Control Theory

# Control Theory

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## Control Theory

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1. ControlTheory D action – Time delays

2. Derivative Action • Whenthere’stoomuchoscillation, thiscansometimesbesolvedbyadding a derivative action. Thisaction willtake into account howfast the error signal is changing. • In the time domain this leads to: • the PD controller: • and the PID controller:

3. Derivative Action PD in the s domain: PID in the s domain:

4. Group Task r(t) z(t) 2/(s2+2s) P or PD? Controller Process Compare a P controller withKc=10 with a PD controller withKc=10 and TD=2 (step response for the servoproblem)

6. Problem(s) introduced by D-control? • Control action in servo problem becomes “infinitely” high. • High frequencybehavior/noise is amplified. • None of the above. • Both A and B [Default] [MC Any] [MC All]

7. Bode plots! Bode plot PD:

8. Derivative Action with filtering We introduce an extra pole to solve the problem at high frequencies: This means we first filter the error with a low pass filter. Off course the time constant of the filtering << TD (Remark: to solve the problem of suddenly changing reference values, often another solution is presented: the derivative action is put on the MEASUREMENT instead of on the ERROR!)

9. Derivative Action with filtering Difference in Bode plot: Blue: D action Green: D action + prefiltering Remark: The positive effect on the phase shift is now lost at higher frequencies!

10. Influence of a time delay 1) What is a time delay? Typical example: measurement comes too late, e.g.: TR TR TR,m TR,m td

11. RelationshipbetweenTR,m(t)andTR(t)? • TR,m(t) = TR(t) • TR,m(t) = TR(t-td) • TR,m(t) = TR(t+td) • None of the above TR [Default] [MC Any] [MC All] TR,m td

12. Which is true? • A time delay = a pure phase shift, B is false • A time delay = a non-linearsubsystem, A is false • Both are true • Both are wrong [Default] [MC Any] [MC All]

13. Influence of a time delay In other words: f(t-td) f(t) Dead time td This is of course a pure phase shift. In Laplace (see table): e-td s F(s) F(s) Dead time td A dead time is given as e-td s in the s domain: it’snon-linear! On the Bodeplot? AR = |e-tdjω|=1 φ = (e-tdjω) = -tdω (in rad) Pure phase shift

14. CHAPTER 3. PID CONTROL Influence of dead time 2) What is the influence on a feedback system? A- Bode plot of a time delay:

15. Influence of a time delay 2) What is the influence on a feedback system? φ = (e-tdjω) = -tdω * 180/π (in degrees) • Thiscan have severe impact on the stability: • The information comestoo late. • How can we seethis in our analysis? • Based on TF: difficult: time delay = non-linearthing • … but it is a pure phase shift… on Bode plot?