An unstable robot system is described by the vector differential equation

!!__ [X1] = [ 1 OJ [X1] + [1] u(t). dt Xz -1 2 xz 1 Both state VaT"iables are measurable, and so the control signal is set as u(t) = -K(x1 + xz).

(a) Design gain K so tho.t the performance inde:r; is minimized.

(b) Eval1wte the minimum value of the performance index.

(c) Determine the sensitivity of the performance to a change inK.

Assume that the initial conditions are x(O) = [n.

(d) Is the system stable without feedback signals due to u(t) 9 Problem