Control SystemsControl Design: Loop shaping L. Lanari Sunday, November 30, 2014 Outline • • • • • specifications open-loop shaping principle lead and lag controllers 4 basic situations sketch of PID controllers Lanari: CS - Loop shaping Sunday, November 30, 2014 2 Specifications (closed-loop) Static • • • • desired behavior w.r.t. order k inputs in terms of a maximum allowed absolute error desired attenuation level w.r.t. constant disturbances acting on the forward loop tracking of a sinusoidal reference attenuation of sinusoidal disturbances and measurement noise Dynamic • • • location of eigenvalues/poles in the complex plane time domain specifications on the step response (mainly on the reference to output behavior) frequency domain specifications through the resonance peak and the bandwidth (mainly on the reference to output behavior) Stability • • location of eigenvalues/poles in the complex plane robustness in terms of gain and phase margin Lanari: CS - Loop shaping Sunday, November 30, 2014 3 Specifications CLOSED-LOOP System Static equivalent to OPEN-LOOP System presence of a sufficient number of poles in 0 and/or at the reference/disturbance angular frequency sufficiently high gain (in absolute value) Necessary conditions require the following structure in the controller � � for sinusoidal 1 Kc reference or 2 2 h s +ω ¯ s disturbance Dynamic bandwidth B3 (and rise time tr) !c crossover frequency resonant peak Mr (and overshoot Mp) PM phase margin taken care by R(s) Kc C(s) = h R(s) s Lanari: CS - Loop shaping Sunday, November 30, 2014 4 d1 d2 u + + C (s) r e + - m+ Kc sh R (s) P (s) + + + necessary part of the controller C (s) controller y n Kc sh P (s) R (s) to be still chosen in order to satisfy the dynamic specifications Pˆ (s) extended plant plant Loop function L (s) R (s) dynamic specifications Lanari: CS - Loop shaping Sunday, November 30, 2014 Kc sh P (s) static specifications 5 Open Loop Shaping Basic idea: being the plant P (s) fixed, choose the controller C (s) to shape the loop function frequency response L (j!) L (j!) C (s) magdB such that with some desired characteristics ωc∗ phase PM∗ C (s) = C1 (s)R (s) C1 (s) static specs Kc sh R (s) dynamic performance specs static specs Lanari: CS - Loop shaping Sunday, November 30, 2014 6 hyp: the necessary part of the controller C (s) has been determined so we have the extended plant Kc ˆ P (s) = C1 (s)P (s) = h P (s) s we need to determine C (s) so to satisfy also the dynamics specifications and ensure stability specifications !c* desired crossover frequency at which we want to have a phase margin of at least PM * from the extended plant frequency response we need to check which action is necessary both in terms of magnitude and phase by comparing the actual value of the magnitude and phase at the future crossover frequency !c* � Amplification: we need to increase the magnitude at some frequency Magnitude Attenuation: we need to decrease the magnitude at some frequency � Lead: we need to increase the phase at some frequency Phase Lag: phase can be decreased at some frequency if necessary we want P M ≥ P M ∗ Lanari: CS - Loop shaping Sunday, November 30, 2014 so if we have extra phase we can keep it 7 ∠Pˆ (jω) attenuation needed !c* ! 