A very interesting and useful velocity control system has been designed for a wheelchair control system. A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure

1.The headgear sensor provides an output proportional to the magnitude of the head movement. There is a sensor mounted at 90° intervals so that forward, left, right, or reverse can be commanded. \text { Typical values for the time constants are } \tau_{1}=0.5 \mathrm{~s}, \mathrm{t}_{2}=1 . \mathrm{s}, \text { and } \tau_{3}=0.25 \mathrm{~s} (a) Determine the limiting gain K = K¡K2K3 for a stable system. (b) When the gain K is set equal to one-third (1/3) of the limiting value, determine whetherthe settling time (to within 2% of the final value of the system) is T§< 4 s. (c) Determine the value of gain that results in a system with a settling time of Ts<4 s. Also,obtain the value of the roots of the characteristic equation when the settling time is Ts<4 s.