Consider the automatic ship-steering system discussed in P8.ll and P9.15.The state

Chapter 0, Problem P11.20

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Consider the automatic ship-steering system discussed in P8.ll and P9.15.The state variable form of the system differential equation is m -0.05 - 6 0 0 -10"3 -0.15 0 0 1 0 0 13 0 1 0 0 x(t) + -0.2 0.03 0 0 5(f), where xr (/) = [v cos v 0]. The state variables are X\ = v - the transverse velocity; x2 = o>s = angular rate of ship's coordinate frame relative to response frame; x = y = deviation distance on an axis perpendicular to the track; x4 = 0 = deviation angle. (a) Determine whether the system is stable, (b) Feedback can be added so that 6(0 = -kxx{ - 3x3. Determine whether this system is stable for suitable values of k} and k$.