Lecture 8. PID control. The role of P, I, and D action 2. PID tuning Indutrial proce control (92... today) Feedback control i ued to improve the proce performance: tatic performance: for contant reference, the output aymptotically converge to the deired value, regardle of load diturbance. dynamic performance: enure proper tranient behavior during tep change Proportional control i not ufficient: feedback gain mut be frequency dependent! controller i itelf a dynamical ytem C() Syt3 lecture 8 Syt3 lecture 8 2 PID control Inight about PID action The magical three-term controller (99% of indutrial controller): u = K P e }{{} proportional Frequency interpretation: e + }{{} K D ė }{{} derivative integral +K I C() = K P + K I + K D K I dominate at low frequencie while K D dominate at high frequencie P I D Syt3 lecture 8 3 Syt3 lecture 8 4
Proportional control Frequency domain interpretation: 2 log H Suppoe P() = τ+ and C() = K. Cloed-loop tranfer function: Y R = K +K + +K τ The tatic error and the time contant are divided by ( + K) Feedback modifie the location of the pole. More generally: rlocu how how cloed-loop pole move with K. 2 log( + K) τ τ ( + K) Feedback ditribute the loop tranfer from u to y: trading gain for bandwidth... ω Increaing the bandwidth typically reduce the phae margin. Proportional control uually increae ocillation and overhoot in tep repone. Syt3 lecture 8 5 Syt3 lecture 8 6 Feedback to reduce tatic error r - Y = d e C() u + CP + CP R + P + CP D P() Aymptotic value of tep repone i given by lim t y(t) = lim Y () if ROC(Y ()) include the imaginary axi. For tep input R() = r and D() = d, we obtain y( ) = C()G() + C()G() r G() + + C()G() d r if + C()G() = S() >> y The role of integral feedback Integral action C() ha a pole at =. y( ) = r + d!! a general feature in feedback loop with contant external ignal: modulo tability (tatic regime may not exit!), block that contain a pole at = aymptotically force their input to zero. the property i independent of ytem parameter and linearity aumption! Syt3 lecture 8 7 Syt3 lecture 8 8
Integral feedback Internal model principle: for perfect tracking (rejection), the controller mut include a model of the reference (diturbance) ignal. For tep ignal, the model i. the controller pole at = caue a phae lag of 9deg in the loop tranfer integral action uually reduce the phae margin and increae overhoot Integral feedback a automatic bia adjutment compenate for tatic error if u(t) = Ke(t) + u ff u ff = + KP() With a PI control u(t) = Ke(t) + K I e, the bia i automatically adjuted. Syt3 lecture 8 9 Syt3 lecture 8 Derivative feedback PD control anticipate future error: u(t) = ke(t) + k d ė(t) = k(e(t) + T d ė(t)) ke(t + T d ) Think of pendulum model: u i a torque, y i a poition derivative feedback act a velocity feedback, i.e. a a friction force derivative control add a phae lead of +9deg in the loop tranfer uually increae the phae margin and damp the ocillation of the tep repone Derivative feedback amplifie noie the derivative action act a a noie amplifier require good enor the derivative i uually filtered: T d i replaced by limit the high-frequency gain amplification T d +T d /N to in a digital implementation, dicretizing the derivative feedback can be interpreted a filtering the derivative action i often not ued in proce control Syt3 lecture 8 Syt3 lecture 8 2
PI controller : + T i Set-point weighting G db G PID control i often implemented in the form T i -9 log ω log ω t u(t) = k(βr(t) y(t)) + k i (r y)(τ)dτ + k d (γṙ(t) ẏ(t)) Good for tatic performance but phae lag chooe T i ω B PD controller : + T d G db T d G +9 log ω log ω which correpond to a controller with two degree of freedom: U() = C()(R Y ) + F()R C() i tuned for diturbance rejection and F() i adjuted for reference tracking Phae lead i beneficial to tability margin but the cloed-loop bandwidth i increaed limit T d Syt3 lecture 8 3 Syt3 lecture 8 4 Antiwindup Due to actuator limitation, integral action i never implemented without an antiwindup mechanim. A general tructure for PID control The antiwindup reet the integral action when the input aturate. Syt3 lecture 8 5 Syt3 lecture 8 6
