ControlTheory Bode StabilityCriterion
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.
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 = ?
Example 1 rad/s
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 in degrees / 100 Example: How big do youthink the overshootwillbeif the open loop TF is
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]
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%?
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?
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
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°
Use of the Bode plot in control Exercise: Drug-induced anasthesia: c) Influence on the step response?