midterm08_sol

midterm08_sol - ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY AND DESIGN F08 Midterm Exam Closed book and notes. No calculators. No cell phones. Show all work. Name: Solution Problem 1: /25 Problem 2: /25 Problem 3: /25 Problem 4: /25 Total: /100 Miscellaneous: b b 2 4 ac 2 a , 1 . 8 n , 4 . 6 , exp parenleftBigg radicalbig 1 2 parenrightBigg , atan([0.1 1/3 0.5 2/3 1 2 5])*180/pi = [5.7 18.4 26.6 33.7 45 63.4 78.7] acos([0.5 0.7071 0.866])*180/pi =[60 45 30] asin([0.5 0.7071 0.866])*180/pi =[30 45 60] When m 11 , m 12 , m 21 and m 22 are scalars: bracketleftBigg m 11 m 12 m 21 m 22 bracketrightBigg- 1 = bracketleftBigg m 22 m 12 m 21 m 11 bracketrightBigg m 11 m 22 m 21 m 12 . When M 11 and M 22 are invertible matrices: bracketleftBigg M 11 M 12 M 22 bracketrightBigg- 1 = bracketleftBigg M- 1 11 M- 1 11 M 12 M- 1 22 M- 1 22 bracketrightBigg . In turn, when the number of columns of C 1 agrees with the size of M 11 and the number of rows of B 2 agrees with the size of M 22 , bracketleftBig C 1 bracketrightBig bracketleftBigg M 11 M 12 M 22 bracketrightBigg- 1 bracketleftBigg B 2 bracketrightBigg = C 1 M- 1 11 M 12 M- 1 22 B 2 . Name: Problem 1 A particular levitated-ball system has dynamics given by the nonlinear equations = 1 z 2 1 + 2 z = z + tildewide u tildewide y = . 1. Find an operating point corresponding to a constant output equal to 3. Is there one or more than one such operating point? We take the state to be x = ( x 1 ,x 2 ,x 3 ) = ( , ,z ) . This gives the equations x 1 = x 2 x 2 = 1 x 2 3 1 + x 2 1 x 3 = x 3 + tildewide u tildewide y = x 1 Operating points with output equal to 3 must satisfy x * 1 = 3 , x * 2 = 0 , ( x * 3 ) 2 1 + 9 = 1 , u * = x * 3 . So there are two operating points, corresponding to x * 3 = 10 and x * 3 = 10 . 2. Compute the transfer function from tildewide u u * to tildewide y 3 where u * is an input that makes the output a constant equal to 3. (You may find helpful the expressions on the first page of the exam.) We consider the operating point with x * 3 = 10 . The transfer function for x * 3 = 10 is the same except for an extra minus sign as a factor. We compute d = 0 , C = bracketleftBig 1 0 0 bracketrightBig , B = bracketleftBig 0 0 1 bracketrightBig T , A = 1 2 x * 1 ( x * 3 ) 2 (1 + ( x * 1 ) 2 ) 2 2 x * 3 1 + ( x * 1 ) 2 1 = 1 3 5 radicalbigg 2 5 1 ....
View Full Document

This note was uploaded on 08/03/2010 for the course ECE PROF. VOLK taught by Professor Volkanrodoplu during the Spring '10 term at UCSB.

Page1 / 9

midterm08_sol - ECE 147A FEEDBACK CONTROL SYSTEMS - THEORY...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online