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16.31 Handout #3 Prof. J. P. How September 21, 2007 T.A. TBD Due: September 28, 2007 16.31 Homework Assignment #3 1. Given the plant G ( s ) = 1 /s 2 , design a lead compensator so that the dominant poles are located at 1 ± 1 j 2. Determine the required compensation for the system K G ( s ) = ( s + 8)( s + 14)( s + 20) to meet the following specifications: Overshoot 10% 10-90% rise time t r 100 msec Simulate the response of this closed-loop system to a step response. Comment on the steady-state error. You should find that it is quite large. Determine what modifications you would need to make to this controller so that the system also has K p > 6 thereby reducing the steady state error. Simulate the response of this new closed-loop system and confirm that all the specifications are met. 3. Develop a state space model for the transfer function (not in modal/diagonal form). Discuss what state vector you chose and why. ( s + 1)( s + 3) G 1 ( s ) = (1) ( s + 2)( s + 4) (a) Develop a “modal” state space model for this transfer function as well. (b) Confirm that both models yield the same transfer function when you compute G ˆ ( s ) = C ( sI A ) 1 B + D 1 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare
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Unformatted text preview: (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 4. A set of state-space equations is given by: x ˙ 1 = x 1 ( u − βx 2 ) x ˙ 2 = x 2 ( − α + βx 1 ) where u is the input and α and β are positive constants. (a) Is this system linear or nonlinear, time-varying or time-invariant? (b) Determine the equilibrium points for this system (constant operating points), assuming a constant input u = 2. (c) Near the positive equilibrium point from (b), Fnd a linearized state-space model of the system. 2 Cite as: Jonathan How, course materials for 16.31 Feedback Control Systems, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]....
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