PID tuning control parameter tuning can be interpreted a pole placement for low-order model control parameter tuning can be interpreted a loop haping Special tuning method are model-free: the tuning i intead baed on a few open-loop experiment Many indutrial controller are mituned becaue they are tuned once forever adaptive control or auto-tuning method PI control of a firt order ytem Proce tranfer function Controller tranfer function P() = K τ + C() = k P + k I Cloed loop tranfer function from reference r to output y: Y () R() = CP + CP = K(k P + k I ) τ 2 + ( + Kk P ) + Kk I Cloed loop ytem of econd order; two parameter allow to place the pole of the cloed loop ytem Syt3 lecture 8 7 Syt3 lecture 8 8 PI control of a firt order ytem Example of deign: cancel the (low) proce pole, i.e. k I = τk P and place the cloed loop pole no overhoot, aignment of time contant, no tatic error. Perhap not the bet choice: one hould alo pay attention to the cloed loop tranfer function from the diturbance d to the output y: Y () D() = P + CP = K τ 2 + ( + Kk P ) + Kk I PD control of a econd order ytem Proce tranfer function Controller tranfer function b P() = 2 + a + a 2 C() = k P + k D Cloed loop tranfer function from reference r to output y: Y () R() = CP + CP = b(k D + k P ) 2 + (a + bk D ) + a 2 + bk P Cloed loop ytem of econd order; two parameter allow to place the pole of the cloed loop ytem. Syt3 lecture 8 9 Syt3 lecture 8 2
PD control of a econd order ytem The derivative action will damp the ocillation ( ζ) bk The cloed loop tatic gain i P bk P +a 2. But increaing k P may be harmful to bandwidth. The zero = k P /k D may caue coniderable overhoot if it i cloer to the origin than the dominant pole. PID control of a econd order ytem Proce tranfer function Controller tranfer function b G() = 2 + a + a 2 C() = k P + k D + k I Cloed loop tranfer function from reference r to output y: Y () R() = CG + CG = b(k D 2 + k P + k I ) 3 + (a + bk D ) 2 + (a 2 + bk P ) + bk I Cloed loop ytem of third order; three parameter allow to place the pole of the cloed loop ytem. Intability may occur for bad choice of control parameter! Syt3 lecture 8 2 Syt3 lecture 8 22 A zero in the tranfer function affect the tep repone H new = (z + )H old The location of a zero i NOT affected by feedback. In contrat, feedforward action change the location of zero. Interet of controller with two degree of freedom..2.8.6.4.2.2.4.5.5 2 2.5 3 3.5 4 new (t) = old (t) + d old z dt (t) zero in left half plane: increae the overhoot a it approache imaginary axi. zero in right-half plane: reponible for invere repone phenomenon. Syt3 lecture 8 23 Syt3 lecture 8 24
Loop haping with PI control Two parameter allow to fix one point on the Nyquit curve. C() = K + T i Chooe K to aign the croover frequency ω c with ufficient phae margin. Maximize T i under the contraint φ m φ min m. Note: A larger T i low down the diturbance rejection. Phae-lag compenation K T i+, > T i + = K T i + T i + = K ( + z), z p + p G db -6-9 ωt i ωt i Static gain i increaed by factor. Phae lag will not affect the phae margin if T i ω C. Synthei guideline: adjut K to aign ω C with ufficient phae margin. Evaluate the neceary reduction of tatic error to chooe ; Maximize T i without degrading the phae margin. Syt3 lecture 8 25 Syt3 lecture 8 26 Loop haping with phae-lead compenator C() = K(T d + ) T d +, < = PD control Low-pa filter = K +z +p, p z G db K K +9 ωt d φ max Limitation of derivative action at high frequency. ωt d Phae-lead compenation Ueful formula: log ω max = 2 φ max = arcin ω max = z p ( log ( )) + log T d T d ( ) + Synthei guideline: Chooe ω c to aign cloed-loop bandwidth. Evaluate the neceary phae lead at ω c (not more than 6 = 6) and adjut T d to place the maximal phae lead at ω C. Chooe K to have L(jω c ) =. Syt3 lecture 8 27 Syt3 lecture 8 28
Summary of lecture PID i widely ued in the indutry. Three parameter roughly correpond to three performance criteria: tatic performance (I), cloed-loop bandwidth (P), and tability margin (D). Implementation of PID control include antiwindup, filtering of the derivative action, and et-point weighting. Syt3 lecture 8 29