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1 Active Control? Contact : Website : Teaching

2 Active Control? Disturbances System Measurement Control Controler. Regulator.,,,

3 Aims of an Active Control Disturbances Reference + Error Controler control Actuators System Outputs Sensors Measure - Stabilisation : Inverted pendulum, Launcher, etc. Control and regulation : Temperature control of a room, robotics, etc. Disturbance rejection: Vibration isolation of a lithography table The 3 objectives are coupled 3

4 Analysis and controler synthesis Stability: Open loop gain Gain margin and phase margin Open Loop : Nyquist, Nichols or Bode diagrams. Closed Loop system poles (Pole Map). Perturbation rejection = following the reference : y 1 Sensitivity function : Maximize GH d 1 GH y GH Complementary Sensitivity function : r 1 GH Maximize GH Stability requierements limit achievable performances! 4

5 Bode Diagram : performance specification Gain variation v.s. frequency Phase variation v.s. frequency Open loop gain GH( j) GH( j) ( GH( j)) Unstable Fréquence 5

6 Nyquist Diagram: Stability analysis Complex plane : Plot of GH(jw) in the complex plane Critical point ( -1, 0) Gain = 1 et Phase =

7 Nichols Diagram: Stability analysis Plot of gain variation in decibel in function of the phase Critical point ( -180, 0db) Gain = 1 et Phase =

8 1. Disturbance rejection: y d 1 1 GH Controler effect Large effect if GH>>1 No effect if GH<<1 Unstability if GH=-1!!. Design trade-off : Dangerous zone! 8

9 3. 1st Bode integral: 4. Ideal design of Bode : Valid for slow variation of G GM -180 PM 9

10 Active control «classic» P.8.10 (a) The error specification of the radial positioning system of a CD is 0. microns. If the system is subjected to a random disturbance of 00 microns around 5 Hz, compute an estimation of the bandwidth of the control system in order to achieve appropriate disturbance rejection with a reasonable phase margin. 10

11 3. 1st Bode integral: 4. Ideal design of Bode : Valid for slow variation of G GM -180 PM 11

12 Solution : P.8.10 (a) The error specification of the radial positioning system of a CD is 0. microns. If the system is subjected to a random disturbance of 00 microns around 5 Hz, compute an estimation of the bandwidth of the control system in order to achieve appropriate disturbance rejection with a reasonable phase margin. d = 00 µm e max = 0. µm GH(5 Hz) = 1000 = 60 db 1. If the slope is -0dB/dec : Φ = -90 MP = 90 ω c = 5 Hz + 3 décades = 5 khz!. If we reduce the phase margin : Φ = -10 MP = 60 ωc = 5 Hz +.5 décades = 4.5kHz 1

13 Periodic disturbance P.8.10 (b) What if the disturbance is known to be periodic? Suppose that the disturbance is the superposition of a perfect sine at 5 Hz and its first harmonic at 50 Hz, and design the appropriate controller with very lightly damped poles (e.g. ξ=0.5%). Next, compute the disturbance rejection if there is a small error in the disturbance frequency. Do the same when the poles of the controller have more damping (e.g. ξ=1%). Same idea: But, we have more information about d and we will take it into consideration. 13

14 Phase (deg) Magnitude (db) Lightly damped poles at the disturbance freq. ~ G -1 (s) G(s) H(s) H( s) 1 s g1 g s 1 s 1 s s Bode Diagram Stable if the model of G(s) is accurate at ± 90 arround the disturbance frequency, Gain stabilization (i.e. the phase of G(s)H(s) does never exceed ±180 ) Frequency (Hz) 14

15 15 Controler synthesis : Open loop GH ) ( s s g s s g s H ) ( s s g s s g s s H Without derivative action With derivative action

16 Sensitivity function: S( s) y d 1 1 G( s) H( s) Without derivative action With derivative action 16

17 What If the disturbance frequency is not exactly 5 Hz? 0.93 if df=0.1% Info : Electrical network frequency (in Belgium ) variation is around 0.1% 17

18 18

19 Impossible to reduce G(s)H(s) of 60 db in only one decade (PM= -90!) One must reduce strongly G(s)H(s) around 1rad/s and 10rad/s, to get an «acceptable» slope at ω c. Possible? Yes, with high order filters. But difficult and not recommended! 19

20 Bode integral or «water bed effect» with: The decrease of S at some frequencies (i.e. disturbance rejection ) is always compensated by an increase of S at other frequencies. There is a direct relationship between the phase margin and the overshoot of the closed loop response, In general, a phase margin >60 is sufficient to avoid the peak of the closed loop response, 0

21 1

22 Solution : 1. amplification if 1+G <1. 1+G < 1 when G(jω) enters the circle with radius of 1 centered at (-1,0) 3. The slope exceeds -0dB/dec at high frequencies: So φ < -90 AND G(jω) 0 if ω >>> So G(jω) crosses the «Amplification circle» at high frequencies.

23 «non-minimum phase» Systems Non-minimum phase = presence of zeros with positive real part These systems cannot be inverted! They are treated with «pass-all» filters : with: System almost identical to G(s) BUT minimum phase. 3

24 Characteristics of A(s) : 1. Magnitude = 1 («all pass»). Phase : ω/a 4

25 Solution : 3 a/5 5

26 Solution: 3 a 6

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