ME461_F11_Prelab8_11_06_11 - ME 461 Homework #8 Fall 2011...

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Unformatted text preview: ME 461 Homework #8 Fall 2011 Due at the beginning of Lab #8 Suggested Reading: Study the Matlab help pages for the following functions: mldivide, sisotool, tf, zpk, pzplot and rltool. Lab 8 manual. 1. Write out the syntax for creating the following transfer functions in Matlab. Use 50ms sampling time. G( z ) = ( z − 0.25 ) 2z 2 − z + 4 H (z) = 5 3 2 z + 5z − z + 1 ( z + 0.5 )( z − 0.7 ) 2. Sketch the pole‐zero plots of the transfer functions in problem 1. Indicate whether each is stable and/or has minimum phase. You may use Matlab to check your answers. K preceded by a zero‐order hold (ZOH) has a discrete τs +1 K ⎡1 − e −T τ ⎤ ⎣ ⎦ 3. Show that a first‐order plant G ( s ) = transfer function given by V (z) U (z) = . z − e −T τ 4. Show that the transfer function derived in problem 3 results in a difference equation (which can be implemented in a microprocessor!) given by v ( n ) = c1v ( n − 1) + c2u ( n − 1) , where c1 and c2 are constants. 5. Derive the transfer function (from error to control input) of a discrete PI controller (using the Tustin rule) as a single rational function in terms of the controller gains Kp and Ki and the sample time Ts. Also determine the corresponding difference equation. 6. Show that the characteristic equation of a first‐order plant preceded by a zero‐order hold and a discrete PI controller is given by the equation found at the bottom of slide 31 in the lecture notes. 7. Show that the characteristic equation determined in problem 6 can be rewritten as shown at the top of slide 32 in the lecture notes. 8. Express Ki (the integral gain) and KP (the proportional gain) in terms of Kloop (the loop gain of the characteristic equation) and zc (the compensator zero). ME 461 1 Prelab #8 ...
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This note was uploaded on 11/07/2011 for the course ME 461 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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