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ps6 - 2.171 Problem Set 6 Assigned Wed Nov 1 2006 Due Wed...

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2.171 Problem Set 6 Assigned: Wed. Nov 1, 2006 Due: Wed. Nov 8, 2006, in class Problem 1 Suppose we have a plant with the transfer function G p ( z ) = ( z 1) 3 ( z 0 . 999) 3 ( z 0 . 99) 2 Assume that this plant is in the forward path of a unity-negative-feedback loop, with a controller transfer function G c ( z ) preceeding the plant. The objective of this problem is to design a controller which stabilizes this plant with as wide a bandwidth as possible. a) Sketch the Bode plot of this plant. Use Matlab to confirm your sketch. b) Design a controller G c ( z ) which stabilizes this plant with as wide a bandwidth as possible. Clearly explain your design thinking. Use a Nyquist analysis to indicate the stability issues and to show that your design results in a stable system. c) For your chosen controller, sketch a root locus as a function of loop gain. Use Matlab to confirm your sketch. Problem 2 FPW 7.13 a–c only. Problem 3 This problem looks at the difficulties associated with the design in Problem 2. Specif- ically, we will see that the sample rate of 50 Hz specified in the problem is too low to allow a robust design. Also, we will see that the rise time spec in Problem 2 is too long to guarantee a sufficiently robust design. a) Plot the response of your system in x and i to an initial position error in x of 1 mm. Comment on the features of this response with respect to the designed damping and natural frequency. b) Sketch a Bode plot of the loop transfer function of your design in Problem 2. You should see that there are gain-increase and gain-decrease margins. By this we mean that the controlled system will go unstable for some gain-increase
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