CDS 101/110: Lecture 9.1 Frequency DomainLoop Shaping

Transcription

1 CDS /: Lecture 9. Frequency DomainLoop Shaping November 3, 6 Goals: Review Basic Loop Shaping Concepts Work through example(s) Reading: Åström and Murray, Feedback Systems -e, Section.,.-.4,.6 I.e., we are not going to cover Section. (feedforward design) and.5 (root locus). Section.6 will be mainly discussed next week.

2 General Loop Transfer Functions rr System outputs yy ηη vv uu ee disturbance ee uu vv F(s) + C(s) + P(s) + Feed forward = Controller PPPPPP +PPPP PPPPPP +PPPP CCCC +PPPP CCCC +PPPP FF +PPPP Gang of Seven Control PP +PPPP PP +PPPP +PPPP PPPP +PPPP PP +PPPP - +PPPP PPPP +PPPP CC +PPPP CC +PPPP +PPPP System inputs rr dd nn η Plant Process TF = CFS = noise PPPPPP +PPPP CCCC +PPPP Response of (y, u) to r yy T = CS = r = reference input e = error u = control v = control + disturbance η = true output (what we want to control!) y = measured output Gang of Six PPPP +PPPP CC +PPPP Response of u to (d,n) PS = PP +PPPP S = +PPPP Response of y to (d,n)

3 Key Loop Transfer Functions rr d n ee uu vv F(s) + C(s) + P(s) + yy η - F(s) = : Four unique transfer functions define performance ( Gang of Four ) Sensitivity: Function Complementary Sensitivity Function: Load Sensitivity Function: Noise Sensitivity Function: GG eeee = S(s) = +LL(ss) GG yyyy = T(s) = GG yyyy = PS(s) = GG yyyy = CS (s) = LL(ss) +LL(ss) PP(ss) +LL(ss) CC(ss) +LL(ss) Gang of Four (the sensitivity functions) Characterize most performance criteria of interest

4 Rough Loop Shaping Design Process A Process: sequence of (nonunique) steps. Start with plant and performance specifications. If plant not stable, first stabilize it (e.g., PID) 3. Adjust/increase simple gains Increase proportional gain for tracking error Introduce integral term for steady-state error Will derivative term improve overshoot? 4. Analyze/adjust for stability and/or phase margin Adjust gains for margin Introduce Lead or Lag Compensators to adjust phase margin at crossover and other critical frequencies Consider PID if you haven t already 4

5 Summary of Specifications Hyr Key Idea: convert closed loop specifications on GG yyyy ss = PP ss CC(ss) + PP ss CC(ss) = LL (ss) + LL ss to equivalent specifications on loop system LL(ss) Time domain spec.s can often be converted to frequency domain spec.s Steady-state tracking error < XXX LL > /XX Tracking error < YYY up to frequency ff tt Hz LL iiii > /YY for ωω < ππff tt Bandwidth of ωω bb rad/sec LL iiωω bb = Usually needed for rise/settling time spec. 5

6 Summary of Specifications Hyr Overshoot < ZZZ Phase Margin > ff(zz) Phase/Gain margins (Specified Directly) For robustness Typically, at least gain margin of (6 db) Usually, phase margin of 3-6 degrees 6

7 Loop Shaping : Design Loop Transfer Function e u η r + C(s) + P(s) + d - BW GM n y Translate specs to loop shape LL ss = PP ss CC(ss) Design L(s) to obey constraints High gain at low frequency - Good Steady-state error - Good disturbance rejection at low freqs. - Decent tracking in bandwidth Low gain at high frequency - Avoid amplifying noise Sufficiently high bandwidth - Good rise/settling time Shallow slope at crossover - Sufficient phase margin for robustness, low overshoot PM Loop shaping is trial and error 7

8 Additional Loop Shaping Concepts Disturbance rejection Would like HH eeee to be small make large L(s) Typically require this in low frequency range High frequency measurement noise HH uuuu = CC(ss) +PP ss CC(ss) Want to make sure that HH uuuu is small (avoid amplifying noise) Typically generates constraints in high frequency range Robustness: gain and phase margin Focus on gain crossover region: make sure the slope is gentle at gain crossover Fundamental tradeoff: transition from high gain to low gain through crossover 8

9 Design Method #: Process Inversion Simple trick: invert out process Write performance specs in terms of desired loop transfer function Choose L(s) to satisfy specfications Choose controller by inverting P(s) Pros Simple design process L(s) = k/s often works very well Can be used as a first cut, with additional tuning Cons High order controllers (at least same order as plant) Requires perfect process model (due to inversion) Can generate non-proper controllers (order(num) > order(den)) - Difficult to implement, plus amplifies noise at high frequency (C( ) = ) - Fix by adding high frequency poles to roll off control response at high frequency Does not work if right half plane poles or zeros (internal instability) 9

