AERATOR MIXING STATION Steady State, Step Response Analysis, Sine and Relay Analysis, Root Locus Green Team: Marc Labrie Matt Baltimore Michael Newman Michael Sherrit University of Tennessee at Chattanooga March 29, 2011 ENGR 3280L
OVERVIEW System Overview SSOC Analysis Step Response Analysis FOPDT Modeling FOPDT Analysis Sine Analysis Relay Analysis Root Locus plot Conclusions
SYSTEM DIAGRAM
SCHEMATIC OF THE SYSTEM Speed Recording Controller Speed Transmitter Speed Controller
EXPERIMENTAL DATA ANALYSIS 75 rpm 6 rpm 75+/- 12 rpm
STEADY STATE OPERATING CURVE Operating Ranges 0-25% lower 25-50% lower mid 50-75% upper mid 75-100% upper
STEP RESPONSE EXPERIMENTAL DATA UP
STEP RESPONSE DATA UP CONTINUED To=.1sec (Black Lines) Tau=.4s (Green Lines)
STEP RESPONSE DATA DOWN
STEP RESPONSE DATA UP CONTINUED To=.1s (Black Lines) Tau=.4s (Green Lines)
RESULTS
CONTINUED 0.12 Step Response Dead Time, t0 (Dead Time) 0.1 0.08 Time (s) 0.06 0.04 0.02 0 0%-25% Up 0%-25% Down 25%-50% Up 25%-50% Down 50%-75% Up 50%-75%Down 75%-100% Up 75%-100% Down
CONTINUED
FOPDT THEORY FOPDT Transfer Function For step functions the Manipulated variable m(t) and the Output c(t) are:
Model Output
Model Output
FOPDT MODEL Model Equation System Output c(t) can be modeled by changing K,τ,to, and td. Parameters for 25-50% Step up td (s) A (%) K (RPM/%) to (s) τ(s) Inbl. (%) Outbl. (RPM) 5.42 25 17.4 0.12 0.20 25 424 Parameters for 25-50% Step-down td (s) A (%) K (RPM/%) to (s) τ(s) Inbl. (%) Outbl. (RPM) 4.15 25 17.4 0.12 0.20 50 861
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 0-25% 25-50% 50-75% 75-100%
Dead Time (sec) Up Experimental Up Model Down Experimental Down Model 0.14 0.12 0.1 Time (sec) 0.08 0.06 0.04 0.02 0 0-25% 25-50% 50-75% 75-100%
Time Constant, τ(sec) Up Experimental Up Model Down Experimental Down Model 0.45 0.4 0.35 0.3 Time (s) 0.25 0.2 0.15 0.1 0.05 0 0-25% 25-50% 50-75% 75-100%
SINE RESPONSE Input Speed (RPM)
Frequency Response (f=0.5) 1300 80 Output (RPM) 1200 1100 1000 900 800 700 600 Ar = 15.69 ± 0.03 PA = -38 ± 5 2 3 4 5 6 7 8 Time (s) Output(RPM) Input Value(%) MWN 3/01/2011 75 70 65 60 55 50 45 40 35 Input (%)
Amplitude Ratio (RPM/%) K = 17.6 RPM/% τ= 0.17 s t 0 = 0.3 s fu = 3.2 Hz Kcu= 0.4 RPM/% Order = 2 Amplitude Ratio (Ar) A 1 0.01 0.1 1 10 Frequency (Hz) 100 10-40 -60-80 -100-120 -140-160 -180-200 -220-240 MWN 3/01/2011 0 0.01 0.1-20 1 10 Phase Angle (degrees) MWN 3/01/2011 Phase Angle (PA) Frequency (Hz)
Sine Response Theory
Amplitude Ratio Frequency (Hz) 0.01 0.1 1 10 100 Experimental MWN Model 3/02/2011 K = 17.4 RPM/% τ= 0.2 s t 0 = 0.1 s 10 Amplitude Ratio(RPM/%) 1 Phase Angle Frequency (Hz) 0.01 0.1 1 10 0 Experimental Model MWN 3/02/2011-60 -120-180 -240 Phase Angle (degrees)
Gain, K (RPM/%) Experimental Model K (RPM/%) 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 50% - 74% 75% - 99% Input (%)
Dead Time, t 0 (s) 0.35 Experimental Model 0.3 0.25 t 0 (s) 0.2 0.15 0.1 0.05 0 50% - 74% 75% - 99% Input (%)
Time Constant, τ (s) Experimental Model 0.2 0.15 Tau (s) 0.1 0.05 0 50% - 74% 75% - 99% Input (%)
Ultimate Frequency, fu (Hz) Experimental 4 3.5 3 2.5 fu (Hz) 2 1.5 1 0.5 0 50% - 74% 75% - 99% Input (%)
Ultimate Controller Gain, K cu (%/RPM) Experimental 0.45 0.4 0.35 0.3 K cu (%/RPM) 0.25 0.2 0.15 0.1 0.05 0 50% - 74% 75% - 99% Input (%)
50-100% Relay Response Trial 1 110 1550 100 1450 1350 90 1250 Input (%) 80 70 1150 1050 950 Output (RPM) 60 850 50 750 650 40 550 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 Time (s)
20 K (RPM/%) 18 16 14 12 10 8 6 4 2 0 Frequency Response Relay Response
0.5 τ (sec) 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Frequency Response Relay Response
0.12 to (sec) 0.1 0.08 0.06 0.04 0.02 0 Frequency Response Relay Response
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 = 0
ROOT LOCUS MODEL ROOT LOCUS PLOT 20 15 10 IMAGINARY AXIS 5 0-14 -12-10 -8-6 -4-2 0 2 4-5 -10-15 REAL AXIS -20
KCU COMPARISON Kcu 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 MSS 3/27/11 Root locus Relay Frequency 0 0-25% 25-50% 50-75% 75-100%
FU COMPARISON Fu Root locus 6 5 MSS 3/27/11 Relay Frequency 4 3 2 1 0 0-25% 25-50% 50-75% 75-100%
USEFUL KCU RANGE GREEN TEAM ROOT LOCUS PLOT MWN 3/27/11 Kc 1/4 40 30 K cu IMAGINARY AXIS 0-25 -20-15 -10-5 0 5 10 15 Over-damped Region Kc 1/500 Kc 1/10 Kcd REAL AXIS 20 10-10 -20-30 -40 Underdamped Region Kcu= 0.29 RPM/% Kc 1/4 = 0.19 RPM/% Kc 1/10 = 0.14 RPM/% Kc 1/500 = 0.06 RPM/% Kcd= 0.02 RPM/%
CONCLUSIONS The SSOC shows a linear correlation between input power and voltage The operating range of the system is 0-100% The gain (K) of the system calculated is equal to the slope of the SSOC and experimental vs. model results match closely. The to (dead time) was consistent throughout the experiments
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) 0.2 t 0 (sec) 0.1
CONCLUSIONS CONTINUED The Kcu from the Root Locus model agreed with what was found from previous experiments. The Fu from the Root Locus model did not agree with what was found from previous experiments. The useful range of Kcu for the speed system varies between 0.06 RPM/% and 0.29 RPM/%.