of the error to drive the sliding mode observer, the observer trajectories become insensitive to many forms of noise. Hence, some sliding mode observers have attractive properties similar to the Kalman filter but with simpler implementation. As suggested by Drakunov, a sliding mode observer can also be designed for a class of non-linear systems. Such an observer can be written in terms of original variable estimate and has the form where: The vector extends the scalar signum function to dimensions. That is, for the vector . The vector has components that are the output function and its repeated Lie derivatives. In particular, where is the ith Lie derivative of output function along the vector field (i.e., along trajectories of the non-linear system). In the special case where the system has no input or has a relative degree of n, is a collection of the output and its derivatives. Because the inverse of the Jacobian linearization of must exist for this observer to be well defined, the transformation is guaranteed to be a local diffeomorphism. The diagonal matrix of gains is such that where, for each , element and suitably large to ensure reachability of the sliding mode. The observer vector is such that where here is the normal signum function defined for scalars, and denotes an "equivalent value operator" of a discontinuous function in sliding mode. The idea can be briefly explained as follows. According to the theory of sliding modes, in order to describe the system behavior, once sliding mode starts, the function should be replaced by equivalent values (see equivalent control in the theory of sliding modes). In practice, it switches (chatters) with high frequency with slow component being equal to the equivalent value. Applying appropriate lowpass filter to get rid of the high frequency component on can obtain the value of the equivalent control, which contains more information about the state of the estimated system. The observer described above uses this method several times to obtain the state of the nonlinear system ideally in finite time. The modified observation error can be written in the transformed states . In particular, and so So: As long as , the first row of the error dynamics, , will meet sufficient conditions to enter the sliding mode in finite time. Along the surface, the corresponding equivalent control will be equal to , and so . Hence, so long as ,