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Experimental Four-Tank Process


  
Figure 3: Schematic of the experimental process

\resizebox*{3in}{!}{\includegraphics{4tank.ps}}


An interacting four-tank process has been implemented at the University of Delaware [9]. This process is currently used in both the elective multidisciplinary undergraduate control laboratory and the advanced graduate control course. The design is inspired by the benchtop apparatus described in [11]. A simple schematic is shown in Figure 3. Two voltage-controlled pumps are used to pump water from a basin into four overhead tanks. The two upper tanks drain freely into the two bottom tanks, and the two bottom tanks drain freely into the reservoir basin. The liquid levels in the bottom two tanks are directly measured with pressure transducers. As can be seen from the schematic, the piping system is configured such that each pump affects the liquid levels of both measured tanks. A portion of the flow from one pump flows directly into one of the lower level tanks where the level is monitored. The rest of the flow from a single pump is diverted into an overhead tank, which drains into the other monitored tank. By adjusting the bypass valves on the system, the amount of interaction between the two pump flow rates (inputs) and the two lower tank level heights (outputs) can be varied. For this work, it is assumed that an external unmeasured disturbance flow may also be present which drains or fills the top tanks. The original work of [11] employed tanks with volume of 0.5L whereas the present work uses 19L (5 gallon) tanks for a more realistic experimental process.

A nonlinear model given by [11] has been modified to accommodate the addition of disturbance flows into and out of the various tanks, di. This model is given in the following equations:

\( \frac{dh_{1}}{dt}=-\frac{a_{1}}{A_{1}}\sqrt{2gh_{1}}+\frac{a_{3}}{A_{1}}\sqrt{2gh_{3}}+\frac{\gamma _{1}k_{1}}{A_{1}}\nu _{1}-\frac{k_{d_{1}}d_{1}}{A_{3}} \)

\( \frac{dh_{2}}{dt}=-\frac{a_{2}}{A_{2}}\sqrt{2gh_{2}}+\frac{a_{4}}{A_{2}}\sqrt{2gh_{4}}+\frac{\gamma _{2}k_{2}}{A_{2}}\nu -\frac{k_{d_{2}}d_{2}}{A_{3}} \)

\( \frac{dh_{3}}{dt}=-\frac{a_{3}}{A_{3}}\sqrt{2gh_{3}}+\frac{(1-\gamma _{2})k_{2}}{A_{3}}\nu _{2}-\frac{k_{d_{3}}d_{3}}{A_{3}} \)

\( \frac{dh_{4}}{dt}=-\frac{a_{4}}{A_{4}}\sqrt{2gh_{4}}+\frac{(1-\gamma _{1})k_{1}}{A_{4}}\nu _{1}-\frac{k_{d_{4}}d_{4}}{A_{4}} \)

Although this fundamental model is a reasonably accurate description of the system dynamics, many of the parameters are not available a priori, which requires the estimation of several model parameters. The tank areas Aican be measured directly from the apparatus. Using tank drainage data, the cross sectional outlet areas ai can also be determined. The steady-state operating point of \( \nu _{1}=55\% \) and \( \nu _{2}=55\% \) were used for subsequent results. The system valves were set for the bypass values \( \gamma _{i} \)such that the operating point exhibits inverse response ( \( \gamma _{1}+\gamma _{2}<1 \)). Time constants, Ti, for the linearized system model are on the order of 40 seconds.


 
Table 1: Model parameters
a1,a2 \( 2.3\, cm^{2} \) k1 \( 6.81\, cm^{3}/s \)
a3,a4 \( 2.3\, cm^{2} \) k2 \( 6.94\, cm^{3}/s \)
A1,A2,A3,A4 \( 730\, cm^{2} \) g \( 981\, \frac{cm}{s^{2}} \)
\( \nu _{1}(0) \) \( 55\% \) \( \gamma _{1} \) 0.102
\( \nu _{2}(0) \) \( 55\% \) \( \gamma _{2} \) 0.202
h1(0) \( 11.3\, cm \) h2(0) \( 16.4\, cm \)
h3(0) \( 8.9\, cm \) h4(0) \( 10.9\, cm \)
y1bias -1.8 y2bias -6.0


A suitable test input sequence to generate data for the estimation of the remaining parameters. In this case, they elected to identify the parameters of the original nonlinear model, requiring the solution of a nonlinear optimization problem. The problem was formulated to minimize the 2-norm of the difference between the nonlinear model and actual measurements, searching over six parameters. Using dynamic data from the experiment, the optimization routine found the optimal pump gains ki, gamma values \( \gamma _{i} \), and measurement bias yibias as depicted in Table 1. A similar routine could be employed to model the characteristics of the disturbance introduced by the submersible pumps, kdi , but this was not done. The models used for estimation are based on a first-principles model without first-hand knowledge of the physical parameters.

