We are going to control a nonlinear system. If we minimize the error vector of a closed loop system to zero we are able to make our controller stable. So, we try to define a function which is called q( e) to make it possible. if we minimize this function to zero we are able to minimize the error vector to zero. In a controller we have desired values and actual values. So, we are able to form the error. If we have just one measurement at output of our plant we have just one error value which is scalar. Think about a system such a manipulator which has a nonlinear transfer function. We are interested in position and velocity as well as acceleration of the end effector during passing trough the desired trajectory. If so we form a vector of errors as a 1 by 3 column vector. This error vector should be converged to zero in order to have stable control over the closed loop. In Sliding mode control, we try to define new measure consists of error vector as a function. We define a kind of properties for this function form the error vector to make it a function of Lyapunov stability criterion direct method. Direct method of Lyapunov stability can be studied at http://www.math.byu.edu/~grant/courses/m634/f99/lec22.pdf . In short we can say, if we define a Lyapunov function which is positive definite it means it is greater than zero for every value in domain except value is equal to zero. If the first partial derivatives are continuous fictions within ad certain region around the position of rest and negative definite then we can say this position of rest is asymptotically stable. This theorem is well defined in book Fuzzy control (by :Kai Michels, Frank Klawonn, Rudolf Kruse, Andreas Nürnberger). |

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