Robust Control 3 The Closed Loop

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1 Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002

2 Outline Closed Loop Transfer Functions Traditional Performance Measures Time domain Frequency domain Robustness Traditional gain/phase margins Multivariable Nyquist

3 Closed Loop Transfer Functons d r - K u G y d 2 ( ) ( ) Y = G U + D U = K R Y D E = R Y 2

4 Transfer Functions Con t Output [ ] [ ] [ I GsKs () ()] GsKsD () () () s [ ] [ ] [ I GsKs () ()] GsKsD () () () s [ ] [ ] Ys () = I+ GsKs () () GsKsRs () () () I+ GsKs () () GsD () () s Error Es () = I+ GsKs () () Rs () I+ GsKs () () GsD () () s Control U() s = K() s I + G() s K() s R() s K() s I + G() s K() s G() s D () s Ks () I+ GsKs () () D() s [ ]

5 Sensitivity Functions [ ] [ ] [ ] E() s = I + L R() s I + L GD () s + I + L LD () s where L: = GK sensitivity function: [ ] S: = I + L 2 { [ L] L [ L] } complementary sensitivity function: For SISO systems Bode derived: dt T dt L = dl L dl T = = [+ L] L + = [ + L] = S L [ + L] 2 [ ] T : = I + L L L

6 Traditional Performance ~ Time Domain rise time, T r, usually defined as the time to get from 0% to 90% of its ultimate (i.e., final) value. settling time, T s, the time at which the trajectory first enters an ε-tolerance of its ultimate value and remains there (ε is often taken as 2% of the ultimate value). peak time, T p, the time at which the trajectory attains its peak value. peak overshoot, OS, the peak or supreme value of the trajectory ordinarily expressed as a percentage of the ultimate value of the trajectory. An overshoot of more than 30% is often considered undesirable. A system without overshoot is overdamped and may be too slow (as measured by rise time and settling time).

7 Traditional Performance ~ Time Domain Cont d y p.4.2 2ε t T r T p T s

8 Traditional Performance ~ Frequency Domain db deg MAGNITUDE rad êsec PHASE rad êsec Complementary sensitivity 2 s +.5s+ 0.5 K s db deg MAGNITUDE rad êsec ( 20s+ )( 0s+ )( 0.5s+ ) PHASE rad êsec Sensitivity V-22 Osprey altitude control K=280

9 Traditional Performance ~ Frequency Domain Sensitivity Function (first crosses / 2=0.707~-3db from below): { v S j v } ω = max : ( ω) < 2 ω [0, ) BS v Complementary Sensitivity Function (highest frequency where T crosses 2 from above) { v T j v } ω = min : ( ω) < 2 ω (, ) BT v Crossover frequency { v L j v } ω = max : ( ω) ω [0, ) c v

10 Traditional Performance ~ Frequency Domain Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. Im L plane M M S T = max S( jω ) ω = max T ( jω ) ω S > S < - a Re S = a L( jω) S = + L

11 A Fundamental Tradeoff Note that Usefull identities: [ ] [ ] I + L + I + L L= I S + T = I [ ] [ ] T = I + L L= I + L = L I + L [ + ] = [ + ] [ + ] = [ + ] [ + ] = [ + ] = [ + ] = ( ) + 2 = ( ) ( ) ( ) ( ) G I KG I GK G K I GK I KG K GK I GK G I KG K I GK GK Es () SsRs () () SsGsD () () s TsD () () s { 2 } U() s K() s S s R() s G s D s D s

12 Bode Waterbed Formula systems with relative degree 2 or greater: (Waterbed effect) 0 0 ln S( jω) dω = π p ln T( jω) dω = π 2 ω ORHP poles ORHP zeros i q i

13 Cauchy Theorem Theorem (Cauchy): Let C be a simple closed curve in the s-plane. F s is a rational function, having neither poles nor zeros on C. If C is the ( ) image of C under the map F s, then Z = P N where N the number of counterclockwise encirclements of the origin by C as s traverses C once in the clockwise direction. ( ) ( ) Z the number of zeros of F s enclosed by C, counting multiplicities. P the number of poles of F s enclosed by C, counting multiplicities. ( )

14 Nyquist Take F(s)=+L(s) (return difference, F=S - ) Choose a C that encloses the entire RHP Map into L-plane instead of F-plane (shift by -) R

15 Nyquist Theorem Theorem (Nyquist): If the plot of L(s) (i.e., the image of the Nyquist contour in the L- plane) encircles the point -+j0 in the counterclockwise direction as many times as there are unstable open loop poles (poles of L(s) within the Nyquist contour) then the feedback system has no poles in the RHP. Z = P N closed loop poles in RHP = open loop poles in RHP - cc encirclements of -

16 Example Hs + L Hs 2 +. s + L MAGNITUDE db rad êsec 0 4. PHASE deg rad êsec

17 Example Hs + L Hs L s Hs s + 2L Hs s + 4L.5 db MAGNITUDE rad êsec PHASE deg rad êsec

18 Gain & Phase Margin γ m Gain, db γ m φ m Phase, deg φ m L( jω ) Nyquist Bode

19 Guaranteed Gain/Phase Margin Theorem: Suppose M s is the sensitivity peak value, then α GM, PM ± 2 sin, α = M ± α 2 S R = S = a a GH-plane - θ = 2sin 2a + a

20 Multivariable Nyquuist Recall that the closed loop poles are the roots of the polynomial ( ) = det I + L( s) ( s) [ ] ds () = ()det s I+ Ls () L If the open loop system is stable, then we need only be concerned with the zeros of F s i.e., SISO MIMO + L ( ) or L( s) ( s) det I + L s det I + L

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