Wireless Ad Hoc and Sensor Networks

Background 75Evaluating the first difference along the closed-loop system trajectoriesyields∆L( xk ( )) ≤−x( k) ( 2 1−kv2 max )− 2xkkdk ( ) v ( ) + d 2 ( k)which is negative, as long ask maxd + k maxd + 1−k max d|( xk)|>2− kwhich, after simplification, results in( )( 1 vmax )2 2 2 2v M v M v M|( xk)|>( 1+kvmax)d21−k( vmax)MThe UUB bound can be made smaller by moving the closed-loop polesnear the origin. Placing the poles at the origin will result in a deadbeatcontroller and it should be avoided under all circumstances.ReferencesGoodwin, C.G. and Sin, K.S., Adaptive Filtering, Prediction, and Control, PrenticeHall, Englewood Cliffs, NJ, 1984.Ioannou, P. and Kokotovic, P., Adaptive Systems with Reduced Models, Springer-Verlag, New York, 1983.Jagannathan, S., Neural Network Control of Nonlinear Discrete-Time Systems, Taylorand Francis (CRC Press), Boca Raton, FL, 2006.Kalkkuhl, J.C. and Hunt, K.J., Discrete-time neural model structures for continuoustimenonlinear systems, Neural Adaptive Control Technology, Zbikowski, R.and Hunt, K.J., Eds., World Scientific, Singapore 1996, chap. 1.Landau, Y.D., Adaptive Control: The Model Reference Approach, Marcel Dekker, Basel,1979.Lewis, F.L. and Syrmos, V.L., Optimal Control, 2nd ed., John Wiley and Sons, NewYork, 1995.Lewis, F.L., Abdallah, C.T., and Dawson, D.M., Control of Robot Manipulators,Macmillan, New York, 1993.Lewis, F.L., Jagannathan, S., and Yesiderek, A., Neural Network Control of RobotManipulators and Nonlinear Systems, Taylor and Francis, London, 1999.Luenberger, D.G., Introduction to Dynamic Systems, Wiley, New York, 1979.Narendra, K.S. and Annaswamy, A.M., Stable Adaptive Systems, Prentice Hall,Englewood Cliffs, NJ, 1989.