An introduction to the theory of superfluidity by Isaac M. Khalatnikov, Isaac Markovich Khalatnikov

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By Isaac M. Khalatnikov, Isaac Markovich Khalatnikov

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3 Design of Nominal Model Suppose the desired trajectory is T d , and the nominal-model tracking error is e T n  T d . e. 3) The controller is designed for the nominal model as P § · B J n ¨ h1e  h2 e  n T d  T d ¸ Jn © ¹ From Eqs. 4) 2 Normal Sliding Mode Control § B · In order to make the system stable, s 2  ¨ h2  n ¸ s  h1 must be Hurwitz . Jn ¹ © For k ! 0, let the eigenvalue be k . e. h2 2k  n , h1 k 2 . Therefore, Jn Jn we can get h1 and h2 . 5) where J m is the minimum value of J, J M is the maximum value of J, Bm is the minimum value of B, BM is the maximum value of B, and d M is the maximum value of d.

28  Fig. 30. 29) instead of switch function (in program M 2), the simulation results are shown in Fig. 31  Fig. 33. 0*sin(t); dx(1)=x(2); dx(2)=-a*x(2)+b*ut+dt; References [1] Itkis U. Control System of Variable Structure. New York: Wiley, 1976 [2] Hung JY, Gao W, Hung JC. Variable Structure Control: A Survey, IEEE Transaction on Industrial Electronics, 1993,40(1): 2  22 [3] Edwards C, Spurgeon S. com Abstract This chapter introduces several normal sliding mode controllers design, including sliding mode control based on nominal model, global sliding mode control, sliding mode control based on linearization feedback technology and sliding mode control based on low pass filter.

25) where f and g are known nonlinear functions, u  R and y T  R are the control input and measurement output respectively. d 0 (t ) is the disturbance, and | d (T ,T , t ) |İ D, D is a positive constant. Let d (T ,T , t ) 'f (T ,T )  'g (T ,T )u  d 0 (t ), therefore, Eq. 2 Controller Design Let desired position input be T d , and e T d  T . 27) where c ! 0. 28) 6elect the Lyapunov function as L 1 2 s 2 Therefore, we have L ss s (T d  f  gu  d  ce) s (T d  f  ( f  T d  ce  K sgn ( s ))  d  ce) s (d  K sgn ( s ))  sd  K | s | If K ı D, then L  sd  K | s |İ 0 19 Advanced Sliding Mode Control for Mechanical Systems: Design, Analysis and MATLAB Simulation In order to restrain the chattering phenomenon, the saturated function sat( s ) is adopted instead of sgn( s ) in Eq.

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