How can one use a single scalar control input to bring a vibrating square membrane as close to rest as possible?
- Write down the controlled wave equation on the unit square.
- State the Dirichlet eigenfunctions and eigenvalues.
- Derive the modal equations
- Explain the meaning of the coupling coefficient
$\beta_{mn}$ .
- Choose a truncation level
$M$ . - Flatten the double indices
$(m,n)$ into a vector of modal amplitudes. - Construct the matrices
$A$ and$B$ for the first-order system. - Verify dimensions carefully.
- Choose
$Q$ and$R$ . - Solve the algebraic Riccati equation.
- Compute the feedback gain
$K$ . - Explain how the choice of
$R$ affects the controller.
- Choose at least one nontrivial initial condition.
- Simulate the open-loop response.
- Simulate the closed-loop response.
- Plot modal energy and control input.
- Reconstruct
$u(x,y,t)$ on a spatial grid from the modal coefficients. - Plot snapshots of the membrane at several times.
- Create an animation or sequence of frames.
- Compare at least two actuator locations.
- Include the center
$\left( \frac{1}{2}, \frac{1}{2} \right)$ as one test case. - Identify modes with zero or very small coupling.
- Discuss how this affects stabilization.
- Short derivation of the modal model
- Clear definition of
$A$ ,$B$ ,$Q$ ,$R$ ,$K$ - At least two plots from the simulation
- At least one membrane reconstruction figure
- Discussion of actuator placement
- Replace the point actuator with a smooth Gaussian patch.
- Add light viscous damping and compare results.
- Track how performance changes as the truncation size grows.
- Estimate which low modes dominate the transient response.