Chapter 8

Chapter 8 - 8-1 Chapter 8 Designing Feedback Controllers...

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Unformatted text preview: 8-1 Chapter 8 Designing Feedback Controllers for Motor Drives Exit 2001 by N. Mohan Print TOC " ! 8-2 Feedback Control Objectives desired (reference) + Σ signal - error error amplifier P P ! #" "U $ Electric Machine Mech Load output ! #" " $ Electrical System Mechanical System measured output signal ❏ Feedback control x makes system insensitive to disturbances and parameter variation ❏ Control Objectives x Zero steady-state error x Good dynamic response - fast - small overshoot Exit 2001 by N. Mohan Controller * X (s) + ∑ − E (s) Plant Gc ( s ) G p ( s) TOC X ( s) " ! 8-3 Definitions 100 ❏ Open loop GOL ( s ) = Gc ( s )G p ( s ) 50 GOL 0 ❏ Closed loop -50 GCL ( s ) = GOL ( s ) /(1 + GOL ( s )) ❏ Crossover frequency fc , ωc -50 -100 ∠GOL f c ,ω c phase margin -150 −180 o -200 ❏ Gain Margin ❏ Phase Margin > 45o for no oscillations 60o preferable ❏ Closed loop bandwidth ! fc 10 -2 10 -1 10 0 10 1 10 2 frequency GCL ( jω ) 0 dB −3dB ω BW desired high for fast response Exit 2001 by N. Mohan TOC " ! 8-4 Example GOL ( s ) = K OL / s ; KOL = 2 × 10 3 GOL ( s ) x* (t ) ( dB ) ω c = 2 × 103 0 log ω 0.632 τ x(t ) GCL ( s ) ( dB ) 0 −20 dB / decade log ω closed loop step response Exit 2001 by N. Mohan TOC " ! 8-5 Cascaded Control speed* position* + − Position controller + − torque* Speed controller + torque Torque controller − Electrical System Mech System speed 1 s position TOC " torque(current) speed position ❏ Torque loop : fastest ❏ Speed loop : slower ❏ Position loop : slowest Exit 2001 by N. Mohan ! 8-6 Steps in Designing the Controller ❏ Assume system is linear about the steady state operating point ➝ design controller using Linear Control Theory ❏ Simulate design under large signal conditions and “tweak” controller as necessary System representation for small signal analysis ❏ Assume x Steady state system operating point = 0 x Highest bandwidth at least an order of magnitude lower than switching frequency ➝ neglect switching frequency components Exit 2001 by N. Mohan TOC " ! 8-7 Averaged Representation of the PPU id (t ) id idA idB iA + Vd − A + B v a − iB − q A (t ) ia (t ) + + i A = ia iB = − ia N vcontrol (t ) ia ea + Vd 1 − + d (t ) va (t ) ea − − vcontrol (t ) 1 ˆ Vtri d (t ) = d A( t) − d B )t ( qB ( t ) va (t ) = k PWM vc (t ) Va ( s ) Vc ( s ) k PWM Va ( s ) = k PWM Vc ( s ) Exit 2001 by N. Mohan TOC " ! 8-8 Modeling of DC Machines and Mechanical Load Combinations d ia (t ) dt ea (t ) = k E ω m (t ) va (t ) = ea (t ) + Ra ia (t ) + La T ia = em kT + Ra va Va ( s ) = Ea ( s ) + ( Ra + s La ) I a ( s ) − La + Tem ea = k E ω m _ ωm TL JM V ( s ) − Ea ( s ) ⇒ I a ( s) = a ( Ra + s La ) JL ; Ea ( s ) = k E ω m ( s ) TL ( s ) Tem ( s ) = kT I a ( s ) ωm (s) = Exit 2001 by N. Mohan Tem ( s ) sJ eq Va ( s ) + − I a (s) 1 Ra + sLa − + kT Tem ( s ) ωm (s ) 1 sJ eq kE TOC " ! 8-9 PI Controller kp vc, p ( s ) G ( s) p %""&""' X * (s) + E (s) − ki s vc,i ( s ) + + X ( s) vcontrol ( s ) !""" #""" $ Gc ( s ) vc ( s ) ki ki s = k p + = 1+ E (s) s s ki / k p ❏ Proportional-Integral (PI) Controller x In the torque and speed loops, proportional control without integral control input leads to steady-state error Exit 2001 by N. Mohan TOC " ! 8-10 Controller Design ❏ Procedure x Design torque loop (fastest) first x Design speed loop assuming torque loop to be ideal x Design position loop (slowest) assuming speed loop to be ideal Exit 2001 by N. Mohan TOC " ! 8-11 Design of the Torque (Current) Loop Simplifying assumptions * I a (s) + Σ − PI 1 / Ra 1 + sτ e V ( s) k PWM a + Σ − I a ( s) kT Tem − Σ + * I a ( s) ❏ Interleaved Σ + loops redrawn − as nested loops Ia (s) ❏ Assuming J high enough, inner loop can be ignored ⇓ PI k PWM Va ( s ) + Σ − 1 / Ra 1 + sτ e I a ( s) kT Tem 1 sJ ωm k E kT sJ ⇓ kiI s 1+ s kiI / k pI * I a ( s) + Σ − k PWM Va ( s ) 1 / Ra 1 + sτ e I a ( s) I a ( s) kiI s 1 / Ra GI ,OL ( s ) = 1 + k PWM # s kiI / k p ! $ #"" PPU 1 + sτ e !#$ !"" $ PI controller 2001 by N. Mohan ωm 1 sJ kE I a (s) Exit TL motor TOC " ! 