According to the above formulas, the current error equation is
From the equivalent control concept [8], if the current trajectories reach the sliding manifold, we have By selecting large enough, the sliding mode will occur , and then it follows that
Combining the results from (5) and (8), the rotor flux observer can be constructed as
where L is the observer gain matrix to be decided such that the observer is asymptotically stable. From (5) and (8), the error equation for the rotor flux is
In order to derive the adaptive scheme, Lyapunov’s stability theorem is utilized. If we consider the rotor speed as a variable parameter, the error equation of flux observer is described by the following equation:
Since the second term is equal to the third term in (14), the time derivative of becomes
It is apparent that (17) is negative definite. From Lyapunov stability theory, the flux observer is asymptotically stable, guaranteeing the observed flux to converge to the real rotor flux.
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