P11.3 An unstable robot system is described by the vec

tor differential equation [9]

10 0 *x() = [ 1 8 ]x(0) + [ 9 ] (1), 00 where x(t) = (x1 (1) = x2 (1)). Both state variables are measurable, and so the control signal is set as u(t) = -k(x1 (1)+ x2 (1)). Design the gain k so that the perfor- mance index J = x(t)x(t) dt is minimized. Evaluate the minimum value of the performance index. Determine the sensitivity of the performance to a change in k. Assume that the initial conditions are x(0)= D) = ( : ). Is the system stable without the feedback signals due to u(t)?