UTC Engineering 329 Proportional Controller Design for Speed System By John Beverly Green Team John Beverly Keith Skiles John Barker 24 Mar 2006
Introdution This experiment is intended test the variable speed mor that powers the aerar at the waste-water treatment plant in Chattanooga TN and design a proportional ontroller. The aerar mixes the water enable steady growth of the bateria that onsume the unwanted waste in the water. The management of the faility require the output of the mixer be variable. Previous tests developed the Steady State Operating Curve (SSOC) from whih the system gain was determined. The Step Response test was also previously performed and baseline values for the system gain, dead time and time onstant were determined. This test will use the data from the previous tests as a basis determine the ontroller gain and the effets it has on the system. This data an then be used tune the system ahieve the desired response for any given input. This report inludes Bakground and Theory, the Proedure used ollet the empirial data, the Results of the tests and the mathematial modeling, a Disussion of the test findings and a Conlusion with Reommendations. An Appendix ontaining illustrations and referenes is loated in the bak. The Bakground and Theory setion is broken down in explanations of the previous SSOC and Step Response tests along with the results of the experiments and the tests performed in this experiment. (Frequeny Response, Root Lous and Proportional Control and Offset) Likewise, the Proedure setion lists eah test that was performed, inluding the previous tests, and is broken in eah setion.
Bakground and Theory The following desription refers the diagram below. (Figure 1) Power input the pump drive mor, (M-247) is ontrolled by the ontroller (SRC 247). Power input is adjusted manually provide a perentage of tal power the mor drive. The power input is provided by a 3-phase drive. (SCZ 247) As the input is adjusted, the mor (M- 247) hanges speed aordingly. The mor speed is monired by an optial sensor (ST 247) whih reports the speed in revolutions per minute. (rpm) As the mor speed hanges the mixer/aerar speed also hanges eventually affeting the oxygenation of the waste water. (Note: The gear-box ontains a gear set that redues the speed of the mixer 1/100 of the mor speed. Normal operating range is between 2 and 17 rpm of the mixer and 200 1700 rpm for the mor.) Figure 1 Shemati of the Aerar/Mixer
Figure 2 is a blok diagram of the Aerar/Mixer System. The manipulated (manual) input is represented by m(t). The Aerar Mixer in the red box is the transfer funtion. The output in rpm is represented by (t). Figure 2 Blok Diagram of the Aerar/Mixer SSOC The graph below illustrates the Steady State Operating Curve (SSOC). It is the response of the mor in revolutions per minute (rpm) a given input power (%). Several tests were performed through a range of input power develop the data neessary form the urve. It allows the output be predited for any given input. The error bars indiate a onfidene range of 95%. The slope (system gain (K)) is onstant throughout the range and is 17.3 rpm/%. (See Table 1 in the Appendix setion of this report for a omplete list of the data.)
Steady State Operating Curve 1800 High Speed Output (RPM) 1600 1400 1200 1000 800 600 400 200 Low Mid 0 0 20 40 60 80 100 Mor Input (%) Figure 3 Steady State Operating Curve Step Response The effet of a stepped input was measured and modeled mathematially. (Step Response Test) The First Order Plus Dead Time (FOPDT) method was used develop the model. Figure 4 is an example of the step response and the derived mathemati model. The model was determined aurately represent the output at any input in the operating range (200-1700 rpm). The equation used model the response urve is shown below (Equation 1) where A is the amplitude of the step input, t is time, t d is the dead time (lag time between input and output), t o is the lag of the model step, τ is the time onstant (length of time reah steady state), K is the system gain at steady state. The