2 possible actions on the magnitude at some frequency |Pˆ (jω)|dB amplification needed |Pˆ (jω)|dB maximum lag we can tolerate lead (phase increase) needed PM * -¼ 2 possible actions on the phase at some frequency ∠Pˆ (jω) Lanari: CS - Loop shaping Sunday, November 30, 2014 8 Remember that the static specifications have already been met (if we ensure stability) and therefore we do not want to alter this first step. Therefore, in general, we are not going to use a • • • zero in s = 0 to obtain a phase lead pole in s = 0 to attenuate at some frequency a gain smaller than 1 in magnitude to attenuate if we have a constraint on the loop gain from the static requirements (while we may use a gain greater than 1 to amplify) elementary functions that can provide these magnitude and phase contributions lead compensator 1 + τa s Ra (s) = τa s 1+ m a lag compensator τi s 1+ m i Ri (s) = 1 + τi s ¿a > 0 ma > 1 ¿i > 0 mi > 1 both have unit gain so no magnitude change in ! = 0 Lanari: CS - Loop shaping Sunday, November 30, 2014 9 Lead compensator Magnitude (dB) 1 + τa s Ra (s) = τa s 1+ m a Phase (deg) pole in ma τa zero 10 0 −10 −20 1 τa ma τa pole zero 45 0 −45 pole −90 Lanari: CS - Loop shaping Sunday, November 30, 2014 ma > 1 20 90 ma = 10 ¿a > 0 zero in 1 τa 10 Lag compensator Magnitude (dB) τi s 1+ m i Ri (s) = 1 + τi s ¿i > 0 mi > 1 20 10 1 τi 0 −10 −20 pole zero in mi τi pole in 1 τi zero mi τi Phase (deg) 90 mi = 10 0 −45 −90 Lanari: CS - Loop shaping Sunday, November 30, 2014 zero 45 pole 11 Choice of R (s) we assume C1 (s) (static specs) has already been chosen. Therefore we need to • by evaluating the actual values of the extended plant magnitude and phase at the desired crossover frequency !c*, understand what action needs to be undertaken: ‣ ‣ • amplification or attenuation at !c* phase lead or maximum allowed phase lag at !c* choose the elementary function(s) Ra(s) and/or Ri(s) (since multiple actions can be combined) needed ‣ ‣ choose ma (or mi) and the normalized frequency !¿ deciding to obtain the desired action at !c* choose ¿a (or ¿i) so that !c*¿a = !¿ frequency chosen at which we want to obtain the desired action Lanari: CS - Loop shaping Sunday, November 30, 2014 (or !c*¿i = !¿) we solve in ¿a 12 Universal diagrams 25 16 1 + τa s Ra (s) = τa s 1+ m a ma > 1 12 10 20 8 magnitude (dB) ¿a > 0 14 6 15 4 3 10 2 5 0 −1 10 for different values of ma 0 10 1 10 normalized frequency 2 10 70 16 60 50 phase (deg) for Ri(s) just change sign to the ordinates 14 12 10 8 6 40 4 30 3 20 2 10 0 −1 10 Lanari: CS - Loop shaping Sunday, November 30, 2014 0 10 1 10 normalized frequency 2 10 13 Case I 40 ωc∗ 20 30 R a (jω) 0 20 −20 10 Magnitude (dB) Magnitude (dB) 40 −40 −60 −2 10 −1 10 0 10 frequency (rad/s) 1 10 Phase (deg) 50 0 −10 −20 L(j ω) −30 −90 −40 PM * −180 −200 −270 −2 10 specs 0 −1 10 0 10 frequency (rad/s) !c* = !c −60 −2 10 1 10 30 actions needed: • 0 10 frequency (rad/s) 1 10 frequency which coincides with the desired one phase: increase the phase, in this case of exactly PM * −90 L(j ω) −180 −200 Pˆ (jω) −270 −2 10 Lanari: CS - Loop shaping R a (jω) 0 Phase (deg) magnitude: as small amplification as possible in order not to move the current crossover Sunday, November 30, 2014 −1 10 PM ≥ PM * 50 • Pˆ (jω) −50 −1 10 0 10 frequency (rad/s) 1 10 14 example we need a phase lead of 25° with the smallest amplification possible 70 16 60 phase (deg) 50 25° Lanari: CS - Loop shaping Sunday, November 30, 2014 6 40 4 30 20 2 10 0 −1 10 we have chosen 0 10 1 2 10 normalized frequency 10 0.5 is the smallest (for these plots) normalized frequency at which we obtain the desired phase lead (for ma = 16) 25 16 14 12 10 20 8 magnitude (dB) with the amplification of 1 dB at !c* = 0.3 rad/s we choose ¿a as ¿a = 0.5/0.3 8 3 desired phase lead ma = 16 !