10 Lead & Lag Compensators Lead: K >, a < b Add phase near crossover Lag: Improve gain & phase margins, increase bandwidth (better transient response). K >, a> b Add gain in low frequencies Improves steady state error r + - e K (ss+aa) (ss+bb) u P(s) Lead/Lag: Better transient and steady state response y C(i ) C(i ) 9 45 Lead PD C(i ) C(i ) 45 Lag PI 9 a b b a Frequency [rad/s] Frequency [rad/s] (a) Lead compensation, a < b (b) Lag compensation, b < a

11 Lead & Lag Compensators Lead: adds phase, φφ mm at: ωω = aaaa r + - e K (ss+aa) (ss+bb) u P(s) y φφ mm = 9 oo tan aa bb C(i ) C(i ) 9 45 a Frequency b [rad/s] (a) Lead compensation, a < b Lead PD Lag: reduces steady state error by factor of a/b C(i ) C(i ) 45 9 b Frequency a [rad/s] (b) Lag compensation, b < a Lag PI

12 Design Method #: Add Lead, Lag, Lead/Lag compensation Lead: increases phase in frequency band Effect: lifts phase by increasing gain at high frequency Increases PM Bode: add phase between zero and pole Nyquist: increase phase margin r + - e u P(s) y

13 Example: Lead Compensation for Second Order System System description Poles: pp =, pp = 5 Control specs Track constant reference with error < % Good tracking up to rad/s (less than % error) Overshoot less than % - Gives PM of ~6 deg Try a lead compensator Want gain cross over at approximately rad/sec => center phase gain there Set zero frequency gain of controller to give small error LL() > a =, b = 5, K =, (gives CC() = LL() = 4) 3

14 Example: Third Order System - - System description PP ss = ss + 3 Poles: pp,,3 = Bode plot for PP(ss) Control specs Steady state error < % <% tracking error up to rad/s.8 sec settling time Overshoot less than % - Gives PM of ~6 deg Step Response.9.8 First Cut: Need to boost low frequency gain Need to increase bandwidth Need to increase phase at higher frequency Amplitude Step Reponse for PP(ss) Time (seconds) 4

15 Example: Third Order System () Gm = -.9 db (at.73 rad/s), Pm = -5.7 deg (at 4.53 rad/s) Start with P of PID control kk pp = Closed loop system is unstable Bandwidth too low Next: PD Control kk pp =, kk dd = 7 Closed loop system is stable Phase margin oo at 8.3 rad/sec - Oscillatory step response Tracking requirement not met Gm = Inf db (at Inf rad/s), Pm =.8 deg (at 8.34 rad/s) Step Response.8.6 Can I in PID help? adds 9 oo phase! Amplitude Time (seconds) 5

16 Example: Third Order System (3) Gm = Inf db (at Inf rad/s), Pm =. deg (at 4. rad/s) Can I in PID help? Can create stable system which meets steady-state spec., but not phase margin (and overshoot) & bandwidth Amplitude Step Response Time (seconds) PD + Lead compensator? kk pp =, kk dd = 7 Lead compensator reduces low frequency -5-5 Gm = Inf db (at Inf rad/s), Pm = 43.7 deg (at 5.7 rad/s) gain (& bandwidth) ss + aa CC llllllll ss = ss + bb aa =.5, bb = Bode plot for Lead Bode plot for LL(ss) 6

17 Example: Third Order System (4) Gm = Inf db (at Inf rad/s), Pm = 5.4 deg (at 6.5 rad/s) Step Response 5. PD + Lead compensator Increase proportional gain to compensate for Lead at low frequency. Not enough bandwidth to meet tracking spec. Live with it, or keep searching Bode plot for LL(ss) 3 Amplitude Time (seconds) Increase D to add gain at high frequency kk pp =, kk dd = 5 Shift phase lead center frequency (a=5, b=4); Almost meets spec.s Amplitude Gm = Inf db (at Inf rad/s), Pm = 46.6 deg (at 4.3 rad/s) Step Response Time (seconds) 7

18 -5 Example: Third Order System (5) Nyquist Diagram Before trying more esoteric designs, check the basics Nyquist Gang of Four 5 5 Imaginary Axis GG eeee = SS = + PP ss CC(ss) GG yyyy = TT = PP ss CC(ss) + PP ss CC(ss) Real Axis CC ss = 5ss + 6ss + 5 ss GG yyyy = PPPP = PP ss + PP ss CC(ss) GG yyyy = GG uuuu = CCCC = CC ss + PP ss CC(ss) This controller amplifies high frequency noise, which will lead to a significant actuator activity Fix by adding roll-off pole(s) at higher freq

19 Summary: Loop Shaping Loop Shaping for Stability & Performance Steady state error, bandwidth, tracking response Specs can be on any input/output response pair BW GM Main ideas Performance specs give bounds on loop transfer function Use controller to shape response Gain/phase relationships constrain design approach Standard compensators: proportional, lead, PI PM Things to remember (for homework and exams) Always plot Nyquist to verify stability/robustness Check gang of 4 to make sure that noise and disturbance responses also look OK u 9