variables A, K and t o were adjusted manually until the urves mathed the shape of the output response urve. (The blue model urve lays on p of the magenta urve and mathes the shape Figure 4) t t o t d Equation 1: ( ) (1 ) τ Y t = AK e The results of the Step Response are as follows: (also Table 2 in Appendix) K = 17.1 rpm/% t o =.14 se τ =.19 se Step Up 50-70% 1300 75 1200 70 1100 Speed (RPM) 1000 900 Output Model Output Input Model Input 65 60 Input (%) 55 800 700 50 600 4 5 6 7 8 9 45 Tim e(s) Figure 4 Step Response
Frequeny Response The aerar mor was subjeted a sine wave input at various frequenies and input power levels throughout the operating range. The response of the mor was reorded and plotted. Figure 5 is a sample of one of the plots. Two important piees of information were gleaned from the sine wave plots; phase angle and amplitude ratio. The phase angle is the degree of lag between the output and input. It was determined using Equation 2, where t is the distane between the peaks of the input and output and T is the distane between the peaks of the input. (See Figure 5) The amplitude ratio is the height of the output sine wave ( Δ ) divided by the height of the input sine wave ( Δ m). (Equation 3) t T 0 Equation 2: PhaseAngle ( PA) = (360 ) Equation 3: AmplitudeRatio( AR) = Δ Δm
Frequeny Response 1800 110 Speed (RPM) 1600 1400 1200 1000 800 600 t Δ T Δm 100 90 80 70 60 50 40 30 Input (%) 400 20 200 10 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Time (se) Figure 5 Sine Response Graph The amplitude ratio and phase angle are then graphed against the frequeny in a Bode Plot. (See Figure 6) The plot for the amplitude ratio is log-log and the phase angle is semi-log. Four piees of information an be taken from the Bode Plots; f u, Ku, Order and K. (See Figure 6) The value for fu an be determined by loating the frequeny at the intersetion of a line drawn horizontally through the phase angle plot at 180 degrees and the data line. The Ku is the inverse of the intersetion point of the f u line (transferred the amplitude ratio plot) and the data line. (Inverse of the point value 1/Ku) The differential order of the plot an be found by the maximum slope of the urved portion. (This is a guideline develop the equation neessary model the urve.) The gain (k) an be found with a horizontal line drawn aross the p of the urve where it no longer inreases. (See Table 3-Frequeny Test Data in Results)
Bode Plot 100 45 degree K Amplitude Ration (rpm/%) 10 1/Ku 0.01 0.1 1 10 Frequeny (Hz) 1 Bode Plot 0 0.01 0.1 1 10-50 fu Phase Angle (deg) -180 d -100-150 -200 Frequeny (Hz) -250 Figure 6 Bode Plots Amplitude Ratio and Phase Angle
Root Lous *NOTE: A detailed derivation of the following equations, along with more explanation is listed in the Appendix at the end of this report. The following diagram (Figure 7) depits the Open Loop System. The value of G (s) is K (ontroller gain) and the G(s) is ke. (System Transfer Funtion) The Open Loop τs + 1 Transfer Funtion (OLTF) is a produt of the transfer funtions of the system. (See Equation 4) Figure 7 Open Loop Blok Diagram Equation 4: OLTF = G( s) G( s) = K Ke τs + 1 The following diagram depits the Closed Loop System. The negative feedbak loop enables the system respond the output. The Closed Loop Transfer Funtion (CLTF) is listed below in Equation 5 in terms of the OLTF and the system values. The CLTF will permit further evaluation of the system mathematially.
