Department of Mechanical Engineering Massachusetts Institute of Technology Modeling, Dynamics and Control III Spring 2002

Transcription

1 Department of Mechanical Engineering Maachuett Intitute of Technology Modeling, Dynamic and Control III Spring 2002 SOLUTIONS: Problem Set # 10 Problem 1 Etimating tranfer function from Bode Plot. Thi i omething that i very ueful when working with experimental data. It come with practice. If you ee how to do it immediately, great, if not here are ome tip on how to do it a little bit more methodological. It long but hopefully worthwhile. How to obtain a tranfer function from the graph (Melia Way, not necearily the bet way) Step 1: Relative order of ytem Look at Phae diagram. The final value will tell you the relative order of the ytem. Think about it. Each pole at high frequencie will contribute 90 to the phae, while each zero will contribute 90. (Complex pole/zero are made of 2 pole/zero o they contribute / 180 ) You can alo do thi by looking at the final lope of the magnitude repone, but I think it i clearer to look at the phae repone. Now for thi problem we ee that the phae diagram end at 180. Thi tell you that there are 2 more pole than zero in the tranfer function.

2 Step 2: Number of Pole and zero Now we need to decide how many pole and zero there are in the ytem. Looking at the magnitude diagram we can ee that the initial lope i 20 db/dec, and the final lope i 40 db/dec. Since we never ee a decreae in lope i.e. from 40 db/dec to 20 db/dec, we can aume that there are no zero. So we know we have two pole and no zero. Step 3: Location of Pole and zero Since the magnitude plot tart at a lope of 20 db/dec we know that thi i a type 1 ytem, o we have a pole at 0. We can alo ee thi from the phae diagram ince it tart at 90. Now we can figure out where the econd pole i. By extending the aymptotic line at the beginning and end of the diagram we can ee that they meet at about 5 rad/. Therefore thi i the break frequency of econd pole. Now we know our tranfer function look like: K G ( ) 1 5 The lat thing we need to do i find the gain that give u thi repone Step 4: Finding the Gain Since thi i a Type 1 ytem, the gain will not imply be the point were it croe a it would for a Type 0 ytem. Intead we can figure out the gain by knowing that the pole at 0 would be at 40 db if there were no gain. Intead we can ee that it i more at about 68 db. Therefore the DC gain mut have contributed thi additional 28 db. The gain i then : 20 log K 28dB K G ( ) ( 5) 1 5 Step 5: MATLAB Check To verify that thi i the correct we can plot it into MATLAB. Indeed the repone i very cloe to our original function.

3 Part b) ame method Step 1: Relative order of ytem Here we ee that the phae plot tart at 0 and drop down to about 90, which implie a relative order of 1. Notice that even though the phae goe up to about 50, what we care about i the final value for high frequencie. Step 2: Number of Pole and zero Now we know that we have one more pole than zero, but we don t know how many or where they are. We need to carefully analyze thi ytem. Again we look at the lope of the repone. Initially the lope i 0, o we know we have a Type 0 ytem. Then the repone goe up, at a lope of 20 db/dec; o there mut be a zero around 1 rad/. Around 10 rad/ the repone drop and goe down at a lope of about 40 db/dec. Which mean that there mut have been a contribution of 3 pole. One to offet the zero, and 2 more bring the lope down to 40. Then around 20 rad/ we ee another change in lope from 40 db/dec to 20 db/dec, which mut be due to another zero. Therefore we can afely ay that we have 3 pole and 2 zero, with a relative order of 1 a expected. Step 3: Location of Pole and zero By drawing the aymptote a hown in the figure we can etimate the break frequency for the firt zero at 1 rad/. We ee though that the repone increae beyond the expected lope of the aymptote. The only other thing that would caue the repone to rie other than a zero i a econd order pole with ζ < So then we know that around a frequency of 10 rad/ we have a complex double pole. In addition around thi area we have an additional imple pole, ince the lope between rad/ i 40 db/dec. Thi pole probably lie omewhere between 6 to 20 rad/. Then the zero the that caue the reduction in lope lie around 25 rad/. So our tranfer

4 function look like: K G( ) 2 1 2ζ Where we till need to find K and ζ. Step 4: Finding the Gain The gain for thi problem will be the y-intercept at low frequencie o: 20 log K 10dB K Since the econd order pole caue a light increae in the repone but doen t caue it to hoot up, we can etimate 0.4 ζ < , we will chooe ζ 0.6 and check whether thi i viable. Finally we have ( 1)( 25) ( ) G 2 2 ( 10)( 6 100) Step 5: MATLAB Check Again, good correlation with our matlab reult

5 Problem 2 You are aked to ketch the bode plot o it hould be done by hand. When K ) 100)( ( 0.5) ( ) ( K G } { GM 30dB Φ 67 M

6 The gain margin of 30 db implie that the entire magnitude curve can be hifted up by 30 db, which i the equivalent of lowering the 0 db mark by 30 db. Since we can hift the curve by up to 30 db that mean that the gain can increae from what it i now by a factor of up to: 20log(gain) 30 db gain 31 time. Therefore our value of K can be: 800 < K Problem 3 Thi i a very imple problem a oon a you undertand what you are looking at, and what you need to do. a) Firt let remember how we obtain the gain margin, we find the point on the phae diagram where the phae i exactly 180 and then we find the magnitude at the ame frequency and the ditance from the 0dB i the gain margin. In thi problem we are not given a bode plot intead we are given the input/output curve which i eentially how the bode plot i formed. Since we are given input/output curve when the phae i 180, all we need to do i find the gain at thi frequency, which i: output input Therefore the gain margin i: 4.08 db 5 5 or 20log 4.08dB 8 8 b) Now we are aked to find the phae margin. With a bode plot we find the point where the repone hit the 0 db line and then find the phae at the ame frequency. In figure (3d) we are given an input/output curve where the gain i 1 which i 0 db. So, all we need to do i find out how much the inuoid ha been hifted. Input e(t) Output y(t)

7 The ditance hown i how much the output would need to hift to be at a phae of 180, in other word thi i our phae margin. By etimating thi hift from the graph and converting it to degree, we get that the hift margin i Φ 40 Problem 4 M. You hould not need to do thi by hand, jut ue matlab. Here are the command you need: >> num [22.5] >> den conv ([1 4],[ ]) >> bode (num,den) Though it i difficult to ee uing matlab we find that when the phae croe 180 at ω 3.55 rad/ the magnitude i 1.2 db, o the phae margin i 1.2 db. The magnitude will cro the 0 db line twice the firt time the phae about 50, which we dimi. The econd time the phae i about 178, o we ay that the phae margin i 2. Thi ytem i very cloe to intability.

8 Problem 5 Thi one pecially you hould ue matlab >> num 1.96*conv([1 1.25],[ ]) >> den conv ([ ],[ ]) >> bode (num,den) If thi i ued a PID controller the frequencie reduced will be thoe of the peak of the magnitude repone. Namely ω 0.75 rad/ and ω 4.1 rad/ If you plot thi by hand, you MUST calculate the M p, the peak of the magnitude, and ω r the frequency at which they occur. b) Thee frequencie will be reduced becaue if the pole and zero of the controller match the pole and zero of the diturbance then there will be pole/zero cancellation and the dynamic of the diturbance will be reduced.