ECE-320 Linear Control Systems. Spring 2014, Exam 1. No calculators or computers allowed, you may leave your answers as fractions.

Transcription

1 ECE-0 Linear Control Sytem Spring 04, Exam No calculator or computer allowed, you may leave your anwer a fraction. All problem are worth point unle noted otherwie. Total /00

2 Problem - refer to the unit tep repone of a ytem, hown below.8.6 Step Repone Input Output.4 Diplacement Time (ec) ) Etimate the teady tate error ) Etimate the percent overhoot ) Etimate the tatic gain

3 4) Etimate the teady tate error for the ramp repone of the ytem hown below: 7 Ramp Repone 6 5 Diplacement 4 Input Output Time (ec) 5) Etimate the teady tate error for the ramp repone of the ytem hown below:.5 Ramp Repone Diplacement Input Output Time (ec)

4 6) (0 point) For thi problem aume the following unity feedback ytem with Gp () = ( + )( + ) and G ( ) = ( + ) c a) Determine the poition error contant K p b) Etimate the teady tate error for a unit tep uing the poition error contant. c) Determine the velocity error contant K v d) Etimate the teady tate error for a unit ramp uing the velocity error contant. 4

5 7) (0 point) For thi problem aume the following unity feedback ytem with Gp () = ( + )( + 4) and () 5( + ) Gc = a) Determine the poition error contant K p b) Etimate the teady tate error for a unit tep uing the poition error contant. c) Determine the velocity error contant K v d) Etimate the teady tate error for a unit ramp uing the velocity error contant. 5

6 Problem 8-0 refer to the following root locu plot (from iotool) Root Locu Editor for Open Loop (OL) Imag Axi Real Axi 8) I it poible for - to be a cloed loop pole for thi ytem? (Ye or No) 9) When the gain i approximately the cloed loop pole are a hown in the figure. If we want the ytem to be table what condition do we need to place on the gain k? 0) I thi a type one ytem? (Ye or No) 6

7 Problem - refer to the following root locu plot (from iotool).5 Root Locu Editor for Open Loop (OL) 0.5 Imag Axi Real Axi ) When k = 0.5 the pole are a they are hown in the figure. Etimate the cloed loop pole. ) Etimate the ettling time a the gain k ) I thi a type one ytem? (Ye or No) 7

8 4) (6 point) For the following two PID controller, determine k p, k i, and k d 0.( ) Gc () = ( + )( + ) Gc () = 5) (0 point) For thi problem aume the cloed loop ytem below and aume Gp () = ( + + j) ( + j) For each of the following problem you hould ketch the root locu to anwer the following quetion. (You will not be graded on your root locu ketche, jut your anwer.) a) Auming a proportional controller Gc() = kp, what i the ettling time a k p? b) Auming a proportional + derivative controller G() = k ( + z), what i the value of z o that the ettling time T = a k c c) Auming a the PI controller a k () k( + z) Gc =, what i the value of z o that the ettling time T = 4 8

9 6) (0 point) For thi problem aume the following unity feedback ytem with Gp () = ( + ) For each of the following problem you hould ketch the root locu to anwer the following quetion. (You will not be graded on your root locu ketche, jut your anwer.) a) Auming a proportional + derivative controller G() = k ( + z), what i the value of z o that the ettling time T = a k c b) Auming an integral controller G i c () k =, i the cloed loop ytem table for all value of k i c) Auming a the PI controller T = 0.5 a k () k( + z) Gc =, what i the value of z o that the ettling time 9

10 7) ( point) For the following block diagram a) Draw the correponding ignal flow graph, labeling each branch and direction. Feel free to inert a many branche with a gain of a you think you may need. b) Determine the ytem tranfer function uing Maon gain rule. You mut clearly indicate all of the path, the loop, the determinant and the cofactor, but you do not need to implify your final anwer (it can be written in term of the Pi, L i, and i ) 0

11 8) (0 point) Determine both the impule repone and the unit tep repone of a ytem with tranfer function H() = ( + ) +