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Control Theory

Control Theory. Bode Stability Criterion. Other view on stability of CL. Where the PHASE of the open loop TF equals -180°(+/-n.360°), we have positive feedback. If the AMPLITUDE RATIO at these frequencies > 0db: unstable closed loop. Two important measures. GAIN MARGIN

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Control Theory

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  1. ControlTheory Bode StabilityCriterion

  2. Other view onstability of CL Where the PHASE of the open loop TF equals -180°(+/-n.360°), we have positive feedback. If the AMPLITUDE RATIO at these frequencies > 0db: unstableclosed loop.

  3. Two important measures GAIN MARGIN = Howmuch dB of amplitude ratio we canstilladd in the open loop before the amplitude ratio goesabove 0dB at a frequencywhere the phase crosses -180° 2. PHASE MARGIN = ?

  4. Phase Margin = • How much the phase can still be increased before it reaches 0° at a frequency where the amplitude ratio is 0dB. • Howmuch the phasecanstillbedecreasedbeforeitreaches -180° at a frequencywhere the amplitude ratio is 0dB. • None of the above makes sense. [Default] [MC Any] [MC All]

  5. Example 1 rad/s

  6. Given the previous Bode plot of the OPEN LOOP, • GM = 50 dB, PM = 40° • GM = 50 dB, PM = 90° • GM = 30 dB, PM = 40° • GM = 30 dB, PM = 90° • None of the above [Default] [MC Any] [MC All]

  7. On the phase margin The bigger the phase margin, the lessovershoot in the closed loop. First approximation: the “damping ratio” of the closed loop = PM/100 Example: How big do youthink the overshootwillbeif the open loop TF is

  8. The estimated overshoot is • ca. 15% • ca. 30% • ca. 45% • ca. 60% [Default] [MC Any] [MC All]

  9. We can now state that the “disadvantage” of the I action is • thatitincreases the OL gain at low frequencies • thatitincreases the OL gain at high frequencies • that it decreases the OL phase at low frequencies • that it both decreases the OL phase and increases the OL gain at low frequencies [Default] [MC Any] [MC All]

  10. We canuse the stabilitycriterion to design controllers as well • GROUP TASK 1: • A second order processwithgain 2, damping ratio 0.5 and naturaleigenfrequency 20 rad/s is controlledwith a P controller. The time delay in the loop is 0.01s. • What is the maximallyallowedcontrolgain • in order for the CL to bestable • in order for the overshoot to be smaller than 50%?

  11. Group Task II Drug-inducedanasthesia Reaction of the patient’s arterial blood pressure to a drug may vary. Therefore a closed loop system is used. However: Amount of drug supplied to the patient Desired pressure Blood pressure 2e-sT/s 2(s+5) Controller Body 2/(s+2) Sensor • Remark: What kind of control? Why? • Whatis the maximum time delay of the body’s response before the system willbecomeunstable? • Determine the PM and the GM when T=0.05s? When T=0.1s? • What is the influence of T on the step response?

  12. Open loop Bode plot

  13. Use of the Bode plot in control Exercise: Drug-induced anasthesia a) Maximum T? Zonder de dode tijd! ωPM= 8.94 rad/s: PM = 73.4° Tmax = 0.1433s

  14. Use of the Bode plot in control Exercise: Drug-induced anasthesia: b) PM and GM when T=0.05s? When T=0.1s? Zonder de dode tijd! A- Without time delay: PM = 73.4° ωPM = 8.94 rad/s B- Influence of T: T = 0.05s: -25.6° T = 0.1s: -51.2°

  15. Use of the Bode plot in control Exercise: Drug-induced anasthesia: c) Influence on the step response?

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