Consider the system:

$x_1^{}=x_2$

$x_2^{}=-2x_1-3x_2+u$

The above system is to be controlled to minimize a performance index and to keep the state $x_l(t)$ close to a ramp function $2t.$ The final time $t_f=15sec.,$ the final state $x(t_t)$ is free and the admissible controls and states are unbounded. Formulate the performance index, obtain the feedback control law. Simulate the closed loop system in MATLAB and plot all the time histories of Riccati coefficients, optimal states and control.