Explain how the time constant of the optimal closed-loop

Question

1. Consider the LQR minimization problem

J = ∫_o^∞ (mx²(t)+u²(t))dt; m>o

and constraint:

x^{.(t) = ax(t)+µ(t); y(t) = x(t)}

a. Is the system controllable? Observable?

b. Find the optimal state feedback. Solve the ARE for two possible solutions. Which is the positive definite solution?

c. Verify that selecting the positive definite solution of ARE in (ii) gives a stable response by analyzing the optimal closed-loop g coefficient for optimal state equation x*(t) = gx*(t).

d. Explain how the time constant of the optimal closed-loop system varies with parameter m. In your answer concentrate on the speed with which the state x(t) decays to zero starting from some arbitrary initial condition x(0) = xo (e.g. the time constant of the response) and on the peak value of the optimal control signal u(t) i.e. max_t≥o |u(t)|.

e. Compare the two parameters (time constant and the max_t≥0 |u(t)| in relation to the real physical limitation on the maximum magnitude of the optimal control.