1、International Journal of Heating,Ventilating, Air-conditioning and Refrigerating Research HVAC nor may any pu 40 - vj Figure 9(a) shows the situation when k = 100 and r = 1 were used. Note that this design vio- lates the stability robustness inequality Equation (23) which may result in instability o
2、f a close loop system. When r = 1 is selected, the control gain is too large compared to the previous design where r = 4. Figure 9(b) shows the situation when k = 100 and r = 10 are used. In this case, the design cannot meet the performance requirement Equation (24). When r = 10 is selected, the con
3、trol gain is too small, which results in low performance in command following and disturbance rejection. For different machines and different operating points, the controller can be tuned as follows: (1) update the model, (2) determine the maximum multiplicative model error OE(jo), (3) select perfor
4、mance bound p,(o) based on the minimum required performance, (4) select initial forms of Q and R with a couple of iterating parameters, (5) iterate the parameters such that the stability robustness criterion and performance criterion are satisfied, (6)implement the designed controller and improve th
5、e design based on the actual performance and stability of the controlled system by repeating the steps (3), (4), and (5). . 1 ._ j ., Implementation and Experimental Results We implemented the MIMO control for a residential air conditioner. Thermocouples were used to measure temperatures at differen
6、t locations. The control inputs (compressor speed and expansion valve) are therefore generated based on the multivariable control law and the feed- back signals of superheat SH and evaporating temperature Te. At the given operating point on which the above control design was based, the compressor sp
7、eed was 70 Hz, and the expansion valve opening was at 120 steps out of total 400 steps. Dur- ing the tests, the indoor and outdoor conditions remained invariant. Indoor air, dry bulb and wet bulb temperatures were 27C and 19“C, respectively. The outdoor temperature was 35C. The corresponding evapora
8、ting temperature was 8SC, and the superheat was 5C. To compare the command following capability of the MIMO control and the SISO control, the SISO control gains were properly tuned. Figure 10 shows the results when the desired evaporating temperature Te was changed from 8.5“C to 7C while the superhe
9、at value SH was kept at 5C. It can be seen from the figure that the MIMO control resulted in a much faster transient, following the step change in reference command. It took about 4 to 6 min for the SISO control to reach steady state, while the MIMO control could reach steady state in about 2 min. A
10、s shown in Figure 10, the compressor speed STDWASHRAE SRCH IJHVAC i-3-ENGL 220 8.5 h SISO I 1778 0757b50 0535755 4T3 I HVAC 110- O 100 200 300 Time (secs) Figure 10. Command following: Desired Te has step change from 8.5“C to 5C 9.5 17 7 O 100 200 300 Time (secs) -90 N o 65 60 O 100 200 300 Time (se
11、cs) I O 100 200 300 Time (secs) 110 I O 100 200 300 Time (secs) Figure 11. Command following: Desired superheat has step change STD-ASHRAE SRCH IJHVAC 4-3-ENGL 1998 0759b50 053575b 33T VOLUME 4, NUMBEX 3, JULY 1998 22 1 and valve opening generated by the MIMO controller undergo a very quick action t
12、o get to the steady state. By using the dynamic model and the coordination of two control inputs, the tran- sient processes under such fast control actions in the MIMO controller are much faster and sta- ble. If such fast control actions are applied in the SISO system, either oscillating or unstable
13、 transient are observed. Figure 11 shows the comparisons when the desired superheat value is changed from 5C to 4C while the evaporating temperature remains at 8.5“C. The figure shows that the MIMO con- trol performs much better than the SISO control in command following. The disturbance rejection c
14、apability was tested both for the MIMO control and the SISO con- trol. In the disturbance rejection tests, the desired superheat and evaporating temperature need to remain the same after the indoor fan speed is changed from loo0 rpm to 1200 rpm. Figure 12 shows the comparison results. It can be obse
15、rved that the MIMO control had much better distur- bance rejection capability than the SISO control, as predicted in the previous section. The effects of high control gains and model errors on MIMO control stability and perfor- mance are also important to investigate. Because a certain level of unce
16、rtainty in the dynamic model used for control design must exist, excessively high control gains may endanger the closed-loop system stability because they magnify the effect of model errors and actuator satura- tion on the control stability. In this study, the stability robustness inequality criteri
