ps 4 604 soln - ECE 604 STATE VARIABLE ANALYSIS Fall 2009...

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ECE 604 STATE VARIABLE ANALYSIS Fall, 2009 Solution to Problem Set #4 Problem 1: (Rugh 9.1) Controllable: The controllability matrix is C = B AB A 2 B ! " # $ = 1 2 + % 2 + 2 1 1 1 1 0 0 ! " # $ Take the determinant: det C = 1 i det 2 + ! 2 + 2 1 1 " # $ % = 2 + ( 2 ( 2 = ( The system is (C) if and only if " 0 . Observable: The observability matrix is: O = C CA CA 2 ! " # # # $ % = 0 1 0 0 2 1 0 1 0 0 2 0 0 1 0 0 2 0 ! " # # # # # # # $ % Since the first column of O is zero, the system is not observable for any . Problem 2: (Rugh 9.2) Let b 1 ( t ) = b 1 be a constant. Then, det W c = det B AB ! " # $ = det b 1 1 1 0 ! " % % # $ = 1 controllable for any b 1 Let b 1 ( t ) be continuous. Then
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W 0,1 ( ) = ! 0, t ( ) B t ( ) B T t ( ) ! T 0, t ( ) dt 0 1 " ! # , $ ( ) = e A % ( ) = 1 % 0 1 ( ) * + , W 0,1 ( ) = 1 % t 0 1 ( ) * + b 1 t ( ) 1 ( ( ) * + + b 1 t ( ) 1 ( ) * + 1 0 % t 1 ( ) * + dt 0 1 " = b 1 t ( ) % t ( ) 2 b 1 t ( ) % t b 1 t ( ) % t 1 ( ( ( ) * + + + dt 0 1 " If b 1 t ( ) = t , then the rank of W 0,1 ( ) is 1 and the system is not controllable. Problem 3:
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ps 4 604 soln - ECE 604 STATE VARIABLE ANALYSIS Fall 2009...

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