Robust Control 3 The Closed Loop Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /2/2002
Outline Closed Loop Transfer Functions Traditional Performance Measures Time domain Frequency domain Robustness Traditional gain/phase margins Multivariable Nyquist
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
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 2 2 2 [ ]
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
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).
Traditional Performance ~ Time Domain Cont d y p.4.2 2ε 0.8 0.6 0.4 0.2 2 4 6 8 0 2 4 t T r T p T s
Traditional Performance ~ Frequency Domain db deg MAGNITUDE 0. 20 0.2 0.5 2 5 0 5 0-5 -0-5 0. 0.2 0.5 2 rad êsec PHASE 0. 0.2 0.5 2 0-25 -50-75 -00-25 -50-75 0. 0.2 0.5 2 rad êsec Complementary sensitivity 2 s +.5s+ 0.5 K s db deg MAGNITUDE 0. 0.2 0.5 2 20 0 0-0 -20-30 -40-50 0. 0.2 0.5 2 rad êsec -200-250 -300-350 ( 20s+ )( 0s+ )( 0.5s+ ) PHASE 0. 0.2 0.5 2 0. 0.2 0.5 2 rad êsec Sensitivity V-22 Osprey altitude control K=280
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
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
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
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 -5-0 -5 2 3 4-20
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. ( )
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
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 -
Example G@sD= Hs + L Hs 2 +. s + L MAGNITUDE 0. 0.2 0.5 2 5 0 db -0-20 0.5-30 -40 0. 0.2 0.5 2 5 rad êsec Im @gd 0 4. PHASE 0. 0.2 0.5 2 5 deg -50-00 -50-200 -250 0. 0.2 0.5 2 5 rad êsec -0.5 - -0.4-0.2 0 0.2 0.4 0.6 0.8 Re @gd 4.
Example 2 G@sD= 0.2 Hs + L Hs 2 + 2 L s Hs 2 + 2 s + 2L Hs 2 + 2 s + 4L.5 db 20 0-20 -40-60 -80 MAGNITUDE 0.0 0.05 0. 0.5 5 0 0.5 0.0 0.05 0. 0.5 5 0 rad êsec Im @gd 0 3-00 PHASE 0.0 0.05 0. 0.5 5 0-0.5-50 deg -200-250 - 3-300 0.0 0.05 0. 0.5 5 0 rad êsec -.2 - -0.8-0.6-0.4-0.2 0 0.2 Re @gd
Gain & Phase Margin γ m Gain, db γ m φ m Phase, deg φ m L( jω ) Nyquist Bode
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
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