Projection Operator (Lectures on Adaptive Control and Learning)

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There are two common approaches in adaptive control theory to ensure the boundedness of the closed-loop system when it is not possible to obtain a negative semi-definite Lyapunov function derivative. This video covers the first approach - the projection operator. Have fun!
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In case 1) x*e >0 because of the (w_hat is bigger than w which leads to compute state values larger that causes error to be positive as e(t) = x(t) - x_r(t) right?

akanguven