University of Tennessee at Chattanooga. Engineering 329. Step Response Characteristics

Transcription

1 University of Tennessee at Chattanooga Engineering 329 Paint Spray Booth Pressure System: Steady-State Operation and Step Response Characteristics Eric L. Young Jonathan Blanco Matthew Chatham-Tombs September 25, 27

2 Introduction: Three spray paint booths at an assembly plant require a feedback control system to maintain a desired pressure. A blower that is powered by a variable speed motor provides the pressure. The purpose of this lab is to learn about the dynamic response of the pressure system and to determine the first order parameters for the mathematical model of the system. The simplest method for evaluating the dynamic response is to provide an abrupt, instantaneous change in the system s input. This is referred to as a step change in the input. The response of the system to this change in input is called the step response of the system. The main objectives of this lab are to observe the time response of the output function of the system to a step change, to observe the steady-state gain, K, the dead time, t, and the time constant, τ, and to observe these parameters in several regions of the steady-state curve. The following report includes a background of the lab discussing the system, the schematics, the steady-state curve, and the methods for determining the system parameters. Also included is the procedure for the lab and the method by which the results were calculated. These results are then presented using tables and charts. The results are then discussed and conclusions are made.

3 Background: A diagram of the blower, booth, and control system is shown below. Figure : Schematic Diagram of the Dunlap Plant Spray-Paint Booths The input function for the blower-booth system is the power sent to the blower, which varies from -% of the rated power of the motor. The output function is the pressure of the booth measured in cm-h2o. The input function is designated m(t) as it represents the manipulated variable while the output function is designated c(t) as it represents the controlled variable. The following diagram shows the input-output relationship. Figure 2: Block diagram of the paint Booth System

4 The aim of this lab is to obtain step response data. Figure 3 shows a typical input step function, m(t). Figure 3: Step Input The input function is initially at a base line input and abruptly steps up the value of the step height. Notice that the input does not take time to reach the upper operating value. The step of the input happens instantaneously. Figure 4 shows a typical response of a system to a step input. Figure 4: Step Response Unlike the instantaneous change of the input, the output takes a certain amount of time to respond to the input step. From the graph in Figure 4 one is able to determine the parameters of the system. These parameters are the steady-state gain, K, the dead time, t, and the time constant, τ. These are also referred to as the First-Order-Plus-Dead-Time (FOPDT) parameters. These parameters are part of the transfer function of the system.

5 The transfer function of a first order system in the Laplace domain can be approximated by the equation, G (s) = K * [(e^t s)/(τs+)] In this lab it is important to observe these parameters for different regions of the steady-state curve. The steady-state curve was developed in a previous lab using average values of the output for given values of the input. The steady-state curve for the pressure system is shown in the graph below. Steady State Operating Curve, Pressure 7 m, Pressure Output (cm-h2) Series c, Input (%) Figure 5: Steady-state operating curve for the paint Booth System This curve was created using the results of experiments conducted online. The pressure outputs presented on the graph are the averages of the steady-state operating values for each input percentage. The uncertainty bars at each data point show two times

6 the actual standard deviation. The operating range for the pressure system input has been determined to be 25% to %. The corresponding range for the output is. cm-h2o to 5.6 cm-h2o. The slope of the steady-state curve is also a way to calculate the gain, K, of the system. The slope of the steady-state curve appears to rise until it reaches an input of 5%. The average slope in the region before 5% is.4 cm-h2o/%. The average slope from 5% to % is around.65 cm-h2o. In order to determine parameter values for several regions of the operating range several experiments must be conducted. A sample of the resulting response to a step input is shown below. Figure 6: Step Up Response

7 This graph shows the response of the pressure system to a step input of 5%. The base line input value is 3% and at a time of 25 seconds the input instantaneously steps up to 45%. It can be seen that ample time was given for the system to reach a steady state before and after the step takes place. The parameters can also be determined by means of a step down response. The pressure systems response to a step down is shown in the graph below. Figure 7: Step Down Response This graph shows the pressure systems response to a step down input. The base line value is 45% with a step down of 5% at a time of 25 seconds. Again it can be seen that enough time has been permitted for the system to reach a steady state before and after the step. The calculations of the first order parameters using a step up or a step down should be equal in the same region of the operating range. Once several

8 experiments have been conducted for each region of the steady-state curve Excel can be used to model the experimental results. A sample model created using Excel is shown below. FOPDT Model Input (%) Output (cm-h2o) Input Value(%) Input Output(cm-H2) Output Time (s) Figure 8: Excel First-Order-Plus-Dead-Time Model In the graph shown above the purple output line is the result of the experimental data. The blue output line is the resulting model created in Excel. If the input function to an FOPDT is a step function, having a step equal to A and occurring at time equal to td, the input function m(t)=a*u(t-td). The time response of this system is then, c(t)=a*u(t-td-t )*K*(-e^-[(t-td-t)/τ] ). The derivation of these equations can be found on page 237 of Principles and Practice of Automatic Process Control by Smith and Corripio. Using Excel and the time response of the system a model of the experimental data can be created. By manipulating the parameters K, the gain, t, the dead time, and τ, the time constant an accurate representation of the experimental data can be obtained.

