Nonlinear observability and observer design for parametric roll

Parametric roll resonance is a nonlinear resonance phenomenon that can cause roll angles of up to 40 for ships. Modern container ships are especially prone to parametric roll due to their particular hull shape, bow flare and stern overhang. Control strategies have been developed to control parametric roll resonance. An especially promising control strategy seems to be speed control. However this control strategy uses both the speed of the ship and the wave encounter frequency and while generally the speed of the ship is known, the wave encounter frequency is not, since this is not trivial to measure at sea. In this work a state-space model to describe the ship will be derived. For this model observability will be checked using two different techniques. Furthermore the model will be used to design a state observer for the wave frequency and wave encounter frequency. Observability will first be checked using the nonlinear observability rank condition, which proves local observability if the observation space of the model has full column rank. A second technique uses the observability Grammian for the linearized system to determine local observability. Using a particular model structure it will be shown that it suffices to analyze the Grammian of a reduced order model. This technique can also only prove local observability since it depends on the particular linearization of the system. After observability of the model is investigated a state observer is designed. However first the theory of Kalman filtering is presented. The first state observer that is then designed is an extended Kalman filter. An unscented Kalman filter to reduce linearization errors is also developed. These two types of Kalman filters are tested and their behavior with respect to initial conditions is analyzed. Finally a fixed gain observer is designed and tested in simulation. This fixed gain observer still requires a mathematical proof of stability. An attempt to prove stability for this observer is made by using Lyapunov methods based techniques

Nonlinear observability and observer design for parametric roll

Parametric roll resonance is a nonlinear resonance phenomenon that can cause roll angles of up to 40 for ships. Modern container ships are especially prone to parametric roll due to their particular hull shape, bow flare and stern overhang. Control strategies have been developed to control parametric roll resonance. An especially promising control strategy seems to be speed control. However this control strategy uses both the speed of the ship and the wave encounter frequency and while generally the speed of the ship is known, the wave encounter frequency is not, since this is not trivial to measure at sea. In this work a state-space model to describe the ship will be derived. For this model observability will be checked using two different techniques. Furthermore the model will be used to design a state observer for the wave frequency and wave encounter frequency. Observability will first be checked using the nonlinear observability rank condition, which proves local observability if the observation space of the model has full column rank. A second technique uses the observability Grammian for the linearized system to determine local observability. Using a particular model structure it will be shown that it suffices to analyze the Grammian of a reduced order model. This technique can also only prove local observability since it depends on the particular linearization of the system. After observability of the model is investigated a state observer is designed. However first the theory of Kalman filtering is presented. The first state observer that is then designed is an extended Kalman filter. An unscented Kalman filter to reduce linearization errors is also developed. These two types of Kalman filters are tested and their behavior with respect to initial conditions is analyzed. Finally a fixed gain observer is designed and tested in simulation. This fixed gain observer still requires a mathematical proof of stability. An attempt to prove stability for this observer is made by using Lyapunov methods based techniques