Figure 8 Closed Loop Blok Diagram Equation 5: CLTF K Ke OLTF = τs + 1 K Ke = 1 + OLTF K Ke 1+ ( τs + 1) + K Ke ( τs + 1) τs + 1 = The following equation (Equation 6) is the Charateristi Equation (CE) used develop the Root Lous urve desribed in the next paragraph. The roots of the equation were determined for several values of K and the value of K=17.3 rpm/%. Table 4 in the Results setion lists the data olleted for eah K seleted. ***NOTE: This Equation inorporates Pade s Approximation in plae of e -. See Equation 6a in the appendix. 2 Equation 6: CE 2 τ s + ( τ + K K ) s + 1+ K K = 0 2 2 The roots of Equation 4 at eah value of K are plotted on a graph in whih the Y-axis represents the imaginary portion of the root and the x-axis represents the real portion of the root. The figure below (Figure 9) shows the method of finding the value of K for a speified Deay Ratio. (Deay Ratio is the initial overshoot divided by the seond
overshoot. Overshoot is the height of the output overage above the set point. See Figure 10) A line drawn through the origin at 78 o from the real axis will ross the root lous urve at the K value for ¼ deay ratio. 1/500 and 1/10 deay ratios are likewise found at 45 o and 70 o respetively. Critial damping, where there is no overshoot, an be found at the point where the root lous urve would ross the real axis. Ku (Ultimate K ) (not shown in the illustration) is the point where the root lous urve rosses the imaginary (vertial) axis. 1/500 (45 deg) K =.023 ROOT LOCUS PLOT 1/10 (70 deg) K =.075 1/4 (78 deg) K =.1 16 14 12 10 8 6 4 IMAGINARY AXIS 2 0-9 -8-7 -6-5 -4-3 -2-1 0 1-2 Critial Damping -4-6 -8-10 -12-14 REAL AXIS -16 Figure 9 Root Lous Plot
Figure 10 Deay Ratio **Note: Smaller deay ratios result in longer times for the system reah steady state while larger ratios result in higher overshoot. Proportional Controller and Offset Offset is the steady state error that ours due a hange in set point. Equation 7 is used alulate the offset for the hosen value of K. (System gain K is fixed at this point a value determined with the development of the SSOC and Step Response tests) Higher values of K result in less offset, but may ause higher overshoot. Also, the K should not be higher than the K u or the system will be unstable and never ahieve a steady state output. Equation 7: Δr offset = 1 + K K Equation 8 desribes the proportional ontroller. The variable m(t) is the output and is onsidered the manipulated variable with respet the ontroller.(src 247 Figure 1) K
is the ontroller gain, m is the bias (input power %) The variable e(t) equals r(t) (t), where r(t) is the hange in rpm from the starting point r ( Δ r ) and (t) refers the transmitter output (ST 247 Figure 1) that detets the mor speed. Equation 8: m( t) = m + k e( t) = m + K ( r( t) ( t)) If the set point is hanged ( Δ r) the proportional ontroller equation will hange aordingly. This means the (t) value from the transmitter will affet the steady state output (m(t)) proportionally aording the new values of the proportional ontroller equation and there will be an offset value per Equation 7. (See Appendix for a sample of the effets of hanging set point) Figure 11 - Offset
**NOTE: Seletion of the K will have signifiant effet on the offset and the deay ratio and must not exeed the value of Ku. Table 5 in the Results setion ontains a list offset values for eah K at the major divisions of deay ratio and eah hange in set point referred as delta r. ( Δr )
Proedure **NOTE: The SSOC and Step Response tests were previously performed but are reviewed here. 1. SSOC: The first set of tests performed on the system was the SSOC development. A set of data was olleted from several tests through the entire operating range of the system at inrements of 10%. The results were plotted. (see Figure 3) Table 1 in the Results setion ontains the average values and error information for eah input. 2. Step Response: The next step of the system evaluation was the Step Response test. FOPDT modeling was performed using an Exel spreadsheet. (See Figure 4) The modeling resulted in the data listed in Table 2 of the Results setion. 3. Frequeny Response: The frequeny test was performed at three baseline input levels. (35%, 65% and 85%) The range of frequenies between.01 Hz 3Hz was tested. The frequeny was varied in inrements of approximately.03 Hz.3 Hz with a minimum of ten tests per baseline. The data for eah test was graphed. (See Figure 5) The Amplitude Ratio and Phase Angle were determined for eah test and plotted in a Bode Plot. (See Figure 6) The data found during these tests is listed in Table 3 of the Results setiont. 4. Root Lous Modeling: The root lous graph was generated using an Exel spreadsheet and the values of K were determined for ¼, 1/10, and 1/500 deay ratios, Ku and ritial damping. The values of K ranged from.008.2. Table 4 in the Results lists the data.
5. Proportional Controller and Offset: The values of offset were alulated at eah of the K values that orrespond Ku, Critial Deay and ¼, 1/10 and 1/500 deay ratio. Table 5 of the Results setion lists the data.