9 The blue output line in the figure above was created with the time response function and the manipulation of the first order parameters. A more detailed description of how this model is created is presented in the procedure section of the lab.

10 Procedure: The main objectives of this laboratory are to observe the dynamic response of the mathematical model for a system and to observe the impact of parameter values on the dynamic response. In order to accomplish these objectives it is necessary to create an Excel file that will allow for the manipulation of the parameters. The first three columns of the file should contain the experimental data obtained from the online step response experiments. This data will provide the graph we would like to model. The fourth column of the Excel file will model the step input. The argument, =IF(A4>td,A,)+inbl is placed in the fourth column cells. This argument states that if the time is less than the time at which the step occurs the value of the cell will be zero plus the input baseline. When the time is equal to and greater than the time at which the step occurs the value of the cell will be A, or the value of the step, plus the baseline input. The input baseline accounts for the fact that our experimental baseline inputs were not equal to zero. The final column of the Excel file will contain the time response function, =A*u(t-td-t )*K*(-e^-[(t-td-t)/τ] ). By defining the parameters, A, td, t, K, and τ in Excel the manipulation of the time response function becomes much easier. Instead of having to change the values of the parameters each time in the time response function they can be contained within their own cells and be changed much more readily. By manipulating these parameters a very accurate model of the experimental data can be created. This modeling should be done for several step response experiments in different regions of the steady-state operating curve. In order to obtain enough data for an error analysis the same experiment should be run at least four times with the same step input value and

11 input baseline. This should be completed in the lower, middle and upper regions of the steady-state operating curve.

12 Results: The results of the Excel modeling are presented in this section of the lab report. The figure below shows the way in which the experimental values of the first-order parameters were obtained in a previous lab. Δm.632(Δc) Δc Figure 9: Fit 2 Method for determining first order parameters. Figure 9 is a representation of the fit 2 method for determining the first order parameters for the system. The gain, K, is calculated using the equation Δc/Δm. The dead time, t, is determined by drawing a line tangent to the steepest part of the rising input and determining how long after the step occurred this tangent line crosses the input baseline. The time constant, τ, is determined by the amount of time after the dead time it takes for the output to reach 63.2% of the value of Δc. A typical example of the modeling done in Excel is shown below.

13 Average FOPDT Gain, Model K 3%-45% Input (%) Gain (cm-h2o) Time (s).7 Experimental Step Input Up Average Value(%).6 Modeling Step Up Input Average.5 Output(cm-H2).4 Output.3.2 Output (cm-h2o) Figure : First-Order-Plus-Dead-Time Excel Model The figure above shows the results of the modeling done in Excel. By manipulating the first order parameters a very accurate representation of the experimental data was created. This was done for several experiments for a range of input operating values. The parameters determined from this modeling were analyzed using the Student s T method. When such a small number of data points are collected the standard deviation is not a desirable way to determine the accuracy of the results. Using the Student s T method the uncertainty= (c(t) max c(t) min )*t/n. A table of the values of t/n, depending on the number of experimental results, is presented on the website

14 The figures below show the results for experimental and modeling values determined for the system parameter gain, K. These results are in the operating ranges from 3%-45%, 5%-6%, and 75%-95%. Average Gain, K 3%-45%.45 Gain (cm-h2o) Figure : Average Gain, K 3%-45% Average Gain, K 5%-6% Gain (cm-h2o) Figure 2: Average Gain, K 5%-6%

15 Average Gain, K 75%-95% Gain (cm-h2o) Figure 3: Average Gain, K 75%-95% These three figures show the average gain calculated in different regions of the steady-state operating curve. The error bars are a representation of the error analysis conducting using the Student s T method. The experimental averages differ from the modeling values due to the methods in which they were obtained. The data falls within the same range when the uncertainty is taken into account for the calculations. The figures below show the results for experimental and modeling values determined for the system parameter dead time, t. These results are in the operating ranges from 3%-45%, 5%-6%, and 75%-95%.

