MEM 355 Performance Enhancement of Dynamical Systems


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1 MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 5/8/25
2 Outline Closed Loop Transfer Functions The gang of four Sensitivity functions Traditional Performance Measures Time domain Frequency domain Robustness Nyquist Traditional gain/phase margins
3 Closed Loop Transfer Functons d r G c u G p y d 2 p c ( ) ( ) Y = G U + D U = G R Y D E = R Y 2
4 Transfer Functions Con t Output Y() s = I+ Gp() sgc() s Gp() sgc() srs () I+ Gp() sgc() s Gp() sd() s + I+ Gp() sgc() s Gp() sgc() sd2() s Error E() s = I + Gp() s Gc() s R() s I + Gp() s Gc() s Gp() s D() s + I+ Gp() sgc() s Gp() sgc() sd2( s) Control U() s = Gc() s I + Gp() s Gc() s R() s Gc() s I + Gp() s Gc() s Gp() s D() s Gc() s I + Gp() s Gc() s D2() s
5 Gang of Four Reference Disturbance Noise Output Error Control GG p c + GG p c Gc + GG p c + GG p Gp Gp GG p c + GG + GG + GG p c GG p c GG p c Gc + GG + GG + GG p c p p c c p p c c c
6 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
7 Traditional Performance ~ Time Domain rise time, T r, usually defined as the time to get from % to 9% of its ultimate (i.e., final) value. settling time, T s, the time at which the trajectory first enters an εtolerance 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 3% is often considered undesirable. A system without overshoot is overdamped and may be too slow (as measured by rise time and settling time).
8 Traditional Performance ~ Time Domain Cont d y p.4 Im.2 α degree of stability, decay rate /α 2ε.8.6 ideal region for closed loop poles θ damping ratio θ = sin Re ρ t T r T p T s Time response parameters Ideal pole locations
9 Traditional Performance ~ Frequency Domain db deg MAGNITUDE rad êsec PHASE rad êsec Complementary sensitivity Desired Altitude  2 s +.5s+.5 K s disturbance db deg MAGNITUDE rad êsec ( 2s+ )( s+ )(.5s+ ) PHASE rad êsec Sensitivity V22 Osprey altitude control K=28
10 Sensitivity Functions, Cont d For unity feed back systems: S is the error response to command transfer function T is the output response to command transfer function ( ) Suppose G s is strictly proper m< n, then lim S( jω ) = lim = ω + G j ω lim S ( jω ) ω ω ( jω ) ω ω ( ω ) typeg = lim = + G j c << typeg = ( ω ) ( ω ) G( jω ) ( ω ) ( ω ) G j lim T( jω ) = lim = ω + ω limt G j typeg = lim = + G j c typeg =
11 Traditional Performance ~ Frequency Domain Bandwidth Definitions Sensitivity Function (first crosses / 2=.77~3db from below): { v S j v } ω = max : ( ω) < 2 ω [, ) BS v Complementary Sensitivity Function (highest frequency where crosses ω BT 2 from above) = min{ v : T( jω) < 2 ω ( v, ) } v Crossover frequency { v L j v } ω = max : ( ω) ω [, ) c v T
12 Example: Osprey Bandwidth Complementary sensitivity Open loop Sensitivity MAGNITUDE db rad êsec
13 Transfer Functions as Maps
14 Traditional Performance ~ Frequency Domain Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. M M S T = = max S( jω ) ω max T( jω ) ω Im L plane S > S < L= + ρ ( ) jθ S = ρe = + L e jθ S  = a a L( jω) Re constant S proscribes circle S = + L
15 A Fundamental Tradeoff [ ] [ ] Note that + L + + L L= S+ T = Making S small improves tracking & disturbance rejection but makes system susceptible to noise S T : ( ) Es () = SsRs () () SsGsD () () s+ TsD () () s Typical design specifications S T ( jω ) < < ω [, ω] ( jω) << ω [ ω, ] And, there are other limitations. 2 2 ω > ω 2
16 Bode Waterbed Formula systems with relative degree 2 or greater: (Waterbed effect) ln S( jω) dω = π p ln T( jω) dω = π 2 ω ORHP poles ORHP zeros i q i ln S( jω) vs. ω for Osprey 2