17、a (Equation (23) were violated, i.e. the maximum singular value of the closed-loop system was able to go beyond the stability robustness bound. Figure 13 shows that if too high gains were used in MIMO control design (where r, the relative magnitude of weighting matrices Q and R, equals 1, only one f
18、ourth of the proper design value), it caused instability of the closed loop system. Therefore, it was necessary to select appropriate control gains for MIMO control that could be reflected by the relative magnitude of weighting matrices Q and R in the linear quadratic regula- 9 v 8.5 1 7.5 I O loo 2
19、00 300 Time (secs) o 100 200 300 Time (secs) 5.5 h s? I5 v) -. O 100 200 300 Time (secs) 11$ loo 200 Time (secs) Figure 12. Disturbance rejection: Indoor fan speed changed from 1000 rpm to 1200 rpm 222 101 I HVAC&R RESEARCH “O 100 200 300 Time (secs) O 100 200 300 Time (secs) A u -90. N a 65, 60 O 1
20、00 200 300 O 100 200 300 Time (secs) Time (secs) Figure 13. Effect of high control gain tor design step Equation (40). The effects of model errors on the MIMO control stability were also tested. The closed loop system was still robustly stable even when the evaporating heat transfer coefficient or c
21、ondensing heat transfer coefficient had 50% change. That means the bandwidth selected for the controller enabled the closed-loop system to be stable with the pres- ence of model uncertainty. MIMO Control Around Two Operating Points The MIMO control discussed above is based on the plant model that wa
22、s obtained by lineariz- ing the original nonlinear model around the given operating point. The plant model is only valid within a local range around the operating point. Therefore the MIMO controller based on this model may only work well around the operating point. When operating conditions change,
23、 for example, if the compressor were run at 50 Hz, a much low speed compared to 70 Hz, the con- troller based on the previous plant model might not be able to work in a stable manner with the same desired performance. Since system nonlinearities in a vapor compression cycle become evident over a wid
24、e range of operating conditions, it is necessary to adapt the plant model and then adapt the control law to different operating ranges to guarantee stability and performance. Control of the vapor compression cycle is desirable over a wide range of operations, because the environment conditions and t
25、hermal load requirement may be varying. The gain scheduling technique is used here to adapt the control law to different operating ranges and then achieve stable MIMO control with good performance over a wide range of operations. The schematic of gain scheduling is shown in Figure 14. Gain-scheduled
26、 control is a kind of adaptive system. When operating conditions change, the gain scheduler adapts the MIMO control law in a lookup table based on the values of variables which are used as operating conditions for switching to different controllers. All MIMO controllers in the lookup table have been
27、 designed off-line, based on linearized models around different operating points. STDIASHRAE SRCH IJHVAC 4-3-ENGL 1778 0759b50 0535758 202 W x Controller Machine VOLUME 4, NUMBER 3, JULY 1998 223 Operating Condition Gain d Schedule The selection of appropriate operating condition variables for switc
28、hing controllers depends greatly on the significance of the nonlinearity of certain control variables or process variables and on how easy it is to obtain the values of these variables. In this study, we selected the com- pressor speed and the expansion valve opening as variables for switching contr
29、ollers, since they are the main operating conditions that cause severe nonlinearity of the system if indoor and out- door air conditions, indoor fan speed and outdoor fan speed have no significant changes. Another issue in the design of a gain scheduling scheme is how one should partition the operat
30、- ing condition space. Controllers have to be switched from one partition to another partition when operating condition variables move in the space. Generally, one has to make sure each MIMO controller is robustly stable in each partition. If one partition is too large, and the system around this ra
31、nge is already quite nonlinear, the linearized model in this partition would have relatively large errors that may influence low performance of the controller. If partitions are made too small, it would take too much effort and time to design. Tradeoffs between effort and performance should be made.
