MEM 355 Performance Enhancement of Dynamical Systems


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1 MEM 355 Performance Enhancement of Dynamical Systems Frequency Domain Design Intro Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University /5/27
2 Outline Closed Loop Transfer Functions The gang of four Sensitivity functions Traditional Performance Measures Time domain Frequency domain Robustness Introduction role of sensitivity functions Nyquist Traditional gain/phase margins
3 Closed Loop Transfer Functions /5/27
4 Closed Loop Transfer Functons error control d disturbance r G c u G p y output d 2 reference p c Y G U D U G RY D E RY 2 noise
5 Transfer Functions Output GG G GG Y() s R() s D () s D () s GG GG GG Error p c p p c 2 p c p c p c G G G E() s R() s D () s D () s GG GG GG Control p p c 2 p c p c p c G GG G U() s R() s D () s D () s GG GG GG c p c c 2 p c p c p c
6 Gang of Four Outputs inputs Output Error Control Reference GG p c GG p c Gc GG p c GG p c Disturbance Noise Gp G GG p 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
7 Sensitivity Functions p 2 E() s I L R() s I L G D () s I L LD () s where L: G G p sensitivity function: S: I L command error complementary sensitivity c function: T : I L L command output For SISO systems Bode derived: dt T dt L change in command output transfer function dl L dl T with respect to change in L (normalized) 2 L [ L] L[ L] [ L] L [ L] L [ L] S
8 Traditional Performance Criteria /5/27
9 Traditional Performance ~ Time Domain ultimate error, limit of e(t) as t approaches infinity 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 tolerance of its ultimate value and remains there ( 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).
10 rise time peak time settling time Traditional Performance ~ Time Domain Cont d y p.4 Im.2 degree of stability, decay rate / ideal region for closed loop poles damping ratio sin Re.4.2 T T r T p s rise time peak time settling time t T r T p T s Time response parameters Ideal pole locations
11 Bode Plot Given any transfer function describing a SISO system Y s G s U s Suppose the input is a sinusoid u t Asin t, then the output is a sinusoid y t Bsin t B G j A, G j
12 Traditional Performance ~ Frequency Domain db deg MAGNITUDE rad sec 2 PHASE rad sec 2 Complementary sensitivity Desired Altitude  2 s.5s.5 K s disturbance db deg MAGNITUDE rad sec ss.5s PHASE rad sec Sensitivity V22 Osprey altitude control K=28
13 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 L s is strictly proper m n, then lim S j lim L j lim S j lim L j c L j lim T j lim L j limt type L type L L j type L j lim L j c type L We would like S= We would like T= A system is of type p if the transfer function L has p free integrators in the denomintor, i.e. s a s a Ls k s s b s b m m m p np np np
14 Traditional Performance ~ Frequency Domain Bandwidth Definitions Sensitivity Function (first crosses / 2=.77~3db from below): max v : S( j) 2 [, v) BS v Complementary Sensitivity Function crosses BT 2 from above) minv : T( j) 2 ( v, ) v Crossover frequency max v : L( j) [, v) c v (highest frequency where T
15 Example: Osprey Bandwidth Complementary sensitivity Open loop Sensitivity MAGNITUDE 5 db rad sec
16 Interpretation of Bode Plot Gs Y nput U: Es GsUs, U j. t e sin t. t e Any transfer function: Output response to i Y j G j U j Y j G j U j Y j G j MAGNITUDE. 4 2 MAGNITUDE. 2 db db rad sec input frequency distribution 2. rad sec
17 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 () () stsd () () s Typical design specifications S T j, j, And, there are other limitations
18 Cauchy Theorem Any function of a complex number can be viewed as a map of points in one complex plane to another 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 PN 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.
19 Nyquist return difference Sensitivity function Take F(s)=+L(s) (note: F=S  ) Choose a C that encloses the entire RHP Map into Lplane instead of Fplane (shift by ) splane splane Ls Lplane R R Typical Nyquist contours Image of Nyquist contour
20 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 
21 Example I splane R II Ls Lplane III L s 2 2 s ps 2s I: s j L j, 2 2 j p 2j j II: s e,, 2 2 j L e III: s j, III I j 2 j2 j 2 e p e 2e *
22 Example 2 I splane R II Ls Lplane L s s s 2 s 2 2 I: s j L j, j j2 2 2 j II: s e,, 2 2 j L e e e 2 e j 2 j2 j 2 * III: sj, III I j IV : s e,, 2 2 j L e e e 2 e j 2 j2 j 2 IV III 2 e j
23 Example 3 G s s s 2 2.s >> s=tf( s ); >> G=/(s*(s^2+2*.*s+)); >> nyquist(g) >> s=tf( s ); >> G=/((s+.)*(s^2+2*.*s+)); >> nyquist(g)
24 Gain & Phase Margin m m m m Nyquist plot L j Assumption: the nominal system is stable. Bode plot
25 Robustness From Sensitivity Functions M M Sensitivity peaks are related to gain and phase margin. Sensitivity peaks are related to overshoot and damping ratio. S T max S( j) max T( j) Unit circle centered at  Im L plane S S j S e L L e j S  a a L( j) Re constant S prescribes circle S L
26 Example: Osprey Sensitivity function plots for K=28, 5, MAGNITUDE db rad sec
27 Example: Osprey 2 K= Increased gain yields higher bandwidth (and reduced settling time) but higher sensitivity peak (lower damping ratio and more overshoot) K=
28 V22 with PID disturbance Desired Altitude K s 2.5s.5 s 2ss.5s
29 V22 with PID db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec db MAGNITUDE rad sec Complementary sensitivity sensitivity
30 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.
31 V22 PID+Lead K.5.5. s s 2 s s s 2ss.5s
32 V22 PID+Lead Complementary sensitivity MAGNITUDE Notice that sensitivity peak is reduced from 2 db to db Sensitivity MAGNITUDE 5 db rad sec db rad sec 6 MAGNITUDE 8 MAGNITUDE 7 6 db 8 db rad sec Disturbance to output K=5 rad sec Command to control
33 V22 PID+Lead Gain margin is infinite because the system is never unstable for any K>.
34 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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