2. Designing a critically damped controller. (a) Consider the following equation of motion. 25x10x = f where x is the position and f is the external force. i. Consider the controller f = -10x-kax. What is the value of ka to ensure a critically damped system? ii. Consider the control partitioning method of deriving a controller. The control is of the form: f = m(10x kax) + cx + kx. What is k, c, m, and kd for a critical damped controller. (b) The equations of motion a two degree of freedom system is given below. 2 1 1 + 1 0.5] [1 [3 0.5 1 x1 + = 4 10 X2 i. Create a controller for the system using control partitioning and using a proportional derivative controller. What is the condition on the proportional and derivative gain for the system to be critically damped? ii. Draw a block diagram for the system.