Results Table 3 list the results of the Frequeny Response Experiment. The values were ahieved using a First Order Plus Dead Time (FOPDT) mathematial model. The urve mathing shown in Figure 5 is representative of all the urves generated for eah of the three baselines tested. (35%, 65% and 85% input power) Table 3 Frequeny Response Data K 17.3 rpm/% t o.2 se.17 se τ The Root Lous testing resulted in the values of K for the important deay ratios. (Critial, Ku, ¼, 1/10 and 1/500) The urve generated an be seen in Figure 9. The table below lists all values of K that were used graph the urve by hand (for verifiation), but many more points were used generate the Exel graph in Figure 9. Table 4 Root Lous Data K Real Root Imaginary Root (+) Deay Ratio.2 2.2 16i.15 -.3 15i.1-2.9 12i ¼ deay.03-6.4 6.9i.01-7.4 3.7i.008-7.5 3.2i.05-5.4 9i.023-6.7 6i 1/500 deay.075-4.1 11i 1/10 deay.002-8 & -8 0 Critial Deay.156 0.156 Ku The table below lists the values of offset for a hange in set point at eah value of K for the orresponding deay ratio. These an be used determine whih value of K needs be used ahieve the needed output with onsideration of the offset. The offset for Ku
provides the least offset but will reate signifiant overshoot and a larger deay time before settling in a steady state. Table 5 Offset Values in rpm for eah deay ratio and Δr Δr Offset Values for hange in set point (r) (rpm) Δr deay K 50 100 150 200 250 300 ¼ 0.1 18 37 55 73 92 110 1/10 0.08 21 42 63 84 105 126 1/500 0.03 33 66 99 132 165 197 Ku 0.156 14 27 41 54 68 81 Critial 0.002 48 97 145 193 242 290 Δr Δr Δr Δr
Disussion This experiment reveals the offset values for eah K. The use of the Ku puts the system at the edge of stability but provided very little offset. This means the reliability of the output is good, but the system may osillate and experiene signifiant overshoot before it ahieves a steady state operation. Less stability problems exist at lower deay ratios but the system experienes offset values that may be o high. That is, the atual output may vary o far from the required output for signifiant hanges in the set point. Critial damping is very stable but produes very high values of offset and may take a o long reah steady state output.
Conlusions and Reommendations The system models throughout these experiments provide an aurate representation of the atual output. The proportional ontroller model allows reasonable preditions of the system response onsidering the offset. The table of offsets an be used adjust the proportional ontroller equations determine the system response. If the output needs be very stable and the deay time is unimportant then the ¼ deay value for K an be used. *NOTE: The K value of Ku is unstable and will not settle in a steady state response. (ontinuos osillation) Some value less than Ku ould be used but speifi values that are lose need be disreetly tested before reliable results an be determined that will not osillate ontinuously. Overshoot at values approahing Ku will be large and settling times will inrease. In general ¼ deay and 1/10 deay should work well for this system sine a small differene in the speed of the aerar will our. For example, at 1/10 deay the speed of the mor offset for hange in set point of 100 is 42 rpm while the aerar is a mere.42 rpm. Likewise for ¼ deay, the aerar hange in speed is very small. (1/100) These are stable values of ontroller gain and should provide aeptable results for all hanges in set point.
Appendies Table 1 lists the data used develop the SSOC. The Mor Input olumn lists the input test values. The Output Speed lists the atual mor speed in revolutions per minute (rpm). The Standard Deviation olumn lists the deviation of the olleted data for eah of the tests. Table 1. Experimental values used reate the SSOC. Mor Input (%) Output Speed (RPM) 10 158.60 6.00 20 333.78 4.16 30 507.99 3.27 40 682.79 2.32 50 856.29 2.10 60 1031.74 2.32 70 1204.65 1.98 80 1374.01 2.87 90 1540.35 4.88 100 1707.56 4.98 Standard Deviation The step response experiments led the development of the baseline values listed below: Table 2 Step Response Data K 17.3 rpm/% t o.2 se.17 se τ Root Lous Equation Derivations The following steps will develop the mathematial model method used in this experiment. The Laplae Domain equivalents will be used. The final values will be expressed in terms of time.
