AERATOR MIXING STATION Green Team: Marc Labrie Matt Baltimore Michael Newman Michael Sherrit University of Tennessee at Chattanooga April 13, 211 ENGR 328L
OVERVIEW System Overview SSOC Analysis Step Response Analysis FOPDT Modeling FOPDT Analysis Sine and Relay Experiments Root Locus model Proportional-only Controller Experiment Proportional Integral Controller Design Conclusions
SCHEMATIC AND BLOCK DIAGRAM OF THE SYSTEM Speed Controller Speed Recording Controller Speed Transmitter
EXPERIMENTAL DATA ANALYSIS 75 rpm 6 rpm 75+/- 12 rpm
STEADY STATE OPERATING CURVE Operating Ranges -25% lower 25-5% lower mid 5-75% upper mid 75-1% upper
CONCLUSION Linear in the operating range of -1% corresponding to an output of -174RPM Small uncertainty
STEP RESPONSE EXPERIMENTAL DATA UP
STEP RESPONSE DATA UP CONTINUED To=.1sec (Black Lines) Tau=.4s (Green Lines)
RESULTS
CONTINUED.12 Step Response Dead Time, t (Dead Time).1.8 Time (s).6.4.2 %-25% Up %-25% Down 25%-5% Up 25%-5% Down 5%-75% Up 5%-75%Down 75%-1% Up 75%-1% Down
CONTINUED
Step Response 5% to 95% 18 16 14 12 Output(RPM) Input Value(%) 1 9 8 7 Output (%) 1 8 6 5 4 Input (%) 6 4 2 Dc = 1549 RPM Dm = 9 K= 17.2 RPM/% t =.1 s τ =.7 s MWN 4/17/11 3 2 1 9 9.5 1 1.5 11 11.5 12 12.5 13 Time (s)
Step Response 95% to 5% 18 16 14 MWN 4/17/11 Output(RPM) Input Value(%) 1 9 8 12 7 Output (RPM) 1 8 6 4 Dc = 1545 RPM Dm = 9 K= 17.2 RPM/% t =.1 s τ =.6 s 6 5 4 3 2 Input (%) 2 1 9 9.5 1 1.5 11 11.5 12 12.5 13 Time (s)
FOPDT THEORY FOPDT Transfer Function For step functions the Manipulated variable m(t) and the Output c(t) are:
Model Output
Step Response Model 5%-95% 18 1 16 Output(RPM) 9 14 Output (Model) 8 12 Input Value(%) 7 Output (RPM) 1 8 6 4 MWN 4/17/11 Input (Model) K= 17.3 RPM/% t =.1 s τ =.7 s 6 5 4 3 2 Input (%) 2 1 5 6 7 8 9 1 11 12 13 14 15 Time (s)
Step Response Model 5% to 95% 18 16 14 MWN 4/17/11 1 9 8 Output (%) 12 1 8 K= 17.4 RPM/% t =.1 s τ =.6 s 7 6 5 4 Input (%) 6 4 Output(RPM) Output (Model) 3 2 2 Input Value(%) 1 Input (Model) 5 6 7 8 9 1 11 12 13 14 15 Time (s)
Gain, K (RPM/%) Up Experimental Up Model Down Experimental Down Model 18 17.8 17.6 17.4 K (RPM/%) 17.2 17 16.8 16.6 16.4 16.2-25% 25-5% 5-75% 75-1% 5%-95% Input
Dead Time, to(sec) Up Experimental Up Model Down Experimental Down Model.14.12.1 K (RPM/%).8.6.4.2-25% 25-5% 5-75% 75-1% 5%-95% Input
Time Constant, τ(sec) Up Experimental Up Model Down Experimental Down Model.8.7.6.5 K (RPM/%).4.3.2.1-25% 25-5% 5-75% 75-1% Input
CONCLUSIONS K and t from the model and graphical method closely agreed while there was a larger difference in tau for the two methods.