8-12 Design of the Torque (Current) Loop Selecting Parameters * I a ( s) + Σ − kiI s s 1+ kiI / k pI k PWM Va ( s ) I a ( s) 1 / Ra 1 + sτ e I a ( s) k pI ❏ Select zero of PI to cancel motor pole ; ⇒ GI ,OL = k I ,OL s kiI k k ; ki,OL = iI PWM Ra =τ e ❏ Choose kiI to achieve desired cross-over frequency 0 Magnitude (dB) Magnitude (dB) 60 40 20 0 -20 -40 0 10 10 1 3 2001 by N. Mohan 4 10 1 2 3 10 10 Frequency (Hz) -20 10 1 0 -90 10 -10 -30 0 10 5 -90.5 open loop Exit 10 P has e (deg) P has e (deg) -89 2 10 10 Frequency (Hz) -89.5 -91 0 10 k I ,OL = ωCI 10 4 10 5 2 3 2 3 10 10 Frequency (Hz) 10 4 10 5 -50 -100 0 10 10 1 10 10 Frequency (Hz) 10 4 10 5 closed loop TOC " ! 8-13 Design of the Speed Loop * ω m (s) + * I a (s) PI 1 I a ( s) Tem ( s ) kT − ω m (s) 1 Js ω m (s) ❏ Assume current loop to be ideal ➝ represent by unity ❏ Choose crossover frequency ωCω an order of magnitude lower than ωCI ❏ Choose a reasonable phase margin φ PM ,ω open loop closed loop 20 Magnitude (dB) Magnitude (dB) 150 100 50 0 -50 -1 10 0 10 1 10 2001 by N. Mohan 4 10 0 1 10 2 3 10 10 Frequency (Hz) 0 10 1 10 0 -150 10 -40 -60 -1 10 5 10 -100 -200 -1 10 Exit 3 Phase (deg) Phase (deg) -50 2 10 10 Frequency (Hz) 0 -20 4 10 5 10 2 3 2 3 10 10 Frequency (Hz) 4 10 5 10 -50 -100 -1 10 0 10 1 10 10 10 Frequency (Hz) 4 10 5 10 TOC " ! 8-14 Design of the Position Loop * θ m (s) + * ω m (s) k pθ 1 ωm (s) θ m (s) 1 s − ❏ Assume speed loop to be ideal ❏ Proportional gain ( k Pθ ) alone is adequate due to presence of k Pθ pure integrator G = ⇒ k =ω θ ,OL Pθ s 0 Magnitude (dB) Magnitude (dB) 50 0 -50 -1 10 10 0 2 2001 by N. Mohan 3 10 0 1 2 10 10 Frequency (Hz) -40 10 0 0 -90.5 10 -20 -60 -1 10 4 -90 open loop Exit 10 P has e (deg) P has e (deg) -89 1 10 10 Frequency (Hz) -89.5 -91 -1 10 CP 10 3 10 4 1 2 1 2 10 10 Frequency (Hz) 10 3 10 4 -50 -100 -1 10 10 0 10 10 Frequency (Hz) 10 3 10 4 closed loop TOC " ! 8-15 Further Issues ❏ Feed-forward: To improve dynamic response Process computer position* torque* ff speed* ff + Position controller + Σ − + Speed controller Torque controller − torque* speed position 1 s torque(current) * Electrical System Mech System ˆ +Vtri + Σ +Vd speed position ❏ Effect of limits - nonlinearity * I a (s) + kiI s − s 1+ kiI / k pI −Vd ˆ −Vtri kp ❏ Anti-windup integration - suspend integration when output saturates 2001 by N. Mohan cp max input co + co ' ki 0 Exit I a ( s) 1 / Ra 1 + sτ e k PWM min ci max − | co ' | TOC " ! 8-16 Summary ❏ What are the various blocks of a motor drive? ❏ What is a cascaded control and what are its advantages? ❏ Draw the average models of a PWM controller and a dc-dc converter. ❏ Draw the dc-motor equivalent circuit and its representation in Laplace domain. Is this representation linear? ❏ What is the transfer function of a proportional-integral (PI) controller? ❏ Draw the block diagram of the torque loop. ❏ What is the rationale for neglecting the feedback from speed in the torque loop? ❏ Draw the simplified block diagram of the torque loop. ❏ Describe the procedure for designing the PI controller in the torque loop. Exit 2001 by N. Mohan TOC " ! 8-17 Summary ❏ How would we have designed the PI controller of the torque loop if the effect of the speed were not ignored? ❏ What allows us to approximate the closed torque loop by unity in the speed loop? ❏ What is the procedure for designing the PI controller in the speed loop? ❏ How would we have designed the PI controller in the speed loop if the closed torque-loop were not approximated by unity? ❏ Draw the position-loop block diagram. ❏ Why do we only need a P controller in the position loop? ❏ What allows us to approximate the closed speed loop by unity in the position loop? ❏ Describe the design procedure for determining the controller in the position loop Exit 2001 by N. Mohan TOC " ! 8-18 Summary ❏ How would we have designed the position controller if the closed speed loop were not approximated by unity? ❏ Draw the block diagram with feed-forward. What are its advantages? ❏ Why are limiters used and what are their effects? ❏ What is the integrator windup and how can it be avoided? Exit 2001 by N. Mohan TOC " ! ...
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This note was uploaded on 02/06/2012 for the course EE 4002 taught by Professor Scalzo during the Fall '06 term at LSU.

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