Linear step response models are derived from the nonlinear process model, assuming a step in the inputs of \( 5\% \) and a sample time of 5 seconds, using 60 coefficients. Aggressive controller tuning can result oscillations or unstable behavior. The controller was tuned with m=2 and p=40. The values for \( \Gamma _{y} \) were \( [1\, \, 1] \) and for \( \Gamma _{u} \) were \( [.1\, .1] \). This tuning results in acceptable setpoint tracking as well as disturbance rejection. The inverse response of the system imposes a bound on the performance of the system.

 
Table 2: Control and estimation algorithm parameters
m 2 samples p 40 samples
\( \Gamma _{y} \) \( [1\, \, \, 1] \) \( \Gamma _{u} \) \( [0.1\, \, \, 0.1] \)
m1 \( [10\, \, \, 10] \) m2 \( [3.5\, 3.5\, 0.9\, 0.9\, 0.6\, 0.6] \)
m3 \( [1.7\, 1.7\, 0.4\, 0.4\, 0.3\, 0.3] \) h 30 samples


Six faults are identified for this system: four flows into or out of the tanks in addition to bias changes on the process inputs. Linear response models were determined for the disturbance flows, assuming a level of disturbance that would result in an open-loop change of approximately \( 5\, cm \) in the process measurements. The input bias models are taken from the models used for the MPC control calculation. Process measurements are available every second. These values are filtered to provide measurements every five seconds. The tuning parameters m1, m2, and m3 are given in Table 2. These values are chosen such that the system can distinguish between the first-order type response of the lower tanks and the second-order response of the upper tanks, assuming that the disturbance is a step disturbance. Other types of disturbances such as ramps and pulses can be estimated, but in some cases the system recognizes an incorrect fault.


  
Figure 4: Experimental process measurements

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Figure 4 shows actual process measurements during four different disturbances. This figure also shows the process residuals for the system. The output values of the nonlinear process model are compared to the actual process measurements for calculation of the process residuals.


  
Figure 5: Estimated disturbance levels for different faults

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Figure 5 shows the estimated values for these four different disturbances. The input bias disturbances should track the solid lines at a normalized level of four units. The flow rate disturbances were never modeled exactly, so the actual level of the disturbance is unknown. A solid line at the normalized level of two shows the times at which the flow disturbances were applied to the system. Note that in some cases the incorrect fault was initially detected. After more information was available, the correct fault was determined.


  
Figure 6: Disturbance estimates for leak in tank 2

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Figures 6 and 7 show the horizon estimates for two different faults as a function of time. The small inset graphs on these figures show the optimal horizon estimates at different times. The estimated from 6 are much smoother than those of 7. The weights on the change of the disturbance were developed so that both faults could be distinguished, even though both faults are very similar in nature. A step-like leak in tank 2 appears as a first-order change in the residuals. A leak in tank 4 appears as a second-order change in the process residuals. The disturbance models for both faults can be used to account for the other fault. As a result, it can become difficult to develop weighting parameters to distinguish between these two faults.
  
Figure 7: Disturbance estimates for leak in tank 4

\resizebox*{4in}{!}{\includegraphics{fig-openloop-dist6.ps}}


The Model Predictive Control algorithm can be extended to included the effects of a measured disturbance on the process. In cases where a disturbance measurement is not available, an estimation method can be used calculate the level of the disturbance. This can be considered a soft sensor. The disturbance estimate is treated as a measured disturbance. In Figure 8, a simulation of the four-tank system is shown where the MPC algorithm rejects the disturbance of a leak in tank four without a measured disturbance update. In this simulation, the 2-norm of the process error is 9.9.

In Figure 9 the same disturbance hits the system, but the value of the estimated disturbance is used for control. The disturbance estimate is filtered with a first-order filter of the form:

\( y(k)=(\lambda -1)y(k-1)+\lambda u(k) \)

where \( \lambda =0.15 \). The 2-norm of the process error in this case is 5.7. This is a significant improvement over the case without a disturbance update. Figure 10 shows a simulation of the run using the non-filtered estimate. The 2-norm of this run is 4.0, but the control moves are erratic.

  
Figure 8: Closed-loop simulation without using estimates for control

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Figure 9: Closed-loop simulation using filtered estimates for control

\resizebox*{4in}{!}{\includegraphics{simcldistwithfilteredest.ps}}



  
Figure 10: Closed-loop simulation using estimates for control

\resizebox*{4in}{!}{\includegraphics{simcldistwithest.ps}}


next up previous
Next: CO Processing Simulation Up: Application and Results Previous: Application and Results
Edward Price Gatzke
1999-10-27