Problem 5. Consider the infinite-horizon LQR problem *1 = x2 *2 = u [x{ + 2vx1x2 + qxz + u] dt 1 roo (a) Explain why we need the constants weightings q and v to satisfy the condition q- v2 > 0 (b) Find the solution to the algebraic Riccati equation (ARE) by hand. (c) Find the optimal control and the corresponding closed-loop system equation (d) Let v=1. Determine the range q for which the optimal closed-loop system is stable. Problem 5. Consider the infinite-horizon LQR problem *1 = x2 *2 = u [x{ + 2vx1x2 + qxz + u] dt 1 roo (a) Explain why we need the constants weightings q and v to satisfy the condition q- v2 > 0 (b) Find the solution to the algebraic Riccati equation (ARE) by hand. (c) Find the optimal control and the corresponding closed-loop system equation (d) Let v=1. Determine the range q for which the optimal closed-loop system is stable