SINE RESPONSE Power Input Speed (RPM)
Frequency Response (f=.5 Hz) 13 8 Output (RPM) 12 11 1 9 8 7 6 Ar = 15.69 ±.3 PA = -38 ± 5 2 3 4 5 6 7 8 Time (s) Output(RPM) Input Value(%) MWN 3/1/211 75 7 65 6 55 5 45 4 35 Input (%)
Amplitude Ratio 1 75-1% range AR A 1 1.1.1 1 1 Frequency (Hz) -2-4 -6-8 -1-12 -14-16 -18-2 -22-24 -26-28 -3-32 -34-36 -38 Frequency (Hz).1.1 1 1 PA (degrees) MSS 2/27/211 K = 17.1 RPM/% τ =.15 s t =.17 s fu = 3.4 Hz Kcu =.37 %/RPM Order = 2 Phase Angle MSS 2/26/211
Amplitude Ratio.1.1 Frequency (Hz) 1 1 MWN MSS 1 Experimental (5%-74%) Model (5%-74%) Experimental (75%-99%) 1 Amplitude Ratio Model (75%-99%).1.1 Frequency (Hz) 1 1 MWN MSS Phase Angle -6 1-12 Experimental (5%-74%) Model (5%-74%) Experimental (75%-99%) Model (75%-99%) 75-1% range K= 17.1 RPM/% t=.8s tau=.17s 5-75% range K= 17.3 RPM/% t=.9s tau=.16s -18-24 -3-36 -42 Phase Angle
Gain, K (RPM/%) Experimental Model K (RPM/%) 18 17 16 15 14 13 12 11 1 9 8 7 6 5 4 3 2 1 5% - 74% 75% - 99% Input (%)
Dead Time, t (s).35.3.25 Experimental Model t (s).2.15.1.5 5% - 74% 75% - 99% Input (%)
Time Constant, τ (s).2 Experimental Model.15 Tau (s).1.5 5% - 74% 75% - 99% Input (%)
SINE WAVE.1HZ sine output 5% baseline 45% amplitude 18 1 output rpm 16 14 12 1 8 Output(RPM) Input Value(%) 9 8 7 6 5 4 input% Sine wave of whole axis experimental model K 15.7 17.6 to.1.13 ta u.8.42 6 3 4 2 2 1 5 1 15 2 25 time
5-1% Relay Response Trial 1 11 155 1 145 135 9 125 Input (%) 8 7 115 15 95 Output (RPM) 6 85 5 75 65 4 55.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 Time (s)
2 K (RPM/%) 18 16 14 K (RPM/%) 12 1 8 6 4 2 Frequency Response Relay Response
.5 Tau, τ (sec).45.4.35.3 Tau (s).25.2.15.1.5 Frequency Response Relay Response
.12 Dead Time, to (sec).1.8 t (sec).6.4.2 Frequency Response Relay Response
CONCLUSION Overall, the frequency experiment more closely agrees with what was found for K, t, and tau in previous experiments than those from the relay experiment.
TRANSFER FUNCTION The transfer function for an FOPDT system is After substituting Pade s approximation and simplifying, the transfer function becomes
TRANSFER FUNCTION (CONT D) The transfer function for a proportional feedback controller is For an FOPDT system with proportional control, the OLTF is And the characteristic equation becomes 1 + OLTF =
ROOT LOCUS MODEL 5-75% 4 MWN 3/27/11 Kc 1/1 Kc 1/4 3 2 K cu IMAGINARY AXIS -25-2 -15-1 -5 5 1 15 Kcd Kc 1/5 1-1 -2-3 -4 Kcu=.26 %/RPM Kc 1/4 =.17 %/RPM Kc 1/1 =.13 %/RPM Kc 1/5 =.36 %/RPM Kcd=.17 %/RPM REAL AXIS
KCU COMPARISON Kcu (%/RPM) 1.9.8.7.6.5.4.3.2.1 MSS 3/27/11 Root locus Relay Frequency -25% 25-5% 5-75% 75-1%
FU COMPARISON Fu (Hz) Root locus 6 5 4 Relay Frequency MSS 3/27/11 3 2 1-25% 25-5% 5-75% 75-1%
