\) such that Riccati equations have stabilizing solutions. **5.** Extract the optimal controller \(F(s)\) from the Riccati solutions. **This procedure gives the dynamic H∞ optimal controller using Riccati equations and bisection algorithm.** --- **If you need MATLAB code or want to see the Riccati equations for this specific plant, please let me know!**