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# Control Theory PowerPoint PPT Presentation

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

Control Theory

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## 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

= ?

### 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]

### 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

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

• 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

• 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

• 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%?

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

Use of the Bode plot in control

Exercise: Drug-induced anasthesia

a) Maximum T?

Zonder de dode tijd!

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°