SYSTEM CONTROLLER RANGE GREEN TEAM ROOT LOCUS PLOT MWN 3/27/11 Kc 1/1 Kc 1/4 4 3 2 K cu IMAGINARY AXIS Kc 1/5-25 -2-15 -1-5 5 1 15 1 Under damped Region Kcd -1 Over-damped Region -2-3 -4 Kcu =.28 %/RPM Kc 1/4 =.18 %/RPM Kc 1/1 =.14 %/RPM Kc 1/5 =.61 %/RPM Kcd =.18 %/RPM REAL AXIS
CONTROLLER GAIN EXPERIMENT FOR THE ULTIMATE GAIN Experimental Kcu 18 175 17 MSS 4-9-11 12 1 165 8 Output (RPM) 16 155 15 Set-P=165RPM M bar=95% Step=-25RPM AR=.52 Offset=34RPM Overshoot=6.% Settling time=n/a Kc=.3%/RPM Output(RPM) SET-P(RPM) Input Value(%) 6 4 Input (%) 145 14 OFFSET 2 135 9 11 13 15 17 19 Time (s)
CONTROLLER GAIN EXPERIMENT FOR ¼ DECAY Experimental 1/4 Decay 169 164 Set-P=165RPM M bar=95% Step=-25RPM MSS 4-9-11 12 1 Output (RPM) 159 154 149 AR=.24 Offset=6RPM Overshoot=3.8% Settling time=1.2s Kc=.16%/RPM Output(RPM) SET-P(RPM) Input Value(%) 8 6 4 Input (%) 144 OFFSET 2 139 9 9.5 1 1.5 11 11.5 12 12.5 13 Time (s)
CONTROLLER GAIN EXPERIMENT FOR 1/1 DECAY Exeperimental 1/1 Decay 17 165 Set-P=165RPM M bar=95% Step=-25RPM MSS 4-9-11 12 1 Output (RPM) 16 155 15 145 14 AR=.6 Offset=82RPM Overshoot=1.7% Settling time=1s Kc=.1%/RPM OFFSET Output(RPM) SET-P(RPM) Input Value(%) 8 6 4 2 Input (%) 135 9.5 9.7 9.9 1.1 1.3 1.5 1.7 1.9 11.1 11.3 11.5 Time (s)
CONTROLLER GAIN EXPERIMENT FOR 1/5 DECAY Experimental 1/5 Decay 17 165 MSS 4-9-11 12 1 16 Output (RPM) 155 15 145 14 Set-P=165RPM M bar=95% Step=-25RPM Offset=11RPM Overshoot=.48% Settling time=.5s Kc=.7%/RPM Output(RPM) SET-P(RPM) Input Value(%) OFFSET 8 6 4 2 Input (%) 135 9 9.2 9.4 9.6 9.8 1 1.2 1.4 1.6 1.8 11 Time (s)
CONTROLLER GAIN EXPERIMENT FOR CRITICAL DAMPING Experimental Kcd 17 165 16 Set-P=165RPM M bar=95% Step=-25RPM MSS 4-9-11 12 1 98 Output (RPM) 155 15 145 Offset=165RPM Overshoot=% Settling time=.5s Kc=.2%/RPM OFFSET Output(RPM) SET-P(RPM) Input Value(%) 96 94 92 Input (%) 14 9 135 88 9 9.5 1 1.5 11 11.5 12 Time (s)
-25% REGION EXPERIMENT Decay for 12.5% and 1 rpm step up 35 25 3 2 25 output rom 2 15 Output(RPM) 15 1 input % 1 SET-P(RPM) 5 Input Value(%) ML Team Green team Kc=.5 Step up= 1rpm 5 9 9.5 1 1.5 11 time
KC=.1
KC RESULTS AND COMPARISON Kcu =.28 %/RPM Kc 1/4 =.18 %/RPM Kc 1/1 =.14 %/RPM Kc 1/5 =.61 %/RPM Kcd =.18 %/RPM
Kcu Across All Regions 18 16 14 12 AR=.5 Offset = 113RPM Overshoot= 8% Settling time=n/a Kc=.29%/RPM 12 1 8 Output (RPM) 1 8 MWN 4/17/11 6 Input (%) 6 4 4 2 Input Value(%) 9 9.5 1 1.5 11 11.5 12 12.5 13 Time (s) Kcu=.28 %/RPM Set Pt = 345 RPM Step = 12 RPM Output(RPM) SET-P(RPM) 2
PROPORTIONAL INTEGRAL CONTROLLER
τ I = 3.33t Tau I =.3 (5%-75%) REAL -2-15 -1-5 5 Kqd =.16 Kcu =.24 3 Kcd =.22 K5 =.55 K1 =.12 fu = 4. Hz 2 1-1 IMAGINARY MWN 4/16/11 K = 17.4 RPM/% τ =.16 s t =.9 s -2-3
Tau I = 1 (5%-75%) REAL -2-15 -1-5 5 MWN 4/16/11 Kcd.18 K5 =.52 K = 17.4 RPM/% τ =.16 s t =.9 s K1 =.13 Kqd =.17 Kcu =.25 fu = 4.3 Hz 4 3 2 1-1 -2-3 -4 IMAGINARY
