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44 FEEDBACK ON A HIGHLY MANEUVERABLE VESSEL 185 44 Feedback on a Highly Maneuverable Vessel The directional behavior of a highly maneuverable vessel is modeled by θ 0 . 1 s + 1 = δ s 2 + 0 . 6 s 0 . 05 where the rudder deflection is δ and the heading is θ . You note that because the transfer function denominator has a negative coefficient, the system is unstable without a control sys- tem. Also, the numerator shows that the hydrodynamic lift on the rudder has a component that scales rudder position as well as one that scales rudder rotation rate . 1. Find the poles (the roots of the denominator polynomial) of this system. Are the roots complex (indicating an oscillatory behavior), or real (indicating two exponential growth/decay modes)? At what rate does the system go unstable - that is, over what time does the response grow by a factor of e ? (You can confirm this with a simulation of the system behavior from some nonzero initial condition.) The poles are given by the roots of the denominator polynomial equation in s , i.e., the roots of s 2 + 0 . 6 s 0 . 05 = 0 . These are -0.674 and 0.074, both real. The negative one pertains to a stable mode whereas the positive one indicates an unstable mode. The time scale (or time constant) of the unstable mode is 1/0.074 or about 13.5 seconds. This is the amount of time it takes for the output to increase by a factor of e = 2 . 718 . It is confirmed in an impulse response, for example, after the transient response of the first, stable mode has died away. 2. Design a proportional-derivative controller for the boat (PD), so as to achieve a closed- loop bandwidth of ω c = 1 rad/s and a damping ratio of ζ = 0 . 5. The idea here is to choose k p and k d so as to put the closed-loop poles - that is, the roots of 1+ P ( s ) C ( s ) = 0 - at the specific locations ω c ζ ± c 1 ζ 2 . The easiest way to carry this out is to write out the polynomial with k d and k p as free variables, and recognize that this same polynomial can be written as s 2 + 2 ζω c s + ω c 2 = 0. Some algebra will give you a set of two equations in two unknowns.
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