# ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 2010/2011 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques

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1 CAIRO UNIVERSITY FACULTY OF ENGINEERING ELECTRONICS & COMMUNICATIONS DEP. 3rd YEAR, 00/0 CONTROL ENGINEERING SHEET 5 Lead-Lag Compensation Techniques [] For the following system, Design a compensator such that the position error constant of the overall system is.667, and the phase margin is greater than or equal 70 degrees and the gain crossover frequency after compensation does not exceed 5 rad\sec. R(S) + _ ks ( 0) 0( S)( S3) Y(S) []An uncompensated unity feedback control system whose open loop transfer function is GS ( ) 00 ( S)( S ), is to be designed to meet the following specifications: The gain crossover frequency after compensation does not exceed 4 rad\sec, and Phase Margin PM 35. (a) Use a Bode plot to determine the values of ω gc, ω pc, GM and PM for the uncompensated system. (b) Design the necessary compensator. (c) What are the resulting values of ω gc, PM and steady-state error e ss for a unit step input.

2 [3] (Final 998) An uncompensated unity feedback control system whose open-loop transfer function is 4K ( S) S( S )( S 6) G is to be designed to meet the following performance specifications: When the input is a ramp with slope (velocity) equal л rad/sec, the steady state error in position must be л / 0 rad. Phase Margin PM 45 o. The new gain crossover frequency ω gc rad/sec. a) Use the Bode plot to design the necessary compensator. b) Synthesize your compensator circuit. c) Determine ω gc and PM for both uncompensated and the resulting compensated systems. [4] (Final 00) An uncompensated unity feedback control system whose open loop K ( S / 3) transfer function is G(S) = 3 is to be designed to meet the following performance specifications: K p = 4, GM db. And PM 40 o. (a) Use the Bode plot to determine the values of ω gc, ω pc, GM and PM for the uncompensated system. (b) Design a lead compensator to meet the performance specifications. Avoid any zero-pole cancellation in your design and synthesize your compensator circuit. (c) What are the resulting values of ω gc, ω pc, GM, PM and steady-state error e ss for a unit step input.

3 [5] (Final 000) An uncompensated unity feedback control system whose open loop transfer function is 60K G ( S) S( S 8)( S 0), is to be designed to meet the following performance specifications: (i) Steady state error = 0.0 due to a unit ramp input. (ii) Gain Margin 0 db. (iii) Phase Margin PM 40. (iv) Gain crossover frequency 0 < ω gc < 0 rad/sec. (a) Use a Bode plot to determine the values of ω gc, ω pc, GM and PM for the uncompensated system. (b) Design the necessary compensator. Avoid any zero-pole cancellation in your design and synthesize your compensator circuit. (c) What are the resulting values of ω gc, ω pc, GM, PM and steady-state error e ss for a unit step input [6](Final 004)A unity feedback system has 0 GS ( ) S( 0 S) : (a) Design a series compensator that will eliminate the steady state error for a unit ramp input signal. The resulting closed loop system should have PM=30.

4 Summary LEAD COMPENSATOR: I. Circuit synthesization: R ( S ) ( S) ( S ) R Where, CR R R C R II. Bode diagram of lead compensator: 0 db III. Compensation steps: m. Assume that the open loop transfer function of the compensated system is: ( S ) ( S) G ( S) KG( S) ( S ) Determine gain K to satisfy the requirement on the given static error constant.. Plot KG( S ) and calculate from the plot,, GM, PM 3. Get PM PM 5 m req old gc old pc old old old 4. Get from sin m,then check that 0. <. 5. Draw the line of magnitude equals 0log on the old plot. 6. The frequency where this line will intersect with the old plot is m 7. Select m. 8. Get from m. 9. Redraw the magnitude and phase, after compensation, and get,, GM, PM pc new new new 0. If the requirements are not satisfied (specially the PM), return to step 3 and increase the correction term from 5 0.

5 IV. Comments on lead compensation: ) Lead compensator essentially yields an improvement in transient response and a small change in the steady state accuracy. ) It increases the gain crossover frequency [ gc 3) It increases the Bandwidth of the system [ B ], so it may accentuate high frequency noise effects. 4) It increases the resonant frequency [ r 5) It decreases the resonant peak [ M r ], which means that it improves the stability. 6) It speeds up the response of the system, so it acts approximately as a PD controller.. 7) The maximum phase lead m shouldn t be larger than 55 (or 0. ), because such values will require an additional gain of excessive values, if more than 55 is needed; two lead compensators may be used in series with an isolating amplifier. LAG COMPENSATOR: I. Circuit synthesization: R ( S ) ( S) ( S ) R R Where, CR R C R II. Bode diagram of lag compensator: 0 db 0log III. Compensation steps:. Assume that the open loop transfer function of the compensated system is: ( S ) ( S) G ( S) KG( S) ( S ) Determine gain K to satisfy the requirement on the given static error constant.. Plot KG( S ) and calculate from the plot,, GM, PM gc old pc old old old 3. From G( S) 80 PM req,you can get analytically.

6 4. Get octave to decade below (usually 0. ). 5. From plot you can get,as G( S) in db 0log. 6. Redraw the magnitude and phase, after compensation, and get,, GM, PM pc new new new 7. If the requirements are not satisfied (specially the PM), return to step 4 and choose another value for. IV. Comments on lag compensation: ) Lag compensator essentially yields an improvement in the steady state accuracy at the expense of increasing the transient response time. ) It decreases the gain crossover frequency [ gc 3) It decreases the Bandwidth of the system [ B ], so it may suppress high frequency noise effects. 4) It decreases the resonant frequency [ r 5) It stabilizes the unstable systems. 6) It speeds down the response of the system, so it acts approximately as a PI controller. LAG-LEAD COMPENSATOR: I. Circuit synthesization: R G ( S) c ( S)( S) ( S)( S) C C Where C R, C R, C R C R C R R II. Bode diagram of lag compensator: 0 db 0log db

7 III. Compensation steps:. Assume that the open loop transfer function of the compensated system is: ( S)( S) ( S) G ( S) KG( S) ( S)( S) Determine gain K to satisfy the requirement on the given static error constant.. Plot KG( S ) and calculate from the plot,, GM, PM. gc old pc old old old 3. Let and 0 (the distance a G( S) in db must be less than 0db ). pc old 4. Get decade below ( 0. ). 5. Draw a straight line of slope 0db/decade, passing through the point (, a ), the intersection of this line and the -0db line determine the corner frequency. 6. Redraw the magnitude and phase, after compensation, and get,, GM, PM pc new new new IV. Comments on lag compensation: ) Lag-Lead compensator essentially yields an improvement in both the steady state accuracy and the transient response time.

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