Handin 4 - Connected Inverted Pendulums (LQG)

Santa Claus is transporting two Christmas trees on a pair of spring-connected carts and needs some help with control design.

The inputs are forces on the two carts. The measured outputs are the cart positions and christmas tree angles. The system thus has two inputs and four outputs.

The system parameters correspond to $1$m christmas trees mounted on $1$ kg carts. Control signal $u = 1$ corresponds to a force of $10$ N. A spring deflection of $0.1$ m gives a spring force of $10$ N (i.e. $k = 100$). There is also a small viscous friction. The mass of the christmas trees can be neglected.

Design specifications:

Design an LQG controller obtaining

• Stable, well-damped closed-loop system
• $M_s = \|(I + CP)^{-1}\|_\infty < 3$
• $M_t = \|(I + CP)^{-1} CP\|_\infty < 3$
• Settling time (to 10 % of peak) < 5 seconds for initial disturbances $x_1(0) = 0.05$ m. Max input signal $|u_i(t)| < 1,\quad \text{for}\ i = 1,2$
• Settling time (to 10 % of peak) < 5 seconds for initial disturbances $\phi_1(0) = 0.05$ rad. Max input signal $|u_i(t)| < 1,\quad \text{for}\ i = 1,2$
• Integral action is not required.

It would be nice with better robustness margins, but it seems to be hard to get e.g. $M_t$ < 1.5 without sacrificing other performance measures (try it if you have time).

Suggest a change of process parameters that would make better robustness margins feasible (pendulum lengths, cart masses, spring constant, viscous friction constant, ...?).

Submission

Submit your code and a small report including your analysis and your design procedure.
To help you get started we've supplied Santa's initial attempt pend.m Download pend.m