Control theory
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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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Control theory

ControlTheory

Bode StabilityCriterion


Other view on stability of cl

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

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

= ?


Phase margin

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]


Example 1

Example 1

rad/s


Given the previous bode plot of the open loop

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]


On the phase margin

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


The estimated overshoot is

The estimated overshoot is

  • ca. 15%

  • ca. 30%

  • ca. 45%

  • ca. 60%

[Default]

[MC Any]

[MC All]


We can now state that the disadvantage of the i action 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 can use the stability criterion to design controllers as well

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

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?


Open loop bode plot

Open loop Bode plot


Control theory

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


Control theory

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°


Control theory

Use of the Bode plot in control

Exercise: Drug-induced anasthesia:

c) Influence on the step response?


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