17 Implication of Waterbed Formula ln S ( jω ) pops up here ω push down here (e.g., by increasing gain)
18 V22 with PID s s +.5 H.5 s + L s H s + L H2 s + L Desired Altitude  2 s +.5s+.5 K s disturbance ( 2s+ )( s+ )(.5s+ )
19 Example: Osprey Sensitivity function plots for K=28, 5, MAGNITUDE db rad êsec
20 Example: Osprey 2 K= Increased gain yields higher bandwidth (and reduced settling time) but higher sensitivity peak (lower damping ratio and more overshoot) K=
21 Cauchy Theorem Theorem (Cauchy): Let C be a simple closed curve in the splane. 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. ( )
22 Nyquist Take F(s)=+L(s) (return difference, F=S  ) Choose a C that encloses the entire RHP Map into Lplane instead of Fplane (shift by ) splane splane L( s) Lplane R R Typical Nyquist contours Image of Nyquist contour
23 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 +j 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 
24 Example I splane R II L( s) Lplane ( ) L s = 2 2 ( s+ p)( s + 2ζωs+ ω ) jθ ( ρ ) ( ) 2 2 ( s+ p)( s + 2ζωs+ ω ) jθ 2 j2θ jθ 2 ( ρe + p )( ρ e + 2ζωρe + ω ) * III I: s = jω L jω =, < ω < jθ π π II: s = ρe, ρ, = 2 2 L e = III: s = jω, III I ρ
25 Example 2 ( ) L s = 2 2 ( + 2ζωs+ ω ) s s X X X I IV III 2 2 ( + j2 + ) ( + + ) * ( + + ) splane R I: s= jω L( jω) =, < ω < jω ω ζω ω ω jθ π π II: s = ρe, ρ, θ = 2 2 jθ L( ρ e ) = jθ j θ jθ ρ ρe ρ e 2ζω ρe ω III: s= jω, III I jθ π π IV : s = εe, ε, θ = 2 2 jθ L( ε e ) = jθ j θ jθ εe ε e 2ζω εe ω εω II ε 2 e L( s) jθ Lplane
26 Example 3 Hs + L Hs 2 +. s + L MAGNITUDE db rad êsec 4. PHASE deg rad êsec .5 The principle part of the Nyquist plot uses the same data as the bode plot
27 Example 4 Hs + L Hs L s Hs s + 2L Hs s + 4L.5 db MAGNITUDE rad êsec 3  PHASE deg rad êsec
28 Example 5 ( ) G s = ( ) ( s+ ) s s >> G=/(s*(s^2+2*.*s+)); >> nyquist(g) >> G=/((s+.)*(s^2+2*.*s+)); >> nyquist(g)
29 Gain & Phase Margin γ m Gain, db γ m φ m Phase, deg φ m Nyquist L( jω ) Assumption: the nominal system is stable. Bode
30 Example 6 ( ) G s = ( s +.5) ( s+ )( s+ )( s 2 +.s+ ) 5 OpenLoop Bode Editor (C) 4 Nyquist Diagram 3 Magnitude (db) G.M.: 2.5 db Freq: 2.42 rad/sec Stable loop Imaginary Axis 2  Phase (deg) P.M.: 27.3 deg Freq:.8 rad/sec Frequency (rad/sec) Real Axis
31 V22 with PID s s +.5 H.5 s + L s H s + L H2 s + L Desired Altitude  2 s +.5s+.5 K s disturbance ( 2s+ )( s+ )(.5s+ )
32 V22 with PID db MAGNITUDE rad êsec db MAGNITUDE rad êsec db MAGNITUDE rad êsec db MAGNITUDE rad êsec Complementary sensitivity sensitivity
33 V22 with PID 28 Hs s +.5 L H.5 s + L s H s + L H2 s + L
34 V22 with PID >> s=tf('s'); >> G=28*((s^2+.5*s+.5)/s) /((2*s+)*(*s+)*(.5*s+)); >> margin(g) Notice that gain margin is negative because system loses stability when gain drops.
35 V22 PID+Lead s s +.5 H.5 s + L s Hs + L H2 s + L
36 V22 PID+Lead Complementary sensitivity MAGNITUDE Notice that sensitivity peak is reduced from 2 db to db Sensitivity MAGNITUDE db rad êsec db rad êsec 6 MAGNITUDE MAGNITUDE db 8 db rad êsec Disturbance to output K= rad êsec Command to control
37 V22 PID+Lead.5 5 Hs s +.5 L H.5 s + L s Hs + L H2 s + L
38 V22 PID+Lead Gain margin is infinite because the system is never unstable for any K>.
39 Summary Need to consider 24 transfer functions to fully evaluate performance Bandwidth is inversely related to settling time Sensitivity function peak is related to overshoot and inversely to damping ratio Gain and phase margins can be determined from Nyquist or Bode plots Sensitivity peak is inversely related to stability margin Design tools: Root locus helps place poles Bode and/or Nyquist helps establish robustness (margins) & performance (sensitivity peaks)
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