¿ = 0.5 therefore to obtain this lead of 25° together several choices of ma are possible (at different normalized frequencies), we need to find one which is compatible with the magnitude requirement 14 12 10 6 15 4 3 10 smallest amplification obtainable (around 1 dB) 2 5 0 −1 10 1 dB 0 10 1 10 normalized frequency 2 10 15 Case II 40 |Pˆ (jωc∗ )|dB 20 ωc∗ 0 −20 Pˆ (jω) L(j ω) 20 10 −40 −60 −4 10 −3 −2 10 −1 10 10 frequency (rad/s) 0 10 1 10 50 Phase (deg) 30 Magnitude (dB) Magnitude (dB) 40 0 R i (j ω) −20 −40 PM * −180 −200 specs −10 −30 −90 −270 −4 10 0 −3 −2 10 −1 10 10 frequency (rad/s) 0 10 −50 −60 −4 10 1 10 −3 10 −2 −1 −2 −1 10 10 frequency (rad/s) 0 10 1 10 PM ≥ PM * !c* 50 30 actions needed: magnitude: attenuation of |Pˆ (jωc∗ )|dB phase: since ∠Pˆ (jωc∗ ) + π > P M ∗ Phase (deg) • • 0 R i (j ω) −90 Pˆ (jω) L(j ω) −180 we can tolerate at most a lag of ∠Pˆ (jωc∗ ) + π − P M ∗ Lanari: CS - Loop shaping Sunday, November 30, 2014 −200 −270 −4 10 −3 10 10 10 frequency (rad/s) 0 10 1 10 16 example we need an attenuation of 17 dB and can tolerate a maximum lag of 7° - 25 negative values required attenuation 16 12 - 20 10 magnitude (dB) 4 - 3 2 5 0 −1 10 0 10 16 8 - 40 6 4 Sunday, November 30, 2014 60 is the smallest (for these plots) normalized frequency at which we obtain the desired attenuation (for mi = 8) 3 - 20 Lanari: CS - Loop shaping 14 12 10 - 50 - 30 maximum allowed lag 2 10 - 70 negative values - 60 we choose ¿i as ¿i = 60/0.1 1 10 normalized frequency phase (deg) together with a lag smaller than 7° at !c* = 0.1 rad/s 6 - 10 mi = 8 !¿ = 60 therefore to obtain this attenuation of 1 dB several choices of mi are possible (at different normalized frequencies), we need to find one which is compatible with the magnitude requirement 8 - 15 we have chosen 14 2 - 10 0 −1 10 0 10 1 10 normalized frequency 2 10 17 on the choice of the normalized frequency (case II) basic consideration (see sensitivity functions): we usually want open-loop high gain at low frequency, therefore anything that gives an excess of attenuation at low frequency should be avoided if possible case II with two alternative choices of the !¿ Ri1(s) mi = 8 !¿ = 70 Ri2(s) mi = 8 !¿ = 200 starts attenuation before strictly needed 80 0 70 R i1(j ω) 60 R i2(j ω) Pˆ (jω) 40 30 L 1 (jω ) Pˆ (jω) L 2 (jω ) 20 Phase (deg) Magnitude (dB) 50 10 −90 L 1 (jω ) L 2 (jω ) 0 −180 R i1(j ω) −10 R i2(j ω) −200 −20 −30 −4 10 −3 10 Lanari: CS - Loop shaping Sunday, November 30, 2014 −2 10 frequency (rad/s) −1 10 0 10 −4 10 −3 10 −2 10 frequency (rad/s) −1 10 0 10 both compensators solve the specifications 18 Case III 40 ωc∗ 20 0 30 |Pˆ (jωc∗ )|dB −20 −40 −60 −4 10 −3 10 −2 −1 10 10 frequency (rad/s) 0 10 1 10 Phase (deg) 50 0 20 −10 L(j ω) −20 R a (jω) Pˆ (jω) −40 PM * −180 −200 specs 0 −30 −90 −270 −4 10 R a (jω) 10 Magnitude (dB) Magnitude (dB) 40 −3 10 !c* −2 −1 10 10 frequency (rad/s) 0 10 Pˆ (jω) −50 −60 −2 10 1 10 −1 10 0 1 10 frequency (rad/s) 10 PM ≥ PM * 50 30 actions needed: magnitude: