CHAPTER 3 LINEAR MODEL FORMULATIONS
3.5 Linear Model Combination
3.5.3 Full System Assembly
The Redheffer Star Product and embedded MATLAB function ‘lft’ are again used to combine the EEV/evaporator and compressor/condenser subsystem models into a linear four- component VCS model. Figure 3.21 depicts this interconnection of the two subsystems. Note
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that the ordering of inputs and outputs of the subsystems has been adjusted in order to group the feedback signals together.
Figure 3.21: Subsystem Interconnection
The resulting linear four-component VCS model takes the form
sys sys sys sys sys sys sys sys sys sys
x A x B u y C x D u (3.55) where
, , , , , , , , / / , , , , , , , , , , , , T sys EEV e a in e a in comp c a in c a inT sys EEV e cond c
T
EEV e e r out e r out e a out e comp c c r out c r out c a out c
u o T m v T m y y y m P h T SH T Q m P h T SC T Q (3.56) and
or T sys e c T sys e c x x x x x x . (3.57)85
The ‘or’ in Eq. (3.57) is determined by whether the static or dynamic compressor model is used, the difference being that xc includes one additional state for the compressor outlet enthalpy dynamic.
3.5.3.1
Nonlinear and Linear ‘3-2’ Model Comparison
We now compare a nonlinear VCS model using the SMB heat exchangers with the full system linear model described above. We first consider the case where the nonlinear and linear heat exchangers operate in Mode 1 at all times. The linear model is therefore generated by the combination of the linear Mode 1 evaporator and linear Mode 1 condenser models with the linear compressor and EEV, and linearized about the nominal steady-state operating conditions of the nonlinear model reached before and after each input step. Because Mode 1 for the condenser includes three refrigerant phase zones (superheated, two-phase, and subcooled) and Mode 1 of the evaporator includes two refrigerant phase zones (two-phase and superheated), we refer to this as the ‘3-2’ VCS mode.
Figure 3.22 shows the sequence of inputs used to compare the nonlinear and linear VCS models. The inputs are chosen such that the nonlinear system at all times remains in the ‘3-2’ mode. Figure 3.23 shows the output response of the first six outputs of each heat exchanger, and Figure 3.24 shows the seventh output. Figure 3.25 and Figure 3.26 show the states of the evaporator and condenser, respectively.
It is clear from these figures that the nonlinear and linear models match fairly well. Although not all gains of the linear model match the nonlinear model, the outputs and states consistently step in the correct direction. The difference between the linear models with and without the compressor enthalpy dynamic is visible in the subplot of h in Figure 3.26. The r,1 former matches the time constant of the response in the nonlinear model, while the latter has a faster response. However, this difference is small enough that for most applications of the linear model, including the estimation and control design of Chapter 6, the added accuracy due to including the compressor enthalpy dynamic is not worth the added model complexity that it brings.
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Figure 3.23 Nonlinear and Linear VCS Comparison—First Six Outputs of Each Heat Exchanger
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Figure 3.24 Nonlinear and Linear VCS Comparison—Seventh Heat Exchanger Output
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Figure 3.26 Nonlinear and Linear VCS Comparison—Condenser States
3.5.3.2
Nonlinear and Linear ‘3-1’ Model Comparison
We next consider the case where the condenser operates in Mode 1 at all times and the evaporator operates in Mode 2 at all times. The linear model is therefore generated by the combination of the linear Mode 1 condenser models and linear Mode 2 evaporator with the linear compressor and EEV, and linearized about the nominal steady-state operating conditions of the nonlinear model reached before and after each input step. Because Mode 1 for the condenser includes three refrigerant phase zones (superheated, two-phase, and subcooled) and Mode 2 of the evaporator includes only a two-phase zone, we refer to this as the ‘3-1’ VCS mode.
Figure 3.27 shows the sequence of inputs used to compare the nonlinear and linear VCS models. The inputs are chosen such that the nonlinear system remains in the ‘3-1’ mode through all the input steps. Figure 3.28 shows the output response of the first six outputs of each heat exchanger, and Figure 3.29 shows the seventh output. Figure 3.30 and Figure 3.31 show the states of the evaporator and condenser, respectively.
We can see from these figures that the linear models with and without the compressor enthalpy dynamic are essentially equivalent. It is also clear that the match between the linear and nonlinear models is worse than for the ‘3-2’ model. The gains of the linear model do not tend to
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match the nonlinear model as closely, although the outputs and states generally step in the correct direction. This increased mismatch is due to highly nonlinear behavior known to be associated with the two-phase refrigerant that has been exposed at the outlet of the evaporator in the absence of a superheated zone. This nonlinearity can be seen, for example, at high and low quality values of refrigerant heat transfer coefficient, plotted in Figure 2.8. Although this might at first imply that the linear ‘3-1’ model is not sufficiently accurate to be of any value, we recall that the motivation for creating the linear models is not for use in open-loop VCS simulations, but instead for their incorporation into model-based state estimation and control designs. We show in Chapter 6 that through the use of designs that include feedback from the plant and are robust to model error, the linear ‘3-1’ model can be quite effective for these purposes.
Figure 3.27 Nonlinear and Linear VCS Comparison—Inputs
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Figure 3.28 Nonlinear and Linear VCS Comparison—First Six Outputs of Each Heat Exchanger
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Figure 3.29 Nonlinear and Linear VCS Comparison—Seventh Heat Exchanger Output
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Figure 3.31 Nonlinear and Linear VCS Comparison—Condenser States
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Equation Chapter (Next) Section 1