The illustration below depits an open loop system. The Open Loop Transfer Funtion (OLTF) in Equation 1, is the produt of the system transfer funtions. Equation 2 is further development of the equation using the values of the transfer funtions. Equation 1a: OLTF = G ( s) G( s) Equation 2a: OLTF = K Ke τs +1 The following illustration depits the Closed Loop system. The blok diagram above an represented algebraially as follows: (Equation 3) Equation 3a: C( s) R( s) G ( s) G( s) = 1+ G ( s) G( s) = CLTF This equation is the Closed Loop Transfer Funtion (CLTF). The equation an be represented in terms of the OLTF as follows in Equation 4: Equation 4a: OLTF CLTF = 1+ OLTF
The values for the OLTF an be substituted develop Equation 5. Equation 5a: CLTF K Ke τs + 1 K Ke 1+ τs + 1 K Ke = ( τs + 1) + K Ke = ( τs + 1) The denominar of the CLTF (Equation 5) will be used with Pade s Approximation (Equation 6) develop an equation that will be used for several possible values of K. The roots of the polynomial equation set equal zero (Equation 8) will be determined using the quadrati equation (Equation 9) and plotted for eah value of K. Pade s Approximation is a substitution for the exponential funtion (e - ) allow the CLTF denominar be expanded in a meaningful polynomial where roots an be found. Equation 6a: Pade s Approximation e 1 s = 2 1+ s 2 Equation 7a: CLTF Denominar with Pade s Approximation 1 s ( τs + 1) + K K 2 ( τs + 1) = 0 2 + s 2
,whih expands the polynomial in Equation 8: 2 2 2 Equation 8a: τ s + ( τ + K K ) s + 1+ K K = 0,whih is the Charateristi equation (CE) in terms of the Laplae Domain that is used develop the Root Lous Curve for several values of K at the predetermined value of K. (17.3 rpm/%) 2 Equation 9a: Quadrati Equation 2 b 4a b + = roots 2a, where the oeffiients a,b and are: a = τ, b = ( τ + K K ), = 1+ K K 2 2 2 Refer Figure 10 Root Lous urve see a plot of the roots found for eah frequeny. Changes in Set Point Proportional Controller m( t) = m + k e( t) = m + K ( r( t) ( t)) r = initial set point and r = the new set point Example: r = 680, m = 40%, r = 400, Δr = 280, K =.07, K = 17 r ( t) = Δr = r r,...680 680 = 0 initial setting m( t) = 40 +.07(680 ( t))
m( t) = 87.5.07( t) where (t) is the speed reported by ST 247 of Figure 1 the ontroller SRC 247. So, when the speed hanges the ontroller hanges the drives output at SCZ 247 aording the equation. If the setpoint hanges r ( t) = Δr = r r,...400 680 = 280 400 where the new set point is Offset Δr 280 = = 128rpm 1 + KK 1+.07(17) The new proportional equation is m ( t) = 40 +.07( 280 ( t)) m ( t) = 20.4.07( t) so aording the new proportional equation values the response will our with an offset of 128 rpm and the speed reported by ST 247 will ause the ontroller adjust the response aordingly with a 128 rpm offset. (differene in atual response) See Figure 11 for a graphial illustration.
List of Figures 1- System Shemati 2- System Blok Diagram 3- SSOC 4- Step Response 5-5Frequeny Response 6- Bode Diagrams 7- Open Loop 8- Closed Loop 9- Root Lous Plot 10- Deay Ratio 11- Offset List of Tables 1 - SSOC Data 2 Step Response Data 3 Frequeny Response Data 4 Root Lous Data 5 Offset Data List of Equations 1 FOPDT 2- Phase Angle 3 - Amplitude Ratio 4 - OLTF 5 - CLTF 6 - Charateristi Equation 7 - Offset 8 - Proportional Controller Referenes Smith and Corripio Priniples and Praties of Aumati Proess Control, Edition 3E, Wiley Publishing 2006 Dr. Henry 328 Website, UTC Blak Board