The Alpha Beta Gamma Extension
It is sometimes useful to extend the assumptions of the alpha beta filter one level. The second state variable v is presumed to be obtained from integrating a third acceleration state, analogous to the way that the first state is obtained by integrating the second. An equation for the a state is added to the equation system. A third multiplier, gamma, is selected for applying corrections to the new a state estimates. This yields the alpha beta gamma update equations.
Similar extensions to additional higher orders are possible, but most systems of higher order tend to have significant interactions among the multiple states, so approximating the system dynamics as a simple integrator chain is less likely to prove useful.
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