32、 The following example illustrates the effectiveness of gain-scheduled MIMO control to regu- late evaporating Te and superheat SH. In this example, two operating points are considered. Besides the previous operating point where compressor speed u1 and valve opening u2 were 70 Hz and 120 steps, respe
33、ctively, another operating point was considered where u1 and u2 were 50 Hz and 100 steps, respectively. The nonlinear fifth order model was linearized around the second operating point, and therefore the second linear model could be obtained. The second LQG-based MIMO controller was designed based o
34、n the second linear plant model, following the same procedures discussed in Section 3. Therefore two LQG-based MIMO controllers were used to control the system around the two operating points based on gain scheduling. In Figure 15 the control results are shown. The control task in Figure 15 is that
35、the desired evaporating temperature needs to be changed from 7C to 10C while the superheat is kept at 4.5“C. The switch of the two controllers happens when the compressor speed moves from the range of 60 to 85 Hz to that of 40 to 60 Hz. Figure 15 shows that the compressor speed was changed by about
36、40 Hz, which is almost 50% of the operating range. However the gain-scheduled MIMO control was able to successfully control the desired evaporating temperature and superheat. The MIMO control was also shown to be much faster than the SISO control. In this situation, the MIMO control and the SISO con
37、trol resulted in significantly different energy efficiency. Since the transient process controlled by the MIMO system is much faster than that of the SISO con- STD-ASHRAE SRCH IJHVAC i-3-ENGL 224 1998 W 0759b50 0535757 047 HVACBrR RESEARCH Time (secs) 1 100 200 300 40 O Time (secs) 3/ 2 O 100 200 30
38、0 Time (secs) 80 o 100 200 300 Time (secs) Figure 15. Control of vapor compression cycle in a wide range I I I I l O 50 100 150 200 250 300 350 Time (secs) 2 O 50 100 150 200 250 300 350 Time (secs) Figure 16. COP and cooling capacity of the MIMO control and the SISO control VOLUME 4, NUMBER 3, JULY
39、 1998 225 trol, the COP (coefficient of performance) under MIMO control was significantly better, as shown in Figure 16. In addition, the desired capacity was reached much faster by use of MIMO control. APPLICATIONS IN HVAC AND REFRIGERATION SYSTEMS The MIMO control discussed above has direct applic
40、ations on a variety of HVAC and refrig- eration systems including air conditioners, heat pumps, and refrigerators, etc. The following aspects illustrate the main applications of this work to HVAC&R systems: Controlling Superheat at a Lower Value to Increase Energy Efficiency. Evaporating effi- cienc
41、y largely depends on the value of superheat. The lower the superheat value, the higher the energy efficiency. Ideally, with a zero superheat, the system has the highest energy efficiency. However, a positive superheat value must be maintained to prevent liquid refrigerant from enter- ing the compres
42、sor. Lower superheat implies lower energy consumption for a given thermal load. More precise values of energy consumption versus superheat depend on operating condi- tions such as indoor and outdoor temperatures, and thermal load etc. One example that we obtained in an experimental test (with indoor
43、 temperature at 27C and outdoor dry and wet bulb temperature at 35“C/29“C) showed 10% energy consumption savings when the superheat value was changed from 10 to 5C. With conventional control schemes (thermostatic expansion valve or any PID control of electronic expansion valve), the desired value of
44、 superheat had to be set relatively high, due to the limited stability and efficacy of SISO control. In this paper, it has been demonstrated that MIMO control techniques can take advantage of cross couplings in vapor compression systems to better and more effectively regulate superheat response by p
45、roperly coordinating the compressor speed and valve opening. This shows that the desired value of superheat can be set relatively low in the systems with MIMO control. Therefore, the overall system energy efficiency can be increased. Quick and Stable Response to Indoor Temperature Setting Change. Ch
46、ange of indoor temperature setting is often made due to reasons such as uncomfortable thermal sensation. We expect a quick response of the air conditioner that can change the indoor air temperature to the new setting for the desired thermal comfort. As has also been shown in this paper, multivariabl
47、e control can quickly respond to the setting change without causing instability of the system. Instability of the system may occur if quick action is taken based on a conventional control. Integrating with Set-Point Optimization to Achieve High Energy Efficiency. In addition to variable-speed compre
48、ssors and adjustable electronic expansion valve, variable-speed indoor and outdoor fans are also useful for improving system performance, as pointed out by Hiller (1976). There exist multiple combinations of compressor speed, indoor fan speed, outdoor fan speed, and expansion valve opening that can
49、satisfy cooling (or heating) load condition or ther- mal comfort requirement. However, the most energy-efficient operation where the COP is max- imized corresponds to a certain combination of these control inputs, i.e. the optimal set point. If the optimal set point can be determined and stabilized under arbitrary indoor and outdoor envi- ronment, these modern variable-speed drives can be exploited to significantly improve the energy efficiency of vapor compression systems. Energy saving in steady state is more important than during a transient state only