16 Average Dead Time, t 3%-45%.7 Dead Time (s) Figure 4: Average Dead Time 3%-45% Average Dead Time, t 5%-6%.6.5 Dead Time (s) Figure 5: Average Dead Time 5%-6%

17 Average Dead Time, t 75%-95% Dead Time (s) Figure 6: Average Dead Time 75%-95% These three figures show the average dead time calculated for different input values. The experimental and modeling values differ due to the method by which each were obtained. The experimental method, the fit 2 method, is very subjective. The tangent line needed in order to obtain dead time is determined by the researcher. The tangent line is placed tangent to the steepest part of the output increase, which may be difficult to determine. The figures below show the results for experimental and modeling values determined for the system parameter time constant, τ. These results are in the operating ranges from 3%-45%, 5%-6%, and 75%-95%.

18 Average Time Constant, τ 3%-45% Time Constant (s) Figure 7: Average Time Constant 3%-45% Average Time Constant, τ 5%-6% Time Constant (s) Figure 8: Average Time Constant 5%-6%

19 Average Time Constant, τ 75%-95% Time Constant (s) Figure 9: Average Time Constant 75%-95% These three figures show the average time constants evaluated for different regions of the steady state curve. The experimental results were determined using the fit 2 method while the modeling results were obtained by creating a model of the step response graph utilizing Excel. These values are fairly consistent throughout the different ranges. The accuracy of the results were analyzed using the Student s T method and are shown using the error bars.

20 Discussion: The results of the experimental data and the modeling data were similar. The variances in these results were due to the methods by which each were obtained. The experimental values were determined using the fit 2 method. This method requires the use of trial and error in fitting the tangent lines to the graph. One researcher may have different values from another on the same graph due to what they see as the steepest part of the output rise. The following description of the average values of the laboratory experiments come directly from the charts presented in the results section of this report. The step up experimental average values for the gain, the dead time, and the time constant for the range from 3%-45% were determined to be.44cm-h2o,.5s, and.53s respectively. The step down experimental average values for the gain, the dead time, and the time constant for the range from 3%-45% were determined to be.43cm-h2o,.55s, and.5s respectively. The step up modeling average values for the gain, the dead time, and the time constant for the range from 3%-45% were determined to be.44cm-h2o,.4s, and.6s respectively. The step down modeling average values for the gain, the dead time, and the time constant for the range from 3%-45% were determined to be.43cm-h2o,.35s, and.6s respectively. The step up experimental average values for the gain, the dead time, and the time constant for the range from 5%-6% were determined to be.69cm-h2o,.42s, and.5s respectively. The step down experimental average values for the gain, the dead time, and the time constant for the range from 5%-6% were determined to be.56cm-h2o,.35s, and.5s respectively. The step up modeling average values for the gain, the dead time, and the time constant for the range from 5%-6% were determined to be.67cm-h2o,

21 .7s, and.6s respectively. The step down modeling average values for the gain, the dead time, and the time constant for the range from 5%-6% were determined to be.54cm-h2o,.352s, and.5s respectively. The step up experimental average values for the gain, the dead time, and the time constant for the range from 75%-95% were determined to be.98cm-h2o,.3s, and.5s respectively. The step down experimental average values for the gain, the dead time, and the time constant for the range from 75%-95% were determined to be.98cm-h2o,.3s, and.5s respectively. The step up modeling average values for the gain, the dead time, and the time constant for the range from 75%-95% were determined to be.96cm-h2,.74s, and.4srespectively. The step down modeling average values for the gain, the dead time, and the time constant for the range from 75%-95% were determined to be.cm- H2O,.82s, and.8s respectively.

22 Conclusion: The purpose of this lab was to learn about the dynamic response of the pressure system and to determine the first order parameters for the mathematical model of the system. The simplest method for evaluating the dynamic response is to provide an abrupt, instantaneous change in the system s input. The response of the system to this change in input is called the step response of the system. The main objectives of this lab were to observe the time response of the output function of the system to a step change, to observe the steady-state gain, K, the dead time, t, and the time constant, τ, and to observe these parameters in several regions of the steady-state curve. These parameters were determined using Excel to create a model of the step response. The manipulation of these parameters were key in developing a graph that accurately modeled the experimental output.

23 Appendix: Principles and Practice of Automatic Process Control by Smith and Corripio

24 Steady State Operating Curve, Pressure 7 m, Pressure Output (cm-h2) Series c, Input (%)

25

26 FOPDT Model Input (%) Output (cm-h2o) Input Value(%) Input Output(cm-H2) Output Time (s)

27 Average Gain, K 3%-45%.45 Gain (cm-h2o) Average Gain, K 5%-6% Gain (cm-h2o)

28 Average Gain, K 75%-95% Gain (cm-h2o) Average Dead Time, t 3%-45%.7 Dead Time (s)

29 Average Dead Time, t 5%-6%.6.5 Dead Time (s) Average Dead Time, t 75%-95% Dead Time (s)

30 Average Time Constant, 3%-45% Time Constant (s) Average Time Constant, 5%-6% Time Constant (s)

31 Average Time Constant, τ 75%-95% Time Constant (s)