ASHRAE国际期刊：1998年第四卷第三号 HVAC/R 研究

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ASHRAE IJHVAC 4-3是ASHRAE（美国采暖、制冷与空调工程师学会）出版的一份国际期刊，专注于供热、通风、空气调节和制冷领域的科研成果。这份期刊在1998年7月的第4卷第3期发表，版权归属于ASHRAE，并由IHS提供授权许可。文章强调了该领域内的专业研究，编委会由知名学者组成，包括来自普渡大学的雷蒙德·科恩博士、来自伦斯勒理工学院的亚瑟·伯格尔斯博士等，他们在各自的领域内拥有深厚的学术背景和实践经验。期刊的主要内容可能涵盖了空调系统的设计、优化、能效分析、新型材料和先进技术的应用、室内环境控制、可持续性策略以及相关标准的制定与研究。文章作者可能是David A. Didion，他作为热力学方面的领导者，可能在文中探讨了热量传递、能量效率提升或者冷却系统的创新技术。 ASHRAE IJHVAC 4-3的内容对于从事 HVAC&R（供热、通风与空调及制冷）行业的研究人员、工程师、建筑师和政策制定者具有重要的参考价值，它提供了最新的学术研究成果，推动着行业的科技进步和标准制定。阅读此期刊可以帮助读者了解行业动态，掌握前沿理论，以及解决实际工程中的挑战。通过深入研究ASHRAE IJHVAC 4-3的内容，专业人士可以了解到如何提高建筑物的能源效率，改善室内空气质量，同时满足舒适性和可持续性目标。对于那些希望在制冷与空调技术领域进行深入学习或进行创新的人来说，这份期刊是一个不可或缺的知识库。

Integral

Control

..................................................................

r................................................................

Plant

Model

......................................................

fLQG

................................................................,

Figure

Block

diagram

LQG

with

integrators

Because zero steady-state errors for step commands are required in the design, and the vapor

compressor cycle model has no integrators, two integrators have to be included in

LQG.

Figure

shows the block diagram of LQG augmented with two integrators (see appendix

for

detailed

derivations).

design a desired compensator

K(s)

with an LQG structure plus integrators, weighting

matrices were iteratively selected to satisfy the stability robustness inequality criterion and per-

formance requirement. Stability robustness inequality is expressed in the following equation

(Doyle

1981):

where

GK(I

C0-I

is the closed-loop transfer function matrix, and

e,(o)

O[E(jo)]

<r[(G(jo)

GeO.o))Ge(jo)-’]

is a bound for the maximum singular value of multiplicative

error.

denotes the “true” model of the system. Inequality

(23)

guarantees the closed-loop sys-

tem stability with the presence of model errors. Performance requirement

expressed as fol-

lows (Doyle 1981):

where

GO-’

the sensitivity transfer function matrix and

p,(w)

the performance bound

that should be specified realistically. The smaller the performance bound, the better the perfor-

mance of command following and disturbance rejection. Making the largest singular value of

the sensitivity transfer function matrix

less than the performance bound guarantees

good command following and disturbance rejection.

Design Procedures

Based on the fifth order lumped-parameter model discussed in section

a MIMO control

with the structure of LQG with integrators discussed above were designed

regulate evaporat-

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No reproduction or networking permitted without license from IHS

--`,`,`,``,,`,`,,,`,``,,,`,`,,``-`-`,,`,,`,`,,`---

STD-ASHRAE SRCH IJHVAC

9-3-ENGL

1998

0759b50

0535752

774

VOLUME

NUMBER

JULY

1998

217

ing temperature and superheat around desired values, using compressor speed and valve open-

ing. The vapor compression machine discussed here is a residential air conditioner. The machine

is of a split type, with cooling capacity rated at

2.8

kW.

The machine was equipped with a vari-

able-speed compressor with operating range from

Hz to

100

Hz,

and a variable-flow elec-

tronic expansion valve and variable-speed indoor fan. Hardware needed for implementation in a

controller included sensors (eight pressure sensors in different locations, sixteen thermocouples

to measure temperatures, and a refrigerant flow meter of turbine type installed at the exit of

indoor heat exchanger); sensor interface (an

.4/D

board for pressure signal conversion,

multi-

plexier board connecting with an

A/D

board for temperature signal conversion, a signal condi-

tioner for the flow meter); a desktop computer that takes the measurement data; and control

actuators (compressor motor, stepping motor for the expansion valve, and indoor fan motor) by

either

RS232C

serial communication commands or D/A signals. Software needed for implemen-

tation included data acquisition programs, model parameter identification programs, a program

for designing the LQG with integrators, and programs for actuator drives.

The following discussion describes design procedures for a normal operating point. The

matrices

and

in the fifth order model were given in the section, Analysis of the Lin-

earized Model. For other operating points, similar design procedures can be applied.

In the design of the LQG with integrators, we needed to obtain the optimal gain matrix

and

the Kalman filter matrix

The design parameters that we had to determine were weighting

matrices

and

for obtaining

KO,

and weighting matrices

and

for obtaining

We selected

and R to be

identity matrices to obtain the Kalman filter matrix

by assuming that the

process and measurement had the same level of noise. In order to obtain the optimal gain matrix

Ka,

iterations were needed to select weighting matrices

and

such that the controller

K(s)

sat-

isfied performance specifications and stability robustness.

and

were initially selected to be

4/k

5/kj

where

represents the relative magnitude of proportional and integral weighting, and

repre-

sents the relative magnitude of

and

that results in a different magnitude

control gain. The

smaller the

the larger the control gain. Note that the off-diagonal element in matrix

was neg-

ative, which means that the opposite direction of compressor speed change and valve opening

change increases the cost function and should therefore be penalized. Appropriate values of

and

had to be selected iteratively,

that the stability robustness criterion and performance cri-

terion could be satisfied.

“True” model

G,íjw)

can be approximately obtained in experimental tests by exciting the sys-

tem with different frequency sinusoidal control signals. By comparing the plant model

Cow)

and the “true” model

G,oo)

from experimental tests, the maximum multiplicative model error

0[E(jo)]

was found to be less than

W0.08.

That is

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-50.

Figure 9(a) shows the situation when

100

and

were used. Note that this design vio-

lates the stability robustness inequality [Equation

(23)]

which may result in instability of a close

loop system. When

is selected, the control gain is too large compared

the previous

design where

Figure 9(b) shows the situation when

100

and

10 are used. In this

case, the design cannot meet the performance requirement [Equation

(24)].

When

selected, the control 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

O[E(jo)],

(3)

select performance bound

p,(o)

based on the minimum required performance,

(4)

select initial

forms

and

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 the design based on the actual performance and stability of the controlled

system by repeating the steps

(3),

(4),

and

(5).

.......

....._............

.....,..,

Implementation and Experimental Results

We implemented the MIMO control for

residential air conditioner. Thermocouples were

used to measure temperatures at different 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

and evaporating temperature

Te.

the given operating point on which the above control design was based, the compressor

speed was

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

27°C

and 19"C, respectively. The outdoor temperature was

35°C.

The

corresponding evaporating temperature was

8S°C,

and the superheat was

5°C.

compare the

command following capability

the

MIMO

control and the SISO control, the SISO control

gains were properly tuned.

Figure 10 shows the results when the desired evaporating temperature

was changed from

8.5"C

7°C

while the superheat value

was kept at

5°C.

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

min

for

the SISO control to reach steady state, while the MIMO

control could reach steady state in about

min.

shown in Figure 10, the compressor speed

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No reproduction or networking permitted without license from IHS

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ASHRAE国际期刊：1998年第四卷第三号 HVAC/R 研究

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