ECE-320 Linear Control Systems. Winter 2013, Exam 1. No calculators or computers allowed, you may leave your answers as fractions.

Transcription

1 ECE-320 Linear Control Systems Winter 2013, Exam 1 No alulators or omputers allowed, you may leave your answers as frations. All problems are worth 3 points unless noted otherwise. Total /100 1

2 Problems 1-3 refer to the unit step response of a system, shown below Step Response Input Output 1.4 Displaement Time (se) 1) Estimate the steady state error 2) Estimate the perent overshoot 3) Estimate the stati gain 2

3 Problems 4-6 refer to the unit step response of a system, shown below 0.4 Step Response 0.2 Displaement Input Output Time (se) 4) Estimate the steady state error 5) Estimate the perent overshoot 6) Estimate the stati gain 3

4 7) Estimate the steady state error for the ramp response of the system shown below: 2 Ramp Response Displaement Input Output Time (se) 8) Estimate the steady state error for the ramp response of the system shown below: 12 Ramp Response 10 8 Displaement Input Output Time (se) 4

5 9) The unit step response of a system is given by yt ut te ut e ut 4 2 () () t t = () + () Estimate the steady state error for this system t 10) The unit ramp response of a system is given by yt () = 2 ut () + tut () + e ut (). Estimate the steady state error for this system 11) (10 points) For this problem assume the following unity feedbak system with Gp 3 () s = ( s+ 2)( s+ 4) and G ( s) = 2( s+ 3) a) Determine the position error onstant K p b) Estimate the steady state error for a unit step using the position error onstant. ) Determine the veloity error onstant K v d) Estimate the steady state error for a unit ramp using the veloity error onstant. 5

6 12) (10 points) For this problem assume the following unity feedbak system with Gp 3 () s = ( s+ 2)( s+ 4) and () 3( s + 5) G s = s a) Determine the position error onstant K p b) Estimate the steady state error for a unit step using the position error onstant. ) Determine the veloity error onstant K v d) Estimate the steady state error for a unit ramp using the veloity error onstant. 6

7 Problems refer to the following root lous plot (from sisotool) 4 Root Lous Editor for Open Loop 1(OL1) Imag Axis Real Axis 13) Is it possible for -1 to be a losed loop pole for this system? (Yes or No) 14) When the gain is approximately 3 the losed loop poles are as shown in the figure. If we want the system to be stable what onditions do we need to plae on the gain k? 15) For this system, is it possible to get a response that is underdamped? (Yes or No) 16) For this system, is it possible to get a response that is overdamped? (Yes or No) 17) Is this a type one system? (Yes or No) 7

8 18) (19 points) For this problem assume the losed loop system below and assume 3 Gp () s = ( s j) ( s+ 2 2 j) For eah of the following problems sketh the root lous, inluding the diretion travelled as the gain inreases and the angle of the asymptotes and entroid of the asymptotes, if neessary. a) Assume the proportional ontroller G () s = k b) Assume the integral ontroller G i () s k = s p ) Assume the PI ontroller () k( s + 5) G s = s d) Assume the PD ontroller G ( s ) = ks ( + 6) e) Assume the PID ontroller () ks ( j)( s+ 6 2 j) G s = s 8

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10 19) (10 points) For this problem assume the following unity feedbak system with Gp 1 () s = ( s + 3) and G( s) = ks ( + 2) a) Determine an expression for the losed loop transfer funtion G () s o b) Determine an expression for the sensitivity of the losed loop transfer funtion G () s to variations in the parameter k. Your answer should be a funtion of s and should be a ratio of two polynomials. o ) Determine an expression for the magnitude of the sensitivity funtion as a funtion of ω. Your final answer must be simplified as muh as possible and annot ontain any j s 10

11 20) (6 points) Consider the following simple feedbak ontrol blok diagram. The plant is 2 Gp () s =. The input is a unit step. s + 4 a) Determine the settling time of the plant alone (assuming there is no feedbak) b) Assuming a proportional ontroller, G () s = k, determine the losed loop transfer funtion, G () s 0 p ) Assuming a proportional ontroller, G () s k,determine the value of k so the settling time is 0.5 seonds = p p 11

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13 Root Lous Constrution One eah pole has been paired with a zero, we are done 1. Loi Branhes poles zeros k = 0 k = Continuous urves, whih omprise the lous, start at eah of the n poles of Gsfor () whih k = 0. As k approahes, the branhes of the lous approah the m zeros of Gs. () Lous branhes for exess poles extend to infinity. The root lous is symmetri about the real axis. 2. Real Axis Segments The root lous inludes all points along the real axis to the left of an odd number of poles plus zeros of Gs. () 3. Asymptoti Angles As k, the branhes of the lous beome asymptoti to straight lines with angles o o i360 θ =, i = 0, ± 1, ± 2,.. n m until all ( n m) angles not differing by multiples of 360 o are obtained. n is the number of poles of Gs () and m is the number of zeros of Gs. () 4. Centroid of the Asymptotes The starting point on the real axis from whih the asymptoti lines radiate is given by σ = i pi n m where p i is the i th pole of Gs, () z j is the j th zero of Gs, () n is the number of poles of Gs () and m is the number of zeros of Gs. () This point is termed the entroid of the asymptotes. 5. Leaving/Entering the Real Axis When two branhes of the root lous leave or enter the real axis, they usually do so at angles of j z j ± 90 o. 13