Dynamic solution using Lyapunov methods

In Johansen (2004) a control-Lyapunov approach has been used to develop an optimal dynamic control allocation algorithm. The proposed algorithm leads to asymptotic optimality. Consequently, the computational complexity compared to a direct nonlinear programming approach is considerably reduced. This is done by constructing the optimizing control allocation algorithm as a dynamic update law which can be used together with a feedback control system. It is shown that the asymptotically optimal control allocation algorithm in interaction with an exponentially stable trajectory-tracking controller guarantees uniform boundedness and uniform global exponential convergence. A case study addressing low-speed maneuvering of an overactuated ship is used to demonstrate the performance of the control allocation algorithm. Extension to the adaptive case where thrust losses are estimated are given in (Tj0nnas & Johansen, 2005), and extension to the case when actuator dynamics are considered explicitly in the control allocation is given in (Tj0nnas & Johansen, 2007).

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