amplification of −|Pˆ (jωc∗ )|dB phase: since ∠Pˆ (jωc∗ ) + π < P M ∗ we need to obtain a phase lead of � � P M ∗ − ∠Pˆ (jωc∗ ) + π Lanari: CS - Loop shaping Sunday, November 30, 2014 0 Phase (deg) • • R a (jω) −90 −180 −200 L(j ω) Pˆ (jω) −270 −2 10 −1 10 0 10 frequency (rad/s) 1 10 19 example we need an amplification of 20 dB and a phase lead of at least 25° two choices • find a lead compensator that will give simultaneously the required amplification and phase lead ‣ ‣ • usually requires a choice of the normalized frequency on the right-hand side of the phase “bell” shape which gives poor robustness w.r.t. increases in the crossover frequency since the phase of both the extended plant and the compensator are decreasing at the chosen frequency may be not easy to find find a lead compensator that gives the required lead and gives some amplification, integrate the required amplification with an additional gain Kc2 greater that 1. This is usually possible since the static requirement (if any) asks for a gain sufficiently high. Lanari: CS - Loop shaping Sunday, November 30, 2014 20 example we need an amplification of 20 dB and a phase lead of at least 25° 70 Kc2Ra(s) 16 60 phase (deg) 50 25° 8 6 40 4 30 3 20 desired phase lead 2 10 we can choose 0 −1 10 ma = 3 !¿ = 0.9 therefore to obtain this lead of 25° together 0 10 1 2 10 normalized frequency 10 Kc2 / [Kc2 ]dB = 20 - 2.5 25 16 14 12 10 20 8 magnitude (dB) with the amplification of 2.5 dB at !c* = 0.2 rad/s we choose ¿a as ¿a = 0.9/0.2 several choices of ma are possible (at different normalized frequencies), we just keep the choice of the normalized frequency on the left of the “bellshape” 14 12 10 6 15 4 we can obtain an amplification of 2.5 dB) 3 10 2 5 2.5 dB Lanari: CS - Loop shaping Sunday, November 30, 2014 0 −1 10 0 10 1 10 normalized frequency 2 10 21 about the robustness issue case III revisited with two alternative solutions (the gains have been chosen appropriately) 70 16 60 Kc1 Ra1 (jω) 14 12 phase (deg) ma = 6 !¿ = 0.8 Kc2 Ra2 (jω) 10 50 8 6 40 ma = 6 !¿ = 7.5 4 30 3 20 2 left-hand side of the bell right-hand side of the bell 10 0 −1 10 0 10 1 10 normalized frequency 10 2 40 30 50 R a 1(jω) 30 R a 2(jω) 20 R a 2(jω) 0 w.r.t. an increase of the crossover frequency R a 1(jω) Phase (deg) Magnitude (dB) 10 0 −10 −20 L 2 (jω ) −90 L 2 (jω ) −30 −180 L 1 (jω ) −40 −1 10 0 10 frequency (rad/s) L 1 (jω ) Pˆ (jω) Pˆ (jω) −50 −60 −2 10 −200 1 10 −270 −2 10 −1 10 0 10 frequency (rad/s) 1 10 solution 1 is more robust w.r.t. uncertainties in the !c Lanari: CS - Loop shaping Sunday, November 30, 2014 22 Case IV 40 |Pˆ (jωc∗ )|dB 20 ωc∗ 0 −20 20 R a (jω) 10 −40 −60 −2 10 −1 10 0 10 frequency (rad/s) 1 10 50 Phase (deg) 30 Magnitude (dB) Magnitude (dB) 40 0 −10 R i (j ω) −20 −30 −90 −40 PM * −180 −200 −270 −2 10 specs 0 −1 10 !c* 0 10 frequency (rad/s) −50 −60 −3 10 1 10 Pˆ R a Pˆ L = R i R a Pˆ −2 10 −1 10 frequency (rad/s) 0 ∠Pˆ (jωc∗ ) + π < P M ∗ we need to obtain a phase lead of at least � � P M ∗ − ∠Pˆ (jωc∗ ) + π Lanari: CS - Loop shaping 0 R i (j ω) Phase (deg) magnitude: attenuation of |Pˆ (jωc∗ )|dB phase: since R a (jω) 30 actions needed: Sunday, November 30, 2014 10 PM ≥ PM * 50 • • 1 10 −90 −180 −200 Pˆ R a Pˆ L = R i R a Pˆ −270 −3 10 −2 10 −1 10 frequency (rad/s) 0 10 1 10 23 example we need an attenuation of 8 dB and a phase