EE451_Chapter2_Notes_F09

EE451_Chapter2_Notes_F09 - EE451/551: Digital Control...

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EE451/551: Digital Control Chapter 2: Discrete Time Systems
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Difference Equations Assume y and u are the system output and input, respectively; a general n th order linear time invariant TI) difference eqn is given as: (LTI) difference eqn. is given as: 11 0 () ( 1 ) ( 1 ) ( ) ) ( 1 ) ( 1 ) ( ) n yk n a yk n ayk k b k b k bk ++ + + + + + + " 01 1 0 where , , for all are constant n ii b u k n u k n b u kb u k ab i = " What does LTI mean? Why is this important? If u(k) is: zero, y(k) is the homogeneous soln. an impulse, y(k) is the impulse response a step, y(k) is the step response
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Solving Difference Equations Solve the diff eqn. with zero ICs, i.e., y( 2)=y( 1)=0, for a unit step input, i.e., u(k)=1 for all k>=0: ( 2 )0 . 8 ( 1 . 1 ( ) ( ) yk uk ++ + + = Using recursion in the time domain: ( ) 0.8 ( 1) 0.1 ( 2) ( 2) ky k y k y k u k =− − − + 0 (0) 0.8 ( 1) 0.1 ( 2) ( 2) 0 1 (1) 0.8 (0) 0.1 ( 1) ( 1) 0 yy y u y u − + − = 2 (2) 0.8 (1) 0.1 (0) (0) 1 3 (3) 0.8 (2) 0.1 (1) 0.2 y u y u + = + = 4 (4) 0.8 (3) 0.1 (2) 0.8(0.2) 0.1(1) y u + 10 . 7 4 += 5 (5) 0.8 (4) 0.1 (3) 0.8(0.74) 0.1(0.2) 1 0.388 y u + + =
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Solving Diff. Eqns. Using Matlab 2 ) 0 8( 1 ) 0 1() () kk k k + + + The recur.m function (on class site) can be used to solve (2 . . y k y k y k u k ++ = diff.
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EE451_Chapter2_Notes_F09 - EE451/551: Digital Control...

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