s = +2.23 s = –1.77 s = –10.67  j0.25


of which the first represents the unstable mode. The


instability may be corrected by making rudder angle depend on roll angle, and proportional control with a gain in excess of about 4 is adequate in this case. But directional control is needed too. The requirement here could be expressed at various levels of sophistication, in terms of control of heading, or course, or of track (e.g. “keep to the right-hand side of the river”), but for the present purpose it will be sufficient to suppose an input of “desired yaw-rate”, rd. A possible control algorithm, to keep the vessel both near-vertical and turning at the desired rate, could be:


δ = Kpp + Kr(r – rd) . (13)


in which the rate of turn of the rudder is made a linear combination of


For the mathematical models studied so far, Kp and Kr have both been positive constants. result of a simulation of


roll-rate and of the error in yaw-rate. Figure 6 shows the


this mathematical model


responding to a 0.2 rad/s step in rd when Kp = 8.5 and Kr =1.1.


increasing the gains and using phase-advance. points may be noted from the graphs of Figure 6:


A faster response may be had, if required, by Two


a) The roll angle is non-zero during the turn. The vessel leans inward, like a bicycle, and for the same reason, by 80 mrad (4.6°).


b) The rudder is turned “the wrong way” initially, to promote the inward roll. This is theoretically necessary if the vessel-plus-rider combination is regarded as a single rigid body, but can be reduced by applying the rd input more gradually, and


probably eliminated


altogether if the rider bends the upper part of his body slightly outwards at the start of the turn, thereby tilting the vessel inwards. A bicycle appears to behave similarly.


In these idealised simulations, a constant desired yaw- rate leads to constant roll angle and constant rudder angle. Reality is somewhat different, because:


a) the sensing of roll-rate in particular is imperfect, which results corrections.


in continuous small The imperfections


rudder can be


modelled, in the absence of better information, as additive random noise;


b) external disturbances such as waves will necessitate further corrections with the rudder. On a river, the wash of a fast overtaking motor- cruiser can be a particularly challenging such disturbance.


These things can of course be included in the simulation, and have been. Another feature of roll-rate sensing, in the case of a human rider, is that it is done partly by the “semicircular canals” in the ears, and partly by the visual system. The semicircular canals have no zero-frequency response [15]; and the visual system, eyes plus brain, seems from simulator experiments to be much too slow on its own. No doubt the brain combines the two sensory signals in some way, so that each compensates for the defects of the other. But


it is possible, with a more


elaborate control system, to run a successful simulation with a rate-sensor mimicking the semicircular canals in having no zero-frequency response.


Figure 7 shows the


results of such a simulation when sensor noise is included too. The mathematical model of the vessel is the same as above, but the desired yaw-rate here is a sinusoid at one cycle every 20s, 0.2rad/s amplitude, which corresponds to a fluctuation in heading of 36º. The variable y1 is the output of a simulated roll-rate sensor as above, to which noise of 0.01 rad/s rms has been added. The control system uses an “observer” [16] to make an estimate of the actual roll-rate, though there are other, simpler solutions.


Figure 7: Simulation of a vessel controlled with a noisy roll-rate sensor


having


Figure 6: Simulated response to a step in desired yaw- rate


no Desired yaw-rate is a sinusoid zero-frequency response.


B-6


©2007: Royal Institution of Naval Architects


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