lead of at least 25° we need to use both lead and a lag compensators but in the proper order • we choose the lead compensator first in such a way to obtain a phase increase of the required 25° plus an extra (for example of 8°) in order to compensate the lag that will be introduced by the lag compensator This lead function will also introduce, at the chosen frequency, an amplification of exactly ˆ a (jω ∗ )|dB |R c • the lag compensator will be chosen so to introduce an attenuation of ˆ a (jω ∗ )|dB 8 dB + |R c and a lag smaller than the extra 8° previously introduced ma = 8 !¿ = 0.8 34° 2.5 dB mi = 3.2 !¿ = 20 -10.5 dB < 8° (these numbers are just for illustration purposes and have not been verified) Lanari: CS - Loop shaping Sunday, November 30, 2014 24 PID controllers 3 basic heuristic actions Proportional: the control action is set to be directly proportional to the system error (present) Integral: the control action is set to be proportional to the system error integral (past) Derivative: the control action is set to be proportional to the system error derivative (future) output of the controller e (s) m(t) = KP e(t) + KI m (s) CPID (s) - � t e(τ )dτ + KD e(t) ˙ 0 m(s) CP ID (s) = e(s) = ideal PID controller transfer function = KI KP + + KD s s � � 1 KP 1 + + TD s TI s with KP TI = KI TD KD = KP widely spread fixed structure with 3 tunable parameters KP KD KI can be tuned automatically even with scarce knowledge of the (simple) plant basic tuning refers only to static specifications Lanari: CS - Loop shaping Sunday, November 30, 2014 25 Derivative action is physical not realizable (improper transfer function) we need to approximate add a high frequency pole in KP N s=− N =− KD TD a CD (s) TD s � =� TD 1+ N s high frequency pole Magnitude (dB) 60 40 tive a v i r l de a e d i 20 0 approx −20 N = 10, 20, 50, 100 −40 −2 10 −1 10 0 1 10 10 frequency (rad/s) 2 10 3 10 Phase (deg) ideal derivative approximated derivative action for various values of N 90 45 N = 10, 20, 50, 100 0 −2 10 Lanari: CS - Loop shaping Sunday, November 30, 2014 −1 10 0 1 10 10 frequency (rad/s) 2 10 3 10 26 CaPID (s) 1 r + e - 1 KI s + + + m KP P (s) TD s � � TD 1+ N s Basic PID feedback control scheme typical configurations • • • proportional proportional + derivative (approximate) proportional + integrative Lanari: CS - Loop shaping Sunday, November 30, 2014 27 PD configuration 30 CPa D (s) = KP sTD 1+ 1 + s TND � = KP 1 � � 1 + TD NN+1 s � N +1 � 1 + N +1 TD N 1/ma Magnitude (dB) � ¿a 20 10 0 −10 −2 10 Phase (deg) equivalent to a Lead compensator KP = TD = 1, N = 10 1 1+ sTI � KP (1 + TI s) = TI s Magnitude (dB) CP I (s) = KP � Sunday, November 30, 2014 0 1 10 10 −1 10 10 frequency (rad/s) 0 1 10 2 10 3 2 10 3 KP = TI = 1 20 0 −2 10 −2 10 10 −1 10 10 frequency (rad/s) 0 1 10 −1 10 10 frequency (rad/s) 0 1 10 2 10 3 2 10 Phase (deg) 0 10 Lanari: CS - Loop shaping 10 10 frequency (rad/s) 40 zero in s = -1/TI + pole in s = 0 compensates the phase lag introduced by the pole in s = 0 −1 45 0 −2 10 PI configuration 10 3 28 Vocabulary English Italiano lead compensator funzione anticipatrice lag compensator funzione attenuatrice phase lead anticipo di fase phase lag ritardo di fase attenuation attenuazione amplification amplificazione open-loop shaping sintesi per tentativi Lanari: CS - Loop shaping Sunday, November 30, 2014 29