## Saturation of rudders

If the control signal generated by the feedback law is larger than possible or permissible for reasons of safety, the actuator will "saturate" at a lower input level. The effect of occasional control saturation is usually not serious: in fact a system which never saturates is very likely overdesigned, having a larger and less efficient actuator than is needed to accomplish even the most demanding tasks. On the other hand, if the control signals produced by the linear control law are so large that the actuator is always saturated, it is not likely that the system behavior will be satisfactory, unless the actuator saturation is explicitly accounted for in an intentionally nonlinear control law design. If such a design is not intended, the gain matrix should be selected to avoid excessively large control signals for the range of states that the control system can encounter during operation (Friedland, 1987).

A conventional value for the saturation of rudders in underwater vehicles is about 30°. (Haghi et. al, 2007) showed that if saturation occurs, the tracking error will not converge to zero, leading to instability of the vehicle. Obviously saturation must be avoided. In attempt to answer "Why saturation occurs?", we overlook the problem definition again. Previously, it was assumed that the vehicle had a constant forward velocity u . The desired trajectory is a sine wave of the form yd = a sin mt , with the amplitude a and the frequency m . Imagine driving in a road full of sharp turns. Intrinsically, the driver will slow down, to avoid turning over the vehicle. Now if the vehicle's forward speed is constant, then there will be a limit to the frequency of the road turns, that the driver can conquer without smashing his car. Same line of reasoning is made for our underwater vehicle. If the frequency of the desired trajectory m is too much, the control signal that would be needed to keep the vehicle on the track will increase. If the control signal increases so that saturation occurs, the underwater vehicle will turn over and smash out of the road! Therefore, we conclude that there should be a margin to the maximum value of frequency m that we can conquer without decreasing the speed, under which saturation will not occur. This value was found for some design parameters \ and X1, by making numerous simulations, utilizing the method known as Bisection method by numerical analyzers. Simulation results are summarized in Table 2. The value of m has been assumed to be the same for both yd and y/d .

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