LINEAR CONTROL SYSTEMS. Ali Karimpour Associate Professor Ferdowsi University of Mashhad


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1 LINEAR CONTROL SYSTEMS Ali Karimpour Associate Professor Ferdowsi University of Mashhad
2 Controller design in the frequency domain Topics to be covered include: Lag controller design 2 Dr. Ali Karimpour Feb 23
3 Phase (deg) Magnitude (db) Phase (deg) A phaselag controller 3 5 G( s) a s s Let a log a sin m a a 45 m 9 3 / / a Dr. Ali Karimpour Feb 23
4 Phase (deg) Design fundamental of a lag controller Analysis Design Consider a minimum phase system. 2 log a 2 4 a PM= PM=25 What is the effect of a lag controller on BW? Speed of system? What is the effect of a lag controller on noise effect? How can the lag controller help us? Is it different from a Proportional controller? 4 Dr. Ali Karimpour Feb 23
5 Design procedure of a phaselag controller in the frequency domain 2 log a as s Step : Consider ( s) with a as a phaselag controller. Note: If the plant has another gain, let ( ) a G s c s s Step 2: Try to fix according to the performance request, otherwise let = Step 3: Setch the Bode plot of the system (with the fixed ) without controller. Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little).? Step 5: Find the required gain by lag controller and derive the parameter a. a Required gain db 2log( a) Step 6: Put the right corner of the controller sufficiently far from new crossover frequency. cross a Step 7: Chec the designed controller. 5 a Dr. Ali Karimpour Feb 23
6 Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. s( s 25) as s Step : Consider ( s) with a as a phaselag controller. Note: If the plant has another gain, let as ( s) a s ( ) a s s s Step 2: Try to fix according to the performance request, otherwise let = v lim sgc ( s) s s( s 25) 25 6 Dr. Ali Karimpour Feb 23
7 Phase (deg) Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. Step 3: Setch the Bode plot of the system (with the fixed ) without controller. s( s 25) G( s) 25 s( s 25) s( s / 25 ) Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little) new 25 2 new croos cross 35 PM= Dr. Ali Karimpour Feb 23 7
8 Phase (deg) Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. Step 5: Find the required gain by lag controller and derive the parameter a. s( s 25) Required gain db 2log( a) 6 9 2log( a) 9 a 2.35 Step 6: Put the right corner of the controller sufficiently far from crossover frequency. new cross a 2. 4 ( ) a s s s.5s.4s Dr. Ali Karimpour Feb 23 8
9 Phase (deg) Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. Step 7: Chec the designed controller. ( j) G( j) G( j) ( s) 2 log a s( s 25) /.7, / a 2 2log a 9db a a s.4.35s s.4s 6 9 / a.9 35 PM sin m m Dr. Ali Karimpour Feb 23
10 Phase (deg) Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. Step 7: Chec the designed controller. s( s 25) ( j) G( j) G( j) PM Dr. Ali Karimpour Feb 23
11 OpenLoop Gain (db) Example : Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Find the M p of overall system. Finding M p Without controller M p =6 db s( s 25) After applying controller 8 6 Nichols Chart ( j) G( j) G( j) M p =3 db db.5 db db 3 db 6 db db db 3 db 6 db 2 db 2 2 db 4 4 db 6 db OpenLoop Phase (deg) Dr. Ali Karimpour Feb 23
12 Example 2: Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. s( s )( s 2) as s Step : Consider ( s) with a as a phaselag controller. Note: If the plant has another gain, let as ( s) a s ( ) a s s s Step 2: Try to fix according to the performance request, otherwise let = v lim sgc ( s) s s( s )( s 2) 2 2 Dr. Ali Karimpour Feb 23
13 Phase (deg) Example 2: Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Step 3: Setch the Bode plot of the system (with the fixed ) without controller. s( s )( s 2) G( s) 2 s( s )( s 2) s( s / )( s / 2 ) Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little) new new cross cross PM= Dr. Ali Karimpour Feb 23
14 Phase (deg) Example 2: Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Step 5: Find the required gain by lag controller and derive the parameter a. s( s )( s 2) Required gain db 2log( a) a 3 2log( a) Step 6: Put the right corner of the controller sufficiently far from crossover frequency. new cross a ( ) a s 2.5s s s 79s Dr. Ali Karimpour Feb 23 4
15 Phase (deg) Example 2: Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Step 7: Chec the designed controller. s( s )( s 2) ( j) G( j) G( j) PM Dr. Ali Karimpour Feb 23
16 OpenLoop Gain (db) Example 2: Design a lag controller for the following system such that the phase margin be 45 and the ramp error constant be. Step 7: Chec the designed controller. s( s )( s 2) 5 Nichols Chart ( j) G( j) 5 G( j) OpenLoop Phase (deg) Dr. Ali Karimpour Feb 23 6
17 Example 3: Design a lag controller for the following system such that the gain margin be db and the ramp error constant be. s( s )( s 2) as s Step : Consider ( s) with a as a phaselag controller. Note: If the plant has another gain, let as ( s) a s ( ) a s s s Step 2: Try to fix according to the performance request, otherwise let = v lim sgc ( s) s s( s )( s 2) 2 7 Dr. Ali Karimpour Feb 23
18 Phase (deg) Example 3: Design a lag controller for the following system such that the gain margin be db and the ramp error constant be. Step 3: Setch the Bode plot of the system (with the fixed ) without controller. s( s )( s 2) G( s) 2 s( s )( s 2) s( s / )( s / 2 ) Step 4: According to desired PM (GM) choose the new gain crossover frequency (Phase crossover frequency). (reduce it a little) GM= db new 5 5 new cross cross Dr. Ali Karimpour Feb 23
19 Phase (deg) Example 3: Design a lag controller for the following system such that the gain margin be db and the ramp error constant be. Step 5: Find the required gain by lag controller and derive the parameter a. s( s )( s 2) Required gain db 2log( a) 25 2log( a) 25 a 2.56 Step 6: Put the right corner of the controller sufficiently far from crossover frequency a ( ) a s s s s 7.9s Dr. Ali Karimpour Feb 23 9
20 Phase (deg) Example 3: Design a lag controller for the following system such that the gain margin be db and the ramp error constant be. 2 log a Step 7: Chec the designed controller G( j) ( j) G( j) GM db ( s) a s s /.6, / a 2 log a 25db / a.567.9s 7.9s sin m m.56 s( s )( s 2) a Dr. Ali Karimpour Feb 23
21 Example 4: Design a lag controller for the following system such that the phase margin be 45 and the open loop bandwidth be rad/sec 2 s It is not possible explain why? 2 Dr. Ali Karimpour Feb 23
22 Exercises Derive the gain margin of the compensated system in example. 2 Derive the phase margin of the compensated system in example Dr. Ali Karimpour Feb 23
23 Phase (deg) Exercises 3 Following is the open loop transfer function of a system. a) What is the velocity error constant. (answer 8) b) Design lag controller such that PM=45. (answer =(.5s+)/(.56s+)) c) Design lag controller such that PM=45 and the velocity error constant be 2. (answer =(.49s+)/(3.75s+)) Dr. Ali Karimpour Feb 23
24 Phase (deg) Exercises 4 Following is the open loop transfer function of a system. a) Design a controller that the GM of system be 5 db. b) Design lag controller such that PM=45 and the velocity error constant be Dr. Ali Karimpour Feb 23
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