Tau I =.7 (5%-75%) REAL -5-4 -3-2 -1 1 2 K5 =.12 Kcd =.7 MWN 4/16/11 K1 =.36 Kqd =.57 Kcu =.11 fu = 2.4 Hz 2 15 1 5-5 -1-15 -2 IMAGINARY
.16.14 Lower τ I Values %-25% (Tau I =.8) 25%-5% (Tau I =.83) 5%-75% (Tau I =.7) 75%-1% (Tau I =.68).12.1 %/RPM.8.6.4.2 Kcd Kc5 Kc1 Kqd Kcu
Middle τ I Values.3.25 %-25% (Tau I =.3) 25%-5% (Tau I =.33) 5%-75% (Tau I =.3) 75%-1% (Tau I =.27).2 %/RPM.15.1.5 Kcd Kc5 Kc1 Kqd Kcu
Large τ I Values.35.3 %-25% (Tau I = 1.3) 25%-5% (Tau I = 1.3) 5%-75% (Tau I = 1) 75%-1% (Tau I = 1.8).25.2 %/RPM.15.1.5 Kcd Kc5 Kc1 Kqd Kcu
CONCLUSIONS The SSOC shows a linear correlation between input power and speed. The operating range of the system is -1% corresponding to an output range of -174 RPM. The gain (K) of the system calculated is equal to the slope of the SSOC and experimental vs. model results match closely.
CONCLUSIONS CONTINUED From the experimental Bode plot the order is 2. The sine response model shows the best estimate of the FOPDT parameters Best FOPDT Estimate K (RPM/%) 17.3 τ (sec).2 t (sec).1
CONCLUSIONS CONTINUED The Kcu of about.3 %/RPM from the Root Locus model agreed with what was found from previous experiments. The ultimate frequency (Fu) from the Root Locus model did not agree with what was found from previous experiments. Experimental results allow for controller gain recommendations to customers. Recommended Controller Gains and corresponding Tau I values found in the table below. Response Tau I =.8 Tau I =.3 Tau I = 1 Value Value Symbol (%/rpm) Response Symbol (%/rpm) Response Symbol Value (%/rpm) Critical Kcd.6 Critical Kcd.28 Critical Kcd.21 1/5 Kc5.13 1/5 Kc5.6 1/5 Kc5.58 1/1 Kc1.41 1/1 Kc1.13 1/1 Kc1.14 1/4 Kqd.63 1/4 Kqd.17 1/4 Kqd.19 Ultimate Kcu.126 Ultimate Kcu.25 Ultimate Kcu.27
WHAT WE CAN GIVE THE CUSTOMER WITH A PROPORTIONAL-INTEGRAL CONTROLLER Quarter Decay: Kc=.17 %/RPM, Tau I =.3 18 11 16 1 14 9 Output (RPM) 12 8 7 Input (%) 1 6 8 MWN 4/18/11 Output(RPM) SET-P(RPM) Input Value(%) 6 14 14.5 15 15.5 16 16.5 17 17.5 18 5 4 Time (s)
ASSUMPTION No offset Least reset windup Fastest to reach set point Least oscillation
PI Controller 18 17 16 Set Pt. = 11 RPM ΔSet Pt. = 25 RPM Tau I =.3 s Kc =.55 %/RPM K5 From P Controller MWN 4/22/11 1 95 9 15 Output (RPM) 14 13 12 85 8 75 Input (%) 11 1 9 Output(RPM) SET-P(RPM) Input Value(%) 7 65 8 6 5 6 7 8 9 1 11 12 13 14 15 Axis Title
ALL REGIONS INCLUDED Optimized Proportional Integral Controller 2 18 16 14 Output(RPM) SET-P(RPM) Input Value(%) 1 9 8 7 output rpm 12 1 8 6 4 2 ML Team Green Kc=.55 TI=.3 Set point = 1 to 165 6 5 4 3 2 1 innput % 5 6 7 8 9 1 11 12 13 14 15 Time (s)
CONCLUSIONS Proportional Integral controller has best output curve PI has no offset Best values for Proportional TI=.3 or higher Kc=.55 %/rpm Kc is for 1/5 th decay in proportional and acts with little overshoot